{ "protocol": "matched-integer-det-minus-two-degree-seven-v1", "seed": 20260922, "question": "Is the Jacobian conjecture true", "repeats": 1, "baseline": [], "baseline_seconds": [], "completed_calls": 500, "total_calls": 500, "complete": true, "audit": { "exact_controls": { "determinant": -2, "coefficient_domain": "Z", "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "serialization": "fully expanded, identical formatter" }, "feature_names": [ "log_mean_coefficient", "log_max_coefficient", "log_median_coefficient", "log_min_coefficient", "log_coefficient_sd", "characters" ], "family_feature_means": { "bijection": [ 3.8290315468126255, 4.724655965209398, 3.4745170446798266, 0.0, 3.9547558427328853, 4982.704 ], "counterexample": [ 3.7282020856649702, 4.630258877741876, 3.381166523794971, 0.0, 3.850637813121421, 4948.552 ] }, "paired_5fold_ridge_accuracy": 0.54, "paired_5fold_stump_accuracy": 0.55, "note": "These surface-feature screens do not establish absence of all family cues." }, "rows": [ { "case": "B105", "pair": 105, "family": "bijection", "mapping": "F(x,y,z)=(8*x^7 - 54*x^6*y - 104*x^6*z + 54*x^6 + 156*x^5*y^2 + 600*x^5*y*z - 312*x^5*y + 576*x^5*z^2 - 600*x^5*z + 188*x^5 - 250*x^4*y^3 - 1440*x^4*y^2*z + 750*x^4*y^2 - 2760*x^4*y*z^2 + 2880*x^4*y*z - 886*x^4*y - 1760*x^4*z^3 + 2760*x^4*z^2 - 1680*x^4*z + 402*x^4 + 240*x^3*y^4 + 1840*x^3*y^3*z - 960*x^3*y^3 + 5280*x^3*y^2*z^2 - 5520*x^3*y^2*z + 1668*x^3*y^2 + 6720*x^3*y*z^3 - 10560*x^3*y*z^2 + 6316*x^3*y*z - 1476*x^3*y + 3200*x^3*z^4 - 6720*x^3*z^3 + 5968*x^3*z^2 - 2748*x^3*z + 689*x^3 - 138*x^2*y^5 - 1320*x^2*y^4*z + 690*x^2*y^4 - 5040*x^2*y^3*z^2 + 5280*x^2*y^3*z - 1568*x^2*y^3 - 9600*x^2*y^2*z^3 + 15120*x^2*y^2*z^2 - 8892*x^2*y^2*z + 2028*x^2*y^2 - 9120*x^2*y*z^4 + 19200*x^2*y*z^3 - 16776*x^2*y*z^2 + 7536*x^2*y*z - 1769*x^2*y - 3456*x^2*z^5 + 9120*x^2*z^4 - 10528*x^2*z^3 + 6984*x^2*z^2 - 3058*x^2*z + 789*x^2 + 44*x*y^6 + 504*x*y^5*z - 264*x*y^5 + 2400*x*y^4*z^2 - 2520*x*y^4*z + 736*x*y^4 + 6080*x*y^3*z^3 - 9600*x*y^3*z^2 + 5556*x*y^3*z - 1236*x*y^3 + 8640*x*y^2*z^4 - 18240*x*y^2*z^3 + 15696*x*y^2*z^2 - 6876*x*y^2*z + 1523*x*y^2 + 6528*x*y*z^5 - 17280*x*y*z^4 + 19664*x*y*z^3 - 12720*x*y*z^2 + 5304*x*y*z - 1266*x*y + 2048*x*z^6 - 6528*x*z^5 + 9216*x*z^4 - 7824*x*z^3 + 4660*x*z^2 - 2016*x*z + 466*x - 6*y^7 - 80*y^6*z + 42*y^6 - 456*y^5*z^2 + 480*y^5*z - 138*y^5 - 1440*y^4*z^3 + 2280*y^4*z^2 - 1300*y^4*z + 282*y^4 - 2720*y^3*z^4 + 5760*y^3*z^3 - 4888*y^3*z^2 + 2088*y^3*z - 439*y^3 - 3072*y^2*z^5 + 8160*y^2*z^4 - 9168*y^2*z^3 + 5784*y^2*z^2 - 2306*y^2*z + 513*y^2 - 1920*y*z^6 + 6144*y*z^5 - 8576*y*z^4 + 7104*y*z^3 - 4066*y*z^2 + 1656*y*z - 359*y - 512*z^7 + 1920*z^6 - 3200*z^5 + 3264*z^4 - 2400*z^3 + 1362*z^2 - 541*z + 1, -8*x^7 + 54*x^6*y + 104*x^6*z - 54*x^6 - 156*x^5*y^2 - 600*x^5*y*z + 312*x^5*y - 576*x^5*z^2 + 600*x^5*z - 188*x^5 + 250*x^4*y^3 + 1440*x^4*y^2*z - 750*x^4*y^2 + 2760*x^4*y*z^2 - 2880*x^4*y*z + 886*x^4*y + 1760*x^4*z^3 - 2760*x^4*z^2 + 1680*x^4*z - 402*x^4 - 240*x^3*y^4 - 1840*x^3*y^3*z + 960*x^3*y^3 - 5280*x^3*y^2*z^2 + 5520*x^3*y^2*z - 1668*x^3*y^2 - 6720*x^3*y*z^3 + 10560*x^3*y*z^2 - 6316*x^3*y*z + 1476*x^3*y - 3200*x^3*z^4 + 6720*x^3*z^3 - 5968*x^3*z^2 + 2748*x^3*z - 689*x^3 + 138*x^2*y^5 + 1320*x^2*y^4*z - 690*x^2*y^4 + 5040*x^2*y^3*z^2 - 5280*x^2*y^3*z + 1568*x^2*y^3 + 9600*x^2*y^2*z^3 - 15120*x^2*y^2*z^2 + 8892*x^2*y^2*z - 2028*x^2*y^2 + 9120*x^2*y*z^4 - 19200*x^2*y*z^3 + 16776*x^2*y*z^2 - 7536*x^2*y*z + 1769*x^2*y + 3456*x^2*z^5 - 9120*x^2*z^4 + 10528*x^2*z^3 - 6984*x^2*z^2 + 3058*x^2*z - 789*x^2 - 44*x*y^6 - 504*x*y^5*z + 264*x*y^5 - 2400*x*y^4*z^2 + 2520*x*y^4*z - 736*x*y^4 - 6080*x*y^3*z^3 + 9600*x*y^3*z^2 - 5556*x*y^3*z + 1236*x*y^3 - 8640*x*y^2*z^4 + 18240*x*y^2*z^3 - 15696*x*y^2*z^2 + 6876*x*y^2*z - 1523*x*y^2 - 6528*x*y*z^5 + 17280*x*y*z^4 - 19664*x*y*z^3 + 12720*x*y*z^2 - 5304*x*y*z + 1266*x*y - 2048*x*z^6 + 6528*x*z^5 - 9216*x*z^4 + 7824*x*z^3 - 4660*x*z^2 + 2016*x*z - 458*x + 6*y^7 + 80*y^6*z - 42*y^6 + 456*y^5*z^2 - 480*y^5*z + 138*y^5 + 1440*y^4*z^3 - 2280*y^4*z^2 + 1300*y^4*z - 282*y^4 + 2720*y^3*z^4 - 5760*y^3*z^3 + 4888*y^3*z^2 - 2088*y^3*z + 439*y^3 + 3072*y^2*z^5 - 8160*y^2*z^4 + 9168*y^2*z^3 - 5784*y^2*z^2 + 2306*y^2*z - 513*y^2 + 1920*y*z^6 - 6144*y*z^5 + 8576*y*z^4 - 7104*y*z^3 + 4066*y*z^2 - 1656*y*z + 353*y + 512*z^7 - 1920*z^6 + 3200*z^5 - 3264*z^4 + 2400*z^3 - 1362*z^2 + 533*z - 1, -16*x^7 + 108*x^6*y + 208*x^6*z - 108*x^6 - 312*x^5*y^2 - 1200*x^5*y*z + 624*x^5*y - 1152*x^5*z^2 + 1200*x^5*z - 376*x^5 + 500*x^4*y^3 + 2880*x^4*y^2*z - 1500*x^4*y^2 + 5520*x^4*y*z^2 - 5760*x^4*y*z + 1772*x^4*y + 3520*x^4*z^3 - 5520*x^4*z^2 + 3360*x^4*z - 804*x^4 - 480*x^3*y^4 - 3680*x^3*y^3*z + 1920*x^3*y^3 - 10560*x^3*y^2*z^2 + 11040*x^3*y^2*z - 3336*x^3*y^2 - 13440*x^3*y*z^3 + 21120*x^3*y*z^2 - 12632*x^3*y*z + 2952*x^3*y - 6400*x^3*z^4 + 13440*x^3*z^3 - 11936*x^3*z^2 + 5496*x^3*z - 1377*x^3 + 276*x^2*y^5 + 2640*x^2*y^4*z - 1380*x^2*y^4 + 10080*x^2*y^3*z^2 - 10560*x^2*y^3*z + 3136*x^2*y^3 + 19200*x^2*y^2*z^3 - 30240*x^2*y^2*z^2 + 17784*x^2*y^2*z - 4056*x^2*y^2 + 18240*x^2*y*z^4 - 38400*x^2*y*z^3 + 33552*x^2*y*z^2 - 15072*x^2*y*z + 3535*x^2*y + 6912*x^2*z^5 - 18240*x^2*z^4 + 21056*x^2*z^3 - 13968*x^2*z^2 + 6110*x^2*z - 1575*x^2 - 88*x*y^6 - 1008*x*y^5*z + 528*x*y^5 - 4800*x*y^4*z^2 + 5040*x*y^4*z - 1472*x*y^4 - 12160*x*y^3*z^3 + 19200*x*y^3*z^2 - 11112*x*y^3*z + 2472*x*y^3 - 17280*x*y^2*z^4 + 36480*x*y^2*z^3 - 31392*x*y^2*z^2 + 13752*x*y^2*z - 3043*x*y^2 - 13056*x*y*z^5 + 34560*x*y*z^4 - 39328*x*y*z^3 + 25440*x*y*z^2 - 10596*x*y*z + 2526*x*y - 4096*x*z^6 + 13056*x*z^5 - 18432*x*z^4 + 15648*x*z^3 - 9308*x*z^2 + 4020*x*z - 935*x + 12*y^7 + 160*y^6*z - 84*y^6 + 912*y^5*z^2 - 960*y^5*z + 276*y^5 + 2880*y^4*z^3 - 4560*y^4*z^2 + 2600*y^4*z - 564*y^4 + 5440*y^3*z^4 - 11520*y^3*z^3 + 9776*y^3*z^2 - 4176*y^3*z + 877*y^3 + 6144*y^2*z^5 - 16320*y^2*z^4 + 18336*y^2*z^3 - 11568*y^2*z^2 + 4606*y^2*z - 1023*y^2 + 3840*y*z^6 - 12288*y*z^5 + 17152*y*z^4 - 14208*y*z^3 + 8120*y*z^2 - 3300*y*z + 720*y + 1024*z^7 - 3840*z^6 + 6400*z^5 - 6528*z^4 + 4792*z^3 - 2712*z^2 + 1083*z - 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ 1, -1, -2 ], [ 2, -1, -1 ], [ 3, -2, -2 ] ], "source_shift": [ 1, 2, 2 ], "target_matrix": [ [ 1, 1, 1 ], [ -1, -1, -2 ], [ -2, -1, -1 ] ], "target_shift": [ -105, 99, 214 ], "inverse_recipe": { "p": 3, "q": 1, "f": "x**3", "g": "x", "h": "y**2*z + z**3", "polynomials": { "f": [ [ [ 3, 0, 0 ], 1 ] ], "g": [ [ [ 1, 0, 0 ], 1 ] ], "h": [ [ [ 0, 2, 1 ], 1 ], [ [ 0, 0, 3 ], 1 ] ] }, "h_multiplier": 2 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "Maybe" ], "seconds": [ 1.1081224139998085 ], "surface_features": [ 3.7323640462725227, 4.59470186120636, 3.4409090820652177, 0.0, 3.824997816343506, 4981 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C008", "pair": 8, "family": "counterexample", "mapping": "F(x,y,z)=(2*x^7 + 19*x^6*y - 13*x^6*z - 28*x^6 + 75*x^5*y^2 - 105*x^5*y*z - 231*x^5*y + 36*x^5*z^2 + 156*x^5*z + 165*x^5 + 159*x^4*y^3 - 342*x^4*y^2*z - 768*x^4*y^2 + 240*x^4*y*z^2 + 1062*x^4*y*z + 1143*x^4*y - 55*x^4*z^3 - 360*x^4*z^2 - 768*x^4*z - 545*x^4 + 195*x^3*y^4 - 573*x^3*y^3*z - 1311*x^3*y^3 + 618*x^3*y^2*z^2 + 2790*x^3*y^2*z + 3054*x^3*y^2 - 290*x^3*y*z^3 - 1938*x^3*y*z^2 - 4209*x^3*y*z - 3037*x^3*y + 50*x^3*z^4 + 440*x^3*z^3 + 1422*x^3*z^2 + 2037*x^3*z + 1098*x^3 + 138*x^2*y^5 - 519*x^2*y^4*z - 1206*x^2*y^4 + 765*x^2*y^3*z^2 + 3516*x^2*y^3*z + 3909*x^2*y^3 - 552*x^2*y^2*z^3 - 3762*x^2*y^2*z^2 - 8313*x^2*y^2*z - 6093*x^2*y^2 + 195*x^2*y*z^4 + 1752*x^2*y*z^3 + 5769*x^2*y*z^2 + 8412*x^2*y*z + 4614*x^2*y - 27*x^2*z^5 - 300*x^2*z^4 - 1308*x^2*z^3 - 2841*x^2*z^2 - 3090*x^2*z - 1354*x^2 + 52*x*y^6 - 240*x*y^5*z - 564*x*y^5 + 453*x*y^4*z^2 + 2112*x*y^4*z + 2379*x*y^4 - 447*x*y^3*z^3 - 3099*x*y^3*z^2 - 6957*x*y^3*z - 5170*x*y^3 + 243*x*y^2*z^4 + 2226*x*y^2*z^3 + 7464*x*y^2*z^2 + 11070*x*y^2*z + 6171*x*y^2 - 69*x*y*z^5 - 783*x*y*z^4 - 3483*x*y*z^3 - 7713*x*y*z^2 - 8547*x*y*z - 3812*x*y + 8*x*z^6 + 108*x*z^5 + 597*x*z^4 + 1751*x*z^3 + 2886*x*z^2 + 2549*x*z + 949*x + 8*y^7 - 44*y^6*z - 104*y^6 + 102*y^5*z^2 + 480*y^5*z + 546*y^5 - 129*y^4*z^3 - 906*y^4*z^2 - 2061*y^4*z - 1550*y^4 + 96*y^3*z^4 + 894*y^3*z^3 + 3048*y^3*z^2 + 4591*y^3*z + 2593*y^3 - 42*y^2*z^5 - 486*y^2*z^4 - 2205*y^2*z^3 - 4977*y^2*z^2 - 5613*y^2*z - 2539*y^2 + 10*y*z^6 + 138*y*z^5 + 780*y*z^4 + 2338*y*z^3 + 3933*y*z^2 + 3539*y*z + 1344*y - z^7 - 16*z^6 - 108*z^5 - 402*z^4 - 894*z^3 - 1195*z^2 - 896*z - 1, -2*x^7 - 19*x^6*y + 13*x^6*z + 28*x^6 - 75*x^5*y^2 + 105*x^5*y*z + 231*x^5*y - 36*x^5*z^2 - 156*x^5*z - 165*x^5 - 159*x^4*y^3 + 342*x^4*y^2*z + 768*x^4*y^2 - 240*x^4*y*z^2 - 1062*x^4*y*z - 1143*x^4*y + 55*x^4*z^3 + 360*x^4*z^2 + 768*x^4*z + 543*x^4 - 195*x^3*y^4 + 573*x^3*y^3*z + 1311*x^3*y^3 - 618*x^3*y^2*z^2 - 2790*x^3*y^2*z - 3054*x^3*y^2 + 290*x^3*y*z^3 + 1938*x^3*y*z^2 + 4209*x^3*y*z + 3024*x^3*y - 50*x^3*z^4 - 440*x^3*z^3 - 1422*x^3*z^2 - 2030*x^3*z - 1088*x^3 - 138*x^2*y^5 + 519*x^2*y^4*z + 1206*x^2*y^4 - 765*x^2*y^3*z^2 - 3516*x^2*y^3*z - 3909*x^2*y^3 + 552*x^2*y^2*z^3 + 3762*x^2*y^2*z^2 + 8313*x^2*y^2*z + 6063*x^2*y^2 - 195*x^2*y*z^4 - 1752*x^2*y*z^3 - 5769*x^2*y*z^2 - 8379*x^2*y*z - 4566*x^2*y + 27*x^2*z^5 + 300*x^2*z^4 + 1308*x^2*z^3 + 2832*x^2*z^2 + 3063*x^2*z + 1333*x^2 - 52*x*y^6 + 240*x*y^5*z + 564*x*y^5 - 453*x*y^4*z^2 - 2112*x*y^4*z - 2379*x*y^4 + 447*x*y^3*z^3 + 3099*x*y^3*z^2 + 6957*x*y^3*z + 5142*x*y^3 - 243*x*y^2*z^4 - 2226*x*y^2*z^3 - 7464*x*y^2*z^2 - 11022*x*y^2*z - 6099*x*y^2 + 69*x*y*z^5 + 783*x*y*z^4 + 3483*x*y*z^3 + 7686*x*y*z^2 + 8463*x*y*z + 3743*x*y - 8*x*z^6 - 108*x*z^5 - 597*x*z^4 - 1746*x*z^3 - 2862*x*z^2 - 2510*x*z - 931*x - 8*y^7 + 44*y^6*z + 104*y^6 - 102*y^5*z^2 - 480*y^5*z - 546*y^5 + 129*y^4*z^3 + 906*y^4*z^2 + 2061*y^4*z + 1542*y^4 - 96*y^3*z^4 - 894*y^3*z^3 - 3048*y^3*z^2 - 4571*y^3*z - 2561*y^3 + 42*y^2*z^5 + 486*y^2*z^4 + 2205*y^2*z^3 + 4959*y^2*z^2 + 5553*y^2*z + 2485*y^2 - 10*y*z^6 - 138*y*z^5 - 780*y*z^4 - 2331*y*z^3 - 3897*y*z^2 - 3476*y*z - 1314*y + z^7 + 16*z^6 + 108*z^5 + 401*z^4 + 887*z^3 + 1177*z^2 + 879*z + 1, 4*x^7 + 38*x^6*y - 26*x^6*z - 50*x^6 + 150*x^5*y^2 - 210*x^5*y*z - 411*x^5*y + 72*x^5*z^2 + 279*x^5*z + 276*x^5 + 318*x^4*y^3 - 684*x^4*y^2*z - 1362*x^4*y^2 + 480*x^4*y*z^2 + 1893*x^4*y*z + 1902*x^4*y - 110*x^4*z^3 - 645*x^4*z^2 - 1287*x^4*z - 873*x^4 + 390*x^3*y^4 - 1146*x^3*y^3*z - 2319*x^3*y^3 + 1236*x^3*y^2*z^2 + 4959*x^3*y^2*z + 5058*x^3*y^2 - 580*x^3*y*z^3 - 3462*x^3*y*z^2 - 7020*x^3*y*z - 4836*x^3*y + 100*x^3*z^4 + 790*x^3*z^3 + 2388*x^3*z^2 + 3268*x^3*z + 1708*x^3 + 276*x^2*y^5 - 1038*x^2*y^4*z - 2130*x^2*y^4 + 1530*x^2*y^3*z^2 + 6237*x^2*y^3*z + 6450*x^2*y^3 - 1104*x^2*y^2*z^3 - 6705*x^2*y^2*z^2 - 13809*x^2*y^2*z - 9651*x^2*y^2 + 390*x^2*y*z^4 + 3138*x^2*y*z^3 + 9648*x^2*y*z^2 + 13422*x^2*y*z + 7128*x^2*y - 54*x^2*z^5 - 540*x^2*z^4 - 2202*x^2*z^3 - 4566*x^2*z^2 - 4812*x^2*z - 2066*x^2 + 104*x*y^6 - 480*x*y^5*z - 996*x*y^5 + 906*x*y^4*z^2 + 3744*x*y^4*z + 3918*x*y^4 - 894*x*y^3*z^3 - 5517*x*y^3*z^2 - 11526*x*y^3*z - 8157*x*y^3 + 486*x*y^2*z^4 + 3981*x*y^2*z^3 + 12444*x*y^2*z^2 + 17583*x*y^2*z + 9474*x*y^2 - 138*x*y*z^5 - 1407*x*y*z^4 - 5844*x*y*z^3 - 12336*x*y*z^2 - 13224*x*y*z - 5770*x*y + 16*x*z^6 + 195*x*z^5 + 1008*x*z^4 + 2820*x*z^3 + 4500*x*z^2 + 3892*x*z + 1432*x + 16*y^7 - 88*y^6*z - 184*y^6 + 204*y^5*z^2 + 852*y^5*z + 900*y^5 - 258*y^4*z^3 - 1614*y^4*z^2 - 3414*y^4*z - 2442*y^4 + 192*y^3*z^4 + 1599*y^3*z^3 + 5076*y^3*z^2 + 7273*y^3*z + 3964*y^3 - 84*y^2*z^5 - 873*y^2*z^4 - 3693*y^2*z^3 - 7932*y^2*z^2 - 8637*y^2*z - 3815*y^2 + 20*y*z^6 + 249*y*z^5 + 1314*y*z^4 + 3750*y*z^3 + 6096*y*z^2 + 5362*y*z + 2006*y - 2*z^7 - 29*z^6 - 183*z^5 - 649*z^4 - 1396*z^3 - 1826*z^2 - 1352*z - 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -1, -2, 1 ], [ -1, -1, 1 ], [ -2, -1, 1 ] ], "source_shift": [ 1, 2, 1 ], "target_matrix": [ [ -1, -1, -1 ], [ 1, 1, 2 ], [ -2, -1, -1 ] ], "target_shift": [ 290, -285, 437 ], "collision_preimages": [ [ "-3/4", "-1", "-15/4" ], [ "-9", "-7/2", "-16" ], [ "-6", "3/2", "-5" ] ], "collision_image": [ "1161/4", "-1141/4", "875/2" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0195416700007627 ], "surface_features": [ 3.342936730294912, 4.245092976101281, 3.000867721531227, 0.0, 3.453454517462562, 4805 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C125", "pair": 125, "family": "counterexample", "mapping": "F(x,y,z)=(-2*x^7 + 20*x^6*y - 15*x^6*z + 34*x^6 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3318*x^2*z^3 - 5910*x^2*z^2 + 5436*x^2*z - 2066*x^2 + 176*x*y^6 - 888*x*y^5*z + 1488*x*y^5 + 1812*x*y^4*z^2 - 5964*x*y^4*z + 5196*x*y^4 - 1914*x*y^3*z^3 + 9396*x*y^3*z^2 - 16182*x*y^3*z + 9708*x*y^3 + 1104*x*y^2*z^4 - 7257*x*y^2*z^3 + 18660*x*y^2*z^2 - 22209*x*y^2*z + 10302*x*y^2 - 330*x*y*z^5 + 2742*x*y*z^4 - 9420*x*y*z^3 + 16764*x*y*z^2 - 15456*x*y*z + 5906*x*y + 40*x*z^6 - 405*x*z^5 + 1752*x*z^4 - 4164*x*z^3 + 5748*x*z^2 - 4372*x*z + 1432*x - 32*y^7 + 192*y^6*z - 328*y^6 - 480*y^5*z^2 + 1596*y^5*z - 1404*y^5 + 648*y^4*z^3 - 3186*y^4*z^2 + 5520*y^4*z - 3330*y^4 - 510*y^3*z^4 + 3333*y^3*z^3 - 8583*y^3*z^2 + 10255*y^3*z - 4775*y^3 + 234*y^2*z^5 - 1923*y^2*z^4 + 6585*y^2*z^3 - 11730*y^2*z^2 + 10851*y^2*z - 4163*y^2 - 58*y*z^6 + 579*y*z^5 - 2487*y*z^4 + 5898*y*z^3 - 8148*y*z^2 + 6218*y*z - 2050*y + 6*z^7 - 71*z^6 + 369*z^5 - 1097*z^4 + 2020*z^3 - 2306*z^2 + 1512*z - 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -1, 1, -1 ], [ -1, 2, -1 ], [ -2, 2, -3 ] ], "source_shift": [ 1, 2, 1 ], "target_matrix": [ [ 1, 2, 1 ], [ -3, -2, -2 ], [ -2, -1, -1 ] ], "target_shift": [ -439, 728, 437 ], "collision_preimages": [ [ "3/4", "-1", "-3/4" ], [ "2", "-7/2", "-11/2" ], [ "13", "3/2", "-19/2" ] ], "collision_image": [ "-1757/4", "2915/4", "875/2" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0818309770002088 ], "surface_features": [ 3.6531637199259426, 4.554416468651753, 3.340741474989978, 0.0, 3.7580953741457215, 4923 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C111", "pair": 111, "family": "counterexample", "mapping": "F(x,y,z)=(16*x^7 + 176*x^6*y - 104*x^6*z - 208*x^6 + 816*x^5*y^2 - 972*x^5*y*z - 1968*x^5*y + 288*x^5*z^2 + 1176*x^5*z + 1044*x^5 + 2060*x^4*y^3 - 3714*x^4*y^2*z - 7656*x^4*y^2 + 2220*x^4*y*z^2 + 9216*x^4*y*z + 8184*x^4*y - 440*x^4*z^3 - 2760*x^4*z^2 - 4950*x^4*z - 2710*x^4 + 3043*x^3*y^4 - 7393*x^3*y^3*z - 15644*x^3*y^3 + 6696*x^3*y^2*z^2 + 28470*x^3*y^2*z + 25359*x^3*y^2 - 2680*x^3*y*z^3 - 17184*x^3*y*z^2 - 30885*x^3*y*z - 16805*x^3*y + 400*x^3*z^4 + 3440*x^3*z^3 + 9360*x^3*z^2 + 10305*x^3*z + 3948*x^3 + 2610*x^2*y^5 - 8031*x^2*y^4*z - 17661*x^2*y^4 + 9819*x^2*y^3*z^2 + 43230*x^2*y^3*z + 38772*x^2*y^3 - 5964*x^2*y^2*z^3 - 39474*x^2*y^2*z^2 - 71340*x^2*y^2*z - 38649*x^2*y^2 + 1800*x^2*y*z^4 + 15936*x^2*y*z^3 + 43551*x^2*y*z^2 + 47703*x^2*y*z + 18075*x^2*y - 216*x^2*z^5 - 2400*x^2*z^4 - 8820*x^2*z^3 - 14655*x^2*z^2 - 11259*x^2*z - 3217*x^2 + 1188*x*y^6 - 4464*x*y^5*z - 10404*x*y^5 + 6933*x*y^4*z^2 + 32154*x*y^4*z + 29196*x*y^4 - 5699*x*y^3*z^3 - 39528*x*y^3*z^2 - 72180*x*y^3*z - 39033*x*y^3 + 2616*x*y^2*z^4 + 24162*x*y^2*z^3 + 66603*x*y^2*z^2 + 72738*x*y^2*z + 27306*x*y^2 - 636*x*y*z^5 - 7344*x*y*z^4 - 27183*x*y*z^3 - 44991*x*y*z^2 - 34209*x*y*z - 9639*x*y + 64*x*z^6 + 888*x*z^5 + 4140*x*z^4 + 9235*x*z^3 + 10674*x*z^2 + 6104*x*z + 1349*x + 216*y^7 - 972*y^6*z - 2484*y^6 + 1854*y^5*z^2 + 9324*y^5*z + 8640*y^5 - 1945*y^4*z^3 - 14493*y^4*z^2 - 26928*y^4*z - 14586*y^4 + 1213*y^3*z^4 + 11942*y^3*z^3 + 33408*y^3*z^2 + 36489*y^3*z + 13602*y^3 - 450*y^2*z^5 - 5502*y^2*z^4 - 20622*y^2*z^3 - 34089*y^2*z^2 - 25704*y^2*z - 7158*y^2 + 92*y*z^6 + 1344*y*z^5 + 6333*y*z^4 + 14093*y*z^3 + 16134*y*z^2 + 9105*y*z + 1984*y - 8*z^7 - 136*z^6 - 774*z^5 - 2175*z^4 - 3363*z^3 - 2887*z^2 - 1276*z - 1, 16*x^7 + 176*x^6*y - 104*x^6*z - 208*x^6 + 816*x^5*y^2 - 972*x^5*y*z - 1968*x^5*y + 288*x^5*z^2 + 1176*x^5*z + 1044*x^5 + 2060*x^4*y^3 - 3714*x^4*y^2*z - 7656*x^4*y^2 + 2220*x^4*y*z^2 + 9216*x^4*y*z + 8184*x^4*y - 440*x^4*z^3 - 2760*x^4*z^2 - 4950*x^4*z - 2708*x^4 + 3043*x^3*y^4 - 7393*x^3*y^3*z - 15644*x^3*y^3 + 6696*x^3*y^2*z^2 + 28470*x^3*y^2*z + 25359*x^3*y^2 - 2680*x^3*y*z^3 - 17184*x^3*y*z^2 - 30885*x^3*y*z - 16792*x^3*y + 400*x^3*z^4 + 3440*x^3*z^3 + 9360*x^3*z^2 + 10298*x^3*z + 3928*x^3 + 2610*x^2*y^5 - 8031*x^2*y^4*z 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7002*y^2 + 92*y*z^6 + 1344*y*z^5 + 6333*y*z^4 + 14086*y*z^3 + 16059*y*z^2 + 8925*y*z + 1872*y - 8*z^7 - 136*z^6 - 774*z^5 - 2174*z^4 - 3349*z^3 - 2836*z^2 - 1210*z - 1, -32*x^7 - 352*x^6*y + 208*x^6*z + 392*x^6 - 1632*x^5*y^2 + 1944*x^5*y*z + 3708*x^5*y - 576*x^5*z^2 - 2220*x^5*z - 1824*x^5 - 4120*x^4*y^3 + 7428*x^4*y^2*z + 14430*x^4*y^2 - 4440*x^4*y*z^2 - 17400*x^4*y*z - 14268*x^4*y + 880*x^4*z^3 + 5220*x^4*z^2 + 8664*x^4*z + 4396*x^4 - 6086*x^3*y^4 + 14786*x^3*y^3*z + 29521*x^3*y^3 - 13392*x^3*y^2*z^2 - 53799*x^3*y^2*z - 44148*x^3*y^2 + 5360*x^3*y*z^3 + 32520*x^3*y*z^2 + 53964*x^3*y*z + 27176*x^3*y - 800*x^3*z^4 - 6520*x^3*z^3 - 16416*x^3*z^2 - 16744*x^3*z - 6008*x^3 - 5220*x^2*y^5 + 16062*x^2*y^4*z + 33408*x^2*y^4 - 19638*x^2*y^3*z^2 - 81849*x^2*y^3*z - 67467*x^2*y^3 + 11928*x^2*y^2*z^3 + 74817*x^2*y^2*z^2 + 124536*x^2*y^2*z + 62361*x^2*y^2 - 3600*x^2*y*z^4 - 30240*x^2*y*z^3 - 76284*x^2*y*z^2 - 77298*x^2*y*z - 27432*x^2*y + 432*x^2*z^5 + 4560*x^2*z^4 + 15504*x^2*z^3 + 23856*x^2*z^2 + 17160*x^2*z + 4658*x^2 - 2376*x*y^6 + 8928*x*y^5*z + 19764*x*y^5 - 13866*x*y^4*z^2 - 61092*x*y^4*z - 50844*x*y^4 + 11398*x*y^3*z^3 + 75135*x*y^3*z^2 + 126030*x*y^3*z + 62902*x*y^3 - 5232*x*y^2*z^4 - 45957*x*y^2*z^3 - 116628*x*y^2*z^2 - 117654*x*y^2*z - 41364*x*y^2 + 1272*x*y*z^5 + 13980*x*y*z^4 + 47748*x*y*z^3 + 73068*x*y*z^2 + 52014*x*y*z + 13935*x*y - 128*x*z^6 - 1692*x*z^5 - 7296*x*z^4 - 15064*x*z^3 - 16296*x*z^2 - 8851*x*z - 1888*x - 432*y^7 + 1944*y^6*z + 4752*y^6 - 3708*y^5*z^2 - 17820*y^5*z - 15084*y^5 + 3890*y^4*z^3 + 27684*y^4*z^2 + 47100*y^4*z + 23504*y^4 - 2426*y^3*z^4 - 22807*y^3*z^3 - 58563*y^3*z^2 - 58982*y^3*z - 20584*y^3 + 900*y^2*z^5 + 10509*y^2*z^4 + 36240*y^2*z^3 + 55293*y^2*z^2 + 39024*y^2*z + 10338*y^2 - 184*y*z^6 - 2568*y*z^5 - 11160*y*z^4 - 22946*y*z^3 - 24582*y*z^2 - 13185*y*z - 2775*y + 16*z^7 + 260*z^6 + 1368*z^5 + 3556*z^4 + 5144*z^3 + 4193*z^2 + 1788*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -1, -2, 1 ], [ -2, -3, 2 ], [ -2, -1, 1 ] ], "source_shift": [ 2, 1, 2 ], "target_matrix": [ [ -1, -1, -1 ], [ -1, -1, -2 ], [ 2, 1, 1 ] ], "target_shift": [ 228, 204, -312 ], "collision_preimages": [ [ "13/4", "3", "29/4" ], [ "-6", "-1/2", "-8" ], [ "-1", "13/2", "9" ] ], "collision_image": [ "913/4", "817/4", "-625/2" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.095219768001698 ], "surface_features": [ 4.218957457978805, 5.1004739362576546, 3.904769625906595, 0.0, 4.336054254160486, 5132 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B141", "pair": 141, "family": "bijection", "mapping": "F(x,y,z)=(12*x^7 - 84*x^6*y - 84*x^6*z + 168*x^6 + 252*x^5*y^2 + 504*x^5*y*z - 1008*x^5*y + 252*x^5*z^2 - 1008*x^5*z + 1044*x^5 - 420*x^4*y^3 - 1260*x^4*y^2*z + 2520*x^4*y^2 - 1260*x^4*y*z^2 + 5040*x^4*y*z - 5232*x^4*y - 420*x^4*z^3 + 2520*x^4*z^2 - 5232*x^4*z + 3697*x^4 + 420*x^3*y^4 + 1680*x^3*y^3*z - 3360*x^3*y^3 + 2520*x^3*y^2*z^2 - 10080*x^3*y^2*z + 10488*x^3*y^2 + 1680*x^3*y*z^3 - 10080*x^3*y*z^2 + 20976*x^3*y*z - 14896*x^3*y + 420*x^3*z^4 - 3360*x^3*z^3 + 10488*x^3*z^2 - 14884*x^3*z + 7974*x^3 - 252*x^2*y^5 - 1260*x^2*y^4*z + 2520*x^2*y^4 - 2520*x^2*y^3*z^2 + 10080*x^2*y^3*z - 10512*x^2*y^3 - 2520*x^2*y^2*z^3 + 15120*x^2*y^2*z^2 - 31536*x^2*y^2*z + 22506*x^2*y^2 - 1260*x^2*y*z^4 + 10080*x^2*y*z^3 - 31536*x^2*y*z^2 + 44976*x^2*y*z - 24282*x^2*y - 252*x^2*z^5 + 2520*x^2*z^4 - 10512*x^2*z^3 + 22470*x^2*z^2 - 24210*x^2*z + 10416*x^2 + 84*x*y^6 + 504*x*y^5*z - 1008*x*y^5 + 1260*x*y^4*z^2 - 5040*x*y^4*z + 5268*x*y^4 + 1680*x*y^3*z^3 - 10080*x*y^3*z^2 + 21072*x*y^3*z - 15112*x*y^3 + 1260*x*y^2*z^4 - 10080*x*y^2*z^3 + 31608*x*y^2*z^2 - 45300*x*y^2*z + 24642*x*y^2 + 504*x*y*z^5 - 5040*x*y*z^4 + 21072*x*y*z^3 - 45264*x*y*z^2 + 49140*x*y*z - 21408*x*y + 84*x*z^6 - 1008*x*z^5 + 5268*x*z^4 - 15076*x*z^3 + 24498*x*z^2 - 21228*x*z + 7601*x - 12*y^7 - 84*y^6*z + 168*y^6 - 252*y^5*z^2 + 1008*y^5*z - 1056*y^5 - 420*y^4*z^3 + 2520*y^4*z^2 - 5280*y^4*z + 3805*y^4 - 420*y^3*z^4 + 3360*y^3*z^3 - 10560*y^3*z^2 + 15208*y^3*z - 8334*y^3 - 252*y^2*z^5 + 2520*y^2*z^4 - 10560*y^2*z^3 + 22794*y^2*z^2 - 24930*y^2*z + 11004*y^2 - 84*y*z^6 + 1008*y*z^5 - 5280*y*z^4 + 15184*y*z^3 - 24858*y*z^2 + 21816*y*z - 7948*y - 12*z^7 + 168*z^6 - 1056*z^5 + 3793*z^4 - 8262*z^3 + 10812*z^2 - 7815*z + 1, -6*x^7 + 42*x^6*y + 42*x^6*z - 84*x^6 - 126*x^5*y^2 - 252*x^5*y*z + 504*x^5*y - 126*x^5*z^2 + 504*x^5*z - 522*x^5 + 210*x^4*y^3 + 630*x^4*y^2*z - 1260*x^4*y^2 + 630*x^4*y*z^2 - 2520*x^4*y*z + 2616*x^4*y + 210*x^4*z^3 - 1260*x^4*z^2 + 2616*x^4*z - 1850*x^4 - 210*x^3*y^4 - 840*x^3*y^3*z + 1680*x^3*y^3 - 1260*x^3*y^2*z^2 + 5040*x^3*y^2*z - 5244*x^3*y^2 - 840*x^3*y*z^3 + 5040*x^3*y*z^2 - 10488*x^3*y*z + 7454*x^3*y - 210*x^3*z^4 + 1680*x^3*z^3 - 5244*x^3*z^2 + 7448*x^3*z - 3998*x^3 + 126*x^2*y^5 + 630*x^2*y^4*z - 1260*x^2*y^4 + 1260*x^2*y^3*z^2 - 5040*x^2*y^3*z + 5256*x^2*y^3 + 1260*x^2*y^2*z^3 - 7560*x^2*y^2*z^2 + 15768*x^2*y^2*z - 11262*x^2*y^2 + 630*x^2*y*z^4 - 5040*x^2*y*z^3 + 15768*x^2*y*z^2 - 22506*x^2*y*z + 12174*x^2*y + 126*x^2*z^5 - 1260*x^2*z^4 + 5256*x^2*z^3 - 11244*x^2*z^2 + 12138*x^2*z - 5238*x^2 - 42*x*y^6 - 252*x*y^5*z + 504*x*y^5 - 630*x*y^4*z^2 + 2520*x*y^4*z - 2634*x*y^4 - 840*x*y^3*z^3 + 5040*x*y^3*z^2 - 10536*x*y^3*z + 7562*x*y^3 - 630*x*y^2*z^4 + 5040*x*y^2*z^3 - 15804*x*y^2*z^2 + 22668*x*y^2*z - 12354*x*y^2 - 252*x*y*z^5 + 2520*x*y*z^4 - 10536*x*y*z^3 + 22650*x*y*z^2 - 24636*x*y*z + 10764*x*y - 42*x*z^6 + 504*x*z^5 - 2634*x*z^4 + 7544*x*z^3 - 12282*x*z^2 + 10674*x*z - 3835*x + 6*y^7 + 42*y^6*z - 84*y^6 + 126*y^5*z^2 - 504*y^5*z + 528*y^5 + 210*y^4*z^3 - 1260*y^4*z^2 + 2640*y^4*z - 1904*y^4 + 210*y^3*z^4 - 1680*y^3*z^3 + 5280*y^3*z^2 - 7610*y^3*z + 4178*y^3 + 126*y^2*z^5 - 1260*y^2*z^4 + 5280*y^2*z^3 - 11406*y^2*z^2 + 12498*y^2*z - 5532*y^2 + 42*y*z^6 - 504*y*z^5 + 2640*y*z^4 - 7598*y*z^3 + 12462*y*z^2 - 10968*y*z + 4009*y + 6*z^7 - 84*z^6 + 528*z^5 - 1898*z^4 + 4142*z^3 - 5436*z^2 + 3941*z - 1, -6*x^7 + 42*x^6*y + 42*x^6*z - 84*x^6 - 126*x^5*y^2 - 252*x^5*y*z + 504*x^5*y - 126*x^5*z^2 + 504*x^5*z - 522*x^5 + 210*x^4*y^3 + 630*x^4*y^2*z - 1260*x^4*y^2 + 630*x^4*y*z^2 - 2520*x^4*y*z + 2616*x^4*y + 210*x^4*z^3 - 1260*x^4*z^2 + 2616*x^4*z - 1849*x^4 - 210*x^3*y^4 - 840*x^3*y^3*z + 1680*x^3*y^3 - 1260*x^3*y^2*z^2 + 5040*x^3*y^2*z - 5244*x^3*y^2 - 840*x^3*y*z^3 + 5040*x^3*y*z^2 - 10488*x^3*y*z + 7450*x^3*y - 210*x^3*z^4 + 1680*x^3*z^3 - 5244*x^3*z^2 + 7444*x^3*z - 3990*x^3 + 126*x^2*y^5 + 630*x^2*y^4*z - 1260*x^2*y^4 + 1260*x^2*y^3*z^2 - 5040*x^2*y^3*z + 5256*x^2*y^3 + 1260*x^2*y^2*z^3 - 7560*x^2*y^2*z^2 + 15768*x^2*y^2*z - 11256*x^2*y^2 + 630*x^2*y*z^4 - 5040*x^2*y*z^3 + 15768*x^2*y*z^2 - 22494*x^2*y*z + 12150*x^2*y + 126*x^2*z^5 - 1260*x^2*z^4 + 5256*x^2*z^3 - 11238*x^2*z^2 + 12114*x^2*z - 5214*x^2 - 42*x*y^6 - 252*x*y^5*z + 504*x*y^5 - 630*x*y^4*z^2 + 2520*x*y^4*z - 2634*x*y^4 - 840*x*y^3*z^3 + 5040*x*y^3*z^2 - 10536*x*y^3*z + 7558*x*y^3 - 630*x*y^2*z^4 + 5040*x*y^2*z^3 - 15804*x*y^2*z^2 + 22656*x*y^2*z - 12330*x*y^2 - 252*x*y*z^5 + 2520*x*y*z^4 - 10536*x*y*z^3 + 22638*x*y*z^2 - 24588*x*y*z + 10716*x*y - 42*x*z^6 + 504*x*z^5 - 2634*x*z^4 + 7540*x*z^3 - 12258*x*z^2 + 10626*x*z - 3802*x + 6*y^7 + 42*y^6*z - 84*y^6 + 126*y^5*z^2 - 504*y^5*z + 528*y^5 + 210*y^4*z^3 - 1260*y^4*z^2 + 2640*y^4*z - 1903*y^4 + 210*y^3*z^4 - 1680*y^3*z^3 + 5280*y^3*z^2 - 7606*y^3*z + 4170*y^3 + 126*y^2*z^5 - 1260*y^2*z^4 + 5280*y^2*z^3 - 11400*y^2*z^2 + 12474*y^2*z - 5508*y^2 + 42*y*z^6 - 504*y*z^5 + 2640*y*z^4 - 7594*y*z^3 + 12438*y*z^2 - 10920*y*z + 3975*y + 6*z^7 - 84*z^6 + 528*z^5 - 1897*z^4 + 4134*z^3 - 5412*z^2 + 3908*z - 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ 1, -1, -1 ], [ 1, -2, -1 ], [ 2, -3, -3 ] ], "source_shift": [ 2, 2, 1 ], "target_matrix": [ [ 2, 1, 1 ], [ -1, -2, -1 ], [ -1, -1, -1 ] ], "target_shift": [ -2375, 1203, 1185 ], "inverse_recipe": { "p": 4, "q": 3, "f": "x**4", "g": "x**3 + x", "h": "y*z", "polynomials": { "f": [ [ [ 4, 0, 0 ], 1 ] ], "g": [ [ [ 3, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ] ] }, "h_multiplier": 6 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0503666909971798 ], "surface_features": [ 3.766322566283044, 4.691435152144062, 3.401400540781544, 0.0, 3.9012186707605476, 4971 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B152", "pair": 152, "family": "bijection", "mapping": "F(x,y,z)=(-128*x^7 - 1344*x^6*y + 896*x^6*z + 896*x^6 - 6048*x^5*y^2 + 8064*x^5*y*z + 8064*x^5*y - 2688*x^5*z^2 - 5376*x^5*z - 2784*x^5 - 15120*x^4*y^3 + 30240*x^4*y^2*z + 30240*x^4*y^2 - 20160*x^4*y*z^2 - 40320*x^4*y*z - 20848*x^4*y + 4480*x^4*z^3 + 13440*x^4*z^2 + 13904*x^4*z + 5016*x^4 - 22680*x^3*y^4 + 60480*x^3*y^3*z + 60480*x^3*y^3 - 60480*x^3*y^2*z^2 - 120960*x^3*y^2*z - 62448*x^3*y^2 + 26880*x^3*y*z^3 + 80640*x^3*y*z^2 + 83296*x^3*y*z + 29964*x^3*y - 4480*x^3*z^4 - 17920*x^3*z^3 - 27776*x^3*z^2 - 20000*x^3*z - 5672*x^3 - 20412*x^2*y^5 + 68040*x^2*y^4*z + 68040*x^2*y^4 - 90720*x^2*y^3*z^2 - 181440*x^2*y^3*z - 93528*x^2*y^3 + 60480*x^2*y^2*z^3 + 181440*x^2*y^2*z^2 + 187128*x^2*y^2*z + 67122*x^2*y^2 - 20160*x^2*y*z^4 - 80640*x^2*y*z^3 - 124800*x^2*y*z^2 - 89604*x^2*y*z - 25312*x^2*y + 2688*x^2*z^5 + 13440*x^2*z^4 + 27744*x^2*z^3 + 29904*x^2*z^2 + 16916*x^2*z + 4028*x^2 - 10206*x*y^6 + 40824*x*y^5*z + 40824*x*y^5 - 68040*x*y^4*z^2 - 136080*x*y^4*z - 70038*x*y^4 + 60480*x*y^3*z^3 + 181440*x*y^3*z^2 + 186840*x*y^3*z + 66825*x*y^3 - 30240*x*y^2*z^4 - 120960*x*y^2*z^3 - 186912*x*y^2*z^2 - 133812*x*y^2*z - 37650*x*y^2 + 8064*x*y*z^5 + 40320*x*y*z^4 + 83104*x*y*z^3 + 89316*x*y*z^2 + 50324*x*y*z + 11920*x*y - 896*x*z^6 - 5376*x*z^5 - 13856*x*z^4 - 19872*x*z^3 - 16816*x*z^2 - 7981*x*z - 1668*x - 2187*y^7 + 10206*y^6*z + 10206*y^6 - 20412*y^5*z^2 - 40824*y^5*z - 20979*y^5 + 22680*y^4*z^3 + 68040*y^4*z^2 + 69957*y^4*z + 24948*y^4 - 15120*y^3*z^4 - 60480*y^3*z^3 - 93312*y^3*z^2 - 66609*y^3*z - 18666*y^3 + 6048*y^2*z^5 + 30240*y^2*z^4 + 62232*y^2*z^3 + 66690*y^2*z^2 + 37425*y^2*z + 8818*y^2 - 1344*y*z^6 - 8064*y*z^5 - 20752*y*z^4 - 29676*y*z^3 - 25012*y*z^2 - 11808*y*z - 2451*y + 128*z^7 + 896*z^6 + 2768*z^5 + 4952*z^4 + 5572*z^3 + 3953*z^2 + 1646*z + 1, -128*x^7 - 1344*x^6*y + 896*x^6*z + 896*x^6 - 6048*x^5*y^2 + 8064*x^5*y*z + 8064*x^5*y - 2688*x^5*z^2 - 5376*x^5*z - 2784*x^5 - 15120*x^4*y^3 + 30240*x^4*y^2*z + 30240*x^4*y^2 - 20160*x^4*y*z^2 - 40320*x^4*y*z - 20848*x^4*y + 4480*x^4*z^3 + 13440*x^4*z^2 + 13904*x^4*z + 5000*x^4 - 22680*x^3*y^4 + 60480*x^3*y^3*z + 60480*x^3*y^3 - 60480*x^3*y^2*z^2 - 120960*x^3*y^2*z - 62448*x^3*y^2 + 26880*x^3*y*z^3 + 80640*x^3*y*z^2 + 83296*x^3*y*z + 29868*x^3*y - 4480*x^3*z^4 - 17920*x^3*z^3 - 27776*x^3*z^2 - 19936*x^3*z - 5592*x^3 - 20412*x^2*y^5 + 68040*x^2*y^4*z + 68040*x^2*y^4 - 90720*x^2*y^3*z^2 - 181440*x^2*y^3*z - 93528*x^2*y^3 + 60480*x^2*y^2*z^3 + 181440*x^2*y^2*z^2 + 187128*x^2*y^2*z + 66906*x^2*y^2 - 20160*x^2*y*z^4 - 80640*x^2*y*z^3 - 124800*x^2*y*z^2 - 89316*x^2*y*z - 24952*x^2*y + 2688*x^2*z^5 + 13440*x^2*z^4 + 27744*x^2*z^3 + 29808*x^2*z^2 + 16676*x^2*z + 3880*x^2 - 10206*x*y^6 + 40824*x*y^5*z + 40824*x*y^5 - 68040*x*y^4*z^2 - 136080*x*y^4*z - 70038*x*y^4 + 60480*x*y^3*z^3 + 181440*x*y^3*z^2 + 186840*x*y^3*z + 66609*x*y^3 - 30240*x*y^2*z^4 - 120960*x*y^2*z^3 - 186912*x*y^2*z^2 - 133380*x*y^2*z - 37110*x*y^2 + 8064*x*y*z^5 + 40320*x*y*z^4 + 83104*x*y*z^3 + 89028*x*y*z^2 + 49604*x*y*z + 11476*x*y - 896*x*z^6 - 5376*x*z^5 - 13856*x*z^4 - 19808*x*z^3 - 16576*x*z^2 - 7685*x*z - 1537*x - 2187*y^7 + 10206*y^6*z + 10206*y^6 - 20412*y^5*z^2 - 40824*y^5*z - 20979*y^5 + 22680*y^4*z^3 + 68040*y^4*z^2 + 69957*y^4*z + 24867*y^4 - 15120*y^3*z^4 - 60480*y^3*z^3 - 93312*y^3*z^2 - 66393*y^3*z - 18396*y^3 + 6048*y^2*z^5 + 30240*y^2*z^4 + 62232*y^2*z^3 + 66474*y^2*z^2 + 36885*y^2*z + 8485*y^2 - 1344*y*z^6 - 8064*y*z^5 - 20752*y*z^4 - 29580*y*z^3 - 24652*y*z^2 - 11364*y*z - 2259*y + 128*z^7 + 896*z^6 + 2768*z^5 + 4936*z^4 + 5492*z^3 + 3805*z^2 + 1517*z + 1, -256*x^7 - 2688*x^6*y + 1792*x^6*z + 1792*x^6 - 12096*x^5*y^2 + 16128*x^5*y*z + 16128*x^5*y - 5376*x^5*z^2 - 10752*x^5*z - 5568*x^5 - 30240*x^4*y^3 + 60480*x^4*y^2*z + 60480*x^4*y^2 - 40320*x^4*y*z^2 - 80640*x^4*y*z - 41696*x^4*y + 8960*x^4*z^3 + 26880*x^4*z^2 + 27808*x^4*z + 10000*x^4 - 45360*x^3*y^4 + 120960*x^3*y^3*z + 120960*x^3*y^3 - 120960*x^3*y^2*z^2 - 241920*x^3*y^2*z - 124896*x^3*y^2 + 53760*x^3*y*z^3 + 161280*x^3*y*z^2 + 166592*x^3*y*z + 59736*x^3*y - 8960*x^3*z^4 - 35840*x^3*z^3 - 55552*x^3*z^2 - 39872*x^3*z - 11200*x^3 - 40824*x^2*y^5 + 136080*x^2*y^4*z + 136080*x^2*y^4 - 181440*x^2*y^3*z^2 - 362880*x^2*y^3*z - 187056*x^2*y^3 + 120960*x^2*y^2*z^3 + 362880*x^2*y^2*z^2 + 374256*x^2*y^2*z + 133812*x^2*y^2 - 40320*x^2*y*z^4 - 161280*x^2*y*z^3 - 249600*x^2*y*z^2 - 178632*x^2*y*z - 49976*x^2*y + 5376*x^2*z^5 + 26880*x^2*z^4 + 55488*x^2*z^3 + 59616*x^2*z^2 + 33400*x^2*z + 7808*x^2 - 20412*x*y^6 + 81648*x*y^5*z + 81648*x*y^5 - 136080*x*y^4*z^2 - 272160*x*y^4*z - 140076*x*y^4 + 120960*x*y^3*z^3 + 362880*x*y^3*z^2 + 373680*x*y^3*z + 133218*x*y^3 - 60480*x*y^2*z^4 - 241920*x*y^2*z^3 - 373824*x*y^2*z^2 - 266760*x*y^2*z - 74328*x*y^2 + 16128*x*y*z^5 + 80640*x*y*z^4 + 166208*x*y*z^3 + 178056*x*y*z^2 + 99352*x*y*z + 23096*x*y - 1792*x*z^6 - 10752*x*z^5 - 27712*x*z^4 - 39616*x*z^3 - 33200*x*z^2 - 15466*x*z - 3130*x - 4374*y^7 + 20412*y^6*z + 20412*y^6 - 40824*y^5*z^2 - 81648*y^5*z - 41958*y^5 + 45360*y^4*z^3 + 136080*y^4*z^2 + 139914*y^4*z + 49734*y^4 - 30240*y^3*z^4 - 120960*y^3*z^3 - 186624*y^3*z^2 - 132786*y^3*z - 36846*y^3 + 12096*y^2*z^5 + 60480*y^2*z^4 + 124464*y^2*z^3 + 132948*y^2*z^2 + 73878*y^2*z + 17078*y^2 - 2688*y*z^6 - 16128*y*z^5 - 41504*y*z^4 - 59160*y*z^3 - 49376*y*z^2 - 22872*y*z - 4598*y + 256*z^7 + 1792*z^6 + 5536*z^5 + 9872*z^4 + 11000*z^3 + 7658*z^2 + 3088*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -2, -3, 2 ], [ -1, -1, 1 ], [ -2, -1, 1 ] ], "source_shift": [ 2, 2, 2 ], "target_matrix": [ [ 1, 2, 1 ], [ 1, 1, 2 ], [ 2, 2, 3 ] ], "target_shift": [ -313, -265, -555 ], "inverse_recipe": { "p": 4, "q": 3, "f": "x**4 + x**2 + x", "g": "x**3 + x", "h": "y*z", "polynomials": { "f": [ [ [ 4, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "g": [ [ [ 3, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0961192909999227 ], "surface_features": [ 4.741053304409384, 5.573168771596262, 4.473370169288795, 0.0, 4.834055412996554, 5354 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B178", "pair": 178, "family": "bijection", "mapping": "F(x,y,z)=(3*x^7 + 21*x^6*y - 21*x^6*z - 42*x^6 + 63*x^5*y^2 - 126*x^5*y*z - 252*x^5*y + 63*x^5*z^2 + 252*x^5*z + 261*x^5 + 105*x^4*y^3 - 315*x^4*y^2*z - 630*x^4*y^2 + 315*x^4*y*z^2 + 1260*x^4*y*z + 1305*x^4*y - 105*x^4*z^3 - 630*x^4*z^2 - 1308*x^4*z - 932*x^4 + 105*x^3*y^4 - 420*x^3*y^3*z - 840*x^3*y^3 + 630*x^3*y^2*z^2 + 2520*x^3*y^2*z + 2610*x^3*y^2 - 420*x^3*y*z^3 - 2520*x^3*y*z^2 - 5232*x^3*y*z - 3725*x^3*y + 105*x^3*z^4 + 840*x^3*z^3 + 2622*x^3*z^2 + 3749*x^3*z + 2057*x^3 + 63*x^2*y^5 - 315*x^2*y^4*z - 630*x^2*y^4 + 630*x^2*y^3*z^2 + 2520*x^2*y^3*z + 2610*x^2*y^3 - 630*x^2*y^2*z^3 - 3780*x^2*y^2*z^2 - 7848*x^2*y^2*z - 5583*x^2*y^2 + 315*x^2*y*z^4 + 2520*x^2*y*z^3 + 7866*x^2*y*z^2 + 11238*x^2*y*z + 6153*x^2*y - 63*x^2*z^5 - 630*x^2*z^4 - 2628*x^2*z^3 - 5655*x^2*z^2 - 6228*x^2*z - 2801*x^2 + 21*x*y^6 - 126*x*y^5*z - 252*x*y^5 + 315*x*y^4*z^2 + 1260*x*y^4*z + 1305*x*y^4 - 420*x*y^3*z^3 - 2520*x*y^3*z^2 - 5232*x*y^3*z - 3719*x*y^3 + 315*x*y^2*z^4 + 2520*x*y^2*z^3 + 7866*x*y^2*z^2 + 11229*x*y^2*z + 6135*x*y^2 - 126*x*y*z^5 - 1260*x*y*z^4 - 5256*x*y*z^3 - 11301*x*y*z^2 - 12420*x*y*z - 5560*x*y + 21*x*z^6 + 252*x*z^5 + 1317*x*z^4 + 3791*x*z^3 + 6285*x*z^2 + 5677*x*z + 2154*x + 3*y^7 - 21*y^6*z - 42*y^6 + 63*y^5*z^2 + 252*y^5*z + 261*y^5 - 105*y^4*z^3 - 630*y^4*z^2 - 1308*y^4*z - 929*y^4 + 105*y^3*z^4 + 840*y^3*z^3 + 2622*y^3*z^2 + 3740*y^3*z + 2039*y^3 - 63*y^2*z^5 - 630*y^2*z^4 - 2628*y^2*z^3 - 5646*y^2*z^2 - 6192*y^2*z - 2759*y^2 + 21*y*z^6 + 252*y*z^5 + 1317*y*z^4 + 3788*y*z^3 + 6267*y*z^2 + 5632*y*z + 2125*y - 3*z^7 - 42*z^6 - 264*z^5 - 953*z^4 - 2114*z^3 - 2873*z^2 - 2198*z - 1, -6*x^7 - 42*x^6*y + 42*x^6*z + 84*x^6 - 126*x^5*y^2 + 252*x^5*y*z + 504*x^5*y - 126*x^5*z^2 - 504*x^5*z - 522*x^5 - 210*x^4*y^3 + 630*x^4*y^2*z + 1260*x^4*y^2 - 630*x^4*y*z^2 - 2520*x^4*y*z - 2610*x^4*y + 210*x^4*z^3 + 1260*x^4*z^2 + 2616*x^4*z + 1866*x^4 - 210*x^3*y^4 + 840*x^3*y^3*z + 1680*x^3*y^3 - 1260*x^3*y^2*z^2 - 5040*x^3*y^2*z - 5220*x^3*y^2 + 840*x^3*y*z^3 + 5040*x^3*y*z^2 + 10464*x^3*y*z + 7458*x^3*y - 210*x^3*z^4 - 1680*x^3*z^3 - 5244*x^3*z^2 - 7506*x^3*z - 4130*x^3 - 126*x^2*y^5 + 630*x^2*y^4*z + 1260*x^2*y^4 - 1260*x^2*y^3*z^2 - 5040*x^2*y^3*z - 5220*x^2*y^3 + 1260*x^2*y^2*z^3 + 7560*x^2*y^2*z^2 + 15696*x^2*y^2*z + 11178*x^2*y^2 - 630*x^2*y*z^4 - 5040*x^2*y*z^3 - 15732*x^2*y*z^2 - 22500*x^2*y*z - 12354*x^2*y + 126*x^2*z^5 + 1260*x^2*z^4 + 5256*x^2*z^3 + 11322*x^2*z^2 + 12504*x^2*z + 5652*x^2 - 42*x*y^6 + 252*x*y^5*z + 504*x*y^5 - 630*x*y^4*z^2 - 2520*x*y^4*z - 2610*x*y^4 + 840*x*y^3*z^3 + 5040*x*y^3*z^2 + 10464*x*y^3*z + 7446*x*y^3 - 630*x*y^2*z^4 - 5040*x*y^2*z^3 - 15732*x*y^2*z^2 - 22482*x*y^2*z - 12318*x*y^2 + 252*x*y*z^5 + 2520*x*y*z^4 + 10512*x*y*z^3 + 22626*x*y*z^2 + 24936*x*y*z + 11220*x*y - 42*x*z^6 - 504*x*z^5 - 2634*x*z^4 - 7590*x*z^3 - 12618*x*z^2 - 11454*x*z - 4386*x - 6*y^7 + 42*y^6*z + 84*y^6 - 126*y^5*z^2 - 504*y^5*z - 522*y^5 + 210*y^4*z^3 + 1260*y^4*z^2 + 2616*y^4*z + 1860*y^4 - 210*y^3*z^4 - 1680*y^3*z^3 - 5244*y^3*z^2 - 7488*y^3*z - 4094*y^3 + 126*y^2*z^5 + 1260*y^2*z^4 + 5256*y^2*z^3 + 11304*y^2*z^2 + 12432*y^2*z + 5568*y^2 - 42*y*z^6 - 504*y*z^5 - 2634*y*z^4 - 7584*y*z^3 - 12582*y*z^2 - 11364*y*z - 4326*y + 6*z^7 + 84*z^6 + 528*z^5 + 1908*z^4 + 4244*z^3 + 5796*z^2 + 4472*z + 1, 3*x^7 + 21*x^6*y - 21*x^6*z - 42*x^6 + 63*x^5*y^2 - 126*x^5*y*z - 252*x^5*y + 63*x^5*z^2 + 252*x^5*z + 261*x^5 + 105*x^4*y^3 - 315*x^4*y^2*z - 630*x^4*y^2 + 315*x^4*y*z^2 + 1260*x^4*y*z + 1305*x^4*y - 105*x^4*z^3 - 630*x^4*z^2 - 1308*x^4*z - 934*x^4 + 105*x^3*y^4 - 420*x^3*y^3*z - 840*x^3*y^3 + 630*x^3*y^2*z^2 + 2520*x^3*y^2*z + 2610*x^3*y^2 - 420*x^3*y*z^3 - 2520*x^3*y*z^2 - 5232*x^3*y*z - 3733*x^3*y + 105*x^3*z^4 + 840*x^3*z^3 + 2622*x^3*z^2 + 3757*x^3*z + 2074*x^3 + 63*x^2*y^5 - 315*x^2*y^4*z - 630*x^2*y^4 + 630*x^2*y^3*z^2 + 2520*x^2*y^3*z + 2610*x^2*y^3 - 630*x^2*y^2*z^3 - 3780*x^2*y^2*z^2 - 7848*x^2*y^2*z - 5595*x^2*y^2 + 315*x^2*y*z^4 + 2520*x^2*y*z^3 + 7866*x^2*y*z^2 + 11262*x^2*y*z + 6204*x^2*y - 63*x^2*z^5 - 630*x^2*z^4 - 2628*x^2*z^3 - 5667*x^2*z^2 - 6279*x^2*z - 2857*x^2 + 21*x*y^6 - 126*x*y^5*z - 252*x*y^5 + 315*x*y^4*z^2 + 1260*x*y^4*z + 1305*x*y^4 - 420*x*y^3*z^3 - 2520*x*y^3*z^2 - 5232*x*y^3*z - 3727*x*y^3 + 315*x*y^2*z^4 + 2520*x*y^2*z^3 + 7866*x*y^2*z^2 + 11253*x*y^2*z + 6186*x*y^2 - 126*x*y*z^5 - 1260*x*y*z^4 - 5256*x*y*z^3 - 11325*x*y*z^2 - 12522*x*y*z - 5672*x*y + 21*x*z^6 + 252*x*z^5 + 1317*x*z^4 + 3799*x*z^3 + 6336*x*z^2 + 5789*x*z + 2246*x + 3*y^7 - 21*y^6*z - 42*y^6 + 63*y^5*z^2 + 252*y^5*z + 261*y^5 - 105*y^4*z^3 - 630*y^4*z^2 - 1308*y^4*z - 931*y^4 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"(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0985208760030218 ], "surface_features": [ 3.469266729298085, 4.396826789268784, 3.100370545117563, 0.0, 3.6029543699521756, 4846 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B176", "pair": 176, "family": "bijection", "mapping": "F(x,y,z)=(8*x^7 - 56*x^6*y - 56*x^6*z + 128*x^6 + 168*x^5*y^2 + 336*x^5*y*z - 776*x^5*y + 168*x^5*z^2 - 768*x^5*z + 868*x^5 - 280*x^4*y^3 - 840*x^4*y^2*z + 1960*x^4*y^2 - 840*x^4*y*z^2 + 3880*x^4*y*z - 4428*x^4*y - 280*x^4*z^3 + 1920*x^4*z^2 - 4340*x^4*z + 3244*x^4 + 280*x^3*y^4 + 1120*x^3*y^3*z - 2640*x^3*y^3 + 1680*x^3*y^2*z^2 - 7840*x^3*y^2*z + 9032*x^3*y^2 + 1120*x^3*y*z^3 - 7760*x^3*y*z^2 + 17712*x^3*y*z - 13368*x^3*y + 280*x^3*z^4 - 2560*x^3*z^3 + 8680*x^3*z^2 - 12976*x^3*z + 7260*x^3 - 168*x^2*y^5 - 840*x^2*y^4*z + 2000*x^2*y^4 - 1680*x^2*y^3*z^2 + 7920*x^2*y^3*z - 9208*x^2*y^3 - 1680*x^2*y^2*z^3 + 11760*x^2*y^2*z^2 - 27096*x^2*y^2*z + 20640*x^2*y^2 - 840*x^2*y*z^4 + 7760*x^2*y*z^3 - 26568*x^2*y*z^2 + 40104*x^2*y*z - 22684*x^2*y - 168*x^2*z^5 + 1920*x^2*z^4 - 8680*x^2*z^3 + 19464*x^2*z^2 - 21788*x^2*z + 9811*x^2 + 56*x*y^6 + 336*x*y^5*z - 808*x*y^5 + 840*x*y^4*z^2 - 4000*x*y^4*z + 4692*x*y^4 + 1120*x*y^3*z^3 - 7920*x*y^3*z^2 + 18416*x*y^3*z - 14152*x*y^3 + 840*x*y^2*z^4 - 7840*x*y^2*z^3 + 27096*x*y^2*z^2 - 41280*x*y^2*z + 23588*x*y^2 + 336*x*y*z^5 - 3880*x*y*z^4 + 17712*x*y*z^3 - 40104*x*y*z^2 + 45384*x*y*z - 20758*x*y + 56*x*z^6 - 768*x*z^5 + 4340*x*z^4 - 12976*x*z^3 + 21796*x*z^2 - 19662*x*z + 7429*x - 8*y^7 - 56*y^6*z + 136*y^6 - 168*y^5*z^2 + 808*y^5*z - 956*y^5 - 280*y^4*z^3 + 2000*y^4*z^2 - 4692*y^4*z + 3636*y^4 - 280*y^3*z^4 + 2640*y^3*z^3 - 9208*y^3*z^2 + 14152*y^3*z - 8164*y^3 - 168*y^2*z^5 + 1960*y^2*z^4 - 9032*y^2*z^3 + 20640*y^2*z^2 - 23596*y^2*z + 10963*y^2 - 56*y*z^6 + 776*y*z^5 - 4428*y*z^4 + 13368*y*z^3 - 22700*y*z^2 + 20806*y*z - 8036*y - 8*z^7 + 128*z^6 - 868*z^5 + 3244*z^4 - 7268*z^3 + 9851*z^2 - 7465*z + 1, -4*x^7 + 28*x^6*y + 28*x^6*z - 64*x^6 - 84*x^5*y^2 - 168*x^5*y*z + 388*x^5*y - 84*x^5*z^2 + 384*x^5*z - 434*x^5 + 140*x^4*y^3 + 420*x^4*y^2*z - 980*x^4*y^2 + 420*x^4*y*z^2 - 1940*x^4*y*z + 2214*x^4*y + 140*x^4*z^3 - 960*x^4*z^2 + 2170*x^4*z - 1622*x^4 - 140*x^3*y^4 - 560*x^3*y^3*z + 1320*x^3*y^3 - 840*x^3*y^2*z^2 + 3920*x^3*y^2*z - 4516*x^3*y^2 - 560*x^3*y*z^3 + 3880*x^3*y*z^2 - 8856*x^3*y*z + 6684*x^3*y - 140*x^3*z^4 + 1280*x^3*z^3 - 4340*x^3*z^2 + 6488*x^3*z - 3630*x^3 + 84*x^2*y^5 + 420*x^2*y^4*z - 1000*x^2*y^4 + 840*x^2*y^3*z^2 - 3960*x^2*y^3*z + 4604*x^2*y^3 + 840*x^2*y^2*z^3 - 5880*x^2*y^2*z^2 + 13548*x^2*y^2*z - 10320*x^2*y^2 + 420*x^2*y*z^4 - 3880*x^2*y*z^3 + 13284*x^2*y*z^2 - 20052*x^2*y*z + 11342*x^2*y + 84*x^2*z^5 - 960*x^2*z^4 + 4340*x^2*z^3 - 9732*x^2*z^2 + 10894*x^2*z - 4906*x^2 - 28*x*y^6 - 168*x*y^5*z + 404*x*y^5 - 420*x*y^4*z^2 + 2000*x*y^4*z - 2346*x*y^4 - 560*x*y^3*z^3 + 3960*x*y^3*z^2 - 9208*x*y^3*z + 7076*x*y^3 - 420*x*y^2*z^4 + 3920*x*y^2*z^3 - 13548*x*y^2*z^2 + 20640*x*y^2*z - 11794*x*y^2 - 168*x*y*z^5 + 1940*x*y*z^4 - 8856*x*y*z^3 + 20052*x*y*z^2 - 22692*x*y*z + 10380*x*y - 28*x*z^6 + 384*x*z^5 - 2170*x*z^4 + 6488*x*z^3 - 10898*x*z^2 + 9832*x*z - 3717*x + 4*y^7 + 28*y^6*z - 68*y^6 + 84*y^5*z^2 - 404*y^5*z + 478*y^5 + 140*y^4*z^3 - 1000*y^4*z^2 + 2346*y^4*z - 1818*y^4 + 140*y^3*z^4 - 1320*y^3*z^3 + 4604*y^3*z^2 - 7076*y^3*z + 4082*y^3 + 84*y^2*z^5 - 980*y^2*z^4 + 4516*y^2*z^3 - 10320*y^2*z^2 + 11798*y^2*z - 5482*y^2 + 28*y*z^6 - 388*y*z^5 + 2214*y*z^4 - 6684*y*z^3 + 11350*y*z^2 - 10404*y*z + 4021*y + 4*z^7 - 64*z^6 + 434*z^5 - 1622*z^4 + 3634*z^3 - 4926*z^2 + 3735*z - 1, -4*x^7 + 28*x^6*y + 28*x^6*z - 64*x^6 - 84*x^5*y^2 - 168*x^5*y*z + 388*x^5*y - 84*x^5*z^2 + 384*x^5*z - 432*x^5 + 140*x^4*y^3 + 420*x^4*y^2*z - 980*x^4*y^2 + 420*x^4*y*z^2 - 1940*x^4*y*z + 2204*x^4*y + 140*x^4*z^3 - 960*x^4*z^2 + 2160*x^4*z - 1600*x^4 - 140*x^3*y^4 - 560*x^3*y^3*z + 1320*x^3*y^3 - 840*x^3*y^2*z^2 + 3920*x^3*y^2*z - 4496*x^3*y^2 - 560*x^3*y*z^3 + 3880*x^3*y*z^2 - 8816*x^3*y*z + 6596*x^3*y - 140*x^3*z^4 + 1280*x^3*z^3 - 4320*x^3*z^2 + 6400*x^3*z - 3532*x^3 + 84*x^2*y^5 + 420*x^2*y^4*z - 1000*x^2*y^4 + 840*x^2*y^3*z^2 - 3960*x^2*y^3*z + 4584*x^2*y^3 + 840*x^2*y^2*z^3 - 5880*x^2*y^2*z^2 + 13488*x^2*y^2*z - 10188*x^2*y^2 + 420*x^2*y*z^4 - 3880*x^2*y*z^3 + 13224*x^2*y*z^2 - 19788*x^2*y*z + 11048*x^2*y + 84*x^2*z^5 - 960*x^2*z^4 + 4320*x^2*z^3 - 9600*x^2*z^2 + 10600*x^2*z - 4685*x^2 - 28*x*y^6 - 168*x*y^5*z + 404*x*y^5 - 420*x*y^4*z^2 + 2000*x*y^4*z - 2336*x*y^4 - 560*x*y^3*z^3 + 3960*x*y^3*z^2 - 9168*x*y^3*z + 6988*x*y^3 - 420*x*y^2*z^4 + 3920*x*y^2*z^3 - 13488*x*y^2*z^2 + 20376*x*y^2*z - 11500*x*y^2 - 168*x*y*z^5 + 1940*x*y*z^4 - 8816*x*y*z^3 + 19788*x*y*z^2 - 22104*x*y*z + 9938*x*y - 28*x*z^6 + 384*x*z^5 - 2160*x*z^4 + 6400*x*z^3 - 10604*x*z^2 + 9390*x*z - 3458*x + 4*y^7 + 28*y^6*z - 68*y^6 + 84*y^5*z^2 - 404*y^5*z + 476*y^5 + 140*y^4*z^3 - 1000*y^4*z^2 + 2336*y^4*z - 1796*y^4 + 140*y^3*z^4 - 1320*y^3*z^3 + 4584*y^3*z^2 - 6988*y^3*z + 3984*y^3 + 84*y^2*z^5 - 980*y^2*z^4 + 4496*y^2*z^3 - 10188*y^2*z^2 + 11504*y^2*z - 5261*y^2 + 28*y*z^6 - 388*y*z^5 + 2204*y*z^4 - 6596*y*z^3 + 11056*y*z^2 - 9962*y*z + 3759*y + 4*z^7 - 64*z^6 + 432*z^5 - 1600*z^4 + 3536*z^3 - 4705*z^2 + 3474*z - 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ 1, -1, -1 ], [ 1, -2, -1 ], [ 2, -3, -3 ] ], "source_shift": [ 2, 1, 2 ], "target_matrix": [ [ 2, 3, 2 ], [ -1, -2, -1 ], [ -1, -1, -2 ] ], "target_shift": [ -2418, 1211, 1086 ], "inverse_recipe": { "p": 2, "q": 5, "f": "x**2", "g": "x**5 + x**4 + x**3 + x", "h": "y**2 + y*z + y", "polynomials": { "f": [ [ [ 2, 0, 0 ], 1 ] ], "g": [ [ [ 5, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ], [ [ 3, 0, 0 ], 1 ], [ [ 4, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 2, 0 ], 1 ], [ [ 0, 1, 0 ], 1 ] ] }, "h_multiplier": 4 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0564136760003748 ], "surface_features": [ 3.7082001083487346, 4.656902770556646, 3.3010299956639813, 0.0, 3.8556396364778514, 4901 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C018", "pair": 18, "family": "counterexample", "mapping": "F(x,y,z)=(8*x^7 + 68*x^6*y - 44*x^6*z - 116*x^6 + 234*x^5*y^2 - 312*x^5*y*z - 828*x^5*y + 102*x^5*z^2 + 540*x^5*z + 696*x^5 + 419*x^4*y^3 - 867*x^4*y^2*z - 2337*x^4*y^2 + 585*x^4*y*z^2 + 3126*x^4*y*z + 4104*x^4*y - 129*x^4*z^3 - 1029*x^4*z^2 - 2676*x^4*z - 2278*x^4 + 419*x^3*y^4 - 1199*x^3*y^3*z - 3326*x^3*y^3 + 1257*x^3*y^2*z^2 + 6846*x^3*y^2*z + 9222*x^3*y^2 - 573*x^3*y*z^3 - 4626*x^3*y*z^2 - 12276*x^3*y*z - 10719*x^3*y + 96*x^3*z^4 + 1026*x^3*z^3 + 4038*x^3*z^2 + 6957*x^3*z + 4432*x^3 + 234*x^2*y^5 - 867*x^2*y^4*z - 2511*x^2*y^4 + 1257*x^2*y^3*z^2 + 7065*x^2*y^3*z + 9816*x^2*y^3 - 891*x^2*y^2*z^3 - 7353*x^2*y^2*z^2 - 20025*x^2*y^2*z - 17967*x^2*y^2 + 309*x^2*y*z^4 + 3351*x^2*y*z^3 + 13470*x^2*y*z^2 + 23793*x^2*y*z + 15579*x^2*y - 42*x^2*z^5 - 564*x^2*z^4 - 2985*x^2*z^3 - 7800*x^2*z^2 - 10071*x^2*z - 5146*x^2 + 68*x*y^6 - 312*x*y^5*z - 948*x*y^5 + 585*x*y^4*z^2 + 3426*x*y^4*z + 4896*x*y^4 - 573*x*y^3*z^3 - 4884*x*y^3*z^2 - 13668*x*y^3*z - 12572*x*y^3 + 309*x*y^2*z^4 + 3432*x*y^2*z^3 + 14136*x*y^2*z^2 + 25584*x*y^2*z + 17205*x*y^2 - 87*x*y*z^5 - 1188*x*y*z^4 - 6420*x*y*z^3 - 17169*x*y*z^2 - 22749*x*y*z - 11968*x*y + 10*x*z^6 + 162*x*z^5 + 1080*x*z^4 + 3799*x*z^3 + 7446*x*z^2 + 7724*x*z + 3324*x + 8*y^7 - 44*y^6*z - 140*y^6 + 102*y^5*z^2 + 624*y^5*z + 912*y^5 - 129*y^4*z^3 - 1143*y^4*z^2 - 3276*y^4*z - 3066*y^4 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624*x^5*y*z + 1812*x^5*y - 204*x^5*z^2 - 1188*x^5*z - 1620*x^5 - 838*x^4*y^3 + 1734*x^4*y^2*z + 5064*x^4*y^2 - 1170*x^4*y*z^2 - 6816*x^4*y*z - 9456*x^4*y + 258*x^4*z^3 + 2256*x^4*z^2 + 6204*x^4*z + 5508*x^4 - 838*x^3*y^4 + 2398*x^3*y^3*z + 7129*x^3*y^3 - 2514*x^3*y^2*z^2 - 14775*x^3*y^2*z - 21009*x^3*y^2 + 1146*x^3*y*z^3 + 10047*x^3*y*z^2 + 28158*x^3*y*z + 25606*x^3*y - 192*x^3*z^4 - 2241*x^3*z^3 - 9321*x^3*z^2 - 16738*x^3*z - 11052*x^3 - 468*x^2*y^5 + 1734*x^2*y^4*z + 5325*x^2*y^4 - 2514*x^2*y^3*z^2 - 15084*x^2*y^3*z - 22095*x^2*y^3 + 1782*x^2*y^2*z^3 + 15804*x^2*y^2*z^2 + 45396*x^2*y^2*z + 42396*x^2*y^2 - 618*x^2*y*z^4 - 7248*x^2*y*z^3 - 30747*x^2*y*z^2 - 56538*x^2*y*z - 38286*x^2*y + 84*x^2*z^5 + 1227*x^2*z^4 + 6858*x^2*z^3 + 18666*x^2*z^2 + 24954*x^2*z + 13178*x^2 - 136*x*y^6 + 624*x*y^5*z + 1992*x*y^5 - 1170*x*y^4*z^2 - 7242*x*y^4*z - 10884*x*y^4 + 1146*x*y^3*z^3 + 10389*x*y^3*z^2 + 30606*x*y^3*z + 29280*x*y^3 - 618*x*y^2*z^4 - 7347*x*y^2*z^3 - 31881*x*y^2*z^2 - 60018*x*y^2*z - 41670*x*y^2 + 174*x*y*z^5 + 2559*x*y*z^4 + 14580*x*y*z^3 + 40566*x*y*z^2 + 55530*x*y*z + 30089*x*y - 20*x*z^6 - 351*x*z^5 - 2469*x*z^4 - 9040*x*z^3 - 18324*x*z^2 - 19609*x*z - 8685*x - 16*y^7 + 88*y^6*z + 292*y^6 - 204*y^5*z^2 - 1308*y^5*z - 2004*y^5 + 258*y^4*z^3 + 2409*y^4*z^2 + 7248*y^4*z + 7046*y^4 - 192*y^3*z^4 - 2334*y^3*z^3 - 10347*y^3*z^2 - 19850*y^3*z - 14022*y^3 + 84*y^2*z^5 + 1254*y^2*z^4 + 7290*y^2*z^3 + 20688*y^2*z^2 + 28884*y^2*z + 15989*y^2 - 20*y*z^6 - 354*y*z^5 - 2535*y*z^4 - 9464*y*z^3 - 19578*y*z^2 - 21416*y*z - 9723*y + 2*z^7 + 41*z^6 + 348*z^5 + 1604*z^4 + 4372*z^3 + 7091*z^2 + 6364*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -2, -1, 1 ], [ -1, -2, 1 ], [ -1, -1, 1 ] ], "source_shift": [ 2, 2, 1 ], "target_matrix": [ [ -1, -1, -2 ], [ -1, -1, -1 ], [ 2, 3, 2 ] ], "target_shift": [ 924, 952, -2441 ], "collision_preimages": [ [ "3/4", "3/4", "1/4" ], [ "13/2", "9", "21" ], [ "17/2", "6", "20" ] ], "collision_image": [ "3697/4", "3809/4", "-4883/2" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0081808709983306 ], "surface_features": [ 3.7960955669270984, 4.77828151918887, 3.3961993470957363, 0.0, 3.9710840171272035, 4932 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C067", "pair": 67, "family": "counterexample", "mapping": "F(x,y,z)=(2*x^7 - 20*x^6*y - 16*x^6*z + 31*x^6 + 84*x^5*y^2 + 138*x^5*y*z - 276*x^5*y + 54*x^5*z^2 - 207*x^5*z + 195*x^5 - 192*x^4*y^3 - 486*x^4*y^2*z + 1008*x^4*y^2 - 390*x^4*y*z^2 + 1539*x^4*y*z - 1482*x^4*y - 100*x^4*z^3 + 570*x^4*z^2 - 1071*x^4*z + 667*x^4 + 258*x^3*y^4 + 894*x^3*y^3*z - 1932*x^3*y^3 + 1104*x^3*y^2*z^2 - 4503*x^3*y^2*z + 4443*x^3*y^2 + 580*x^3*y*z^3 - 3396*x^3*y*z^2 + 6516*x^3*y*z - 4138*x^3*y + 110*x^3*z^4 - 830*x^3*z^3 + 2334*x^3*z^2 - 2910*x^3*z + 1352*x^3 - 204*x^2*y^5 - 906*x^2*y^4*z + 2049*x^2*y^4 - 1530*x^2*y^3*z^2 + 6477*x^2*y^3*z - 6564*x^2*y^3 - 1236*x^2*y^2*z^3 + 7461*x^2*y^2*z^2 - 14649*x^2*y^2*z + 9501*x^2*y^2 - 480*x^2*y*z^4 + 3714*x^2*y*z^3 - 10656*x^2*y*z^2 + 13542*x^2*y*z - 6414*x^2*y - 72*x^2*z^5 + 675*x^2*z^4 - 2526*x^2*z^3 + 4728*x^2*z^2 - 4410*x^2*z + 1631*x^2 + 88*x*y^6 + 480*x*y^5*z - 1140*x*y^5 + 1038*x*y^4*z^2 - 4578*x*y^4*z + 4776*x*y^4 + 1146*x*y^3*z^3 - 7158*x*y^3*z^2 + 14412*x*y^3*z - 9562*x*y^3 + 684*x*y^2*z^4 - 5445*x*y^2*z^3 + 15969*x*y^2*z^2 - 20712*x*y^2*z + 10014*x*y^2 + 210*x*y*z^5 - 2016*x*y*z^4 + 7692*x*y*z^3 - 14670*x*y*z^2 + 13950*x*y*z - 5264*x*y + 26*x*z^6 - 291*x*z^5 + 1359*x*z^4 - 3394*x*z^3 + 4764*x*z^2 - 3547*x*z + 1087*x - 16*y^7 - 104*y^6*z + 260*y^6 - 276*y^5*z^2 + 1272*y^5*z - 1368*y^5 - 390*y^4*z^3 + 2529*y^4*z^2 - 5232*y^4*z + 3556*y^4 - 318*y^3*z^4 + 2613*y^3*z^3 - 7848*y^3*z^2 + 10402*y^3*z - 5144*y^3 - 150*y^2*z^5 + 1479*y^2*z^4 - 5763*y^2*z^3 + 11211*y^2*z^2 - 10878*y^2*z + 4193*y^2 - 38*y*z^6 + 435*y*z^5 - 2070*y*z^4 + 5266*y*z^3 - 7536*y*z^2 + 5729*y*z - 1788*y - 4*z^7 + 52*z^6 - 291*z^5 + 909*z^4 - 1706*z^3 + 1916*z^2 - 1187*z + 1, -x^7 + 10*x^6*y + 8*x^6*z - 20*x^6 - 42*x^5*y^2 - 69*x^5*y*z + 174*x^5*y - 27*x^5*z^2 + 135*x^5*z - 156*x^5 + 96*x^4*y^3 + 243*x^4*y^2*z - 621*x^4*y^2 + 195*x^4*y*z^2 - 981*x^4*y*z + 1155*x^4*y + 50*x^4*z^3 - 375*x^4*z^2 + 864*x^4*z - 621*x^4 - 129*x^3*y^4 - 447*x^3*y^3*z + 1164*x^3*y^3 - 552*x^3*y^2*z^2 + 2805*x^3*y^2*z - 3369*x^3*y^2 - 290*x^3*y*z^3 + 2184*x^3*y*z^2 - 5121*x^3*y*z + 3759*x^3*y - 55*x^3*z^4 + 550*x^3*z^3 - 1896*x^3*z^2 + 2726*x^3*z - 1404*x^3 + 102*x^2*y^5 + 453*x^2*y^4*z - 1209*x^2*y^4 + 765*x^2*y^3*z^2 - 3945*x^2*y^3*z + 4839*x^2*y^3 + 618*x^2*y^2*z^3 - 4689*x^2*y^2*z^2 + 11199*x^2*y^2*z - 8406*x^2*y^2 + 240*x^2*y*z^4 - 2406*x^2*y*z^3 + 8433*x^2*y*z^2 - 12375*x^2*y*z + 6510*x^2*y + 36*x^2*z^5 - 450*x^2*z^4 + 2064*x^2*z^3 - 4452*x^2*z^2 + 4602*x^2*z - 1840*x^2 - 44*x*y^6 - 240*x*y^5*z + 660*x*y^5 - 519*x*y^4*z^2 + 2730*x*y^4*z 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129*x^3*y^4 - 447*x^3*y^3*z + 1032*x^3*y^3 - 552*x^3*y^2*z^2 + 2436*x^3*y^2*z - 2604*x^3*y^2 - 290*x^3*y*z^3 + 1860*x^3*y*z^2 - 3879*x^3*y*z + 2631*x^3*y - 55*x^3*z^4 + 460*x^3*z^3 - 1410*x^3*z^2 + 1877*x^3*z - 915*x^3 + 102*x^2*y^5 + 453*x^2*y^4*z - 1086*x^2*y^4 + 765*x^2*y^3*z^2 - 3474*x^2*y^3*z + 3801*x^2*y^3 + 618*x^2*y^2*z^3 - 4050*x^2*y^2*z^2 + 8616*x^2*y^2*z - 5967*x^2*y^2 + 240*x^2*y*z^4 - 2040*x^2*y*z^3 + 6363*x^2*y*z^2 - 8634*x^2*y*z + 4296*x^2*y + 36*x^2*z^5 - 375*x^2*z^4 + 1530*x^2*z^3 - 3057*x^2*z^2 + 2991*x^2*z - 1144*x^2 - 44*x*y^6 - 240*x*y^5*z + 600*x*y^5 - 519*x*y^4*z^2 + 2436*x*y^4*z - 2733*x*y^4 - 573*x*y^3*z^3 + 3852*x*y^3*z^2 - 8373*x*y^3*z + 5928*x*y^3 - 342*x*y^2*z^4 + 2964*x*y^2*z^3 - 9420*x*y^2*z^2 + 13041*x*y^2*z - 6630*x*y^2 - 105*x*y*z^5 + 1110*x*y*z^4 - 4605*x*y*z^3 + 9375*x*y*z^2 - 9363*x*y*z + 3661*x*y - 13*x*z^6 + 162*x*z^5 - 825*x*z^4 + 2199*x*z^3 - 3237*x*z^2 + 2492*x*z - 781*x + 8*y^7 + 52*y^6*z - 136*y^6 + 138*y^5*z^2 - 672*y^5*z + 774*y^5 + 195*y^4*z^3 - 1350*y^4*z^2 + 3003*y^4*z - 2175*y^4 + 159*y^3*z^4 - 1410*y^3*z^3 + 4572*y^3*z^2 - 6463*y^3*z + 3363*y^3 + 75*y^2*z^5 - 807*y^2*z^4 + 3408*y^2*z^3 - 7074*y^2*z^2 + 7215*y^2*z - 2887*y^2 + 19*y*z^6 - 240*y*z^5 + 1242*y*z^4 - 3372*y*z^3 + 5067*y*z^2 - 3991*y*z + 1276*y + 2*z^7 - 29*z^6 + 177*z^5 - 590*z^4 + 1161*z^3 - 1348*z^2 + 854*z - 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ 1, -1, -1 ], [ 1, -2, -1 ], [ 1, -1, -2 ] ], "source_shift": [ 2, 1, 2 ], "target_matrix": [ [ 2, 1, 1 ], [ -1, -2, -1 ], [ -1, -1, -1 ] ], "target_shift": [ -312, 397, 228 ], "collision_preimages": [ [ "-11/4", "-1", "1/4" ], [ "-5", "3/2", "-11/2" ], [ "-14", "-7/2", "-15/2" ] ], "collision_image": [ "-625/2", "1589/4", "913/4" ] }, "probe_target": "No", "jev_answers": [ "Maybe" ], "seconds": [ 0.9686365119996481 ], "surface_features": [ 3.43705524099582, 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6989*y + 6*z^7 + 102*z^6 + 728*z^5 + 2836*z^4 + 6540*z^3 + 9001*z^2 + 6945*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ 1, 1, 1 ], [ 2, 1, 1 ], [ 1, 2, 1 ] ], "source_shift": [ 2, 1, 2 ], "target_matrix": [ [ 1, 2, 1 ], [ 1, 1, 1 ], [ 3, 3, 2 ] ], "target_shift": [ -757, -750, -2356 ], "inverse_recipe": { "p": 2, "q": 5, "f": "x**2 + x", "g": "x**5 + x**4 + x", "h": "y*z + y + z", "polynomials": { "f": [ [ [ 2, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "g": [ [ [ 5, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ], [ [ 4, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 0, 1 ], 1 ], [ [ 0, 1, 0 ], 1 ] ] }, "h_multiplier": 2 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 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3318*x^2*y*z^4 + 28494*x^2*y*z^3 - 63366*x^2*y*z^2 + 56661*x^2*y*z - 18225*x^2*y - 492*x^2*z^5 + 5376*x^2*z^4 - 15978*x^2*z^3 + 21330*x^2*z^2 - 13593*x^2*z + 3420*x^2 + 702*x*y^6 + 3294*x*y^5*z - 6534*x*y^5 + 6372*x*y^4*z^2 - 25434*x*y^4*z + 18576*x*y^4 + 6502*x*y^3*z^3 - 39234*x*y^3*z^2 + 57312*x*y^3*z - 25595*x*y^3 + 3690*x*y^2*z^4 - 29982*x*y^2*z^3 + 65730*x*y^2*z^2 - 58416*x*y^2*z + 18744*x*y^2 + 1104*x*y*z^5 - 11352*x*y*z^4 + 33222*x*y*z^3 - 44061*x*y*z^2 + 28008*x*y*z - 7047*x*y + 136*x*z^6 - 1704*x*z^5 + 6246*x*z^4 - 10990*x*z^3 + 10380*x*z^2 - 5148*x*z + 1056*x - 108*y^7 - 594*y^6*z + 1134*y^6 - 1386*y^5*z^2 + 5310*y^5*z - 3834*y^5 - 1778*y^4*z^3 + 10266*y^4*z^2 - 14814*y^4*z + 6580*y^4 - 1354*y^3*z^4 + 10490*y^3*z^3 - 22698*y^3*z^2 + 20057*y^3*z - 6421*y^3 - 612*y^2*z^5 + 5976*y^2*z^4 - 17244*y^2*z^3 + 22731*y^2*z^2 - 14415*y^2*z + 3627*y^2 - 152*y*z^6 + 1800*y*z^5 - 6498*y*z^4 + 11359*y*z^3 - 10701*y*z^2 + 5307*y*z - 1093*y - 16*z^7 + 224*z^6 - 972*z^5 + 2113*z^4 - 2629*z^3 + 1928*z^2 - 783*z + 1, 54*x^7 - 432*x^6*y - 324*x^6*z + 837*x^6 + 1458*x^5*y^2 + 2214*x^5*y*z - 5265*x^5*y + 828*x^5*z^2 - 4050*x^5*z + 2844*x^5 - 2700*x^4*y^3 - 6210*x^4*y^2*z + 13770*x^4*y^2 - 4698*x^4*y*z^2 + 21249*x^4*y*z - 14715*x^4*y - 1168*x^4*z^3 + 8118*x^4*z^2 - 11229*x^4*z + 4774*x^4 + 2970*x^3*y^4 + 9180*x^3*y^3*z - 19170*x^3*y^3 + 10512*x^3*y^2*z^2 - 44496*x^3*y^2*z + 30420*x^3*y^2 + 5282*x^3*y*z^3 - 34101*x^3*y*z^2 + 46524*x^3*y*z - 19667*x^3*y + 982*x^3*z^4 - 8630*x^3*z^3 + 17634*x^3*z^2 - 14824*x^3*z + 4568*x^3 - 1944*x^2*y^5 - 7560*x^2*y^4*z + 14985*x^2*y^4 - 11628*x^2*y^3*z^2 + 46494*x^2*y^3*z - 31410*x^2*y^3 - 8838*x^2*y^2*z^3 + 53595*x^2*y^2*z^2 - 72198*x^2*y^2*z + 30357*x^2*y^2 - 3318*x^2*y*z^4 + 27207*x^2*y*z^3 - 54843*x^2*y*z^2 + 45843*x^2*y*z - 14097*x^2*y - 492*x^2*z^5 + 5133*x^2*z^4 - 13773*x^2*z^3 + 17166*x^2*z^2 - 10455*x^2*z + 2562*x^2 + 702*x*y^6 + 3294*x*y^5*z - 6237*x*y^5 + 6372*x*y^4*z^2 - 24246*x*y^4*z + 16200*x*y^4 + 6502*x*y^3*z^3 - 37359*x*y^3*z^2 + 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18612*x^3*y^2*z^2 + 58932*x^3*y^2*z - 37494*x^3*y^2 + 17448*x^3*y*z^3 - 82584*x^3*y*z^2 + 107307*x^3*y*z - 41659*x^3*y - 6000*x^3*z^4 + 37680*x^3*z^3 - 74916*x^3*z^2 + 59572*x^3*z - 16729*x^3 + 738*x^2*y^5 - 5364*x^2*y^4*z + 8235*x^2*y^4 + 15282*x^2*y^3*z^2 - 46791*x^2*y^3*z + 29691*x^2*y^3 - 21312*x^2*y^2*z^3 + 97524*x^2*y^2*z^2 - 126108*x^2*y^2*z + 49191*x^2*y^2 + 14544*x^2*y*z^4 - 88344*x^2*y*z^3 + 174294*x^2*y*z^2 - 138999*x^2*y*z + 39339*x^2*y - 3888*x^2*z^5 + 29376*x^2*z^4 - 78336*x^2*z^3 + 95652*x^2*z^2 - 55413*x^2*z + 12459*x^2 - 204*x*y^6 + 1764*x*y^5*z - 2628*x*y^5 - 6228*x*y^4*z^2 + 18504*x*y^4*z - 11700*x*y^4 + 11484*x*y^3*z^3 - 51012*x*y^3*z^2 + 65592*x*y^3*z - 25678*x*y^3 - 11664*x*y^2*z^4 + 68832*x*y^2*z^3 - 134712*x*y^2*z^2 + 107616*x*y^2*z - 30651*x*y^2 + 6192*x*y*z^5 - 45504*x*y*z^4 + 120132*x*y*z^3 - 146592*x*y*z^2 + 85263*x*y*z - 19314*x*y - 1344*x*z^6 + 11808*x*z^5 - 39312*x*z^4 + 64928*x*z^3 - 57660*x*z^2 + 26670*x*z - 5081*x + 24*y^7 - 240*y^6*z + 348*y^6 + 1008*y^5*z^2 - 2916*y^5*z + 1836*y^5 - 2304*y^4*z^3 + 9972*y^4*z^2 - 12744*y^4*z + 5002*y^4 + 3096*y^3*z^4 - 17820*y^3*z^3 + 34596*y^3*z^2 - 27658*y^3*z + 7918*y^3 - 2448*y^2*z^5 + 17568*y^2*z^4 - 45936*y^2*z^3 + 55980*y^2*z^2 - 32658*y^2*z + 7443*y^2 + 1056*y*z^6 - 9072*y*z^5 + 29880*y*z^4 - 49196*y*z^3 + 43734*y*z^2 - 20301*y*z + 3893*y - 192*z^7 + 1920*z^6 - 7632*z^5 + 15872*z^4 - 19052*z^3 + 13452*z^2 - 5250*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -1, 1, -2 ], [ -3, 2, -2 ], [ -1, 1, -1 ] ], "source_shift": [ 1, 2, 1 ], "target_matrix": [ [ 1, 1, 1 ], [ 2, 1, 1 ], [ 3, 3, 2 ] ], "target_shift": [ -290, -437, -877 ], "collision_preimages": [ [ "-1/2", "-2", "-1/4" ], [ "29/2", "51/2", "11/2" ], [ "23/2", "49/2", "15/2" ] ], "collision_image": [ "-1161/4", "-875/2", "-3511/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0336801829980686 ], "surface_features": [ 4.283587271212706, 5.241282436956567, 3.9450498455597853, 0.0, 4.424179069825559, 5160 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C058", "pair": 58, "family": "counterexample", "mapping": "F(x,y,z)=(-8*x^7 + 68*x^6*y + 44*x^6*z - 88*x^6 - 234*x^5*y^2 - 312*x^5*y*z + 636*x^5*y - 102*x^5*z^2 + 408*x^5*z - 378*x^5 + 419*x^4*y^3 + 867*x^4*y^2*z - 1818*x^4*y^2 + 585*x^4*y*z^2 - 2388*x^4*y*z + 2259*x^4*y + 129*x^4*z^3 - 774*x^4*z^2 + 1437*x^4*z - 826*x^4 - 419*x^3*y^4 - 1199*x^3*y^3*z + 2617*x^3*y^3 - 1257*x^3*y^2*z^2 + 5286*x^3*y^2*z - 5103*x^3*y^2 - 573*x^3*y*z^3 + 3513*x^3*y*z^2 - 6651*x^3*y*z + 3918*x^3*y - 96*x^3*z^4 + 768*x^3*z^3 - 2142*x^3*z^2 + 2481*x^3*z - 1025*x^3 + 234*x^2*y^5 + 867*x^2*y^4*z - 1992*x^2*y^4 + 1257*x^2*y^3*z^2 - 5505*x^2*y^3*z + 5400*x^2*y^3 + 891*x^2*y^2*z^3 - 5634*x^2*y^2*z^2 + 10854*x^2*y^2*z - 6510*x^2*y^2 + 309*x^2*y*z^4 - 2529*x^2*y*z^3 + 7179*x^2*y*z^2 - 8496*x^2*y*z + 3642*x^2*y + 42*x^2*z^5 - 420*x^2*z^4 + 1563*x^2*z^3 - 2733*x^2*z^2 + 2298*x^2*z - 766*x^2 - 68*x*y^6 - 312*x*y^5*z + 756*x*y^5 - 585*x*y^4*z^2 + 2688*x*y^4*z - 2655*x*y^4 - 573*x*y^3*z^3 + 3771*x*y^3*z^2 - 7347*x*y^3*z + 4457*x*y^3 - 309*x*y^2*z^4 + 2610*x*y^2*z^3 - 7512*x*y^2*z^2 + 9018*x*y^2*z - 3960*x*y^2 - 87*x*y*z^5 + 891*x*y*z^4 - 3366*x*y*z^3 + 5991*x*y*z^2 - 5190*x*y*z + 1801*x*y - 10*x*z^6 + 120*x*z^5 - 558*x*z^4 + 1307*x*z^3 - 1671*x*z^2 + 1139*x*z - 343*x + 8*y^7 + 44*y^6*z - 112*y^6 + 102*y^5*z^2 - 492*y^5*z + 486*y^5 + 129*y^4*z^3 - 888*y^4*z^2 + 1737*y^4*z - 1056*y^4 + 96*y^3*z^4 - 843*y^3*z^3 + 2445*y^3*z^2 - 2955*y^3*z + 1309*y^3 + 42*y^2*z^5 - 444*y^2*z^4 + 1695*y^2*z^3 - 3048*y^2*z^2 + 2682*y^2*z - 952*y^2 + 10*y*z^6 - 123*y*z^5 + 579*y*z^4 - 1375*y*z^3 + 1797*y*z^2 - 1261*y*z + 383*y + z^7 - 14*z^6 + 78*z^5 - 229*z^4 + 394*z^3 - 409*z^2 + 248*z - 1, -16*x^7 + 136*x^6*y + 88*x^6*z - 152*x^6 - 468*x^5*y^2 - 624*x^5*y*z + 1116*x^5*y - 204*x^5*z^2 + 708*x^5*z - 588*x^5 + 838*x^4*y^3 + 1734*x^4*y^2*z - 3246*x^4*y^2 + 1170*x^4*y*z^2 - 4212*x^4*y*z + 3570*x^4*y + 258*x^4*z^3 - 1350*x^4*z^2 + 2250*x^4*z - 1226*x^4 - 838*x^3*y^4 - 2398*x^3*y^3*z + 4757*x^3*y^3 - 2514*x^3*y^2*z^2 + 9489*x^3*y^2*z - 8208*x^3*y^2 - 1146*x^3*y*z^3 + 6231*x^3*y*z^2 - 10590*x^3*y*z + 5883*x^3*y - 192*x^3*z^4 + 1347*x^3*z^3 - 3378*x^3*z^2 + 3705*x^3*z - 1522*x^3 + 468*x^2*y^5 + 1734*x^2*y^4*z - 3681*x^2*y^4 + 2514*x^2*y^3*z^2 - 10056*x^2*y^3*z + 8853*x^2*y^3 + 1782*x^2*y^2*z^3 - 10170*x^2*y^2*z^2 + 17604*x^2*y^2*z - 9942*x^2*y^2 + 618*x^2*y*z^4 - 4512*x^2*y*z^3 + 11523*x^2*y*z^2 - 12864*x^2*y*z + 5409*x^2*y + 84*x^2*z^5 - 741*x^2*z^4 + 2484*x^2*z^3 - 4110*x^2*z^2 + 3414*x^2*z - 1157*x^2 - 136*x*y^6 - 624*x*y^5*z + 1416*x*y^5 - 1170*x*y^4*z^2 + 4986*x*y^4*z - 4440*x*y^4 - 1146*x*y^3*z^3 + 6921*x*y^3*z^2 - 12150*x*y^3*z + 6946*x*y^3 - 618*x*y^2*z^4 + 4737*x*y^2*z^3 - 12288*x*y^2*z^2 + 13908*x*y^2*z - 5946*x*y^2 - 174*x*y*z^5 + 1599*x*y*z^4 - 5448*x*y*z^3 + 9153*x*y*z^2 - 7752*x*y*z + 2693*x*y - 20*x*z^6 + 213*x*z^5 - 894*x*z^4 + 1981*x*z^3 - 2490*x*z^2 + 1711*x*z - 514*x + 16*y^7 + 88*y^6*z - 212*y^6 + 204*y^5*z^2 - 924*y^5*z + 828*y^5 + 258*y^4*z^3 - 1653*y^4*z^2 + 2928*y^4*z - 1680*y^4 + 192*y^3*z^4 - 1554*y^3*z^3 + 4077*y^3*z^2 - 4650*y^3*z + 2000*y^3 + 84*y^2*z^5 - 810*y^2*z^4 + 2796*y^2*z^3 - 4746*y^2*z^2 + 4062*y^2*z - 1433*y^2 + 20*y*z^6 - 222*y*z^5 + 945*y*z^4 - 2120*y*z^3 + 2703*y*z^2 - 1892*y*z + 575*y + 2*z^7 - 25*z^6 + 126*z^5 - 350*z^4 + 590*z^3 - 614*z^2 + 372*z - 1, 24*x^7 - 204*x^6*y - 132*x^6*z + 240*x^6 + 702*x^5*y^2 + 936*x^5*y*z - 1752*x^5*y + 306*x^5*z^2 - 1116*x^5*z + 966*x^5 - 1257*x^4*y^3 - 2601*x^4*y^2*z + 5064*x^4*y^2 - 1755*x^4*y*z^2 + 6600*x^4*y*z - 5829*x^4*y - 387*x^4*z^3 + 2124*x^4*z^2 - 3687*x^4*z + 2060*x^4 + 1257*x^3*y^4 + 3597*x^3*y^3*z - 7374*x^3*y^3 + 3771*x^3*y^2*z^2 - 14775*x^3*y^2*z + 13311*x^3*y^2 + 1719*x^3*y*z^3 - 9744*x^3*y*z^2 + 17241*x^3*y*z - 9821*x^3*y + 288*x^3*z^4 - 2115*x^3*z^3 + 5520*x^3*z^2 - 6206*x^3*z + 2587*x^3 - 702*x^2*y^5 - 2601*x^2*y^4*z + 5673*x^2*y^4 - 3771*x^2*y^3*z^2 + 15561*x^2*y^3*z - 14253*x^2*y^3 - 2673*x^2*y^2*z^3 + 15804*x^2*y^2*z^2 - 28458*x^2*y^2*z + 16470*x^2*y^2 - 927*x^2*y*z^4 + 7041*x^2*y*z^3 - 18702*x^2*y*z^2 + 21396*x^2*y*z - 9135*x^2*y - 126*x^2*z^5 + 1161*x^2*z^4 - 4047*x^2*z^3 + 6861*x^2*z^2 - 5784*x^2*z + 1977*x^2 + 204*x*y^6 + 936*x*y^5*z - 2172*x*y^5 + 1755*x*y^4*z^2 - 7674*x*y^4*z + 7095*x*y^4 + 1719*x*y^3*z^3 - 10692*x*y^3*z^2 + 19497*x*y^3*z - 11410*x*y^3 + 927*x*y^2*z^4 - 7347*x*y^2*z^3 + 19800*x*y^2*z^2 - 22947*x*y^2*z + 9960*x*y^2 + 261*x*y*z^5 - 2490*x*y*z^4 + 8814*x*y*z^3 - 15165*x*y*z^2 + 13038*x*y*z - 4569*x*y + 30*x*z^6 - 333*x*z^5 + 1452*x*z^4 - 3295*x*z^3 + 4203*x*z^2 - 2913*x*z + 881*x - 24*y^7 - 132*y^6*z + 324*y^6 - 306*y^5*z^2 + 1416*y^5*z - 1314*y^5 - 387*y^4*z^3 + 2541*y^4*z^2 - 4665*y^4*z + 2737*y^4 - 288*y^3*z^4 + 2397*y^3*z^3 - 6522*y^3*z^2 + 7609*y^3*z - 3320*y^3 - 126*y^2*z^5 + 1254*y^2*z^4 - 4491*y^2*z^3 + 7800*y^2*z^2 - 6774*y^2*z + 2409*y^2 - 30*y*z^6 + 345*y*z^5 - 1524*y*z^4 + 3499*y*z^3 - 4527*y*z^2 + 3195*y*z - 975*y - 3*z^7 + 39*z^6 - 204*z^5 + 580*z^4 - 992*z^3 + 1041*z^2 - 634*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ 2, -1, -1 ], [ 1, -2, -1 ], [ 1, -1, -1 ] ], "source_shift": [ 1, 1, 2 ], "target_matrix": [ [ -1, -1, -2 ], [ -2, -1, -1 ], [ 3, 2, 2 ] ], "target_shift": [ 69, 102, -175 ], "collision_preimages": [ [ "5/4", "-5/4", "19/4" ], [ "-9/2", "7", "-16" ], [ "-13/2", "4", "-15" ] ], "collision_image": [ "277/4", "205/2", "-703/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0117185240014805 ], "surface_features": [ 3.504925233101967, 4.454204375035515, 3.182699903336043, 0.0, 3.634074331331018, 4881 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B218", "pair": 218, "family": "bijection", "mapping": "F(x,y,z)=(2*x^7 - 28*x^6*y - 14*x^6*z + 36*x^6 + 168*x^5*y^2 + 168*x^5*y*z - 430*x^5*y + 42*x^5*z^2 - 216*x^5*z + 281*x^5 - 560*x^4*y^3 - 840*x^4*y^2*z + 2140*x^4*y^2 - 420*x^4*y*z^2 + 2150*x^4*y*z - 2782*x^4*y - 70*x^4*z^3 + 540*x^4*z^2 - 1405*x^4*z + 1235*x^4 + 1120*x^3*y^4 + 2240*x^3*y^3*z - 5680*x^3*y^3 + 1680*x^3*y^2*z^2 - 8560*x^3*y^2*z + 11016*x^3*y^2 + 560*x^3*y*z^3 - 4300*x^3*y*z^2 + 11128*x^3*y*z - 9712*x^3*y + 70*x^3*z^4 - 720*x^3*z^3 + 2810*x^3*z^2 - 4940*x^3*z + 3300*x^3 - 1344*x^2*y^5 - 3360*x^2*y^4*z + 8480*x^2*y^4 - 3360*x^2*y^3*z^2 + 17040*x^2*y^3*z - 21808*x^2*y^3 - 1680*x^2*y^2*z^3 + 12840*x^2*y^2*z^2 - 33048*x^2*y^2*z + 28638*x^2*y^2 - 420*x^2*y*z^4 + 4300*x^2*y*z^3 - 16692*x^2*y*z^2 + 29136*x^2*y*z - 19278*x^2*y - 42*x^2*z^5 + 540*x^2*z^4 - 2810*x^2*z^3 + 7410*x^2*z^2 - 9902*x^2*z + 5352*x^2 + 896*x*y^6 + 2688*x*y^5*z - 6752*x*y^5 + 3360*x*y^4*z^2 - 16960*x*y^4*z + 21584*x*y^4 + 2240*x*y^3*z^3 - 17040*x*y^3*z^2 + 43616*x*y^3*z - 37528*x*y^3 + 840*x*y^2*z^4 - 8560*x*y^2*z^3 + 33048*x*y^2*z^2 - 57276*x*y^2*z + 37548*x*y^2 + 168*x*y*z^5 - 2150*x*y*z^4 + 11128*x*y*z^3 - 29136*x*y*z^2 + 38564*x*y*z - 20592*x*y + 14*x*z^6 - 216*x*z^5 + 1405*x*z^4 - 4940*x*z^3 + 9904*x*z^2 - 10716*x*z + 4868*x - 256*y^7 - 896*y^6*z + 2240*y^6 - 1344*y^5*z^2 + 6752*y^5*z - 8544*y^5 - 1120*y^4*z^3 + 8480*y^4*z^2 - 21584*y^4*z + 18440*y^4 - 560*y^3*z^4 + 5680*y^3*z^3 - 21808*y^3*z^2 + 37528*y^3*z - 24382*y^3 - 168*y^2*z^5 + 2140*y^2*z^4 - 11016*y^2*z^3 + 28638*y^2*z^2 - 37556*y^2*z + 19826*y^2 - 28*y*z^6 + 430*y*z^5 - 2782*y*z^4 + 9712*y*z^3 - 19286*y*z^2 + 20614*y*z - 9231*y - 2*z^7 + 36*z^6 - 281*z^5 + 1235*z^4 - 3302*z^3 + 5364*z^2 - 4885*z + 1, 3*x^7 - 42*x^6*y - 21*x^6*z + 54*x^6 + 252*x^5*y^2 + 252*x^5*y*z - 645*x^5*y + 63*x^5*z^2 - 324*x^5*z + 423*x^5 - 840*x^4*y^3 - 1260*x^4*y^2*z + 3210*x^4*y^2 - 630*x^4*y*z^2 + 3225*x^4*y*z - 4188*x^4*y - 105*x^4*z^3 + 810*x^4*z^2 - 2115*x^4*z + 1869*x^4 + 1680*x^3*y^4 + 3360*x^3*y^3*z - 8520*x^3*y^3 + 2520*x^3*y^2*z^2 - 12840*x^3*y^2*z + 16584*x^3*y^2 + 840*x^3*y*z^3 - 6450*x^3*y*z^2 + 16752*x^3*y*z - 14700*x^3*y + 105*x^3*z^4 - 1080*x^3*z^3 + 4230*x^3*z^2 - 7476*x^3*z + 5022*x^3 - 2016*x^2*y^5 - 5040*x^2*y^4*z + 12720*x^2*y^4 - 5040*x^2*y^3*z^2 + 25560*x^2*y^3*z - 32832*x^2*y^3 - 2520*x^2*y^2*z^3 + 19260*x^2*y^2*z^2 - 49752*x^2*y^2*z + 43353*x^2*y^2 - 630*x^2*y*z^4 + 6450*x^2*y*z^3 - 25128*x^2*y*z^2 + 44100*x^2*y*z - 29349*x^2*y - 63*x^2*z^5 + 810*x^2*z^4 - 4230*x^2*z^3 + 11214*x^2*z^2 - 15069*x^2*z + 8183*x^2 + 1344*x*y^6 + 4032*x*y^5*z - 10128*x*y^5 + 5040*x*y^4*z^2 - 25440*x*y^4*z + 32496*x*y^4 + 3360*x*y^3*z^3 - 25560*x*y^3*z^2 + 65664*x*y^3*z - 56820*x*y^3 + 1260*x*y^2*z^4 - 12840*x*y^2*z^3 + 49752*x*y^2*z^2 - 86706*x*y^2*z + 57186*x*y^2 + 252*x*y*z^5 - 3225*x*y*z^4 + 16752*x*y*z^3 - 44100*x*y*z^2 + 58710*x*y*z - 31508*x*y + 21*x*z^6 - 324*x*z^5 + 2115*x*z^4 - 7476*x*z^3 + 15072*x*z^2 - 16384*x*z + 7467*x - 384*y^7 - 1344*y^6*z + 3360*y^6 - 2016*y^5*z^2 + 10128*y^5*z - 12864*y^5 - 1680*y^4*z^3 + 12720*y^4*z^2 - 32496*y^4*z + 27924*y^4 - 840*y^3*z^4 + 8520*y^3*z^3 - 32832*y^3*z^2 + 56820*y^3*z - 37149*y^3 - 252*y^2*z^5 + 3210*y^2*z^4 - 16584*y^2*z^3 + 43353*y^2*z^2 - 57198*y^2*z + 30359*y^2 - 42*y*z^6 + 645*y*z^5 - 4188*y*z^4 + 14700*y*z^3 - 29361*y*z^2 + 31541*y*z - 14176*y - 3*z^7 + 54*z^6 - 423*z^5 + 1869*z^4 - 5025*z^3 + 8201*z^2 - 7494*z + 1, x^7 - 14*x^6*y - 7*x^6*z + 18*x^6 + 84*x^5*y^2 + 84*x^5*y*z - 215*x^5*y + 21*x^5*z^2 - 108*x^5*z + 141*x^5 - 280*x^4*y^3 - 420*x^4*y^2*z + 1070*x^4*y^2 - 210*x^4*y*z^2 + 1075*x^4*y*z - 1396*x^4*y - 35*x^4*z^3 + 270*x^4*z^2 - 705*x^4*z + 623*x^4 + 560*x^3*y^4 + 1120*x^3*y^3*z - 2840*x^3*y^3 + 840*x^3*y^2*z^2 - 4280*x^3*y^2*z + 5528*x^3*y^2 + 280*x^3*y*z^3 - 2150*x^3*y*z^2 + 5584*x^3*y*z - 4900*x^3*y + 35*x^3*z^4 - 360*x^3*z^3 + 1410*x^3*z^2 - 2492*x^3*z + 1674*x^3 - 672*x^2*y^5 - 1680*x^2*y^4*z + 4240*x^2*y^4 - 1680*x^2*y^3*z^2 + 8520*x^2*y^3*z - 10944*x^2*y^3 - 840*x^2*y^2*z^3 + 6420*x^2*y^2*z^2 - 16584*x^2*y^2*z + 14451*x^2*y^2 - 210*x^2*y*z^4 + 2150*x^2*y*z^3 - 8376*x^2*y*z^2 + 14700*x^2*y*z - 9783*x^2*y - 21*x^2*z^5 + 270*x^2*z^4 - 1410*x^2*z^3 + 3738*x^2*z^2 - 5023*x^2*z + 2727*x^2 + 448*x*y^6 + 1344*x*y^5*z - 3376*x*y^5 + 1680*x*y^4*z^2 - 8480*x*y^4*z + 10832*x*y^4 + 1120*x*y^3*z^3 - 8520*x*y^3*z^2 + 21888*x*y^3*z - 18940*x*y^3 + 420*x*y^2*z^4 - 4280*x*y^2*z^3 + 16584*x*y^2*z^2 - 28902*x*y^2*z + 19062*x*y^2 + 84*x*y*z^5 - 1075*x*y*z^4 + 5584*x*y*z^3 - 14700*x*y*z^2 + 19570*x*y*z - 10500*x*y + 7*x*z^6 - 108*x*z^5 + 705*x*z^4 - 2492*x*z^3 + 5024*x*z^2 - 5460*x*z + 2485*x - 128*y^7 - 448*y^6*z + 1120*y^6 - 672*y^5*z^2 + 3376*y^5*z - 4288*y^5 - 560*y^4*z^3 + 4240*y^4*z^2 - 10832*y^4*z + 9308*y^4 - 280*y^3*z^4 + 2840*y^3*z^3 - 10944*y^3*z^2 + 18940*y^3*z - 12383*y^3 - 84*y^2*z^5 + 1070*y^2*z^4 - 5528*y^2*z^3 + 14451*y^2*z^2 - 19066*y^2*z + 10117*y^2 - 14*y*z^6 + 215*y*z^5 - 1396*y*z^4 + 4900*y*z^3 - 9787*y*z^2 + 10511*y*z - 4718*y - z^7 + 18*z^6 - 141*z^5 + 623*z^4 - 1675*z^3 + 2733*z^2 - 2494*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ 1, -2, -1 ], [ 1, -1, -2 ], [ 1, -1, -1 ] ], "source_shift": [ 2, 2, 2 ], "target_matrix": [ [ 2, 1, 1 ], [ 3, 3, 2 ], [ 1, 1, 1 ] ], "target_shift": [ -1911, -2937, -973 ], "inverse_recipe": { "p": 5, "q": 2, "f": "x**5 + x**4 + x", "g": "x**2 + x", "h": "y*z + z**3 + z", "polynomials": { "f": [ [ [ 5, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ], [ [ 4, 0, 0 ], 1 ] ], "g": [ [ [ 2, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 0, 1 ], 1 ], [ [ 0, 0, 3 ], 1 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0768325839999306 ], "surface_features": [ 3.952115027806251, 4.93804915141774, 3.5085297189712867, 0.0, 4.121913814360413, 4979 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C140", "pair": 140, "family": "counterexample", "mapping": "F(x,y,z)=(8*x^7 + 52*x^6*y + 44*x^6*z + 112*x^6 + 138*x^5*y^2 + 240*x^5*y*z + 540*x^5*y + 102*x^5*z^2 + 492*x^5*z + 486*x^5 + 195*x^4*y^3 + 519*x^4*y^2*z + 1068*x^4*y^2 + 453*x^4*y*z^2 + 1956*x^4*y*z + 1863*x^4*y + 129*x^4*z^3 + 888*x^4*z^2 + 1737*x^4*z + 1056*x^4 + 159*x^3*y^4 + 573*x^3*y^3*z + 1107*x^3*y^3 + 765*x^3*y^2*z^2 + 3057*x^3*y^2*z + 2820*x^3*y^2 + 447*x^3*y*z^3 + 2793*x^3*y*z^2 + 5265*x^3*y*z + 3139*x^3*y + 96*x^3*z^4 + 843*x^3*z^3 + 2445*x^3*z^2 + 2955*x^3*z + 1309*x^3 + 75*x^2*y^5 + 342*x^2*y^4*z + 633*x^2*y^4 + 618*x^2*y^3*z^2 + 2343*x^2*y^3*z + 2103*x^2*y^3 + 552*x^2*y^2*z^3 + 3231*x^2*y^2*z^2 + 5901*x^2*y^2*z + 3453*x^2*y^2 + 243*x^2*y*z^4 + 1965*x^2*y*z^3 + 5493*x^2*y*z^2 + 6501*x^2*y*z + 2841*x^2*y + 42*x^2*z^5 + 444*x^2*z^4 + 1695*x^2*z^3 + 3048*x^2*z^2 + 2682*x^2*z + 952*x^2 + 19*x*y^6 + 105*x*y^5*z + 189*x*y^5 + 240*x*y^4*z^2 + 879*x*y^4*z + 771*x*y^4 + 290*x*y^3*z^3 + 1626*x*y^3*z^2 + 2892*x*y^3*z + 1663*x*y^3 + 195*x*y^2*z^4 + 1494*x*y^2*z^3 + 4050*x*y^2*z^2 + 4701*x*y^2*z + 2025*x*y^2 + 69*x*y*z^5 + 681*x*y*z^4 + 2508*x*y*z^3 + 4413*x*y*z^2 + 3822*x*y*z + 1339*x*y + 10*x*z^6 + 123*x*z^5 + 579*x*z^4 + 1375*x*z^3 + 1797*x*z^2 + 1261*x*z + 383*x + 2*y^7 + 13*y^6*z + 23*y^6 + 36*y^5*z^2 + 129*y^5*z + 111*y^5 + 55*y^4*z^3 + 300*y^4*z^2 + 522*y^4*z + 295*y^4 + 50*y^3*z^4 + 370*y^3*z^3 + 978*y^3*z^2 + 1114*y^3*z + 472*y^3 + 27*y^2*z^5 + 255*y^2*z^4 + 912*y^2*z^3 + 1572*y^2*z^2 + 1338*y^2*z + 463*y^2 + 8*y*z^6 + 93*y*z^5 + 423*y*z^4 + 982*y*z^3 + 1260*y*z^2 + 872*y*z + 266*y + z^7 + 14*z^6 + 78*z^5 + 229*z^4 + 394*z^3 + 409*z^2 + 248*z + 1, -8*x^7 - 52*x^6*y - 44*x^6*z - 112*x^6 - 138*x^5*y^2 - 240*x^5*y*z - 540*x^5*y - 102*x^5*z^2 - 492*x^5*z - 486*x^5 - 195*x^4*y^3 - 519*x^4*y^2*z - 1068*x^4*y^2 - 453*x^4*y*z^2 - 1956*x^4*y*z - 1863*x^4*y - 129*x^4*z^3 - 888*x^4*z^2 - 1737*x^4*z - 1057*x^4 - 159*x^3*y^4 - 573*x^3*y^3*z - 1107*x^3*y^3 - 765*x^3*y^2*z^2 - 3057*x^3*y^2*z - 2820*x^3*y^2 - 447*x^3*y*z^3 - 2793*x^3*y*z^2 - 5265*x^3*y*z - 3144*x^3*y - 96*x^3*z^4 - 843*x^3*z^3 - 2445*x^3*z^2 - 2959*x^3*z - 1320*x^3 - 75*x^2*y^5 - 342*x^2*y^4*z - 633*x^2*y^4 - 618*x^2*y^3*z^2 - 2343*x^2*y^3*z - 2103*x^2*y^3 - 552*x^2*y^2*z^3 - 3231*x^2*y^2*z^2 - 5901*x^2*y^2*z - 3462*x^2*y^2 - 243*x^2*y*z^4 - 1965*x^2*y*z^3 - 5493*x^2*y*z^2 - 6516*x^2*y*z - 2874*x^2*y - 42*x^2*z^5 - 444*x^2*z^4 - 1695*x^2*z^3 - 3054*x^2*z^2 - 2712*x^2*z - 976*x^2 - 19*x*y^6 - 105*x*y^5*z - 189*x*y^5 - 240*x*y^4*z^2 - 879*x*y^4*z - 771*x*y^4 - 290*x*y^3*z^3 - 1626*x*y^3*z^2 - 2892*x*y^3*z - 1670*x*y^3 - 195*x*y^2*z^4 - 1494*x*y^2*z^3 - 4050*x*y^2*z^2 - 4719*x*y^2*z - 2058*x*y^2 - 69*x*y*z^5 - 681*x*y*z^4 - 2508*x*y*z^3 - 4428*x*y*z^2 - 3882*x*y*z - 1384*x*y - 10*x*z^6 - 123*x*z^5 - 579*x*z^4 - 1379*x*z^3 - 1824*x*z^2 - 1303*x*z - 400*x - 2*y^7 - 13*y^6*z - 23*y^6 - 36*y^5*z^2 - 129*y^5*z - 111*y^5 - 55*y^4*z^3 - 300*y^4*z^2 - 522*y^4*z - 297*y^4 - 50*y^3*z^4 - 370*y^3*z^3 - 978*y^3*z^2 - 1121*y^3*z - 483*y^3 - 27*y^2*z^5 - 255*y^2*z^4 - 912*y^2*z^3 - 1581*y^2*z^2 - 1368*y^2*z - 484*y^2 - 8*y*z^6 - 93*y*z^5 - 423*y*z^4 - 987*y*z^3 - 1287*y*z^2 - 911*y*z - 281*y - z^7 - 14*z^6 - 78*z^5 - 230*z^4 - 402*z^3 - 427*z^2 - 262*z - 1, -8*x^7 - 52*x^6*y - 44*x^6*z - 124*x^6 - 138*x^5*y^2 - 240*x^5*y*z - 612*x^5*y - 102*x^5*z^2 - 552*x^5*z - 630*x^5 - 195*x^4*y^3 - 519*x^4*y^2*z - 1239*x^4*y^2 - 453*x^4*y*z^2 - 2250*x^4*y*z - 2457*x^4*y - 129*x^4*z^3 - 1011*x^4*z^2 - 2283*x^4*z - 1492*x^4 - 159*x^3*y^4 - 573*x^3*y^3*z - 1314*x^3*y^3 - 765*x^3*y^2*z^2 - 3603*x^3*y^2*z - 3783*x^3*y^2 - 447*x^3*y*z^3 - 3264*x^3*y*z^2 - 7041*x^3*y*z - 4503*x^3*y - 96*x^3*z^4 - 975*x^3*z^3 - 3258*x^3*z^2 - 4231*x^3*z - 1971*x^3 - 75*x^2*y^5 - 342*x^2*y^4*z - 768*x^2*y^4 - 618*x^2*y^3*z^2 - 2826*x^2*y^3*z - 2868*x^2*y^3 - 552*x^2*y^2*z^3 - 3870*x^2*y^2*z^2 - 8025*x^2*y^2*z - 5031*x^2*y^2 - 243*x^2*y*z^4 - 2334*x^2*y*z^3 - 7446*x^2*y*z^2 - 9453*x^2*y*z - 4353*x^2*y - 42*x^2*z^5 - 522*x^2*z^4 - 2289*x^2*z^3 - 4422*x^2*z^2 - 4104*x^2*z - 1519*x^2 - 19*x*y^6 - 105*x*y^5*z - 234*x*y^5 - 240*x*y^4*z^2 - 1083*x*y^4*z - 1068*x*y^4 - 290*x*y^3*z^3 - 1992*x*y^3*z^2 - 3996*x*y^3*z - 2462*x*y^3 - 195*x*y^2*z^4 - 1818*x*y^2*z^3 - 5580*x*y^2*z^2 - 6945*x*y^2*z - 3162*x*y^2 - 69*x*y*z^5 - 822*x*y*z^4 - 3444*x*y*z^3 - 6504*x*y*z^2 - 5958*x*y*z - 2188*x*y - 10*x*z^6 - 147*x*z^5 - 792*x*z^4 - 2021*x*z^3 - 2796*x*z^2 - 2059*x*z - 642*x - 2*y^7 - 13*y^6*z - 29*y^6 - 36*y^5*z^2 - 162*y^5*z - 156*y^5 - 55*y^4*z^3 - 375*y^4*z^2 - 732*y^4*z - 444*y^4 - 50*y^3*z^4 - 460*y^3*z^3 - 1368*y^3*z^2 - 1673*y^3*z - 753*y^3 - 27*y^2*z^5 - 315*y^2*z^4 - 1272*y^2*z^3 - 2355*y^2*z^2 - 2130*y^2*z - 778*y^2 - 8*y*z^6 - 114*y*z^5 - 588*y*z^4 - 1467*y*z^3 - 2001*y*z^2 - 1463*y*z - 459*y - z^7 - 17*z^6 - 108*z^5 - 341*z^4 - 624*z^3 - 685*z^2 - 428*z - 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ 1, 1, 1 ], [ 2, 1, 1 ], [ 1, 2, 1 ] ], "source_shift": [ 1, 1, 2 ], "target_matrix": [ [ 1, 1, 2 ], [ -1, -1, -1 ], [ -1, -2, -1 ] ], "target_shift": [ -69, 72, 118 ], "collision_preimages": [ [ "0", "-5/4", "1/4" ], [ "-5/2", "9/2", "-2" ], [ "5/2", "13/2", "-11" ] ], "collision_image": [ "-277/4", "289/4", "473/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0155459640009212 ], "surface_features": [ 3.1145025638591606, 3.975569657893662, 2.7781512503836434, 0.0, 3.209288718487139, 4743 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C049", "pair": 49, "family": "counterexample", "mapping": "F(x,y,z)=(-x^7 - 10*x^6*y + 10*x^6*z + 16*x^6 - 42*x^5*y^2 + 87*x^5*y*z + 138*x^5*y - 42*x^5*z^2 - 138*x^5*z - 108*x^5 - 96*x^4*y^3 + 309*x^4*y^2*z + 486*x^4*y^2 - 309*x^4*y*z^2 - 1005*x^4*y*z - 780*x^4*y + 96*x^4*z^3 + 489*x^4*z^2 + 783*x^4*z + 401*x^4 - 129*x^3*y^4 + 573*x^3*y^3*z + 894*x^3*y^3 - 891*x^3*y^2*z^2 - 2868*x^3*y^2*z - 2205*x^3*y^2 + 573*x^3*y*z^3 + 2886*x^3*y*z^2 + 4587*x^3*y*z + 2331*x^3*y - 129*x^3*z^4 - 912*x^3*z^3 - 2244*x^3*z^2 - 2339*x^3*z - 887*x^3 - 102*x^2*y^5 + 585*x^2*y^4*z + 906*x^2*y^4 - 1257*x^2*y^3*z^2 - 4005*x^2*y^3*z - 3048*x^2*y^3 + 1257*x^2*y^2*z^3 + 6255*x^2*y^2*z^2 + 9852*x^2*y^2*z + 4959*x^2*y^2 - 585*x^2*y*z^4 - 4089*x^2*y*z^3 - 10005*x^2*y*z^2 - 10368*x^2*y*z - 3897*x^2*y + 102*x^2*z^5 + 945*x^2*z^4 + 3177*x^2*z^3 + 5049*x^2*z^2 + 3885*x^2*z + 1177*x^2 - 44*x*y^6 + 312*x*y^5*z + 480*x*y^5 - 867*x*y^4*z^2 - 2736*x*y^4*z - 2061*x*y^4 + 1199*x*y^3*z^3 + 5892*x*y^3*z^2 + 9183*x*y^3*z + 4571*x*y^3 - 867*x*y^2*z^4 - 5982*x*y^2*z^3 - 14526*x*y^2*z^2 - 14937*x*y^2*z - 5553*x*y^2 + 312*x*y*z^5 + 2862*x*y*z^4 + 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12168*y^2*z^3 - 18078*y^2*z^2 - 13182*y^2*z - 3815*y^2 + 136*y*z^6 + 1488*y*z^5 + 5940*y*z^4 + 12051*y*z^3 + 13512*y*z^2 + 8048*y*z + 2006*y - 16*z^7 - 220*z^6 - 1080*z^5 - 2783*z^4 - 4226*z^3 - 3851*z^2 - 1970*z - 1, x^7 + 10*x^6*y - 10*x^6*z - 16*x^6 + 42*x^5*y^2 - 87*x^5*y*z - 138*x^5*y + 42*x^5*z^2 + 138*x^5*z + 108*x^5 + 96*x^4*y^3 - 309*x^4*y^2*z - 486*x^4*y^2 + 309*x^4*y*z^2 + 1005*x^4*y*z + 780*x^4*y - 96*x^4*z^3 - 489*x^4*z^2 - 783*x^4*z - 402*x^4 + 129*x^3*y^4 - 573*x^3*y^3*z - 894*x^3*y^3 + 891*x^3*y^2*z^2 + 2868*x^3*y^2*z + 2205*x^3*y^2 - 573*x^3*y*z^3 - 2886*x^3*y*z^2 - 4587*x^3*y*z - 2338*x^3*y + 129*x^3*z^4 + 912*x^3*z^3 + 2244*x^3*z^2 + 2343*x^3*z + 894*x^3 + 102*x^2*y^5 - 585*x^2*y^4*z - 906*x^2*y^4 + 1257*x^2*y^3*z^2 + 4005*x^2*y^3*z + 3048*x^2*y^3 - 1257*x^2*y^2*z^3 - 6255*x^2*y^2*z^2 - 9852*x^2*y^2*z - 4977*x^2*y^2 + 585*x^2*y*z^4 + 4089*x^2*y*z^3 + 10005*x^2*y*z^2 + 10389*x^2*y*z + 3933*x^2*y - 102*x^2*z^5 - 945*x^2*z^4 - 3177*x^2*z^3 - 5055*x^2*z^2 - 3909*x^2*z - 1195*x^2 + 44*x*y^6 - 312*x*y^5*z - 480*x*y^5 + 867*x*y^4*z^2 + 2736*x*y^4*z + 2061*x*y^4 - 1199*x*y^3*z^3 - 5892*x*y^3*z^2 - 9183*x*y^3*z - 4591*x*y^3 + 867*x*y^2*z^4 + 5982*x*y^2*z^3 + 14526*x*y^2*z^2 + 14973*x*y^2*z + 5613*x*y^2 - 312*x*y*z^5 - 2862*x*y*z^4 - 9594*x*y*z^3 - 15213*x*y*z^2 - 11691*x*y*z - 3539*x*y + 44*x*z^6 + 516*x*z^5 + 2220*x*z^4 + 4778*x*z^3 + 5610*x*z^2 + 3472*x*z + 896*x + 8*y^7 - 68*y^6*z - 104*y^6 + 234*y^5*z^2 + 732*y^5*z + 546*y^5 - 419*y^4*z^3 - 2034*y^4*z^2 - 3135*y^4*z - 1550*y^4 + 419*y^3*z^4 + 2849*y^3*z^3 + 6849*y^3*z^2 + 6994*y^3*z + 2593*y^3 - 234*y^2*z^5 - 2121*y^2*z^4 - 7077*y^2*z^3 - 11163*y^2*z^2 - 8502*y^2*z - 2539*y^2 + 68*y*z^6 + 792*y*z^5 + 3408*y*z^4 + 7327*y*z^3 + 8571*y*z^2 + 5266*y*z + 1344*y - 8*z^7 - 116*z^6 - 612*z^5 - 1667*z^4 - 2636*z^3 - 2473*z^2 - 1291*z - 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -1, -2, 1 ], [ -1, -1, 2 ], [ -1, -1, 1 ] ], "source_shift": [ 1, 2, 1 ], "target_matrix": [ [ 1, 1, 2 ], [ -2, -1, -1 ], [ -1, -1, -1 ] ], "target_shift": [ -285, 437, 290 ], "collision_preimages": [ [ "3/4", "-1/4", "-3/4" ], [ "-20", "11/2", "-9" ], [ "-19", "15/2", "-6" ] ], "collision_image": [ "-1141/4", "875/2", "1161/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 0.9640664760008804 ], "surface_features": [ 3.5289088480710666, 4.394574246649326, 3.1971426649725627, 0.0, 3.642764365497977, 4870 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C131", "pair": 131, "family": "counterexample", "mapping": "F(x,y,z)=(2*x^7 - 20*x^6*y - 20*x^6*z + 37*x^6 + 84*x^5*y^2 + 174*x^5*y*z - 321*x^5*y + 84*x^5*z^2 - 318*x^5*z + 276*x^5 - 192*x^4*y^3 - 618*x^4*y^2*z + 1137*x^4*y^2 - 618*x^4*y*z^2 + 2331*x^4*y*z - 2016*x^4*y - 192*x^4*z^3 + 1122*x^4*z^2 - 1989*x^4*z + 1094*x^4 + 258*x^3*y^4 + 1146*x^3*y^3*z - 2103*x^3*y^3 + 1782*x^3*y^2*z^2 - 6696*x^3*y^2*z + 5766*x^3*y^2 + 1146*x^3*y*z^3 - 6669*x^3*y*z^2 + 11796*x^3*y*z - 6466*x^3*y + 258*x^3*z^4 - 2082*x^3*z^3 + 5658*x^3*z^2 - 6314*x^3*z + 2528*x^3 - 204*x^2*y^5 - 1170*x^2*y^4*z + 2142*x^2*y^4 - 2514*x^2*y^3*z^2 + 9411*x^2*y^3*z - 8064*x^2*y^3 - 2514*x^2*y^2*z^3 + 14562*x^2*y^2*z^2 - 25671*x^2*y^2*z + 14007*x^2*y^2 - 1170*x^2*y*z^4 + 9405*x^2*y*z^3 - 25554*x^2*y*z^2 + 28470*x^2*y*z - 11346*x^2*y - 204*x^2*z^5 + 2145*x^2*z^4 - 7941*x^2*z^3 + 13467*x^2*z^2 - 10902*x^2*z + 3458*x^2 + 88*x*y^6 + 624*x*y^5*z - 1140*x*y^5 + 1734*x*y^4*z^2 - 6468*x*y^4*z + 5514*x*y^4 + 2398*x*y^3*z^3 - 13821*x*y^3*z^2 + 24258*x*y^3*z - 13161*x*y^3 + 1734*x*y^2*z^4 - 13872*x*y^2*z^3 + 37638*x*y^2*z^2 - 41814*x*y^2*z + 16560*x*y^2 + 624*x*y*z^5 - 6546*x*y*z^4 + 24294*x*y*z^3 - 41211*x*y*z^2 + 33264*x*y*z - 10486*x*y + 88*x*z^6 - 1164*x*z^5 + 5496*x*z^4 - 12570*x*z^3 + 15408*x*z^2 - 9866*x*z + 2628*x - 16*y^7 - 136*y^6*z + 248*y^6 - 468*y^5*z^2 + 1740*y^5*z - 1476*y^5 - 838*y^4*z^3 + 4806*y^4*z^2 - 8394*y^4*z + 4526*y^4 - 838*y^3*z^4 + 6665*y^3*z^3 - 18030*y^3*z^2 + 19947*y^3*z - 7840*y^3 - 468*y^2*z^5 + 4893*y^2*z^4 - 18189*y^2*z^3 + 30831*y^2*z^2 - 24765*y^2*z + 7736*y^2 - 136*y*z^6 + 1800*y*z^5 - 8544*y*z^4 + 19585*y*z^3 - 23988*y*z^2 + 15302*y*z - 4054*y - 16*z^7 + 260*z^6 - 1500*z^5 + 4329*z^4 - 7127*z^3 + 6896*z^2 - 3698*z + 1, -x^7 + 10*x^6*y + 10*x^6*z - 17*x^6 - 42*x^5*y^2 - 87*x^5*y*z + 147*x^5*y - 42*x^5*z^2 + 147*x^5*z - 120*x^5 + 96*x^4*y^3 + 309*x^4*y^2*z - 519*x^4*y^2 + 309*x^4*y*z^2 - 1074*x^4*y*z + 873*x^4*y + 96*x^4*z^3 - 522*x^4*z^2 + 870*x^4*z - 458*x^4 - 129*x^3*y^4 - 573*x^3*y^3*z + 957*x^3*y^3 - 891*x^3*y^2*z^2 + 3075*x^3*y^2*z - 2487*x^3*y^2 - 573*x^3*y*z^3 + 3093*x^3*y*z^2 - 5139*x^3*y*z + 2694*x^3*y - 129*x^3*z^4 + 975*x^3*z^3 - 2490*x^3*z^2 + 2660*x^3*z - 1029*x^3 + 102*x^2*y^5 + 585*x^2*y^4*z - 972*x^2*y^4 + 1257*x^2*y^3*z^2 - 4308*x^2*y^3*z + 3465*x^2*y^3 + 1257*x^2*y^2*z^3 - 6732*x^2*y^2*z^2 + 11139*x^2*y^2*z - 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- 6729*x*y^2 - 312*x*y*z^5 + 3078*x*y*z^4 - 10716*x*y*z^3 + 17409*x*y*z^2 - 13653*x*y*z + 4220*x*y - 44*x*z^6 + 552*x*z^5 - 2448*x*z^4 + 5366*x*z^3 - 6393*x*z^2 + 4015*x*z - 1055*x + 8*y^7 + 68*y^6*z - 112*y^6 + 234*y^5*z^2 - 792*y^5*z + 630*y^5 + 419*y^4*z^3 - 2208*y^4*z^2 + 3615*y^4*z - 1868*y^4 + 419*y^3*z^4 - 3094*y^3*z^3 + 7842*y^3*z^2 - 8305*y^3*z + 3172*y^3 + 234*y^2*z^5 - 2295*y^2*z^4 + 7992*y^2*z^3 - 12963*y^2*z^2 + 10113*y^2*z - 3097*y^2 + 68*y*z^6 - 852*y*z^5 + 3792*y*z^4 - 8323*y*z^3 + 9903*y*z^2 - 6193*y*z + 1617*y + 8*z^7 - 124*z^6 + 672*z^5 - 1859*z^4 + 2975*z^3 - 2824*z^2 + 1495*z - 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ 1, -2, -1 ], [ 1, -1, -2 ], [ 1, -1, -1 ] ], "source_shift": [ 1, 2, 2 ], "target_matrix": [ [ 2, 3, 3 ], [ -1, -1, -2 ], [ -1, -1, -1 ] ], "target_shift": [ -857, 337, 343 ], "collision_preimages": [ [ "-15/4", "-5/4", "-1/4" ], [ "17", "9/2", "8" ], [ "16", "13/2", "5" ] ], "collision_image": [ "-1715/2", "1349/4", "1373/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.050128973001847 ], "surface_features": [ 3.653671765153687, 4.621321714903542, 3.2704459080179626, 0.0, 3.818395042183231, 4902 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C091", "pair": 91, "family": "counterexample", "mapping": "F(x,y,z)=(-24*x^7 + 276*x^6*y - 192*x^6*z + 336*x^6 - 1350*x^5*y^2 + 1908*x^5*y*z - 3312*x^5*y - 648*x^5*z^2 + 2268*x^5*z - 1890*x^5 + 3639*x^4*y^3 - 7848*x^4*y^2*z + 13536*x^4*y^2 + 5400*x^4*y*z^2 - 18720*x^4*y*z + 15579*x^4*y - 1200*x^4*z^3 + 6300*x^4*z^2 - 10404*x^4*z + 5612*x^4 - 5835*x^3*y^4 + 17097*x^3*y^3*z - 29364*x^3*y^3 - 17892*x^3*y^2*z^2 + 61515*x^3*y^2*z - 51228*x^3*y^2 + 8040*x^3*y*z^3 - 41760*x^3*y*z^2 + 68715*x^3*y*z - 37079*x^3*y - 1320*x^3*z^4 + 9240*x^3*z^3 - 22716*x^3*z^2 + 24193*x^3*z - 9556*x^3 + 5562*x^2*y^5 - 20799*x^2*y^4*z + 35667*x^2*y^4 + 29457*x^2*y^3*z^2 - 100611*x^2*y^3*z + 84024*x^2*y^3 - 20088*x^2*y^2*z^3 + 103329*x^2*y^2*z^2 - 169704*x^2*y^2*z + 91701*x^2*y^2 + 6660*x^2*y*z^4 - 46080*x^2*y*z^3 + 112671*x^2*y*z^2 - 119832*x^2*y*z + 47241*x^2*y - 864*x^2*z^5 + 7560*x^2*z^4 - 24624*x^2*z^3 + 38907*x^2*z^2 - 30354*x^2*z + 9357*x^2 - 2916*x*y^6 + 13392*x*y^5*z - 23004*x*y^5 - 24093*x*y^4*z^2 + 81918*x*y^4*z - 68760*x*y^4 + 22179*x*y^3*z^3 - 113130*x*y^3*z^2 + 185796*x*y^3*z - 100620*x*y^3 - 11142*x*y^2*z^4 + 76257*x*y^2*z^3 - 185724*x*y^2*z^2 + 197460*x*y^2*z - 77697*x*y^2 + 2916*x*y*z^5 - 25200*x*y*z^4 + 81513*x*y*z^3 - 128427*x*y*z^2 + 99909*x*y*z - 30624*x*y - 312*x*z^6 + 3276*x*z^5 - 13266*x*z^4 + 27683*x*z^3 - 32040*x*z^2 + 19539*x*z - 4887*x + 648*y^7 - 3564*y^6*z + 6156*y^6 + 7830*y^5*z^2 - 26568*y^5*z + 22464*y^5 - 9129*y^4*z^3 + 46251*y^4*z^2 - 76104*y^4*z + 41334*y^4 + 6180*y^3*z^4 - 41883*y^3*z^3 + 101772*y^3*z^2 - 108258*y^3*z + 42516*y^3 - 2448*y^2*z^5 + 20907*y^2*z^4 - 67248*y^2*z^3 + 105759*y^2*z^2 - 82053*y^2*z + 25005*y^2 + 528*y*z^6 - 5472*y*z^5 + 21978*y*z^4 - 45674*y*z^3 + 52668*y*z^2 - 31929*y*z + 7912*y - 48*z^7 + 588*z^6 - 2844*z^5 + 7357*z^4 - 11242*z^3 + 10182*z^2 - 5047*z + 1, 8*x^7 - 92*x^6*y + 64*x^6*z - 112*x^6 + 450*x^5*y^2 - 636*x^5*y*z + 1104*x^5*y + 216*x^5*z^2 - 756*x^5*z + 630*x^5 - 1213*x^4*y^3 + 2616*x^4*y^2*z - 4512*x^4*y^2 - 1800*x^4*y*z^2 + 6240*x^4*y*z - 5193*x^4*y + 400*x^4*z^3 - 2100*x^4*z^2 + 3468*x^4*z - 1868*x^4 + 1945*x^3*y^4 - 5699*x^3*y^3*z + 9788*x^3*y^3 + 5964*x^3*y^2*z^2 - 20505*x^3*y^2*z + 17076*x^3*y^2 - 2680*x^3*y*z^3 + 13920*x^3*y*z^2 - 22905*x^3*y*z + 12345*x^3*y + 440*x^3*z^4 - 3080*x^3*z^3 + 7572*x^3*z^2 - 8051*x^3*z + 3172*x^3 - 1854*x^2*y^5 + 6933*x^2*y^4*z - 11889*x^2*y^4 - 9819*x^2*y^3*z^2 + 33537*x^2*y^3*z - 28008*x^2*y^3 + 6696*x^2*y^2*z^3 - 34443*x^2*y^2*z^2 + 56568*x^2*y^2*z - 30537*x^2*y^2 - 2220*x^2*y*z^4 + 15360*x^2*y*z^3 - 37557*x^2*y*z^2 + 39888*x^2*y*z - 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+ 102720*x*y^2*z^2 - 104262*x*y^2*z + 39732*x*y^2 - 1944*x*y*z^5 + 14844*x*y*z^4 - 44754*x*y*z^3 + 67383*x*y*z^2 - 50832*x*y*z + 15290*x*y + 208*x*z^6 - 1920*x*z^5 + 7236*x*z^4 - 14436*x*z^3 + 16218*x*z^2 - 9712*x*z + 2414*x - 432*y^7 + 2376*y^6*z - 3780*y^6 - 5220*y^5*z^2 + 16092*y^5*z - 12708*y^5 + 6086*y^4*z^3 - 27729*y^4*z^2 + 42636*y^4*z - 22057*y^4 - 4120*y^3*z^4 + 24910*y^3*z^3 - 56535*y^3*z^2 + 57394*y^3*z - 21825*y^3 + 1632*y^2*z^5 - 12354*y^2*z^4 + 37077*y^2*z^3 - 55719*y^2*z^2 + 41913*y^2*z - 12536*y^2 - 352*y*z^6 + 3216*y*z^5 - 12036*y*z^4 + 23918*y*z^3 - 26769*y*z^2 + 15938*y*z - 3926*y + 32*z^7 - 344*z^6 + 1548*z^5 - 3830*z^4 + 5685*z^3 - 5060*z^2 + 2494*z - 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -2, 3, -2 ], [ -1, 2, -1 ], [ -1, 1, -2 ] ], "source_shift": [ 1, 2, 2 ], "target_matrix": [ [ 3, 3, 2 ], [ -1, -1, -1 ], [ -2, -1, -1 ] ], "target_shift": [ -1037, 343, 517 ], "collision_preimages": [ [ "-29/4", "-3", "13/4" ], [ "-19/2", "-7", "-1" ], [ "17/2", "1", "-6" ] ], "collision_image": [ "-4151/4", "1373/4", "1035/2" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0803470889986784 ], "surface_features": [ 4.354917823027693, 5.295479132678017, 4.0046437500057115, 0.0, 4.496766597201882, 5181 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B137", "pair": 137, "family": "bijection", "mapping": "F(x,y,z)=(10*x^7 + 70*x^6*y - 70*x^6*z - 110*x^6 + 210*x^5*y^2 - 420*x^5*y*z - 660*x^5*y + 210*x^5*z^2 + 680*x^5*z + 560*x^5 + 350*x^4*y^3 - 1050*x^4*y^2*z - 1650*x^4*y^2 + 1050*x^4*y*z^2 + 3400*x^4*y*z + 2810*x^4*y - 350*x^4*z^3 - 1750*x^4*z^2 - 2940*x^4*z - 1690*x^4 + 350*x^3*y^4 - 1400*x^3*y^3*z - 2200*x^3*y^3 + 2100*x^3*y^2*z^2 + 6800*x^3*y^2*z + 5640*x^3*y^2 - 1400*x^3*y*z^3 - 7000*x^3*y*z^2 - 11800*x^3*y*z - 6840*x^3*y + 350*x^3*z^4 + 2400*x^3*z^3 + 6170*x^3*z^2 + 7220*x^3*z + 3252*x^3 + 210*x^2*y^5 - 1050*x^2*y^4*z - 1650*x^2*y^4 + 2100*x^2*y^3*z^2 + 6800*x^2*y^3*z + 5660*x^2*y^3 - 2100*x^2*y^2*z^3 - 10500*x^2*y^2*z^2 - 17760*x^2*y^2*z - 10380*x^2*y^2 + 1050*x^2*y*z^4 + 7200*x^2*y*z^3 + 18570*x^2*y*z^2 + 21920*x^2*y*z + 10016*x^2*y - 210*x^2*z^5 - 1850*x^2*z^4 - 6470*x^2*z^3 - 11550*x^2*z^2 - 10596*x^2*z - 3934*x^2 + 70*x*y^6 - 420*x*y^5*z - 660*x*y^5 + 1050*x*y^4*z^2 + 3400*x*y^4*z + 2840*x*y^4 - 1400*x*y^3*z^3 - 7000*x*y^3*z^2 - 11880*x*y^3*z - 7000*x*y^3 + 1050*x*y^2*z^4 + 7200*x*y^2*z^3 + 18630*x*y^2*z^2 + 22180*x*y^2*z + 10276*x*y^2 - 420*x*y*z^5 - 3700*x*y*z^4 - 12980*x*y*z^3 - 23380*x*y*z^2 - 21792*x*y*z - 8268*x*y + 70*x*z^6 + 760*x*z^5 + 3390*x*z^4 + 8200*x*z^3 + 11496*x*z^2 + 8708*x*z + 2705*x + 10*y^7 - 70*y^6*z - 110*y^6 + 210*y^5*z^2 + 680*y^5*z + 570*y^5 - 350*y^4*z^3 - 1750*y^4*z^2 - 2980*y^4*z - 1770*y^4 + 350*y^3*z^4 + 2400*y^3*z^3 + 6230*y^3*z^2 + 7480*y^3*z + 3512*y^3 - 210*y^2*z^5 - 1850*y^2*z^4 - 6510*y^2*z^3 - 11830*y^2*z^2 - 11196*y^2*z - 4334*y^2 + 70*y*z^6 + 760*y*z^5 + 3400*y*z^4 + 8300*y*z^3 + 11846*y*z^2 + 9208*y*z + 2957*y - 10*z^7 - 130*z^6 - 710*z^5 - 2180*z^4 - 4152*z^3 - 4804*z^2 - 3003*z - 1, 30*x^7 + 210*x^6*y - 210*x^6*z - 330*x^6 + 630*x^5*y^2 - 1260*x^5*y*z - 1980*x^5*y + 630*x^5*z^2 + 2040*x^5*z + 1680*x^5 + 1050*x^4*y^3 - 3150*x^4*y^2*z - 4950*x^4*y^2 + 3150*x^4*y*z^2 + 10200*x^4*y*z + 8430*x^4*y - 1050*x^4*z^3 - 5250*x^4*z^2 - 8820*x^4*z - 5070*x^4 + 1050*x^3*y^4 - 4200*x^3*y^3*z - 6600*x^3*y^3 + 6300*x^3*y^2*z^2 + 20400*x^3*y^2*z + 16920*x^3*y^2 - 4200*x^3*y*z^3 - 21000*x^3*y*z^2 - 35400*x^3*y*z - 20520*x^3*y + 1050*x^3*z^4 + 7200*x^3*z^3 + 18510*x^3*z^2 + 21660*x^3*z + 9752*x^3 + 630*x^2*y^5 - 3150*x^2*y^4*z - 4950*x^2*y^4 + 6300*x^2*y^3*z^2 + 20400*x^2*y^3*z + 16980*x^2*y^3 - 6300*x^2*y^2*z^3 - 31500*x^2*y^2*z^2 - 53280*x^2*y^2*z - 31140*x^2*y^2 + 3150*x^2*y*z^4 + 21600*x^2*y*z^3 + 55710*x^2*y*z^2 + 65760*x^2*y*z + 30036*x^2*y - 630*x^2*z^5 - 5550*x^2*z^4 - 19410*x^2*z^3 - 34650*x^2*z^2 - 31776*x^2*z - 11792*x^2 + 210*x*y^6 - 1260*x*y^5*z - 1980*x*y^5 + 3150*x*y^4*z^2 + 10200*x*y^4*z + 8520*x*y^4 - 4200*x*y^3*z^3 - 21000*x*y^3*z^2 - 35640*x*y^3*z - 21000*x*y^3 + 3150*x*y^2*z^4 + 21600*x*y^2*z^3 + 55890*x*y^2*z^2 + 66540*x*y^2*z + 30816*x*y^2 - 1260*x*y*z^5 - 11100*x*y*z^4 - 38940*x*y*z^3 - 70140*x*y*z^2 - 65352*x*y*z - 24784*x*y + 210*x*z^6 + 2280*x*z^5 + 10170*x*z^4 + 24600*x*z^3 + 34476*x*z^2 + 26104*x*z + 8099*x + 30*y^7 - 210*y^6*z - 330*y^6 + 630*y^5*z^2 + 2040*y^5*z + 1710*y^5 - 1050*y^4*z^3 - 5250*y^4*z^2 - 8940*y^4*z - 5310*y^4 + 1050*y^3*z^4 + 7200*y^3*z^3 + 18690*y^3*z^2 + 22440*y^3*z + 10532*y^3 - 630*y^2*z^5 - 5550*y^2*z^4 - 19530*y^2*z^3 - 35490*y^2*z^2 - 33576*y^2*z - 12992*y^2 + 210*y*z^6 + 2280*y*z^5 + 10200*y*z^4 + 24900*y*z^3 + 35526*y*z^2 + 27604*y*z + 8851*y - 30*z^7 - 390*z^6 - 2130*z^5 - 6540*z^4 - 12452*z^3 - 14402*z^2 - 8995*z - 1, -20*x^7 - 140*x^6*y + 140*x^6*z + 220*x^6 - 420*x^5*y^2 + 840*x^5*y*z + 1320*x^5*y - 420*x^5*z^2 - 1360*x^5*z - 1120*x^5 - 700*x^4*y^3 + 2100*x^4*y^2*z + 3300*x^4*y^2 - 2100*x^4*y*z^2 - 6800*x^4*y*z - 5620*x^4*y + 700*x^4*z^3 + 3500*x^4*z^2 + 5880*x^4*z + 3380*x^4 - 700*x^3*y^4 + 2800*x^3*y^3*z + 4400*x^3*y^3 - 4200*x^3*y^2*z^2 - 13600*x^3*y^2*z - 11280*x^3*y^2 + 2800*x^3*y*z^3 + 14000*x^3*y*z^2 + 23600*x^3*y*z + 13680*x^3*y - 700*x^3*z^4 - 4800*x^3*z^3 - 12340*x^3*z^2 - 14440*x^3*z - 6501*x^3 - 420*x^2*y^5 + 2100*x^2*y^4*z + 3300*x^2*y^4 - 4200*x^2*y^3*z^2 - 13600*x^2*y^3*z - 11320*x^2*y^3 + 4200*x^2*y^2*z^3 + 21000*x^2*y^2*z^2 + 35520*x^2*y^2*z + 20760*x^2*y^2 - 2100*x^2*y*z^4 - 14400*x^2*y*z^3 - 37140*x^2*y*z^2 - 43840*x^2*y*z - 20023*x^2*y + 420*x^2*z^5 + 3700*x^2*z^4 + 12940*x^2*z^3 + 23100*x^2*z^2 + 21183*x^2*z + 7861*x^2 - 140*x*y^6 + 840*x*y^5*z + 1320*x*y^5 - 2100*x*y^4*z^2 - 6800*x*y^4*z - 5680*x*y^4 + 2800*x*y^3*z^3 + 14000*x*y^3*z^2 + 23760*x*y^3*z + 14000*x*y^3 - 2100*x*y^2*z^4 - 14400*x*y^2*z^3 - 37260*x*y^2*z^2 - 44360*x*y^2*z - 20543*x*y^2 + 840*x*y*z^5 + 7400*x*y*z^4 + 25960*x*y*z^3 + 46760*x*y*z^2 + 43566*x*y*z + 16522*x*y - 140*x*z^6 - 1520*x*z^5 - 6780*x*z^4 - 16400*x*z^3 - 22983*x*z^2 - 17402*x*z - 5400*x - 20*y^7 + 140*y^6*z + 220*y^6 - 420*y^5*z^2 - 1360*y^5*z - 1140*y^5 + 700*y^4*z^3 + 3500*y^4*z^2 + 5960*y^4*z + 3540*y^4 - 700*y^3*z^4 - 4800*y^3*z^3 - 12460*y^3*z^2 - 14960*y^3*z - 7021*y^3 + 420*y^2*z^5 + 3700*y^2*z^4 + 13020*y^2*z^3 + 23660*y^2*z^2 + 22383*y^2*z + 8661*y^2 - 140*y*z^6 - 1520*y*z^5 - 6800*y*z^4 - 16600*y*z^3 - 23683*y*z^2 - 18402*y*z - 5901*y + 20*z^7 + 260*z^6 + 1420*z^5 + 4360*z^4 + 8301*z^3 + 9601*z^2 + 5998*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -1, -1, 1 ], [ -2, -3, 2 ], [ -1, -1, 2 ] ], "source_shift": [ 1, 1, 2 ], "target_matrix": [ [ -1, -2, -1 ], [ -3, -2, -2 ], [ 2, 1, 1 ] ], "target_shift": [ 748, 2242, -1496 ], "inverse_recipe": { "p": 3, "q": 2, "f": "x**3 + x", "g": "x**2 + x", "h": "y*z**2 + 2*y*z + y", "polynomials": { "f": [ [ [ 3, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "g": [ [ [ 2, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 2 ], 1 ], [ [ 0, 1, 1 ], 2 ], [ [ 0, 1, 0 ], 1 ] ] }, "h_multiplier": 10 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0810613419998845 ], "surface_features": [ 3.9648290423079504, 4.845965761545484, 3.630122642859312, 0.0, 4.0876889949557835, 5054 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C145", "pair": 145, "family": "counterexample", "mapping": "F(x,y,z)=(108*x^7 + 594*x^6*y + 702*x^6*z + 1134*x^6 + 1386*x^5*y^2 + 3294*x^5*y*z + 5301*x^5*y + 1944*x^5*z^2 + 6507*x^5*z + 4266*x^5 + 1778*x^4*y^3 + 6372*x^4*y^2*z + 10227*x^4*y^2 + 7560*x^4*y*z^2 + 25272*x^4*y*z + 16506*x^4*y + 2970*x^4*z^3 + 15525*x^4*z^2 + 20601*x^4*z + 8469*x^4 + 1354*x^3*y^4 + 6502*x^3*y^3*z + 10423*x^3*y^3 + 11628*x^3*y^2*z^2 + 38877*x^3*y^2*z + 25308*x^3*y^2 + 9180*x^3*y*z^3 + 48078*x^3*y*z^2 + 63612*x^3*y*z + 26004*x^3*y + 2700*x^3*z^4 + 19710*x^3*z^3 + 39744*x^3*z^2 + 32868*x^3*z + 9900*x^3 + 612*x^2*y^5 + 3690*x^2*y^4*z + 5919*x^2*y^4 + 8838*x^2*y^3*z^2 + 29610*x^2*y^3*z + 19224*x^2*y^3 + 10512*x^2*y^2*z^3 + 55269*x^2*y^2*z^2 + 72957*x^2*y^2*z + 29661*x^2*y^2 + 6210*x^2*y*z^4 + 45612*x^2*y*z^3 + 91800*x^2*y*z^2 + 75522*x^2*y*z + 22572*x^2*y + 1458*x^2*z^5 + 14040*x^2*z^4 + 38286*x^2*z^3 + 47790*x^2*z^2 + 28872*x^2*z + 6906*x^2 + 152*x*y^6 + 1104*x*y^5*z + 1776*x*y^5 + 3318*x*y^4*z^2 + 11166*x*y^4*z + 7236*x*y^4 + 5282*x*y^3*z^3 + 27951*x*y^3*z^2 + 36840*x*y^3*z + 14901*x*y^3 + 4698*x*y^2*z^4 + 34815*x*y^2*z^3 + 69990*x*y^2*z^2 + 57291*x*y^2*z + 16992*x*y^2 + 2214*x*y*z^5 + 21573*x*y*z^4 + 58788*x*y*z^3 + 73032*x*y*z^2 + 43788*x*y*z + 10362*x*y + 432*x*z^6 + 5319*x*z^5 + 18414*x*z^4 + 30852*x*z^3 + 28044*x*z^2 + 13428*x*z + 2668*x + 16*y^7 + 136*y^6*z + 220*y^6 + 492*y^5*z^2 + 1668*y^5*z + 1080*y^5 + 982*y^4*z^3 + 5247*y^4*z^2 + 6912*y^4*z + 2783*y^4 + 1168*y^3*z^4 + 8764*y^3*z^3 + 17616*y^3*z^2 + 14354*y^3*z + 4226*y^3 + 828*y^2*z^5 + 8196*y^2*z^4 + 22341*y^2*z^3 + 27630*y^2*z^2 + 16443*y^2*z + 3851*y^2 + 324*y*z^6 + 4068*y*z^5 + 14094*y*z^4 + 23514*y*z^3 + 21216*y*z^2 + 10050*y*z + 1970*y + 54*z^7 + 837*z^6 + 3537*z^5 + 7461*z^4 + 9072*z^3 + 6522*z^2 + 2588*z + 1, -54*x^7 - 297*x^6*y - 351*x^6*z - 594*x^6 - 693*x^5*y^2 - 1647*x^5*y*z - 2781*x^5*y - 972*x^5*z^2 - 3402*x^5*z - 2385*x^5 - 889*x^4*y^3 - 3186*x^4*y^2*z - 5373*x^4*y^2 - 3780*x^4*y*z^2 - 13230*x^4*y*z - 9258*x^4*y - 1485*x^4*z^3 - 8100*x^4*z^2 - 11502*x^4*z - 4997*x^4 - 677*x^3*y^4 - 3251*x^3*y^3*z - 5483*x^3*y^3 - 5814*x^3*y^2*z^2 - 20376*x^3*y^2*z - 14238*x^3*y^2 - 4590*x^3*y*z^3 - 25110*x^3*y*z^2 - 35625*x^3*y*z - 15409*x^3*y - 1350*x^3*z^4 - 10260*x^3*z^3 - 22158*x^3*z^2 - 19375*x^3*z - 6094*x^3 - 306*x^2*y^5 - 1845*x^2*y^4*z - 3117*x^2*y^4 - 4419*x^2*y^3*z^2 - 15534*x^2*y^3*z - 10845*x^2*y^3 - 5256*x^2*y^2*z^3 - 28890*x^2*y^2*z^2 - 40974*x^2*y^2*z - 17646*x^2*y^2 - 3105*x^2*y*z^4 - 23760*x^2*y*z^3 - 51327*x^2*y*z^2 - 44703*x^2*y*z - 13962*x^2*y - 729*x^2*z^5 - 7290*x^2*z^4 - 21312*x^2*z^3 - 28143*x^2*z^2 - 17760*x^2*z - 4392*x^2 - 76*x*y^6 - 552*x*y^5*z - 936*x*y^5 - 1659*x*y^4*z^2 - 5862*x*y^4*z - 4092*x*y^4 - 2641*x*y^3*z^3 - 14619*x*y^3*z^2 - 20742*x*y^3*z - 8897*x*y^3 - 2349*x*y^2*z^4 - 18144*x*y^2*z^3 - 39234*x*y^2*z^2 - 34041*x*y^2*z - 10557*x*y^2 - 1107*x*y*z^5 - 11205*x*y*z^4 - 32811*x*y*z^3 - 43179*x*y*z^2 - 27063*x*y*z - 6624*x*y - 216*x*z^6 - 2754*x*z^5 - 10233*x*z^4 - 18149*x*z^3 - 17238*x*z^2 - 8535*x*z - 1739*x - 8*y^7 - 68*y^6*z - 116*y^6 - 246*y^5*z^2 - 876*y^5*z - 612*y^5 - 491*y^4*z^3 - 2745*y^4*z^2 - 3900*y^4*z - 1667*y^4 - 584*y^3*z^4 - 4568*y^3*z^3 - 9897*y^3*z^2 - 8558*y^3*z - 2636*y^3 - 414*y^2*z^5 - 4257*y^2*z^4 - 12498*y^2*z^3 - 16395*y^2*z^2 - 10206*y^2*z - 2473*y^2 - 162*y*z^6 - 2106*y*z^5 - 7851*y*z^4 - 13885*y*z^3 - 13101*y*z^2 - 6420*y*z - 1291*y - 27*z^7 - 432*z^6 - 1962*z^5 - 4384*z^4 - 5572*z^3 - 4143*z^2 - 1686*z - 1, -162*x^7 - 891*x^6*y - 1053*x^6*z - 1728*x^6 - 2079*x^5*y^2 - 4941*x^5*y*z - 8082*x^5*y - 2916*x^5*z^2 - 9909*x^5*z - 6651*x^5 - 2667*x^4*y^3 - 9558*x^4*y^2*z - 15600*x^4*y^2 - 11340*x^4*y*z^2 - 38502*x^4*y*z - 25764*x^4*y - 4455*x^4*z^3 - 23625*x^4*z^2 - 32103*x^4*z - 13464*x^4 - 2031*x^3*y^4 - 9753*x^3*y^3*z - 15906*x^3*y^3 - 17442*x^3*y^2*z^2 - 59253*x^3*y^2*z - 39546*x^3*y^2 - 13770*x^3*y*z^3 - 73188*x^3*y*z^2 - 99237*x^3*y*z - 41406*x^3*y - 4050*x^3*z^4 - 29970*x^3*z^3 - 61902*x^3*z^2 - 52236*x^3*z - 15978*x^3 - 918*x^2*y^5 - 5535*x^2*y^4*z - 9036*x^2*y^4 - 13257*x^2*y^3*z^2 - 45144*x^2*y^3*z - 30069*x^2*y^3 - 15768*x^2*y^2*z^3 - 84159*x^2*y^2*z^2 - 113931*x^2*y^2*z - 47298*x^2*y^2 - 9315*x^2*y*z^4 - 69372*x^2*y*z^3 - 143127*x^2*y*z^2 - 120207*x^2*y*z - 36492*x^2*y - 2187*x^2*z^5 - 21330*x^2*z^4 - 59598*x^2*z^3 - 75924*x^2*z^2 - 46587*x^2*z - 11265*x^2 - 228*x*y^6 - 1656*x*y^5*z - 2712*x*y^5 - 4977*x*y^4*z^2 - 17028*x*y^4*z - 11328*x*y^4 - 7923*x*y^3*z^3 - 42570*x*y^3*z^2 - 57582*x*y^3*z - 23793*x*y^3 - 7047*x*y^2*z^4 - 52959*x*y^2*z^3 - 109224*x*y^2*z^2 - 91317*x*y^2*z - 27513*x*y^2 - 3321*x*y*z^5 - 32778*x*y*z^4 - 91599*x*y*z^3 - 116196*x*y*z^2 - 70773*x*y*z - 16929*x*y - 648*x*z^6 - 8073*x*z^5 - 28647*x*z^4 - 48996*x*z^3 - 45240*x*z^2 - 21900*x*z - 4383*x - 24*y^7 - 204*y^6*z - 336*y^6 - 738*y^5*z^2 - 2544*y^5*z - 1692*y^5 - 1473*y^4*z^3 - 7992*y^4*z^2 - 10812*y^4*z - 4449*y^4 - 1752*y^3*z^4 - 13332*y^3*z^3 - 27513*y^3*z^2 - 22908*y^3*z - 6852*y^3 - 1242*y^2*z^5 - 12453*y^2*z^4 - 34839*y^2*z^3 - 44019*y^2*z^2 - 26616*y^2*z - 6300*y^2 - 486*y*z^6 - 6174*y*z^5 - 21945*y*z^4 - 37395*y*z^3 - 34281*y*z^2 - 16416*y*z - 3241*y - 81*z^7 - 1269*z^6 - 5499*z^5 - 11844*z^4 - 14631*z^3 - 10635*z^2 - 4251*z - 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ 1, 1, 1 ], [ 3, 2, 3 ], [ 2, 1, 1 ] ], "source_shift": [ 1, 2, 1 ], "target_matrix": [ [ 2, 1, 1 ], [ -1, -1, -1 ], [ -3, -2, -3 ] ], "target_shift": [ -437, 290, 723 ], "collision_preimages": [ [ "-1/4", "-1", "1/4" ], [ "11/2", "7/2", "-9" ], [ "15/2", "-11/2", "-4" ] ], "collision_image": [ "-875/2", "1161/4", "2895/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.027344512000127 ], "surface_features": [ 4.229831102727802, 5.155721568385414, 3.9110510203281263, 0.0, 4.356283564796642, 5145 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C220", "pair": 220, "family": "counterexample", "mapping": "F(x,y,z)=(4*x^7 + 26*x^6*y - 38*x^6*z - 50*x^6 + 72*x^5*y^2 - 210*x^5*y*z - 279*x^5*y + 150*x^5*z^2 + 417*x^5*z + 270*x^5 + 110*x^4*y^3 - 480*x^4*y^2*z - 645*x^4*y^2 + 684*x^4*y*z^2 + 1926*x^4*y*z + 1254*x^4*y - 318*x^4*z^3 - 1413*x^4*z^2 - 1911*x^4*z - 813*x^4 + 100*x^3*y^4 - 580*x^3*y^3*z - 790*x^3*y^3 + 1236*x^3*y^2*z^2 + 3534*x^3*y^2*z + 2316*x^3*y^2 - 1146*x^3*y*z^3 - 5181*x^3*y*z^2 - 7056*x^3*y*z - 3010*x^3*y + 390*x^3*z^4 + 2487*x^3*z^3 + 5292*x^3*z^2 + 4678*x^3*z + 1464*x^3 + 54*x^2*y^5 - 390*x^2*y^4*z - 540*x^2*y^4 + 1104*x^2*y^3*z^2 + 3216*x^2*y^3*z + 2124*x^2*y^3 - 1530*x^2*y^2*z^3 - 7065*x^2*y^2*z^2 - 9702*x^2*y^2*z - 4152*x^2*y^2 + 1038*x^2*y*z^4 + 6780*x^2*y*z^3 + 14556*x^2*y*z^2 + 12906*x^2*y*z + 4038*x^2*y - 276*x^2*z^5 - 2397*x^2*z^4 - 7167*x^2*z^3 - 9891*x^2*z^2 - 6408*x^2*z - 1562*x^2 + 16*x*y^6 - 138*x*y^5*z - 195*x*y^5 + 486*x*y^4*z^2 + 1449*x*y^4*z + 966*x*y^4 - 894*x*y^3*z^3 - 4239*x*y^3*z^2 - 5880*x*y^3*z - 2526*x*y^3 + 906*x*y^2*z^4 + 6099*x*y^2*z^3 + 13236*x*y^2*z^2 + 11778*x*y^2*z + 3684*x*y^2 - 480*x*y*z^5 - 4314*x*y*z^4 - 13050*x*y*z^3 - 18072*x*y*z^2 - 11694*x*y*z - 2839*x*y + 104*x*z^6 + 1200*x*z^5 + 4752*x*z^4 + 9118*x*z^3 + 9168*x*z^2 + 4607*x*z + 902*x + 2*y^7 - 20*y^6*z - 29*y^6 + 84*y^5*z^2 + 258*y^5*z + 174*y^5 - 192*y^4*z^3 - 942*y^4*z^2 - 1323*y^4*z - 571*y^4 + 258*y^3*z^4 + 1806*y^3*z^3 + 3972*y^3*z^2 + 3550*y^3*z + 1110*y^3 - 204*y^2*z^5 - 1917*y^2*z^4 - 5883*y^2*z^3 - 8181*y^2*z^2 - 5286*y^2*z - 1277*y^2 + 88*y*z^6 + 1068*y*z^5 + 4296*y*z^4 + 8278*y*z^3 + 8304*y*z^2 + 4142*y*z + 802*y - 16*z^7 - 244*z^6 - 1236*z^5 - 3100*z^4 - 4304*z^3 - 3329*z^2 - 1323*z - 1, 2*x^7 + 13*x^6*y - 19*x^6*z - 28*x^6 + 36*x^5*y^2 - 105*x^5*y*z - 156*x^5*y + 75*x^5*z^2 + 231*x^5*z + 165*x^5 + 55*x^4*y^3 - 240*x^4*y^2*z - 360*x^4*y^2 + 342*x^4*y*z^2 + 1065*x^4*y*z + 765*x^4*y - 159*x^4*z^3 - 774*x^4*z^2 - 1152*x^4*z - 530*x^4 + 50*x^3*y^4 - 290*x^3*y^3*z - 440*x^3*y^3 + 618*x^3*y^2*z^2 + 1950*x^3*y^2*z + 1410*x^3*y^2 - 573*x^3*y*z^3 - 2832*x^3*y*z^2 - 4245*x^3*y*z - 1959*x^3*y + 195*x^3*z^4 + 1347*x^3*z^3 + 3144*x^3*z^2 + 3009*x^3*z + 1000*x^3 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5157*y*z^3 + 5523*y*z^2 + 2890*y*z + 577*y - 8*z^7 - 128*z^6 - 702*z^5 - 1905*z^4 - 2828*z^3 - 2302*z^2 - 946*z - 1, -2*x^7 - 13*x^6*y + 19*x^6*z + 28*x^6 - 36*x^5*y^2 + 105*x^5*y*z + 156*x^5*y - 75*x^5*z^2 - 231*x^5*z - 165*x^5 - 55*x^4*y^3 + 240*x^4*y^2*z + 360*x^4*y^2 - 342*x^4*y*z^2 - 1065*x^4*y*z - 765*x^4*y + 159*x^4*z^3 + 774*x^4*z^2 + 1152*x^4*z + 528*x^4 - 50*x^3*y^4 + 290*x^3*y^3*z + 440*x^3*y^3 - 618*x^3*y^2*z^2 - 1950*x^3*y^2*z - 1410*x^3*y^2 + 573*x^3*y*z^3 + 2832*x^3*y*z^2 + 4245*x^3*y*z + 1952*x^3*y - 195*x^3*z^4 - 1347*x^3*z^3 - 3144*x^3*z^2 - 3002*x^3*z - 984*x^3 - 27*x^2*y^5 + 195*x^2*y^4*z + 300*x^2*y^4 - 552*x^2*y^3*z^2 - 1770*x^2*y^3*z - 1290*x^2*y^3 + 765*x^2*y^2*z^3 + 3852*x^2*y^2*z^2 + 5823*x^2*y^2*z + 2688*x^2*y^2 - 519*x^2*y*z^4 - 3663*x^2*y*z^3 - 8628*x^2*y*z^2 - 8268*x^2*y*z - 2712*x^2*y + 138*x^2*z^5 + 1284*x^2*z^4 + 4194*x^2*z^3 + 6261*x^2*z^2 + 4281*x^2*z + 1057*x^2 - 8*x*y^6 + 69*x*y^5*z + 108*x*y^5 - 243*x*y^4*z^2 - 795*x*y^4*z - 585*x*y^4 + 447*x*y^3*z^3 + 2304*x*y^3*z^2 + 3519*x*y^3*z + 1632*x*y^3 - 453*x*y^2*z^4 - 3285*x*y^2*z^3 - 7824*x*y^2*z^2 - 7530*x*y^2*z - 2472*x*y^2 + 240*x*y*z^5 + 2304*x*y*z^4 + 7617*x*y*z^3 + 11418*x*y*z^2 + 7803*x*y*z + 1922*x*y - 52*x*z^6 - 636*x*z^5 - 2739*x*z^4 - 5687*x*z^3 - 6066*x*z^2 - 3130*x*z - 600*x - y^7 + 10*y^6*z + 16*y^6 - 42*y^5*z^2 - 141*y^5*z - 105*y^5 + 96*y^4*z^3 + 510*y^4*z^2 + 789*y^4*z + 368*y^4 - 129*y^3*z^4 - 969*y^3*z^3 - 2340*y^3*z^2 - 2264*y^3*z - 744*y^3 + 102*y^2*z^5 + 1020*y^2*z^4 + 3423*y^2*z^3 + 5157*y^2*z^2 + 3522*y^2*z + 865*y^2 - 44*y*z^6 - 564*y*z^5 - 2469*y*z^4 - 5153*y*z^3 - 5487*y*z^2 - 2812*y*z - 535*y + 8*z^7 + 128*z^6 + 702*z^5 + 1904*z^4 + 2815*z^3 + 2257*z^2 + 892*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -1, -1, 1 ], [ -1, -1, 2 ], [ -2, -1, 1 ] ], "source_shift": [ 2, 1, 1 ], "target_matrix": [ [ -2, -1, -1 ], [ -1, -1, -1 ], [ 1, 1, 2 ] ], "target_shift": [ 212, 155, -139 ], "collision_preimages": [ [ "-3/4", "15/4", "1" ], [ "-13/2", "6", "-3/2" ], [ "-17/2", "15", "7/2" ] ], "collision_image": [ "425/2", "621/4", "-557/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 0.9834322339993378 ], "surface_features": [ 3.343213779599929, 4.2570062179123065, 2.985650973690949, 0.0, 3.4601585420967256, 4802 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C006", "pair": 6, "family": "counterexample", "mapping": "F(x,y,z)=(-32*x^7 + 176*x^6*y + 288*x^6*z - 320*x^6 - 408*x^5*y^2 - 1320*x^5*y*z + 1440*x^5*y - 1056*x^5*z^2 + 2304*x^5*z - 1224*x^5 + 516*x^4*y^3 + 2472*x^4*y^2*z - 2664*x^4*y^2 + 3900*x^4*y*z^2 - 8424*x^4*y*z + 4476*x^4*y + 2024*x^4*z^3 - 6576*x^4*z^2 + 7056*x^4*z - 2482*x^4 - 384*x^3*y^4 - 2418*x^3*y^3*z + 2592*x^3*y^3 - 5634*x^3*y^2*z^2 + 12144*x^3*y^2*z - 6456*x^3*y^2 - 5750*x^3*y*z^3 + 18720*x^3*y*z^2 - 20082*x^3*y*z + 7096*x^3*y - 2166*x^3*z^4 + 9488*x^3*z^3 - 15390*x^3*z^2 + 10991*x^3*z - 2950*x^3 + 168*x^2*y^5 + 1302*x^2*y^4*z - 1398*x^2*y^4 + 3978*x^2*y^3*z^2 - 8622*x^2*y^3*z + 4590*x^2*y^3 + 5982*x^2*y^2*z^3 - 19674*x^2*y^2*z^2 + 21126*x^2*y^2*z - 7494*x^2*y^2 + 4422*x^2*y*z^4 - 19674*x^2*y*z^3 + 31932*x^2*y*z^2 - 22863*x^2*y*z + 6177*x^2*y + 1284*x^2*z^5 - 7272*x^2*z^4 + 15840*x^2*z^3 - 17142*x^2*z^2 + 9360*x^2*z - 2087*x^2 - 40*x*y^6 - 366*x*y^5*z + 396*x*y^5 - 1374*x*y^4*z^2 + 3012*x*y^4*z - 1608*x*y^4 - 2706*x*y^3*z^3 + 9036*x*y^3*z^2 - 9732*x*y^3*z + 3466*x*y^3 - 2946*x*y^2*z^4 + 13356*x*y^2*z^3 - 21756*x*y^2*z^2 + 15627*x*y^2*z - 4242*x*y^2 - 1680*x*y*z^5 + 9720*x*y*z^4 - 21282*x*y*z^3 + 23094*x*y*z^2 - 12636*x*y*z + 2834*x*y - 392*x*z^6 + 2784*x*z^5 - 7686*x*z^4 + 11186*x*z^3 - 9222*x*z^2 + 4169*x*z - 820*x + 4*y^7 + 42*y^6*z - 46*y^6 + 186*y^5*z^2 - 414*y^5*z + 222*y^5 + 450*y^4*z^3 - 1530*y^4*z^2 + 1656*y^4*z - 592*y^4 + 642*y^3*z^4 - 2970*y^3*z^3 + 4866*y^3*z^2 - 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+ 6848*y^4*z + 4680*y^4 + 280*y^3*z^4 + 2240*y^3*z^3 + 6856*y^3*z^2 + 9396*y^3*z + 4808*y^3 + 84*y^2*z^5 + 840*y^2*z^4 + 3432*y^2*z^3 + 7074*y^2*z^2 + 7272*y^2*z + 2961*y^2 + 14*y*z^6 + 168*y*z^5 + 859*y*z^4 + 2367*y*z^3 + 3666*y*z^2 + 3008*y*z + 1011*y + z^7 + 14*z^6 + 86*z^5 + 297*z^4 + 616*z^3 + 764*z^2 + 519*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ 1, 2, 1 ], [ 1, 1, 1 ], [ 3, 3, 2 ] ], "source_shift": [ 2, 1, 1 ], "target_matrix": [ [ 2, 1, 1 ], [ 2, 2, 3 ], [ 1, 1, 2 ] ], "target_shift": [ -398, -341, -153 ], "inverse_recipe": { "p": 3, "q": 4, "f": "x**3 + x", "g": "x**4", "h": "y*z + 2*y", "polynomials": { "f": [ [ [ 3, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "g": [ [ [ 4, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 1, 0 ], 2 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.1130254010022327 ], "surface_features": [ 3.608434505938868, 4.456487497115823, 3.225309281725863, 0.0, 3.731256652871072, 4891 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B184", "pair": 184, "family": "bijection", "mapping": "F(x,y,z)=(896*x^7 - 3136*x^6*y - 3136*x^6*z + 6496*x^6 + 4704*x^5*y^2 + 9408*x^5*y*z - 19600*x^5*y + 4704*x^5*z^2 - 19600*x^5*z + 20416*x^5 - 3920*x^4*y^3 - 11760*x^4*y^2*z + 24640*x^4*y^2 - 11760*x^4*y*z^2 + 49280*x^4*y*z - 51600*x^4*y - 3920*x^4*z^3 + 24640*x^4*z^2 - 51600*x^4*z + 36112*x^4 + 1960*x^3*y^4 + 7840*x^3*y^3*z - 16520*x^3*y^3 + 11760*x^3*y^2*z^2 - 49560*x^3*y^2*z + 52160*x^3*y^2 + 7840*x^3*y*z^3 - 49560*x^3*y*z^2 + 104320*x^3*y*z - 73344*x^3*y + 1960*x^3*z^4 - 16520*x^3*z^3 + 52160*x^3*z^2 - 73344*x^3*z + 39016*x^3 - 588*x^2*y^5 - 2940*x^2*y^4*z + 6230*x^2*y^4 - 5880*x^2*y^3*z^2 + 24920*x^2*y^3*z - 26360*x^2*y^3 - 5880*x^2*y^2*z^3 + 37380*x^2*y^2*z^2 - 79080*x^2*y^2*z + 55848*x^2*y^2 - 2940*x^2*y*z^4 + 24920*x^2*y*z^3 - 79080*x^2*y*z^2 + 111696*x^2*y*z - 59700*x^2*y - 588*x^2*z^5 + 6230*x^2*z^4 - 26360*x^2*z^3 + 55848*x^2*z^2 - 59672*x^2*z + 25943*x^2 + 98*x*y^6 + 588*x*y^5*z - 1253*x*y^5 + 1470*x*y^4*z^2 - 6265*x*y^4*z + 6660*x*y^4 + 1960*x*y^3*z^3 - 12530*x*y^3*z^2 + 26640*x*y^3*z - 18896*x*y^3 + 1470*x*y^2*z^4 - 12530*x*y^2*z^3 + 39960*x*y^2*z^2 - 56688*x*y^2*z + 30438*x*y^2 + 588*x*y*z^5 - 6265*x*y*z^4 + 26640*x*y*z^3 - 56688*x*y*z^2 + 60848*x*y*z - 26643*x*y + 98*x*z^6 - 1253*x*z^5 + 6660*x*z^4 - 18896*x*z^3 + 30410*x*z^2 - 26580*x*z + 9943*x - 7*y^7 - 49*y^6*z + 105*y^6 - 147*y^5*z^2 + 630*y^5*z - 673*y^5 - 245*y^4*z^3 + 1575*y^4*z^2 - 3365*y^4*z + 2397*y^4 - 245*y^3*z^4 + 2100*y^3*z^3 - 6730*y^3*z^2 + 9588*y^3*z - 5171*y^3 - 147*y^2*z^5 + 1575*y^2*z^4 - 6730*y^2*z^3 + 14382*y^2*z^2 - 15506*y^2*z + 6841*y^2 - 49*y*z^6 + 630*y*z^5 - 3365*y*z^4 + 9588*y*z^3 - 15499*y*z^2 + 13647*y*z - 5175*y - 7*z^7 + 105*z^6 - 673*z^5 + 2397*z^4 - 5164*z^3 + 6806*z^2 - 5132*z + 1, -1792*x^7 + 6272*x^6*y + 6272*x^6*z - 12992*x^6 - 9408*x^5*y^2 - 18816*x^5*y*z + 39200*x^5*y - 9408*x^5*z^2 + 39200*x^5*z - 40864*x^5 + 7840*x^4*y^3 + 23520*x^4*y^2*z - 49280*x^4*y^2 + 23520*x^4*y*z^2 - 98560*x^4*y*z + 103280*x^4*y + 7840*x^4*z^3 - 49280*x^4*z^2 + 103280*x^4*z - 72384*x^4 - 3920*x^3*y^4 - 15680*x^3*y^3*z + 33040*x^3*y^3 - 23520*x^3*y^2*z^2 + 99120*x^3*y^2*z - 104400*x^3*y^2 - 15680*x^3*y*z^3 + 99120*x^3*y*z^2 - 208800*x^3*y*z + 147008*x^3*y - 3920*x^3*z^4 + 33040*x^3*z^3 - 104400*x^3*z^2 + 147008*x^3*z - 78352*x^3 + 1176*x^2*y^5 + 5880*x^2*y^4*z - 12460*x^2*y^4 + 11760*x^2*y^3*z^2 - 49840*x^2*y^3*z + 52760*x^2*y^3 + 11760*x^2*y^2*z^3 - 74760*x^2*y^2*z^2 + 158280*x^2*y^2*z - 111936*x^2*y^2 + 5880*x^2*y*z^4 - 49840*x^2*y*z^3 + 158280*x^2*y*z^2 - 223872*x^2*y*z + 119880*x^2*y + 1176*x^2*z^5 - 12460*x^2*z^4 + 52760*x^2*z^3 - 111936*x^2*z^2 + 119824*x^2*z - 52202*x^2 - 196*x*y^6 - 1176*x*y^5*z + 2506*x*y^5 - 2940*x*y^4*z^2 + 12530*x*y^4*z - 13330*x*y^4 - 3920*x*y^3*z^3 + 25060*x*y^3*z^2 - 53320*x*y^3*z + 37872*x*y^3 - 2940*x*y^2*z^4 + 25060*x*y^2*z^3 - 79980*x*y^2*z^2 + 113616*x*y^2*z - 61116*x*y^2 - 1176*x*y*z^5 + 12530*x*y*z^4 - 53320*x*y*z^3 + 113616*x*y*z^2 - 122176*x*y*z + 53602*x*y - 196*x*z^6 + 2506*x*z^5 - 13330*x*z^4 + 37872*x*z^3 - 61060*x*z^2 + 53476*x*z - 20041*x + 14*y^7 + 98*y^6*z - 210*y^6 + 294*y^5*z^2 - 1260*y^5*z + 1347*y^5 + 490*y^4*z^3 - 3150*y^4*z^2 + 6735*y^4*z - 4804*y^4 + 490*y^3*z^4 - 4200*y^3*z^3 + 13470*y^3*z^2 - 19216*y^3*z + 10382*y^3 + 294*y^2*z^5 - 3150*y^2*z^4 + 13470*y^2*z^3 - 28824*y^2*z^2 + 31132*y^2*z - 13761*y^2 + 98*y*z^6 - 1260*y*z^5 + 6735*y*z^4 - 19216*y*z^3 + 31118*y*z^2 - 27452*y*z + 10428*y + 14*z^7 - 210*z^6 + 1347*z^5 - 4804*z^4 + 10368*z^3 - 13691*z^2 + 10341*z - 1, -896*x^7 + 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126*x^2*y^5 + 597*x^2*y^4*z - 1188*x^2*y^4 + 1128*x^2*y^3*z^2 - 4506*x^2*y^3*z + 4452*x^2*y^3 + 1062*x^2*y^2*z^3 - 6390*x^2*y^2*z^2 + 12696*x^2*y^2*z - 8457*x^2*y^2 + 498*x^2*y*z^4 - 4014*x^2*y*z^3 + 12036*x^2*y*z^2 - 16131*x^2*y*z + 8181*x^2*y + 93*x^2*z^5 - 942*x^2*z^4 + 3792*x^2*z^3 - 7674*x^2*z^2 + 7830*x^2*z - 3232*x^2 - 30*x*y^6 - 171*x*y^5*z + 345*x*y^5 - 405*x*y^4*z^2 + 1638*x*y^4*z - 1626*x*y^4 - 510*x*y^3*z^3 + 3102*x*y^3*z^2 - 6183*x*y^3*z + 4126*x*y^3 - 360*x*y^2*z^4 + 2928*x*y^2*z^3 - 8793*x*y^2*z^2 + 11790*x*y^2*z - 5991*x*y^2 - 135*x*y*z^5 + 1377*x*y*z^4 - 5541*x*y*z^3 + 11202*x*y*z^2 - 11439*x*y*z + 4741*x*y - 21*x*z^6 + 258*x*z^5 - 1305*x*z^4 + 3538*x*z^3 - 5448*x*z^2 + 4537*x*z - 1601*x + 3*y^7 + 20*y^6*z - 41*y^6 + 57*y^5*z^2 - 234*y^5*z + 234*y^5 + 90*y^4*z^3 - 555*y^4*z^2 + 1113*y^4*z - 745*y^4 + 85*y^3*z^4 - 700*y^3*z^3 + 2112*y^3*z^2 - 2837*y^3*z + 1445*y^3 + 48*y^2*z^5 - 495*y^2*z^4 + 1998*y^2*z^3 - 4041*y^2*z^2 + 4131*y^2*z - 1717*y^2 + 15*y*z^6 - 186*y*z^5 + 942*y*z^4 - 2551*y*z^3 + 3927*y*z^2 - 3275*y*z + 1161*y + 2*z^7 - 29*z^6 + 177*z^5 - 602*z^4 + 1241*z^3 - 1558*z^2 + 1108*z - 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ 1, -1, -1 ], [ 2, -1, -1 ], [ 3, -3, -2 ] ], "source_shift": [ 1, 2, 2 ], "target_matrix": [ [ -1, -2, -1 ], [ 2, 3, 3 ], [ -1, -1, -1 ] ], "target_shift": [ 519, -857, 343 ], "collision_preimages": [ [ "-1", "-3/4", "3/4" ], [ "-7/2", "-8", "9/2" ], [ "3/2", "-7", "21/2" ] ], "collision_image": [ "2077/4", "-1715/2", "1373/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0339369739995163 ], "surface_features": [ 3.631023307839171, 4.582801717665471, 3.298307137328508, 0.0, 3.7610856687549523, 4918 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C069", "pair": 69, "family": "counterexample", "mapping": "F(x,y,z)=(-54*x^7 - 297*x^6*y + 351*x^6*z + 567*x^6 - 693*x^5*y^2 + 1647*x^5*y*z + 2655*x^5*y - 972*x^5*z^2 - 3267*x^5*z - 1917*x^5 - 889*x^4*y^3 + 3186*x^4*y^2*z + 5133*x^4*y^2 - 3780*x^4*y*z^2 - 12717*x^4*y*z - 7407*x^4*y + 1485*x^4*z^3 + 7830*x^4*z^2 + 9288*x^4*z + 3289*x^4 - 677*x^3*y^4 + 3251*x^3*y^3*z + 5245*x^3*y^3 - 5814*x^3*y^2*z^2 - 19617*x^3*y^2*z - 11349*x^3*y^2 + 4590*x^3*y*z^3 + 24318*x^3*y*z^2 + 28656*x^3*y*z + 10025*x^3*y - 1350*x^3*z^4 - 9990*x^3*z^3 - 17982*x^3*z^2 - 12794*x^3*z - 3202*x^3 - 306*x^2*y^5 + 1845*x^2*y^4*z + 2988*x^2*y^4 - 4419*x^2*y^3*z^2 - 14991*x^2*y^3*z - 8622*x^2*y^3 + 5256*x^2*y^2*z^3 + 28053*x^2*y^2*z^2 + 32865*x^2*y^2*z + 11361*x^2*y^2 - 3105*x^2*y*z^4 - 23202*x^2*y*z^3 - 41526*x^2*y*z^2 - 29199*x^2*y*z - 7185*x^2*y + 729*x^2*z^5 + 7155*x^2*z^4 + 17388*x^2*z^3 + 18648*x^2*z^2 + 9348*x^2*z + 1797*x^2 - 76*x*y^6 + 552*x*y^5*z + 900*x*y^5 - 1659*x*y^4*z^2 - 5676*x*y^4*z - 3249*x*y^4 + 2641*x*y^3*z^3 + 14247*x*y^3*z^2 + 16611*x*y^3*z + 5677*x*y^3 - 2349*x*y^2*z^4 - 17787*x*y^2*z^3 - 31683*x*y^2*z^2 - 22023*x*y^2*z - 5331*x*y^2 + 1107*x*y*z^5 + 11043*x*y*z^4 + 26712*x*y*z^3 + 28323*x*y*z^2 + 13962*x*y*z + 2625*x*y - 216*x*z^6 - 2727*x*z^5 - 8397*x*z^4 - 12070*x*z^3 - 9090*x*z^2 - 3492*x*z - 536*x - 8*y^7 + 68*y^6*z + 112*y^6 - 246*y^5*z^2 - 852*y^5*z - 486*y^5 + 491*y^4*z^3 + 2688*y^4*z^2 + 3123*y^4*z + 1056*y^4 - 584*y^3*z^4 - 4501*y^3*z^3 - 7989*y^3*z^2 - 5493*y^3*z - 1309*y^3 + 414*y^2*z^5 + 4218*y^2*z^4 + 10167*y^2*z^3 + 10662*y^2*z^2 + 5172*y^2*z + 952*y^2 - 162*y*z^6 - 2097*y*z^5 - 6435*y*z^4 - 9149*y*z^3 - 6777*y*z^2 - 2547*y*z - 383*y + 27*z^7 + 432*z^6 + 1620*z^5 + 2927*z^4 + 2944*z^3 + 1695*z^2 + 518*z + 1, -108*x^7 - 594*x^6*y + 702*x^6*z + 1080*x^6 - 1386*x^5*y^2 + 3294*x^5*y*z + 5049*x^5*y - 1944*x^5*z^2 - 6237*x^5*z - 3339*x^5 - 1778*x^4*y^3 + 6372*x^4*y^2*z + 9747*x^4*y^2 - 7560*x^4*y*z^2 - 24246*x^4*y*z - 12837*x^4*y + 2970*x^4*z^3 + 14985*x^4*z^2 + 16200*x^4*z + 5345*x^4 - 1354*x^3*y^4 + 6502*x^3*y^3*z + 9947*x^3*y^3 - 11628*x^3*y^2*z^2 - 37359*x^3*y^2*z - 19575*x^3*y^2 + 9180*x^3*y*z^3 + 46494*x^3*y*z^2 + 49740*x^3*y*z + 16195*x^3*y - 2700*x^3*z^4 - 19170*x^3*z^3 - 31410*x^3*z^2 - 20809*x^3*z - 4961*x^3 - 612*x^2*y^5 + 3690*x^2*y^4*z + 5661*x^2*y^4 - 8838*x^2*y^3*z^2 - 28524*x^2*y^3*z - 14805*x^2*y^3 + 10512*x^2*y^2*z^3 + 53595*x^2*y^2*z^2 + 56784*x^2*y^2*z + 18252*x^2*y^2 - 6210*x^2*y*z^4 - 44496*x^2*y*z^3 - 72198*x^2*y*z^2 - 47214*x^2*y*z - 11073*x^2*y + 1458*x^2*z^5 + 13770*x^2*z^4 + 30420*x^2*z^3 + 30357*x^2*z^2 + 14490*x^2*z + 2715*x^2 - 152*x*y^6 + 1104*x*y^5*z + 1704*x*y^5 - 3318*x*y^4*z^2 - 10794*x*y^4*z - 5556*x*y^4 + 5282*x*y^3*z^3 + 27207*x*y^3*z^2 + 28578*x*y^3*z + 9074*x*y^3 - 4698*x*y^2*z^4 - 34101*x*y^2*z^3 - 54843*x*y^2*z^2 - 35418*x*y^2*z - 8178*x*y^2 + 2214*x*y*z^5 + 21249*x*y*z^4 + 46524*x*y*z^3 + 45843*x*y*z^2 + 21528*x*y*z + 3957*x*y - 432*x*z^6 - 5265*x*z^5 - 14715*x*z^4 - 19667*x*z^3 - 14097*x*z^2 - 5277*x*z - 805*x - 16*y^7 + 136*y^6*z + 212*y^6 - 492*y^5*z^2 - 1620*y^5*z - 828*y^5 + 982*y^4*z^3 + 5133*y^4*z^2 + 5352*y^4*z + 1680*y^4 - 1168*y^3*z^4 - 8630*y^3*z^3 - 13773*y^3*z^2 - 8790*y^3*z - 2000*y^3 + 828*y^2*z^5 + 8118*y^2*z^4 + 17634*y^2*z^3 + 17166*y^2*z^2 + 7938*y^2*z + 1433*y^2 - 324*y*z^6 - 4050*y*z^5 - 11229*y*z^4 - 14824*y*z^3 - 10455*y*z^2 - 3840*y*z - 575*y + 54*z^7 + 837*z^6 + 2844*z^5 + 4774*z^4 + 4568*z^3 + 2562*z^2 + 778*z + 1, -54*x^7 - 297*x^6*y + 351*x^6*z + 567*x^6 - 693*x^5*y^2 + 1647*x^5*y*z + 2655*x^5*y - 972*x^5*z^2 - 3267*x^5*z - 1917*x^5 - 889*x^4*y^3 + 3186*x^4*y^2*z + 5133*x^4*y^2 - 3780*x^4*y*z^2 - 12717*x^4*y*z - 7407*x^4*y + 1485*x^4*z^3 + 7830*x^4*z^2 + 9288*x^4*z + 3291*x^4 - 677*x^3*y^4 + 3251*x^3*y^3*z + 5245*x^3*y^3 - 5814*x^3*y^2*z^2 - 19617*x^3*y^2*z - 11349*x^3*y^2 + 4590*x^3*y*z^3 + 24318*x^3*y*z^2 + 28656*x^3*y*z + 10032*x^3*y - 1350*x^3*z^4 - 9990*x^3*z^3 - 17982*x^3*z^2 - 12801*x^3*z - 3219*x^3 - 306*x^2*y^5 + 1845*x^2*y^4*z + 2988*x^2*y^4 - 4419*x^2*y^3*z^2 - 14991*x^2*y^3*z - 8622*x^2*y^3 + 5256*x^2*y^2*z^3 + 28053*x^2*y^2*z^2 + 32865*x^2*y^2*z + 11370*x^2*y^2 - 3105*x^2*y*z^4 - 23202*x^2*y*z^3 - 41526*x^2*y*z^2 - 29217*x^2*y*z - 7230*x^2*y + 729*x^2*z^5 + 7155*x^2*z^4 + 17388*x^2*z^3 + 18657*x^2*z^2 + 9396*x^2*z + 1830*x^2 - 76*x*y^6 + 552*x*y^5*z + 900*x*y^5 - 1659*x*y^4*z^2 - 5676*x*y^4*z - 3249*x*y^4 + 2641*x*y^3*z^3 + 14247*x*y^3*z^2 + 16611*x*y^3*z + 5682*x*y^3 - 2349*x*y^2*z^4 - 17787*x*y^2*z^3 - 31683*x*y^2*z^2 - 22038*x*y^2*z - 5370*x*y^2 + 1107*x*y*z^5 + 11043*x*y*z^4 + 26712*x*y*z^3 + 28338*x*y*z^2 + 14046*x*y*z + 2682*x*y - 216*x*z^6 - 2727*x*z^5 - 8397*x*z^4 - 12075*x*z^3 - 9135*x*z^2 - 3555*x*z - 557*x - 8*y^7 + 68*y^6*z + 112*y^6 - 246*y^5*z^2 - 852*y^5*z - 486*y^5 + 491*y^4*z^3 + 2688*y^4*z^2 + 3123*y^4*z + 1057*y^4 - 584*y^3*z^4 - 4501*y^3*z^3 - 7989*y^3*z^2 - 5497*y^3*z - 1320*y^3 + 414*y^2*z^5 + 4218*y^2*z^4 + 10167*y^2*z^3 + 10668*y^2*z^2 + 5208*y^2*z + 976*y^2 - 162*y*z^6 - 2097*y*z^5 - 6435*y*z^4 - 9153*y*z^3 - 6816*y*z^2 - 2601*y*z - 400*y + 27*z^7 + 432*z^6 + 1620*z^5 + 2928*z^4 + 2958*z^3 + 1725*z^2 + 538*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -1, -1, 1 ], [ -3, -2, 3 ], [ -2, -1, 1 ] ], "source_shift": [ 1, 1, 2 ], "target_matrix": [ [ 1, 1, 2 ], [ 2, 1, 1 ], [ 1, 1, 1 ] ], "target_shift": [ -69, -102, -72 ], "collision_preimages": [ [ "5/4", "2", "9/4" ], [ "-9/2", "-5/2", "-7" ], [ "-13/2", "13/2", "-2" ] ], "collision_image": [ "-277/4", "-205/2", "-289/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0970162240009813 ], "surface_features": [ 3.9457387725149093, 4.858525167084953, 3.6565294062334983, 0.0, 4.054954073712669, 5051 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C217", "pair": 217, "family": "counterexample", "mapping": "F(x,y,z)=(16*x^7 - 152*x^6*y - 160*x^6*z + 220*x^6 + 612*x^5*y^2 + 1320*x^5*y*z - 1776*x^5*y + 672*x^5*z^2 - 1836*x^5*z + 1008*x^5 - 1354*x^4*y^3 - 4488*x^4*y^2*z + 5919*x^4*y^2 - 4680*x^4*y*z^2 + 12516*x^4*y*z - 6768*x^4*y - 1536*x^4*z^3 + 6252*x^4*z^2 - 6924*x^4*z + 2249*x^4 + 1778*x^3*y^4 + 8050*x^3*y^3*z - 10423*x^3*y^3 + 12900*x^3*y^2*z^2 - 33825*x^3*y^2*z + 18024*x^3*y^2 + 8664*x^3*y*z^3 - 34536*x^3*y*z^2 + 37692*x^3*y*z - 12159*x^3*y + 2064*x^3*z^4 - 11124*x^3*z^3 + 18600*x^3*z^2 - 12123*x^3*z + 2834*x^3 - 1386*x^2*y^5 - 8034*x^2*y^4*z + 10227*x^2*y^4 - 17592*x^2*y^3*z^2 + 45291*x^2*y^3*z - 23790*x^2*y^3 - 18144*x^2*y^2*z^3 + 70938*x^2*y^2*z^2 - 76335*x^2*y^2*z + 24450*x^2*y^2 - 8832*x^2*y*z^4 + 46644*x^2*y*z^3 - 76908*x^2*y*z^2 + 49833*x^2*y*z - 11616*x^2*y - 1632*x^2*z^5 + 10920*x^2*z^4 - 24444*x^2*z^3 + 23922*x^2*z^2 - 11187*x^2*z + 2084*x^2 + 594*x*y^6 + 4230*x*y^5*z - 5301*x*y^5 + 11868*x*y^4*z^2 - 30039*x*y^4*z + 15558*x*y^4 + 16720*x*y^3*z^3 - 64200*x*y^3*z^2 + 68136*x*y^3*z - 21660*x*y^3 + 12480*x*y^2*z^4 - 64680*x*y^2*z^3 + 105240*x*y^2*z^2 - 67788*x*y^2*z + 15744*x*y^2 + 4704*x*y*z^5 - 30864*x*y*z^4 + 68160*x*y*z^3 - 66360*x*y*z^2 + 30996*x*y*z - 5784*x*y + 704*x*z^6 - 5616*x*z^5 + 15744*x*z^4 - 20528*x*z^3 + 14352*x*z^2 - 5314*x*z + 832*x - 108*y^7 - 918*y^6*z + 1134*y^6 - 3168*y^5*z^2 + 7893*y^5*z - 4032*y^5 - 5720*y^4*z^3 + 21594*y^4*z^2 - 22605*y^4*z + 7128*y^4 - 5824*y^3*z^4 + 29656*y^3*z^3 - 47640*y^3*z^2 + 30498*y^3*z - 7050*y^3 - 3360*y^2*z^5 + 21648*y^2*z^4 - 47208*y^2*z^3 + 45732*y^2*z^2 - 21330*y^2*z + 3984*y^2 - 1024*y*z^6 + 8016*y*z^5 - 22176*y*z^4 + 28776*y*z^3 - 20112*y*z^2 + 7470*y*z - 1174*y - 128*z^7 + 1184*z^6 - 3984*z^5 + 6480*z^4 - 6008*z^3 + 3308*z^2 - 1030*z + 1, 8*x^7 - 76*x^6*y - 80*x^6*z + 116*x^6 + 306*x^5*y^2 + 660*x^5*y*z - 936*x^5*y + 336*x^5*z^2 - 972*x^5*z + 570*x^5 - 677*x^4*y^3 - 2244*x^4*y^2*z + 3117*x^4*y^2 - 2340*x^4*y*z^2 + 6624*x^4*y*z - 3819*x^4*y - 768*x^4*z^3 + 3324*x^4*z^2 - 3936*x^4*z + 1318*x^4 + 889*x^3*y^4 + 4025*x^3*y^3*z - 5483*x^3*y^3 + 6450*x^3*y^2*z^2 - 17889*x^3*y^2*z + 10146*x^3*y^2 + 4332*x^3*y*z^3 - 18360*x^3*y*z^2 + 21387*x^3*y*z - 7110*x^3*y + 1032*x^3*z^4 - 5940*x^3*z^3 + 10626*x^3*z^2 - 7140*x^3*z + 1699*x^3 - 693*x^2*y^5 - 4017*x^2*y^4*z + 5373*x^2*y^4 - 8796*x^2*y^3*z^2 + 23928*x^2*y^3*z - 13356*x^2*y^3 - 9072*x^2*y^2*z^3 + 37692*x^2*y^2*z^2 - 43221*x^2*y^2*z + 14262*x^2*y^2 - 4416*x^2*y*z^4 + 24912*x^2*y*z^3 - 43872*x^2*y*z^2 + 29295*x^2*y*z - 6948*x^2*y - 816*x^2*z^5 + 5856*x^2*z^4 - 14028*x^2*z^3 + 14154*x^2*z^2 - 6741*x^2*z + 1270*x^2 + 297*x*y^6 + 2115*x*y^5*z - 2781*x*y^5 + 5934*x*y^4*z^2 - 15849*x*y^4*z + 8709*x*y^4 + 8360*x*y^3*z^3 - 34080*x*y^3*z^2 + 38484*x*y^3*z - 12600*x*y^3 + 6240*x*y^2*z^4 - 34536*x*y^2*z^3 + 59928*x*y^2*z^2 - 39765*x*y^2*z + 9393*x*y^2 + 2352*x*y*z^5 - 16560*x*y*z^4 + 39072*x*y*z^3 - 39204*x*y*z^2 + 18642*x*y*z - 3516*x*y + 352*x*z^6 - 3024*x*z^5 + 9072*x*z^4 - 12196*x*z^3 + 8688*x*z^2 - 3257*x*z + 513*x - 54*y^7 - 459*y^6*z + 594*y^6 - 1584*y^5*z^2 + 4158*y^5*z - 2250*y^5 - 2860*y^4*z^3 + 11448*y^4*z^2 - 12732*y^4*z + 4134*y^4 - 2912*y^3*z^4 + 15824*y^3*z^3 - 27072*y^3*z^2 + 17847*y^3*z - 4194*y^3 - 1680*y^2*z^5 + 11616*y^2*z^4 - 27024*y^2*z^3 + 26970*y^2*z^2 - 12801*y^2*z + 2415*y^2 - 512*y*z^6 + 4320*y*z^5 - 12768*y*z^4 + 17076*y*z^3 - 12156*y*z^2 + 4569*y*z - 722*y - 64*z^7 + 640*z^6 - 2304*z^5 + 3864*z^4 - 3652*z^3 + 2038*z^2 - 639*z + 1, 8*x^7 - 76*x^6*y - 80*x^6*z + 104*x^6 + 306*x^5*y^2 + 660*x^5*y*z - 840*x^5*y + 336*x^5*z^2 - 864*x^5*z + 438*x^5 - 677*x^4*y^3 - 2244*x^4*y^2*z + 2802*x^4*y^2 - 2340*x^4*y*z^2 + 5892*x^4*y*z - 2949*x^4*y - 768*x^4*z^3 + 2928*x^4*z^2 - 2988*x^4*z + 930*x^4 + 889*x^3*y^4 + 4025*x^3*y^3*z - 4940*x^3*y^3 + 6450*x^3*y^2*z^2 - 15936*x^3*y^2*z + 7878*x^3*y^2 + 4332*x^3*y*z^3 - 16176*x^3*y*z^2 + 16305*x^3*y*z - 5044*x^3*y + 1032*x^3*z^4 - 5184*x^3*z^3 + 7974*x^3*z^2 - 4976*x^3*z + 1125*x^3 - 693*x^2*y^5 - 4017*x^2*y^4*z + 4854*x^2*y^4 - 8796*x^2*y^3*z^2 + 21363*x^2*y^3*z - 10434*x^2*y^3 - 9072*x^2*y^2*z^3 + 33246*x^2*y^2*z^2 - 33114*x^2*y^2*z + 10179*x^2*y^2 - 4416*x^2*y*z^4 + 21732*x^2*y*z^3 - 33036*x^2*y*z^2 + 20511*x^2*y*z - 4632*x^2*y - 816*x^2*z^5 + 5064*x^2*z^4 - 10416*x^2*z^3 + 9750*x^2*z^2 - 4395*x^2*z + 793*x^2 + 297*x*y^6 + 2115*x*y^5*z - 2520*x*y^5 + 5934*x*y^4*z^2 - 14190*x*y^4*z + 6849*x*y^4 + 8360*x*y^3*z^3 - 30120*x*y^3*z^2 + 29652*x*y^3*z - 9053*x*y^3 + 6240*x*y^2*z^4 - 30144*x*y^2*z^3 + 45312*x*y^2*z^2 - 27990*x*y^2*z + 6309*x*y^2 + 2352*x*y*z^5 - 14304*x*y*z^4 + 29088*x*y*z^3 - 27108*x*y*z^2 + 12228*x*y*z - 2217*x*y + 352*x*z^6 - 2592*x*z^5 + 6672*x*z^4 - 8312*x*z^3 + 5580*x*z^2 - 1988*x*z + 305*x - 54*y^7 - 459*y^6*z + 540*y^6 - 1584*y^5*z^2 + 3735*y^5*z - 1782*y^5 - 2860*y^4*z^3 + 10146*y^4*z^2 - 9873*y^4*z + 2992*y^4 - 2912*y^3*z^4 + 13832*y^3*z^3 - 20568*y^3*z^2 + 12638*y^3*z - 2840*y^3 - 1680*y^2*z^5 + 10032*y^2*z^4 - 20184*y^2*z^3 + 18732*y^2*z^2 - 8454*y^2*z + 1539*y^2 - 512*y*z^6 + 3696*y*z^5 - 9408*y*z^4 + 11672*y*z^3 - 7848*y*z^2 + 2814*y*z - 434*y - 64*z^7 + 544*z^6 - 1680*z^5 + 2608*z^4 - 2312*z^3 + 1216*z^2 - 370*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ 1, -1, -2 ], [ 2, -3, -2 ], [ 1, -2, -1 ] ], "source_shift": [ 1, 1, 1 ], "target_matrix": [ [ 2, 3, 2 ], [ 1, 2, 1 ], [ 1, 1, 2 ] ], "target_shift": [ -141, -87, -51 ], "collision_preimages": [ [ "3", "3/2", "5/4" ], [ "-59/2", "-27/2", "-8" ], [ "-37/2", "-21/2", "-3" ] ], "collision_image": [ "-283/2", "-349/4", "-205/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.022490473002108 ], "surface_features": [ 4.06114952341433, 5.022180839413665, 3.7598188773748262, 0.0, 4.188065670044849, 5086 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B055", "pair": 55, "family": "bijection", "mapping": "F(x,y,z)=(-x^7 - 14*x^6*y + 7*x^6*z + 19*x^6 - 84*x^5*y^2 + 84*x^5*y*z + 226*x^5*y - 21*x^5*z^2 - 116*x^5*z - 153*x^5 - 280*x^4*y^3 + 420*x^4*y^2*z + 1120*x^4*y^2 - 210*x^4*y*z^2 - 1150*x^4*y*z - 1503*x^4*y + 35*x^4*z^3 + 295*x^4*z^2 + 791*x^4*z + 678*x^4 - 560*x^3*y^4 + 1120*x^3*y^3*z + 2960*x^3*y^3 - 840*x^3*y^2*z^2 - 4560*x^3*y^2*z - 5905*x^3*y^2 + 280*x^3*y*z^3 + 2340*x^3*y*z^2 + 6218*x^3*y*z + 5278*x^3*y - 35*x^3*z^4 - 400*x^3*z^3 - 1635*x^3*z^2 - 2846*x^3*z - 1784*x^3 - 672*x^2*y^5 + 1680*x^2*y^4*z + 4400*x^2*y^4 - 1680*x^2*y^3*z^2 - 9040*x^2*y^3*z - 11598*x^2*y^3 + 840*x^2*y^2*z^3 + 6960*x^2*y^2*z^2 + 18327*x^2*y^2*z + 15405*x^2*y^2 - 210*x^2*y*z^4 - 2380*x^2*y*z^3 - 9642*x^2*y*z^2 - 16622*x^2*y*z - 10308*x^2*y + 21*x^2*z^5 + 305*x^2*z^4 + 1689*x^2*z^3 + 4477*x^2*z^2 + 5696*x^2*z + 2781*x^2 - 448*x*y^6 + 1344*x*y^5*z + 3488*x*y^5 - 1680*x*y^4*z^2 - 8960*x*y^4*z - 11388*x*y^4 + 1120*x*y^3*z^3 + 9200*x*y^3*z^2 + 24004*x*y^3*z + 19980*x*y^3 - 420*x*y^2*z^4 - 4720*x*y^2*z^3 - 18951*x*y^2*z^2 - 32354*x*y^2*z - 19853*x*y^2 + 84*x*y*z^5 + 1210*x*y*z^4 + 6642*x*y*z^3 + 17438*x*y*z^2 + 21946*x*y*z + 10588*x*y - 7*x*z^6 - 124*x*z^5 - 872*x*z^4 - 3128*x*z^3 - 6057*x*z^2 - 6002*x*z - 2371*x - 128*y^7 + 448*y^6*z + 1152*y^6 - 672*y^5*z^2 - 3552*y^5*z - 4472*y^5 + 560*y^4*z^3 + 4560*y^4*z^2 + 11788*y^4*z + 9716*y^4 - 280*y^3*z^4 - 3120*y^3*z^3 - 12414*y^3*z^2 - 20988*y^3*z - 12745*y^3 + 84*y^2*z^5 + 1200*y^2*z^4 + 6529*y^2*z^3 + 16977*y^2*z^2 + 21139*y^2*z + 10082*y^2 - 14*y*z^6 - 246*y*z^5 - 1715*y*z^4 - 6094*y*z^3 - 11671*y*z^2 - 11424*y*z - 4457*y + z^7 + 21*z^6 + 180*z^5 + 819*z^4 + 2145*z^3 + 3235*z^2 + 2596*z + 1, -2*x^7 - 28*x^6*y + 14*x^6*z + 38*x^6 - 168*x^5*y^2 + 168*x^5*y*z + 452*x^5*y - 42*x^5*z^2 - 232*x^5*z - 306*x^5 - 560*x^4*y^3 + 840*x^4*y^2*z + 2240*x^4*y^2 - 420*x^4*y*z^2 - 2300*x^4*y*z - 3006*x^4*y + 70*x^4*z^3 + 590*x^4*z^2 + 1582*x^4*z + 1356*x^4 - 1120*x^3*y^4 + 2240*x^3*y^3*z + 5920*x^3*y^3 - 1680*x^3*y^2*z^2 - 9120*x^3*y^2*z - 11810*x^3*y^2 + 560*x^3*y*z^3 + 4680*x^3*y*z^2 + 12436*x^3*y*z + 10556*x^3*y - 70*x^3*z^4 - 800*x^3*z^3 - 3270*x^3*z^2 - 5692*x^3*z - 3574*x^3 - 1344*x^2*y^5 + 3360*x^2*y^4*z + 8800*x^2*y^4 - 3360*x^2*y^3*z^2 - 18080*x^2*y^3*z - 23196*x^2*y^3 + 1680*x^2*y^2*z^3 + 13920*x^2*y^2*z^2 + 36654*x^2*y^2*z + 30810*x^2*y^2 - 420*x^2*y*z^4 - 4760*x^2*y*z^3 - 19284*x^2*y*z^2 - 33244*x^2*y*z - 20652*x^2*y + 42*x^2*z^5 + 610*x^2*z^4 + 3378*x^2*z^3 + 8954*x^2*z^2 + 11410*x^2*z + 5603*x^2 - 896*x*y^6 + 2688*x*y^5*z + 6976*x*y^5 - 3360*x*y^4*z^2 - 17920*x*y^4*z - 22776*x*y^4 + 2240*x*y^3*z^3 + 18400*x*y^3*z^2 + 48008*x*y^3*z + 39960*x*y^3 - 840*x*y^2*z^4 - 9440*x*y^2*z^3 - 37902*x*y^2*z^2 - 64708*x*y^2*z - 39778*x*y^2 + 168*x*y*z^5 + 2420*x*y*z^4 + 13284*x*y*z^3 + 34876*x*y*z^2 + 43964*x*y*z + 21340*x*y - 14*x*z^6 - 248*x*z^5 - 1744*x*z^4 - 6256*x*z^3 - 12132*x*z^2 - 12086*x*z - 4838*x - 256*y^7 + 896*y^6*z + 2304*y^6 - 1344*y^5*z^2 - 7104*y^5*z - 8944*y^5 + 1120*y^4*z^3 + 9120*y^4*z^2 + 23576*y^4*z + 19432*y^4 - 560*y^3*z^4 - 6240*y^3*z^3 - 24828*y^3*z^2 - 41976*y^3*z - 25538*y^3 + 168*y^2*z^5 + 2400*y^2*z^4 + 13058*y^2*z^3 + 33954*y^2*z^2 + 42350*y^2*z + 20328*y^2 - 28*y*z^6 - 492*y*z^5 - 3430*y*z^4 - 12188*y*z^3 - 23378*y*z^2 - 23012*y*z - 9101*y + 2*z^7 + 42*z^6 + 360*z^5 + 1638*z^4 + 4296*z^3 + 6511*z^2 + 5287*z + 1, -2*x^7 - 28*x^6*y + 14*x^6*z + 38*x^6 - 168*x^5*y^2 + 168*x^5*y*z + 452*x^5*y - 42*x^5*z^2 - 232*x^5*z - 306*x^5 - 560*x^4*y^3 + 840*x^4*y^2*z + 2240*x^4*y^2 - 420*x^4*y*z^2 - 2300*x^4*y*z - 3006*x^4*y + 70*x^4*z^3 + 590*x^4*z^2 + 1582*x^4*z + 1356*x^4 - 1120*x^3*y^4 + 2240*x^3*y^3*z + 5920*x^3*y^3 - 1680*x^3*y^2*z^2 - 9120*x^3*y^2*z - 11810*x^3*y^2 + 560*x^3*y*z^3 + 4680*x^3*y*z^2 + 12436*x^3*y*z + 10556*x^3*y - 70*x^3*z^4 - 800*x^3*z^3 - 3270*x^3*z^2 - 5692*x^3*z - 3570*x^3 - 1344*x^2*y^5 + 3360*x^2*y^4*z + 8800*x^2*y^4 - 3360*x^2*y^3*z^2 - 18080*x^2*y^3*z - 23196*x^2*y^3 + 1680*x^2*y^2*z^3 + 13920*x^2*y^2*z^2 + 36654*x^2*y^2*z + 30810*x^2*y^2 - 420*x^2*y*z^4 - 4760*x^2*y*z^3 - 19284*x^2*y*z^2 - 33244*x^2*y*z - 20628*x^2*y + 42*x^2*z^5 + 610*x^2*z^4 + 3378*x^2*z^3 + 8954*x^2*z^2 + 11398*x^2*z + 5576*x^2 - 896*x*y^6 + 2688*x*y^5*z + 6976*x*y^5 - 3360*x*y^4*z^2 - 17920*x*y^4*z - 22776*x*y^4 + 2240*x*y^3*z^3 + 18400*x*y^3*z^2 + 48008*x*y^3*z + 39960*x*y^3 - 840*x*y^2*z^4 - 9440*x*y^2*z^3 - 37902*x*y^2*z^2 - 64708*x*y^2*z - 39730*x*y^2 + 168*x*y*z^5 + 2420*x*y*z^4 + 13284*x*y*z^3 + 34876*x*y*z^2 + 43916*x*y*z + 21232*x*y - 14*x*z^6 - 248*x*z^5 - 1744*x*z^4 - 6256*x*z^3 - 12120*x*z^2 - 12032*x*z - 4776*x - 256*y^7 + 896*y^6*z + 2304*y^6 - 1344*y^5*z^2 - 7104*y^5*z - 8944*y^5 + 1120*y^4*z^3 + 9120*y^4*z^2 + 23576*y^4*z + 19432*y^4 - 560*y^3*z^4 - 6240*y^3*z^3 - 24828*y^3*z^2 - 41976*y^3*z - 25506*y^3 + 168*y^2*z^5 + 2400*y^2*z^4 + 13058*y^2*z^3 + 33954*y^2*z^2 + 42302*y^2*z + 20220*y^2 - 28*y*z^6 - 492*y*z^5 - 3430*y*z^4 - 12188*y*z^3 - 23354*y*z^2 - 22904*y*z - 8980*y + 2*z^7 + 42*z^6 + 360*z^5 + 1638*z^4 + 4292*z^3 + 6484*z^2 + 5226*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -1, -2, 1 ], [ -1, -1, 2 ], [ -1, -1, 1 ] ], "source_shift": [ 2, 2, 2 ], "target_matrix": [ [ 1, 1, 2 ], [ 2, 1, 1 ], [ 2, 2, 3 ] ], "target_shift": [ -849, -1775, -1727 ], "inverse_recipe": { "p": 2, "q": 3, "f": "x**2 + x", "g": "x**3 + x**2", "h": "y**2*z", "polynomials": { "f": [ [ [ 2, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "g": [ [ [ 3, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "h": [ [ [ 0, 2, 1 ], 1 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0390606930013746 ], "surface_features": [ 3.9140237400811646, 4.810957976821329, 3.4861469968065726, 0.0, 4.061660741936897, 4972 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B243", "pair": 243, "family": "bijection", "mapping": "F(x,y,z)=(6*x^7 - 42*x^6*y + 84*x^6*z - 48*x^6 + 126*x^5*y^2 - 504*x^5*y*z + 288*x^5*y + 504*x^5*z^2 - 576*x^5*z + 174*x^5 - 210*x^4*y^3 + 1260*x^4*y^2*z - 720*x^4*y^2 - 2520*x^4*y*z^2 + 2880*x^4*y*z - 870*x^4*y + 1680*x^4*z^3 - 2880*x^4*z^2 + 1734*x^4*z - 391*x^4 + 210*x^3*y^4 - 1680*x^3*y^3*z + 960*x^3*y^3 + 5040*x^3*y^2*z^2 - 5760*x^3*y^2*z + 1740*x^3*y^2 - 6720*x^3*y*z^3 + 11520*x^3*y*z^2 - 6936*x^3*y*z + 1558*x^3*y + 3360*x^3*z^4 - 7680*x^3*z^3 + 6912*x^3*z^2 - 3074*x^3*z + 584*x^3 - 126*x^2*y^5 + 1260*x^2*y^4*z - 720*x^2*y^4 - 5040*x^2*y^3*z^2 + 5760*x^2*y^3*z - 1740*x^2*y^3 + 10080*x^2*y^2*z^3 - 17280*x^2*y^2*z^2 + 10404*x^2*y^2*z - 2328*x^2*y^2 - 10080*x^2*y*z^4 + 23040*x^2*y*z^3 - 20736*x^2*y*z^2 + 9186*x^2*y*z - 1734*x^2*y + 4032*x^2*z^5 - 11520*x^2*z^4 + 13776*x^2*z^3 - 9060*x^2*z^2 + 3372*x^2*z - 577*x^2 + 42*x*y^6 - 504*x*y^5*z + 288*x*y^5 + 2520*x*y^4*z^2 - 2880*x*y^4*z + 870*x*y^4 - 6720*x*y^3*z^3 + 11520*x*y^3*z^2 - 6936*x*y^3*z + 1546*x*y^3 + 10080*x*y^2*z^4 - 23040*x*y^2*z^3 + 20736*x*y^2*z^2 - 9150*x*y^2*z + 1716*x*y^2 - 8064*x*y*z^5 + 23040*x*y*z^4 - 27552*x*y*z^3 + 18048*x*y*z^2 - 6672*x*y*z + 1130*x*y + 2688*x*z^6 - 9216*x*z^5 + 13728*x*z^4 - 11864*x*z^3 + 6480*x*z^2 - 2140*x*z + 362*x - 6*y^7 + 84*y^6*z - 48*y^6 - 504*y^5*z^2 + 576*y^5*z - 174*y^5 + 1680*y^4*z^3 - 2880*y^4*z^2 + 1734*y^4*z - 385*y^4 - 3360*y^3*z^4 + 7680*y^3*z^3 - 6912*y^3*z^2 + 3038*y^3*z - 566*y^3 + 4032*y^2*z^5 - 11520*y^2*z^4 + 13776*y^2*z^3 - 8988*y^2*z^2 + 3300*y^2*z - 553*y^2 - 2688*y*z^6 + 9216*y*z^5 - 13728*y*z^4 + 11816*y*z^3 - 6408*y*z^2 + 2098*y*z - 343*y + 768*z^7 - 3072*z^6 + 5472*z^5 - 5824*z^4 + 4144*z^3 - 1996*z^2 + 615*z - 1, -3*x^7 + 21*x^6*y - 42*x^6*z + 24*x^6 - 63*x^5*y^2 + 252*x^5*y*z - 144*x^5*y - 252*x^5*z^2 + 288*x^5*z - 87*x^5 + 105*x^4*y^3 - 630*x^4*y^2*z + 360*x^4*y^2 + 1260*x^4*y*z^2 - 1440*x^4*y*z + 435*x^4*y - 840*x^4*z^3 + 1440*x^4*z^2 - 867*x^4*z + 197*x^4 - 105*x^3*y^4 + 840*x^3*y^3*z - 480*x^3*y^3 - 2520*x^3*y^2*z^2 + 2880*x^3*y^2*z - 870*x^3*y^2 + 3360*x^3*y*z^3 - 5760*x^3*y*z^2 + 3468*x^3*y*z - 785*x^3*y - 1680*x^3*z^4 + 3840*x^3*z^3 - 3456*x^3*z^2 + 1549*x^3*z - 297*x^3 + 63*x^2*y^5 - 630*x^2*y^4*z + 360*x^2*y^4 + 2520*x^2*y^3*z^2 - 2880*x^2*y^3*z + 870*x^2*y^3 - 5040*x^2*y^2*z^3 + 8640*x^2*y^2*z^2 - 5202*x^2*y^2*z + 1173*x^2*y^2 + 5040*x^2*y*z^4 - 11520*x^2*y*z^3 + 10368*x^2*y*z^2 - 4629*x^2*y*z + 882*x^2*y - 2016*x^2*z^5 + 5760*x^2*z^4 - 6888*x^2*z^3 + 4566*x^2*z^2 - 1716*x^2*z + 296*x^2 - 21*x*y^6 + 252*x*y^5*z - 144*x*y^5 - 1260*x*y^4*z^2 + 1440*x*y^4*z - 435*x*y^4 + 3360*x*y^3*z^3 - 5760*x*y^3*z^2 + 3468*x*y^3*z - 779*x*y^3 - 5040*x*y^2*z^4 + 11520*x*y^2*z^3 - 10368*x*y^2*z^2 + 4611*x*y^2*z - 873*x*y^2 + 4032*x*y*z^5 - 11520*x*y*z^4 + 13776*x*y*z^3 - 9096*x*y*z^2 + 3396*x*y*z - 580*x*y - 1344*x*z^6 + 4608*x*z^5 - 6864*x*z^4 + 5980*x*z^3 - 3300*x*z^2 + 1100*x*z - 189*x + 3*y^7 - 42*y^6*z + 24*y^6 + 252*y^5*z^2 - 288*y^5*z + 87*y^5 - 840*y^4*z^3 + 1440*y^4*z^2 - 867*y^4*z + 194*y^4 + 1680*y^3*z^4 - 3840*y^3*z^3 + 3456*y^3*z^2 - 1531*y^3*z + 288*y^3 - 2016*y^2*z^5 + 5760*y^2*z^4 - 6888*y^2*z^3 + 4530*y^2*z^2 - 1680*y^2*z + 284*y^2 + 1344*y*z^6 - 4608*y*z^5 + 6864*y*z^4 - 5956*y*z^3 + 3264*y*z^2 - 1079*y*z + 178*y - 384*z^7 + 1536*z^6 - 2736*z^5 + 2936*z^4 - 2112*z^3 + 1028*z^2 - 320*z + 1, 3*x^7 - 21*x^6*y + 42*x^6*z - 24*x^6 + 63*x^5*y^2 - 252*x^5*y*z + 144*x^5*y + 252*x^5*z^2 - 288*x^5*z + 87*x^5 - 105*x^4*y^3 + 630*x^4*y^2*z - 360*x^4*y^2 - 1260*x^4*y*z^2 + 1440*x^4*y*z - 435*x^4*y + 840*x^4*z^3 - 1440*x^4*z^2 + 867*x^4*z - 196*x^4 + 105*x^3*y^4 - 840*x^3*y^3*z + 480*x^3*y^3 + 2520*x^3*y^2*z^2 - 2880*x^3*y^2*z + 870*x^3*y^2 - 3360*x^3*y*z^3 + 5760*x^3*y*z^2 - 3468*x^3*y*z + 781*x^3*y + 1680*x^3*z^4 - 3840*x^3*z^3 + 3456*x^3*z^2 - 1541*x^3*z + 293*x^3 - 63*x^2*y^5 + 630*x^2*y^4*z - 360*x^2*y^4 - 2520*x^2*y^3*z^2 + 2880*x^2*y^3*z - 870*x^2*y^3 + 5040*x^2*y^2*z^3 - 8640*x^2*y^2*z^2 + 5202*x^2*y^2*z - 1167*x^2*y^2 - 5040*x^2*y*z^4 + 11520*x^2*y*z^3 - 10368*x^2*y*z^2 + 4605*x^2*y*z - 870*x^2*y + 2016*x^2*z^5 - 5760*x^2*z^4 + 6888*x^2*z^3 - 4542*x^2*z^2 + 1692*x^2*z - 289*x^2 + 21*x*y^6 - 252*x*y^5*z + 144*x*y^5 + 1260*x*y^4*z^2 - 1440*x*y^4*z + 435*x*y^4 - 3360*x*y^3*z^3 + 5760*x*y^3*z^2 - 3468*x*y^3*z + 775*x*y^3 + 5040*x*y^2*z^4 - 11520*x*y^2*z^3 + 10368*x*y^2*z^2 - 4587*x*y^2*z + 861*x*y^2 - 4032*x*y*z^5 + 11520*x*y*z^4 - 13776*x*y*z^3 + 9048*x*y*z^2 - 3348*x*y*z + 566*x*y + 1344*x*z^6 - 4608*x*z^5 + 6864*x*z^4 - 5948*x*z^3 + 3252*x*z^2 - 1072*x*z + 181*x - 3*y^7 + 42*y^6*z - 24*y^6 - 252*y^5*z^2 + 288*y^5*z - 87*y^5 + 840*y^4*z^3 - 1440*y^4*z^2 + 867*y^4*z - 193*y^4 - 1680*y^3*z^4 + 3840*y^3*z^3 - 3456*y^3*z^2 + 1523*y^3*z - 284*y^3 + 2016*y^2*z^5 - 5760*y^2*z^4 + 6888*y^2*z^3 - 4506*y^2*z^2 + 1656*y^2*z - 277*y^2 - 1344*y*z^6 + 4608*y*z^5 - 6864*y*z^4 + 5924*y*z^3 - 3216*y*z^2 + 1051*y*z - 171*y + 384*z^7 - 1536*z^6 + 2736*z^5 - 2920*z^4 + 2080*z^3 - 1000*z^2 + 307*z - 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -1, 1, -2 ], [ -2, 1, -1 ], [ -1, 1, -1 ] ], "source_shift": [ 1, 2, 1 ], "target_matrix": [ [ -2, -1, -1 ], [ 1, 2, 1 ], [ -1, -1, -1 ] ], "target_shift": [ 109, -58, 54 ], "inverse_recipe": { "p": 4, "q": 3, "f": "x**4 + x**2", "g": "x**3", "h": "y*z + y + z**2 + z", "polynomials": { "f": [ [ [ 4, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "g": [ [ [ 3, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 1, 0 ], 1 ], [ [ 0, 0, 2 ], 1 ], [ [ 0, 0, 1 ], 1 ] ] }, "h_multiplier": 3 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0726225890030037 ], "surface_features": [ 3.5322761654558583, 4.44015312977356, 3.225309281725863, 0.0, 3.639335926435654, 4895 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B096", "pair": 96, "family": "bijection", "mapping": "F(x,y,z)=(x^7 - 7*x^6*y + 14*x^6*z - 15*x^6 + 21*x^5*y^2 - 84*x^5*y*z + 90*x^5*y + 84*x^5*z^2 - 180*x^5*z + 99*x^5 - 35*x^4*y^3 + 210*x^4*y^2*z - 225*x^4*y^2 - 420*x^4*y*z^2 + 900*x^4*y*z - 496*x^4*y + 280*x^4*z^3 - 900*x^4*z^2 + 989*x^4*z - 374*x^4 + 35*x^3*y^4 - 280*x^3*y^3*z + 300*x^3*y^3 + 840*x^3*y^2*z^2 - 1800*x^3*y^2*z + 994*x^3*y^2 - 1120*x^3*y*z^3 + 3600*x^3*y*z^2 - 3964*x^3*y*z + 1504*x^3*y + 560*x^3*z^4 - 2400*x^3*z^3 + 3952*x^3*z^2 - 2983*x^3*z + 872*x^3 - 21*x^2*y^5 + 210*x^2*y^4*z - 225*x^2*y^4 - 840*x^2*y^3*z^2 + 1800*x^2*y^3*z - 996*x^2*y^3 + 1680*x^2*y^2*z^3 - 5400*x^2*y^2*z^2 + 5958*x^2*y^2*z - 2268*x^2*y^2 - 1680*x^2*y*z^4 + 7200*x^2*y*z^3 - 11880*x^2*y*z^2 + 8997*x^2*y*z - 2641*x^2*y + 672*x^2*z^5 - 3600*x^2*z^4 + 7896*x^2*z^3 - 8922*x^2*z^2 + 5200*x^2*z - 1249*x^2 + 7*x*y^6 - 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280*x^4*z^3 + 900*x^4*z^2 - 989*x^4*z + 374*x^4 - 35*x^3*y^4 + 280*x^3*y^3*z - 300*x^3*y^3 - 840*x^3*y^2*z^2 + 1800*x^3*y^2*z - 994*x^3*y^2 + 1120*x^3*y*z^3 - 3600*x^3*y*z^2 + 3964*x^3*y*z - 1504*x^3*y - 560*x^3*z^4 + 2400*x^3*z^3 - 3952*x^3*z^2 + 2983*x^3*z - 870*x^3 + 21*x^2*y^5 - 210*x^2*y^4*z + 225*x^2*y^4 + 840*x^2*y^3*z^2 - 1800*x^2*y^3*z + 996*x^2*y^3 - 1680*x^2*y^2*z^3 + 5400*x^2*y^2*z^2 - 5958*x^2*y^2*z + 2268*x^2*y^2 + 1680*x^2*y*z^4 - 7200*x^2*y*z^3 + 11880*x^2*y*z^2 - 8997*x^2*y*z + 2635*x^2*y - 672*x^2*z^5 + 3600*x^2*z^4 - 7896*x^2*z^3 + 8922*x^2*z^2 - 5188*x^2*z + 1235*x^2 - 7*x*y^6 + 84*x*y^5*z - 90*x*y^5 - 420*x*y^4*z^2 + 900*x*y^4*z - 499*x*y^4 + 1120*x*y^3*z^3 - 3600*x*y^3*z^2 + 3980*x*y^3*z - 1520*x*y^3 - 1680*x*y^2*z^4 + 7200*x*y^2*z^3 - 11904*x*y^2*z^2 + 9045*x*y^2*z - 2660*x*y^2 + 1344*x*y*z^5 - 7200*x*y*z^4 + 15824*x*y*z^3 - 17940*x*y*z^2 + 10476*x*y*z - 2507*x*y - 448*x*z^6 + 2880*x*z^5 - 7888*x*z^4 + 11860*x*z^3 - 10312*x*z^2 + 4885*x*z - 973*x + y^7 - 14*y^6*z 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10800*x^2*y^2*z^2 + 11916*x^2*y^2*z - 4530*x^2*y^2 - 3360*x^2*y*z^4 + 14400*x^2*y*z^3 - 23760*x^2*y*z^2 + 17970*x^2*y*z - 5264*x^2*y + 1344*x^2*z^5 - 7200*x^2*z^4 + 15792*x^2*z^3 - 17820*x^2*z^2 + 10364*x^2*z - 2487*x^2 + 14*x*y^6 - 168*x*y^5*z + 180*x*y^5 + 840*x*y^4*z^2 - 1800*x*y^4*z + 998*x*y^4 - 2240*x*y^3*z^3 + 7200*x*y^3*z^2 - 7960*x*y^3*z + 3036*x*y^3 + 3360*x*y^2*z^4 - 14400*x*y^2*z^3 + 23808*x*y^2*z^2 - 18066*x*y^2*z + 5314*x*y^2 - 2688*x*y*z^5 + 14400*x*y*z^4 - 31648*x*y*z^3 + 35832*x*y*z^2 - 20928*x*y*z + 5048*x*y + 896*x*z^6 - 5760*x*z^5 + 15776*x*z^4 - 23688*x*z^3 + 20600*x*z^2 - 9838*x*z + 2017*x - 2*y^7 + 28*y^6*z - 30*y^6 - 168*y^5*z^2 + 360*y^5*z - 200*y^5 + 560*y^4*z^3 - 1800*y^4*z^2 + 1994*y^4*z - 763*y^4 - 1120*y^3*z^4 + 4800*y^3*z^3 - 7952*y^3*z^2 + 6054*y^3*z - 1788*y^3 + 1344*y^2*z^5 - 7200*y^2*z^4 + 15856*y^2*z^3 - 18012*y^2*z^2 + 10564*y^2*z - 2561*y^2 - 896*y*z^6 + 5760*y*z^5 - 15808*y*z^4 + 23816*y*z^3 - 20800*y*z^2 + 9984*y*z - 2057*y + 256*z^7 - 1920*z^6 + 6304*z^5 - 11808*z^4 + 13648*z^3 - 9730*z^2 + 3959*z - 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -1, 1, -2 ], [ -1, 1, -1 ], [ -1, 2, -1 ] ], "source_shift": [ 2, 1, 1 ], "target_matrix": [ [ -1, -1, -1 ], [ 1, 1, 2 ], [ -2, -1, -1 ] ], "target_shift": [ 349, -319, 708 ], "inverse_recipe": { "p": 4, "q": 3, "f": "x**4 + x**2", "g": "x**3 + x**2 + x", "h": "y*z + 2*y", "polynomials": { "f": [ [ [ 4, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "g": [ [ [ 3, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 1, 0 ], 2 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0524147739997716 ], 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20307*x*y^2 + 87*x*y*z^5 - 1257*x*y*z^4 + 7110*x*y*z^3 - 19737*x*y*z^2 + 27009*x*y*z - 14647*x*y - 10*x*z^6 + 171*x*z^5 - 1194*x*z^4 + 4369*x*z^3 - 8868*x*z^2 + 9509*x*z - 4236*x + 8*y^7 - 44*y^6*z + 148*y^6 + 102*y^5*z^2 - 660*y^5*z + 996*y^5 - 129*y^4*z^3 + 1209*y^4*z^2 - 3588*y^4*z + 3438*y^4 + 96*y^3*z^4 - 1164*y^3*z^3 + 5097*y^3*z^2 - 9666*y^3*z + 6762*y^3 - 42*y^2*z^5 + 621*y^2*z^4 - 3570*y^2*z^3 + 10044*y^2*z^2 - 13929*y^2*z + 7669*y^2 + 10*y*z^6 - 174*y*z^5 + 1233*y*z^4 - 4575*y*z^3 + 9426*y*z^2 - 10282*y*z + 4667*y - z^7 + 20*z^6 - 168*z^5 + 771*z^4 - 2097*z^3 + 3397*z^2 - 3054*z + 1, -16*x^7 + 136*x^6*y - 88*x^6*z + 224*x^6 - 468*x^5*y^2 + 624*x^5*y*z - 1620*x^5*y - 204*x^5*z^2 + 1044*x^5*z - 1332*x^5 + 838*x^4*y^3 - 1734*x^4*y^2*z + 4632*x^4*y^2 + 1170*x^4*y*z^2 - 6120*x^4*y*z + 7956*x^4*y - 258*x^4*z^3 + 1992*x^4*z^2 - 5112*x^4*z + 4364*x^4 - 838*x^3*y^4 + 2398*x^3*y^3*z - 6665*x^3*y^3 - 2514*x^3*y^2*z^2 + 13563*x^3*y^2*z - 18045*x^3*y^2 + 1146*x^3*y*z^3 - 9063*x^3*y*z^2 + 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16350*y^3*z + 11010*y^3 - 84*y^2*z^5 + 1164*y^2*z^4 - 6291*y^2*z^3 + 16830*y^2*z^2 - 22467*y^2*z + 12041*y^2 + 20*y*z^6 - 324*y*z^5 + 2154*y*z^4 - 7596*y*z^3 + 15072*y*z^2 - 16022*y*z + 7154*y - 2*z^7 + 37*z^6 - 291*z^5 + 1269*z^4 - 3327*z^3 + 5261*z^2 - 4661*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -2, 1, -1 ], [ -1, 2, -1 ], [ -1, 1, -1 ] ], "source_shift": [ 2, 2, 2 ], "target_matrix": [ [ 2, 2, 3 ], [ 1, 1, 2 ], [ 2, 1, 1 ] ], "target_shift": [ -2403, -1183, -1789 ], "collision_preimages": [ [ "-1/4", "1/4", "11/4" ], [ "11/2", "-8", "-18" ], [ "15/2", "-5", "-17" ] ], "collision_image": [ "-4807/2", "-4733/4", "-3579/2" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0802219479992345 ], "surface_features": [ 3.887701988252947, 4.768734271659541, 3.4319263739116437, 0.0, 4.027509337320764, 4979 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C244", "pair": 244, "family": "counterexample", "mapping": "F(x,y,z)=(16*x^7 - 128*x^6*y - 184*x^6*z + 260*x^6 + 432*x^5*y^2 + 1272*x^5*y*z - 1692*x^5*y + 900*x^5*z^2 - 2568*x^5*z + 1368*x^5 - 800*x^4*y^3 - 3600*x^4*y^2*z + 4560*x^4*y^2 - 5232*x^4*y*z^2 + 13980*x^4*y*z - 7296*x^4*y - 2426*x^4*z^3 + 10509*x^4*z^2 - 11160*x^4*z + 3556*x^4 + 880*x^3*y^4 + 5360*x^3*y^3*z - 6520*x^3*y^3 + 11928*x^3*y^2*z^2 - 30240*x^3*y^2*z + 15504*x^3*y^2 + 11398*x^3*y*z^3 - 45957*x^3*y*z^2 + 47748*x^3*y*z - 15064*x^3*y + 3890*x^3*z^4 - 22807*x^3*z^3 + 36240*x^3*z^2 - 22946*x^3*z + 5144*x^3 - 576*x^2*y^5 - 4440*x^2*y^4*z + 5220*x^2*y^4 - 13392*x^2*y^3*z^2 + 32520*x^2*y^3*z - 16416*x^2*y^3 - 19638*x^2*y^2*z^3 + 74817*x^2*y^2*z^2 - 76284*x^2*y^2*z + 23856*x^2*y^2 - 13866*x^2*y*z^4 + 75135*x^2*y*z^3 - 116628*x^2*y*z^2 + 73068*x^2*y*z - 16296*x^2*y - 3708*x^2*z^5 + 27684*x^2*z^4 - 58563*x^2*z^3 + 55293*x^2*z^2 - 24582*x^2*z + 4193*x^2 + 208*x*y^6 + 1944*x*y^5*z - 2220*x*y^5 + 7428*x*y^4*z^2 - 17400*x*y^4*z + 8664*x*y^4 + 14786*x*y^3*z^3 - 53799*x*y^3*z^2 + 53964*x*y^3*z - 16744*x*y^3 + 16062*x*y^2*z^4 - 81849*x*y^2*z^3 + 124536*x*y^2*z^2 - 77298*x*y^2*z + 17160*x*y^2 + 8928*x*y*z^5 - 61092*x*y*z^4 + 126030*x*y*z^3 - 117654*x*y*z^2 + 52014*x*y*z - 8851*x*y + 1944*x*z^6 - 17820*x*z^5 + 47100*x*z^4 - 58982*x*z^3 + 39024*x*z^2 - 13185*x*z + 1788*x - 32*y^7 - 352*y^6*z + 392*y^6 - 1632*y^5*z^2 + 3708*y^5*z - 1824*y^5 - 4120*y^4*z^3 + 14430*y^4*z^2 - 14268*y^4*z + 4396*y^4 - 6086*y^3*z^4 + 29521*y^3*z^3 - 44148*y^3*z^2 + 27176*y^3*z - 6008*y^3 - 5220*y^2*z^5 + 33408*y^2*z^4 - 67467*y^2*z^3 + 62361*y^2*z^2 - 27432*y^2*z + 4658*y^2 - 2376*y*z^6 + 19764*y*z^5 - 50844*y*z^4 + 62902*y*z^3 - 41364*y*z^2 + 13935*y*z - 1888*y - 432*z^7 + 4752*z^6 - 15084*z^5 + 23504*z^4 - 20584*z^3 + 10338*z^2 - 2775*z + 1, -8*x^7 + 64*x^6*y + 92*x^6*z - 136*x^6 - 216*x^5*y^2 - 636*x^5*y*z + 888*x^5*y - 450*x^5*z^2 + 1344*x^5*z - 774*x^5 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9230*x^3*y - 1945*x^3*z^4 + 11942*x^3*z^3 - 20622*x^3*z^2 + 14086*x^3*z - 3349*x^3 + 288*x^2*y^5 + 2220*x^2*y^4*z - 2760*x^2*y^4 + 6696*x^2*y^3*z^2 - 17184*x^2*y^3*z + 9360*x^2*y^3 + 9819*x^2*y^2*z^3 - 39474*x^2*y^2*z^2 + 43551*x^2*y^2*z - 14646*x^2*y^2 + 6933*x^2*y*z^4 - 39528*x^2*y*z^3 + 66603*x^2*y*z^2 - 44964*x^2*y*z + 10626*x^2*y + 1854*x^2*z^5 - 14493*x^2*z^4 + 33408*x^2*z^3 - 34071*x^2*z^2 + 16059*x^2*z - 2836*x^2 - 104*x*y^6 - 972*x*y^5*z + 1176*x*y^5 - 3714*x*y^4*z^2 + 9216*x*y^4*z - 4950*x*y^4 - 7393*x*y^3*z^3 + 28470*x*y^3*z^2 - 30885*x*y^3*z + 10298*x*y^3 - 8031*x*y^2*z^4 + 43230*x*y^2*z^3 - 71340*x*y^2*z^2 + 47670*x*y^2*z - 11205*x*y^2 - 4464*x*y*z^5 + 32154*x*y*z^4 - 72180*x*y*z^3 + 72690*x*y*z^2 - 34035*x*y*z + 5990*x*y - 972*x*z^6 + 9324*x*z^5 - 26928*x*z^4 + 36469*x*z^3 - 25572*x*z^2 + 8925*x*z - 1210*x + 16*y^7 + 176*y^6*z - 208*y^6 + 816*y^5*z^2 - 1968*y^5*z + 1044*y^5 + 2060*y^4*z^3 - 7656*y^4*z^2 + 8184*y^4*z - 2708*y^4 + 3043*y^3*z^4 - 15644*y^3*z^3 + 25359*y^3*z^2 - 16792*y^3*z + 3928*y^3 + 2610*y^2*z^5 - 17661*y^2*z^4 + 38772*y^2*z^3 - 38619*y^2*z^2 + 17976*y^2*z - 3154*y^2 + 1188*y*z^6 - 10404*y*z^5 + 29196*y*z^4 - 39005*y*z^3 + 27150*y*z^2 - 9435*y*z + 1275*y + 216*z^7 - 2484*z^6 + 8640*z^5 - 14578*z^4 + 13526*z^3 - 7002*z^2 + 1872*z - 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ 1, -1, -2 ], [ 2, -2, -3 ], [ 1, -2, -1 ] ], "source_shift": [ 2, 1, 2 ], "target_matrix": [ [ 2, 1, 1 ], [ -1, -1, -1 ], [ -1, -1, -2 ] ], "target_shift": [ -312, 228, 204 ], "collision_preimages": [ [ "29/4", "13/4", "3" ], [ "-8", "-6", "-1/2" ], [ "9", "-1", "13/2" ] ], "collision_image": [ "-625/2", "913/4", "817/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0773349810006039 ], "surface_features": [ 4.218957457978805, 5.1004739362576546, 3.904769625906595, 0.0, 4.336054254160486, 5133 ], 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3072*x*y^5 + 3360*x*y^4*z^2 + 7680*x*y^4*z + 4448*x*y^4 - 2240*x*y^3*z^3 - 7680*x*y^3*z^2 - 8928*x*y^3*z - 3624*x*y^3 + 840*x*y^2*z^4 + 3840*x*y^2*z^3 + 6720*x*y^2*z^2 + 5496*x*y^2*z + 1816*x*y^2 - 168*x*y*z^5 - 960*x*y*z^4 - 2248*x*y*z^3 - 2778*x*y*z^2 - 1856*x*y*z - 550*x*y + 14*x*z^6 + 96*x*z^5 + 282*x*z^4 + 468*x*z^3 + 474*x*z^2 + 288*x*z + 85*x + 256*y^7 - 896*y^6*z - 1024*y^6 + 1344*y^5*z^2 + 3072*y^5*z + 1760*y^5 - 1120*y^4*z^3 - 3840*y^4*z^2 - 4416*y^4*z - 1760*y^4 + 560*y^3*z^4 + 2560*y^3*z^3 + 4432*y^3*z^2 + 3560*y^3*z + 1152*y^3 - 168*y^2*z^5 - 960*y^2*z^4 - 2224*y^2*z^3 - 2700*y^2*z^2 - 1768*y^2*z - 506*y^2 + 28*y*z^6 + 192*y*z^5 + 558*y*z^4 + 910*y*z^3 + 904*y*z^2 + 528*y*z + 145*y - 2*z^7 - 16*z^6 - 56*z^5 - 115*z^4 - 154*z^3 - 138*z^2 - 79*z - 1, -2*x^7 - 28*x^6*y + 14*x^6*z + 16*x^6 - 168*x^5*y^2 + 168*x^5*y*z + 192*x^5*y - 42*x^5*z^2 - 96*x^5*z - 58*x^5 - 560*x^4*y^3 + 840*x^4*y^2*z + 960*x^4*y^2 - 420*x^4*y*z^2 - 960*x^4*y*z - 574*x^4*y + 70*x^4*z^3 + 240*x^4*z^2 + 288*x^4*z + 123*x^4 - 1120*x^3*y^4 + 2240*x^3*y^3*z + 2560*x^3*y^3 - 1680*x^3*y^2*z^2 - 3840*x^3*y^2*z - 2272*x^3*y^2 + 560*x^3*y*z^3 + 1920*x^3*y*z^2 + 2280*x^3*y*z + 958*x^3*y - 70*x^3*z^4 - 320*x^3*z^3 - 572*x^3*z^2 - 484*x^3*z - 164*x^3 - 1344*x^2*y^5 + 3360*x^2*y^4*z + 3840*x^2*y^4 - 3360*x^2*y^3*z^2 - 7680*x^2*y^3*z - 4496*x^2*y^3 + 1680*x^2*y^2*z^3 + 5760*x^2*y^2*z^2 + 6768*x^2*y^2*z + 2796*x^2*y^2 - 420*x^2*y*z^4 - 1920*x^2*y*z^3 - 3396*x^2*y*z^2 - 2826*x^2*y*z - 940*x^2*y + 42*x^2*z^5 + 240*x^2*z^4 + 568*x^2*z^3 + 714*x^2*z^2 + 480*x^2*z + 142*x^2 - 896*x*y^6 + 2688*x*y^5*z + 3072*x*y^5 - 3360*x*y^4*z^2 - 7680*x*y^4*z - 4448*x*y^4 + 2240*x*y^3*z^3 + 7680*x*y^3*z^2 + 8928*x*y^3*z + 3624*x*y^3 - 840*x*y^2*z^4 - 3840*x*y^2*z^3 - 6720*x*y^2*z^2 - 5496*x*y^2*z - 1792*x*y^2 + 168*x*y*z^5 + 960*x*y*z^4 + 2248*x*y*z^3 + 2778*x*y*z^2 + 1832*x*y*z + 518*x*y - 14*x*z^6 - 96*x*z^5 - 282*x*z^4 - 468*x*z^3 - 468*x*z^2 - 272*x*z - 71*x - 256*y^7 + 896*y^6*z + 1024*y^6 - 1344*y^5*z^2 - 3072*y^5*z - 1760*y^5 + 1120*y^4*z^3 + 3840*y^4*z^2 + 4416*y^4*z + 1760*y^4 - 560*y^3*z^4 - 2560*y^3*z^3 - 4432*y^3*z^2 - 3560*y^3*z - 1136*y^3 + 168*y^2*z^5 + 960*y^2*z^4 + 2224*y^2*z^3 + 2700*y^2*z^2 + 1744*y^2*z + 474*y^2 - 28*y*z^6 - 192*y*z^5 - 558*y*z^4 - 910*y*z^3 - 892*y*z^2 - 496*y*z - 123*y + 2*z^7 + 16*z^6 + 56*z^5 + 115*z^4 + 152*z^3 + 130*z^2 + 67*z + 1, 4*x^7 + 56*x^6*y - 28*x^6*z - 32*x^6 + 336*x^5*y^2 - 336*x^5*y*z - 384*x^5*y + 84*x^5*z^2 + 192*x^5*z + 116*x^5 + 1120*x^4*y^3 - 1680*x^4*y^2*z - 1920*x^4*y^2 + 840*x^4*y*z^2 + 1920*x^4*y*z + 1148*x^4*y - 140*x^4*z^3 - 480*x^4*z^2 - 576*x^4*z - 245*x^4 + 2240*x^3*y^4 - 4480*x^3*y^3*z - 5120*x^3*y^3 + 3360*x^3*y^2*z^2 + 7680*x^3*y^2*z + 4544*x^3*y^2 - 1120*x^3*y*z^3 - 3840*x^3*y*z^2 - 4560*x^3*y*z - 1908*x^3*y + 140*x^3*z^4 + 640*x^3*z^3 + 1144*x^3*z^2 + 964*x^3*z + 330*x^3 + 2688*x^2*y^5 - 6720*x^2*y^4*z - 7680*x^2*y^4 + 6720*x^2*y^3*z^2 + 15360*x^2*y^3*z + 8992*x^2*y^3 - 3360*x^2*y^2*z^3 - 11520*x^2*y^2*z^2 - 13536*x^2*y^2*z - 5568*x^2*y^2 + 840*x^2*y*z^4 + 3840*x^2*y*z^3 + 6792*x^2*y*z^2 + 5628*x^2*y*z + 1892*x^2*y - 84*x^2*z^5 - 480*x^2*z^4 - 1136*x^2*z^3 - 1422*x^2*z^2 - 966*x^2*z - 302*x^2 + 1792*x*y^6 - 5376*x*y^5*z - 6144*x*y^5 + 6720*x*y^4*z^2 + 15360*x*y^4*z + 8896*x*y^4 - 4480*x*y^3*z^3 - 15360*x*y^3*z^2 - 17856*x*y^3*z - 7216*x*y^3 + 1680*x*y^2*z^4 + 7680*x*y^2*z^3 + 13440*x*y^2*z^2 + 10944*x*y^2*z + 3608*x*y^2 - 336*x*y*z^5 - 1920*x*y*z^4 - 4496*x*y*z^3 - 5532*x*y*z^2 - 3688*x*y*z - 1108*x*y + 28*x*z^6 + 192*x*z^5 + 564*x*z^4 + 932*x*z^3 + 942*x*z^2 + 580*x*z + 178*x + 512*y^7 - 1792*y^6*z - 2048*y^6 + 2688*y^5*z^2 + 6144*y^5*z + 3520*y^5 - 2240*y^4*z^3 - 7680*y^4*z^2 - 8832*y^4*z - 3504*y^4 + 1120*y^3*z^4 + 5120*y^3*z^3 + 8864*y^3*z^2 + 7088*y^3*z + 2288*y^3 - 336*y^2*z^5 - 1920*y^2*z^4 - 4448*y^2*z^3 - 5376*y^2*z^2 - 3512*y^2*z - 1020*y^2 + 56*y*z^6 + 384*y*z^5 + 1116*y*z^4 + 1812*y*z^3 + 1796*y*z^2 + 1064*y*z + 301*y - 4*z^7 - 32*z^6 - 112*z^5 - 229*z^4 - 306*z^3 - 278*z^2 - 164*z - 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -1, -2, 1 ], [ -1, -1, 1 ], [ -2, -1, 1 ] ], "source_shift": [ 1, 2, 1 ], "target_matrix": [ [ -1, -1, -1 ], [ 1, 1, 2 ], [ -2, -1, -1 ] ], "target_shift": [ 22, -16, 47 ], "inverse_recipe": { "p": 4, "q": 3, "f": "x**4 + x", "g": "x**3 + x**2", "h": "y*z", "polynomials": { "f": [ [ [ 4, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "g": [ [ [ 3, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ] ] }, "h_multiplier": 2 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.010657666996849 ], "surface_features": [ 3.340223716722268, 4.251784177257485, 2.9845273133437926, 0.0, 3.4544216499813314, 4816 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C053", "pair": 53, "family": "counterexample", "mapping": "F(x,y,z)=(-8*x^7 + 44*x^6*y + 52*x^6*z - 124*x^6 - 102*x^5*y^2 - 240*x^5*y*z + 576*x^5*y - 138*x^5*z^2 + 672*x^5*z - 708*x^5 + 129*x^4*y^3 + 453*x^4*y^2*z - 1095*x^4*y^2 + 519*x^4*y*z^2 - 2538*x^4*y*z + 2724*x^4*y + 195*x^4*z^3 - 1443*x^4*z^2 + 3096*x^4*z - 2026*x^4 - 96*x^3*y^4 - 447*x^3*y^3*z + 1089*x^3*y^3 - 765*x^3*y^2*z^2 + 3759*x^3*y^2*z - 4113*x^3*y^2 - 573*x^3*y*z^3 + 4251*x^3*y*z^2 - 9294*x^3*y*z + 6225*x^3*y - 159*x^3*z^4 + 1581*x^3*z^3 - 5181*x^3*z^2 + 6953*x^3*z - 3306*x^3 + 42*x^2*y^5 + 243*x^2*y^4*z - 597*x^2*y^4 + 552*x^2*y^3*z^2 - 2727*x^2*y^3*z + 3042*x^2*y^3 + 618*x^2*y^2*z^3 - 4599*x^2*y^2*z^2 + 10245*x^2*y^2*z - 7029*x^2*y^2 + 342*x^2*y*z^4 - 3405*x^2*y*z^3 + 11364*x^2*y*z^2 - 15618*x^2*y*z + 7626*x^2*y + 75*x^2*z^5 - 936*x^2*z^4 + 4161*x^2*z^3 - 8589*x^2*z^2 + 8424*x^2*z - 3172*x^2 - 10*x*y^6 - 69*x*y^5*z + 171*x*y^5 - 195*x*y^4*z^2 + 969*x*y^4*z - 1101*x*y^4 - 290*x*y^3*z^3 + 2166*x*y^3*z^2 - 4911*x*y^3*z + 3451*x*y^3 - 240*x*y^2*z^4 + 2394*x*y^2*z^3 - 8127*x*y^2*z^2 + 11433*x*y^2*z - 5736*x*y^2 - 105*x*y*z^5 + 1311*x*y*z^4 - 5925*x*y*z^3 + 12513*x*y*z^2 - 12609*x*y*z + 4892*x*y - 19*x*z^6 + 285*x*z^5 - 1608*x*z^4 + 4531*x*z^3 - 6873*x*z^2 + 5372*x*z - 1697*x + y^7 + 8*y^6*z - 20*y^6 + 27*y^5*z^2 - 135*y^5*z + 156*y^5 + 50*y^4*z^3 - 375*y^4*z^2 + 864*y^4*z - 621*y^4 + 55*y^3*z^4 - 550*y^3*z^3 + 1896*y^3*z^2 - 2726*y^3*z + 1404*y^3 + 36*y^2*z^5 - 450*y^2*z^4 + 2064*y^2*z^3 - 4452*y^2*z^2 + 4602*y^2*z - 1840*y^2 + 13*y*z^6 - 195*y*z^5 + 1116*y*z^4 - 3210*y*z^3 + 4992*y*z^2 - 4019*y*z + 1310*y + 2*z^7 - 35*z^6 + 240*z^5 - 863*z^4 + 1794*z^3 - 2179*z^2 + 1437*z - 1, -8*x^7 + 44*x^6*y + 52*x^6*z - 100*x^6 - 102*x^5*y^2 - 240*x^5*y*z + 468*x^5*y - 138*x^5*z^2 + 540*x^5*z - 504*x^5 + 129*x^4*y^3 + 453*x^4*y^2*z - 897*x^4*y^2 + 519*x^4*y*z^2 - 2058*x^4*y*z + 1956*x^4*y + 195*x^4*z^3 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27*y^5*z^2 - 114*y^5*z + 117*y^5 + 50*y^4*z^3 - 315*y^4*z^2 + 645*y^4*z - 429*y^4 + 55*y^3*z^4 - 460*y^3*z^3 + 1410*y^3*z^2 - 1877*y^3*z + 915*y^3 + 36*y^2*z^5 - 375*y^2*z^4 + 1530*y^2*z^3 - 3057*y^2*z^2 + 2991*y^2*z - 1144*y^2 + 13*y*z^6 - 162*y*z^5 + 825*y*z^4 - 2199*y*z^3 + 3237*y*z^2 - 2492*y*z + 781*y + 2*z^7 - 29*z^6 + 177*z^5 - 590*z^4 + 1161*z^3 - 1348*z^2 + 854*z - 1, 16*x^7 - 88*x^6*y - 104*x^6*z + 176*x^6 + 204*x^5*y^2 + 480*x^5*y*z - 828*x^5*y + 276*x^5*z^2 - 948*x^5*z + 804*x^5 - 258*x^4*y^3 - 906*x^4*y^2*z + 1596*x^4*y^2 - 1038*x^4*y*z^2 + 3636*x^4*y*z - 3144*x^4*y - 390*x^4*z^3 + 2040*x^4*z^2 - 3528*x^4*z + 2008*x^4 + 192*x^3*y^4 + 894*x^3*y^3*z - 1611*x^3*y^3 + 1530*x^3*y^2*z^2 - 5475*x^3*y^2*z + 4833*x^3*y^2 + 1146*x^3*y*z^3 - 6117*x^3*y*z^2 + 10794*x^3*y*z - 6278*x^3*y + 318*x^3*z^4 - 2253*x^3*z^3 + 5961*x^3*z^2 - 6950*x^3*z + 2992*x^3 - 84*x^2*y^5 - 486*x^2*y^4*z + 897*x^2*y^4 - 1104*x^2*y^3*z^2 + 4041*x^2*y^3*z - 3645*x^2*y^3 - 1236*x^2*y^2*z^3 + 6741*x^2*y^2*z^2 - 12147*x^2*y^2*z + 7227*x^2*y^2 - 684*x^2*y*z^4 + 4947*x^2*y*z^3 - 13359*x^2*y*z^2 + 15930*x^2*y*z - 7032*x^2*y - 150*x^2*z^5 + 1350*x^2*z^4 - 4857*x^2*z^3 + 8703*x^2*z^2 - 7716*x^2*z + 2690*x^2 + 20*x*y^6 + 138*x*y^5*z - 261*x*y^5 + 390*x*y^4*z^2 - 1461*x*y^4*z + 1347*x*y^4 + 580*x*y^3*z^3 - 3234*x*y^3*z^2 + 5952*x*y^3*z - 3624*x*y^3 + 480*x*y^2*z^4 - 3546*x*y^2*z^3 + 9774*x*y^2*z^2 - 11922*x*y^2*z + 5400*x*y^2 + 210*x*y*z^5 - 1929*x*y*z^4 + 7080*x*y*z^3 - 12972*x*y*z^2 + 11802*x*y*z - 4237*x*y + 38*x*z^6 - 417*x*z^5 + 1911*x*z^4 - 4674*x*z^3 + 6402*x*z^2 - 4627*x*z + 1373*x - 2*y^7 - 16*y^6*z + 31*y^6 - 54*y^5*z^2 + 207*y^5*z - 195*y^5 - 100*y^4*z^3 + 570*y^4*z^2 - 1071*y^4*z + 667*y^4 - 110*y^3*z^4 + 830*y^3*z^3 - 2334*y^3*z^2 + 2910*y^3*z - 1352*y^3 - 72*y^2*z^5 + 675*y^2*z^4 - 2526*y^2*z^3 + 4728*y^2*z^2 - 4410*y^2*z + 1631*y^2 - 26*y*z^6 + 291*y*z^5 - 1359*y*z^4 + 3394*y*z^3 - 4764*y*z^2 + 3547*y*z - 1087*y - 4*z^7 + 52*z^6 - 291*z^5 + 909*z^4 - 1706*z^3 + 1916*z^2 - 1187*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ 2, -1, -1 ], [ 1, -1, -1 ], [ 1, -1, -2 ] ], "source_shift": [ 2, 1, 2 ], "target_matrix": [ [ -1, -2, -1 ], [ -1, -1, -1 ], [ 2, 1, 1 ] ], "target_shift": [ 397, 228, -312 ], "collision_preimages": [ [ "-1", "-5/4", "5/4" ], [ "3/2", "11", "-7" ], [ "-7/2", "0", "-4" ] ], "collision_image": [ "1589/4", "913/4", "-625/2" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0438335300023027 ], "surface_features": [ 3.3721358570059925, 4.202215775801132, 3.040602340114073, 0.0, 3.4769867231472777, 4817 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B139", "pair": 139, "family": "bijection", "mapping": "F(x,y,z)=(2187*x^7 - 15309*x^6*y - 10206*x^6*z + 5346*x^6 + 45927*x^5*y^2 + 61236*x^5*y*z - 32076*x^5*y + 20412*x^5*z^2 - 21465*x^5*z + 5751*x^5 - 76545*x^4*y^3 - 153090*x^4*y^2*z + 80190*x^4*y^2 - 102060*x^4*y*z^2 + 107325*x^4*y*z - 28755*x^4*y - 22680*x^4*z^3 + 35910*x^4*z^2 - 19305*x^4*z + 3699*x^4 + 76545*x^3*y^4 + 204120*x^3*y^3*z - 106920*x^3*y^3 + 204120*x^3*y^2*z^2 - 214650*x^3*y^2*z + 57510*x^3*y^2 + 90720*x^3*y*z^3 - 143640*x^3*y*z^2 + 77220*x^3*y*z - 14796*x^3*y + 15120*x^3*z^4 - 32040*x^3*z^3 + 25920*x^3*z^2 - 9963*x^3*z + 1620*x^3 - 45927*x^2*y^5 - 153090*x^2*y^4*z + 80190*x^2*y^4 - 204120*x^2*y^3*z^2 + 214650*x^2*y^3*z - 57510*x^2*y^3 - 136080*x^2*y^2*z^3 + 215460*x^2*y^2*z^2 - 115830*x^2*y^2*z + 22194*x^2*y^2 - 45360*x^2*y*z^4 + 96120*x^2*y*z^3 - 77760*x^2*y*z^2 + 29889*x^2*y*z - 4851*x^2*y - 6048*x^2*z^5 + 16080*x^2*z^4 - 17400*x^2*z^3 + 10062*x^2*z^2 - 3285*x^2*z + 546*x^2 + 15309*x*y^6 + 61236*x*y^5*z - 32076*x*y^5 + 102060*x*y^4*z^2 - 107325*x*y^4*z + 28755*x*y^4 + 90720*x*y^3*z^3 - 143640*x*y^3*z^2 + 77220*x*y^3*z - 14796*x*y^3 + 45360*x*y^2*z^4 - 96120*x*y^2*z^3 + 77760*x*y^2*z^2 - 29889*x*y^2*z + 4842*x*y^2 + 12096*x*y*z^5 - 32160*x*y*z^4 + 34800*x*y*z^3 - 20124*x*y*z^2 + 6558*x*y*z - 1085*x*y + 1344*x*z^6 - 4304*x*z^5 + 5840*x*z^4 - 4516*x*z^3 + 2220*x*z^2 - 742*x*z + 136*x - 2187*y^7 - 10206*y^6*z + 5346*y^6 - 20412*y^5*z^2 + 21465*y^5*z - 5751*y^5 - 22680*y^4*z^3 + 35910*y^4*z^2 - 19305*y^4*z + 3699*y^4 - 15120*y^3*z^4 + 32040*y^3*z^3 - 25920*y^3*z^2 + 9963*y^3*z - 1611*y^3 - 6048*y^2*z^5 + 16080*y^2*z^4 - 17400*y^2*z^3 + 10062*y^2*z^2 - 3273*y^2*z + 539*y^2 - 1344*y*z^6 + 4304*y*z^5 - 5840*y*z^4 + 4516*y*z^3 - 2216*y*z^2 + 737*y*z - 135*y - 128*z^7 + 480*z^6 - 784*z^5 + 760*z^4 - 500*z^3 + 252*z^2 - 95*z + 1, -2187*x^7 + 15309*x^6*y + 10206*x^6*z - 5346*x^6 - 45927*x^5*y^2 - 61236*x^5*y*z + 32076*x^5*y - 20412*x^5*z^2 + 21465*x^5*z - 5751*x^5 + 76545*x^4*y^3 + 153090*x^4*y^2*z - 80190*x^4*y^2 + 102060*x^4*y*z^2 - 107325*x^4*y*z + 28755*x^4*y + 22680*x^4*z^3 - 35910*x^4*z^2 + 19305*x^4*z - 3699*x^4 - 76545*x^3*y^4 - 204120*x^3*y^3*z + 106920*x^3*y^3 - 204120*x^3*y^2*z^2 + 214650*x^3*y^2*z - 57510*x^3*y^2 - 90720*x^3*y*z^3 + 143640*x^3*y*z^2 - 77220*x^3*y*z + 14796*x^3*y - 15120*x^3*z^4 + 32040*x^3*z^3 - 25920*x^3*z^2 + 9963*x^3*z - 1620*x^3 + 45927*x^2*y^5 + 153090*x^2*y^4*z - 80190*x^2*y^4 + 204120*x^2*y^3*z^2 - 214650*x^2*y^3*z + 57510*x^2*y^3 + 136080*x^2*y^2*z^3 - 215460*x^2*y^2*z^2 + 115830*x^2*y^2*z - 22194*x^2*y^2 + 45360*x^2*y*z^4 - 96120*x^2*y*z^3 + 77760*x^2*y*z^2 - 29889*x^2*y*z + 4851*x^2*y + 6048*x^2*z^5 - 16080*x^2*z^4 + 17400*x^2*z^3 - 10062*x^2*z^2 + 3285*x^2*z - 537*x^2 - 15309*x*y^6 - 61236*x*y^5*z + 32076*x*y^5 - 102060*x*y^4*z^2 + 107325*x*y^4*z - 28755*x*y^4 - 90720*x*y^3*z^3 + 143640*x*y^3*z^2 - 77220*x*y^3*z + 14796*x*y^3 - 45360*x*y^2*z^4 + 96120*x*y^2*z^3 - 77760*x*y^2*z^2 + 29889*x*y^2*z - 4842*x*y^2 - 12096*x*y*z^5 + 32160*x*y*z^4 - 34800*x*y*z^3 + 20124*x*y*z^2 - 6558*x*y*z + 1067*x*y - 1344*x*z^6 + 4304*x*z^5 - 5840*x*z^4 + 4516*x*z^3 - 2220*x*z^2 + 730*x*z - 129*x + 2187*y^7 + 10206*y^6*z - 5346*y^6 + 20412*y^5*z^2 - 21465*y^5*z + 5751*y^5 + 22680*y^4*z^3 - 35910*y^4*z^2 + 19305*y^4*z - 3699*y^4 + 15120*y^3*z^4 - 32040*y^3*z^3 + 25920*y^3*z^2 - 9963*y^3*z + 1611*y^3 + 6048*y^2*z^5 - 16080*y^2*z^4 + 17400*y^2*z^3 - 10062*y^2*z^2 + 3273*y^2*z - 530*y^2 + 1344*y*z^6 - 4304*y*z^5 + 5840*y*z^4 - 4516*y*z^3 + 2216*y*z^2 - 725*y*z + 128*y + 128*z^7 - 480*z^6 + 784*z^5 - 760*z^4 + 500*z^3 - 248*z^2 + 90*z - 1, -4374*x^7 + 30618*x^6*y + 20412*x^6*z - 10692*x^6 - 91854*x^5*y^2 - 122472*x^5*y*z + 64152*x^5*y - 40824*x^5*z^2 + 42930*x^5*z - 11988*x^5 + 153090*x^4*y^3 + 306180*x^4*y^2*z - 160380*x^4*y^2 + 204120*x^4*y*z^2 - 214650*x^4*y*z + 59940*x^4*y + 45360*x^4*z^3 - 71820*x^4*z^2 + 40230*x^4*z - 8208*x^4 - 153090*x^3*y^4 - 408240*x^3*y^3*z + 213840*x^3*y^3 - 408240*x^3*y^2*z^2 + 429300*x^3*y^2*z - 119880*x^3*y^2 - 181440*x^3*y*z^3 + 287280*x^3*y*z^2 - 160920*x^3*y*z + 32832*x^3*y - 30240*x^3*z^4 + 64080*x^3*z^3 - 54000*x^3*z^2 + 22086*x^3*z - 3834*x^3 + 91854*x^2*y^5 + 306180*x^2*y^4*z - 160380*x^2*y^4 + 408240*x^2*y^3*z^2 - 429300*x^2*y^3*z + 119880*x^2*y^3 + 272160*x^2*y^2*z^3 - 430920*x^2*y^2*z^2 + 241380*x^2*y^2*z - 49248*x^2*y^2 + 90720*x^2*y*z^4 - 192240*x^2*y*z^3 + 162000*x^2*y*z^2 - 66258*x^2*y*z + 11484*x^2*y + 12096*x^2*z^5 - 32160*x^2*z^4 + 36240*x^2*z^3 - 22284*x^2*z^2 + 7758*x^2*z - 1317*x^2 - 30618*x*y^6 - 122472*x*y^5*z + 64152*x*y^5 - 204120*x*y^4*z^2 + 214650*x*y^4*z - 59940*x*y^4 - 181440*x*y^3*z^3 + 287280*x*y^3*z^2 - 160920*x*y^3*z + 32832*x*y^3 - 90720*x*y^2*z^4 + 192240*x*y^2*z^3 - 162000*x*y^2*z^2 + 66258*x*y^2*z - 11466*x*y^2 - 24192*x*y*z^5 + 64320*x*y*z^4 - 72480*x*y*z^3 + 44568*x*y*z^2 - 15492*x*y*z + 2620*x*y - 2688*x*z^6 + 8608*x*z^5 - 12160*x*z^4 + 9992*x*z^3 - 5232*x*z^2 + 1784*x*z - 321*x + 4374*y^7 + 20412*y^6*z - 10692*y^6 + 40824*y^5*z^2 - 42930*y^5*z + 11988*y^5 + 45360*y^4*z^3 - 71820*y^4*z^2 + 40230*y^4*z - 8208*y^4 + 30240*y^3*z^4 - 64080*y^3*z^3 + 54000*y^3*z^2 - 22086*y^3*z + 3816*y^3 + 12096*y^2*z^5 - 32160*y^2*z^4 + 36240*y^2*z^3 - 22284*y^2*z^2 + 7734*y^2*z - 1303*y^2 + 2688*y*z^6 - 8608*y*z^5 + 12160*y*z^4 - 9992*y*z^3 + 5224*y*z^2 - 1774*y*z + 317*y + 256*z^7 - 960*z^6 + 1632*z^5 - 1680*z^4 + 1176*z^3 - 604*z^2 + 221*z - 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ 3, -3, -2 ], [ 1, -1, -1 ], [ 2, -1, -1 ] ], "source_shift": [ 1, 2, 1 ], "target_matrix": [ [ 1, 2, 1 ], [ -1, -1, -1 ], [ -2, -1, -1 ] ], "target_shift": [ -23, 20, 48 ], "inverse_recipe": { "p": 2, "q": 5, "f": "x**2", "g": "x**5 + x**3 + x**2 + x", "h": "y**2 + y*z + y", "polynomials": { "f": [ [ [ 2, 0, 0 ], 1 ] ], "g": [ [ [ 5, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ], [ [ 3, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 1, 0 ], 1 ], [ [ 0, 2, 0 ], 1 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "Maybe" ], "seconds": [ 1.0015400779993797 ], "surface_features": [ 4.7504932024212385, 5.634396651173698, 4.3471152548414045, 0.0, 4.9008673058583625, 5313 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C104", "pair": 104, "family": "counterexample", "mapping": "F(x,y,z)=(24*x^7 - 204*x^6*y + 132*x^6*z - 348*x^6 + 702*x^5*y^2 - 936*x^5*y*z + 2484*x^5*y + 306*x^5*z^2 - 1620*x^5*z + 2088*x^5 - 1257*x^4*y^3 + 2601*x^4*y^2*z - 7011*x^4*y^2 - 1755*x^4*y*z^2 + 9378*x^4*y*z - 12312*x^4*y + 387*x^4*z^3 - 3087*x^4*z^2 + 8028*x^4*z - 6866*x^4 + 1257*x^3*y^4 - 3597*x^3*y^3*z + 9978*x^3*y^3 + 3771*x^3*y^2*z^2 - 20538*x^3*y^2*z + 27666*x^3*y^2 - 1719*x^3*y*z^3 + 13878*x^3*y*z^2 - 36828*x^3*y*z + 32237*x^3*y + 288*x^3*z^4 - 3078*x^3*z^3 + 12114*x^3*z^2 - 20951*x^3*z + 13472*x^3 - 702*x^2*y^5 + 2601*x^2*y^4*z - 7533*x^2*y^4 - 3771*x^2*y^3*z^2 + 21195*x^2*y^3*z - 29448*x^2*y^3 + 2673*x^2*y^2*z^3 - 22059*x^2*y^2*z^2 + 60075*x^2*y^2*z - 53973*x^2*y^2 - 927*x^2*y*z^4 + 10053*x^2*y*z^3 - 40410*x^2*y*z^2 + 71523*x^2*y*z - 47121*x^2*y + 126*x^2*z^5 - 1692*x^2*z^4 + 8955*x^2*z^3 - 23472*x^2*z^2 + 30549*x^2*z - 15822*x^2 + 204*x*y^6 - 936*x*y^5*z + 2844*x*y^5 + 1755*x*y^4*z^2 - 10278*x*y^4*z + 14688*x*y^4 - 1719*x*y^3*z^3 + 14652*x*y^3*z^2 - 41004*x*y^3*z + 37744*x*y^3 + 927*x*y^2*z^4 - 10296*x*y^2*z^3 + 42408*x*y^2*z^2 - 76836*x*y^2*z + 51867*x*y^2 - 261*x*y*z^5 + 3564*x*y*z^4 - 19260*x*y*z^3 + 51591*x*y*z^2 - 68703*x*y*z + 36480*x*y + 30*x*z^6 - 486*x*z^5 + 3240*x*z^4 - 11425*x*z^3 + 22542*x*z^2 - 23652*x*z + 10324*x - 24*y^7 + 132*y^6*z - 420*y^6 - 306*y^5*z^2 + 1872*y^5*z - 2736*y^5 + 387*y^4*z^3 - 3429*y^4*z^2 + 9828*y^4*z - 9202*y^4 - 288*y^3*z^4 + 3303*y^3*z^3 - 13932*y^3*z^2 + 25738*y^3*z - 17680*y^3 + 126*y^2*z^5 - 1764*y^2*z^4 + 9747*y^2*z^3 - 26637*y^2*z^2 + 36171*y^2*z - 19617*y^2 - 30*y*z^6 + 495*y*z^5 - 3366*y*z^4 + 12103*y*z^3 - 24360*y*z^2 + 26115*y*z - 11682*y + 3*z^7 - 57*z^6 + 459*z^5 - 2038*z^4 + 5407*z^3 - 8598*z^2 + 7605*z - 1, 8*x^7 - 68*x^6*y + 44*x^6*z - 140*x^6 + 234*x^5*y^2 - 312*x^5*y*z + 984*x^5*y + 102*x^5*z^2 - 648*x^5*z + 924*x^5 - 419*x^4*y^3 + 867*x^4*y^2*z - 2727*x^4*y^2 - 585*x^4*y*z^2 + 3690*x^4*y*z - 5352*x^4*y + 129*x^4*z^3 - 1227*x^4*z^2 + 3528*x^4*z - 3222*x^4 + 419*x^3*y^4 - 1199*x^3*y^3*z + 3803*x^3*y^3 + 1257*x^3*y^2*z^2 - 7929*x^3*y^2*z + 11787*x^3*y^2 - 573*x^3*y*z^3 + 5421*x^3*y*z^2 - 15882*x^3*y*z + 14867*x^3*y + 96*x^3*z^4 - 1215*x^3*z^3 + 5283*x^3*z^2 - 9761*x^3*z + 6576*x^3 - 234*x^2*y^5 + 867*x^2*y^4*z - 2814*x^2*y^4 - 1257*x^2*y^3*z^2 + 8019*x^2*y^3*z - 12279*x^2*y^3 + 891*x^2*y^2*z^3 - 8451*x^2*y^2*z^2 + 25371*x^2*y^2*z - 24411*x^2*y^2 - 309*x^2*y*z^4 + 3897*x^2*y*z^3 - 17277*x^2*y*z^2 + 32709*x^2*y*z - 22611*x^2*y + 42*x^2*z^5 - 663*x^2*z^4 + 3873*x^2*z^3 - 10848*x^2*z^2 + 14799*x^2*z - 7936*x^2 + 68*x*y^6 - 312*x*y^5*z + 1044*x*y^5 + 585*x*y^4*z^2 - 3816*x*y^4*z + 5988*x*y^4 - 573*x*y^3*z^3 + 5505*x*y^3*z^2 - 16938*x*y^3*z + 16701*x*y^3 + 309*x*y^2*z^4 - 3915*x*y^2*z^3 + 17745*x*y^2*z^2 - 34413*x*y^2*z + 24402*x*y^2 - 87*x*y*z^5 + 1371*x*y*z^4 - 8160*x*y*z^3 + 23376*x*y*z^2 - 32667*x*y*z + 17977*x*y + 10*x*z^6 - 189*x*z^5 + 1389*x*z^4 - 5234*x*z^3 + 10827*x*z^2 - 11765*x*z + 5273*x - 8*y^7 + 44*y^6*z - 152*y^6 - 102*y^5*z^2 + 684*y^5*z - 1092*y^5 + 129*y^4*z^3 - 1266*y^4*z^2 + 3972*y^4*z - 3979*y^4 - 96*y^3*z^4 + 1233*y^3*z^3 - 5703*y^3*z^2 + 11272*y^3*z - 8133*y^3 + 42*y^2*z^5 - 666*y^2*z^4 + 4041*y^2*z^3 - 11811*y^2*z^2 + 16839*y^2*z - 9466*y^2 - 10*y*z^6 + 189*y*z^5 - 1413*y*z^4 + 5431*y*z^3 - 11469*y*z^2 + 12739*y*z - 5851*y + z^7 - 22*z^6 + 195*z^5 - 925*z^4 + 2573*z^3 - 4237*z^2 + 3847*z - 1, -8*x^7 + 68*x^6*y - 44*x^6*z + 116*x^6 - 234*x^5*y^2 + 312*x^5*y*z - 828*x^5*y - 102*x^5*z^2 + 540*x^5*z - 696*x^5 + 419*x^4*y^3 - 867*x^4*y^2*z + 2337*x^4*y^2 + 585*x^4*y*z^2 - 3126*x^4*y*z + 4104*x^4*y - 129*x^4*z^3 + 1029*x^4*z^2 - 2676*x^4*z + 2286*x^4 - 419*x^3*y^4 + 1199*x^3*y^3*z - 3326*x^3*y^3 - 1257*x^3*y^2*z^2 + 6846*x^3*y^2*z - 9222*x^3*y^2 + 573*x^3*y*z^3 - 4626*x^3*y*z^2 + 12276*x^3*y*z - 10739*x^3*y - 96*x^3*z^4 + 1026*x^3*z^3 - 4038*x^3*z^2 + 6977*x^3*z - 4476*x^3 + 234*x^2*y^5 - 867*x^2*y^4*z + 2511*x^2*y^4 + 1257*x^2*y^3*z^2 - 7065*x^2*y^3*z + 9816*x^2*y^3 - 891*x^2*y^2*z^3 + 7353*x^2*y^2*z^2 - 20025*x^2*y^2*z + 17985*x^2*y^2 + 309*x^2*y*z^4 - 3351*x^2*y*z^3 + 13470*x^2*y*z^2 - 23829*x^2*y*z + 15675*x^2*y - 42*x^2*z^5 + 564*x^2*z^4 - 2985*x^2*z^3 + 7818*x^2*z^2 - 10155*x^2*z + 5242*x^2 - 68*x*y^6 + 312*x*y^5*z - 948*x*y^5 - 585*x*y^4*z^2 + 3426*x*y^4*z - 4896*x*y^4 + 573*x*y^3*z^3 - 4884*x*y^3*z^2 + 13668*x*y^3*z - 12579*x*y^3 - 309*x*y^2*z^4 + 3432*x*y^2*z^3 - 14136*x*y^2*z^2 + 25605*x*y^2*z - 17268*x*y^2 + 87*x*y*z^5 - 1188*x*y*z^4 + 6420*x*y*z^3 - 17190*x*y*z^2 + 22863*x*y*z - 12112*x*y - 10*x*z^6 + 162*x*z^5 - 1080*x*z^4 + 3806*x*z^3 - 7497*x*z^2 + 7844*x*z - 3412*x + 8*y^7 - 44*y^6*z + 140*y^6 + 102*y^5*z^2 - 624*y^5*z + 912*y^5 - 129*y^4*z^3 + 1143*y^4*z^2 - 3276*y^4*z + 3067*y^4 + 96*y^3*z^4 - 1101*y^3*z^3 + 4644*y^3*z^2 - 8578*y^3*z + 5889*y^3 - 42*y^2*z^5 + 588*y^2*z^4 - 3249*y^2*z^3 + 8877*y^2*z^2 - 12045*y^2*z + 6523*y^2 + 10*y*z^6 - 165*y*z^5 + 1122*y*z^4 - 4033*y*z^3 + 8109*y*z^2 - 8677*y*z + 3872*y - z^7 + 19*z^6 - 153*z^5 + 679*z^4 - 1799*z^3 + 2854*z^2 - 2517*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -2, 1, -1 ], [ -1, 2, -1 ], [ -1, 1, -1 ] ], "source_shift": [ 2, 2, 1 ], "target_matrix": [ [ -3, -3, -2 ], [ -1, -2, -1 ], [ 1, 1, 1 ] ], "target_shift": [ 2886, 1488, -952 ], "collision_preimages": [ [ "3/4", "-3/4", "-1/4" ], [ "13/2", "-9", "-21" ], [ "17/2", "-6", "-20" ] ], "collision_image": [ "11547/4", "5953/4", "-3809/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.029559162001533 ], "surface_features": [ 3.886890680959507, 4.8855647478717925, 3.4889030852677694, 0.0, 4.072944536155137, 4958 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B171", "pair": 171, "family": "bijection", "mapping": "F(x,y,z)=(128*x^7 - 1344*x^6*y - 896*x^6*z + 960*x^6 + 6048*x^5*y^2 + 8064*x^5*y*z - 8576*x^5*y + 2688*x^5*z^2 - 5728*x^5*z + 2944*x^5 - 15120*x^4*y^3 - 30240*x^4*y^2*z + 31920*x^4*y^2 - 20160*x^4*y*z^2 + 42640*x^4*y*z - 21760*x^4*y - 4480*x^4*z^3 + 14240*x^4*z^2 - 14560*x^4*z + 4800*x^4 + 22680*x^3*y^4 + 60480*x^3*y^3*z - 63360*x^3*y^3 + 60480*x^3*y^2*z^2 - 126960*x^3*y^2*z + 64320*x^3*y^2 + 26880*x^3*y*z^3 - 84800*x^3*y*z^2 + 86080*x^3*y*z - 28160*x^3*y + 4480*x^3*z^4 - 18880*x^3*z^3 + 28800*x^3*z^2 - 18880*x^3*z + 4484*x^3 - 20412*x^2*y^5 - 68040*x^2*y^4*z + 70740*x^2*y^4 - 90720*x^2*y^3*z^2 + 189000*x^2*y^3*z - 95040*x^2*y^3 - 60480*x^2*y^2*z^3 + 189360*x^2*y^2*z^2 - 190800*x^2*y^2*z + 61920*x^2*y^2 - 20160*x^2*y*z^4 + 84320*x^2*y*z^3 - 127680*x^2*y*z^2 + 83040*x^2*y*z - 19540*x^2*y - 2688*x^2*z^5 + 14080*x^2*z^4 - 28480*x^2*z^3 + 27840*x^2*z^2 - 13132*x^2*z + 2394*x^2 + 10206*x*y^6 + 40824*x*y^5*z - 42120*x*y^5 + 68040*x*y^4*z^2 - 140670*x*y^4*z + 70200*x*y^4 + 60480*x*y^3*z^3 - 187920*x*y^3*z^2 + 187920*x*y^3*z - 60480*x*y^3 + 30240*x*y^2*z^4 - 125520*x*y^2*z^3 + 188640*x*y^2*z^2 - 121680*x*y^2*z + 28353*x*y^2 + 8064*x*y*z^5 - 41920*x*y*z^4 + 84160*x*y*z^3 - 81600*x*y*z^2 + 38120*x*y*z - 6865*x*y + 896*x*z^6 - 5600*x*z^5 + 14080*x*z^4 - 18240*x*z^3 + 12812*x*z^2 - 4627*x*z + 684*x - 2187*y^7 - 10206*y^6*z + 10449*y^6 - 20412*y^5*z^2 + 41877*y^5*z - 20736*y^5 - 22680*y^4*z^3 + 69930*y^4*z^2 - 69390*y^4*z + 22140*y^4 - 15120*y^3*z^4 + 62280*y^3*z^3 - 92880*y^3*z^2 + 59400*y^3*z - 13698*y^3 - 6048*y^2*z^5 + 31200*y^2*z^4 - 62160*y^2*z^3 + 59760*y^2*z^2 - 27633*y^2*z + 4910*y^2 - 1344*y*z^6 + 8336*y*z^5 - 20800*y*z^4 + 26720*y*z^3 - 18580*y*z^2 + 6623*y*z - 955*y - 128*z^7 + 928*z^6 - 2784*z^5 + 4480*z^4 - 4164*z^3 + 2233*z^2 - 648*z + 1, 256*x^7 - 2688*x^6*y - 1792*x^6*z + 1920*x^6 + 12096*x^5*y^2 + 16128*x^5*y*z - 17152*x^5*y + 5376*x^5*z^2 - 11456*x^5*z + 6080*x^5 - 30240*x^4*y^3 - 60480*x^4*y^2*z + 63840*x^4*y^2 - 40320*x^4*y*z^2 + 85280*x^4*y*z - 44960*x^4*y - 8960*x^4*z^3 + 28480*x^4*z^2 - 30080*x^4*z + 10560*x^4 + 45360*x^3*y^4 + 120960*x^3*y^3*z - 126720*x^3*y^3 + 120960*x^3*y^2*z^2 - 253920*x^3*y^2*z + 132960*x^3*y^2 + 53760*x^3*y*z^3 - 169600*x^3*y*z^2 + 177920*x^3*y*z - 62080*x^3*y + 8960*x^3*z^4 - 37760*x^3*z^3 + 59520*x^3*z^2 - 41600*x^3*z + 10888*x^3 - 40824*x^2*y^5 - 136080*x^2*y^4*z + 141480*x^2*y^4 - 181440*x^2*y^3*z^2 + 378000*x^2*y^3*z - 196560*x^2*y^3 - 120960*x^2*y^2*z^3 + 378720*x^2*y^2*z^2 - 394560*x^2*y^2*z + 136800*x^2*y^2 - 40320*x^2*y*z^4 + 168640*x^2*y*z^3 - 264000*x^2*y*z^2 + 183360*x^2*y*z - 47720*x^2*y - 5376*x^2*z^5 + 28160*x^2*z^4 - 58880*x^2*z^3 + 61440*x^2*z^2 - 32024*x^2*z + 6704*x^2 + 20412*x*y^6 + 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5728*x^5*z + 3008*x^5 - 15120*x^4*y^3 - 30240*x^4*y^2*z + 31920*x^4*y^2 - 20160*x^4*y*z^2 + 42640*x^4*y*z - 22240*x^4*y - 4480*x^4*z^3 + 14240*x^4*z^2 - 14880*x^4*z + 5120*x^4 + 22680*x^3*y^4 + 60480*x^3*y^3*z - 63360*x^3*y^3 + 60480*x^3*y^2*z^2 - 126960*x^3*y^2*z + 65760*x^3*y^2 + 26880*x^3*y*z^3 - 84800*x^3*y*z^2 + 88000*x^3*y*z - 30080*x^3*y + 4480*x^3*z^4 - 18880*x^3*z^3 + 29440*x^3*z^2 - 20160*x^3*z + 5124*x^3 - 20412*x^2*y^5 - 68040*x^2*y^4*z + 70740*x^2*y^4 - 90720*x^2*y^3*z^2 + 189000*x^2*y^3*z - 97200*x^2*y^3 - 60480*x^2*y^2*z^3 + 189360*x^2*y^2*z^2 - 195120*x^2*y^2*z + 66240*x^2*y^2 - 20160*x^2*y*z^4 + 84320*x^2*y*z^3 - 130560*x^2*y*z^2 + 88800*x^2*y*z - 22420*x^2*y - 2688*x^2*z^5 + 14080*x^2*z^4 - 29120*x^2*z^3 + 29760*x^2*z^2 - 15052*x^2*z + 3034*x^2 + 10206*x*y^6 + 40824*x*y^5*z - 42120*x*y^5 + 68040*x*y^4*z^2 - 140670*x*y^4*z + 71820*x*y^4 + 60480*x*y^3*z^3 - 187920*x*y^3*z^2 + 192240*x*y^3*z - 64800*x*y^3 + 30240*x*y^2*z^4 - 125520*x*y^2*z^3 + 192960*x*y^2*z^2 - 130320*x*y^2*z + 32673*x*y^2 + 8064*x*y*z^5 - 41920*x*y*z^4 + 86080*x*y*z^3 - 87360*x*y*z^2 + 43880*x*y*z - 8785*x*y + 896*x*z^6 - 5600*x*z^5 + 14400*x*z^4 - 19520*x*z^3 + 14732*x*z^2 - 5907*x*z + 1006*x - 2187*y^7 - 10206*y^6*z + 10449*y^6 - 20412*y^5*z^2 + 41877*y^5*z - 21222*y^5 - 22680*y^4*z^3 + 69930*y^4*z^2 - 71010*y^4*z + 23760*y^4 - 15120*y^3*z^4 + 62280*y^3*z^3 - 95040*y^3*z^2 + 63720*y^3*z - 15858*y^3 - 6048*y^2*z^5 + 31200*y^2*z^4 - 63600*y^2*z^3 + 64080*y^2*z^2 - 31953*y^2*z + 6350*y^2 - 1344*y*z^6 + 8336*y*z^5 - 21280*y*z^4 + 28640*y*z^3 - 21460*y*z^2 + 8543*y*z - 1439*y - 128*z^7 + 928*z^6 - 2848*z^5 + 4800*z^4 - 4804*z^3 + 2873*z^2 - 970*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ 2, -3, -2 ], [ 2, -1, -1 ], [ 1, -2, -1 ] ], "source_shift": [ 2, 2, 2 ], "target_matrix": [ [ 1, 1, 2 ], [ 2, 1, 1 ], [ 1, 1, 1 ] ], "target_shift": [ -81, -361, -149 ], "inverse_recipe": { "p": 2, "q": 5, "f": "x**2", "g": "x**5", "h": "y*z + y", "polynomials": { "f": [ [ [ 2, 0, 0 ], 1 ] ], "g": [ [ [ 5, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 1, 0 ], 1 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0463762569997925 ], "surface_features": [ 4.746434258118333, 5.596113054915637, 4.46179855752511, 0.0, 4.850490764043363, 5343 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B136", "pair": 136, "family": "bijection", "mapping": "F(x,y,z)=(22*x^7 - 154*x^6*y - 308*x^6*z + 220*x^6 + 462*x^5*y^2 + 1848*x^5*y*z - 1320*x^5*y + 1848*x^5*z^2 - 2618*x^5*z + 927*x^5 - 770*x^4*y^3 - 4620*x^4*y^2*z + 3300*x^4*y^2 - 9240*x^4*y*z^2 + 13090*x^4*y*z - 4635*x^4*y - 6160*x^4*z^3 + 12980*x^4*z^2 - 9138*x^4*z + 2130*x^4 + 770*x^3*y^4 + 6160*x^3*y^3*z - 4400*x^3*y^3 + 18480*x^3*y^2*z^2 - 26180*x^3*y^2*z + 9270*x^3*y^2 + 24640*x^3*y*z^3 - 51920*x^3*y*z^2 + 36552*x^3*y*z - 8520*x^3*y + 12320*x^3*z^4 - 34320*x^3*z^3 + 36024*x^3*z^2 - 16732*x^3*z + 2968*x^3 - 462*x^2*y^5 - 4620*x^2*y^4*z + 3300*x^2*y^4 - 18480*x^2*y^3*z^2 + 26180*x^2*y^3*z - 9270*x^2*y^3 - 36960*x^2*y^2*z^3 + 77880*x^2*y^2*z^2 - 54828*x^2*y^2*z + 12780*x^2*y^2 - 36960*x^2*y*z^4 + 102960*x^2*y*z^3 - 108072*x^2*y*z^2 + 50196*x^2*y*z - 8882*x^2*y - 14784*x^2*z^5 + 51040*x^2*z^4 - 70992*x^2*z^3 + 49272*x^2*z^2 - 17368*x^2*z + 2728*x^2 + 154*x*y^6 + 1848*x*y^5*z - 1320*x*y^5 + 9240*x*y^4*z^2 - 13090*x*y^4*z + 4635*x*y^4 + 24640*x*y^3*z^3 - 51920*x*y^3*z^2 + 36552*x*y^3*z - 8520*x*y^3 + 36960*x*y^2*z^4 - 102960*x*y^2*z^3 + 108072*x*y^2*z^2 - 50196*x*y^2*z + 8860*x*y^2 + 29568*x*y*z^5 - 102080*x*y*z^4 + 141984*x*y*z^3 - 98544*x*y*z^2 + 34648*x*y*z - 5368*x*y + 9856*x*z^6 - 40480*x*z^5 + 69936*x*z^4 - 64464*x*z^3 + 33856*x*z^2 - 10274*x*z + 1697*x - 22*y^7 - 308*y^6*z + 220*y^6 - 1848*y^5*z^2 + 2618*y^5*z - 927*y^5 - 6160*y^4*z^3 + 12980*y^4*z^2 - 9138*y^4*z + 2130*y^4 - 12320*y^3*z^4 + 34320*y^3*z^3 - 36024*y^3*z^2 + 16732*y^3*z - 2946*y^3 - 14784*y^2*z^5 + 51040*y^2*z^4 - 70992*y^2*z^3 + 49272*y^2*z^2 - 17280*y^2*z + 2640*y^2 - 9856*y*z^6 + 40480*y*z^5 - 69936*y*z^4 + 64464*y*z^3 - 33768*y*z^2 + 10120*y*z - 1606*y - 2816*z^7 + 13376*z^6 - 27552*z^5 + 31616*z^4 - 21984*z^3 + 9724*z^2 - 2902*z + 1, -11*x^7 + 77*x^6*y + 154*x^6*z - 110*x^6 - 231*x^5*y^2 - 924*x^5*y*z + 660*x^5*y - 924*x^5*z^2 + 1309*x^5*z - 463*x^5 + 385*x^4*y^3 + 2310*x^4*y^2*z - 1650*x^4*y^2 + 4620*x^4*y*z^2 - 6545*x^4*y*z + 2315*x^4*y + 3080*x^4*z^3 - 6490*x^4*z^2 + 4564*x^4*z - 1062*x^4 - 385*x^3*y^4 - 3080*x^3*y^3*z + 2200*x^3*y^3 - 9240*x^3*y^2*z^2 + 13090*x^3*y^2*z - 4630*x^3*y^2 - 12320*x^3*y*z^3 + 25960*x^3*y*z^2 - 18256*x^3*y*z + 4248*x^3*y - 6160*x^3*z^4 + 17160*x^3*z^3 - 17992*x^3*z^2 + 8342*x^3*z - 1477*x^3 + 231*x^2*y^5 + 2310*x^2*y^4*z - 1650*x^2*y^4 + 9240*x^2*y^3*z^2 - 13090*x^2*y^3*z + 4630*x^2*y^3 + 18480*x^2*y^2*z^3 - 38940*x^2*y^2*z^2 + 27384*x^2*y^2*z - 6372*x^2*y^2 + 18480*x^2*y*z^4 - 51480*x^2*y*z^3 + 53976*x^2*y*z^2 - 25026*x^2*y*z + 4420*x^2*y + 7392*x^2*z^5 - 25520*x^2*z^4 + 35456*x^2*z^3 - 24564*x^2*z^2 + 8642*x^2*z - 1354*x^2 - 77*x*y^6 - 924*x*y^5*z + 660*x*y^5 - 4620*x*y^4*z^2 + 6545*x*y^4*z - 2315*x*y^4 - 12320*x*y^3*z^3 + 25960*x*y^3*z^2 - 18256*x*y^3*z + 4248*x*y^3 - 18480*x*y^2*z^4 + 51480*x*y^2*z^3 - 53976*x*y^2*z^2 + 25026*x*y^2*z - 4409*x*y^2 - 14784*x*y*z^5 + 51040*x*y*z^4 - 70912*x*y*z^3 + 49128*x*y*z^2 - 17240*x*y*z + 2664*x*y - 4928*x*z^6 + 20240*x*z^5 - 34928*x*z^4 + 32136*x*z^3 - 16844*x*z^2 + 5097*x*z - 834*x + 11*y^7 + 154*y^6*z - 110*y^6 + 924*y^5*z^2 - 1309*y^5*z + 463*y^5 + 3080*y^4*z^3 - 6490*y^4*z^2 + 4564*y^4*z - 1062*y^4 + 6160*y^3*z^4 - 17160*y^3*z^3 + 17992*y^3*z^2 - 8342*y^3*z + 1466*y^3 + 7392*y^2*z^5 - 25520*y^2*z^4 + 35456*y^2*z^3 - 24564*y^2*z^2 + 8598*y^2*z - 1310*y^2 + 4928*y*z^6 - 20240*y*z^5 + 34928*y*z^4 - 32136*y*z^3 + 16800*y*z^2 - 5020*y*z + 789*y + 1408*z^7 - 6688*z^6 + 13760*z^5 - 15760*z^4 + 10936*z^3 - 4822*z^2 + 1426*z - 1, -11*x^7 + 77*x^6*y + 154*x^6*z - 110*x^6 - 231*x^5*y^2 - 924*x^5*y*z + 660*x^5*y - 924*x^5*z^2 + 1309*x^5*z - 463*x^5 + 385*x^4*y^3 + 2310*x^4*y^2*z - 1650*x^4*y^2 + 4620*x^4*y*z^2 - 6545*x^4*y*z + 2315*x^4*y + 3080*x^4*z^3 - 6490*x^4*z^2 + 4564*x^4*z - 1062*x^4 - 385*x^3*y^4 - 3080*x^3*y^3*z + 2200*x^3*y^3 - 9240*x^3*y^2*z^2 + 13090*x^3*y^2*z - 4630*x^3*y^2 - 12320*x^3*y*z^3 + 25960*x^3*y*z^2 - 18256*x^3*y*z + 4248*x^3*y - 6160*x^3*z^4 + 17160*x^3*z^3 - 17992*x^3*z^2 + 8342*x^3*z - 1477*x^3 + 231*x^2*y^5 + 2310*x^2*y^4*z - 1650*x^2*y^4 + 9240*x^2*y^3*z^2 - 13090*x^2*y^3*z + 4630*x^2*y^3 + 18480*x^2*y^2*z^3 - 38940*x^2*y^2*z^2 + 27384*x^2*y^2*z - 6372*x^2*y^2 + 18480*x^2*y*z^4 - 51480*x^2*y*z^3 + 53976*x^2*y*z^2 - 25026*x^2*y*z + 4420*x^2*y + 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explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ 1, -1, -2 ], [ 3, -2, -2 ], [ 1, -1, -1 ] ], "source_shift": [ 1, 2, 2 ], "target_matrix": [ [ 2, 3, 2 ], [ -1, -1, -2 ], [ -1, -1, -1 ] ], "target_shift": [ -528, 253, 261 ], "inverse_recipe": { "p": 5, "q": 2, "f": "x**5 + x**4 + x", "g": "x**2 + x", "h": "y*z + z", "polynomials": { "f": [ [ [ 5, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ], [ [ 4, 0, 0 ], 1 ] ], "g": [ [ [ 2, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 0, 1 ], 1 ] ] }, "h_multiplier": 11 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.1247722530024475 ], "surface_features": [ 4.208288143862554, 5.152239407036461, 3.8687619582120503, 0.0, 4.329224826046123, 5130 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B012", "pair": 12, "family": "bijection", "mapping": "F(x,y,z)=(-4*x^7 + 28*x^6*y + 30*x^6*z - 33*x^6 - 84*x^5*y^2 - 180*x^5*y*z + 198*x^5*y - 96*x^5*z^2 + 212*x^5*z - 179*x^5 + 140*x^4*y^3 + 450*x^4*y^2*z - 495*x^4*y^2 + 480*x^4*y*z^2 - 1060*x^4*y*z + 895*x^4*y + 170*x^4*z^3 - 565*x^4*z^2 + 1095*x^4*z - 544*x^4 - 140*x^3*y^4 - 600*x^3*y^3*z + 660*x^3*y^3 - 960*x^3*y^2*z^2 + 2120*x^3*y^2*z - 1790*x^3*y^2 - 680*x^3*y*z^3 + 2260*x^3*y*z^2 - 4380*x^3*y*z + 2176*x^3*y - 180*x^3*z^4 + 800*x^3*z^3 - 2750*x^3*z^2 + 2878*x^3*z - 911*x^3 + 84*x^2*y^5 + 450*x^2*y^4*z - 495*x^2*y^4 + 960*x^2*y^3*z^2 - 2120*x^2*y^3*z + 1790*x^2*y^3 + 1020*x^2*y^2*z^3 - 3390*x^2*y^2*z^2 + 6570*x^2*y^2*z - 3264*x^2*y^2 + 540*x^2*y*z^4 - 2400*x^2*y*z^3 + 8250*x^2*y*z^2 - 8634*x^2*y*z + 2733*x^2*y + 114*x^2*z^5 - 635*x^2*z^4 + 3550*x^2*z^3 - 5850*x^2*z^2 + 3751*x^2*z - 853*x^2 - 28*x*y^6 - 180*x*y^5*z + 198*x*y^5 - 480*x*y^4*z^2 + 1060*x*y^4*z - 895*x*y^4 - 680*x*y^3*z^3 + 2260*x*y^3*z^2 - 4380*x*y^3*z + 2176*x*y^3 - 540*x*y^2*z^4 + 2400*x*y^2*z^3 - 8250*x*y^2*z^2 + 8634*x*y^2*z - 2733*x*y^2 - 228*x*y*z^5 + 1270*x*y*z^4 - 7100*x*y*z^3 + 11700*x*y*z^2 - 7502*x*y*z + 1710*x*y - 40*x*z^6 + 268*x*z^5 - 2355*x*z^4 + 5402*x*z^3 - 5249*x*z^2 + 2380*x*z - 422*x + 4*y^7 + 30*y^6*z - 33*y^6 + 96*y^5*z^2 - 212*y^5*z + 179*y^5 + 170*y^4*z^3 - 565*y^4*z^2 + 1095*y^4*z - 544*y^4 + 180*y^3*z^4 - 800*y^3*z^3 + 2750*y^3*z^2 - 2878*y^3*z + 911*y^3 + 114*y^2*z^5 - 635*y^2*z^4 + 3550*y^2*z^3 - 5850*y^2*z^2 + 3751*y^2*z - 857*y^2 + 40*y*z^6 - 268*y*z^5 + 2355*y*z^4 - 5402*y*z^3 + 5249*y*z^2 - 2386*y*z + 427*y + 6*z^7 - 47*z^6 + 641*z^5 - 1906*z^4 + 2489*z^3 - 1687*z^2 + 590*z - 1, -8*x^7 + 56*x^6*y + 60*x^6*z - 67*x^6 - 168*x^5*y^2 - 360*x^5*y*z + 402*x^5*y - 192*x^5*z^2 + 430*x^5*z - 365*x^5 + 280*x^4*y^3 + 900*x^4*y^2*z - 1005*x^4*y^2 + 960*x^4*y*z^2 - 2150*x^4*y*z + 1825*x^4*y + 340*x^4*z^3 - 1145*x^4*z^2 + 2225*x^4*z - 1108*x^4 - 280*x^3*y^4 - 1200*x^3*y^3*z + 1340*x^3*y^3 - 1920*x^3*y^2*z^2 + 4300*x^3*y^2*z - 3650*x^3*y^2 - 1360*x^3*y*z^3 + 4580*x^3*y*z^2 - 8900*x^3*y*z + 4432*x^3*y - 360*x^3*z^4 + 1620*x^3*z^3 - 5570*x^3*z^2 + 5836*x^3*z - 1853*x^3 + 168*x^2*y^5 + 900*x^2*y^4*z - 1005*x^2*y^4 + 1920*x^2*y^3*z^2 - 4300*x^2*y^3*z + 3650*x^2*y^3 + 2040*x^2*y^2*z^3 - 6870*x^2*y^2*z^2 + 13350*x^2*y^2*z - 6648*x^2*y^2 + 1080*x^2*y*z^4 - 4860*x^2*y*z^3 + 16710*x^2*y*z^2 - 17508*x^2*y*z + 5559*x^2*y + 228*x^2*z^5 - 1285*x^2*z^4 + 7170*x^2*z^3 - 11820*x^2*z^2 + 7595*x^2*z - 1735*x^2 - 56*x*y^6 - 360*x*y^5*z + 402*x*y^5 - 960*x*y^4*z^2 + 2150*x*y^4*z - 1825*x*y^4 - 1360*x*y^3*z^3 + 4580*x*y^3*z^2 - 8900*x*y^3*z + 4432*x*y^3 - 1080*x*y^2*z^4 + 4860*x*y^2*z^3 - 16710*x*y^2*z^2 + 17508*x*y^2*z - 5559*x*y^2 - 456*x*y*z^5 + 2570*x*y*z^4 - 14340*x*y*z^3 + 23640*x*y*z^2 - 15190*x*y*z + 3478*x*y - 80*x*z^6 + 542*x*z^5 - 4745*x*z^4 + 10884*x*z^3 - 10591*x*z^2 + 4818*x*z - 869*x + 8*y^7 + 60*y^6*z - 67*y^6 + 192*y^5*z^2 - 430*y^5*z + 365*y^5 + 340*y^4*z^3 - 1145*y^4*z^2 + 2225*y^4*z - 1108*y^4 + 360*y^3*z^4 - 1620*y^3*z^3 + 5570*y^3*z^2 - 5836*y^3*z + 1853*y^3 + 228*y^2*z^5 - 1285*y^2*z^4 + 7170*y^2*z^3 - 11820*y^2*z^2 + 7595*y^2*z - 1743*y^2 + 80*y*z^6 - 542*y*z^5 + 4745*y*z^4 - 10884*y*z^3 + 10591*y*z^2 - 4830*y*z + 880*y + 12*z^7 - 95*z^6 + 1289*z^5 - 3832*z^4 + 5009*z^3 - 3403*z^2 + 1209*z - 1, 4*x^7 - 28*x^6*y - 30*x^6*z + 34*x^6 + 84*x^5*y^2 + 180*x^5*y*z - 204*x^5*y + 96*x^5*z^2 - 218*x^5*z + 186*x^5 - 140*x^4*y^3 - 450*x^4*y^2*z + 510*x^4*y^2 - 480*x^4*y*z^2 + 1090*x^4*y*z - 930*x^4*y - 170*x^4*z^3 + 580*x^4*z^2 - 1130*x^4*z + 564*x^4 + 140*x^3*y^4 + 600*x^3*y^3*z - 680*x^3*y^3 + 960*x^3*y^2*z^2 - 2180*x^3*y^2*z + 1860*x^3*y^2 + 680*x^3*y*z^3 - 2320*x^3*y*z^2 + 4520*x^3*y*z - 2256*x^3*y + 180*x^3*z^4 - 820*x^3*z^3 + 2820*x^3*z^2 - 2958*x^3*z + 942*x^3 - 84*x^2*y^5 - 450*x^2*y^4*z + 510*x^2*y^4 - 960*x^2*y^3*z^2 + 2180*x^2*y^3*z - 1860*x^2*y^3 - 1020*x^2*y^2*z^3 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613*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ 1, -1, -1 ], [ 1, -2, -1 ], [ 1, -1, -2 ] ], "source_shift": [ 1, 2, 1 ], "target_matrix": [ [ -1, -1, -2 ], [ -2, -3, -2 ], [ 1, 2, 1 ] ], "target_shift": [ 86, 187, -96 ], "inverse_recipe": { "p": 6, "q": 1, "f": "x**6 + x**5 + x**3 + x**2", "g": "x", "h": "y*z + z**5", "polynomials": { "f": [ [ [ 6, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ], [ [ 3, 0, 0 ], 1 ], [ [ 5, 0, 0 ], 1 ] ], "g": [ [ [ 1, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 0, 5 ], 1 ] ] }, "h_multiplier": 2 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 0.9678009829985967 ], "surface_features": [ 3.387200423948289, 4.373647472209218, 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"mapping": "F(x,y,z)=(256*x^7 + 2688*x^6*y - 1792*x^6*z - 1024*x^6 + 12096*x^5*y^2 - 16128*x^5*y*z - 9088*x^5*y + 5376*x^5*z^2 + 6080*x^5*z + 1728*x^5 + 30240*x^4*y^3 - 60480*x^4*y^2*z - 33600*x^4*y^2 + 40320*x^4*y*z^2 + 44960*x^4*y*z + 12640*x^4*y - 8960*x^4*z^3 - 15040*x^4*z^2 - 8480*x^4*z - 1632*x^4 + 45360*x^3*y^4 - 120960*x^3*y^3*z - 66240*x^3*y^3 + 120960*x^3*y^2*z^2 + 132960*x^3*y^2*z + 36960*x^3*y^2 - 53760*x^3*y*z^3 - 88960*x^3*y*z^2 - 49600*x^3*y*z - 9472*x^3*y + 8960*x^3*z^4 + 19840*x^3*z^3 + 16640*x^3*z^2 + 6368*x^3*z + 984*x^3 + 40824*x^2*y^5 - 136080*x^2*y^4*z - 73440*x^2*y^4 + 181440*x^2*y^3*z^2 + 196560*x^2*y^3*z + 54000*x^2*y^3 - 120960*x^2*y^2*z^3 - 197280*x^2*y^2*z^2 - 108720*x^2*y^2*z - 20592*x^2*y^2 + 40320*x^2*y*z^4 + 88000*x^2*y*z^3 + 72960*x^2*y*z^2 + 27696*x^2*y*z + 4256*x^2*y - 5376*x^2*z^5 - 14720*x^2*z^4 - 16320*x^2*z^3 - 9312*x^2*z^2 - 2864*x^2*z - 432*x^2 + 20412*x*y^6 - 81648*x*y^5*z - 43416*x*y^5 + 136080*x*y^4*z^2 + 145260*x*y^4*z + 39420*x*y^4 - 120960*x*y^3*z^3 - 194400*x*y^3*z^2 - 105840*x*y^3*z - 19872*x*y^3 + 60480*x*y^2*z^4 + 130080*x*y^2*z^3 + 106560*x*y^2*z^2 + 40104*x*y^2*z + 6126*x*y^2 - 16128*x*y*z^5 - 43520*x*y*z^4 - 47680*x*y*z^3 - 26976*x*y*z^2 - 8248*x*y*z - 1234*x*y + 1792*x*z^6 + 5824*x*z^5 + 8000*x*z^4 + 6048*x*z^3 + 2776*x*z^2 + 830*x*z + 150*x + 4374*y^7 - 20412*y^6*z - 10692*y^6 + 40824*y^5*z^2 + 42930*y^5*z + 11502*y^5 - 45360*y^4*z^3 - 71820*y^4*z^2 - 38610*y^4*z - 7182*y^4 + 30240*y^3*z^4 + 64080*y^3*z^3 + 51840*y^3*z^2 + 19332*y^3*z + 2934*y^3 - 12096*y^2*z^5 - 32160*y^2*z^4 - 34800*y^2*z^3 - 19512*y^2*z^2 - 5928*y^2*z - 877*y^2 + 2688*y*z^6 + 8608*y*z^5 + 11680*y*z^4 + 8752*y*z^3 + 3992*y*z^2 + 1182*y*z + 201*y - 256*z^7 - 960*z^6 - 1568*z^5 - 1472*z^4 - 896*z^3 - 398*z^2 - 137*z - 1, 256*x^7 + 2688*x^6*y - 1792*x^6*z - 1024*x^6 + 12096*x^5*y^2 - 16128*x^5*y*z - 9088*x^5*y + 5376*x^5*z^2 + 6080*x^5*z + 1792*x^5 + 30240*x^4*y^3 - 60480*x^4*y^2*z - 33600*x^4*y^2 + 40320*x^4*y*z^2 + 44960*x^4*y*z + 13120*x^4*y - 8960*x^4*z^3 - 15040*x^4*z^2 - 8800*x^4*z - 1792*x^4 + 45360*x^3*y^4 - 120960*x^3*y^3*z - 66240*x^3*y^3 + 120960*x^3*y^2*z^2 + 132960*x^3*y^2*z + 38400*x^3*y^2 - 53760*x^3*y*z^3 - 88960*x^3*y*z^2 - 51520*x^3*y*z - 10432*x^3*y + 8960*x^3*z^4 + 19840*x^3*z^3 + 17280*x^3*z^2 + 7008*x^3*z + 1144*x^3 + 40824*x^2*y^5 - 136080*x^2*y^4*z - 73440*x^2*y^4 + 181440*x^2*y^3*z^2 + 196560*x^2*y^3*z + 56160*x^2*y^3 - 120960*x^2*y^2*z^3 - 197280*x^2*y^2*z^2 - 113040*x^2*y^2*z - 22752*x^2*y^2 + 40320*x^2*y*z^4 + 88000*x^2*y*z^3 + 75840*x^2*y*z^2 + 30576*x^2*y*z + 4976*x^2*y - 5376*x^2*z^5 - 14720*x^2*z^4 - 16960*x^2*z^3 - 10272*x^2*z^2 - 3344*x^2*z - 512*x^2 + 20412*x*y^6 - 81648*x*y^5*z - 43416*x*y^5 + 136080*x*y^4*z^2 + 145260*x*y^4*z + 41040*x*y^4 - 120960*x*y^3*z^3 - 194400*x*y^3*z^2 - 110160*x*y^3*z - 22032*x*y^3 + 60480*x*y^2*z^4 + 130080*x*y^2*z^3 + 110880*x*y^2*z^2 + 44424*x*y^2*z + 7206*x*y^2 - 16128*x*y*z^5 - 43520*x*y*z^4 - 49600*x*y*z^3 - 29856*x*y*z^2 - 9688*x*y*z - 1474*x*y + 1792*x*z^6 + 5824*x*z^5 + 8320*x*z^4 + 6688*x*z^3 + 3256*x*z^2 + 990*x*z + 172*x + 4374*y^7 - 20412*y^6*z - 10692*y^6 + 40824*y^5*z^2 + 42930*y^5*z + 11988*y^5 - 45360*y^4*z^3 - 71820*y^4*z^2 - 40230*y^4*z - 7992*y^4 + 30240*y^3*z^4 + 64080*y^3*z^3 + 54000*y^3*z^2 + 21492*y^3*z + 3474*y^3 - 12096*y^2*z^5 - 32160*y^2*z^4 - 36240*y^2*z^3 - 21672*y^2*z^2 - 7008*y^2*z - 1057*y^2 + 2688*y*z^6 + 8608*y*z^5 + 12160*y*z^4 + 9712*y*z^3 + 4712*y*z^2 + 1422*y*z + 239*y - 256*z^7 - 960*z^6 - 1632*z^5 - 1632*z^4 - 1056*z^3 - 478*z^2 - 161*z - 1, -128*x^7 - 1344*x^6*y + 896*x^6*z + 512*x^6 - 6048*x^5*y^2 + 8064*x^5*y*z + 4544*x^5*y - 2688*x^5*z^2 - 3040*x^5*z - 864*x^5 - 15120*x^4*y^3 + 30240*x^4*y^2*z + 16800*x^4*y^2 - 20160*x^4*y*z^2 - 22480*x^4*y*z - 6320*x^4*y + 4480*x^4*z^3 + 7520*x^4*z^2 + 4240*x^4*z + 816*x^4 - 22680*x^3*y^4 + 60480*x^3*y^3*z + 33120*x^3*y^3 - 60480*x^3*y^2*z^2 - 66480*x^3*y^2*z - 18480*x^3*y^2 + 26880*x^3*y*z^3 + 44480*x^3*y*z^2 + 24800*x^3*y*z + 4736*x^3*y - 4480*x^3*z^4 - 9920*x^3*z^3 - 8320*x^3*z^2 - 3184*x^3*z - 492*x^3 - 20412*x^2*y^5 + 68040*x^2*y^4*z + 36720*x^2*y^4 - 90720*x^2*y^3*z^2 - 98280*x^2*y^3*z - 27000*x^2*y^3 + 60480*x^2*y^2*z^3 + 98640*x^2*y^2*z^2 + 54360*x^2*y^2*z + 10296*x^2*y^2 - 20160*x^2*y*z^4 - 44000*x^2*y*z^3 - 36480*x^2*y*z^2 - 13848*x^2*y*z - 2128*x^2*y + 2688*x^2*z^5 + 7360*x^2*z^4 + 8160*x^2*z^3 + 4656*x^2*z^2 + 1432*x^2*z + 218*x^2 - 10206*x*y^6 + 40824*x*y^5*z + 21708*x*y^5 - 68040*x*y^4*z^2 - 72630*x*y^4*z - 19710*x*y^4 + 60480*x*y^3*z^3 + 97200*x*y^3*z^2 + 52920*x*y^3*z + 9936*x*y^3 - 30240*x*y^2*z^4 - 65040*x*y^2*z^3 - 53280*x*y^2*z^2 - 20052*x*y^2*z - 3063*x*y^2 + 8064*x*y*z^5 + 21760*x*y*z^4 + 23840*x*y*z^3 + 13488*x*y*z^2 + 4124*x*y*z + 623*x*y - 896*x*z^6 - 2912*x*z^5 - 4000*x*z^4 - 3024*x*z^3 - 1388*x*z^2 - 419*x*z - 78*x - 2187*y^7 + 10206*y^6*z + 5346*y^6 - 20412*y^5*z^2 - 21465*y^5*z - 5751*y^5 + 22680*y^4*z^3 + 35910*y^4*z^2 + 19305*y^4*z + 3591*y^4 - 15120*y^3*z^4 - 32040*y^3*z^3 - 25920*y^3*z^2 - 9666*y^3*z - 1467*y^3 + 6048*y^2*z^5 + 16080*y^2*z^4 + 17400*y^2*z^3 + 9756*y^2*z^2 + 2964*y^2*z + 443*y^2 - 1344*y*z^6 - 4304*y*z^5 - 5840*y*z^4 - 4376*y*z^3 - 1996*y*z^2 - 597*y*z - 104*y + 128*z^7 + 480*z^6 + 784*z^5 + 736*z^4 + 448*z^3 + 201*z^2 + 71*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -2, -3, 2 ], [ -2, -1, 1 ], [ -1, -2, 1 ] ], "source_shift": [ 1, 2, 2 ], "target_matrix": [ [ -2, -3, -2 ], [ -2, -1, -1 ], [ 1, 2, 1 ] ], "target_shift": [ 30, 34, -16 ], "inverse_recipe": { "p": 2, "q": 5, "f": "x**2", "g": "x**5 + x**2 + x", "h": "y*z + z", "polynomials": { "f": [ [ [ 2, 0, 0 ], 1 ] ], "g": [ [ [ 5, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 0, 1 ], 1 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0313812650019827 ], "surface_features": [ 4.5005239419087735, 5.295083059251656, 4.154849841520744, 0.0, 4.6221446400913715, 5225 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B053", "pair": 53, "family": "bijection", "mapping": "F(x,y,z)=(-128*x^7 + 448*x^6*y + 448*x^6*z - 1024*x^6 - 672*x^5*y^2 - 1344*x^5*y*z + 3072*x^5*y - 672*x^5*z^2 + 3072*x^5*z - 3504*x^5 + 560*x^4*y^3 + 1680*x^4*y^2*z - 3840*x^4*y^2 + 1680*x^4*y*z^2 - 7680*x^4*y*z + 8768*x^4*y + 560*x^4*z^3 - 3840*x^4*z^2 + 8768*x^4*z - 6720*x^4 - 280*x^3*y^4 - 1120*x^3*y^3*z + 2560*x^3*y^3 - 1680*x^3*y^2*z^2 + 7680*x^3*y^2*z - 8776*x^3*y^2 - 1120*x^3*y*z^3 + 7680*x^3*y*z^2 - 17552*x^3*y*z + 13488*x^3*y - 280*x^3*z^4 + 2560*x^3*z^3 - 8776*x^3*z^2 + 13496*x^3*z - 7856*x^3 + 84*x^2*y^5 + 420*x^2*y^4*z - 960*x^2*y^4 + 840*x^2*y^3*z^2 - 3840*x^2*y^3*z + 4392*x^2*y^3 + 840*x^2*y^2*z^3 - 5760*x^2*y^2*z^2 + 13176*x^2*y^2*z - 10152*x^2*y^2 + 420*x^2*y*z^4 - 3840*x^2*y*z^3 + 13176*x^2*y*z^2 - 20316*x^2*y*z + 11880*x^2*y + 84*x^2*z^5 - 960*x^2*z^4 + 4392*x^2*z^3 - 10164*x^2*z^2 + 11904*x^2*z - 5634*x^2 - 14*x*y^6 - 84*x*y^5*z + 192*x*y^5 - 210*x*y^4*z^2 + 960*x*y^4*z - 1099*x*y^4 - 280*x*y^3*z^3 + 1920*x*y^3*z^2 - 4396*x*y^3*z + 3396*x*y^3 - 210*x*y^2*z^4 + 1920*x*y^2*z^3 - 6594*x*y^2*z^2 + 10194*x*y^2*z - 5988*x*y^2 - 84*x*y*z^5 + 960*x*y*z^4 - 4396*x*y*z^3 + 10200*x*y*z^2 - 12000*x*y*z + 5720*x*y - 14*x*z^6 + 192*x*z^5 - 1099*x*z^4 + 3402*x*z^3 - 6012*x*z^2 + 5747*x*z - 2316*x + y^7 + 7*y^6*z - 16*y^6 + 21*y^5*z^2 - 96*y^5*z + 110*y^5 + 35*y^4*z^3 - 240*y^4*z^2 + 550*y^4*z - 426*y^4 + 35*y^3*z^4 - 320*y^3*z^3 + 1100*y^3*z^2 - 1705*y^3*z + 1006*y^3 + 21*y^2*z^5 - 240*y^2*z^4 + 1100*y^2*z^3 - 2559*y^2*z^2 + 3024*y^2*z - 1452*y^2 + 7*y*z^6 - 96*y*z^5 + 550*y*z^4 - 1707*y*z^3 + 3030*y*z^2 - 2918*y*z + 1189*y + z^7 - 16*z^6 + 110*z^5 - 427*z^4 + 1012*z^3 - 1466*z^2 + 1199*z - 1, -128*x^7 + 448*x^6*y + 448*x^6*z - 1024*x^6 - 672*x^5*y^2 - 1344*x^5*y*z + 3072*x^5*y - 672*x^5*z^2 + 3072*x^5*z - 3504*x^5 + 560*x^4*y^3 + 1680*x^4*y^2*z - 3840*x^4*y^2 + 1680*x^4*y*z^2 - 7680*x^4*y*z + 8768*x^4*y + 560*x^4*z^3 - 3840*x^4*z^2 + 8768*x^4*z - 6720*x^4 - 280*x^3*y^4 - 1120*x^3*y^3*z + 2560*x^3*y^3 - 1680*x^3*y^2*z^2 + 7680*x^3*y^2*z - 8776*x^3*y^2 - 1120*x^3*y*z^3 + 7680*x^3*y*z^2 - 17552*x^3*y*z + 13488*x^3*y - 280*x^3*z^4 + 2560*x^3*z^3 - 8776*x^3*z^2 + 13496*x^3*z - 7848*x^3 + 84*x^2*y^5 + 420*x^2*y^4*z - 960*x^2*y^4 + 840*x^2*y^3*z^2 - 3840*x^2*y^3*z + 4392*x^2*y^3 + 840*x^2*y^2*z^3 - 5760*x^2*y^2*z^2 + 13176*x^2*y^2*z - 10152*x^2*y^2 + 420*x^2*y*z^4 - 3840*x^2*y*z^3 + 13176*x^2*y*z^2 - 20316*x^2*y*z + 11868*x^2*y + 84*x^2*z^5 - 960*x^2*z^4 + 4392*x^2*z^3 - 10164*x^2*z^2 + 11892*x^2*z - 5610*x^2 - 14*x*y^6 - 84*x*y^5*z + 192*x*y^5 - 210*x*y^4*z^2 + 960*x*y^4*z - 1099*x*y^4 - 280*x*y^3*z^3 + 1920*x*y^3*z^2 - 4396*x*y^3*z + 3396*x*y^3 - 210*x*y^2*z^4 + 1920*x*y^2*z^3 - 6594*x*y^2*z^2 + 10194*x*y^2*z - 5982*x*y^2 - 84*x*y*z^5 + 960*x*y*z^4 - 4396*x*y*z^3 + 10200*x*y*z^2 - 11988*x*y*z + 5696*x*y - 14*x*z^6 + 192*x*z^5 - 1099*x*z^4 + 3402*x*z^3 - 6006*x*z^2 + 5723*x*z - 2289*x + y^7 + 7*y^6*z - 16*y^6 + 21*y^5*z^2 - 96*y^5*z + 110*y^5 + 35*y^4*z^3 - 240*y^4*z^2 + 550*y^4*z - 426*y^4 + 35*y^3*z^4 - 320*y^3*z^3 + 1100*y^3*z^2 - 1705*y^3*z + 1005*y^3 + 21*y^2*z^5 - 240*y^2*z^4 + 1100*y^2*z^3 - 2559*y^2*z^2 + 3021*y^2*z - 1446*y^2 + 7*y*z^6 - 96*y*z^5 + 550*y*z^4 - 1707*y*z^3 + 3027*y*z^2 - 2906*y*z + 1175*y + z^7 - 16*z^6 + 110*z^5 - 427*z^4 + 1011*z^3 - 1460*z^2 + 1185*z - 1, 256*x^7 - 896*x^6*y - 896*x^6*z + 2048*x^6 + 1344*x^5*y^2 + 2688*x^5*y*z - 6144*x^5*y + 1344*x^5*z^2 - 6144*x^5*z + 7008*x^5 - 1120*x^4*y^3 - 3360*x^4*y^2*z + 7680*x^4*y^2 - 3360*x^4*y*z^2 + 15360*x^4*y*z - 17536*x^4*y - 1120*x^4*z^3 + 7680*x^4*z^2 - 17536*x^4*z + 13472*x^4 + 560*x^3*y^4 + 2240*x^3*y^3*z - 5120*x^3*y^3 + 3360*x^3*y^2*z^2 - 15360*x^3*y^2*z + 17552*x^3*y^2 + 2240*x^3*y*z^3 - 15360*x^3*y*z^2 + 35104*x^3*y*z - 27040*x^3*y + 560*x^3*z^4 - 5120*x^3*z^3 + 17552*x^3*z^2 - 27056*x^3*z + 15832*x^3 - 168*x^2*y^5 - 840*x^2*y^4*z + 1920*x^2*y^4 - 1680*x^2*y^3*z^2 + 7680*x^2*y^3*z - 8784*x^2*y^3 - 1680*x^2*y^2*z^3 + 11520*x^2*y^2*z^2 - 26352*x^2*y^2*z + 20352*x^2*y^2 - 840*x^2*y*z^4 + 7680*x^2*y*z^3 - 26352*x^2*y*z^2 + 40728*x^2*y*z - 23940*x^2*y - 168*x^2*z^5 + 1920*x^2*z^4 - 8784*x^2*z^3 + 20376*x^2*z^2 - 23988*x^2*z + 11436*x^2 + 28*x*y^6 + 168*x*y^5*z - 384*x*y^5 + 420*x*y^4*z^2 - 1920*x*y^4*z + 2198*x*y^4 + 560*x*y^3*z^3 - 3840*x*y^3*z^2 + 8792*x*y^3*z - 6808*x*y^3 + 420*x*y^2*z^4 - 3840*x*y^2*z^3 + 13188*x*y^2*z^2 - 20436*x*y^2*z + 12066*x*y^2 + 168*x*y*z^5 - 1920*x*y*z^4 + 8792*x*y*z^3 - 20448*x*y*z^2 + 24180*x*y*z - 11608*x*y + 28*x*z^6 - 384*x*z^5 + 2198*x*z^4 - 6820*x*z^3 + 12114*x*z^2 - 11662*x*z + 4733*x - 2*y^7 - 14*y^6*z + 32*y^6 - 42*y^5*z^2 + 192*y^5*z - 220*y^5 - 70*y^4*z^3 + 480*y^4*z^2 - 1100*y^4*z + 854*y^4 - 70*y^3*z^4 + 640*y^3*z^3 - 2200*y^3*z^2 + 3418*y^3*z - 2027*y^3 - 42*y^2*z^5 + 480*y^2*z^4 - 2200*y^2*z^3 + 5130*y^2*z^2 - 6093*y^2*z + 2946*y^2 - 14*y*z^6 + 192*y*z^5 - 1100*y*z^4 + 3422*y*z^3 - 6105*y*z^2 + 5920*y*z - 2428*y - 2*z^7 + 32*z^6 - 220*z^5 + 856*z^4 - 2039*z^3 + 2974*z^2 - 2450*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ 2, -1, -1 ], [ 1, -1, -1 ], [ 1, -1, -2 ] ], "source_shift": [ 2, 1, 2 ], "target_matrix": [ [ -1, -2, -1 ], [ -1, -1, -1 ], [ 2, 1, 1 ] ], "target_shift": [ 424, 413, -872 ], "inverse_recipe": { "p": 3, "q": 4, "f": "x**3 + x", "g": "x**4 + x**3 + x", "h": "y**2 + y*z + z", "polynomials": { "f": [ [ [ 3, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "g": [ [ [ 4, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ], [ [ 3, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 0, 1 ], 1 ], [ [ 0, 2, 0 ], 1 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.121422968000843 ], "surface_features": [ 3.6418515578861315, 4.609893084029282, 3.2285286773571817, 0.0, 3.7882669065936803, 4893 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C189", "pair": 189, "family": "counterexample", "mapping": "F(x,y,z)=(16*x^7 + 88*x^6*y - 104*x^6*z - 200*x^6 + 204*x^5*y^2 - 480*x^5*y*z - 924*x^5*y + 276*x^5*z^2 + 1068*x^5*z + 1044*x^5 + 258*x^4*y^3 - 906*x^4*y^2*z - 1746*x^4*y^2 + 1038*x^4*y*z^2 + 4008*x^4*y*z + 3930*x^4*y - 390*x^4*z^3 - 2262*x^4*z^2 - 4434*x^4*z - 2946*x^4 + 192*x^3*y^4 - 894*x^3*y^3*z - 1725*x^3*y^3 + 1530*x^3*y^2*z^2 + 5895*x^3*y^2*z + 5796*x^3*y^2 - 1146*x^3*y*z^3 - 6615*x^3*y*z^2 - 12966*x^3*y*z - 8647*x^3*y + 318*x^3*z^4 + 2445*x^3*z^3 + 7170*x^3*z^2 + 9531*x^3*z + 4848*x^3 + 84*x^2*y^5 - 486*x^2*y^4*z - 939*x^2*y^4 + 1104*x^2*y^3*z^2 + 4245*x^2*y^3*z + 4182*x^2*y^3 - 1236*x^2*y^2*z^3 - 7101*x^2*y^2*z^2 - 13911*x^2*y^2*z - 9297*x^2*y^2 + 684*x^2*y*z^4 + 5223*x^2*y*z^3 + 15276*x^2*y*z^2 + 20313*x^2*y*z + 10356*x^2*y - 150*x^2*z^5 - 1428*x^2*z^4 - 5547*x^2*z^3 - 11016*x^2*z^2 - 11193*x^2*z - 4652*x^2 + 20*x*y^6 - 138*x*y^5*z - 267*x*y^5 + 390*x*y^4*z^2 + 1497*x*y^4*z + 1476*x*y^4 - 580*x*y^3*z^3 - 3318*x*y^3*z^2 - 6492*x*y^3*z - 4338*x*y^3 + 480*x*y^2*z^4 + 3642*x*y^2*z^3 + 10620*x*y^2*z^2 + 14100*x*y^2*z + 7182*x*y^2 - 210*x*y*z^5 - 1983*x*y*z^4 - 7668*x*y*z^3 - 15186*x*y*z^2 - 15396*x*y*z - 6394*x*y + 38*x*z^6 + 429*x*z^5 + 2064*x*z^4 + 5424*x*z^3 + 8214*x*z^2 + 6802*x*z + 2414*x + 2*y^7 - 16*y^6*z - 31*y^6 + 54*y^5*z^2 + 207*y^5*z + 204*y^5 - 100*y^4*z^3 - 570*y^4*z^2 - 1113*y^4*z - 742*y^4 + 110*y^3*z^4 + 830*y^3*z^3 + 2412*y^3*z^2 + 3192*y^3*z + 1620*y^3 - 72*y^2*z^5 - 675*y^2*z^4 - 2598*y^2*z^3 - 5124*y^2*z^2 - 5172*y^2*z - 2138*y^2 + 26*y*z^6 + 291*y*z^5 + 1392*y*z^4 + 3640*y*z^3 + 5484*y*z^2 + 4516*y*z + 1592*y - 4*z^7 - 52*z^6 - 297*z^5 - 966*z^4 - 1932*z^3 - 2378*z^2 - 1672*z - 1, -8*x^7 - 44*x^6*y + 52*x^6*z + 136*x^6 - 102*x^5*y^2 + 240*x^5*y*z + 624*x^5*y - 138*x^5*z^2 - 732*x^5*z - 846*x^5 - 129*x^4*y^3 + 453*x^4*y^2*z + 1170*x^4*y^2 - 519*x^4*y*z^2 - 2724*x^4*y*z - 3153*x^4*y + 195*x^4*z^3 + 1554*x^4*z^2 + 3603*x^4*z + 2666*x^4 - 96*x^3*y^4 + 447*x^3*y^3*z + 1146*x^3*y^3 - 765*x^3*y^2*z^2 - 3969*x^3*y^2*z - 4599*x^3*y^2 + 573*x^3*y*z^3 + 4500*x^3*y*z^2 + 10407*x^3*y*z + 7729*x^3*y - 159*x^3*z^4 - 1677*x^3*z^3 - 5808*x^3*z^2 - 8590*x^3*z - 4708*x^3 - 42*x^2*y^5 + 243*x^2*y^4*z + 618*x^2*y^4 - 552*x^2*y^3*z^2 - 2829*x^2*y^3*z - 3279*x^2*y^3 + 618*x^2*y^2*z^3 + 4779*x^2*y^2*z^2 + 11019*x^2*y^2*z + 8199*x^2*y^2 - 342*x^2*y*z^4 - 3543*x^2*y*z^3 - 12201*x^2*y*z^2 - 18039*x^2*y*z - 9903*x^2*y + 75*x^2*z^5 + 975*x^2*z^4 + 4461*x^2*z^3 + 9840*x^2*z^2 + 10752*x^2*z + 4705*x^2 - 10*x*y^6 + 69*x*y^5*z + 174*x*y^5 - 195*x*y^4*z^2 - 987*x*y^4*z - 1143*x*y^4 + 290*x*y^3*z^3 + 2208*x*y^3*z^2 + 5073*x*y^3*z + 3772*x*y^3 - 240*x*y^2*z^4 - 2442*x*y^2*z^3 - 8361*x*y^2*z^2 - 12333*x*y^2*z - 6756*x*y^2 + 105*x*y*z^5 + 1338*x*y*z^4 + 6075*x*y*z^3 + 13350*x*y*z^2 + 14535*x*y*z + 6341*x*y - 19*x*z^6 - 291*x*z^5 - 1644*x*z^4 - 4789*x*z^3 - 7779*x*z^2 - 6758*x*z - 2473*x - y^7 + 8*y^6*z + 20*y^6 - 27*y^5*z^2 - 135*y^5*z - 156*y^5 + 50*y^4*z^3 + 375*y^4*z^2 + 858*y^4*z + 636*y^4 - 55*y^3*z^4 - 550*y^3*z^3 - 1872*y^3*z^2 - 2750*y^3*z - 1499*y^3 + 36*y^2*z^5 + 450*y^2*z^4 + 2028*y^2*z^3 + 4434*y^2*z^2 + 4800*y^2*z + 2080*y^2 - 13*y*z^6 - 195*y*z^5 - 1092*y*z^4 - 3162*y*z^3 - 5103*y*z^2 - 4400*y*z - 1593*y + 2*z^7 + 35*z^6 + 234*z^5 + 842*z^4 + 1802*z^3 + 2320*z^2 + 1673*z + 1, 16*x^7 + 88*x^6*y - 104*x^6*z - 248*x^6 + 204*x^5*y^2 - 480*x^5*y*z - 1140*x^5*y + 276*x^5*z^2 + 1332*x^5*z + 1476*x^5 + 258*x^4*y^3 - 906*x^4*y^2*z - 2142*x^4*y^2 + 1038*x^4*y*z^2 + 4968*x^4*y*z + 5514*x^4*y - 390*x^4*z^3 - 2826*x^4*z^2 - 6282*x^4*z - 4534*x^4 + 192*x^3*y^4 - 894*x^3*y^3*z - 2103*x^3*y^3 + 1530*x^3*y^2*z^2 + 7257*x^3*y^2*z + 8064*x^3*y^2 - 1146*x^3*y*z^3 - 8205*x^3*y*z^2 - 18198*x^3*y*z - 13181*x^3*y + 318*x^3*z^4 + 3051*x^3*z^3 + 10134*x^3*z^2 + 14621*x^3*z + 7880*x^3 + 84*x^2*y^5 - 486*x^2*y^4*z - 1137*x^2*y^4 + 1104*x^2*y^3*z^2 + 5187*x^2*y^3*z + 5766*x^2*y^3 - 1236*x^2*y^2*z^3 - 8739*x^2*y^2*z^2 - 19329*x^2*y^2*z - 14025*x^2*y^2 + 684*x^2*y*z^4 + 6465*x^2*y*z^3 + 21360*x^2*y*z^2 + 30807*x^2*y*z + 16632*x^2*y - 150*x^2*z^5 - 1776*x^2*z^4 - 7797*x^2*z^3 - 16782*x^2*z^2 - 18039*x^2*z - 7802*x^2 + 20*x*y^6 - 138*x*y^5*z - 321*x*y^5 + 390*x*y^4*z^2 + 1815*x*y^4*z + 2016*x*y^4 - 580*x*y^3*z^3 - 4050*x*y^3*z^2 - 8928*x*y^3*z - 6473*x*y^3 + 480*x*y^2*z^4 + 4470*x*y^2*z^3 + 14688*x*y^2*z^2 + 21135*x*y^2*z + 11388*x*y^2 - 210*x*y*z^5 - 2445*x*y*z^4 - 10656*x*y*z^3 - 22851*x*y*z^2 - 24480*x*y*z - 10561*x*y + 38*x*z^6 + 531*x*z^5 + 2880*x*z^4 + 8189*x*z^3 + 13092*x*z^2 + 11251*x*z + 4090*x + 2*y^7 - 16*y^6*z - 37*y^6 + 54*y^5*z^2 + 249*y^5*z + 276*y^5 - 100*y^4*z^3 - 690*y^4*z^2 - 1515*y^4*z - 1095*y^4 + 110*y^3*z^4 + 1010*y^3*z^3 + 3300*y^3*z^2 + 4729*y^3*z + 2536*y^3 - 72*y^2*z^5 - 825*y^2*z^4 - 3570*y^2*z^3 - 7617*y^2*z^2 - 8115*y^2*z - 3479*y^2 + 26*y*z^6 + 357*y*z^5 + 1920*y*z^4 + 5427*y*z^3 + 8622*y*z^2 + 7357*y*z + 2648*y - 4*z^7 - 64*z^6 - 411*z^5 - 1444*z^4 - 3043*z^3 - 3878*z^2 - 2781*z - 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -2, -1, 1 ], [ -1, -1, 1 ], [ -1, -1, 2 ] ], "source_shift": [ 1, 2, 2 ], "target_matrix": [ [ -2, -1, -1 ], [ 1, 2, 1 ], [ -2, -3, -2 ] ], "target_shift": [ 517, -519, 863 ], "collision_preimages": [ [ "-1", "11/4", "-1/4" ], [ "-7/2", "15", "8" ], [ "3/2", "4", "5" ] ], "collision_image": [ "1035/2", "-2077/4", "1727/2" ] }, "probe_target": "No", "jev_answers": [ "Maybe" ], "seconds": [ 1.099347018996923 ], "surface_features": [ 3.564432944788032, 4.488649408577202, 3.188084373714938, 0.0, 3.6889545880488788, 4876 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B186", "pair": 186, "family": "bijection", "mapping": "F(x,y,z)=(-2*x^7 + 14*x^6*y + 28*x^6*z - 42*x^6 - 42*x^5*y^2 - 168*x^5*y*z + 248*x^5*y - 168*x^5*z^2 + 492*x^5*z - 348*x^5 + 70*x^4*y^3 + 420*x^4*y^2*z - 610*x^4*y^2 + 840*x^4*y*z^2 - 2420*x^4*y*z + 1688*x^4*y + 560*x^4*z^3 - 2400*x^4*z^2 + 3322*x^4*z - 1512*x^4 - 70*x^3*y^4 - 560*x^3*y^3*z + 800*x^3*y^3 - 1680*x^3*y^2*z^2 + 4760*x^3*y^2*z - 3274*x^3*y^2 - 2240*x^3*y*z^3 + 9440*x^3*y*z^2 - 12884*x^3*y*z + 5796*x^3*y - 1120*x^3*z^4 + 6240*x^3*z^3 - 12674*x^3*z^2 + 11308*x^3*z - 3794*x^3 + 42*x^2*y^5 + 420*x^2*y^4*z - 590*x^2*y^4 + 1680*x^2*y^3*z^2 - 4680*x^2*y^3*z + 3174*x^2*y^3 + 3360*x^2*y^2*z^3 - 13920*x^2*y^2*z^2 + 18732*x^2*y^2*z - 8328*x^2*y^2 + 3360*x^2*y*z^4 - 18400*x^2*y*z^3 + 36846*x^2*y*z^2 - 32488*x^2*y*z + 10786*x^2*y + 1344*x^2*z^5 - 9120*x^2*z^4 + 24156*x^2*z^3 - 31680*x^2*z^2 + 20822*x^2*z - 5559*x^2 - 14*x*y^6 - 168*x*y^5*z + 232*x*y^5 - 840*x*y^4*z^2 + 2300*x*y^4*z - 1538*x*y^4 - 2240*x*y^3*z^3 + 9120*x*y^3*z^2 - 12100*x*y^3*z + 5316*x*y^3 - 3360*x*y^2*z^4 + 18080*x*y^2*z^3 - 35694*x*y^2*z^2 + 31100*x*y^2*z - 10216*x*y^2 - 2688*x*y*z^5 + 17920*x*y*z^4 - 46792*x*y*z^3 + 60640*x*y*z^2 - 39444*x*y*z + 10422*x*y - 896*x*z^6 + 7104*x*z^5 - 23000*x*z^4 + 39408*x*z^3 - 38078*x*z^2 + 19872*x*z - 4433*x + 2*y^7 + 28*y^6*z - 38*y^6 + 168*y^5*z^2 - 452*y^5*z + 298*y^5 + 560*y^4*z^3 - 2240*y^4*z^2 + 2930*y^4*z - 1272*y^4 + 1120*y^3*z^4 - 5920*y^3*z^3 + 11522*y^3*z^2 - 9920*y^3*z + 3224*y^3 + 1344*y^2*z^5 - 8800*y^2*z^4 + 22652*y^2*z^3 - 29008*y^2*z^2 + 18672*y^2*z - 4883*y^2 + 896*y*z^6 - 6976*y*z^5 + 22264*y*z^4 - 37696*y*z^3 + 36050*y*z^2 - 18632*y*z + 4112*y + 256*z^7 - 2304*z^6 + 8752*z^5 - 18368*z^4 + 23202*z^3 - 17786*z^2 + 7735*z - 1, -x^7 + 7*x^6*y + 14*x^6*z - 21*x^6 - 21*x^5*y^2 - 84*x^5*y*z + 124*x^5*y - 84*x^5*z^2 + 246*x^5*z - 174*x^5 + 35*x^4*y^3 + 210*x^4*y^2*z - 305*x^4*y^2 + 420*x^4*y*z^2 - 1210*x^4*y*z + 844*x^4*y + 280*x^4*z^3 - 1200*x^4*z^2 + 1661*x^4*z - 756*x^4 - 35*x^3*y^4 - 280*x^3*y^3*z + 400*x^3*y^3 - 840*x^3*y^2*z^2 + 2380*x^3*y^2*z - 1637*x^3*y^2 - 1120*x^3*y*z^3 + 4720*x^3*y*z^2 - 6442*x^3*y*z + 2898*x^3*y - 560*x^3*z^4 + 3120*x^3*z^3 - 6337*x^3*z^2 + 5654*x^3*z - 1896*x^3 + 21*x^2*y^5 + 210*x^2*y^4*z - 295*x^2*y^4 + 840*x^2*y^3*z^2 - 2340*x^2*y^3*z + 1587*x^2*y^3 + 1680*x^2*y^2*z^3 - 6960*x^2*y^2*z^2 + 9366*x^2*y^2*z - 4164*x^2*y^2 + 1680*x^2*y*z^4 - 9200*x^2*y*z^3 + 18423*x^2*y*z^2 - 16244*x^2*y*z + 5390*x^2*y + 672*x^2*z^5 - 4560*x^2*z^4 + 12078*x^2*z^3 - 15840*x^2*z^2 + 10405*x^2*z - 2774*x^2 - 7*x*y^6 - 84*x*y^5*z + 116*x*y^5 - 420*x*y^4*z^2 + 1150*x*y^4*z - 769*x*y^4 - 1120*x*y^3*z^3 + 4560*x*y^3*z^2 - 6050*x*y^3*z + 2658*x*y^3 - 1680*x*y^2*z^4 + 9040*x*y^2*z^3 - 17847*x*y^2*z^2 + 15550*x*y^2*z - 5105*x*y^2 - 1344*x*y*z^5 + 8960*x*y*z^4 - 23396*x*y*z^3 + 30320*x*y*z^2 - 19710*x*y*z + 5200*x*y - 448*x*z^6 + 3552*x*z^5 - 11500*x*z^4 + 19704*x*z^3 - 19027*x*z^2 + 9914*x*z - 2207*x + y^7 + 14*y^6*z - 19*y^6 + 84*y^5*z^2 - 226*y^5*z + 149*y^5 + 280*y^4*z^3 - 1120*y^4*z^2 + 1465*y^4*z - 636*y^4 + 560*y^3*z^4 - 2960*y^3*z^3 + 5761*y^3*z^2 - 4960*y^3*z + 1611*y^3 + 672*y^2*z^5 - 4400*y^2*z^4 + 11326*y^2*z^3 - 14504*y^2*z^2 + 9330*y^2*z - 2436*y^2 + 448*y*z^6 - 3488*y*z^5 + 11132*y*z^4 - 18848*y*z^3 + 18013*y*z^2 - 9294*y*z + 2046*y + 128*z^7 - 1152*z^6 + 4376*z^5 - 9184*z^4 + 11593*z^3 - 8871*z^2 + 3848*z - 1, 2*x^7 - 14*x^6*y - 28*x^6*z + 42*x^6 + 42*x^5*y^2 + 168*x^5*y*z - 248*x^5*y + 168*x^5*z^2 - 492*x^5*z + 348*x^5 - 70*x^4*y^3 - 420*x^4*y^2*z + 610*x^4*y^2 - 840*x^4*y*z^2 + 2420*x^4*y*z - 1688*x^4*y - 560*x^4*z^3 + 2400*x^4*z^2 - 3322*x^4*z + 1512*x^4 + 70*x^3*y^4 + 560*x^3*y^3*z - 800*x^3*y^3 + 1680*x^3*y^2*z^2 - 4760*x^3*y^2*z + 3274*x^3*y^2 + 2240*x^3*y*z^3 - 9440*x^3*y*z^2 + 12884*x^3*y*z - 5796*x^3*y + 1120*x^3*z^4 - 6240*x^3*z^3 + 12674*x^3*z^2 - 11308*x^3*z + 3790*x^3 - 42*x^2*y^5 - 420*x^2*y^4*z + 590*x^2*y^4 - 1680*x^2*y^3*z^2 + 4680*x^2*y^3*z - 3174*x^2*y^3 - 3360*x^2*y^2*z^3 + 13920*x^2*y^2*z^2 - 18732*x^2*y^2*z + 8328*x^2*y^2 - 3360*x^2*y*z^4 + 18400*x^2*y*z^3 - 36846*x^2*y*z^2 + 32488*x^2*y*z - 10774*x^2*y - 1344*x^2*z^5 + 9120*x^2*z^4 - 24156*x^2*z^3 + 31680*x^2*z^2 - 20798*x^2*z + 5536*x^2 + 14*x*y^6 + 168*x*y^5*z - 232*x*y^5 + 840*x*y^4*z^2 - 2300*x*y^4*z + 1538*x*y^4 + 2240*x*y^3*z^3 - 9120*x*y^3*z^2 + 12100*x*y^3*z - 5316*x*y^3 + 3360*x*y^2*z^4 - 18080*x*y^2*z^3 + 35694*x*y^2*z^2 - 31100*x*y^2*z + 10204*x*y^2 + 2688*x*y*z^5 - 17920*x*y*z^4 + 46792*x*y*z^3 - 60640*x*y*z^2 + 39396*x*y*z - 10376*x*y + 896*x*z^6 - 7104*x*z^5 + 23000*x*z^4 - 39408*x*z^3 + 38030*x*z^2 - 19780*x*z + 4388*x - 2*y^7 - 28*y^6*z + 38*y^6 - 168*y^5*z^2 + 452*y^5*z - 298*y^5 - 560*y^4*z^3 + 2240*y^4*z^2 - 2930*y^4*z + 1272*y^4 - 1120*y^3*z^4 + 5920*y^3*z^3 - 11522*y^3*z^2 + 9920*y^3*z - 3220*y^3 - 1344*y^2*z^5 + 8800*y^2*z^4 - 22652*y^2*z^3 + 29008*y^2*z^2 - 18648*y^2*z + 4860*y^2 - 896*y*z^6 + 6976*y*z^5 - 22264*y*z^4 + 37696*y*z^3 - 36002*y*z^2 + 18540*y*z - 4066*y - 256*z^7 + 2304*z^6 - 8752*z^5 + 18368*z^4 - 23170*z^3 + 17694*z^2 - 7646*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ 1, -1, -2 ], [ 2, -1, -1 ], [ 1, -1, -1 ] ], "source_shift": [ 2, 2, 2 ], "target_matrix": [ [ -2, -1, -1 ], [ -1, -1, -1 ], [ 2, 2, 3 ] ], "target_shift": [ 1491, 739, -1459 ], "inverse_recipe": { "p": 2, "q": 3, "f": "x**2 + x", "g": "x**3", "h": "y**2*z + z**2 + z", "polynomials": { "f": [ [ [ 2, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "g": [ [ [ 3, 0, 0 ], 1 ] ], "h": [ [ [ 0, 2, 1 ], 1 ], [ [ 0, 0, 1 ], 1 ], [ [ 0, 0, 2 ], 1 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0826700559991878 ], "surface_features": [ 3.888608925316117, 4.782759192623997, 3.4668676203541096, 0.0, 4.03346663102796, 4965 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B127", "pair": 127, "family": "bijection", "mapping": "F(x,y,z)=(-2187*x^7 - 10206*x^6*y - 15309*x^6*z - 5589*x^6 - 20412*x^5*y^2 - 61236*x^5*y*z - 22275*x^5*y - 45927*x^5*z^2 - 33291*x^5*z - 6156*x^5 - 22680*x^4*y^3 - 102060*x^4*y^2*z - 36990*x^4*y^2 - 153090*x^4*y*z^2 - 110565*x^4*y*z - 20385*x^4*y - 76545*x^4*z^3 - 82620*x^4*z^2 - 30375*x^4*z - 3996*x^4 - 15120*x^3*y^4 - 90720*x^3*y^3*z - 32760*x^3*y^3 - 204120*x^3*y^2*z^2 - 146880*x^3*y^2*z - 27000*x^3*y^2 - 204120*x^3*y*z^3 - 219510*x^3*y*z^2 - 80460*x^3*y*z - 10557*x^3*y - 76545*x^3*z^4 - 109350*x^3*z^3 - 59940*x^3*z^2 - 15687*x^3*z - 1755*x^3 - 6048*x^2*y^5 - 45360*x^2*y^4*z - 16320*x^2*y^4 - 136080*x^2*y^3*z^2 - 97560*x^2*y^3*z - 17880*x^2*y^3 - 204120*x^2*y^2*z^3 - 218700*x^2*y^2*z^2 - 79920*x^2*y^2*z - 10458*x^2*y^2 - 153090*x^2*y*z^4 - 217890*x^2*y*z^3 - 119070*x^2*y*z^2 - 31077*x^2*y*z - 3465*x^2*y - 45927*x^2*z^5 - 81405*x^2*z^4 - 59130*x^2*z^3 - 23085*x^2*z^2 - 5121*x^2*z - 588*x^2 - 1344*x*y^6 - 12096*x*y^5*z - 4336*x*y^5 - 45360*x*y^4*z^2 - 32400*x*y^4*z - 5920*x*y^4 - 90720*x*y^3*z^3 - 96840*x*y^3*z^2 - 35280*x*y^3*z - 4604*x*y^3 - 102060*x*y^2*z^4 - 144720*x*y^2*z^3 - 78840*x*y^2*z^2 - 20520*x*y^2*z - 2280*x*y^2 - 61236*x*y*z^5 - 108135*x*y*z^4 - 78300*x*y*z^3 - 30483*x*y*z^2 - 6738*x*y*z - 770*x*y - 15309*x*z^6 - 32319*x*z^5 - 29160*x*z^4 - 15093*x*z^3 - 4977*x*z^2 - 1126*x*z - 149*x - 128*y^7 - 1344*y^6*z - 480*y^6 - 6048*y^5*z^2 - 4304*y^5*z - 784*y^5 - 15120*y^4*z^3 - 16080*y^4*z^2 - 5840*y^4*z - 760*y^4 - 22680*y^3*z^4 - 32040*y^3*z^3 - 17400*y^3*z^2 - 4516*y^3*z - 500*y^3 - 20412*y^2*z^5 - 35910*y^2*z^4 - 25920*y^2*z^3 - 10062*y^2*z^2 - 2216*y^2*z - 252*y^2 - 10206*y*z^6 - 21465*y*z^5 - 19305*y*z^4 - 9963*y*z^3 - 3273*y*z^2 - 737*y*z - 95*y - 2187*z^7 - 5346*z^6 - 5751*z^5 - 3699*z^4 - 1611*z^3 - 539*z^2 - 135*z - 1, 4374*x^7 + 20412*x^6*y + 30618*x^6*z + 11178*x^6 + 40824*x^5*y^2 + 122472*x^5*y*z + 44550*x^5*y + 91854*x^5*z^2 + 66582*x^5*z + 11826*x^5 + 45360*x^4*y^3 + 204120*x^4*y^2*z + 73980*x^4*y^2 + 306180*x^4*y*z^2 + 221130*x^4*y*z + 39150*x^4*y + 153090*x^4*z^3 + 165240*x^4*z^2 + 58320*x^4*z + 7182*x^4 + 30240*x^3*y^4 + 181440*x^3*y^3*z + 65520*x^3*y^3 + 408240*x^3*y^2*z^2 + 293760*x^3*y^2*z + 51840*x^3*y^2 + 408240*x^3*y*z^3 + 439020*x^3*y*z^2 + 154440*x^3*y*z + 18954*x^3*y + 153090*x^3*z^4 + 218700*x^3*z^3 + 115020*x^3*z^2 + 28134*x^3*z + 2916*x^3 + 12096*x^2*y^5 + 90720*x^2*y^4*z + 32640*x^2*y^4 + 272160*x^2*y^3*z^2 + 195120*x^2*y^3*z + 34320*x^2*y^3 + 408240*x^2*y^2*z^3 + 437400*x^2*y^2*z^2 + 153360*x^2*y^2*z + 18756*x^2*y^2 + 306180*x^2*y*z^4 + 435780*x^2*y*z^3 + 228420*x^2*y*z^2 + 55674*x^2*y*z + 5742*x^2*y + 91854*x^2*z^5 + 162810*x^2*z^4 + 113400*x^2*z^3 + 41310*x^2*z^2 + 8460*x^2*z + 906*x^2 + 2688*x*y^6 + 24192*x*y^5*z + 8672*x*y^5 + 90720*x*y^4*z^2 + 64800*x*y^4*z + 11360*x*y^4 + 181440*x*y^3*z^3 + 193680*x*y^3*z^2 + 67680*x*y^3*z + 8248*x*y^3 + 204120*x*y^2*z^4 + 289440*x*y^2*z^3 + 151200*x*y^2*z^2 + 36720*x*y^2*z + 3768*x*y^2 + 122472*x*y*z^5 + 216270*x*y*z^4 + 150120*x*y*z^3 + 54486*x*y*z^2 + 11100*x*y*z + 1180*x*y + 30618*x*z^6 + 64638*x*z^5 + 55890*x*z^4 + 26946*x*z^3 + 8172*x*z^2 + 1712*x*z + 214*x + 256*y^7 + 2688*y^6*z + 960*y^6 + 12096*y^5*z^2 + 8608*y^5*z + 1504*y^5 + 30240*y^4*z^3 + 32160*y^4*z^2 + 11200*y^4*z + 1360*y^4 + 45360*y^3*z^4 + 64080*y^3*z^3 + 33360*y^3*z^2 + 8072*y^3*z + 824*y^3 + 40824*y^2*z^5 + 71820*y^2*z^4 + 49680*y^2*z^3 + 17964*y^2*z^2 + 3640*y^2*z + 384*y^2 + 20412*y*z^6 + 42930*y*z^5 + 36990*y*z^4 + 17766*y*z^3 + 5358*y*z^2 + 1114*y*z + 134*y + 4374*z^7 + 10692*z^6 + 11016*z^5 + 6588*z^4 + 2628*z^3 + 808*z^2 + 188*z + 1, -2187*x^7 - 10206*x^6*y - 15309*x^6*z - 5589*x^6 - 20412*x^5*y^2 - 61236*x^5*y*z - 22275*x^5*y - 45927*x^5*z^2 - 33291*x^5*z - 6156*x^5 - 22680*x^4*y^3 - 102060*x^4*y^2*z - 36990*x^4*y^2 - 153090*x^4*y*z^2 - 110565*x^4*y*z - 20385*x^4*y - 76545*x^4*z^3 - 82620*x^4*z^2 - 30375*x^4*z - 3996*x^4 - 15120*x^3*y^4 - 90720*x^3*y^3*z - 32760*x^3*y^3 - 204120*x^3*y^2*z^2 - 146880*x^3*y^2*z - 27000*x^3*y^2 - 204120*x^3*y*z^3 - 219510*x^3*y*z^2 - 80460*x^3*y*z - 10557*x^3*y - 76545*x^3*z^4 - 109350*x^3*z^3 - 59940*x^3*z^2 - 15687*x^3*z - 1755*x^3 - 6048*x^2*y^5 - 45360*x^2*y^4*z - 16320*x^2*y^4 - 136080*x^2*y^3*z^2 - 97560*x^2*y^3*z - 17880*x^2*y^3 - 204120*x^2*y^2*z^3 - 218700*x^2*y^2*z^2 - 79920*x^2*y^2*z - 10458*x^2*y^2 - 153090*x^2*y*z^4 - 217890*x^2*y*z^3 - 119070*x^2*y*z^2 - 31077*x^2*y*z - 3465*x^2*y - 45927*x^2*z^5 - 81405*x^2*z^4 - 59130*x^2*z^3 - 23085*x^2*z^2 - 5121*x^2*z - 579*x^2 - 1344*x*y^6 - 12096*x*y^5*z - 4336*x*y^5 - 45360*x*y^4*z^2 - 32400*x*y^4*z - 5920*x*y^4 - 90720*x*y^3*z^3 - 96840*x*y^3*z^2 - 35280*x*y^3*z - 4604*x*y^3 - 102060*x*y^2*z^4 - 144720*x*y^2*z^3 - 78840*x*y^2*z^2 - 20520*x*y^2*z - 2280*x*y^2 - 61236*x*y*z^5 - 108135*x*y*z^4 - 78300*x*y*z^3 - 30483*x*y*z^2 - 6738*x*y*z - 758*x*y - 15309*x*z^6 - 32319*x*z^5 - 29160*x*z^4 - 15093*x*z^3 - 4977*x*z^2 - 1108*x*z - 141*x - 128*y^7 - 1344*y^6*z - 480*y^6 - 6048*y^5*z^2 - 4304*y^5*z - 784*y^5 - 15120*y^4*z^3 - 16080*y^4*z^2 - 5840*y^4*z - 760*y^4 - 22680*y^3*z^4 - 32040*y^3*z^3 - 17400*y^3*z^2 - 4516*y^3*z - 500*y^3 - 20412*y^2*z^5 - 35910*y^2*z^4 - 25920*y^2*z^3 - 10062*y^2*z^2 - 2216*y^2*z - 248*y^2 - 10206*y*z^6 - 21465*y*z^5 - 19305*y*z^4 - 9963*y*z^3 - 3273*y*z^2 - 725*y*z - 90*y - 2187*z^7 - 5346*z^6 - 5751*z^5 - 3699*z^4 - 1611*z^3 - 530*z^2 - 128*z - 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ 3, 2, 3 ], [ 2, 1, 1 ], [ 1, 1, 1 ] ], "source_shift": [ 1, 2, 2 ], "target_matrix": [ [ -1, -2, -1 ], [ 2, 2, 3 ], [ -1, -1, -1 ] ], "target_shift": [ 21, -25, 18 ], "inverse_recipe": { "p": 2, "q": 5, "f": "x**2", "g": "x**5 + x**3 + x**2 + x", "h": "y**2 + y*z", "polynomials": { "f": [ [ [ 2, 0, 0 ], 1 ] ], "g": [ [ [ 5, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ], [ [ 3, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 2, 0 ], 1 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0343086100001528 ], "surface_features": [ 4.75050318791471, 5.642484305417037, 4.351748006337345, 0.0, 4.903727573269929, 5310 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B182", "pair": 182, "family": "bijection", "mapping": "F(x,y,z)=(-3*x^7 + 42*x^6*y + 21*x^6*z - 27*x^6 - 252*x^5*y^2 - 252*x^5*y*z + 321*x^5*y - 63*x^5*z^2 + 162*x^5*z - 104*x^5 + 840*x^4*y^3 + 1260*x^4*y^2*z - 1590*x^4*y^2 + 630*x^4*y*z^2 - 1605*x^4*y*z + 1016*x^4*y + 105*x^4*z^3 - 405*x^4*z^2 + 520*x^4*z - 229*x^4 - 1680*x^3*y^4 - 3360*x^3*y^3*z + 4200*x^3*y^3 - 2520*x^3*y^2*z^2 + 6360*x^3*y^2*z - 3968*x^3*y^2 - 840*x^3*y*z^3 + 3210*x^3*y*z^2 - 4064*x^3*y*z + 1748*x^3*y - 105*x^3*z^4 + 540*x^3*z^3 - 1040*x^3*z^2 + 916*x^3*z - 323*x^3 + 2016*x^2*y^5 + 5040*x^2*y^4*z - 6240*x^2*y^4 + 5040*x^2*y^3*z^2 - 12600*x^2*y^3*z + 7744*x^2*y^3 + 2520*x^2*y^2*z^3 - 9540*x^2*y^2*z^2 + 11904*x^2*y^2*z - 5001*x^2*y^2 + 630*x^2*y*z^4 - 3210*x^2*y*z^3 + 6096*x^2*y*z^2 - 5244*x^2*y*z + 1782*x^2*y + 63*x^2*z^5 - 405*x^2*z^4 + 1040*x^2*z^3 - 1374*x^2*z^2 + 966*x^2*z - 310*x^2 - 1344*x*y^6 - 4032*x*y^5*z + 4944*x*y^5 - 5040*x*y^4*z^2 + 12480*x*y^4*z - 7552*x*y^4 - 3360*x*y^3*z^3 + 12600*x*y^3*z^2 - 15488*x*y^3*z + 6356*x*y^3 - 1260*x*y^2*z^4 + 6360*x*y^2*z^3 - 11904*x*y^2*z^2 + 10002*x*y^2*z - 3279*x*y^2 - 252*x*y*z^5 + 1605*x*y*z^4 - 4064*x*y*z^3 + 5244*x*y*z^2 - 3552*x*y*z + 1081*x*y - 21*x*z^6 + 162*x*z^5 - 520*x*z^4 + 916*x*z^3 - 963*x*z^2 + 611*x*z - 187*x + 384*y^7 + 1344*y^6*z - 1632*y^6 + 2016*y^5*z^2 - 4944*y^5*z + 2944*y^5 + 1680*y^4*z^3 - 6240*y^4*z^2 + 7552*y^4*z - 3028*y^4 + 840*y^3*z^4 - 4200*y^3*z^3 + 7744*y^3*z^2 - 6356*y^3*z + 2011*y^3 + 252*y^2*z^5 - 1590*y^2*z^4 + 3968*y^2*z^3 - 5001*y^2*z^2 + 3267*y^2*z - 949*y^2 + 42*y*z^6 - 321*y*z^5 + 1016*y*z^4 - 1748*y*z^3 + 1770*y*z^2 - 1066*y*z + 312*y + 3*z^7 - 27*z^6 + 104*z^5 - 229*z^4 + 320*z^3 - 301*z^2 + 185*z - 1, x^7 - 14*x^6*y - 7*x^6*z + 9*x^6 + 84*x^5*y^2 + 84*x^5*y*z - 107*x^5*y + 21*x^5*z^2 - 54*x^5*z + 32*x^5 - 280*x^4*y^3 - 420*x^4*y^2*z + 530*x^4*y^2 - 210*x^4*y*z^2 + 535*x^4*y*z - 312*x^4*y - 35*x^4*z^3 + 135*x^4*z^2 - 160*x^4*z + 63*x^4 + 560*x^3*y^4 + 1120*x^3*y^3*z - 1400*x^3*y^3 + 840*x^3*y^2*z^2 - 2120*x^3*y^2*z + 1216*x^3*y^2 + 280*x^3*y*z^3 - 1070*x^3*y*z^2 + 1248*x^3*y*z - 476*x^3*y + 35*x^3*z^4 - 180*x^3*z^3 + 320*x^3*z^2 - 252*x^3*z + 81*x^3 - 672*x^2*y^5 - 1680*x^2*y^4*z + 2080*x^2*y^4 - 1680*x^2*y^3*z^2 + 4200*x^2*y^3*z - 2368*x^2*y^3 - 840*x^2*y^2*z^3 + 3180*x^2*y^2*z^2 - 3648*x^2*y^2*z + 1347*x^2*y^2 - 210*x^2*y*z^4 + 1070*x^2*y*z^3 - 1872*x^2*y*z^2 + 1428*x^2*y*z - 434*x^2*y - 21*x^2*z^5 + 135*x^2*z^4 - 320*x^2*z^3 + 378*x^2*z^2 - 242*x^2*z + 77*x^2 + 448*x*y^6 + 1344*x*y^5*z - 1648*x*y^5 + 1680*x*y^4*z^2 - 4160*x*y^4*z + 2304*x*y^4 + 1120*x*y^3*z^3 - 4200*x*y^3*z^2 + 4736*x*y^3*z - 1692*x*y^3 + 420*x*y^2*z^4 - 2120*x*y^2*z^3 + 3648*x*y^2*z^2 - 2694*x*y^2*z + 773*x*y^2 + 84*x*y*z^5 - 535*x*y*z^4 + 1248*x*y*z^3 - 1428*x*y*z^2 + 864*x*y*z - 255*x*y + 7*x*z^6 - 54*x*z^5 + 160*x*z^4 - 252*x*z^3 + 241*x*z^2 - 151*x*z + 42*x - 128*y^7 - 448*y^6*z + 544*y^6 - 672*y^5*z^2 + 1648*y^5*z - 896*y^5 - 560*y^4*z^3 + 2080*y^4*z^2 - 2304*y^4*z + 796*y^4 - 280*y^3*z^4 + 1400*y^3*z^3 - 2368*y^3*z^2 + 1692*y^3*z - 457*y^3 - 84*y^2*z^5 + 530*y^2*z^4 - 1216*y^2*z^3 + 1347*y^2*z^2 - 769*y^2*z + 211*y^2 - 14*y*z^6 + 107*y*z^5 - 312*y*z^4 + 476*y*z^3 - 430*y*z^2 + 250*y*z - 71*y - z^7 + 9*z^6 - 32*z^5 + 63*z^4 - 80*z^3 + 74*z^2 - 44*z + 1, -2*x^7 + 28*x^6*y + 14*x^6*z - 18*x^6 - 168*x^5*y^2 - 168*x^5*y*z + 214*x^5*y - 42*x^5*z^2 + 108*x^5*z - 70*x^5 + 560*x^4*y^3 + 840*x^4*y^2*z - 1060*x^4*y^2 + 420*x^4*y*z^2 - 1070*x^4*y*z + 684*x^4*y + 70*x^4*z^3 - 270*x^4*z^2 + 350*x^4*z - 156*x^4 - 1120*x^3*y^4 - 2240*x^3*y^3*z + 2800*x^3*y^3 - 1680*x^3*y^2*z^2 + 4240*x^3*y^2*z - 2672*x^3*y^2 - 560*x^3*y*z^3 + 2140*x^3*y*z^2 - 2736*x^3*y*z + 1192*x^3*y - 70*x^3*z^4 + 360*x^3*z^3 - 700*x^3*z^2 + 624*x^3*z - 222*x^3 + 1344*x^2*y^5 + 3360*x^2*y^4*z - 4160*x^2*y^4 + 3360*x^2*y^3*z^2 - 8400*x^2*y^3*z + 5216*x^2*y^3 + 1680*x^2*y^2*z^3 - 6360*x^2*y^2*z^2 + 8016*x^2*y^2*z - 3414*x^2*y^2 + 420*x^2*y*z^4 - 2140*x^2*y*z^3 + 4104*x^2*y*z^2 - 3576*x^2*y*z + 1228*x^2*y + 42*x^2*z^5 - 270*x^2*z^4 + 700*x^2*z^3 - 936*x^2*z^2 + 664*x^2*z - 213*x^2 - 896*x*y^6 - 2688*x*y^5*z + 3296*x*y^5 - 3360*x*y^4*z^2 + 8320*x*y^4*z - 5088*x*y^4 - 2240*x*y^3*z^3 + 8400*x*y^3*z^2 - 10432*x*y^3*z + 4344*x*y^3 - 840*x*y^2*z^4 + 4240*x*y^2*z^3 - 8016*x*y^2*z^2 + 6828*x*y^2*z - 2266*x*y^2 - 168*x*y*z^5 + 1070*x*y*z^4 - 2736*x*y*z^3 + 3576*x*y*z^2 - 2448*x*y*z + 746*x*y - 14*x*z^6 + 108*x*z^5 - 350*x*z^4 + 624*x*z^3 - 662*x*z^2 + 420*x*z - 129*x + 256*y^7 + 896*y^6*z - 1088*y^6 + 1344*y^5*z^2 - 3296*y^5*z + 1984*y^5 + 1120*y^4*z^3 - 4160*y^4*z^2 + 5088*y^4*z - 2072*y^4 + 560*y^3*z^4 - 2800*y^3*z^3 + 5216*y^3*z^2 - 4344*y^3*z + 1394*y^3 + 168*y^2*z^5 - 1060*y^2*z^4 + 2672*y^2*z^3 - 3414*y^2*z^2 + 2258*y^2*z - 658*y^2 + 28*y*z^6 - 214*y*z^5 + 684*y*z^4 - 1192*y*z^3 + 1220*y*z^2 - 736*y*z + 215*y + 2*z^7 - 18*z^6 + 70*z^5 - 156*z^4 + 220*z^3 - 207*z^2 + 127*z - 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ 1, -2, -1 ], [ 1, -1, -1 ], [ 2, -1, -1 ] ], "source_shift": [ 1, 1, 2 ], "target_matrix": [ [ -3, -2, -2 ], [ 1, 1, 2 ], [ -2, -1, -1 ] ], "target_shift": [ 50, -6, 35 ], "inverse_recipe": { "p": 2, "q": 5, "f": "x**2", "g": "x**5 + x", "h": "y**3 + y*z + 2*y", "polynomials": { "f": [ [ [ 2, 0, 0 ], 1 ] ], "g": [ [ [ 5, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 3, 0 ], 1 ], [ [ 0, 1, 0 ], 2 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "Maybe" ], "seconds": [ 1.0427100829983829 ], "surface_features": [ 3.263925895967294, 4.189995339964319, 2.9304395947667, 0.0, 3.3959127591008653, 4793 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C247", "pair": 247, "family": "counterexample", "mapping": "F(x,y,z)=(4*x^7 - 30*x^6*y + 42*x^6*z - 52*x^6 + 96*x^5*y^2 - 270*x^5*y*z + 333*x^5*y + 186*x^5*z^2 - 471*x^5*z + 297*x^5 - 170*x^4*y^3 + 720*x^4*y^2*z - 885*x^4*y^2 - 996*x^4*y*z^2 + 2517*x^4*y*z - 1578*x^4*y + 450*x^4*z^3 - 1752*x^4*z^2 + 2250*x^4*z - 966*x^4 + 180*x^3*y^4 - 1020*x^3*y^3*z + 1250*x^3*y^3 + 2124*x^3*y^2*z^2 - 5358*x^3*y^2*z + 3342*x^3*y^2 - 1926*x^3*y*z^3 + 7497*x^3*y*z^2 - 9588*x^3*y*z + 4088*x^3*y + 642*x^3*z^4 - 3423*x^3*z^3 + 6723*x^3*z^2 - 5872*x^3*z + 1932*x^3 - 114*x^2*y^5 + 810*x^2*y^4*z - 990*x^2*y^4 - 2256*x^2*y^3*z^2 + 5682*x^2*y^3*z - 3528*x^2*y^3 + 3078*x^2*y^2*z^3 - 11979*x^2*y^2*z^2 + 15264*x^2*y^2*z - 6468*x^2*y^2 - 2058*x^2*y*z^4 + 10989*x^2*y*z^3 - 21534*x^2*y*z^2 + 18702*x^2*y*z - 6108*x^2*y + 540*x^2*z^5 - 3702*x^2*z^4 + 9900*x^2*z^3 - 13188*x^2*z^2 + 8838*x^2*z - 2378*x^2 + 40*x*y^6 - 342*x*y^5*z + 417*x*y^5 + 1194*x*y^4*z^2 - 3003*x*y^4*z + 1857*x*y^4 - 2178*x*y^3*z^3 + 8475*x*y^3*z^2 - 10764*x*y^3*z + 4536*x*y^3 + 2190*x*y^2*z^4 - 11709*x*y^2*z^3 + 22899*x*y^2*z^2 - 19788*x*y^2*z + 6420*x*y^2 - 1152*x*y*z^5 + 7920*x*y*z^4 - 21174*x*y*z^3 + 28095*x*y*z^2 - 18708*x*y*z + 4996*x*y + 248*x*z^6 - 2100*x*z^5 + 7182*x*z^4 - 12969*x*z^3 + 13257*x*z^2 - 7282*x*z + 1672*x - 6*y^7 + 60*y^6*z - 73*y^6 - 252*y^5*z^2 + 633*y^5*z - 390*y^5 + 576*y^4*z^3 - 2241*y^4*z^2 + 2838*y^4*z - 1190*y^4 - 774*y^3*z^4 + 4143*y^3*z^3 - 8088*y^3*z^2 + 6958*y^3*z - 2244*y^3 + 612*y^2*z^5 - 4218*y^2*z^4 + 11274*y^2*z^3 - 14907*y^2*z^2 + 9870*y^2*z - 2618*y^2 - 264*y*z^6 + 2244*y*z^5 - 7686*y*z^4 + 13853*y*z^3 - 14094*y*z^2 + 7690*y*z - 1752*y + 48*z^7 - 488*z^6 + 2052*z^5 - 4714*z^4 + 6522*z^3 - 5468*z^2 + 2574*z - 1, -2*x^7 + 15*x^6*y - 21*x^6*z + 35*x^6 - 48*x^5*y^2 + 135*x^5*y*z - 225*x^5*y - 93*x^5*z^2 + 321*x^5*z - 234*x^5 + 85*x^4*y^3 - 360*x^4*y^2*z + 600*x^4*y^2 + 498*x^4*y*z^2 - 1722*x^4*y*z + 1248*x^4*y - 225*x^4*z^3 + 1209*x^4*z^2 - 1800*x^4*z + 842*x^4 - 90*x^3*y^4 + 510*x^3*y^3*z - 850*x^3*y^3 - 1062*x^3*y^2*z^2 + 3678*x^3*y^2*z - 2652*x^3*y^2 + 963*x^3*y*z^3 - 5193*x^3*y*z^2 + 7701*x^3*y*z - 3574*x^3*y - 321*x^3*z^4 + 2391*x^3*z^3 - 5463*x^3*z^2 + 5201*x^3*z - 1802*x^3 + 57*x^2*y^5 - 405*x^2*y^4*z + 675*x^2*y^4 + 1128*x^2*y^3*z^2 - 3912*x^2*y^3*z + 2808*x^2*y^3 - 1539*x^2*y^2*z^3 + 8325*x^2*y^2*z^2 - 12303*x^2*y^2*z + 5670*x^2*y^2 + 1029*x^2*y*z^4 - 7704*x^2*y*z^3 + 17571*x^2*y*z^2 - 16620*x^2*y*z + 5709*x^2*y - 270*x^2*z^5 + 2616*x^2*z^4 - 8172*x^2*z^3 + 11874*x^2*z^2 - 8385*x^2*z + 2320*x^2 - 20*x*y^6 + 171*x*y^5*z - 285*x*y^5 - 597*x*y^4*z^2 + 2073*x*y^4*z - 1482*x*y^4 + 1089*x*y^3*z^3 - 5907*x*y^3*z^2 + 8703*x*y^3*z - 3986*x*y^3 - 1095*x*y^2*z^4 + 8235*x*y^2*z^3 - 18753*x*y^2*z^2 + 17637*x*y^2*z - 6012*x*y^2 + 576*x*y*z^5 - 5616*x*y*z^4 + 17553*x*y*z^3 - 25389*x*y*z^2 + 17793*x*y*z - 4880*x*y - 124*x*z^6 + 1500*x*z^5 - 6021*x*z^4 + 11872*x*z^3 - 12798*x*z^2 + 7238*x*z - 1673*x + 3*y^7 - 30*y^6*z + 50*y^6 + 126*y^5*z^2 - 438*y^5*z + 312*y^5 - 288*y^4*z^3 + 1566*y^4*z^2 - 2301*y^4*z + 1048*y^4 + 387*y^3*z^4 - 2922*y^3*z^3 + 6645*y^3*z^2 - 6218*y^3*z + 2105*y^3 - 306*y^2*z^5 + 3000*y^2*z^4 - 9381*y^2*z^3 + 13515*y^2*z^2 - 9408*y^2*z + 2560*y^2 + 132*y*z^6 - 1608*y*z^5 + 6471*y*z^4 - 12733*y*z^3 + 13647*y*z^2 - 7655*y*z + 1753*y - 24*z^7 + 352*z^6 - 1746*z^5 + 4388*z^4 - 6406*z^3 + 5539*z^2 - 2633*z + 1, 2*x^7 - 15*x^6*y + 21*x^6*z - 29*x^6 + 48*x^5*y^2 - 135*x^5*y*z + 186*x^5*y + 93*x^5*z^2 - 264*x^5*z + 177*x^5 - 85*x^4*y^3 + 360*x^4*y^2*z - 495*x^4*y^2 - 498*x^4*y*z^2 + 1413*x^4*y*z - 942*x^4*y + 225*x^4*z^3 - 987*x^4*z^2 + 1350*x^4*z - 602*x^4 + 90*x^3*y^4 - 510*x^3*y^3*z + 700*x^3*y^3 + 1062*x^3*y^2*z^2 - 3012*x^3*y^2*z + 1998*x^3*y^2 - 963*x^3*y*z^3 + 4230*x^3*y*z^2 - 5763*x^3*y*z + 2551*x^3*y + 321*x^3*z^4 - 1938*x^3*z^3 + 4062*x^3*z^2 - 3686*x^3*z + 1241*x^3 - 57*x^2*y^5 + 405*x^2*y^4*z - 555*x^2*y^4 - 1128*x^2*y^3*z^2 + 3198*x^2*y^3*z - 2112*x^2*y^3 + 1539*x^2*y^2*z^3 - 6768*x^2*y^2*z^2 + 9189*x^2*y^2*z - 4041*x^2*y^2 - 1029*x^2*y*z^4 + 6231*x^2*y*z^3 - 13035*x^2*y*z^2 + 11757*x^2*y*z - 3927*x^2*y + 270*x^2*z^5 - 2106*x^2*z^4 + 6024*x^2*z^3 - 8340*x^2*z^2 + 5721*x^2*z - 1558*x^2 + 20*x*y^6 - 171*x*y^5*z + 234*x*y^5 + 597*x*y^4*z^2 - 1692*x*y^4*z + 1113*x*y^4 - 1089*x*y^3*z^3 + 4794*x*y^3*z^2 - 6489*x*y^3*z + 2837*x*y^3 + 1095*x*y^2*z^4 - 6648*x*y^2*z^3 + 13884*x*y^2*z^2 - 12456*x*y^2*z + 4131*x*y^2 - 576*x*y*z^5 + 4512*x*y*z^4 - 12909*x*y*z^3 + 17796*x*y*z^2 - 12123*x*y*z + 3275*x*y + 124*x*z^6 - 1200*x*z^5 + 4401*x*z^4 - 8263*x*z^3 + 8649*x*z^2 - 4811*x*z + 1108*x - 3*y^7 + 30*y^6*z - 41*y^6 - 126*y^5*z^2 + 357*y^5*z - 234*y^5 + 288*y^4*z^3 - 1269*y^4*z^2 + 1713*y^4*z - 745*y^4 - 387*y^3*z^4 + 2355*y^3*z^3 - 4911*y^3*z^2 + 4385*y^3*z - 1445*y^3 + 306*y^2*z^5 - 2406*y^2*z^4 + 6885*y^2*z^3 - 9456*y^2*z^2 + 6402*y^2*z - 1717*y^2 - 132*y*z^6 + 1284*y*z^5 - 4719*y*z^4 + 8842*y*z^3 - 9207*y*z^2 + 5084*y*z - 1161*y + 24*z^7 - 280*z^6 + 1266*z^5 - 3026*z^4 + 4288*z^3 - 3643*z^2 + 1723*z - 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -1, 1, -2 ], [ -1, 1, -1 ], [ -2, 3, -3 ] ], "source_shift": [ 1, 2, 2 ], "target_matrix": [ [ -2, -1, -1 ], [ 1, 2, 1 ], [ -1, -1, -1 ] ], "target_shift": [ 517, -519, 343 ], "collision_preimages": [ [ "15/4", "3/4", "-1" ], [ "15", "8", "-7/2" ], [ "6", "7", "3/2" ] ], "collision_image": [ "1035/2", "-2077/4", "1373/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0029856330002076 ], "surface_features": [ 3.59791934295132, 4.448629036430918, 3.262332413822626, 0.0, 3.6979142057582406, 4907 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C044", "pair": 44, "family": "counterexample", "mapping": "F(x,y,z)=(-27*x^7 - 162*x^6*y + 243*x^6*z + 513*x^6 - 414*x^5*y^2 + 1233*x^5*y*z + 2511*x^5*y - 900*x^5*z^2 - 3645*x^5*z - 2907*x^5 - 584*x^4*y^3 + 2589*x^4*y^2*z + 5094*x^4*y^2 - 3750*x^4*y*z^2 - 14643*x^4*y*z - 11532*x^4*y + 1772*x^4*z^3 + 10278*x^4*z^2 + 16176*x^4*z + 7557*x^4 - 491*x^3*y^4 + 2879*x^3*y^3*z + 5483*x^3*y^3 - 6204*x^3*y^2*z^2 - 23415*x^3*y^2*z - 18204*x^3*y^2 + 5816*x^3*y*z^3 + 32568*x^3*y*z^2 + 50529*x^3*y*z + 23346*x^3*y - 2000*x^3*z^4 - 14744*x^3*z^3 - 34188*x^3*z^2 - 31302*x^3*z - 10002*x^3 - 246*x^2*y^5 + 1788*x^2*y^4*z + 3303*x^2*y^4 - 5094*x^2*y^3*z^2 - 18633*x^2*y^3*z - 14298*x^2*y^3 + 7104*x^2*y^2*z^3 + 38538*x^2*y^2*z^2 + 58938*x^2*y^2*z + 26916*x^2*y^2 - 4848*x^2*y*z^4 - 34632*x^2*y*z^3 - 79050*x^2*y*z^2 - 71418*x^2*y*z - 22524*x^2*y + 1296*x^2*z^5 + 11424*x^2*z^4 + 34536*x^2*z^3 + 46140*x^2*z^2 + 28512*x^2*z + 6726*x^2 - 68*x*y^6 + 588*x*y^5*z + 1056*x*y^5 - 2076*x*y^4*z^2 - 7380*x*y^4*z - 5589*x*y^4 + 3828*x*y^3*z^3 + 20184*x*y^3*z^2 + 30429*x*y^3*z + 13731*x*y^3 - 3888*x*y^2*z^4 - 27012*x*y^2*z^3 - 60708*x*y^2*z^2 - 54126*x*y^2*z - 16857*x*y^2 + 2064*x*y*z^5 + 17712*x*y*z^4 + 52668*x*y*z^3 + 69408*x*y*z^2 + 42375*x*y*z + 9882*x*y 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27540*x^3*z^2 - 22988*x^3*z - 6872*x^3 - 246*x^2*y^5 + 1788*x^2*y^4*z + 3060*x^2*y^4 - 5094*x^2*y^3*z^2 - 17148*x^2*y^3*z - 11751*x^2*y^3 + 7104*x^2*y^2*z^3 + 35208*x^2*y^2*z^2 + 47742*x^2*y^2*z + 19935*x^2*y^2 - 4848*x^2*y*z^4 - 31392*x^2*y*z^3 - 63102*x^2*y*z^2 - 52014*x^2*y*z - 15354*x^2*y + 1296*x^2*z^5 + 10272*x^2*z^4 + 27192*x^2*z^3 + 33120*x^2*z^2 + 19146*x^2*z + 4269*x^2 - 68*x*y^6 + 588*x*y^5*z + 972*x*y^5 - 2076*x*y^4*z^2 - 6744*x*y^4*z - 4551*x*y^4 + 3828*x*y^3*z^3 + 18300*x*y^3*z^2 + 24423*x*y^3*z + 10079*x*y^3 - 3888*x*y^2*z^4 - 24288*x*y^2*z^3 - 48036*x*y^2*z^2 - 39108*x*y^2*z - 11403*x*y^2 + 2064*x*y*z^5 + 15792*x*y*z^4 + 41124*x*y*z^3 + 49464*x*y*z^2 + 28251*x*y*z + 6216*x*y - 448*x*z^6 - 4032*x*z^5 - 12960*x*z^4 - 20464*x*z^3 - 17172*x*z^2 - 7344*x*z - 1249*x - 8*y^7 + 80*y^6*z + 128*y^6 - 336*y^5*z^2 - 1056*y^5*z - 702*y^5 + 768*y^4*z^3 + 3552*y^4*z^2 + 4668*y^4*z + 1904*y^4 - 1032*y^3*z^4 - 6240*y^3*z^3 - 12150*y^3*z^2 - 9772*y^3*z - 2815*y^3 + 816*y^2*z^5 + 6048*y^2*z^4 + 15504*y^2*z^3 + 18420*y^2*z^2 + 10395*y^2*z + 2257*y^2 - 352*y*z^6 - 3072*y*z^5 - 9720*y*z^4 - 15160*y*z^3 - 12570*y*z^2 - 5306*y*z - 892*y + 64*z^7 + 640*z^6 + 2400*z^5 + 4608*z^4 + 4992*z^3 + 3076*z^2 + 1005*z + 1, -27*x^7 - 162*x^6*y + 243*x^6*z + 486*x^6 - 414*x^5*y^2 + 1233*x^5*y*z + 2367*x^5*y - 900*x^5*z^2 - 3420*x^5*z - 2457*x^5 - 584*x^4*y^3 + 2589*x^4*y^2*z + 4776*x^4*y^2 - 3750*x^4*y*z^2 - 13656*x^4*y*z - 9657*x^4*y + 1772*x^4*z^3 + 9528*x^4*z^2 + 13356*x^4*z + 5709*x^4 - 491*x^3*y^4 + 2879*x^3*y^3*z + 5111*x^3*y^3 - 6204*x^3*y^2*z^2 - 21696*x^3*y^2*z - 15102*x^3*y^2 + 5816*x^3*y*z^3 + 29976*x^3*y*z^2 + 41319*x^3*y*z + 17460*x^3*y - 2000*x^3*z^4 - 13472*x^3*z^3 - 27540*x^3*z^2 - 22995*x^3*z - 6888*x^3 - 246*x^2*y^5 + 1788*x^2*y^4*z + 3060*x^2*y^4 - 5094*x^2*y^3*z^2 - 17148*x^2*y^3*z - 11751*x^2*y^3 + 7104*x^2*y^2*z^3 + 35208*x^2*y^2*z^2 + 47742*x^2*y^2*z + 19941*x^2*y^2 - 4848*x^2*y*z^4 - 31392*x^2*y*z^3 - 63102*x^2*y*z^2 - 52035*x^2*y*z - 15399*x^2*y + 1296*x^2*z^5 + 10272*x^2*z^4 + 27192*x^2*z^3 + 33138*x^2*z^2 + 19224*x^2*z + 4326*x^2 - 68*x*y^6 + 588*x*y^5*z + 972*x*y^5 - 2076*x*y^4*z^2 - 6744*x*y^4*z - 4551*x*y^4 + 3828*x*y^3*z^3 + 18300*x*y^3*z^2 + 24423*x*y^3*z + 10083*x*y^3 - 3888*x*y^2*z^4 - 24288*x*y^2*z^3 - 48036*x*y^2*z^2 - 39129*x*y^2*z - 11445*x*y^2 + 2064*x*y*z^5 + 15792*x*y*z^4 + 41124*x*y*z^3 + 49500*x*y*z^2 + 28395*x*y*z + 6318*x*y - 448*x*z^6 - 4032*x*z^5 - 12960*x*z^4 - 20484*x*z^3 - 17292*x*z^2 - 7512*x*z - 1315*x - 8*y^7 + 80*y^6*z + 128*y^6 - 336*y^5*z^2 - 1056*y^5*z - 702*y^5 + 768*y^4*z^3 + 3552*y^4*z^2 + 4668*y^4*z + 1905*y^4 - 1032*y^3*z^4 - 6240*y^3*z^3 - 12150*y^3*z^2 - 9779*y^3*z - 2828*y^3 + 816*y^2*z^5 + 6048*y^2*z^4 + 15504*y^2*z^3 + 18438*y^2*z^2 + 10461*y^2*z + 2302*y^2 - 352*y*z^6 - 3072*y*z^5 - 9720*y*z^4 - 15180*y*z^3 - 12678*y*z^2 - 5450*y*z - 946*y + 64*z^7 + 640*z^6 + 2400*z^5 + 4616*z^4 + 5048*z^3 + 3184*z^2 + 1081*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -1, -1, 2 ], [ -3, -2, 2 ], [ -1, -1, 1 ] ], "source_shift": [ 2, 1, 1 ], "target_matrix": [ [ 1, 2, 1 ], [ 1, 1, 2 ], [ 1, 1, 1 ] ], "target_shift": [ -270, -139, -155 ], "collision_preimages": [ [ "-3/2", "2", "-3/4" ], [ "27/2", "-51/2", "-13/2" ], [ "21/2", "-49/2", "-17/2" ] ], "collision_image": [ "-1081/4", "-557/4", "-621/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.048048116001155 ], "surface_features": [ 4.0638593819831845, 4.897901874268228, 3.71717102683231, 0.0, 4.163310684815346, 5087 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C046", "pair": 46, "family": "counterexample", "mapping": "F(x,y,z)=(-x^7 - 10*x^6*y + 8*x^6*z + 19*x^6 - 42*x^5*y^2 + 69*x^5*y*z + 165*x^5*y - 27*x^5*z^2 - 129*x^5*z - 153*x^5 - 96*x^4*y^3 + 243*x^4*y^2*z + 588*x^4*y^2 - 195*x^4*y*z^2 - 936*x^4*y*z - 1122*x^4*y + 50*x^4*z^3 + 360*x^4*z^2 + 858*x^4*z + 678*x^4 - 129*x^3*y^4 + 447*x^3*y^3*z + 1101*x^3*y^3 - 552*x^3*y^2*z^2 - 2673*x^3*y^2*z - 3249*x^3*y^2 + 290*x^3*y*z^3 + 2094*x^3*y*z^2 + 5034*x^3*y*z + 4029*x^3*y - 55*x^3*z^4 - 530*x^3*z^3 - 1902*x^3*z^2 - 3020*x^3*z - 1789*x^3 - 102*x^2*y^5 + 453*x^2*y^4*z + 1143*x^2*y^4 - 765*x^2*y^3*z^2 - 3756*x^2*y^3*z - 4644*x^2*y^3 + 618*x^2*y^2*z^3 + 4491*x^2*y^2*z^2 + 10920*x^2*y^2*z + 8871*x^2*y^2 - 240*x^2*y*z^4 - 2316*x^2*y*z^3 - 8370*x^2*y*z^2 - 13437*x^2*y*z - 8076*x^2*y + 36*x^2*z^5 + 435*x^2*z^4 + 2088*x^2*z^3 + 4992*x^2*z^2 + 5946*x^2*z + 2818*x^2 - 44*x*y^6 + 240*x*y^5*z + 624*x*y^5 - 519*x*y^4*z^2 - 2598*x*y^4*z - 3276*x*y^4 + 573*x*y^3*z^3 + 4209*x*y^3*z^2 + 10380*x*y^3*z + 8574*x*y^3 - 342*x*y^2*z^4 - 3315*x*y^2*z^3 - 12093*x*y^2*z^2 - 19662*x*y^2*z - 12009*x*y^2 + 105*x*y*z^5 + 1269*x*y*z^4 + 6126*x*y*z^3 + 14787*x*y*z^2 + 17841*x*y*z + 8593*x*y - 13*x*z^6 - 189*x*z^5 - 1137*x*z^4 - 3636*x*z^3 - 6525*x*z^2 - 6227*x*z - 2463*x - 8*y^7 + 52*y^6*z + 140*y^6 - 138*y^5*z^2 - 708*y^5*z - 912*y^5 + 195*y^4*z^3 + 1455*y^4*z^2 + 3648*y^4*z + 3066*y^4 - 159*y^3*z^4 - 1554*y^3*z^3 - 5736*y^3*z^2 - 9460*y^3*z - 5876*y^3 + 75*y^2*z^5 + 909*y^2*z^4 + 4422*y^2*z^3 + 10791*y^2*z^2 + 13203*y^2*z + 6475*y^2 - 19*y*z^6 - 276*y*z^5 - 1668*y*z^4 - 5379*y*z^3 - 9765*y*z^2 - 9454*y*z - 3806*y + 2*z^7 + 34*z^6 + 246*z^5 + 986*z^4 + 2368*z^3 + 3409*z^2 + 2722*z + 1, 2*x^7 + 20*x^6*y - 16*x^6*z - 35*x^6 + 84*x^5*y^2 - 138*x^5*y*z - 306*x^5*y + 54*x^5*z^2 + 237*x^5*z + 264*x^5 + 192*x^4*y^3 - 486*x^4*y^2*z - 1098*x^4*y^2 + 390*x^4*y*z^2 + 1731*x^4*y*z + 1953*x^4*y - 100*x^4*z^3 - 660*x^4*z^2 - 1476*x^4*z - 1113*x^4 + 258*x^3*y^4 - 894*x^3*y^3*z - 2070*x^3*y^3 + 1104*x^3*y^2*z^2 + 4977*x^3*y^2*z + 5706*x^3*y^2 - 580*x^3*y*z^3 - 3864*x^3*y*z^2 - 8736*x^3*y*z - 6672*x^3*y + 110*x^3*z^4 + 970*x^3*z^3 + 3264*x^3*z^2 + 4945*x^3*z + 2834*x^3 + 204*x^2*y^5 - 906*x^2*y^4*z - 2163*x^2*y^4 + 1530*x^2*y^3*z^2 + 7041*x^2*y^3*z + 8229*x^2*y^3 - 1236*x^2*y^2*z^3 - 8343*x^2*y^2*z^2 - 19122*x^2*y^2*z - 14826*x^2*y^2 + 480*x^2*y*z^4 + 4266*x^2*y*z^3 + 14490*x^2*y*z^2 + 22197*x^2*y*z + 12891*x^2*y - 72*x^2*z^5 - 795*x^2*z^4 - 3576*x^2*z^3 - 8157*x^2*z^2 - 9402*x^2*z - 4361*x^2 + 88*x*y^6 - 480*x*y^5*z - 1188*x*y^5 + 1038*x*y^4*z^2 + 4902*x*y^4*z + 5856*x*y^4 - 1146*x*y^3*z^3 - 7872*x*y^3*z^2 - 18342*x*y^3*z - 14466*x*y^3 + 684*x*y^2*z^4 + 6147*x*y^2*z^3 + 21126*x*y^2*z^2 + 32787*x*y^2*z + 19332*x*y^2 - 210*x*y*z^5 - 2334*x*y*z^4 - 10584*x*y*z^3 - 24378*x*y*z^2 - 28428*x*y*z - 13376*x*y + 26*x*z^6 + 345*x*z^5 + 1944*x*z^4 + 5931*x*z^3 + 10302*x*z^2 + 9625*x*z + 3758*x + 16*y^7 - 104*y^6*z - 268*y^6 + 276*y^5*z^2 + 1344*y^5*z + 1644*y^5 - 390*y^4*z^3 - 2739*y^4*z^2 - 6504*y^4*z - 5223*y^4 + 318*y^3*z^4 + 2901*y^3*z^3 + 10113*y^3*z^2 + 15929*y^3*z + 9547*y^3 - 150*y^2*z^5 - 1683*y^2*z^4 - 7710*y^2*z^3 - 17961*y^2*z^2 - 21222*y^2*z - 10151*y^2 + 38*y*z^6 + 507*y*z^5 + 2877*y*z^4 + 8853*y*z^3 + 15537*y*z^2 + 14699*y*z + 5831*y - 4*z^7 - 62*z^6 - 420*z^5 - 1606*z^4 - 3734*z^3 - 5264*z^2 - 4150*z - 1, 3*x^7 + 30*x^6*y - 24*x^6*z - 54*x^6 + 126*x^5*y^2 - 207*x^5*y*z - 471*x^5*y + 81*x^5*z^2 + 366*x^5*z + 417*x^5 + 288*x^4*y^3 - 729*x^4*y^2*z - 1686*x^4*y^2 + 585*x^4*y*z^2 + 2667*x^4*y*z + 3075*x^4*y - 150*x^4*z^3 - 1020*x^4*z^2 - 2334*x^4*z - 1792*x^4 + 387*x^3*y^4 - 1341*x^3*y^3*z - 3171*x^3*y^3 + 1656*x^3*y^2*z^2 + 7650*x^3*y^2*z + 8955*x^3*y^2 - 870*x^3*y*z^3 - 5958*x^3*y*z^2 - 13770*x^3*y*z - 10705*x^3*y + 165*x^3*z^4 + 1500*x^3*z^3 + 5166*x^3*z^2 + 7970*x^3*z + 4633*x^3 + 306*x^2*y^5 - 1359*x^2*y^4*z - 3306*x^2*y^4 + 2295*x^2*y^3*z^2 + 10797*x^2*y^3*z + 12873*x^2*y^3 - 1854*x^2*y^2*z^3 - 12834*x^2*y^2*z^2 - 30042*x^2*y^2*z - 23703*x^2*y^2 + 720*x^2*y*z^4 + 6582*x^2*y*z^3 + 22860*x^2*y*z^2 + 35649*x^2*y*z + 21000*x^2*y - 108*x^2*z^5 - 1230*x^2*z^4 - 5664*x^2*z^3 - 13158*x^2*z^2 - 15384*x^2*z - 7215*x^2 + 132*x*y^6 - 720*x*y^5*z - 1812*x*y^5 + 1557*x*y^4*z^2 + 7500*x*y^4*z + 9132*x*y^4 - 1719*x*y^3*z^3 - 12081*x*y^3*z^2 - 28722*x*y^3*z - 23044*x*y^3 + 1026*x*y^2*z^4 + 9462*x*y^2*z^3 + 33219*x*y^2*z^2 + 52464*x*y^2*z + 31377*x*y^2 - 315*x*y*z^5 - 3603*x*y*z^4 - 16710*x*y*z^3 - 39183*x*y*z^2 - 46347*x*y*z - 22053*x*y + 39*x*z^6 + 534*x*z^5 + 3081*x*z^4 + 9574*x*z^3 + 16869*x*z^2 + 15936*x*z + 6275*x + 24*y^7 - 156*y^6*z - 408*y^6 + 414*y^5*z^2 + 2052*y^5*z + 2556*y^5 - 585*y^4*z^3 - 4194*y^4*z^2 - 10152*y^4*z - 8290*y^4 + 477*y^3*z^4 + 4455*y^3*z^3 + 15849*y^3*z^2 + 25394*y^3*z + 15436*y^3 - 225*y^2*z^5 - 2592*y^2*z^4 - 12132*y^2*z^3 - 28761*y^2*z^2 - 34467*y^2*z - 16674*y^2 + 57*y*z^6 + 783*y*z^5 + 4545*y*z^4 + 14239*y*z^3 + 25347*y*z^2 + 24249*y*z + 9703*y - 6*z^7 - 96*z^6 - 666*z^5 - 2594*z^4 - 6118*z^3 - 8721*z^2 - 6934*z - 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -1, -1, 1 ], [ -1, -2, 1 ], [ -1, -1, 2 ] ], "source_shift": [ 2, 2, 1 ], "target_matrix": [ [ 1, 1, 2 ], [ -2, -1, -1 ], [ -3, -2, -2 ] ], "target_shift": [ -924, 1397, 2350 ], "collision_preimages": [ [ "11/4", "0", "3/4" ], [ "5", "5/2", "13/2" ], [ "14", "-5/2", "17/2" ] ], "collision_image": [ "-3697/4", "2795/2", "9403/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0141588620026596 ], "surface_features": [ 3.7457083263077395, 4.71986139932529, 3.3140779917792127, 0.0, 3.9064971416056737, 4916 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B157", "pair": 157, "family": "bijection", "mapping": "F(x,y,z)=(-3*x^7 - 21*x^6*y + 21*x^6*z + 39*x^6 - 63*x^5*y^2 + 126*x^5*y*z + 234*x^5*y - 63*x^5*z^2 - 240*x^5*z - 231*x^5 - 105*x^4*y^3 + 315*x^4*y^2*z + 585*x^4*y^2 - 315*x^4*y*z^2 - 1200*x^4*y*z - 1152*x^4*y + 105*x^4*z^3 + 615*x^4*z^2 + 1209*x^4*z + 795*x^4 - 105*x^3*y^4 + 420*x^3*y^3*z + 780*x^3*y^3 - 630*x^3*y^2*z^2 - 2400*x^3*y^2*z - 2298*x^3*y^2 + 420*x^3*y*z^3 + 2460*x^3*y*z^2 + 4824*x^3*y*z + 3156*x^3*y - 105*x^3*z^4 - 840*x^3*z^3 - 2529*x^3*z^2 - 3399*x^3*z - 1690*x^3 - 63*x^2*y^5 + 315*x^2*y^4*z + 585*x^2*y^4 - 630*x^2*y^3*z^2 - 2400*x^2*y^3*z - 2292*x^2*y^3 + 630*x^2*y^2*z^3 + 3690*x^2*y^2*z^2 + 7218*x^2*y^2*z + 4698*x^2*y^2 - 315*x^2*y*z^4 - 2520*x^2*y*z^3 - 7569*x^2*y*z^2 - 10119*x^2*y*z - 4995*x^2*y + 63*x^2*z^5 + 645*x^2*z^4 + 2643*x^2*z^3 + 5442*x^2*z^2 + 5541*x^2*z + 2196*x^2 - 21*x*y^6 + 126*x*y^5*z + 234*x*y^5 - 315*x*y^4*z^2 - 1200*x*y^4*z - 1143*x*y^4 + 420*x*y^3*z^3 + 2460*x*y^3*z^2 + 4800*x*y^3*z + 3108*x*y^3 - 315*x*y^2*z^4 - 2520*x*y^2*z^3 - 7551*x*y^2*z^2 - 10041*x*y^2*z - 4920*x*y^2 + 126*x*y*z^5 + 1290*x*y*z^4 + 5274*x*y*z^3 + 10800*x*y*z^2 + 10908*x*y*z + 4284*x*y - 21*x*z^6 - 264*x*z^5 - 1380*x*z^4 - 3867*x*z^3 - 6048*x*z^2 - 4908*x*z - 1601*x - 3*y^7 + 21*y^6*z + 39*y^6 - 63*y^5*z^2 - 240*y^5*z - 228*y^5 + 105*y^4*z^3 + 615*y^4*z^2 + 1197*y^4*z + 771*y^4 - 105*y^3*z^4 - 840*y^3*z^3 - 2511*y^3*z^2 - 3321*y^3*z - 1615*y^3 + 63*y^2*z^5 + 645*y^2*z^4 + 2631*y^2*z^3 + 5358*y^2*z^2 + 5367*y^2*z + 2088*y^2 - 21*y*z^6 - 264*y*z^5 - 1377*y*z^4 - 3837*y*z^3 - 5946*y*z^2 - 4773*y*z - 1543*y + 3*z^7 + 45*z^6 + 288*z^5 + 1029*z^4 + 2197*z^3 + 2736*z^2 + 1827*z + 1, -3*x^7 - 21*x^6*y + 21*x^6*z + 39*x^6 - 63*x^5*y^2 + 126*x^5*y*z + 234*x^5*y - 63*x^5*z^2 - 240*x^5*z - 231*x^5 - 105*x^4*y^3 + 315*x^4*y^2*z + 585*x^4*y^2 - 315*x^4*y*z^2 - 1200*x^4*y*z - 1152*x^4*y + 105*x^4*z^3 + 615*x^4*z^2 + 1209*x^4*z + 795*x^4 - 105*x^3*y^4 + 420*x^3*y^3*z + 780*x^3*y^3 - 630*x^3*y^2*z^2 - 2400*x^3*y^2*z - 2298*x^3*y^2 + 420*x^3*y*z^3 + 2460*x^3*y*z^2 + 4824*x^3*y*z + 3156*x^3*y - 105*x^3*z^4 - 840*x^3*z^3 - 2529*x^3*z^2 - 3399*x^3*z - 1690*x^3 - 63*x^2*y^5 + 315*x^2*y^4*z + 585*x^2*y^4 - 630*x^2*y^3*z^2 - 2400*x^2*y^3*z - 2292*x^2*y^3 + 630*x^2*y^2*z^3 + 3690*x^2*y^2*z^2 + 7218*x^2*y^2*z + 4698*x^2*y^2 - 315*x^2*y*z^4 - 2520*x^2*y*z^3 - 7569*x^2*y*z^2 - 10119*x^2*y*z - 4995*x^2*y + 63*x^2*z^5 + 645*x^2*z^4 + 2643*x^2*z^3 + 5442*x^2*z^2 + 5541*x^2*z + 2195*x^2 - 21*x*y^6 + 126*x*y^5*z + 234*x*y^5 - 315*x*y^4*z^2 - 1200*x*y^4*z - 1143*x*y^4 + 420*x*y^3*z^3 + 2460*x*y^3*z^2 + 4800*x*y^3*z + 3108*x*y^3 - 315*x*y^2*z^4 - 2520*x*y^2*z^3 - 7551*x*y^2*z^2 - 10041*x*y^2*z - 4920*x*y^2 + 126*x*y*z^5 + 1290*x*y*z^4 + 5274*x*y*z^3 + 10800*x*y*z^2 + 10908*x*y*z + 4282*x*y - 21*x*z^6 - 264*x*z^5 - 1380*x*z^4 - 3867*x*z^3 - 6048*x*z^2 - 4906*x*z - 1597*x - 3*y^7 + 21*y^6*z + 39*y^6 - 63*y^5*z^2 - 240*y^5*z - 228*y^5 + 105*y^4*z^3 + 615*y^4*z^2 + 1197*y^4*z + 771*y^4 - 105*y^3*z^4 - 840*y^3*z^3 - 2511*y^3*z^2 - 3321*y^3*z - 1615*y^3 + 63*y^2*z^5 + 645*y^2*z^4 + 2631*y^2*z^3 + 5358*y^2*z^2 + 5367*y^2*z + 2087*y^2 - 21*y*z^6 - 264*y*z^5 - 1377*y*z^4 - 3837*y*z^3 - 5946*y*z^2 - 4771*y*z - 1539*y + 3*z^7 + 45*z^6 + 288*z^5 + 1029*z^4 + 2197*z^3 + 2735*z^2 + 1822*z + 1, -6*x^7 - 42*x^6*y + 42*x^6*z + 78*x^6 - 126*x^5*y^2 + 252*x^5*y*z + 468*x^5*y - 126*x^5*z^2 - 480*x^5*z - 462*x^5 - 210*x^4*y^3 + 630*x^4*y^2*z + 1170*x^4*y^2 - 630*x^4*y*z^2 - 2400*x^4*y*z - 2304*x^4*y + 210*x^4*z^3 + 1230*x^4*z^2 + 2418*x^4*z + 1590*x^4 - 210*x^3*y^4 + 840*x^3*y^3*z + 1560*x^3*y^3 - 1260*x^3*y^2*z^2 - 4800*x^3*y^2*z - 4596*x^3*y^2 + 840*x^3*y*z^3 + 4920*x^3*y*z^2 + 9648*x^3*y*z + 6312*x^3*y - 210*x^3*z^4 - 1680*x^3*z^3 - 5058*x^3*z^2 - 6798*x^3*z - 3382*x^3 - 126*x^2*y^5 + 630*x^2*y^4*z + 1170*x^2*y^4 - 1260*x^2*y^3*z^2 - 4800*x^2*y^3*z - 4584*x^2*y^3 + 1260*x^2*y^2*z^3 + 7380*x^2*y^2*z^2 + 14436*x^2*y^2*z + 9396*x^2*y^2 - 630*x^2*y*z^4 - 5040*x^2*y*z^3 - 15138*x^2*y*z^2 - 20238*x^2*y*z - 9996*x^2*y + 126*x^2*z^5 + 1290*x^2*z^4 + 5286*x^2*z^3 + 10884*x^2*z^2 + 11088*x^2*z + 4397*x^2 - 42*x*y^6 + 252*x*y^5*z + 468*x*y^5 - 630*x*y^4*z^2 - 2400*x*y^4*z - 2286*x*y^4 + 840*x*y^3*z^3 + 4920*x*y^3*z^2 + 9600*x*y^3*z + 6216*x*y^3 - 630*x*y^2*z^4 - 5040*x*y^2*z^3 - 15102*x*y^2*z^2 - 20082*x*y^2*z - 9846*x*y^2 + 252*x*y*z^5 + 2580*x*y*z^4 + 10548*x*y*z^3 + 21600*x*y*z^2 + 21828*x*y*z + 8578*x*y - 42*x*z^6 - 528*x*z^5 - 2760*x*z^4 - 7734*x*z^3 - 12102*x*z^2 - 9826*x*z - 3206*x - 6*y^7 + 42*y^6*z + 78*y^6 - 126*y^5*z^2 - 480*y^5*z - 456*y^5 + 210*y^4*z^3 + 1230*y^4*z^2 + 2394*y^4*z + 1542*y^4 - 210*y^3*z^4 - 1680*y^3*z^3 - 5022*y^3*z^2 - 6642*y^3*z - 3232*y^3 + 126*y^2*z^5 + 1290*y^2*z^4 + 5262*y^2*z^3 + 10716*y^2*z^2 + 10740*y^2*z + 4181*y^2 - 42*y*z^6 - 528*y*z^5 - 2754*y*z^4 - 7674*y*z^3 - 11898*y*z^2 - 9556*y*z - 3088*y + 6*z^7 + 90*z^6 + 576*z^5 + 2058*z^4 + 4396*z^3 + 5477*z^2 + 3653*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -1, -1, 1 ], [ -1, -1, 2 ], [ -2, -1, 1 ] ], "source_shift": [ 1, 2, 2 ], "target_matrix": [ [ 1, 2, 1 ], [ 1, 1, 1 ], [ 2, 1, 1 ] ], "target_shift": [ -502, -498, -1003 ], "inverse_recipe": { "p": 2, "q": 3, "f": "x**2 + x", "g": "x**3 + x**2 + x", "h": "y**3 + y**2*z + y*z + y", "polynomials": { "f": [ [ [ 2, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "g": [ [ [ 3, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "h": [ [ [ 0, 2, 1 ], 1 ], [ [ 0, 1, 0 ], 1 ], [ [ 0, 1, 1 ], 1 ], [ [ 0, 3, 0 ], 1 ] ] }, "h_multiplier": 3 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0293189589974645 ], "surface_features": [ 3.4291064574663292, 4.339013945111108, 3.080806804334362, 0.0, 3.5600513517906056, 4842 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C147", "pair": 147, "family": "counterexample", "mapping": "F(x,y,z)=(108*x^7 - 594*x^6*y - 702*x^6*z + 1458*x^6 + 1386*x^5*y^2 + 3294*x^5*y*z - 6759*x^5*y + 1944*x^5*z^2 - 8289*x^5*z + 7182*x^5 - 1778*x^4*y^3 - 6372*x^4*y^2*z + 12927*x^4*y^2 - 7560*x^4*y*z^2 + 31914*x^4*y*z - 27225*x^4*y - 2970*x^4*z^3 + 19575*x^4*z^2 - 34290*x^4*z + 17946*x^4 + 1354*x^3*y^4 + 6502*x^3*y^3*z - 13057*x^3*y^3 + 11628*x^3*y^2*z^2 - 48651*x^3*y^2*z + 40896*x^3*y^2 + 9180*x^3*y*z^3 - 60066*x^3*y*z^2 + 103680*x^3*y*z - 53373*x^3*y + 2700*x^3*z^4 - 24570*x^3*z^3 + 65340*x^3*z^2 - 68751*x^3*z + 25164*x^3 - 612*x^2*y^5 - 3690*x^2*y^4*z + 7347*x^2*y^4 - 8838*x^2*y^3*z^2 + 36708*x^2*y^3*z - 30441*x^2*y^3 - 10512*x^2*y^2*z^3 + 68391*x^2*y^2*z^2 - 116436*x^2*y^2*z + 59025*x^2*y^2 - 6210*x^2*y*z^4 + 56304*x^2*y*z^3 - 147690*x^2*y*z^2 + 152922*x^2*y*z - 55041*x^2*y - 1458*x^2*z^5 + 17280*x^2*z^4 - 62100*x^2*z^3 + 98577*x^2*z^2 - 72324*x^2*z + 19956*x^2 + 152*x*y^6 + 1104*x*y^5*z - 2184*x*y^5 + 3318*x*y^4*z^2 - 13710*x*y^4*z + 11232*x*y^4 + 5282*x*y^3*z^3 - 34245*x*y^3*z^2 + 57582*x*y^3*z - 28781*x*y^3 + 4698*x*y^2*z^4 - 42537*x*y^2*z^3 + 110184*x*y^2*z^2 - 112407*x*y^2*z + 39840*x*y^2 + 2214*x*y*z^5 - 26271*x*y*z^4 + 93240*x*y*z^3 - 145725*x*y*z^2 + 105168*x*y*z - 28551*x*y + 432*x*z^6 - 6453*x*z^5 + 29430*x*z^4 - 62685*x*z^3 + 69156*x*z^2 - 38169*x*z + 8296*x - 16*y^7 - 136*y^6*z + 268*y^6 - 492*y^5*z^2 + 2028*y^5*z - 1644*y^5 - 982*y^4*z^3 + 6363*y^4*z^2 - 10584*y^4*z + 5223*y^4 - 1168*y^3*z^4 + 10594*y^3*z^3 - 27141*y^3*z^2 + 27318*y^3*z - 9547*y^3 - 828*y^2*z^5 + 9870*y^2*z^4 - 34644*y^2*z^3 + 53382*y^2*z^2 - 37950*y^2*z + 10151*y^2 - 324*y*z^6 + 4878*y*z^5 - 22005*y*z^4 + 46176*y*z^3 - 50127*y*z^2 + 27228*y*z - 5831*y - 54*z^7 + 999*z^6 - 5562*z^5 + 14913*z^4 - 21996*z^3 + 18213*z^2 - 7904*z + 1, 54*x^7 - 297*x^6*y - 351*x^6*z + 810*x^6 + 693*x^5*y^2 + 1647*x^5*y*z - 3771*x^5*y + 972*x^5*z^2 - 4590*x^5*z + 4590*x^5 - 889*x^4*y^3 - 3186*x^4*y^2*z + 7242*x^4*y^2 - 3780*x^4*y*z^2 + 17739*x^4*y*z - 17559*x^4*y - 1485*x^4*z^3 + 10800*x^4*z^2 - 21843*x^4*z + 13166*x^4 + 677*x^3*y^4 + 3251*x^3*y^3*z - 7343*x^3*y^3 + 5814*x^3*y^2*z^2 - 27138*x^3*y^2*z + 26604*x^3*y^2 + 4590*x^3*y*z^3 - 33246*x^3*y*z^2 + 66618*x^3*y*z - 39598*x^3*y + 1350*x^3*z^4 - 13500*x^3*z^3 + 41472*x^3*z^2 - 50293*x^3*z + 20860*x^3 - 306*x^2*y^5 - 1845*x^2*y^4*z + 4146*x^2*y^4 - 4419*x^2*y^3*z^2 + 20541*x^2*y^3*z - 19959*x^2*y^3 - 5256*x^2*y^2*z^3 + 37962*x^2*y^2*z^2 - 75417*x^2*y^2*z + 44232*x^2*y^2 - 3105*x^2*y*z^4 + 31014*x^2*y*z^3 - 94500*x^2*y*z^2 + 113073*x^2*y*z - 46104*x^2*y - 729*x^2*z^5 + 9450*x^2*z^4 - 39258*x^2*z^3 + 71883*x^2*z^2 - 59793*x^2*z + 18321*x^2 + 76*x*y^6 + 552*x*y^5*z - 1236*x*y^5 + 1659*x*y^4*z^2 - 7692*x*y^4*z + 7416*x*y^4 + 2641*x*y^3*z^3 - 19053*x*y^3*z^2 + 37566*x*y^3*z - 21759*x*y^3 + 2349*x*y^2*z^4 - 23478*x*y^2*z^3 + 71022*x*y^2*z^2 - 83907*x*y^2*z + 33675*x*y^2 + 1107*x*y*z^5 - 14391*x*y*z^4 + 59382*x*y*z^3 - 107352*x*y*z^2 + 87813*x*y*z - 26433*x*y + 216*x*z^6 - 3510*x*z^5 + 18522*x*z^4 - 45551*x*z^3 + 57006*x*z^2 - 34944*x*z + 8272*x - 8*y^7 - 68*y^6*z + 152*y^6 - 246*y^5*z^2 + 1140*y^5*z - 1092*y^5 - 491*y^4*z^3 + 3546*y^4*z^2 - 6948*y^4*z + 3979*y^4 - 584*y^3*z^4 + 5855*y^3*z^3 - 17607*y^3*z^2 + 20560*y^3*z - 8133*y^3 - 414*y^2*z^5 + 5412*y^2*z^4 - 22209*y^2*z^3 + 39675*y^2*z^2 - 31959*y^2*z + 9466*y^2 - 162*y*z^6 + 2655*y*z^5 - 13941*y*z^4 + 33877*y*z^3 - 41709*y*z^2 + 25125*y*z - 5851*y - 27*z^7 + 540*z^6 - 3483*z^5 + 10795*z^4 - 18073*z^3 + 16623*z^2 - 7855*z + 1, 54*x^7 - 297*x^6*y - 351*x^6*z + 756*x^6 + 693*x^5*y^2 + 1647*x^5*y*z - 3510*x^5*y + 972*x^5*z^2 - 4293*x^5*z + 3924*x^5 - 889*x^4*y^3 - 3186*x^4*y^2*z + 6723*x^4*y^2 - 3780*x^4*y*z^2 + 16551*x^4*y*z - 14928*x^4*y - 1485*x^4*z^3 + 10125*x^4*z^2 - 18711*x^4*z + 10370*x^4 + 677*x^3*y^4 + 3251*x^3*y^3*z - 6800*x^3*y^3 + 5814*x^3*y^2*z^2 - 25263*x^3*y^2*z + 22500*x^3*y^2 + 4590*x^3*y*z^3 - 31104*x^3*y*z^2 + 56766*x^3*y*z - 30988*x^3*y + 1350*x^3*z^4 - 12690*x^3*z^3 + 35604*x^3*z^2 - 39679*x^3*z + 15334*x^3 - 306*x^2*y^5 - 1845*x^2*y^4*z + 3831*x^2*y^4 - 4419*x^2*y^3*z^2 + 19083*x^2*y^3*z - 16800*x^2*y^3 - 5256*x^2*y^2*z^3 + 35451*x^2*y^2*z^2 - 63951*x^2*y^2*z + 34416*x^2*y^2 - 3105*x^2*y*z^4 + 29106*x^2*y*z^3 - 80730*x^2*y*z^2 + 88659*x^2*y*z - 33696*x^2*y - 729*x^2*z^5 + 8910*x^2*z^4 - 33786*x^2*z^3 + 56817*x^2*z^2 - 44019*x^2*z + 12735*x^2 + 76*x*y^6 + 552*x*y^5*z - 1140*x*y^5 + 1659*x*y^4*z^2 - 7134*x*y^4*z + 6216*x*y^4 + 2641*x*y^3*z^3 - 17766*x*y^3*z^2 + 31716*x*y^3*z - 16845*x*y^3 + 2349*x*y^2*z^4 - 22005*x*y^2*z^3 + 60402*x*y^2*z^2 - 65433*x*y^2*z + 24489*x*y^2 + 1107*x*y*z^5 - 13554*x*y*z^4 + 50874*x*y*z^3 - 84354*x*y*z^2 + 64293*x*y*z - 18288*x*y + 216*x*z^6 - 3321*x*z^5 + 15984*x*z^4 - 36077*x*z^3 + 42036*x*z^2 - 24327*x*z + 5494*x - 8*y^7 - 68*y^6*z + 140*y^6 - 246*y^5*z^2 + 1056*y^5*z - 912*y^5 - 491*y^4*z^3 + 3303*y^4*z^2 - 5844*y^4*z + 3067*y^4 - 584*y^3*z^4 + 5483*y^3*z^3 - 14916*y^3*z^2 + 15958*y^3*z - 5889*y^3 - 414*y^2*z^5 + 5094*y^2*z^4 - 18951*y^2*z^3 + 31017*y^2*z^2 - 23289*y^2*z + 6523*y^2 - 162*y*z^6 + 2511*y*z^5 - 11982*y*z^4 + 26683*y*z^3 - 30597*y*z^2 + 17415*y*z - 3872*y - 27*z^7 + 513*z^6 - 3015*z^5 + 8569*z^4 - 13351*z^3 + 11592*z^2 - 5227*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ 1, -1, -1 ], [ 3, -2, -3 ], [ 2, -1, -1 ] ], "source_shift": [ 2, 2, 1 ], "target_matrix": [ [ 2, 1, 1 ], [ 1, 2, 1 ], [ 1, 1, 1 ] ], "target_shift": [ -1397, -1488, -952 ], "collision_preimages": [ [ "3/4", "4", "-5/4" ], [ "13/2", "-1/2", "8" ], [ "17/2", "17/2", "3" ] ], "collision_image": [ "-2795/2", "-5953/4", "-3809/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 0.9932253629995103 ], "surface_features": [ 4.318238919614732, 5.184469969331844, 3.948119424380536, 0.0, 4.436420768075769, 5145 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C099", "pair": 99, "family": "counterexample", "mapping": "F(x,y,z)=(-8*x^7 + 44*x^6*y + 52*x^6*z - 92*x^6 - 102*x^5*y^2 - 240*x^5*y*z + 432*x^5*y - 138*x^5*z^2 + 504*x^5*z - 432*x^5 + 129*x^4*y^3 + 453*x^4*y^2*z - 831*x^4*y^2 + 519*x^4*y*z^2 - 1926*x^4*y*z + 1692*x^4*y + 195*x^4*z^3 - 1095*x^4*z^2 + 1944*x^4*z - 1082*x^4 - 96*x^3*y^4 - 447*x^3*y^3*z + 837*x^3*y^3 - 765*x^3*y^2*z^2 + 2889*x^3*y^2*z - 2604*x^3*y^2 - 573*x^3*y*z^3 + 3267*x^3*y*z^2 - 5940*x^3*y*z + 3409*x^3*y - 159*x^3*z^4 + 1215*x^3*z^3 - 3336*x^3*z^2 + 3865*x^3*z - 1574*x^3 + 42*x^2*y^5 + 243*x^2*y^4*z - 465*x^2*y^4 + 552*x^2*y^3*z^2 - 2124*x^2*y^3*z + 1965*x^2*y^3 + 618*x^2*y^2*z^3 - 3582*x^2*y^2*z^2 + 6672*x^2*y^2*z - 3951*x^2*y^2 + 342*x^2*y*z^4 - 2652*x^2*y*z^3 + 7449*x^2*y*z^2 - 8898*x^2*y*z + 3762*x^2*y + 75*x^2*z^5 - 729*x^2*z^4 + 2742*x^2*z^3 - 4947*x^2*z^2 + 4224*x^2*z - 1348*x^2 - 10*x*y^6 - 69*x*y^5*z + 135*x*y^5 - 195*x*y^4*z^2 + 765*x*y^4*z - 726*x*y^4 - 290*x*y^3*z^3 + 1710*x*y^3*z^2 - 3261*x*y^3*z + 1992*x*y^3 - 240*x*y^2*z^4 + 1890*x*y^2*z^3 - 5427*x*y^2*z^2 + 6678*x*y^2*z - 2934*x*y^2 - 105*x*y*z^5 + 1035*x*y*z^4 - 3975*x*y*z^3 + 7380*x*y*z^2 - 6549*x*y*z + 2186*x*y - 19*x*z^6 + 225*x*z^5 - 1083*x*z^4 + 2694*x*z^3 - 3615*x*z^2 + 2444*x*z - 648*x + y^7 + 8*y^6*z - 16*y^6 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7443*x*y^2*z^2 - 9912*x*y^2*z + 4656*x*y^2 + 105*x*y*z^5 - 1266*x*y*z^4 + 5469*x*y*z^3 - 10992*x*y*z^2 + 10449*x*y*z - 3755*x*y + 19*x*z^6 - 276*x*z^5 + 1494*x*z^4 - 4024*x*z^3 + 5793*x*z^2 - 4235*x*z + 1217*x - y^7 - 8*y^6*z + 19*y^6 - 27*y^5*z^2 + 129*y^5*z - 141*y^5 - 50*y^4*z^3 + 360*y^4*z^2 - 789*y^4*z + 537*y^4 - 55*y^3*z^4 + 530*y^3*z^3 - 1746*y^3*z^2 + 2390*y^3*z - 1162*y^3 - 36*y^2*z^5 + 435*y^2*z^4 - 1914*y^2*z^3 + 3948*y^2*z^2 - 3876*y^2*z + 1450*y^2 - 13*y*z^6 + 189*y*z^5 - 1041*y*z^4 + 2874*y*z^3 - 4266*y*z^2 + 3239*y*z - 971*y - 2*z^7 + 34*z^6 - 225*z^5 + 779*z^4 - 1552*z^3 + 1789*z^2 - 1098*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ 2, -1, -1 ], [ 1, -1, -1 ], [ 1, -1, -2 ] ], "source_shift": [ 2, 1, 1 ], "target_matrix": [ [ -1, -1, -2 ], [ -2, -3, -2 ], [ 1, 2, 1 ] ], "target_shift": [ 139, 426, -270 ], "collision_preimages": [ [ "-1", "-1/4", "1/4" ], [ "3/2", "12", "-8" ], [ "-7/2", "1", "-5" ] ], "collision_image": [ "557/4", "853/2", "-1081/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.106086405001406 ], "surface_features": [ 3.395334645848296, 4.348810920405178, 3.018076063645795, 0.0, 3.5364609989576214, 4816 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C020", "pair": 20, "family": "counterexample", "mapping": "F(x,y,z)=(-27*x^7 + 243*x^6*y - 162*x^6*z + 243*x^6 - 900*x^5*y^2 + 1233*x^5*y*z - 1827*x^5*y - 414*x^5*z^2 + 1233*x^5*z - 837*x^5 + 1772*x^4*y^3 - 3750*x^4*y^2*z + 5514*x^4*y^2 + 2589*x^4*y*z^2 - 7617*x^4*y*z + 5193*x^4*y - 584*x^4*z^3 + 2589*x^4*z^2 - 3501*x^4*z + 1474*x^4 - 2000*x^3*y^4 + 5816*x^3*y^3*z - 8552*x^3*y^3 - 6204*x^3*y^2*z^2 + 18120*x^3*y^2*z - 12432*x^3*y^2 + 2879*x^3*y*z^3 - 12609*x^3*y*z^2 + 17112*x^3*y*z - 7302*x^3*y - 491*x^3*z^4 + 2879*x^3*z^3 - 5814*x^3*z^2 + 4893*x^3*z - 1478*x^3 + 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+ 768*x*z^5 - 2859*x*z^4 + 5167*x*z^3 - 5052*x*z^2 + 2630*x*z - 587*x + 64*y^7 - 352*y^6*z + 640*y^6 + 816*y^5*z^2 - 2928*y^5*z + 2304*y^5 - 1032*y^4*z^3 + 5520*y^4*z^2 - 8592*y^4*z + 3864*y^4 + 768*y^3*z^4 - 5484*y^3*z^3 + 12636*y^3*z^2 - 11400*y^3*z + 3652*y^3 - 336*y^2*z^5 + 3024*y^2*z^4 - 9162*y^2*z^3 + 12408*y^2*z^2 - 8004*y^2*z + 2038*y^2 + 80*y*z^6 - 876*y*z^5 + 3276*y*z^4 - 5907*y*z^3 + 5736*y*z^2 - 2950*y*z + 639*y - 8*z^7 + 104*z^6 - 462*z^5 + 1038*z^4 - 1346*z^3 + 1045*z^2 - 461*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -3, 2, -2 ], [ -1, 2, -1 ], [ -1, 1, -1 ] ], "source_shift": [ 1, 1, 1 ], "target_matrix": [ [ 1, 1, 2 ], [ 1, 1, 1 ], [ 1, 2, 1 ] ], "target_shift": [ -51, -53, -87 ], "collision_preimages": [ [ "-3/2", "1/4", "3" ], [ "11", "-8", "-49/2" ], [ "13", "-5", "-47/2" ] ], "collision_image": [ "-205/4", "-213/4", "-349/4" ] }, 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1678*z - 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ 1, -1, -1 ], [ 1, -2, -1 ], [ 1, -1, -2 ] ], "source_shift": [ 2, 1, 2 ], "target_matrix": [ [ 2, 1, 1 ], [ -1, -2, -1 ], [ -1, -1, -1 ] ], "target_shift": [ -988, 491, 486 ], "inverse_recipe": { "p": 2, "q": 3, "f": "x**2", "g": "x**3", "h": "y**2*z", "polynomials": { "f": [ [ [ 2, 0, 0 ], 1 ] ], "g": [ [ [ 3, 0, 0 ], 1 ] ], "h": [ [ [ 0, 2, 1 ], 1 ] ] }, "h_multiplier": 2 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0476822679993347 ], "surface_features": [ 3.4170973946854746, 4.3637623588693755, 3.0170333392987803, 0.0, 3.566553952550106, 4811 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C175", "pair": 175, "family": "counterexample", "mapping": "F(x,y,z)=(16*x^7 - 88*x^6*y + 104*x^6*z - 224*x^6 + 204*x^5*y^2 - 480*x^5*y*z + 1044*x^5*y + 276*x^5*z^2 - 1212*x^5*z + 1212*x^5 - 258*x^4*y^3 + 906*x^4*y^2*z - 1992*x^4*y^2 - 1038*x^4*y*z^2 + 4596*x^4*y*z - 4680*x^4*y + 390*x^4*z^3 - 2604*x^4*z^2 + 5304*x^4*z - 3360*x^4 + 192*x^3*y^4 - 894*x^3*y^3*z + 1989*x^3*y^3 + 1530*x^3*y^2*z^2 - 6837*x^3*y^2*z + 7095*x^3*y^2 - 1146*x^3*y*z^3 + 7707*x^3*y*z^2 - 15990*x^3*y*z + 10366*x^3*y + 318*x^3*z^4 - 2859*x^3*z^3 + 8895*x^3*z^2 - 11550*x^3*z + 5336*x^3 - 84*x^2*y^5 + 486*x^2*y^4*z - 1095*x^2*y^4 - 1104*x^2*y^3*z^2 + 4983*x^2*y^3*z - 5271*x^2*y^3 + 1236*x^2*y^2*z^3 - 8379*x^2*y^2*z^2 + 17709*x^2*y^2*z - 11757*x^2*y^2 - 684*x^2*y*z^4 + 6189*x^2*y*z^3 - 19605*x^2*y*z^2 + 26070*x^2*y*z - 12384*x^2*y + 150*x^2*z^5 - 1698*x^2*z^4 + 7167*x^2*z^3 - 14313*x^2*z^2 + 13644*x^2*z - 4982*x^2 + 20*x*y^6 - 138*x*y^5*z + 315*x*y^5 + 390*x*y^4*z^2 - 1779*x*y^4*z + 1917*x*y^4 - 580*x*y^3*z^3 + 3966*x*y^3*z^2 - 8532*x*y^3*z + 5800*x*y^3 + 480*x*y^2*z^4 - 4374*x*y^2*z^3 + 14094*x*y^2*z^2 - 19182*x*y^2*z + 9372*x*y^2 - 210*x*y*z^5 + 2391*x*y*z^4 - 10260*x*y*z^3 + 20964*x*y*z^2 - 20562*x*y*z + 7759*x*y + 38*x*z^6 - 519*x*z^5 + 2781*x*z^4 - 7582*x*z^3 + 11190*x*z^2 - 8497*x*z + 2603*x - 2*y^7 + 16*y^6*z - 37*y^6 - 54*y^5*z^2 + 249*y^5*z - 273*y^5 + 100*y^4*z^3 - 690*y^4*z^2 + 1509*y^4*z - 1049*y^4 - 110*y^3*z^4 + 1010*y^3*z^3 - 3306*y^3*z^2 + 4598*y^3*z - 2308*y^3 + 72*y^2*z^5 - 825*y^2*z^4 + 3594*y^2*z^3 - 7500*y^2*z^2 + 7554*y^2*z - 2945*y^2 - 26*y*z^6 + 357*y*z^5 - 1941*y*z^4 + 5402*y*z^3 - 8184*y*z^2 + 6421*y*z - 2037*y + 4*z^7 - 64*z^6 + 417*z^5 - 1451*z^4 + 2938*z^3 - 3476*z^2 + 2229*z - 1, -8*x^7 + 44*x^6*y - 52*x^6*z + 100*x^6 - 102*x^5*y^2 + 240*x^5*y*z - 468*x^5*y - 138*x^5*z^2 + 540*x^5*z - 504*x^5 + 129*x^4*y^3 - 453*x^4*y^2*z + 897*x^4*y^2 + 519*x^4*y*z^2 - 2058*x^4*y*z + 1956*x^4*y - 195*x^4*z^3 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27*y^5*z^2 - 114*y^5*z + 117*y^5 - 50*y^4*z^3 + 315*y^4*z^2 - 645*y^4*z + 428*y^4 + 55*y^3*z^4 - 460*y^3*z^3 + 1410*y^3*z^2 - 1872*y^3*z + 904*y^3 - 36*y^2*z^5 + 375*y^2*z^4 - 1530*y^2*z^3 + 3048*y^2*z^2 - 2952*y^2*z + 1105*y^2 + 13*y*z^6 - 162*y*z^5 + 825*y*z^4 - 2192*y*z^3 + 3192*y*z^2 - 2402*y*z + 727*y - 2*z^7 + 29*z^6 - 177*z^5 + 588*z^4 - 1144*z^3 + 1297*z^2 - 792*z + 1, 8*x^7 - 44*x^6*y + 52*x^6*z - 100*x^6 + 102*x^5*y^2 - 240*x^5*y*z + 468*x^5*y + 138*x^5*z^2 - 540*x^5*z + 504*x^5 - 129*x^4*y^3 + 453*x^4*y^2*z - 897*x^4*y^2 - 519*x^4*y*z^2 + 2058*x^4*y*z - 1956*x^4*y + 195*x^4*z^3 - 1161*x^4*z^2 + 2208*x^4*z - 1342*x^4 + 96*x^3*y^4 - 447*x^3*y^3*z + 900*x^3*y^3 + 765*x^3*y^2*z^2 - 3078*x^3*y^2*z + 2982*x^3*y^2 - 573*x^3*y*z^3 + 3456*x^3*y*z^2 - 6696*x^3*y*z + 4161*x^3*y + 159*x^3*z^4 - 1278*x^3*z^3 + 3714*x^3*z^2 - 4625*x^3*z + 2082*x^3 - 42*x^2*y^5 + 243*x^2*y^4*z - 498*x^2*y^4 - 552*x^2*y^3*z^2 + 2256*x^2*y^3*z - 2229*x^2*y^3 + 618*x^2*y^2*z^3 - 3780*x^2*y^2*z^2 + 7464*x^2*y^2*z - 4746*x^2*y^2 - 342*x^2*y*z^4 + 2784*x^2*y*z^3 - 8241*x^2*y*z^2 + 10500*x^2*y*z - 4854*x^2*y + 75*x^2*z^5 - 762*x^2*z^4 + 3006*x^2*z^3 - 5754*x^2*z^2 + 5340*x^2*z - 1918*x^2 + 10*x*y^6 - 69*x*y^5*z + 144*x*y^5 + 195*x*y^4*z^2 - 810*x*y^4*z + 816*x*y^4 - 290*x*y^3*z^3 + 1800*x*y^3*z^2 - 3621*x*y^3*z + 2356*x*y^3 + 240*x*y^2*z^4 - 1980*x*y^2*z^3 + 5967*x*y^2*z^2 - 7776*x*y^2*z + 3693*x*y^2 - 105*x*y*z^5 + 1080*x*y*z^4 - 4335*x*y*z^3 + 8484*x*y*z^2 - 8091*x*y*z + 2999*x*y + 19*x*z^6 - 234*x*z^5 + 1173*x*z^4 - 3064*x*z^3 + 4398*x*z^2 - 3281*x*z + 994*x - y^7 + 8*y^6*z - 17*y^6 - 27*y^5*z^2 + 114*y^5*z - 117*y^5 + 50*y^4*z^3 - 315*y^4*z^2 + 645*y^4*z - 429*y^4 - 55*y^3*z^4 + 460*y^3*z^3 - 1410*y^3*z^2 + 1877*y^3*z - 915*y^3 + 36*y^2*z^5 - 375*y^2*z^4 + 1530*y^2*z^3 - 3057*y^2*z^2 + 2991*y^2*z - 1144*y^2 - 13*y*z^6 + 162*y*z^5 - 825*y*z^4 + 2199*y*z^3 - 3237*y*z^2 + 2492*y*z - 781*y + 2*z^7 - 29*z^6 + 177*z^5 - 590*z^4 + 1161*z^3 - 1348*z^2 + 854*z - 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -2, 1, -1 ], [ -1, 1, -1 ], [ -1, 1, -2 ] ], "source_shift": [ 2, 1, 2 ], "target_matrix": [ [ -2, -3, -3 ], [ 1, 1, 2 ], [ -1, -1, -1 ] ], "target_shift": [ 602, -204, 228 ], "collision_preimages": [ [ "1", "5/4", "5/4" ], [ "-3/2", "-11", "-7" ], [ "7/2", "0", "-4" ] ], "collision_image": [ "1205/2", "-817/4", "913/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 0.9709862659983628 ], "surface_features": [ 3.4132205740412673, 4.416141031168329, 3.03342375548695, 0.0, 3.5757977841962547, 4819 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C052", "pair": 52, "family": "counterexample", "mapping": "F(x,y,z)=(-16*x^7 + 104*x^6*y - 176*x^6*z + 236*x^6 - 288*x^5*y^2 + 972*x^5*y*z - 1308*x^5*y - 816*x^5*z^2 + 2172*x^5*z - 1188*x^5 + 440*x^4*y^3 - 2220*x^4*y^2*z + 3000*x^4*y^2 + 3714*x^4*y*z^2 - 9936*x^4*y*z + 5490*x^4*y - 2060*x^4*z^3 + 8187*x^4*z^2 - 9018*x^4*z + 3077*x^4 - 400*x^3*y^4 + 2680*x^3*y^3*z - 3640*x^3*y^3 - 6696*x^3*y^2*z^2 + 18024*x^3*y^2*z - 10080*x^3*y^2 + 7393*x^3*y*z^3 - 29601*x^3*y*z^2 + 33063*x^3*y*z - 11447*x^3*y - 3043*x^3*z^4 + 16126*x^3*z^3 - 27024*x^3*z^2 + 18747*x^3*z - 4696*x^3 + 216*x^2*y^5 - 1800*x^2*y^4*z + 2460*x^2*y^4 + 5964*x^2*y^3*z^2 - 16176*x^2*y^3*z + 9180*x^2*y^3 - 9819*x^2*y^2*z^3 + 39681*x^2*y^2*z^2 - 45081*x^2*y^2*z + 15879*x^2*y^2 + 8031*x^2*y*z^4 - 43044*x^2*y*z^3 + 73563*x^2*y*z^2 - 52017*x^2*y*z + 13239*x^2*y - 2610*x^2*z^5 + 17424*x^2*z^4 - 39903*x^2*z^3 + 42531*x^2*z^2 - 21675*x^2*z + 4273*x^2 - 64*x*y^6 + 636*x*y^5*z - 876*x*y^5 - 2616*x*y^4*z^2 + 7164*x*y^4*z - 4140*x*y^4 + 5699*x*y^3*z^3 - 23307*x*y^3*z^2 + 27045*x*y^3*z - 9725*x*y^3 - 6933*x*y^2*z^4 + 37710*x*y^2*z^3 - 66054*x*y^2*z^2 + 47793*x*y^2*z - 12390*x*y^2 + 4464*x*y*z^5 - 30348*x*y*z^4 + 71514*x*y*z^3 - 78180*x*y*z^2 + 40629*x*y*z - 8129*x*y - 1188*x*z^6 + 9720*x*z^5 - 28968*x*z^4 + 42572*x*z^3 - 33258*x*z^2 + 13270*x*z - 2136*x + 8*y^7 - 92*y^6*z + 128*y^6 + 450*y^5*z^2 - 1248*y^5*z + 738*y^5 - 1213*y^4*z^3 + 5040*y^4*z^2 - 6009*y^4*z + 2216*y^4 + 1945*y^3*z^4 - 10792*y^3*z^3 + 19515*y^3*z^2 - 14523*y^3*z + 3847*y^3 - 1854*y^2*z^5 + 12924*y^2*z^4 - 31611*y^2*z^3 + 35649*y^2*z^2 - 18954*y^2*z + 3856*y^2 + 972*y*z^6 - 8208*y*z^5 + 25548*y*z^4 - 38847*y*z^3 + 31083*y*z^2 - 12613*y*z + 2056*y - 216*z^7 + 2160*z^6 - 8244*z^5 + 15857*z^4 - 16966*z^3 + 10294*z^2 - 3329*z + 1, 16*x^7 - 104*x^6*y + 176*x^6*z - 188*x^6 + 288*x^5*y^2 - 972*x^5*y*z + 1044*x^5*y + 816*x^5*z^2 - 1740*x^5*z + 864*x^5 - 440*x^4*y^3 + 2220*x^4*y^2*z - 2400*x^4*y^2 - 3714*x^4*y*z^2 + 7980*x^4*y*z - 4002*x^4*y + 2060*x^4*z^3 - 6603*x^4*z^2 + 6606*x^4*z - 2113*x^4 + 400*x^3*y^4 - 2680*x^3*y^3*z + 2920*x^3*y^3 + 6696*x^3*y^2*z^2 - 14520*x^3*y^2*z + 7368*x^3*y^2 - 7393*x^3*y*z^3 + 23955*x^3*y*z^2 - 24291*x^3*y*z + 7881*x^3*y + 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"certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -2, 2, -3 ], [ -1, 1, -2 ], [ -2, 1, -1 ] ], "source_shift": [ 1, 2, 1 ], "target_matrix": [ [ 1, 2, 1 ], [ -1, -1, -2 ], [ -1, -1, -1 ] ], "target_shift": [ -439, 285, 290 ], "collision_preimages": [ [ "9/4", "25/4", "3" ], [ "-2", "17/2", "7" ], [ "-7", "-19/2", "-1" ] ], "collision_image": [ "-1757/4", "1141/4", "1161/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 0.9939502990018809 ], "surface_features": [ 4.066547986288036, 4.893095666096229, 3.8199385693553953, 0.0, 4.154661939858609, 5089 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C075", "pair": 75, "family": "counterexample", "mapping": "F(x,y,z)=(-54*x^7 + 459*x^6*y + 297*x^6*z - 594*x^6 - 1584*x^5*y^2 - 2115*x^5*y*z + 4158*x^5*y - 693*x^5*z^2 + 2781*x^5*z - 2385*x^5 + 2860*x^4*y^3 + 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1291*z - 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ 1, -2, -1 ], [ 3, -2, -2 ], [ 2, -1, -1 ] ], "source_shift": [ 1, 2, 1 ], "target_matrix": [ [ -1, -1, -2 ], [ 2, 1, 1 ], [ -1, -1, -1 ] ], "target_shift": [ 285, -437, 290 ], "collision_preimages": [ [ "-1/2", "1/4", "0" ], [ "29/2", "-9", "65/2" ], [ "23/2", "-4", "43/2" ] ], "collision_image": [ "1141/4", "-875/2", "1161/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0215654610001366 ], "surface_features": [ 4.142231774551217, 5.054521130653083, 3.8211203237768236, 0.0, 4.245043274281669, 5117 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C141", "pair": 141, "family": "counterexample", "mapping": "F(x,y,z)=(4*x^7 - 42*x^6*y - 30*x^6*z + 62*x^6 + 186*x^5*y^2 + 270*x^5*y*z - 561*x^5*y + 96*x^5*z^2 - 399*x^5*z + 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"certificate_data": { "source_matrix": [ [ 1, -1, -1 ], [ 1, -2, -1 ], [ 2, -3, -3 ] ], "source_shift": [ 2, 2, 1 ], "target_matrix": [ [ 2, 1, 1 ], [ -1, -2, -1 ], [ -1, -1, -1 ] ], "target_shift": [ -1397, 1488, 952 ], "collision_preimages": [ [ "-19/4", "0", "-11/4" ], [ "-17/2", "5/2", "-10" ], [ "-29/2", "-5/2", "-9" ] ], "collision_image": [ "-2795/2", "5953/4", "3809/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0753651140003058 ], "surface_features": [ 3.7872171636792946, 4.679255234780482, 3.412124406173317, 0.0, 3.9082827845099155, 4957 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B099", "pair": 99, "family": "bijection", "mapping": "F(x,y,z)=(-128*x^7 + 448*x^6*y + 448*x^6*z - 960*x^6 - 672*x^5*y^2 - 1344*x^5*y*z + 2880*x^5*y - 672*x^5*z^2 + 2880*x^5*z - 3088*x^5 + 560*x^4*y^3 + 1680*x^4*y^2*z - 3600*x^4*y^2 + 1680*x^4*y*z^2 - 7200*x^4*y*z + 7728*x^4*y + 560*x^4*z^3 - 3600*x^4*z^2 + 7744*x^4*z - 5576*x^4 - 280*x^3*y^4 - 1120*x^3*y^3*z + 2400*x^3*y^3 - 1680*x^3*y^2*z^2 + 7200*x^3*y^2*z - 7736*x^3*y^2 - 1120*x^3*y*z^3 + 7200*x^3*y*z^2 - 15504*x^3*y*z + 11196*x^3*y - 280*x^3*z^4 + 2400*x^3*z^3 - 7768*x^3*z^2 + 11276*x^3*z - 6120*x^3 + 84*x^2*y^5 + 420*x^2*y^4*z - 900*x^2*y^4 + 840*x^2*y^3*z^2 - 3600*x^2*y^3*z + 3872*x^2*y^3 + 840*x^2*y^2*z^3 - 5400*x^2*y^2*z^2 + 11640*x^2*y^2*z - 8430*x^2*y^2 + 420*x^2*y*z^4 - 3600*x^2*y*z^3 + 11664*x^2*y*z^2 - 16980*x^2*y*z + 9264*x^2*y + 84*x^2*z^5 - 900*x^2*z^4 + 3896*x^2*z^3 - 8550*x^2*z^2 + 9408*x^2*z - 4074*x^2 - 14*x*y^6 - 84*x*y^5*z + 180*x*y^5 - 210*x*y^4*z^2 + 900*x*y^4*z - 969*x*y^4 - 280*x*y^3*z^3 + 1800*x*y^3*z^2 - 3884*x*y^3*z + 2821*x*y^3 - 210*x*y^2*z^4 + 1800*x*y^2*z^3 - 5838*x*y^2*z^2 + 8523*x*y^2*z - 4674*x*y^2 - 84*x*y*z^5 + 900*x*y*z^4 - 3900*x*y*z^3 + 8583*x*y*z^2 - 9492*x*y*z + 4144*x*y - 14*x*z^6 + 180*x*z^5 - 977*x*z^4 + 2881*x*z^3 - 4818*x*z^2 + 4259*x*z - 1517*x + y^7 + 7*y^6*z - 15*y^6 + 21*y^5*z^2 - 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2276*z^2 + 1713*z - 1, 128*x^7 - 448*x^6*y - 448*x^6*z + 960*x^6 + 672*x^5*y^2 + 1344*x^5*y*z - 2880*x^5*y + 672*x^5*z^2 - 2880*x^5*z + 3088*x^5 - 560*x^4*y^3 - 1680*x^4*y^2*z + 3600*x^4*y^2 - 1680*x^4*y*z^2 + 7200*x^4*y*z - 7728*x^4*y - 560*x^4*z^3 + 3600*x^4*z^2 - 7744*x^4*z + 5592*x^4 + 280*x^3*y^4 + 1120*x^3*y^3*z - 2400*x^3*y^3 + 1680*x^3*y^2*z^2 - 7200*x^3*y^2*z + 7736*x^3*y^2 + 1120*x^3*y*z^3 - 7200*x^3*y*z^2 + 15504*x^3*y*z - 11228*x^3*y + 280*x^3*z^4 - 2400*x^3*z^3 + 7768*x^3*z^2 - 11308*x^3*z + 6200*x^3 - 84*x^2*y^5 - 420*x^2*y^4*z + 900*x^2*y^4 - 840*x^2*y^3*z^2 + 3600*x^2*y^3*z - 3872*x^2*y^3 - 840*x^2*y^2*z^3 + 5400*x^2*y^2*z^2 - 11640*x^2*y^2*z + 8454*x^2*y^2 - 420*x^2*y*z^4 + 3600*x^2*y*z^3 - 11664*x^2*y*z^2 + 17028*x^2*y*z - 9384*x^2*y - 84*x^2*z^5 + 900*x^2*z^4 - 3896*x^2*z^3 + 8574*x^2*z^2 - 9528*x^2*z + 4218*x^2 + 14*x*y^6 + 84*x*y^5*z - 180*x*y^5 + 210*x*y^4*z^2 - 900*x*y^4*z + 969*x*y^4 + 280*x*y^3*z^3 - 1800*x*y^3*z^2 + 3884*x*y^3*z - 2829*x*y^3 + 210*x*y^2*z^4 - 1800*x*y^2*z^3 + 5838*x*y^2*z^2 - 8547*x*y^2*z + 4734*x*y^2 + 84*x*y*z^5 - 900*x*y*z^4 + 3900*x*y*z^3 - 8607*x*y*z^2 + 9612*x*y*z - 4288*x*y + 14*x*z^6 - 180*x*z^5 + 977*x*z^4 - 2889*x*z^3 + 4878*x*z^2 - 4403*x*z + 1632*x - y^7 - 7*y^6*z + 15*y^6 - 21*y^5*z^2 + 90*y^5*z - 97*y^5 - 35*y^4*z^3 + 225*y^4*z^2 - 486*y^4*z + 355*y^4 - 35*y^3*z^4 + 300*y^3*z^3 - 974*y^3*z^2 + 1430*y^3*z - 796*y^3 - 21*y^2*z^5 + 225*y^2*z^4 - 976*y^2*z^3 + 2160*y^2*z^2 - 2424*y^2*z + 1090*y^2 - 7*y*z^6 + 90*y*z^5 - 489*y*z^4 + 1450*y*z^3 - 2460*y*z^2 + 2239*y*z - 839*y - z^7 + 15*z^6 - 98*z^5 + 365*z^4 - 832*z^3 + 1150*z^2 - 873*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ 2, -1, -1 ], [ 1, -1, -1 ], [ 1, -1, -2 ] ], "source_shift": [ 2, 1, 1 ], "target_matrix": [ [ -1, -1, -2 ], [ -2, -3, -2 ], [ 1, 2, 1 ] ], "target_shift": [ 242, 538, -277 ], "inverse_recipe": { "p": 4, "q": 3, "f": "x**4", "g": "x**3", "h": "y*z + y + z**2 + z", "polynomials": { "f": [ [ [ 4, 0, 0 ], 1 ] ], "g": [ [ [ 3, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 1, 0 ], 1 ], [ [ 0, 0, 2 ], 1 ], [ [ 0, 0, 1 ], 1 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0218613380020543 ], "surface_features": [ 3.5757024945425466, 4.531581091959066, 3.21761552866633, 0.0, 3.716164786362491, 4850 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B143", "pair": 143, "family": "bijection", "mapping": "F(x,y,z)=(-3*x^7 + 21*x^6*y + 21*x^6*z - 51*x^6 - 63*x^5*y^2 - 126*x^5*y*z + 303*x^5*y - 63*x^5*z^2 + 303*x^5*z - 352*x^5 + 105*x^4*y^3 + 315*x^4*y^2*z - 750*x^4*y^2 + 315*x^4*y*z^2 - 1500*x^4*y*z + 1730*x^4*y + 105*x^4*z^3 - 750*x^4*z^2 + 1730*x^4*z - 1303*x^4 - 105*x^3*y^4 - 420*x^3*y^3*z + 990*x^3*y^3 - 630*x^3*y^2*z^2 + 2970*x^3*y^2*z - 3400*x^3*y^2 - 420*x^3*y*z^3 + 2970*x^3*y*z^2 - 6800*x^3*y*z + 5092*x^3*y - 105*x^3*z^4 + 990*x^3*z^3 - 3400*x^3*z^2 + 5092*x^3*z - 2836*x^3 + 63*x^2*y^5 + 315*x^2*y^4*z - 735*x^2*y^4 + 630*x^2*y^3*z^2 - 2940*x^2*y^3*z + 3340*x^2*y^3 + 630*x^2*y^2*z^3 - 4410*x^2*y^2*z^2 + 10020*x^2*y^2*z - 7458*x^2*y^2 + 315*x^2*y*z^4 - 2940*x^2*y*z^3 + 10020*x^2*y*z^2 - 14916*x^2*y*z + 8268*x^2*y + 63*x^2*z^5 - 735*x^2*z^4 + 3340*x^2*z^3 - 7458*x^2*z^2 + 8265*x^2*z - 3679*x^2 - 21*x*y^6 - 126*x*y^5*z + 291*x*y^5 - 315*x*y^4*z^2 + 1455*x*y^4*z - 1640*x*y^4 - 420*x*y^3*z^3 + 2910*x*y^3*z^2 - 6560*x*y^3*z + 4852*x*y^3 - 315*x*y^2*z^4 + 2910*x*y^2*z^3 - 9840*x*y^2*z^2 + 14556*x*y^2*z - 8028*x*y^2 - 126*x*y*z^5 + 1455*x*y*z^4 - 6560*x*y*z^3 + 14556*x*y*z^2 - 16050*x*y*z + 7124*x*y - 21*x*z^6 + 291*x*z^5 - 1640*x*z^4 + 4852*x*z^3 - 8022*x*z^2 + 7103*x*z - 2655*x + 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630*x^2*y^2*z^3 - 4410*x^2*y^2*z^2 + 9960*x^2*y^2*z - 7338*x^2*y^2 + 315*x^2*y*z^4 - 2940*x^2*y*z^3 + 9960*x^2*y*z^2 - 14676*x^2*y*z + 8028*x^2*y + 63*x^2*z^5 - 735*x^2*z^4 + 3320*x^2*z^3 - 7338*x^2*z^2 + 8025*x^2*z - 3517*x^2 - 21*x*y^6 - 126*x*y^5*z + 291*x*y^5 - 315*x*y^4*z^2 + 1455*x*y^4*z - 1630*x*y^4 - 420*x*y^3*z^3 + 2910*x*y^3*z^2 - 6520*x*y^3*z + 4772*x*y^3 - 315*x*y^2*z^4 + 2910*x*y^2*z^3 - 9780*x*y^2*z^2 + 14316*x*y^2*z - 7788*x*y^2 - 126*x*y*z^5 + 1455*x*y*z^4 - 6520*x*y*z^3 + 14316*x*y*z^2 - 15570*x*y*z + 6800*x*y - 21*x*z^6 + 291*x*z^5 - 1630*x*z^4 + 4772*x*z^3 - 7782*x*z^2 + 6779*x*z - 2485*x + 3*y^7 + 21*y^6*z - 48*y^6 + 63*y^5*z^2 - 288*y^5*z + 320*y^5 + 105*y^4*z^3 - 720*y^4*z^2 + 1600*y^4*z - 1163*y^4 + 105*y^3*z^4 - 960*y^3*z^3 + 3200*y^3*z^2 - 4652*y^3*z + 2516*y^3 + 63*y^2*z^5 - 720*y^2*z^4 + 3200*y^2*z^3 - 6978*y^2*z^2 + 7545*y^2*z - 3280*y^2 + 21*y*z^6 - 288*y*z^5 + 1600*y*z^4 - 4652*y*z^3 + 7542*y*z^2 - 6542*y*z + 2393*y + 3*z^7 - 48*z^6 + 320*z^5 - 1163*z^4 + 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-1, -1, -2 ], [ 2, 1, 1 ] ], "target_shift": [ 823, 749, -1713 ], "inverse_recipe": { "p": 2, "q": 5, "f": "x**2 + x", "g": "x**5 + x**2", "h": "y*z", "polynomials": { "f": [ [ [ 2, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "g": [ [ [ 5, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ] ] }, "h_multiplier": 3 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0721459710002819 ], "surface_features": [ 3.5690365793469114, 4.51295107997249, 3.1760912590556813, 0.0, 3.7156641692693144, 4858 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C108", "pair": 108, "family": "counterexample", "mapping": "F(x,y,z)=(8*x^7 - 52*x^6*y + 44*x^6*z - 116*x^6 + 138*x^5*y^2 - 240*x^5*y*z + 576*x^5*y + 102*x^5*z^2 - 516*x^5*z + 612*x^5 - 195*x^4*y^3 + 519*x^4*y^2*z - 1173*x^4*y^2 - 453*x^4*y*z^2 + 2118*x^4*y*z - 2412*x^4*y + 129*x^4*z^3 - 945*x^4*z^2 + 2220*x^4*z - 1667*x^4 + 159*x^3*y^4 - 573*x^3*y^3*z + 1251*x^3*y^3 + 765*x^3*y^2*z^2 - 3414*x^3*y^2*z + 3765*x^3*y^2 - 447*x^3*y*z^3 + 3075*x^3*y*z^2 - 6942*x^3*y*z + 5117*x^3*y + 96*x^3*z^4 - 912*x^3*z^3 + 3177*x^3*z^2 - 4778*x^3*z + 2636*x^3 - 75*x^2*y^5 + 342*x^2*y^4*z - 735*x^2*y^4 - 618*x^2*y^3*z^2 + 2694*x^2*y^3*z - 2904*x^2*y^3 + 552*x^2*y^2*z^3 - 3672*x^2*y^2*z^2 + 8052*x^2*y^2*z - 5838*x^2*y^2 - 243*x^2*y*z^4 + 2202*x^2*y*z^3 - 7392*x^2*y*z^2 + 10893*x^2*y*z - 5961*x^2*y + 42*x^2*z^5 - 489*x^2*z^4 + 2244*x^2*z^3 - 5055*x^2*z^2 + 5610*x^2*z - 2473*x^2 + 19*x*y^6 - 105*x*y^5*z + 225*x*y^5 + 240*x*y^4*z^2 - 1038*x*y^4*z + 1104*x*y^4 - 290*x*y^3*z^3 + 1902*x*y^3*z^2 - 4095*x*y^3*z + 2931*x*y^3 + 195*x*y^2*z^4 - 1728*x*y^2*z^3 + 5661*x*y^2*z^2 - 8205*x*y^2*z + 4452*x*y^2 - 69*x*y*z^5 + 777*x*y*z^4 - 3453*x*y*z^3 + 7617*x*y*z^2 - 8361*x*y*z + 3676*x*y + 10*x*z^6 - 138*x*z^5 + 783*x*z^4 - 2343*x*z^3 + 3909*x*z^2 - 3472*x*z + 1291*x - 2*y^7 + 13*y^6*z - 28*y^6 - 36*y^5*z^2 + 156*y^5*z - 165*y^5 + 55*y^4*z^3 - 360*y^4*z^2 + 768*y^4*z - 545*y^4 - 50*y^3*z^4 + 440*y^3*z^3 - 1422*y^3*z^2 + 2037*y^3*z - 1098*y^3 + 27*y^2*z^5 - 300*y^2*z^4 + 1308*y^2*z^3 - 2841*y^2*z^2 + 3090*y^2*z - 1354*y^2 - 8*y*z^6 + 108*y*z^5 - 597*y*z^4 + 1751*y*z^3 - 2886*y*z^2 + 2549*y*z - 949*y + z^7 - 16*z^6 + 108*z^5 - 402*z^4 + 894*z^3 - 1195*z^2 + 896*z - 1, 24*x^7 - 156*x^6*y + 132*x^6*z - 336*x^6 + 414*x^5*y^2 - 720*x^5*y*z + 1656*x^5*y + 306*x^5*z^2 - 1488*x^5*z + 1692*x^5 - 585*x^4*y^3 + 1557*x^4*y^2*z - 3348*x^4*y^2 - 1359*x^4*y*z^2 + 6060*x^4*y*z - 6624*x^4*y + 387*x^4*z^3 - 2712*x^4*z^2 + 6108*x^4*z - 4449*x^4 + 477*x^3*y^4 - 1719*x^3*y^3*z + 3546*x^3*y^3 + 2295*x^3*y^2*z^2 - 9696*x^3*y^2*z + 10269*x^3*y^2 - 1341*x^3*y*z^3 + 8754*x^3*y*z^2 - 18966*x^3*y*z + 13569*x^3*y + 288*x^3*z^4 - 2604*x^3*z^3 + 8697*x^3*z^2 - 12684*x^3*z + 6852*x^3 - 225*x^2*y^5 + 1026*x^2*y^4*z - 2070*x^2*y^4 - 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7362*y*z^2 + 6402*y*z - 2363*y + 3*z^7 - 45*z^6 + 291*z^5 - 1050*z^4 + 2283*z^3 - 3003*z^2 + 2231*z - 1, -16*x^7 + 104*x^6*y - 88*x^6*z + 220*x^6 - 276*x^5*y^2 + 480*x^5*y*z - 1080*x^5*y - 204*x^5*z^2 + 972*x^5*z - 1080*x^5 + 390*x^4*y^3 - 1038*x^4*y^2*z + 2175*x^4*y^2 + 906*x^4*y*z^2 - 3942*x^4*y*z + 4212*x^4*y - 258*x^4*z^3 + 1767*x^4*z^2 - 3888*x^4*z + 2783*x^4 - 318*x^3*y^4 + 1146*x^3*y^3*z - 2295*x^3*y^3 - 1530*x^3*y^2*z^2 + 6282*x^3*y^2*z - 6504*x^3*y^2 + 894*x^3*y*z^3 - 5679*x^3*y*z^2 + 12024*x^3*y*z - 8457*x^3*y - 192*x^3*z^4 + 1692*x^3*z^3 - 5520*x^3*z^2 + 7910*x^3*z - 4226*x^3 + 150*x^2*y^5 - 684*x^2*y^4*z + 1335*x^2*y^4 + 1236*x^2*y^3*z^2 - 4905*x^2*y^3*z + 4962*x^2*y^3 - 1104*x^2*y^2*z^3 + 6705*x^2*y^2*z^2 - 13791*x^2*y^2*z + 9549*x^2*y^2 + 486*x^2*y*z^4 - 4035*x^2*y*z^3 + 12696*x^2*y*z^2 - 17847*x^2*y*z + 9462*x^2*y - 84*x^2*z^5 + 900*x^2*z^4 - 3867*x^2*z^3 + 8298*x^2*z^2 - 8913*x^2*z + 3851*x^2 - 38*x*y^6 + 210*x*y^5*z - 405*x*y^5 - 480*x*y^4*z^2 + 1872*x*y^4*z - 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[ [ -1, -1, -1 ], [ -3, -2, -3 ], [ 2, 1, 1 ] ], "target_shift": [ 290, 723, -437 ], "collision_preimages": [ [ "1", "-1/4", "-1/4" ], [ "7/2", "11/2", "2" ], [ "-3/2", "15/2", "11" ] ], "collision_image": [ "1161/4", "2895/4", "-875/2" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0429574369991315 ], "surface_features": [ 3.506674868615791, 4.458260037360323, 3.147521743537597, 0.0, 3.640973982410656, 4864 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C199", "pair": 199, "family": "counterexample", "mapping": "F(x,y,z)=(-8*x^7 - 44*x^6*y - 52*x^6*z - 104*x^6 - 102*x^5*y^2 - 240*x^5*y*z - 480*x^5*y - 138*x^5*z^2 - 564*x^5*z - 462*x^5 - 129*x^4*y^3 - 453*x^4*y^2*z - 906*x^4*y^2 - 519*x^4*y*z^2 - 2112*x^4*y*z - 1761*x^4*y - 195*x^4*z^3 - 1206*x^4*z^2 - 2007*x^4*z - 1038*x^4 - 96*x^3*y^4 - 447*x^3*y^3*z - 894*x^3*y^3 - 765*x^3*y^2*z^2 - 3099*x^3*y^2*z - 2631*x^3*y^2 - 573*x^3*y*z^3 - 3516*x^3*y*z^2 - 5943*x^3*y*z - 3137*x^3*y - 159*x^3*z^4 - 1311*x^3*z^3 - 3312*x^3*z^2 - 3504*x^3*z - 1346*x^3 - 42*x^2*y^5 - 243*x^2*y^4*z - 486*x^2*y^4 - 552*x^2*y^3*z^2 - 2226*x^2*y^3*z - 1923*x^2*y^3 - 618*x^2*y^2*z^3 - 3762*x^2*y^2*z^2 - 6456*x^2*y^2*z - 3474*x^2*y^2 - 342*x^2*y*z^4 - 2790*x^2*y*z^3 - 7143*x^2*y*z^2 - 7692*x^2*y*z - 3024*x^2*y - 75*x^2*z^5 - 768*x^2*z^4 - 2610*x^2*z^3 - 4218*x^2*z^2 - 3336*x^2*z - 1045*x^2 - 10*x*y^6 - 69*x*y^5*z - 138*x*y^5 - 195*x*y^4*z^2 - 783*x*y^4*z - 687*x*y^4 - 290*x*y^3*z^3 - 1752*x*y^3*z^2 - 3048*x*y^3*z - 1668*x*y^3 - 240*x*y^2*z^4 - 1938*x*y^2*z^3 - 5022*x*y^2*z^2 - 5490*x*y^2*z - 2205*x*y^2 - 105*x*y*z^5 - 1062*x*y*z^4 - 3648*x*y*z^3 - 5976*x*y*z^2 - 4818*x*y*z - 1550*x*y - 19*x*z^6 - 231*x*z^5 - 987*x*z^4 - 2154*x*z^3 - 2613*x*z^2 - 1700*x*z - 461*x - y^7 - 8*y^6*z - 16*y^6 - 27*y^5*z^2 - 108*y^5*z - 96*y^5 - 50*y^4*z^3 - 300*y^4*z^2 - 528*y^4*z - 293*y^4 - 55*y^3*z^4 - 440*y^3*z^3 - 1152*y^3*z^2 - 1275*y^3*z - 521*y^3 - 36*y^2*z^5 - 360*y^2*z^4 - 1248*y^2*z^3 - 2067*y^2*z^2 - 1692*y^2*z - 556*y^2 - 13*y*z^6 - 156*y*z^5 - 672*y*z^4 - 1481*y*z^3 - 1821*y*z^2 - 1205*y*z - 335*y - 2*z^7 - 28*z^6 - 144*z^5 - 396*z^4 - 650*z^3 - 649*z^2 - 366*z - 1, -16*x^7 - 88*x^6*y - 104*x^6*z - 184*x^6 - 204*x^5*y^2 - 480*x^5*y*z - 852*x^5*y - 276*x^5*z^2 - 996*x^5*z - 780*x^5 - 258*x^4*y^3 - 906*x^4*y^2*z - 1614*x^4*y^2 - 1038*x^4*y*z^2 - 3744*x^4*y*z - 2982*x^4*y - 390*x^4*z^3 - 2130*x^4*z^2 - 3390*x^4*z - 1710*x^4 - 192*x^3*y^4 - 894*x^3*y^3*z - 1599*x^3*y^3 - 1530*x^3*y^2*z^2 - 5517*x^3*y^2*z - 4470*x^3*y^2 - 1146*x^3*y*z^3 - 6237*x^3*y*z^2 - 10074*x^3*y*z - 5185*x^3*y - 318*x^3*z^4 - 2319*x^3*z^3 - 5604*x^3*z^2 - 5781*x^3*z - 2182*x^3 - 84*x^2*y^5 - 486*x^2*y^4*z - 873*x^2*y^4 - 1104*x^2*y^3*z^2 - 3981*x^2*y^3*z - 3279*x^2*y^3 - 1236*x^2*y^2*z^3 - 6705*x^2*y^2*z^2 - 10986*x^2*y^2*z - 5763*x^2*y^2 - 684*x^2*y*z^4 - 4959*x^2*y*z^3 - 12135*x^2*y*z^2 - 12741*x^2*y*z - 4923*x^2*y - 150*x^2*z^5 - 1362*x^2*z^4 - 4428*x^2*z^3 - 6978*x^2*z^2 - 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1477*x*z + 388*x + 2*y^7 + 16*y^6*z + 23*y^6 + 54*y^5*z^2 + 153*y^5*z + 111*y^5 + 100*y^4*z^3 + 420*y^4*z^2 + 606*y^4*z + 299*y^4 + 110*y^3*z^4 + 610*y^3*z^3 + 1314*y^3*z^2 + 1295*y^3*z + 491*y^3 + 72*y^2*z^5 + 495*y^2*z^4 + 1416*y^2*z^3 + 2091*y^2*z^2 + 1590*y^2*z + 497*y^2 + 26*y*z^6 + 213*y*z^5 + 759*y*z^4 + 1493*y*z^3 + 1707*y*z^2 + 1075*y*z + 291*y + 4*z^7 + 38*z^6 + 162*z^5 + 398*z^4 + 608*z^3 + 578*z^2 + 318*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ 2, 1, 1 ], [ 1, 1, 1 ], [ 1, 1, 2 ] ], "source_shift": [ 1, 1, 1 ], "target_matrix": [ [ -1, -2, -1 ], [ -2, -3, -2 ], [ 2, 1, 1 ] ], "target_shift": [ 87, 141, -75 ], "collision_preimages": [ [ "0", "-3/4", "-1/4" ], [ "5/2", "-13", "8" ], [ "-5/2", "-2", "5" ] ], "collision_image": [ "349/4", "283/2", "-151/2" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0699683580023702 ], 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5170*x^3*z + 2593*x^3 - 42*x^2*y^5 + 243*x^2*y^4*z - 486*x^2*y^4 - 552*x^2*y^3*z^2 + 2226*x^2*y^3*z - 2205*x^2*y^3 + 618*x^2*y^2*z^3 - 3762*x^2*y^2*z^2 + 7464*x^2*y^2*z - 4977*x^2*y^2 - 342*x^2*y*z^4 + 2790*x^2*y*z^3 - 8313*x^2*y*z^2 + 11070*x^2*y*z - 5613*x^2*y + 75*x^2*z^5 - 768*x^2*z^4 + 3054*x^2*z^3 - 6093*x^2*z^2 + 6171*x^2*z - 2539*x^2 + 10*x*y^6 - 69*x*y^5*z + 138*x*y^5 + 195*x*y^4*z^2 - 783*x*y^4*z + 780*x*y^4 - 290*x*y^3*z^3 + 1752*x*y^3*z^2 - 3483*x*y^3*z + 2338*x*y^3 + 240*x*y^2*z^4 - 1938*x*y^2*z^3 + 5769*x*y^2*z^2 - 7713*x*y^2*z + 3933*x*y^2 - 105*x*y*z^5 + 1062*x*y*z^4 - 4209*x*y*z^3 + 8412*x*y*z^2 - 8547*x*y*z + 3539*x*y + 19*x*z^6 - 231*x*z^5 + 1143*x*z^4 - 3037*x*z^3 + 4614*x*z^2 - 3812*x*z + 1344*x - y^7 + 8*y^6*z - 16*y^6 - 27*y^5*z^2 + 108*y^5*z - 108*y^5 + 50*y^4*z^3 - 300*y^4*z^2 + 597*y^4*z - 402*y^4 - 55*y^3*z^4 + 440*y^3*z^3 - 1308*y^3*z^2 + 1751*y^3*z - 894*y^3 + 36*y^2*z^5 - 360*y^2*z^4 + 1422*y^2*z^3 - 2841*y^2*z^2 + 2886*y^2*z - 1195*y^2 - 13*y*z^6 + 156*y*z^5 - 768*y*z^4 + 2037*y*z^3 - 3090*y*z^2 + 2549*y*z - 896*y + 2*z^7 - 28*z^6 + 165*z^5 - 545*z^4 + 1098*z^3 - 1354*z^2 + 949*z - 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -2, 1, -1 ], [ -1, 1, -1 ], [ -1, 1, -2 ] ], "source_shift": [ 1, 2, 1 ], "target_matrix": [ [ -1, -2, -1 ], [ 2, 3, 3 ], [ -1, -1, -1 ] ], "target_shift": [ 439, -725, 290 ], "collision_preimages": [ [ "-1", "-15/4", "-3/4" ], [ "-7/2", "-16", "-9" ], [ "3/2", "-5", "-6" ] ], "collision_image": [ "1757/4", "-1451/2", "1161/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.053051923001476 ], "surface_features": [ 3.45758587706111, 4.42646246149016, 3.068371418032643, 0.0, 3.603024462001543, 4829 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C235", "pair": 235, "family": "counterexample", "mapping": "F(x,y,z)=(8*x^7 - 52*x^6*y - 44*x^6*z + 128*x^6 + 138*x^5*y^2 + 240*x^5*y*z - 648*x^5*y + 102*x^5*z^2 - 576*x^5*z + 756*x^5 - 195*x^4*y^3 - 519*x^4*y^2*z + 1344*x^4*y^2 - 453*x^4*y*z^2 + 2412*x^4*y*z - 3024*x^4*y - 129*x^4*z^3 + 1068*x^4*z^2 - 2772*x^4*z + 2219*x^4 + 159*x^3*y^4 + 573*x^3*y^3*z - 1458*x^3*y^3 + 765*x^3*y^2*z^2 - 3960*x^3*y^2*z + 4791*x^3*y^2 + 447*x^3*y*z^3 - 3546*x^3*y*z^2 + 8802*x^3*y*z - 6899*x^3*y + 96*x^3*z^4 - 1044*x^3*z^3 + 4011*x^3*z^2 - 6428*x^3*z + 3692*x^3 - 75*x^2*y^5 - 342*x^2*y^4*z + 870*x^2*y^4 - 618*x^2*y^3*z^2 + 3177*x^2*y^3*z - 3750*x^2*y^3 - 552*x^2*y^2*z^3 + 4311*x^2*y^2*z^2 - 10365*x^2*y^2*z + 7974*x^2*y^2 - 243*x^2*y*z^4 + 2571*x^2*y*z^3 - 9480*x^2*y*z^2 + 14847*x^2*y*z - 8451*x^2*y - 42*x^2*z^5 + 567*x^2*z^4 - 2865*x^2*z^3 + 6873*x^2*z^2 - 7944*x^2*z + 3592*x^2 + 19*x*y^6 + 105*x*y^5*z - 270*x*y^5 + 240*x*y^4*z^2 - 1242*x*y^4*z + 1446*x*y^4 + 290*x*y^3*z^3 - 2268*x*y^3*z^2 + 5349*x*y^3*z - 4056*x*y^3 + 195*x*y^2*z^4 - 2052*x*y^2*z^3 + 7371*x*y^2*z^2 - 11331*x*y^2*z + 6390*x*y^2 + 69*x*y*z^5 - 918*x*y*z^4 + 4479*x*y*z^3 - 10494*x*y*z^2 + 11985*x*y*z - 5407*x*y + 10*x*z^6 - 162*x*z^5 + 1011*x*z^4 - 3219*x*z^3 + 5595*x*z^2 - 5104*x*z + 1923*x - 2*y^7 - 13*y^6*z + 34*y^6 - 36*y^5*z^2 + 189*y^5*z - 219*y^5 - 55*y^4*z^3 + 435*y^4*z^2 - 1017*y^4*z + 764*y^4 - 50*y^3*z^4 + 530*y^3*z^3 - 1878*y^3*z^2 + 2850*y^3*z - 1596*y^3 - 27*y^2*z^5 + 360*y^2*z^4 - 1722*y^2*z^3 + 3966*y^2*z^2 - 4485*y^2*z + 2017*y^2 - 8*y*z^6 + 129*y*z^5 - 783*y*z^4 + 2438*y*z^3 - 4182*y*z^2 + 3794*y*z - 1433*y - z^7 + 19*z^6 - 141*z^5 + 558*z^4 - 1293*z^3 + 1777*z^2 - 1353*z + 1, -8*x^7 + 52*x^6*y + 44*x^6*z - 116*x^6 - 138*x^5*y^2 - 240*x^5*y*z + 576*x^5*y - 102*x^5*z^2 + 516*x^5*z - 612*x^5 + 195*x^4*y^3 + 519*x^4*y^2*z - 1173*x^4*y^2 + 453*x^4*y*z^2 - 2118*x^4*y*z + 2412*x^4*y + 129*x^4*z^3 - 945*x^4*z^2 + 2220*x^4*z - 1667*x^4 - 159*x^3*y^4 - 573*x^3*y^3*z + 1251*x^3*y^3 - 765*x^3*y^2*z^2 + 3414*x^3*y^2*z - 3765*x^3*y^2 - 447*x^3*y*z^3 + 3075*x^3*y*z^2 - 6942*x^3*y*z + 5117*x^3*y - 96*x^3*z^4 + 912*x^3*z^3 - 3177*x^3*z^2 + 4778*x^3*z - 2636*x^3 + 75*x^2*y^5 + 342*x^2*y^4*z - 735*x^2*y^4 + 618*x^2*y^3*z^2 - 2694*x^2*y^3*z + 2904*x^2*y^3 + 552*x^2*y^2*z^3 - 3672*x^2*y^2*z^2 + 8052*x^2*y^2*z - 5838*x^2*y^2 + 243*x^2*y*z^4 - 2202*x^2*y*z^3 + 7392*x^2*y*z^2 - 10893*x^2*y*z + 5961*x^2*y + 42*x^2*z^5 - 489*x^2*z^4 + 2244*x^2*z^3 - 5055*x^2*z^2 + 5610*x^2*z - 2473*x^2 - 19*x*y^6 - 105*x*y^5*z + 225*x*y^5 - 240*x*y^4*z^2 + 1038*x*y^4*z - 1104*x*y^4 - 290*x*y^3*z^3 + 1902*x*y^3*z^2 - 4095*x*y^3*z + 2931*x*y^3 - 195*x*y^2*z^4 + 1728*x*y^2*z^3 - 5661*x*y^2*z^2 + 8205*x*y^2*z - 4452*x*y^2 - 69*x*y*z^5 + 777*x*y*z^4 - 3453*x*y*z^3 + 7617*x*y*z^2 - 8361*x*y*z + 3676*x*y - 10*x*z^6 + 138*x*z^5 - 783*x*z^4 + 2343*x*z^3 - 3909*x*z^2 + 3472*x*z - 1291*x + 2*y^7 + 13*y^6*z - 28*y^6 + 36*y^5*z^2 - 156*y^5*z + 165*y^5 + 55*y^4*z^3 - 360*y^4*z^2 + 768*y^4*z - 545*y^4 + 50*y^3*z^4 - 440*y^3*z^3 + 1422*y^3*z^2 - 2037*y^3*z + 1098*y^3 + 27*y^2*z^5 - 300*y^2*z^4 + 1308*y^2*z^3 - 2841*y^2*z^2 + 3090*y^2*z - 1354*y^2 + 8*y*z^6 - 108*y*z^5 + 597*y*z^4 - 1751*y*z^3 + 2886*y*z^2 - 2549*y*z + 949*y + z^7 - 16*z^6 + 108*z^5 - 402*z^4 + 894*z^3 - 1195*z^2 + 896*z - 1, -16*x^7 + 104*x^6*y + 88*x^6*z - 220*x^6 - 276*x^5*y^2 - 480*x^5*y*z + 1080*x^5*y - 204*x^5*z^2 + 972*x^5*z - 1080*x^5 + 390*x^4*y^3 + 1038*x^4*y^2*z - 2175*x^4*y^2 + 906*x^4*y*z^2 - 3942*x^4*y*z + 4212*x^4*y + 258*x^4*z^3 - 1767*x^4*z^2 + 3888*x^4*z - 2783*x^4 - 318*x^3*y^4 - 1146*x^3*y^3*z + 2295*x^3*y^3 - 1530*x^3*y^2*z^2 + 6282*x^3*y^2*z - 6504*x^3*y^2 - 894*x^3*y*z^3 + 5679*x^3*y*z^2 - 12024*x^3*y*z + 8457*x^3*y - 192*x^3*z^4 + 1692*x^3*z^3 - 5520*x^3*z^2 + 7910*x^3*z - 4226*x^3 + 150*x^2*y^5 + 684*x^2*y^4*z - 1335*x^2*y^4 + 1236*x^2*y^3*z^2 - 4905*x^2*y^3*z + 4962*x^2*y^3 + 1104*x^2*y^2*z^3 - 6705*x^2*y^2*z^2 + 13791*x^2*y^2*z - 9549*x^2*y^2 + 486*x^2*y*z^4 - 4035*x^2*y*z^3 + 12696*x^2*y*z^2 - 17847*x^2*y*z + 9462*x^2*y + 84*x^2*z^5 - 900*x^2*z^4 + 3867*x^2*z^3 - 8298*x^2*z^2 + 8913*x^2*z - 3851*x^2 - 38*x*y^6 - 210*x*y^5*z + 405*x*y^5 - 480*x*y^4*z^2 + 1872*x*y^4*z - 1866*x*y^4 - 580*x*y^3*z^3 + 3438*x*y^3*z^2 - 6936*x*y^3*z + 4744*x*y^3 - 390*x*y^2*z^4 + 3132*x*y^2*z^3 - 9612*x*y^2*z^2 + 13302*x*y^2*z - 6996*x*y^2 - 138*x*y*z^5 + 1413*x*y*z^4 - 5880*x*y*z^3 + 12372*x*y*z^2 - 13152*x*y*z + 5666*x*y - 20*x*z^6 + 252*x*z^5 - 1338*x*z^4 + 3814*x*z^3 - 6156*x*z^2 + 5354*x*z - 1970*x + 4*y^7 + 26*y^6*z - 50*y^6 + 72*y^5*z^2 - 279*y^5*z + 276*y^5 + 110*y^4*z^3 - 645*y^4*z^2 + 1287*y^4*z - 873*y^4 + 100*y^3*z^4 - 790*y^3*z^3 + 2388*y^3*z^2 - 3268*y^3*z + 1708*y^3 + 54*y^2*z^5 - 540*y^2*z^4 + 2202*y^2*z^3 - 4566*y^2*z^2 + 4812*y^2*z - 2066*y^2 + 16*y*z^6 - 195*y*z^5 + 1008*y*z^4 - 2820*y*z^3 + 4500*y*z^2 - 3892*y*z + 1432*y + 2*z^7 - 29*z^6 + 183*z^5 - 649*z^4 + 1396*z^3 - 1826*z^2 + 1352*z - 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ 1, -1, -1 ], [ 2, -1, -1 ], [ 1, -2, -1 ] ], "source_shift": [ 1, 2, 1 ], "target_matrix": [ [ 1, 2, 1 ], [ -1, -1, -1 ], [ -2, -1, -1 ] ], "target_shift": [ -439, 290, 437 ], "collision_preimages": [ [ "-1", "1/4", "-1/4" ], [ "-7/2", "-11/2", "2" ], [ "3/2", "-15/2", "11" ] ], "collision_image": [ "-1757/4", "1161/4", "875/2" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0368282409981475 ], "surface_features": [ 3.3804554636766864, 4.2515652236446275, 3.037027879755775, 0.0, 3.4870310149574992, 4819 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B003", "pair": 3, "family": "bijection", "mapping": "F(x,y,z)=(4*x^7 + 28*x^6*y - 28*x^6*z - 64*x^6 + 84*x^5*y^2 - 168*x^5*y*z - 384*x^5*y + 84*x^5*z^2 + 388*x^5*z + 426*x^5 + 140*x^4*y^3 - 420*x^4*y^2*z - 960*x^4*y^2 + 420*x^4*y*z^2 + 1940*x^4*y*z + 2130*x^4*y - 140*x^4*z^3 - 980*x^4*z^2 - 2170*x^4*z - 1540*x^4 + 140*x^3*y^4 - 560*x^3*y^3*z - 1280*x^3*y^3 + 840*x^3*y^2*z^2 + 3880*x^3*y^2*z + 4260*x^3*y^2 - 560*x^3*y*z^3 - 3920*x^3*y*z^2 - 8680*x^3*y*z - 6160*x^3*y + 140*x^3*z^4 + 1320*x^3*z^3 + 4420*x^3*z^2 + 6320*x^3*z + 3292*x^3 + 84*x^2*y^5 - 420*x^2*y^4*z - 960*x^2*y^4 + 840*x^2*y^3*z^2 + 3880*x^2*y^3*z + 4260*x^2*y^3 - 840*x^2*y^2*z^3 - 5880*x^2*y^2*z^2 - 13020*x^2*y^2*z - 9240*x^2*y^2 + 420*x^2*y*z^4 + 3960*x^2*y*z^3 + 13260*x^2*y*z^2 + 18960*x^2*y*z + 9872*x^2*y - 84*x^2*z^5 - 1000*x^2*z^4 - 4500*x^2*z^3 - 9720*x^2*z^2 - 10192*x^2*z - 4221*x^2 + 28*x*y^6 - 168*x*y^5*z - 384*x*y^5 + 420*x*y^4*z^2 + 1940*x*y^4*z + 2130*x*y^4 - 560*x*y^3*z^3 - 3920*x*y^3*z^2 - 8680*x*y^3*z - 6160*x*y^3 + 420*x*y^2*z^4 + 3960*x*y^2*z^3 + 13260*x*y^2*z^2 + 18960*x*y^2*z + 9868*x*y^2 - 168*x*y*z^5 - 2000*x*y*z^4 - 9000*x*y*z^3 - 19440*x*y*z^2 - 20376*x*y*z - 8418*x*y + 28*x*z^6 + 404*x*z^5 + 2290*x*z^4 + 6640*x*z^3 + 10508*x*z^2 + 8750*x*z + 3054*x + 4*y^7 - 28*y^6*z - 64*y^6 + 84*y^5*z^2 + 388*y^5*z + 426*y^5 - 140*y^4*z^3 - 980*y^4*z^2 - 2170*y^4*z - 1540*y^4 + 140*y^3*z^4 + 1320*y^3*z^3 + 4420*y^3*z^2 + 6320*y^3*z + 3288*y^3 - 84*y^2*z^5 - 1000*y^2*z^4 - 4500*y^2*z^3 - 9720*y^2*z^2 - 10184*y^2*z - 4197*y^2 + 28*y*z^6 + 404*y*z^5 + 2290*y*z^4 + 6640*y*z^3 + 10504*y*z^2 + 8722*y*z + 3028*y - 4*z^7 - 68*z^6 - 466*z^5 - 1700*z^4 - 3608*z^3 - 4525*z^2 - 3177*z - 1, 2*x^7 + 14*x^6*y - 14*x^6*z - 32*x^6 + 42*x^5*y^2 - 84*x^5*y*z - 192*x^5*y + 42*x^5*z^2 + 194*x^5*z + 212*x^5 + 70*x^4*y^3 - 210*x^4*y^2*z - 480*x^4*y^2 + 210*x^4*y*z^2 + 970*x^4*y*z + 1060*x^4*y - 70*x^4*z^3 - 490*x^4*z^2 - 1080*x^4*z - 760*x^4 + 70*x^3*y^4 - 280*x^3*y^3*z - 640*x^3*y^3 + 420*x^3*y^2*z^2 + 1940*x^3*y^2*z + 2120*x^3*y^2 - 280*x^3*y*z^3 - 1960*x^3*y*z^2 - 4320*x^3*y*z - 3040*x^3*y + 70*x^3*z^4 + 660*x^3*z^3 + 2200*x^3*z^2 + 3120*x^3*z + 1606*x^3 + 42*x^2*y^5 - 210*x^2*y^4*z - 480*x^2*y^4 + 420*x^2*y^3*z^2 + 1940*x^2*y^3*z + 2120*x^2*y^3 - 420*x^2*y^2*z^3 - 2940*x^2*y^2*z^2 - 6480*x^2*y^2*z - 4560*x^2*y^2 + 210*x^2*y*z^4 + 1980*x^2*y*z^3 + 6600*x^2*y*z^2 + 9360*x^2*y*z + 4816*x^2*y - 42*x^2*z^5 - 500*x^2*z^4 - 2240*x^2*z^3 - 4800*x^2*z^2 - 4976*x^2*z - 2032*x^2 + 14*x*y^6 - 84*x*y^5*z - 192*x*y^5 + 210*x*y^4*z^2 + 970*x*y^4*z + 1060*x*y^4 - 280*x*y^3*z^3 - 1960*x*y^3*z^2 - 4320*x*y^3*z - 3040*x*y^3 + 210*x*y^2*z^4 + 1980*x*y^2*z^3 + 6600*x*y^2*z^2 + 9360*x*y^2*z + 4814*x*y^2 - 84*x*y*z^5 - 1000*x*y*z^4 - 4480*x*y*z^3 - 9600*x*y*z^2 - 9948*x*y*z - 4052*x*y + 14*x*z^6 + 202*x*z^5 + 1140*x*z^4 + 3280*x*z^3 + 5134*x*z^2 + 4218*x*z + 1453*x + 2*y^7 - 14*y^6*z - 32*y^6 + 42*y^5*z^2 + 194*y^5*z + 212*y^5 - 70*y^4*z^3 - 490*y^4*z^2 - 1080*y^4*z - 760*y^4 + 70*y^3*z^4 + 660*y^3*z^3 + 2200*y^3*z^2 + 3120*y^3*z + 1604*y^3 - 42*y^2*z^5 - 500*y^2*z^4 - 2240*y^2*z^3 - 4800*y^2*z^2 - 4972*y^2*z - 2020*y^2 + 14*y*z^6 + 202*y*z^5 + 1140*y*z^4 + 3280*y*z^3 + 5132*y*z^2 + 4204*y*z + 1441*y - 2*z^7 - 34*z^6 - 232*z^5 - 840*z^4 - 1764*z^3 - 2184*z^2 - 1517*z - 1, -2*x^7 - 14*x^6*y + 14*x^6*z + 32*x^6 - 42*x^5*y^2 + 84*x^5*y*z + 192*x^5*y - 42*x^5*z^2 - 194*x^5*z - 212*x^5 - 70*x^4*y^3 + 210*x^4*y^2*z + 480*x^4*y^2 - 210*x^4*y*z^2 - 970*x^4*y*z - 1060*x^4*y + 70*x^4*z^3 + 490*x^4*z^2 + 1080*x^4*z + 760*x^4 - 70*x^3*y^4 + 280*x^3*y^3*z + 640*x^3*y^3 - 420*x^3*y^2*z^2 - 1940*x^3*y^2*z - 2120*x^3*y^2 + 280*x^3*y*z^3 + 1960*x^3*y*z^2 + 4320*x^3*y*z + 3040*x^3*y - 70*x^3*z^4 - 660*x^3*z^3 - 2200*x^3*z^2 - 3120*x^3*z - 1606*x^3 - 42*x^2*y^5 + 210*x^2*y^4*z + 480*x^2*y^4 - 420*x^2*y^3*z^2 - 1940*x^2*y^3*z - 2120*x^2*y^3 + 420*x^2*y^2*z^3 + 2940*x^2*y^2*z^2 + 6480*x^2*y^2*z + 4560*x^2*y^2 - 210*x^2*y*z^4 - 1980*x^2*y*z^3 - 6600*x^2*y*z^2 - 9360*x^2*y*z - 4816*x^2*y + 42*x^2*z^5 + 500*x^2*z^4 + 2240*x^2*z^3 + 4800*x^2*z^2 + 4976*x^2*z + 2031*x^2 - 14*x*y^6 + 84*x*y^5*z + 192*x*y^5 - 210*x*y^4*z^2 - 970*x*y^4*z - 1060*x*y^4 + 280*x*y^3*z^3 + 1960*x*y^3*z^2 + 4320*x*y^3*z + 3040*x*y^3 - 210*x*y^2*z^4 - 1980*x*y^2*z^3 - 6600*x*y^2*z^2 - 9360*x*y^2*z - 4814*x*y^2 + 84*x*y*z^5 + 1000*x*y*z^4 + 4480*x*y*z^3 + 9600*x*y*z^2 + 9948*x*y*z + 4050*x*y - 14*x*z^6 - 202*x*z^5 - 1140*x*z^4 - 3280*x*z^3 - 5134*x*z^2 - 4216*x*z - 1447*x - 2*y^7 + 14*y^6*z + 32*y^6 - 42*y^5*z^2 - 194*y^5*z - 212*y^5 + 70*y^4*z^3 + 490*y^4*z^2 + 1080*y^4*z + 760*y^4 - 70*y^3*z^4 - 660*y^3*z^3 - 2200*y^3*z^2 - 3120*y^3*z - 1604*y^3 + 42*y^2*z^5 + 500*y^2*z^4 + 2240*y^2*z^3 + 4800*y^2*z^2 + 4972*y^2*z + 2019*y^2 - 14*y*z^6 - 202*y*z^5 - 1140*y*z^4 - 3280*y*z^3 - 5132*y*z^2 - 4202*y*z - 1435*y + 2*z^7 + 34*z^6 + 232*z^5 + 840*z^4 + 1764*z^3 + 2183*z^2 + 1510*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -1, -1, 1 ], [ -1, -1, 2 ], [ -2, -1, 1 ] ], "source_shift": [ 2, 1, 1 ], "target_matrix": [ [ -2, -1, -1 ], [ -1, -2, -1 ], [ 1, 1, 1 ] ], "target_shift": [ 976, 463, -456 ], "inverse_recipe": { "p": 2, "q": 5, "f": "x**2 + x", "g": "x**5 + x", "h": "y*z + 2*y", "polynomials": { "f": [ [ [ 2, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "g": [ [ [ 5, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 1, 0 ], 2 ] ] }, "h_multiplier": 2 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.001016342001094 ], "surface_features": [ 3.377091135400481, 4.309118931955559, 3.0, 0.0, 3.521931176053257, 4792 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B160", "pair": 160, "family": "bijection", "mapping": "F(x,y,z)=(x^7 - 7*x^6*y + 14*x^6*z - 15*x^6 + 21*x^5*y^2 - 84*x^5*y*z + 90*x^5*y + 84*x^5*z^2 - 179*x^5*z + 92*x^5 - 35*x^4*y^3 + 210*x^4*y^2*z - 225*x^4*y^2 - 420*x^4*y*z^2 + 895*x^4*y*z - 460*x^4*y + 280*x^4*z^3 - 890*x^4*z^2 + 910*x^4*z - 300*x^4 + 35*x^3*y^4 - 280*x^3*y^3*z + 300*x^3*y^3 + 840*x^3*y^2*z^2 - 1790*x^3*y^2*z + 920*x^3*y^2 - 1120*x^3*y*z^3 + 3560*x^3*y*z^2 - 3640*x^3*y*z + 1200*x^3*y + 560*x^3*z^4 - 2360*x^3*z^3 + 3600*x^3*z^2 - 2360*x^3*z + 562*x^3 - 21*x^2*y^5 + 210*x^2*y^4*z - 225*x^2*y^4 - 840*x^2*y^3*z^2 + 1790*x^2*y^3*z - 920*x^2*y^3 + 1680*x^2*y^2*z^3 - 5340*x^2*y^2*z^2 + 5460*x^2*y^2*z - 1800*x^2*y^2 - 1680*x^2*y*z^4 + 7080*x^2*y*z^3 - 10800*x^2*y*z^2 + 7080*x^2*y*z - 1687*x^2*y + 672*x^2*z^5 - 3520*x^2*z^4 + 7120*x^2*z^3 - 6960*x^2*z^2 + 3291*x^2*z - 606*x^2 + 7*x*y^6 - 84*x*y^5*z + 90*x*y^5 + 420*x*y^4*z^2 - 895*x*y^4*z + 460*x*y^4 - 1120*x*y^3*z^3 + 3560*x*y^3*z^2 - 3640*x*y^3*z + 1200*x*y^3 + 1680*x*y^2*z^4 - 7080*x*y^2*z^3 + 10800*x*y^2*z^2 - 7080*x*y^2*z + 1688*x*y^2 - 1344*x*y*z^5 + 7040*x*y*z^4 - 14240*x*y*z^3 + 13920*x*y*z^2 - 6586*x*y*z + 1217*x*y + 448*x*z^6 - 2800*x*z^5 + 7040*x*z^4 - 9120*x*z^3 + 6420*x*z^2 - 2337*x*z + 345*x - y^7 + 14*y^6*z - 15*y^6 - 84*y^5*z^2 + 179*y^5*z - 92*y^5 + 280*y^4*z^3 - 890*y^4*z^2 + 910*y^4*z - 300*y^4 - 560*y^3*z^4 + 2360*y^3*z^3 - 3600*y^3*z^2 + 2360*y^3*z - 563*y^3 + 672*y^2*z^5 - 3520*y^2*z^4 + 7120*y^2*z^3 - 6960*y^2*z^2 + 3295*y^2*z - 611*y^2 - 448*y*z^6 + 2800*y*z^5 - 7040*y*z^4 + 9120*y*z^3 - 6424*y*z^2 + 2346*y*z - 347*y + 128*z^7 - 928*z^6 + 2784*z^5 - 4480*z^4 + 4172*z^3 - 2251*z^2 + 652*z - 1, 2*x^7 - 14*x^6*y + 28*x^6*z - 30*x^6 + 42*x^5*y^2 - 168*x^5*y*z + 180*x^5*y + 168*x^5*z^2 - 358*x^5*z + 190*x^5 - 70*x^4*y^3 + 420*x^4*y^2*z - 450*x^4*y^2 - 840*x^4*y*z^2 + 1790*x^4*y*z - 950*x^4*y + 560*x^4*z^3 - 1780*x^4*z^2 + 1880*x^4*z - 660*x^4 + 70*x^3*y^4 - 560*x^3*y^3*z + 600*x^3*y^3 + 1680*x^3*y^2*z^2 - 3580*x^3*y^2*z + 1900*x^3*y^2 - 2240*x^3*y*z^3 + 7120*x^3*y*z^2 - 7520*x^3*y*z + 2640*x^3*y + 1120*x^3*z^4 - 4720*x^3*z^3 + 7440*x^3*z^2 - 5200*x^3*z + 1364*x^3 - 42*x^2*y^5 + 420*x^2*y^4*z - 450*x^2*y^4 - 1680*x^2*y^3*z^2 + 3580*x^2*y^3*z - 1900*x^2*y^3 + 3360*x^2*y^2*z^3 - 10680*x^2*y^2*z^2 + 11280*x^2*y^2*z - 3960*x^2*y^2 - 3360*x^2*y*z^4 + 14160*x^2*y*z^3 - 22320*x^2*y*z^2 + 15600*x^2*y*z - 4094*x^2*y + 1344*x^2*z^5 - 7040*x^2*z^4 + 14720*x^2*z^3 - 15360*x^2*z^2 + 8022*x^2*z - 1691*x^2 + 14*x*y^6 - 168*x*y^5*z + 180*x*y^5 + 840*x*y^4*z^2 - 1790*x*y^4*z + 950*x*y^4 - 2240*x*y^3*z^3 + 7120*x*y^3*z^2 - 7520*x*y^3*z + 2640*x*y^3 + 3360*x*y^2*z^4 - 14160*x*y^2*z^3 + 22320*x*y^2*z^2 - 15600*x*y^2*z + 4096*x*y^2 - 2688*x*y*z^5 + 14080*x*y*z^4 - 29440*x*y*z^3 + 30720*x*y*z^2 - 16052*x*y*z + 3392*x*y + 896*x*z^6 - 5600*x*z^5 + 14560*x*z^4 - 20160*x*z^3 + 15720*x*z^2 - 6590*x*z + 1177*x - 2*y^7 + 28*y^6*z - 30*y^6 - 168*y^5*z^2 + 358*y^5*z - 190*y^5 + 560*y^4*z^3 - 1780*y^4*z^2 + 1880*y^4*z - 660*y^4 - 1120*y^3*z^4 + 4720*y^3*z^3 - 7440*y^3*z^2 + 5200*y^3*z - 1366*y^3 + 1344*y^2*z^5 - 7040*y^2*z^4 + 14720*y^2*z^3 - 15360*y^2*z^2 + 8030*y^2*z - 1701*y^2 - 896*y*z^6 + 5600*y*z^5 - 14560*y*z^4 + 20160*y*z^3 - 15728*y*z^2 + 6608*y*z - 1187*y + 256*z^7 - 1856*z^6 + 5760*z^5 - 9920*z^4 + 10264*z^3 - 6418*z^2 + 2273*z - 1, -x^7 + 7*x^6*y - 14*x^6*z + 15*x^6 - 21*x^5*y^2 + 84*x^5*y*z - 90*x^5*y - 84*x^5*z^2 + 179*x^5*z - 94*x^5 + 35*x^4*y^3 - 210*x^4*y^2*z + 225*x^4*y^2 + 420*x^4*y*z^2 - 895*x^4*y*z + 470*x^4*y - 280*x^4*z^3 + 890*x^4*z^2 - 930*x^4*z + 320*x^4 - 35*x^3*y^4 + 280*x^3*y^3*z - 300*x^3*y^3 - 840*x^3*y^2*z^2 + 1790*x^3*y^2*z - 940*x^3*y^2 + 1120*x^3*y*z^3 - 3560*x^3*y*z^2 + 3720*x^3*y*z - 1280*x^3*y - 560*x^3*z^4 + 2360*x^3*z^3 - 3680*x^3*z^2 + 2520*x^3*z - 642*x^3 + 21*x^2*y^5 - 210*x^2*y^4*z + 225*x^2*y^4 + 840*x^2*y^3*z^2 - 1790*x^2*y^3*z + 940*x^2*y^3 - 1680*x^2*y^2*z^3 + 5340*x^2*y^2*z^2 - 5580*x^2*y^2*z + 1920*x^2*y^2 + 1680*x^2*y*z^4 - 7080*x^2*y*z^3 + 11040*x^2*y*z^2 - 7560*x^2*y*z + 1927*x^2*y - 672*x^2*z^5 + 3520*x^2*z^4 - 7280*x^2*z^3 + 7440*x^2*z^2 - 3771*x^2*z + 766*x^2 - 7*x*y^6 + 84*x*y^5*z - 90*x*y^5 - 420*x*y^4*z^2 + 895*x*y^4*z - 470*x*y^4 + 1120*x*y^3*z^3 - 3560*x*y^3*z^2 + 3720*x*y^3*z - 1280*x*y^3 - 1680*x*y^2*z^4 + 7080*x*y^2*z^3 - 11040*x*y^2*z^2 + 7560*x*y^2*z - 1928*x*y^2 + 1344*x*y*z^5 - 7040*x*y*z^4 + 14560*x*y*z^3 - 14880*x*y*z^2 + 7546*x*y*z - 1537*x*y - 448*x*z^6 + 2800*x*z^5 - 7200*x*z^4 + 9760*x*z^3 - 7380*x*z^2 + 2977*x*z - 509*x + y^7 - 14*y^6*z + 15*y^6 + 84*y^5*z^2 - 179*y^5*z + 94*y^5 - 280*y^4*z^3 + 890*y^4*z^2 - 930*y^4*z + 320*y^4 + 560*y^3*z^4 - 2360*y^3*z^3 + 3680*y^3*z^2 - 2520*y^3*z + 643*y^3 - 672*y^2*z^5 + 3520*y^2*z^4 - 7280*y^2*z^3 + 7440*y^2*z^2 - 3775*y^2*z + 771*y^2 + 448*y*z^6 - 2800*y*z^5 + 7200*y*z^4 - 9760*y*z^3 + 7384*y*z^2 - 2986*y*z + 513*y - 128*z^7 + 928*z^6 - 2848*z^5 + 4800*z^4 - 4812*z^3 + 2891*z^2 - 978*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -1, 1, -2 ], [ -1, 1, -1 ], [ -1, 2, -1 ] ], "source_shift": [ 2, 2, 1 ], "target_matrix": [ [ -1, -1, -2 ], [ -2, -1, -1 ], [ 1, 1, 1 ] ], "target_shift": [ 77, 359, -147 ], "inverse_recipe": { "p": 2, "q": 5, "f": "x**2", "g": "x**5 + x", "h": "y*z", "polynomials": { "f": [ [ [ 2, 0, 0 ], 1 ] ], "g": [ [ [ 5, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0420589559980726 ], "surface_features": [ 3.5280363694612635, 4.487421211359474, 3.1283992687178066, 0.0, 3.6738262883659694, 4838 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B247", "pair": 247, "family": "bijection", "mapping": "F(x,y,z)=(8*x^7 - 56*x^6*y + 108*x^6*z - 62*x^6 + 168*x^5*y^2 - 648*x^5*y*z + 372*x^5*y + 624*x^5*z^2 - 720*x^5*z + 204*x^5 - 280*x^4*y^3 + 1620*x^4*y^2*z - 930*x^4*y^2 - 3120*x^4*y*z^2 + 3600*x^4*y*z - 1020*x^4*y + 2000*x^4*z^3 - 3480*x^4*z^2 + 1980*x^4*z - 442*x^4 + 280*x^3*y^4 - 2160*x^3*y^3*z + 1240*x^3*y^3 + 6240*x^3*y^2*z^2 - 7200*x^3*y^2*z + 2040*x^3*y^2 - 8000*x^3*y*z^3 + 13920*x^3*y*z^2 - 7920*x^3*y*z + 1768*x^3*y + 3840*x^3*z^4 - 8960*x^3*z^3 + 7680*x^3*z^2 - 3324*x^3*z + 822*x^3 - 168*x^2*y^5 + 1620*x^2*y^4*z - 930*x^2*y^4 - 6240*x^2*y^3*z^2 + 7200*x^2*y^3*z - 2040*x^2*y^3 + 12000*x^2*y^2*z^3 - 20880*x^2*y^2*z^2 + 11880*x^2*y^2*z - 2652*x^2*y^2 - 11520*x^2*y*z^4 + 26880*x^2*y*z^3 - 23040*x^2*y*z^2 + 9972*x^2*y*z - 2466*x^2*y + 4416*x^2*z^5 - 12960*x^2*z^4 + 14880*x^2*z^3 - 9432*x^2*z^2 + 4284*x^2*z - 1204*x^2 + 56*x*y^6 - 648*x*y^5*z + 372*x*y^5 + 3120*x*y^4*z^2 - 3600*x*y^4*z + 1020*x*y^4 - 8000*x*y^3*z^3 + 13920*x*y^3*z^2 - 7920*x*y^3*z + 1768*x*y^3 + 11520*x*y^2*z^4 - 26880*x*y^2*z^3 + 23040*x*y^2*z^2 - 9972*x*y^2*z + 2466*x*y^2 - 8832*x*y*z^5 + 25920*x*y*z^4 - 29760*x*y*z^3 + 18864*x*y*z^2 - 8568*x*y*z + 2416*x*y + 2816*x*z^6 - 9984*x*z^5 + 14400*x*z^4 - 11952*x*z^3 + 7560*x*z^2 - 3900*x*z + 1062*x - 8*y^7 + 108*y^6*z - 62*y^6 - 624*y^5*z^2 + 720*y^5*z - 204*y^5 + 2000*y^4*z^3 - 3480*y^4*z^2 + 1980*y^4*z - 442*y^4 - 3840*y^3*z^4 + 8960*y^3*z^3 - 7680*y^3*z^2 + 3324*y^3*z - 822*y^3 + 4416*y^2*z^5 - 12960*y^2*z^4 + 14880*y^2*z^3 - 9432*y^2*z^2 + 4284*y^2*z - 1212*y^2 - 2816*y*z^6 + 9984*y*z^5 - 14400*y*z^4 + 11952*y*z^3 - 7560*y*z^2 + 3912*y*z - 1076*y + 768*z^7 - 3200*z^6 + 5568*z^5 - 5700*z^4 + 4512*z^3 - 3204*z^2 + 1657*z - 1, -4*x^7 + 28*x^6*y - 54*x^6*z + 30*x^6 - 84*x^5*y^2 + 324*x^5*y*z - 180*x^5*y - 312*x^5*z^2 + 348*x^5*z - 96*x^5 + 140*x^4*y^3 - 810*x^4*y^2*z + 450*x^4*y^2 + 1560*x^4*y*z^2 - 1740*x^4*y*z + 480*x^4*y - 1000*x^4*z^3 + 1680*x^4*z^2 - 930*x^4*z + 206*x^4 - 140*x^3*y^4 + 1080*x^3*y^3*z - 600*x^3*y^3 - 3120*x^3*y^2*z^2 + 3480*x^3*y^2*z - 960*x^3*y^2 + 4000*x^3*y*z^3 - 6720*x^3*y*z^2 + 3720*x^3*y*z - 824*x^3*y - 1920*x^3*z^4 + 4320*x^3*z^3 - 3600*x^3*z^2 + 1542*x^3*z - 390*x^3 + 84*x^2*y^5 - 810*x^2*y^4*z + 450*x^2*y^4 + 3120*x^2*y^3*z^2 - 3480*x^2*y^3*z + 960*x^2*y^3 - 6000*x^2*y^2*z^3 + 10080*x^2*y^2*z^2 - 5580*x^2*y^2*z + 1236*x^2*y^2 + 5760*x^2*y*z^4 - 12960*x^2*y*z^3 + 10800*x^2*y*z^2 - 4626*x^2*y*z + 1170*x^2*y - 2208*x^2*z^5 + 6240*x^2*z^4 - 6960*x^2*z^3 + 4356*x^2*z^2 - 2016*x^2*z + 584*x^2 - 28*x*y^6 + 324*x*y^5*z - 180*x*y^5 - 1560*x*y^4*z^2 + 1740*x*y^4*z - 480*x*y^4 + 4000*x*y^3*z^3 - 6720*x*y^3*z^2 + 3720*x*y^3*z - 824*x*y^3 - 5760*x*y^2*z^4 + 12960*x*y^2*z^3 - 10800*x*y^2*z^2 + 4626*x*y^2*z - 1170*x*y^2 + 4416*x*y*z^5 - 12480*x*y*z^4 + 13920*x*y*z^3 - 8712*x*y*z^2 + 4032*x*y*z - 1172*x*y - 1408*x*z^6 + 4800*x*z^5 - 6720*x*z^4 + 5496*x*z^3 - 3528*x*z^2 + 1878*x*z - 523*x + 4*y^7 - 54*y^6*z + 30*y^6 + 312*y^5*z^2 - 348*y^5*z + 96*y^5 - 1000*y^4*z^3 + 1680*y^4*z^2 - 930*y^4*z + 206*y^4 + 1920*y^3*z^4 - 4320*y^3*z^3 + 3600*y^3*z^2 - 1542*y^3*z + 390*y^3 - 2208*y^2*z^5 + 6240*y^2*z^4 - 6960*y^2*z^3 + 4356*y^2*z^2 - 2016*y^2*z + 588*y^2 + 1408*y*z^6 - 4800*y*z^5 + 6720*y*z^4 - 5496*y*z^3 + 3528*y*z^2 - 1884*y*z + 529*y - 384*z^7 + 1536*z^6 - 2592*z^5 + 2610*z^4 - 2088*z^3 + 1530*z^2 - 812*z + 1, 4*x^7 - 28*x^6*y + 54*x^6*z - 30*x^6 + 84*x^5*y^2 - 324*x^5*y*z + 180*x^5*y + 312*x^5*z^2 - 348*x^5*z + 96*x^5 - 140*x^4*y^3 + 810*x^4*y^2*z - 450*x^4*y^2 - 1560*x^4*y*z^2 + 1740*x^4*y*z - 480*x^4*y + 1000*x^4*z^3 - 1680*x^4*z^2 + 930*x^4*z - 206*x^4 + 140*x^3*y^4 - 1080*x^3*y^3*z + 600*x^3*y^3 + 3120*x^3*y^2*z^2 - 3480*x^3*y^2*z + 960*x^3*y^2 - 4000*x^3*y*z^3 + 6720*x^3*y*z^2 - 3720*x^3*y*z + 824*x^3*y + 1920*x^3*z^4 - 4320*x^3*z^3 + 3600*x^3*z^2 - 1542*x^3*z + 390*x^3 - 84*x^2*y^5 + 810*x^2*y^4*z - 450*x^2*y^4 - 3120*x^2*y^3*z^2 + 3480*x^2*y^3*z - 960*x^2*y^3 + 6000*x^2*y^2*z^3 - 10080*x^2*y^2*z^2 + 5580*x^2*y^2*z - 1236*x^2*y^2 - 5760*x^2*y*z^4 + 12960*x^2*y*z^3 - 10800*x^2*y*z^2 + 4626*x^2*y*z - 1170*x^2*y + 2208*x^2*z^5 - 6240*x^2*z^4 + 6960*x^2*z^3 - 4356*x^2*z^2 + 2016*x^2*z - 584*x^2 + 28*x*y^6 - 324*x*y^5*z + 180*x*y^5 + 1560*x*y^4*z^2 - 1740*x*y^4*z + 480*x*y^4 - 4000*x*y^3*z^3 + 6720*x*y^3*z^2 - 3720*x*y^3*z + 824*x*y^3 + 5760*x*y^2*z^4 - 12960*x*y^2*z^3 + 10800*x*y^2*z^2 - 4626*x*y^2*z + 1170*x*y^2 - 4416*x*y*z^5 + 12480*x*y*z^4 - 13920*x*y*z^3 + 8712*x*y*z^2 - 4032*x*y*z + 1172*x*y + 1408*x*z^6 - 4800*x*z^5 + 6720*x*z^4 - 5496*x*z^3 + 3528*x*z^2 - 1878*x*z + 521*x - 4*y^7 + 54*y^6*z - 30*y^6 - 312*y^5*z^2 + 348*y^5*z - 96*y^5 + 1000*y^4*z^3 - 1680*y^4*z^2 + 930*y^4*z - 206*y^4 - 1920*y^3*z^4 + 4320*y^3*z^3 - 3600*y^3*z^2 + 1542*y^3*z - 390*y^3 + 2208*y^2*z^5 - 6240*y^2*z^4 + 6960*y^2*z^3 - 4356*y^2*z^2 + 2016*y^2*z - 588*y^2 - 1408*y*z^6 + 4800*y*z^5 - 6720*y*z^4 + 5496*y*z^3 - 3528*y*z^2 + 1884*y*z - 527*y + 384*z^7 - 1536*z^6 + 2592*z^5 - 2610*z^4 + 2088*z^3 - 1530*z^2 + 809*z - 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -1, 1, -2 ], [ -1, 1, -1 ], [ -2, 3, -3 ] ], "source_shift": [ 1, 2, 2 ], "target_matrix": [ [ -2, -1, -1 ], [ 1, 2, 1 ], [ -1, -1, -1 ] ], "target_shift": [ 396, -198, 195 ], "inverse_recipe": { "p": 1, "q": 6, "f": "x", "g": "x**6 + x**3", "h": "y**4 + y*z + y + z", "polynomials": { "f": [ [ [ 1, 0, 0 ], 1 ] ], "g": [ [ [ 6, 0, 0 ], 1 ], [ [ 3, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 0, 1 ], 1 ], [ [ 0, 4, 0 ], 1 ], [ [ 0, 1, 0 ], 1 ] ] }, "h_multiplier": 2 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "Maybe" ], "seconds": [ 1.0356677220006532 ], "surface_features": [ 3.5809474560709007, 4.473632926873841, 3.290034611362518, 0.0, 3.679717564851442, 4919 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B151", "pair": 151, "family": "bijection", "mapping": "F(x,y,z)=(-128*x^7 - 448*x^6*y + 448*x^6*z + 512*x^6 - 672*x^5*y^2 + 1344*x^5*y*z + 1536*x^5*y - 672*x^5*z^2 - 1536*x^5*z - 912*x^5 - 560*x^4*y^3 + 1680*x^4*y^2*z + 1920*x^4*y^2 - 1680*x^4*y*z^2 - 3840*x^4*y*z - 2288*x^4*y + 560*x^4*z^3 + 1920*x^4*z^2 + 2304*x^4*z + 1008*x^4 - 280*x^3*y^4 + 1120*x^3*y^3*z + 1280*x^3*y^3 - 1680*x^3*y^2*z^2 - 3840*x^3*y^2*z - 2296*x^3*y^2 + 1120*x^3*y*z^3 + 3840*x^3*y*z^2 + 4624*x^3*y*z + 2040*x^3*y - 280*x^3*z^4 - 1280*x^3*z^3 - 2328*x^3*z^2 - 2080*x^3*z - 752*x^3 - 84*x^2*y^5 + 420*x^2*y^4*z + 480*x^2*y^4 - 840*x^2*y^3*z^2 - 1920*x^2*y^3*z - 1152*x^2*y^3 + 840*x^2*y^2*z^3 + 2880*x^2*y^2*z^2 + 3480*x^2*y^2*z + 1548*x^2*y^2 - 420*x^2*y*z^4 - 1920*x^2*y*z^3 - 3504*x^2*y*z^2 - 3156*x^2*y*z - 1152*x^2*y + 84*x^2*z^5 + 480*x^2*z^4 + 1176*x^2*z^3 + 1608*x^2*z^2 + 1188*x^2*z + 369*x^2 - 14*x*y^6 + 84*x*y^5*z + 96*x*y^5 - 210*x*y^4*z^2 - 480*x*y^4*z - 289*x*y^4 + 280*x*y^3*z^3 + 960*x*y^3*z^2 + 1164*x*y^3*z + 522*x*y^3 - 210*x*y^2*z^4 - 960*x*y^2*z^3 - 1758*x*y^2*z^2 - 1596*x*y^2*z - 588*x*y^2 + 84*x*y*z^5 + 480*x*y*z^4 + 1180*x*y*z^3 + 1626*x*y*z^2 + 1212*x*y*z + 382*x*y - 14*x*z^6 - 96*x*z^5 - 297*x*z^4 - 552*x*z^3 - 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by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -1, 2, -1 ], [ -2, 2, -3 ], [ -1, 1, -1 ] ], "source_shift": [ 1, 1, 2 ], "target_matrix": [ [ 1, 1, 2 ], [ -2, -1, -1 ], [ -1, -1, -1 ] ], "target_shift": [ -88, 204, 98 ], "inverse_recipe": { "p": 3, "q": 4, "f": "x**3 + x**2", "g": "x**4 + x**3 + x", "h": "y*z", "polynomials": { "f": [ [ [ 3, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "g": [ [ [ 4, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ], [ [ 3, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ] ] }, "h_multiplier": 7 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 0.982610922997992 ], "surface_features": [ 3.9695536138547616, 4.901698016307038, 3.665534088433329, 0.0, 4.084959603935063, 5042 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C038", "pair": 38, "family": "counterexample", "mapping": "F(x,y,z)=(-x^7 + 10*x^6*y - 8*x^6*z + 19*x^6 - 42*x^5*y^2 + 69*x^5*y*z - 165*x^5*y - 27*x^5*z^2 + 129*x^5*z - 153*x^5 + 96*x^4*y^3 - 243*x^4*y^2*z + 588*x^4*y^2 + 195*x^4*y*z^2 - 936*x^4*y*z + 1122*x^4*y - 50*x^4*z^3 + 360*x^4*z^2 - 858*x^4*z + 679*x^4 - 129*x^3*y^4 + 447*x^3*y^3*z - 1101*x^3*y^3 - 552*x^3*y^2*z^2 + 2673*x^3*y^2*z - 3249*x^3*y^2 + 290*x^3*y*z^3 - 2094*x^3*y*z^2 + 5034*x^3*y*z - 4033*x^3*y - 55*x^3*z^4 + 530*x^3*z^3 - 1902*x^3*z^2 + 3025*x^3*z - 1799*x^3 + 102*x^2*y^5 - 453*x^2*y^4*z + 1143*x^2*y^4 + 765*x^2*y^3*z^2 - 3756*x^2*y^3*z + 4644*x^2*y^3 - 618*x^2*y^2*z^3 + 4491*x^2*y^2*z^2 - 10920*x^2*y^2*z + 8877*x^2*y^2 + 240*x^2*y*z^4 - 2316*x^2*y*z^3 + 8370*x^2*y*z^2 - 13452*x^2*y*z + 8109*x^2*y - 36*x^2*z^5 + 435*x^2*z^4 - 2088*x^2*z^3 + 5001*x^2*z^2 - 5982*x^2*z + 2854*x^2 - 44*x*y^6 + 240*x*y^5*z - 624*x*y^5 - 519*x*y^4*z^2 + 2598*x*y^4*z - 3276*x*y^4 + 573*x*y^3*z^3 - 4209*x*y^3*z^2 + 10380*x*y^3*z - 8578*x*y^3 - 342*x*y^2*z^4 + 3315*x*y^2*z^3 - 12093*x*y^2*z^2 + 19677*x*y^2*z - 12045*x*y^2 + 105*x*y*z^5 - 1269*x*y*z^4 + 6126*x*y*z^3 - 14805*x*y*z^2 + 17919*x*y*z - 8677*x*y - 13*x*z^6 + 189*x*z^5 - 1137*x*z^4 + 3643*x*z^3 - 6567*x*z^2 + 6311*x*z - 2517*x + 8*y^7 - 52*y^6*z + 140*y^6 + 138*y^5*z^2 - 708*y^5*z + 912*y^5 - 195*y^4*z^3 + 1455*y^4*z^2 - 3648*y^4*z + 3067*y^4 + 159*y^3*z^4 - 1554*y^3*z^3 + 5736*y^3*z^2 - 9465*y^3*z + 5889*y^3 - 75*y^2*z^5 + 909*y^2*z^4 - 4422*y^2*z^3 + 10800*y^2*z^2 - 13245*y^2*z + 6523*y^2 + 19*y*z^6 - 276*y*z^5 + 1668*y*z^4 - 5386*y*z^3 + 9810*y*z^2 - 9550*y*z + 3872*y - 2*z^7 + 34*z^6 - 246*z^5 + 988*z^4 - 2384*z^3 + 3457*z^2 - 2784*z + 1, x^7 - 10*x^6*y + 8*x^6*z - 19*x^6 + 42*x^5*y^2 - 69*x^5*y*z + 165*x^5*y + 27*x^5*z^2 - 129*x^5*z + 153*x^5 - 96*x^4*y^3 + 243*x^4*y^2*z - 588*x^4*y^2 - 195*x^4*y*z^2 + 936*x^4*y*z - 1122*x^4*y + 50*x^4*z^3 - 360*x^4*z^2 + 858*x^4*z - 678*x^4 + 129*x^3*y^4 - 447*x^3*y^3*z + 1101*x^3*y^3 + 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3066*y^4 - 159*y^3*z^4 + 1554*y^3*z^3 - 5736*y^3*z^2 + 9460*y^3*z - 5876*y^3 + 75*y^2*z^5 - 909*y^2*z^4 + 4422*y^2*z^3 - 10791*y^2*z^2 + 13203*y^2*z - 6475*y^2 - 19*y*z^6 + 276*y*z^5 - 1668*y*z^4 + 5379*y*z^3 - 9765*y*z^2 + 9454*y*z - 3806*y + 2*z^7 - 34*z^6 + 246*z^5 - 986*z^4 + 2368*z^3 - 3409*z^2 + 2722*z - 1, 2*x^7 - 20*x^6*y + 16*x^6*z - 35*x^6 + 84*x^5*y^2 - 138*x^5*y*z + 306*x^5*y + 54*x^5*z^2 - 237*x^5*z + 264*x^5 - 192*x^4*y^3 + 486*x^4*y^2*z - 1098*x^4*y^2 - 390*x^4*y*z^2 + 1731*x^4*y*z - 1953*x^4*y + 100*x^4*z^3 - 660*x^4*z^2 + 1476*x^4*z - 1113*x^4 + 258*x^3*y^4 - 894*x^3*y^3*z + 2070*x^3*y^3 + 1104*x^3*y^2*z^2 - 4977*x^3*y^2*z + 5706*x^3*y^2 - 580*x^3*y*z^3 + 3864*x^3*y*z^2 - 8736*x^3*y*z + 6672*x^3*y + 110*x^3*z^4 - 970*x^3*z^3 + 3264*x^3*z^2 - 4945*x^3*z + 2834*x^3 - 204*x^2*y^5 + 906*x^2*y^4*z - 2163*x^2*y^4 - 1530*x^2*y^3*z^2 + 7041*x^2*y^3*z - 8229*x^2*y^3 + 1236*x^2*y^2*z^3 - 8343*x^2*y^2*z^2 + 19122*x^2*y^2*z - 14826*x^2*y^2 - 480*x^2*y*z^4 + 4266*x^2*y*z^3 - 14490*x^2*y*z^2 + 22197*x^2*y*z - 12891*x^2*y + 72*x^2*z^5 - 795*x^2*z^4 + 3576*x^2*z^3 - 8157*x^2*z^2 + 9402*x^2*z - 4361*x^2 + 88*x*y^6 - 480*x*y^5*z + 1188*x*y^5 + 1038*x*y^4*z^2 - 4902*x*y^4*z + 5856*x*y^4 - 1146*x*y^3*z^3 + 7872*x*y^3*z^2 - 18342*x*y^3*z + 14466*x*y^3 + 684*x*y^2*z^4 - 6147*x*y^2*z^3 + 21126*x*y^2*z^2 - 32787*x*y^2*z + 19332*x*y^2 - 210*x*y*z^5 + 2334*x*y*z^4 - 10584*x*y*z^3 + 24378*x*y*z^2 - 28428*x*y*z + 13376*x*y + 26*x*z^6 - 345*x*z^5 + 1944*x*z^4 - 5931*x*z^3 + 10302*x*z^2 - 9625*x*z + 3758*x - 16*y^7 + 104*y^6*z - 268*y^6 - 276*y^5*z^2 + 1344*y^5*z - 1644*y^5 + 390*y^4*z^3 - 2739*y^4*z^2 + 6504*y^4*z - 5223*y^4 - 318*y^3*z^4 + 2901*y^3*z^3 - 10113*y^3*z^2 + 15929*y^3*z - 9547*y^3 + 150*y^2*z^5 - 1683*y^2*z^4 + 7710*y^2*z^3 - 17961*y^2*z^2 + 21222*y^2*z - 10151*y^2 - 38*y*z^6 + 507*y*z^5 - 2877*y*z^4 + 8853*y*z^3 - 15537*y*z^2 + 14699*y*z - 5831*y + 4*z^7 - 62*z^6 + 420*z^5 - 1606*z^4 + 3734*z^3 - 5264*z^2 + 4150*z - 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -1, 1, -1 ], [ -1, 2, -1 ], [ -1, 1, -2 ] ], "source_shift": [ 2, 2, 1 ], "target_matrix": [ [ 1, 1, 1 ], [ -1, -1, -2 ], [ -2, -1, -1 ] ], "target_shift": [ -952, 924, 1397 ], "collision_preimages": [ [ "11/4", "0", "-3/4" ], [ "5", "-5/2", "-13/2" ], [ "14", "5/2", "-17/2" ] ], "collision_image": [ "-3809/4", "3697/4", "2795/2" ] }, "probe_target": "No", "jev_answers": [ "Maybe" ], "seconds": [ 1.0911471439976594 ], "surface_features": [ 3.5824503668650416, 4.515701680681241, 3.1659858227744544, 0.0, 3.72006452520477, 4858 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B131", "pair": 131, "family": "bijection", "mapping": "F(x,y,z)=(4*x^7 - 56*x^6*y - 28*x^6*z + 52*x^6 + 336*x^5*y^2 + 336*x^5*y*z - 616*x^5*y + 84*x^5*z^2 - 312*x^5*z + 280*x^5 - 1120*x^4*y^3 - 1680*x^4*y^2*z + 3040*x^4*y^2 - 840*x^4*y*z^2 + 3080*x^4*y*z - 2732*x^4*y - 140*x^4*z^3 + 780*x^4*z^2 - 1404*x^4*z + 832*x^4 + 2240*x^3*y^4 + 4480*x^3*y^3*z - 8000*x^3*y^3 + 3360*x^3*y^2*z^2 - 12160*x^3*y^2*z + 10660*x^3*y^2 + 1120*x^3*y*z^3 - 6160*x^3*y*z^2 + 10960*x^3*y*z - 6408*x^3*y + 140*x^3*z^4 - 1040*x^3*z^3 + 2816*x^3*z^2 - 3368*x^3*z + 1507*x^3 - 2688*x^2*y^5 - 6720*x^2*y^4*z + 11840*x^2*y^4 - 6720*x^2*y^3*z^2 + 24000*x^2*y^3*z - 20792*x^2*y^3 - 3360*x^2*y^2*z^3 + 18240*x^2*y^2*z^2 - 32076*x^2*y^2*z + 18504*x^2*y^2 - 840*x^2*y*z^4 + 6160*x^2*y*z^3 - 16488*x^2*y*z^2 + 19456*x^2*y*z - 8562*x^2*y - 84*x^2*z^5 + 780*x^2*z^4 - 2824*x^2*z^3 + 5112*x^2*z^2 - 4649*x^2*z + 1686*x^2 + 1792*x*y^6 + 5376*x*y^5*z - 9344*x*y^5 + 6720*x*y^4*z^2 - 23680*x*y^4*z + 20272*x*y^4 + 4480*x*y^3*z^3 - 24000*x*y^3*z^2 + 41712*x*y^3*z - 23744*x*y^3 + 1680*x*y^2*z^4 - 12160*x*y^2*z^3 + 32172*x*y^2*z^2 - 37456*x*y^2*z + 16220*x*y^2 + 336*x*y*z^5 - 3080*x*y*z^4 + 11024*x*y*z^3 - 19688*x*y*z^2 + 17604*x*y*z - 6248*x*y + 28*x*z^6 - 312*x*z^5 + 1416*x*z^4 - 3448*x*z^3 + 4777*x*z^2 - 3548*x*z + 1084*x - 512*y^7 - 1792*y^6*z + 3072*y^6 - 2688*y^5*z^2 + 9344*y^5*z - 7904*y^5 - 2240*y^4*z^3 + 11840*y^4*z^2 - 20336*y^4*z + 11424*y^4 - 1120*y^3*z^4 + 8000*y^3*z^3 - 20920*y^3*z^2 + 24032*y^3*z - 10244*y^3 - 336*y^2*z^5 + 3040*y^2*z^4 - 10756*y^2*z^3 + 18952*y^2*z^2 - 16672*y^2*z + 5804*y^2 - 56*y*z^6 + 616*y*z^5 - 2764*y*z^4 + 6640*y*z^3 - 9042*y*z^2 + 6560*y*z - 1947*y - 4*z^7 + 52*z^6 - 284*z^5 + 872*z^4 - 1635*z^3 + 1866*z^2 - 1183*z + 1, -2*x^7 + 28*x^6*y + 14*x^6*z - 26*x^6 - 168*x^5*y^2 - 168*x^5*y*z + 308*x^5*y - 42*x^5*z^2 + 156*x^5*z - 140*x^5 + 560*x^4*y^3 + 840*x^4*y^2*z - 1520*x^4*y^2 + 420*x^4*y*z^2 - 1540*x^4*y*z + 1366*x^4*y + 70*x^4*z^3 - 390*x^4*z^2 + 702*x^4*z - 416*x^4 - 1120*x^3*y^4 - 2240*x^3*y^3*z + 4000*x^3*y^3 - 1680*x^3*y^2*z^2 + 6080*x^3*y^2*z - 5330*x^3*y^2 - 560*x^3*y*z^3 + 3080*x^3*y*z^2 - 5480*x^3*y*z + 3204*x^3*y - 70*x^3*z^4 + 520*x^3*z^3 - 1408*x^3*z^2 + 1684*x^3*z - 753*x^3 + 1344*x^2*y^5 + 3360*x^2*y^4*z - 5920*x^2*y^4 + 3360*x^2*y^3*z^2 - 12000*x^2*y^3*z + 10396*x^2*y^3 + 1680*x^2*y^2*z^3 - 9120*x^2*y^2*z^2 + 16038*x^2*y^2*z - 9252*x^2*y^2 + 420*x^2*y*z^4 - 3080*x^2*y*z^3 + 8244*x^2*y*z^2 - 9728*x^2*y*z + 4278*x^2*y + 42*x^2*z^5 - 390*x^2*z^4 + 1412*x^2*z^3 - 2556*x^2*z^2 + 2323*x^2*z - 840*x^2 - 896*x*y^6 - 2688*x*y^5*z + 4672*x*y^5 - 3360*x*y^4*z^2 + 11840*x*y^4*z - 10136*x*y^4 - 2240*x*y^3*z^3 + 12000*x*y^3*z^2 - 20856*x*y^3*z + 11872*x*y^3 - 840*x*y^2*z^4 + 6080*x*y^2*z^3 - 16086*x*y^2*z^2 + 18728*x*y^2*z - 8104*x*y^2 - 168*x*y*z^5 + 1540*x*y*z^4 - 5512*x*y*z^3 + 9844*x*y*z^2 - 8796*x*y*z + 3112*x*y - 14*x*z^6 + 156*x*z^5 - 708*x*z^4 + 1724*x*z^3 - 2387*x*z^2 + 1768*x*z - 535*x + 256*y^7 + 896*y^6*z - 1536*y^6 + 1344*y^5*z^2 - 4672*y^5*z + 3952*y^5 + 1120*y^4*z^3 - 5920*y^4*z^2 + 10168*y^4*z - 5712*y^4 + 560*y^3*z^4 - 4000*y^3*z^3 + 10460*y^3*z^2 - 12016*y^3*z + 5118*y^3 + 168*y^2*z^5 - 1520*y^2*z^4 + 5378*y^2*z^3 - 9476*y^2*z^2 + 8330*y^2*z - 2890*y^2 + 28*y*z^6 - 308*y*z^5 + 1382*y*z^4 - 3320*y*z^3 + 4518*y*z^2 - 3268*y*z + 961*y + 2*z^7 - 26*z^6 + 142*z^5 - 436*z^4 + 817*z^3 - 930*z^2 + 584*z - 1, -2*x^7 + 28*x^6*y + 14*x^6*z - 26*x^6 - 168*x^5*y^2 - 168*x^5*y*z + 308*x^5*y - 42*x^5*z^2 + 156*x^5*z - 140*x^5 + 560*x^4*y^3 + 840*x^4*y^2*z - 1520*x^4*y^2 + 420*x^4*y*z^2 - 1540*x^4*y*z + 1366*x^4*y + 70*x^4*z^3 - 390*x^4*z^2 + 702*x^4*z - 416*x^4 - 1120*x^3*y^4 - 2240*x^3*y^3*z + 4000*x^3*y^3 - 1680*x^3*y^2*z^2 + 6080*x^3*y^2*z - 5330*x^3*y^2 - 560*x^3*y*z^3 + 3080*x^3*y*z^2 - 5480*x^3*y*z + 3204*x^3*y - 70*x^3*z^4 + 520*x^3*z^3 - 1408*x^3*z^2 + 1684*x^3*z - 753*x^3 + 1344*x^2*y^5 + 3360*x^2*y^4*z - 5920*x^2*y^4 + 3360*x^2*y^3*z^2 - 12000*x^2*y^3*z + 10396*x^2*y^3 + 1680*x^2*y^2*z^3 - 9120*x^2*y^2*z^2 + 16038*x^2*y^2*z - 9252*x^2*y^2 + 420*x^2*y*z^4 - 3080*x^2*y*z^3 + 8244*x^2*y*z^2 - 9728*x^2*y*z + 4278*x^2*y + 42*x^2*z^5 - 390*x^2*z^4 + 1412*x^2*z^3 - 2556*x^2*z^2 + 2323*x^2*z - 842*x^2 - 896*x*y^6 - 2688*x*y^5*z + 4672*x*y^5 - 3360*x*y^4*z^2 + 11840*x*y^4*z - 10136*x*y^4 - 2240*x*y^3*z^3 + 12000*x*y^3*z^2 - 20856*x*y^3*z + 11872*x*y^3 - 840*x*y^2*z^4 + 6080*x*y^2*z^3 - 16086*x*y^2*z^2 + 18728*x*y^2*z - 8104*x*y^2 - 168*x*y*z^5 + 1540*x*y*z^4 - 5512*x*y*z^3 + 9844*x*y*z^2 - 8796*x*y*z + 3120*x*y - 14*x*z^6 + 156*x*z^5 - 708*x*z^4 + 1724*x*z^3 - 2387*x*z^2 + 1772*x*z - 543*x + 256*y^7 + 896*y^6*z - 1536*y^6 + 1344*y^5*z^2 - 4672*y^5*z + 3952*y^5 + 1120*y^4*z^3 - 5920*y^4*z^2 + 10168*y^4*z - 5712*y^4 + 560*y^3*z^4 - 4000*y^3*z^3 + 10460*y^3*z^2 - 12016*y^3*z + 5118*y^3 + 168*y^2*z^5 - 1520*y^2*z^4 + 5378*y^2*z^3 - 9476*y^2*z^2 + 8330*y^2*z - 2898*y^2 + 28*y*z^6 - 308*y*z^5 + 1382*y*z^4 - 3320*y*z^3 + 4518*y*z^2 - 3276*y*z + 975*y + 2*z^7 - 26*z^6 + 142*z^5 - 436*z^4 + 817*z^3 - 932*z^2 + 592*z - 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ 1, -2, -1 ], [ 1, -1, -2 ], [ 1, -1, -1 ] ], "source_shift": [ 1, 2, 2 ], "target_matrix": [ [ 2, 3, 3 ], [ -1, -1, -2 ], [ -1, -1, -1 ] ], "target_shift": [ -309, 148, 156 ], "inverse_recipe": { "p": 3, "q": 2, "f": "x**3 + x**2", "g": "x**2 + x", "h": "y**2 + y*z**2", "polynomials": { "f": [ [ [ 3, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "g": [ [ [ 2, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 2 ], 1 ], [ [ 0, 2, 0 ], 1 ] ] }, "h_multiplier": 2 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0671069029995124 ], "surface_features": [ 3.6800470465000594, 4.620261013824254, 3.3502480183341627, 0.0, 3.8042035313977958, 4936 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B092", "pair": 92, "family": "bijection", "mapping": "F(x,y,z)=(27*x^7 + 162*x^6*y + 162*x^6*z + 324*x^6 + 414*x^5*y^2 + 828*x^5*y*z + 1647*x^5*y + 414*x^5*z^2 + 1647*x^5*z + 1620*x^5 + 584*x^4*y^3 + 1752*x^4*y^2*z + 3465*x^4*y^2 + 1752*x^4*y*z^2 + 6930*x^4*y*z + 6777*x^4*y + 584*x^4*z^3 + 3465*x^4*z^2 + 6777*x^4*z + 4430*x^4 + 491*x^3*y^4 + 1964*x^3*y^3*z + 3861*x^3*y^3 + 2946*x^3*y^2*z^2 + 11583*x^3*y^2*z + 11259*x^3*y^2 + 1964*x^3*y*z^3 + 11583*x^3*y*z^2 + 22518*x^3*y*z + 14615*x^3*y + 491*x^3*z^4 + 3861*x^3*z^3 + 11259*x^3*z^2 + 14642*x^3*z + 7119*x^3 + 246*x^2*y^5 + 1230*x^2*y^4*z + 2403*x^2*y^4 + 2460*x^2*y^3*z^2 + 9612*x^2*y^3*z + 9285*x^2*y^3 + 2460*x^2*y^2*z^3 + 14418*x^2*y^2*z^2 + 27855*x^2*y^2*z + 17949*x^2*y^2 + 1230*x^2*y*z^4 + 9612*x^2*y*z^3 + 27855*x^2*y*z^2 + 35952*x^2*y*z + 17361*x^2*y + 246*x^2*z^5 + 2403*x^2*z^4 + 9285*x^2*z^3 + 18003*x^2*z^2 + 17442*x^2*z + 6702*x^2 + 68*x*y^6 + 408*x*y^5*z + 792*x*y^5 + 1020*x*y^4*z^2 + 3960*x*y^4*z + 3801*x*y^4 + 1360*x*y^3*z^3 + 7920*x*y^3*z^2 + 15204*x*y^3*z + 9727*x*y^3 + 1020*x*y^2*z^4 + 7920*x*y^2*z^3 + 22806*x*y^2*z^2 + 29217*x*y^2*z + 14013*x*y^2 + 408*x*y*z^5 + 3960*x*y*z^4 + 15204*x*y*z^3 + 29253*x*y*z^2 + 28134*x*y*z + 10758*x*y + 68*x*z^6 + 792*x*z^5 + 3801*x*z^4 + 9763*x*z^3 + 14121*x*z^2 + 10839*x*z + 3429*x + 8*y^7 + 56*y^6*z + 108*y^6 + 168*y^5*z^2 + 648*y^5*z + 618*y^5 + 280*y^4*z^3 + 1620*y^4*z^2 + 3090*y^4*z + 1963*y^4 + 280*y^3*z^4 + 2160*y^3*z^3 + 6180*y^3*z^2 + 7860*y^3*z + 3745*y^3 + 168*y^2*z^5 + 1620*y^2*z^4 + 6180*y^2*z^3 + 11802*y^2*z^2 + 11271*y^2*z + 4290*y^2 + 56*y*z^6 + 648*y*z^5 + 3090*y*z^4 + 7876*y*z^3 + 11307*y*z^2 + 8634*y*z + 2731*y + 8*z^7 + 108*z^6 + 618*z^5 + 1971*z^4 + 3781*z^3 + 4344*z^2 + 2760*z + 1, -54*x^7 - 324*x^6*y - 324*x^6*z - 648*x^6 - 828*x^5*y^2 - 1656*x^5*y*z - 3294*x^5*y - 828*x^5*z^2 - 3294*x^5*z - 3240*x^5 - 1168*x^4*y^3 - 3504*x^4*y^2*z - 6930*x^4*y^2 - 3504*x^4*y*z^2 - 13860*x^4*y*z - 13554*x^4*y - 1168*x^4*z^3 - 6930*x^4*z^2 - 13554*x^4*z - 8859*x^4 - 982*x^3*y^4 - 3928*x^3*y^3*z - 7722*x^3*y^3 - 5892*x^3*y^2*z^2 - 23166*x^3*y^2*z - 22518*x^3*y^2 - 3928*x^3*y*z^3 - 23166*x^3*y*z^2 - 45036*x^3*y*z - 29226*x^3*y - 982*x^3*z^4 - 7722*x^3*z^3 - 22518*x^3*z^2 - 29280*x^3*z - 14229*x^3 - 492*x^2*y^5 - 2460*x^2*y^4*z - 4806*x^2*y^4 - 4920*x^2*y^3*z^2 - 19224*x^2*y^3*z - 18570*x^2*y^3 - 4920*x^2*y^2*z^3 - 28836*x^2*y^2*z^2 - 55710*x^2*y^2*z - 35892*x^2*y^2 - 2460*x^2*y*z^4 - 19224*x^2*y*z^3 - 55710*x^2*y*z^2 - 71892*x^2*y*z - 34695*x^2*y - 492*x^2*z^5 - 4806*x^2*z^4 - 18570*x^2*z^3 - 36000*x^2*z^2 - 34857*x^2*z - 13374*x^2 - 136*x*y^6 - 816*x*y^5*z - 1584*x*y^5 - 2040*x*y^4*z^2 - 7920*x*y^4*z - 7602*x*y^4 - 2720*x*y^3*z^3 - 15840*x*y^3*z^2 - 30408*x*y^3*z - 19450*x*y^3 - 2040*x*y^2*z^4 - 15840*x*y^2*z^3 - 45612*x*y^2*z^2 - 58422*x*y^2*z - 27999*x*y^2 - 816*x*y*z^5 - 7920*x*y*z^4 - 30408*x*y*z^3 - 58494*x*y*z^2 - 56214*x*y*z - 21456*x*y - 136*x*z^6 - 1584*x*z^5 - 7602*x*z^4 - 19522*x*z^3 - 28215*x*z^2 - 21618*x*z - 6830*x - 16*y^7 - 112*y^6*z - 216*y^6 - 336*y^5*z^2 - 1296*y^5*z - 1236*y^5 - 560*y^4*z^3 - 3240*y^4*z^2 - 6180*y^4*z - 3925*y^4 - 560*y^3*z^4 - 4320*y^3*z^3 - 12360*y^3*z^2 - 15716*y^3*z - 7481*y^3 - 336*y^2*z^5 - 3240*y^2*z^4 - 12360*y^2*z^3 - 23598*y^2*z^2 - 22515*y^2*z - 8550*y^2 - 112*y*z^6 - 1296*y*z^5 - 6180*y*z^4 - 15748*y*z^3 - 22587*y*z^2 - 17208*y*z - 5428*y - 16*z^7 - 216*z^6 - 1236*z^5 - 3941*z^4 - 7553*z^3 - 8658*z^2 - 5485*z - 1, -27*x^7 - 162*x^6*y - 162*x^6*z - 324*x^6 - 414*x^5*y^2 - 828*x^5*y*z - 1647*x^5*y - 414*x^5*z^2 - 1647*x^5*z - 1620*x^5 - 584*x^4*y^3 - 1752*x^4*y^2*z - 3465*x^4*y^2 - 1752*x^4*y*z^2 - 6930*x^4*y*z - 6777*x^4*y - 584*x^4*z^3 - 3465*x^4*z^2 - 6777*x^4*z - 4430*x^4 - 491*x^3*y^4 - 1964*x^3*y^3*z - 3861*x^3*y^3 - 2946*x^3*y^2*z^2 - 11583*x^3*y^2*z - 11259*x^3*y^2 - 1964*x^3*y*z^3 - 11583*x^3*y*z^2 - 22518*x^3*y*z - 14615*x^3*y - 491*x^3*z^4 - 3861*x^3*z^3 - 11259*x^3*z^2 - 14642*x^3*z - 7119*x^3 - 246*x^2*y^5 - 1230*x^2*y^4*z - 2403*x^2*y^4 - 2460*x^2*y^3*z^2 - 9612*x^2*y^3*z - 9285*x^2*y^3 - 2460*x^2*y^2*z^3 - 14418*x^2*y^2*z^2 - 27855*x^2*y^2*z - 17949*x^2*y^2 - 1230*x^2*y*z^4 - 9612*x^2*y*z^3 - 27855*x^2*y*z^2 - 35952*x^2*y*z - 17361*x^2*y - 246*x^2*z^5 - 2403*x^2*z^4 - 9285*x^2*z^3 - 18003*x^2*z^2 - 17442*x^2*z - 6702*x^2 - 68*x*y^6 - 408*x*y^5*z - 792*x*y^5 - 1020*x*y^4*z^2 - 3960*x*y^4*z - 3801*x*y^4 - 1360*x*y^3*z^3 - 7920*x*y^3*z^2 - 15204*x*y^3*z - 9727*x*y^3 - 1020*x*y^2*z^4 - 7920*x*y^2*z^3 - 22806*x*y^2*z^2 - 29217*x*y^2*z - 14013*x*y^2 - 408*x*y*z^5 - 3960*x*y*z^4 - 15204*x*y*z^3 - 29253*x*y*z^2 - 28134*x*y*z - 10758*x*y - 68*x*z^6 - 792*x*z^5 - 3801*x*z^4 - 9763*x*z^3 - 14121*x*z^2 - 10839*x*z - 3435*x - 8*y^7 - 56*y^6*z - 108*y^6 - 168*y^5*z^2 - 648*y^5*z - 618*y^5 - 280*y^4*z^3 - 1620*y^4*z^2 - 3090*y^4*z - 1963*y^4 - 280*y^3*z^4 - 2160*y^3*z^3 - 6180*y^3*z^2 - 7860*y^3*z - 3745*y^3 - 168*y^2*z^5 - 1620*y^2*z^4 - 6180*y^2*z^3 - 11802*y^2*z^2 - 11271*y^2*z - 4290*y^2 - 56*y*z^6 - 648*y*z^5 - 3090*y*z^4 - 7876*y*z^3 - 11307*y*z^2 - 8634*y*z - 2735*y - 8*z^7 - 108*z^6 - 618*z^5 - 1971*z^4 - 3781*z^3 - 4344*z^2 - 2764*z - 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ 1, 1, 1 ], [ 2, 2, 3 ], [ 2, 1, 1 ] ], "source_shift": [ 2, 1, 1 ], "target_matrix": [ [ 1, 1, 2 ], [ -2, -1, -1 ], [ -1, -1, -1 ] ], "target_shift": [ -741, 1476, 747 ], "inverse_recipe": { "p": 4, "q": 1, "f": "x**4 + x**3", "g": "x", "h": "y*z**3 + y + z**3", "polynomials": { "f": [ [ [ 4, 0, 0 ], 1 ], [ [ 3, 0, 0 ], 1 ] ], "g": [ [ [ 1, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 3 ], 1 ], [ [ 0, 1, 0 ], 1 ], [ [ 0, 0, 3 ], 1 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.054113658999995 ], "surface_features": [ 3.926325579937923, 4.85668056563799, 3.5833121519830775, 0.0, 4.0485282557566205, 5030 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B244", "pair": 244, "family": "bijection", "mapping": "F(x,y,z)=(12*x^7 - 84*x^6*y - 164*x^6*z + 156*x^6 + 252*x^5*y^2 + 984*x^5*y*z - 936*x^5*y + 960*x^5*z^2 - 1824*x^5*z + 912*x^5 - 420*x^4*y^3 - 2460*x^4*y^2*z + 2340*x^4*y^2 - 4800*x^4*y*z^2 + 9120*x^4*y*z - 4584*x^4*y - 3120*x^4*z^3 + 8880*x^4*z^2 - 8840*x^4*z + 3072*x^4 + 420*x^3*y^4 + 3280*x^3*y^3*z - 3120*x^3*y^3 + 9600*x^3*y^2*z^2 - 18240*x^3*y^2*z + 9216*x^3*y^2 + 12480*x^3*y*z^3 - 35520*x^3*y*z^2 + 35544*x^3*y*z - 12456*x^3*y + 6080*x^3*z^4 - 23040*x^3*z^3 + 34248*x^3*z^2 - 23640*x^3*z + 6382*x^3 - 252*x^2*y^5 - 2460*x^2*y^4*z + 2340*x^2*y^4 - 9600*x^2*y^3*z^2 + 18240*x^2*y^3*z - 9264*x^2*y^3 - 18720*x^2*y^2*z^3 + 53280*x^2*y^2*z^2 - 53592*x^2*y^2*z + 18936*x^2*y^2 - 18240*x^2*y*z^4 + 69120*x^2*y*z^3 - 103272*x^2*y*z^2 + 71880*x^2*y*z - 19626*x^2*y - 7104*x^2*z^5 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"(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.020949236000888 ], "surface_features": [ 4.20840223711824, 5.140332026976173, 3.8515029527705447, 0.0, 4.334904999771651, 5102 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B025", "pair": 25, "family": "bijection", "mapping": "F(x,y,z)=(-9*x^7 - 114*x^6*y - 69*x^6*z - 123*x^6 - 616*x^5*y^2 - 752*x^5*y*z - 1328*x^5*y - 226*x^5*z^2 - 812*x^5*z - 719*x^5 - 1840*x^4*y^3 - 3400*x^4*y^2*z - 5944*x^4*y^2 - 2060*x^4*y*z^2 - 7336*x^4*y*z - 6426*x^4*y - 410*x^4*z^3 - 2226*x^4*z^2 - 3977*x^4*z - 2329*x^4 - 3280*x^3*y^4 - 8160*x^3*y^3*z - 14112*x^3*y^3 - 7480*x^3*y^2*z^2 - 26384*x^3*y^2*z - 22844*x^3*y^2 - 3000*x^3*y*z^3 - 16152*x^3*y*z^2 - 28564*x^3*y*z - 16516*x^3*y - 445*x^3*z^4 - 3244*x^3*z^3 - 8767*x^3*z^2 - 10374*x^3*z - 4511*x^3 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1140*x^5*y*z + 2292*x^5*y + 570*x^5*z^2 - 2292*x^5*z + 2334*x^5 - 900*x^4*y^3 + 2700*x^4*y^2*z - 5430*x^4*y^2 - 2700*x^4*y*z^2 + 10860*x^4*y*z - 11060*x^4*y + 900*x^4*z^3 - 5430*x^4*z^2 + 11060*x^4*z - 7632*x^4 + 850*x^3*y^4 - 3400*x^3*y^3*z + 6840*x^3*y^3 + 5100*x^3*y^2*z^2 - 20520*x^3*y^2*z + 20900*x^3*y^2 - 3400*x^3*y*z^3 + 20520*x^3*y*z^2 - 41800*x^3*y*z + 28838*x^3*y + 850*x^3*z^4 - 6840*x^3*z^3 + 20900*x^3*z^2 - 28838*x^3*z + 15198*x^3 - 480*x^2*y^5 + 2400*x^2*y^4*z - 4830*x^2*y^4 - 4800*x^2*y^3*z^2 + 19320*x^2*y^3*z - 19680*x^2*y^3 + 4800*x^2*y^2*z^3 - 28980*x^2*y^2*z^2 + 59040*x^2*y^2*z - 40722*x^2*y^2 - 2400*x^2*y*z^4 + 19320*x^2*y*z^3 - 59040*x^2*y*z^2 + 81444*x^2*y*z - 42894*x^2*y + 480*x^2*z^5 - 4830*x^2*z^4 + 19680*x^2*z^3 - 40722*x^2*z^2 + 42894*x^2*z - 18600*x^2 + 150*x*y^6 - 900*x*y^5*z + 1812*x*y^5 + 2250*x*y^4*z^2 - 9060*x*y^4*z + 9230*x*y^4 - 3000*x*y^3*z^3 + 18120*x*y^3*z^2 - 36920*x*y^3*z + 25458*x*y^3 + 2250*x*y^2*z^4 - 18120*x*y^2*z^3 + 55380*x*y^2*z^2 - 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39792*x^2*y*z^3 - 79242*x^2*y*z^2 + 60972*x^2*y*z - 15642*x^2*y + 1584*x^2*z^5 - 13476*x^2*z^4 + 36396*x^2*z^3 - 42726*x^2*z^2 + 22272*x^2*z - 4168*x^2 + 80*x*y^6 - 660*x*y^5*z + 1128*x*y^5 + 2244*x*y^4*z^2 - 7728*x*y^4*z + 5196*x*y^4 - 4025*x*y^3*z^3 + 21006*x*y^3*z^2 - 28677*x*y^3*z + 11321*x*y^3 + 4017*x*y^2*z^4 - 28308*x*y^2*z^3 + 58962*x*y^2*z^2 - 47379*x*y^2*z + 12675*x*y^2 - 2115*x*y*z^5 + 18906*x*y*z^4 - 53505*x*y*z^3 + 65745*x*y*z^2 - 35829*x*y*z + 6992*x*y + 459*x*z^6 - 5004*x*z^5 + 18072*x*z^4 - 30237*x*z^3 + 25218*x*z^2 - 10020*x*z + 1489*x - 8*y^7 + 76*y^6*z - 136*y^6 - 306*y^5*z^2 + 1104*y^5*z - 774*y^5 + 677*y^4*z^3 - 3702*y^4*z^2 + 5277*y^4*z - 2175*y^4 - 889*y^3*z^4 + 6562*y^3*z^3 - 14286*y^3*z^2 + 12007*y^3*z - 3363*y^3 + 693*y^2*z^5 - 6483*y^2*z^4 + 19188*y^2*z^3 - 24702*y^2*z^2 + 14133*y^2*z - 2887*y^2 - 297*y*z^6 + 3384*y*z^5 - 12780*y*z^4 + 22434*y*z^3 - 19701*y*z^2 + 8217*y*z - 1276*y + 54*z^7 - 729*z^6 + 3375*z^5 - 7584*z^4 + 9105*z^3 - 5826*z^2 + 1844*z - 1, 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4332*x^3*y^3*z + 6744*x^3*y^3 + 9072*x^3*y^2*z^2 - 28368*x^3*y^2*z + 17616*x^3*y^2 - 8360*x^3*y*z^3 + 39504*x^3*y*z^2 - 49596*x^3*y*z + 18064*x^3*y + 2860*x^3*z^4 - 18208*x^3*z^3 + 34728*x^3*z^2 - 25632*x^3*z + 6272*x^3 - 336*x^2*y^5 + 2340*x^2*y^4*z - 3816*x^2*y^4 - 6450*x^2*y^3*z^2 + 21168*x^2*y^3*z - 13662*x^2*y^3 + 8796*x^2*y^2*z^3 - 43704*x^2*y^2*z^2 + 57156*x^2*y^2*z - 21636*x^2*y^2 - 5934*x^2*y*z^4 + 39792*x^2*y*z^3 - 79242*x^2*y*z^2 + 60924*x^2*y*z - 15522*x^2*y + 1584*x^2*z^5 - 13476*x^2*z^4 + 36396*x^2*z^3 - 42696*x^2*z^2 + 22116*x^2*z - 4036*x^2 + 80*x*y^6 - 660*x*y^5*z + 1128*x*y^5 + 2244*x*y^4*z^2 - 7728*x*y^4*z + 5196*x*y^4 - 4025*x*y^3*z^3 + 21006*x*y^3*z^2 - 28677*x*y^3*z + 11314*x*y^3 + 4017*x*y^2*z^4 - 28308*x*y^2*z^3 + 58962*x*y^2*z^2 - 47352*x*y^2*z + 12603*x*y^2 - 2115*x*y*z^5 + 18906*x*y*z^4 - 53505*x*y*z^3 + 65712*x*y*z^2 - 35649*x*y*z + 6824*x*y + 459*x*z^6 - 5004*x*z^5 + 18072*x*z^4 - 30224*x*z^3 + 25110*x*z^2 - 9804*x*z + 1389*x - 8*y^7 + 76*y^6*z - 136*y^6 - 306*y^5*z^2 + 1104*y^5*z - 774*y^5 + 677*y^4*z^3 - 3702*y^4*z^2 + 5277*y^4*z - 2174*y^4 - 889*y^3*z^4 + 6562*y^3*z^3 - 14286*y^3*z^2 + 12002*y^3*z - 3349*y^3 + 693*y^2*z^5 - 6483*y^2*z^4 + 19188*y^2*z^3 - 24693*y^2*z^2 + 14082*y^2*z - 2836*y^2 - 297*y*z^6 + 3384*y*z^5 - 12780*y*z^4 + 22427*y*z^3 - 19641*y*z^2 + 8091*y*z - 1210*y + 54*z^7 - 729*z^6 + 3375*z^5 - 7582*z^4 + 9082*z^3 - 5751*z^2 + 1758*z - 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -2, 1, -1 ], [ -2, 2, -3 ], [ -1, 1, -2 ] ], "source_shift": [ 2, 1, 2 ], "target_matrix": [ [ -1, -1, -1 ], [ 1, 2, 1 ], [ -1, -1, -2 ] ], "target_shift": [ 228, -397, 204 ], "collision_preimages": [ [ "13/4", "8", "7/2" ], [ "-6", "-49/2", "-23/2" ], [ "-1", "-27/2", "-17/2" ] ], "collision_image": [ "913/4", "-1589/4", "817/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0146504119984456 ], "surface_features": [ 4.161359099938838, 4.9978667262391445, 3.8174992618677583, 0.0, 4.259062381438747, 5104 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B042", "pair": 42, "family": "bijection", "mapping": "F(x,y,z)=(256*x^7 + 896*x^6*y - 896*x^6*z - 1024*x^6 + 1344*x^5*y^2 - 2688*x^5*y*z - 3072*x^5*y + 1344*x^5*z^2 + 3072*x^5*z + 1888*x^5 + 1120*x^4*y^3 - 3360*x^4*y^2*z - 3840*x^4*y^2 + 3360*x^4*y*z^2 + 7680*x^4*y*z + 4736*x^4*y - 1120*x^4*z^3 - 3840*x^4*z^2 - 4736*x^4*z - 2096*x^4 + 560*x^3*y^4 - 2240*x^3*y^3*z - 2560*x^3*y^3 + 3360*x^3*y^2*z^2 + 7680*x^3*y^2*z + 4752*x^3*y^2 - 2240*x^3*y*z^3 - 7680*x^3*y*z^2 - 9504*x^3*y*z - 4232*x^3*y + 560*x^3*z^4 + 2560*x^3*z^3 + 4752*x^3*z^2 + 4248*x^3*z + 1552*x^3 + 168*x^2*y^5 - 840*x^2*y^4*z - 960*x^2*y^4 + 1680*x^2*y^3*z^2 + 3840*x^2*y^3*z + 2384*x^2*y^3 - 1680*x^2*y^2*z^3 - 5760*x^2*y^2*z^2 - 7152*x^2*y^2*z - 3204*x^2*y^2 + 840*x^2*y*z^4 + 3840*x^2*y*z^3 + 7152*x^2*y*z^2 + 6432*x^2*y*z + 2372*x^2*y - 168*x^2*z^5 - 960*x^2*z^4 - 2384*x^2*z^3 - 3228*x^2*z^2 - 2404*x^2*z - 790*x^2 + 28*x*y^6 - 168*x*y^5*z - 192*x*y^5 + 420*x*y^4*z^2 + 960*x*y^4*z + 598*x*y^4 - 560*x*y^3*z^3 - 1920*x*y^3*z^2 - 2392*x*y^3*z - 1078*x*y^3 + 420*x*y^2*z^4 + 1920*x*y^2*z^3 + 3588*x*y^2*z^2 + 3246*x*y^2*z + 1208*x*y^2 - 168*x*y*z^5 - 960*x*y*z^4 - 2392*x*y*z^3 - 3258*x*y*z^2 - 2448*x*y*z - 818*x*y + 28*x*z^6 + 192*x*z^5 + 598*x*z^4 + 1090*x*z^3 + 1240*x*z^2 + 844*x*z + 265*x + 2*y^7 - 14*y^6*z - 16*y^6 + 42*y^5*z^2 + 96*y^5*z + 60*y^5 - 70*y^4*z^3 - 240*y^4*z^2 - 300*y^4*z - 136*y^4 + 70*y^3*z^4 + 320*y^3*z^3 + 600*y^3*z^2 + 546*y^3*z + 205*y^3 - 42*y^2*z^5 - 240*y^2*z^4 - 600*y^2*z^3 - 822*y^2*z^2 - 623*y^2*z - 212*y^2 + 14*y*z^6 + 96*y*z^5 + 300*y*z^4 + 550*y*z^3 + 631*y*z^2 + 438*y*z + 142*y - 2*z^7 - 16*z^6 - 60*z^5 - 138*z^4 - 213*z^3 - 226*z^2 - 150*z - 1, -768*x^7 - 2688*x^6*y + 2688*x^6*z + 3072*x^6 - 4032*x^5*y^2 + 8064*x^5*y*z + 9216*x^5*y - 4032*x^5*z^2 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2880*x*y*z^4 + 7176*x*y*z^3 + 9822*x*y*z^2 + 7428*x*y*z + 2494*x*y - 84*x*z^6 - 576*x*z^5 - 1794*x*z^4 - 3286*x*z^3 - 3762*x*z^2 - 2572*x*z - 812*x - 6*y^7 + 42*y^6*z + 48*y^6 - 126*y^5*z^2 - 288*y^5*z - 180*y^5 + 210*y^4*z^3 + 720*y^4*z^2 + 900*y^4*z + 410*y^4 - 210*y^3*z^4 - 960*y^3*z^3 - 1800*y^3*z^2 - 1646*y^3*z - 622*y^3 + 126*y^2*z^5 + 720*y^2*z^4 + 1800*y^2*z^3 + 2478*y^2*z^2 + 1890*y^2*z + 646*y^2 - 42*y*z^6 - 288*y*z^5 - 900*y*z^4 - 1658*y*z^3 - 1914*y*z^2 - 1334*y*z - 435*y + 6*z^7 + 48*z^6 + 180*z^5 + 416*z^4 + 646*z^3 + 688*z^2 + 461*z + 1, 256*x^7 + 896*x^6*y - 896*x^6*z - 1024*x^6 + 1344*x^5*y^2 - 2688*x^5*y*z - 3072*x^5*y + 1344*x^5*z^2 + 3072*x^5*z + 1888*x^5 + 1120*x^4*y^3 - 3360*x^4*y^2*z - 3840*x^4*y^2 + 3360*x^4*y*z^2 + 7680*x^4*y*z + 4736*x^4*y - 1120*x^4*z^3 - 3840*x^4*z^2 - 4736*x^4*z - 2096*x^4 + 560*x^3*y^4 - 2240*x^3*y^3*z - 2560*x^3*y^3 + 3360*x^3*y^2*z^2 + 7680*x^3*y^2*z + 4752*x^3*y^2 - 2240*x^3*y*z^3 - 7680*x^3*y*z^2 - 9504*x^3*y*z - 4232*x^3*y + 560*x^3*z^4 + 2560*x^3*z^3 + 4752*x^3*z^2 + 4248*x^3*z + 1560*x^3 + 168*x^2*y^5 - 840*x^2*y^4*z - 960*x^2*y^4 + 1680*x^2*y^3*z^2 + 3840*x^2*y^3*z + 2384*x^2*y^3 - 1680*x^2*y^2*z^3 - 5760*x^2*y^2*z^2 - 7152*x^2*y^2*z - 3204*x^2*y^2 + 840*x^2*y*z^4 + 3840*x^2*y*z^3 + 7152*x^2*y*z^2 + 6432*x^2*y*z + 2384*x^2*y - 168*x^2*z^5 - 960*x^2*z^4 - 2384*x^2*z^3 - 3228*x^2*z^2 - 2416*x^2*z - 806*x^2 + 28*x*y^6 - 168*x*y^5*z - 192*x*y^5 + 420*x*y^4*z^2 + 960*x*y^4*z + 598*x*y^4 - 560*x*y^3*z^3 - 1920*x*y^3*z^2 - 2392*x*y^3*z - 1078*x*y^3 + 420*x*y^2*z^4 + 1920*x*y^2*z^3 + 3588*x*y^2*z^2 + 3246*x*y^2*z + 1214*x*y^2 - 168*x*y*z^5 - 960*x*y*z^4 - 2392*x*y*z^3 - 3258*x*y*z^2 - 2460*x*y*z - 834*x*y + 28*x*z^6 + 192*x*z^5 + 598*x*z^4 + 1090*x*z^3 + 1246*x*z^2 + 860*x*z + 278*x + 2*y^7 - 14*y^6*z - 16*y^6 + 42*y^5*z^2 + 96*y^5*z + 60*y^5 - 70*y^4*z^3 - 240*y^4*z^2 - 300*y^4*z - 136*y^4 + 70*y^3*z^4 + 320*y^3*z^3 + 600*y^3*z^2 + 546*y^3*z + 206*y^3 - 42*y^2*z^5 - 240*y^2*z^4 - 600*y^2*z^3 - 822*y^2*z^2 - 626*y^2*z - 216*y^2 + 14*y*z^6 + 96*y*z^5 + 300*y*z^4 + 550*y*z^3 + 634*y*z^2 + 446*y*z + 149*y - 2*z^7 - 16*z^6 - 60*z^5 - 138*z^4 - 214*z^3 - 230*z^2 - 157*z - 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -2, -1, 1 ], [ -1, -1, 1 ], [ -1, -1, 2 ] ], "source_shift": [ 1, 2, 1 ], "target_matrix": [ [ -1, -1, -1 ], [ 3, 2, 2 ], [ -1, -2, -1 ] ], "target_shift": [ 47, -146, 52 ], "inverse_recipe": { "p": 3, "q": 4, "f": "x**3 + x**2 + x", "g": "x**4 + x**2 + x", "h": "y*z + y", "polynomials": { "f": [ [ [ 3, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "g": [ [ [ 4, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 1, 0 ], 1 ] ] }, "h_multiplier": 2 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 0.9816017129996908 ], "surface_features": [ 3.466084624919151, 4.455027682356781, 3.09795107099415, 0.0, 3.624127004841935, 4851 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B212", "pair": 212, "family": "bijection", "mapping": "F(x,y,z)=(-3*x^7 + 21*x^6*y + 42*x^6*z - 27*x^6 - 63*x^5*y^2 - 252*x^5*y*z + 159*x^5*y - 252*x^5*z^2 + 315*x^5*z - 94*x^5 + 105*x^4*y^3 + 630*x^4*y^2*z - 390*x^4*y^2 + 1260*x^4*y*z^2 - 1545*x^4*y*z + 455*x^4*y + 840*x^4*z^3 - 1530*x^4*z^2 + 895*x^4*z - 170*x^4 - 105*x^3*y^4 - 840*x^3*y^3*z + 510*x^3*y^3 - 2520*x^3*y^2*z^2 + 3030*x^3*y^2*z - 880*x^3*y^2 - 3360*x^3*y*z^3 + 6000*x^3*y*z^2 - 3460*x^3*y*z + 650*x^3*y - 1680*x^3*z^4 + 3960*x^3*z^3 - 3400*x^3*z^2 + 1270*x^3*z - 181*x^3 + 63*x^2*y^5 + 630*x^2*y^4*z - 375*x^2*y^4 + 2520*x^2*y^3*z^2 - 2970*x^2*y^3*z + 850*x^2*y^3 + 5040*x^2*y^2*z^3 - 8820*x^2*y^2*z^2 + 5010*x^2*y^2*z - 930*x^2*y^2 + 5040*x^2*y*z^4 - 11640*x^2*y*z^3 + 9840*x^2*y*z^2 - 3630*x^2*y*z + 513*x^2*y + 2016*x^2*z^5 - 5760*x^2*z^4 + 6440*x^2*z^3 - 3540*x^2*z^2 + 993*x^2*z - 144*x^2 - 21*x*y^6 - 252*x*y^5*z + 147*x*y^5 - 1260*x*y^4*z^2 + 1455*x*y^4*z - 410*x*y^4 - 3360*x*y^3*z^3 + 5760*x*y^3*z^2 - 3220*x*y^3*z + 590*x*y^3 - 5040*x*y^2*z^4 + 11400*x*y^2*z^3 - 9480*x*y^2*z^2 + 3450*x*y^2*z - 483*x*y^2 - 4032*x*y*z^5 + 11280*x*y*z^4 - 12400*x*y*z^3 + 6720*x*y*z^2 - 1866*x*y*z + 267*x*y - 1344*x*z^6 + 4464*x*z^5 - 6080*x*z^4 + 4360*x*z^3 - 1800*x*z^2 + 501*x*z - 109*x + 3*y^7 + 42*y^6*z - 24*y^6 + 252*y^5*z^2 - 285*y^5*z + 79*y^5 + 840*y^4*z^3 - 1410*y^4*z^2 + 775*y^4*z - 140*y^4 + 1680*y^3*z^4 - 3720*y^3*z^3 + 3040*y^3*z^2 - 1090*y^3*z + 151*y^3 + 2016*y^2*z^5 - 5520*y^2*z^4 + 5960*y^2*z^3 - 3180*y^2*z^2 + 873*y^2*z - 123*y^2 + 1344*y*z^6 - 4368*y*z^5 + 5840*y*z^4 - 4120*y*z^3 + 1680*y*z^2 - 462*y*z + 89*y + 384*z^7 - 1440*z^6 + 2288*z^5 - 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3360*x^3*y*z^3 - 6000*x^3*y*z^2 + 3460*x^3*y*z - 650*x^3*y + 1680*x^3*z^4 - 3960*x^3*z^3 + 3400*x^3*z^2 - 1270*x^3*z + 181*x^3 - 63*x^2*y^5 - 630*x^2*y^4*z + 375*x^2*y^4 - 2520*x^2*y^3*z^2 + 2970*x^2*y^3*z - 850*x^2*y^3 - 5040*x^2*y^2*z^3 + 8820*x^2*y^2*z^2 - 5010*x^2*y^2*z + 930*x^2*y^2 - 5040*x^2*y*z^4 + 11640*x^2*y*z^3 - 9840*x^2*y*z^2 + 3630*x^2*y*z - 513*x^2*y - 2016*x^2*z^5 + 5760*x^2*z^4 - 6440*x^2*z^3 + 3540*x^2*z^2 - 993*x^2*z + 143*x^2 + 21*x*y^6 + 252*x*y^5*z - 147*x*y^5 + 1260*x*y^4*z^2 - 1455*x*y^4*z + 410*x*y^4 + 3360*x*y^3*z^3 - 5760*x*y^3*z^2 + 3220*x*y^3*z - 590*x*y^3 + 5040*x*y^2*z^4 - 11400*x*y^2*z^3 + 9480*x*y^2*z^2 - 3450*x*y^2*z + 483*x*y^2 + 4032*x*y*z^5 - 11280*x*y*z^4 + 12400*x*y*z^3 - 6720*x*y*z^2 + 1866*x*y*z - 265*x*y + 1344*x*z^6 - 4464*x*z^5 + 6080*x*z^4 - 4360*x*z^3 + 1800*x*z^2 - 497*x*z + 105*x - 3*y^7 - 42*y^6*z + 24*y^6 - 252*y^5*z^2 + 285*y^5*z - 79*y^5 - 840*y^4*z^3 + 1410*y^4*z^2 - 775*y^4*z + 140*y^4 - 1680*y^3*z^4 + 3720*y^3*z^3 - 3040*y^3*z^2 + 1090*y^3*z - 151*y^3 - 2016*y^2*z^5 + 5520*y^2*z^4 - 5960*y^2*z^3 + 3180*y^2*z^2 - 873*y^2*z + 122*y^2 - 1344*y*z^6 + 4368*y*z^5 - 5840*y*z^4 + 4120*y*z^3 - 1680*y*z^2 + 458*y*z - 86*y - 384*z^7 + 1440*z^6 - 2288*z^5 + 2000*z^4 - 1076*z^3 + 431*z^2 - 149*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ 1, -1, -2 ], [ 2, -1, -1 ], [ 1, -1, -1 ] ], "source_shift": [ 1, 1, 2 ], "target_matrix": [ [ -1, -2, -1 ], [ -1, -1, -2 ], [ 1, 1, 1 ] ], "target_shift": [ 32, 22, -30 ], "inverse_recipe": { "p": 2, "q": 5, "f": "x**2", "g": "x**5 + x", "h": "y*z + 2*y", "polynomials": { "f": [ [ [ 2, 0, 0 ], 1 ] ], "g": [ [ [ 5, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 1, 0 ], 2 ] ] }, "h_multiplier": 3 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0739513989974512 ], "surface_features": [ 3.358548711071293, 4.093421685162235, 3.0569048513364727, 0.0, 3.4441211655502846, 4833 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C151", "pair": 151, "family": "counterexample", "mapping": "F(x,y,z)=(-8*x^7 - 44*x^6*y + 52*x^6*z + 104*x^6 - 102*x^5*y^2 + 240*x^5*y*z + 480*x^5*y - 138*x^5*z^2 - 564*x^5*z - 462*x^5 - 129*x^4*y^3 + 453*x^4*y^2*z + 906*x^4*y^2 - 519*x^4*y*z^2 - 2112*x^4*y*z - 1761*x^4*y + 195*x^4*z^3 + 1206*x^4*z^2 + 2007*x^4*z + 1038*x^4 - 96*x^3*y^4 + 447*x^3*y^3*z + 894*x^3*y^3 - 765*x^3*y^2*z^2 - 3099*x^3*y^2*z - 2631*x^3*y^2 + 573*x^3*y*z^3 + 3516*x^3*y*z^2 + 5943*x^3*y*z + 3137*x^3*y - 159*x^3*z^4 - 1311*x^3*z^3 - 3312*x^3*z^2 - 3504*x^3*z - 1346*x^3 - 42*x^2*y^5 + 243*x^2*y^4*z + 486*x^2*y^4 - 552*x^2*y^3*z^2 - 2226*x^2*y^3*z - 1923*x^2*y^3 + 618*x^2*y^2*z^3 + 3762*x^2*y^2*z^2 + 6456*x^2*y^2*z + 3474*x^2*y^2 - 342*x^2*y*z^4 - 2790*x^2*y*z^3 - 7143*x^2*y*z^2 - 7692*x^2*y*z - 3024*x^2*y + 75*x^2*z^5 + 768*x^2*z^4 + 2610*x^2*z^3 + 4218*x^2*z^2 + 3336*x^2*z + 1045*x^2 - 10*x*y^6 + 69*x*y^5*z + 138*x*y^5 - 195*x*y^4*z^2 - 783*x*y^4*z - 687*x*y^4 + 290*x*y^3*z^3 + 1752*x*y^3*z^2 + 3048*x*y^3*z + 1668*x*y^3 - 240*x*y^2*z^4 - 1938*x*y^2*z^3 - 5022*x*y^2*z^2 - 5490*x*y^2*z - 2205*x*y^2 + 105*x*y*z^5 + 1062*x*y*z^4 + 3648*x*y*z^3 + 5976*x*y*z^2 + 4818*x*y*z + 1550*x*y - 19*x*z^6 - 231*x*z^5 - 987*x*z^4 - 2154*x*z^3 - 2613*x*z^2 - 1700*x*z - 461*x - y^7 + 8*y^6*z + 16*y^6 - 27*y^5*z^2 - 108*y^5*z - 96*y^5 + 50*y^4*z^3 + 300*y^4*z^2 + 528*y^4*z + 293*y^4 - 55*y^3*z^4 - 440*y^3*z^3 - 1152*y^3*z^2 - 1275*y^3*z - 521*y^3 + 36*y^2*z^5 + 360*y^2*z^4 + 1248*y^2*z^3 + 2067*y^2*z^2 + 1692*y^2*z + 556*y^2 - 13*y*z^6 - 156*y*z^5 - 672*y*z^4 - 1481*y*z^3 - 1821*y*z^2 - 1205*y*z - 335*y + 2*z^7 + 28*z^6 + 144*z^5 + 396*z^4 + 650*z^3 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2886*x^3*y^2 - 1146*x^3*y*z^3 - 4647*x^3*y*z^2 - 6450*x^3*y*z - 3027*x^3*y + 318*x^3*z^4 + 1713*x^3*z^3 + 3564*x^3*z^2 + 3355*x^3*z + 1194*x^3 + 84*x^2*y^5 - 486*x^2*y^4*z - 675*x^2*y^4 + 1104*x^2*y^3*z^2 + 3039*x^2*y^3*z + 2145*x^2*y^3 - 1236*x^2*y^2*z^3 - 5067*x^2*y^2*z^2 - 7134*x^2*y^2*z - 3411*x^2*y^2 + 684*x^2*y*z^4 + 3717*x^2*y*z^3 + 7833*x^2*y*z^2 + 7503*x^2*y*z + 2733*x^2*y - 150*x^2*z^5 - 1014*x^2*z^4 - 2844*x^2*z^3 - 4092*x^2*z^2 - 3000*x^2*z - 890*x^2 + 20*x*y^6 - 138*x*y^5*z - 195*x*y^5 + 390*x*y^4*z^2 + 1089*x*y^4*z + 780*x*y^4 - 580*x*y^3*z^3 - 2406*x*y^3*z^2 - 3432*x*y^3*z - 1669*x*y^3 + 480*x*y^2*z^4 + 2634*x*y^2*z^3 + 5616*x*y^2*z^2 + 5463*x*y^2*z + 2034*x*y^2 - 210*x*y*z^5 - 1431*x*y*z^4 - 4056*x*y*z^3 - 5919*x*y*z^2 - 4428*x*y*z - 1351*x*y + 38*x*z^6 + 309*x*z^5 + 1092*x*z^4 + 2125*x*z^3 + 2394*x*z^2 + 1477*x*z + 388*x + 2*y^7 - 16*y^6*z - 23*y^6 + 54*y^5*z^2 + 153*y^5*z + 111*y^5 - 100*y^4*z^3 - 420*y^4*z^2 - 606*y^4*z - 299*y^4 + 110*y^3*z^4 + 610*y^3*z^3 + 1314*y^3*z^2 + 1295*y^3*z + 491*y^3 - 72*y^2*z^5 - 495*y^2*z^4 - 1416*y^2*z^3 - 2091*y^2*z^2 - 1590*y^2*z - 497*y^2 + 26*y*z^6 + 213*y*z^5 + 759*y*z^4 + 1493*y*z^3 + 1707*y*z^2 + 1075*y*z + 291*y - 4*z^7 - 38*z^6 - 162*z^5 - 398*z^4 - 608*z^3 - 578*z^2 - 318*z - 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -2, -1, 1 ], [ -1, -1, 1 ], [ -1, -1, 2 ] ], "source_shift": [ 1, 1, 1 ], "target_matrix": [ [ 1, 2, 1 ], [ -3, -2, -2 ], [ -2, -1, -1 ] ], "target_shift": [ -87, 129, 75 ], "collision_preimages": [ [ "0", "3/4", "-1/4" ], [ "-5/2", "13", "8" ], [ "5/2", "2", "5" ] ], "collision_image": [ "-349/4", "519/4", "151/2" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0037483320011233 ], "surface_features": [ 3.2533231946838694, 4.108057373783854, 2.9923325590474645, 0.0, 3.346376250593239, 4809 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B107", "pair": 107, "family": "bijection", "mapping": "F(x,y,z)=(-2*x^7 + 14*x^6*y - 28*x^6*z + 32*x^6 - 42*x^5*y^2 + 168*x^5*y*z - 192*x^5*y - 168*x^5*z^2 + 382*x^5*z - 218*x^5 + 70*x^4*y^3 - 420*x^4*y^2*z + 480*x^4*y^2 + 840*x^4*y*z^2 - 1910*x^4*y*z + 1090*x^4*y - 560*x^4*z^3 + 1900*x^4*z^2 - 2158*x^4*z + 820*x^4 - 70*x^3*y^4 + 560*x^3*y^3*z - 640*x^3*y^3 - 1680*x^3*y^2*z^2 + 3820*x^3*y^2*z - 2180*x^3*y^2 + 2240*x^3*y*z^3 - 7600*x^3*y*z^2 + 8632*x^3*y*z - 3280*x^3*y - 1120*x^3*z^4 + 5040*x^3*z^3 - 8544*x^3*z^2 + 6462*x^3*z - 1842*x^3 + 42*x^2*y^5 - 420*x^2*y^4*z + 480*x^2*y^4 + 1680*x^2*y^3*z^2 - 3820*x^2*y^3*z + 2180*x^2*y^3 - 3360*x^2*y^2*z^3 + 11400*x^2*y^2*z^2 - 12948*x^2*y^2*z + 4920*x^2*y^2 + 3360*x^2*y*z^4 - 15120*x^2*y*z^3 + 25632*x^2*y*z^2 - 19386*x^2*y*z + 5528*x^2*y - 1344*x^2*z^5 + 7520*x^2*z^4 - 16912*x^2*z^3 + 19092*x^2*z^2 - 10830*x^2*z + 2481*x^2 - 14*x*y^6 + 168*x*y^5*z - 192*x*y^5 - 840*x*y^4*z^2 + 1910*x*y^4*z - 1090*x*y^4 + 2240*x*y^3*z^3 - 7600*x*y^3*z^2 + 8632*x*y^3*z - 3280*x*y^3 - 3360*x*y^2*z^4 + 15120*x*y^2*z^3 - 25632*x*y^2*z^2 + 19386*x*y^2*z - 5530*x*y^2 + 2688*x*y*z^5 - 15040*x*y*z^4 + 33824*x*y*z^3 - 38184*x*y*z^2 + 21668*x*y*z - 4972*x*y - 896*x*z^6 + 5984*x*z^5 - 16736*x*z^4 + 25064*x*z^3 - 21216*x*z^2 + 9662*x*z - 1867*x + 2*y^7 - 28*y^6*z + 32*y^6 + 168*y^5*z^2 - 382*y^5*z + 218*y^5 - 560*y^4*z^3 + 1900*y^4*z^2 - 2158*y^4*z + 820*y^4 + 1120*y^3*z^4 - 5040*y^3*z^3 + 8544*y^3*z^2 - 6462*y^3*z + 1844*y^3 - 1344*y^2*z^5 + 7520*y^2*z^4 - 16912*y^2*z^3 + 19092*y^2*z^2 - 10838*y^2*z + 2491*y^2 + 896*y*z^6 - 5984*y*z^5 + 16736*y*z^4 - 25064*y*z^3 + 21224*y*z^2 - 9680*y*z + 1877*y - 256*z^7 + 1984*z^6 - 6624*z^5 + 12336*z^4 - 13848*z^3 + 9402*z^2 - 3603*z + 1, -2*x^7 + 14*x^6*y - 28*x^6*z + 32*x^6 - 42*x^5*y^2 + 168*x^5*y*z - 192*x^5*y - 168*x^5*z^2 + 382*x^5*z - 214*x^5 + 70*x^4*y^3 - 420*x^4*y^2*z + 480*x^4*y^2 + 840*x^4*y*z^2 - 1910*x^4*y*z + 1070*x^4*y 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7906*x*z - 1368*x + 2*y^7 - 28*y^6*z + 32*y^6 + 168*y^5*z^2 - 382*y^5*z + 214*y^5 - 560*y^4*z^3 + 1900*y^4*z^2 - 2118*y^4*z + 776*y^4 + 1120*y^3*z^4 - 5040*y^3*z^3 + 8384*y^3*z^2 - 6110*y^3*z + 1648*y^3 - 1344*y^2*z^5 + 7520*y^2*z^4 - 16592*y^2*z^3 + 18036*y^2*z^2 - 9662*y^2*z + 2052*y^2 + 896*y*z^6 - 5984*y*z^5 + 16416*y*z^4 - 23656*y*z^3 + 18872*y*z^2 - 7924*y*z + 1374*y - 256*z^7 + 1984*z^6 - 6496*z^5 + 11632*z^4 - 12280*z^3 + 7646*z^2 - 2608*z + 1, -x^7 + 7*x^6*y - 14*x^6*z + 16*x^6 - 21*x^5*y^2 + 84*x^5*y*z - 96*x^5*y - 84*x^5*z^2 + 191*x^5*z - 106*x^5 + 35*x^4*y^3 - 210*x^4*y^2*z + 240*x^4*y^2 + 420*x^4*y*z^2 - 955*x^4*y*z + 530*x^4*y - 280*x^4*z^3 + 950*x^4*z^2 - 1049*x^4*z + 377*x^4 - 35*x^3*y^4 + 280*x^3*y^3*z - 320*x^3*y^3 - 840*x^3*y^2*z^2 + 1910*x^3*y^2*z - 1060*x^3*y^2 + 1120*x^3*y*z^3 - 3800*x^3*y*z^2 + 4196*x^3*y*z - 1508*x^3*y - 560*x^3*z^4 + 2520*x^3*z^3 - 4152*x^3*z^2 + 2967*x^3*z - 774*x^3 + 21*x^2*y^5 - 210*x^2*y^4*z + 240*x^2*y^4 + 840*x^2*y^3*z^2 - 1910*x^2*y^3*z 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- 3216*z^5 + 5640*z^4 - 5748*z^3 + 3383*z^2 - 1053*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -1, 1, -2 ], [ -1, 1, -1 ], [ -1, 2, -1 ] ], "source_shift": [ 2, 2, 2 ], "target_matrix": [ [ 2, 1, 1 ], [ 2, 2, 3 ], [ 1, 1, 2 ] ], "target_shift": [ -609, -375, -127 ], "inverse_recipe": { "p": 2, "q": 5, "f": "x**2", "g": "x**5 + x**4 + x**3 + x", "h": "y*z", "polynomials": { "f": [ [ [ 2, 0, 0 ], 1 ] ], "g": [ [ [ 5, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ], [ [ 3, 0, 0 ], 1 ], [ [ 4, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 0.9661601269981475 ], "surface_features": [ 3.69751582253322, 4.5818814213581875, 3.2810333672477277, 0.0, 3.832896792429814, 4912 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B020", "pair": 20, "family": "bijection", "mapping": "F(x,y,z)=(-2187*x^7 + 10206*x^6*y - 10206*x^6*z + 5103*x^6 - 20412*x^5*y^2 + 40824*x^5*y*z - 20412*x^5*y - 20412*x^5*z^2 + 20412*x^5*z - 5670*x^5 + 22680*x^4*y^3 - 68040*x^4*y^2*z + 34020*x^4*y^2 + 68040*x^4*y*z^2 - 68040*x^4*y*z + 18927*x^4*y - 22680*x^4*z^3 + 34020*x^4*z^2 - 18927*x^4*z + 4104*x^4 - 15120*x^3*y^4 + 60480*x^3*y^3*z - 30240*x^3*y^3 - 90720*x^3*y^2*z^2 + 90720*x^3*y^2*z - 25272*x^3*y^2 + 60480*x^3*y*z^3 - 90720*x^3*y*z^2 + 50544*x^3*y*z - 11016*x^3*y - 15120*x^3*z^4 + 30240*x^3*z^3 - 25272*x^3*z^2 + 10989*x^3*z - 1953*x^3 + 6048*x^2*y^5 - 30240*x^2*y^4*z + 15120*x^2*y^4 + 60480*x^2*y^3*z^2 - 60480*x^2*y^3*z + 16872*x^2*y^3 - 60480*x^2*y^2*z^3 + 90720*x^2*y^2*z^2 - 50616*x^2*y^2*z + 11088*x^2*y^2 + 30240*x^2*y*z^4 - 60480*x^2*y*z^3 + 50616*x^2*y*z^2 - 22122*x^2*y*z + 3963*x^2*y - 6048*x^2*z^5 + 15120*x^2*z^4 - 16872*x^2*z^3 + 11034*x^2*z^2 - 3936*x^2*z + 598*x^2 - 1344*x*y^6 + 8064*x*y^5*z - 4032*x*y^5 - 20160*x*y^4*z^2 + 20160*x*y^4*z - 5632*x*y^4 + 26880*x*y^3*z^3 - 40320*x*y^3*z^2 + 22528*x*y^3*z - 4960*x*y^3 - 20160*x*y^2*z^4 + 40320*x*y^2*z^3 - 33792*x*y^2*z^2 + 14844*x*y^2*z - 2680*x*y^2 + 8064*x*y*z^5 - 20160*x*y*z^4 + 22528*x*y*z^3 - 14808*x*y*z^2 + 5324*x*y*z - 821*x*y - 1344*x*z^6 + 4032*x*z^5 - 5632*x*z^4 + 4924*x*z^3 - 2644*x*z^2 + 808*x*z - 104*x + 128*y^7 - 896*y^6*z + 448*y^6 + 2688*y^5*z^2 - 2688*y^5*z + 752*y^5 - 4480*y^4*z^3 + 6720*y^4*z^2 - 3760*y^4*z + 832*y^4 + 4480*y^3*z^4 - 8960*y^3*z^3 + 7520*y^3*z^2 - 3320*y^3*z + 604*y^3 - 2688*y^2*z^5 + 6720*y^2*z^4 - 7520*y^2*z^3 + 4968*y^2*z^2 - 1800*y^2*z + 282*y^2 + 896*y*z^6 - 2688*y*z^5 + 3760*y*z^4 - 3304*y*z^3 + 1788*y*z^2 - 555*y*z + 77*y - 128*z^7 + 448*z^6 - 752*z^5 + 824*z^4 - 592*z^3 + 273*z^2 - 71*z + 1, -2187*x^7 + 10206*x^6*y - 10206*x^6*z + 5103*x^6 - 20412*x^5*y^2 + 40824*x^5*y*z - 20412*x^5*y - 20412*x^5*z^2 + 20412*x^5*z - 5670*x^5 + 22680*x^4*y^3 - 68040*x^4*y^2*z + 34020*x^4*y^2 + 68040*x^4*y*z^2 - 68040*x^4*y*z + 18927*x^4*y - 22680*x^4*z^3 + 34020*x^4*z^2 - 18927*x^4*z + 4104*x^4 - 15120*x^3*y^4 + 60480*x^3*y^3*z - 30240*x^3*y^3 - 90720*x^3*y^2*z^2 + 90720*x^3*y^2*z - 25272*x^3*y^2 + 60480*x^3*y*z^3 - 90720*x^3*y*z^2 + 50544*x^3*y*z - 11016*x^3*y - 15120*x^3*z^4 + 30240*x^3*z^3 - 25272*x^3*z^2 + 10989*x^3*z - 2007*x^3 + 6048*x^2*y^5 - 30240*x^2*y^4*z + 15120*x^2*y^4 + 60480*x^2*y^3*z^2 - 60480*x^2*y^3*z + 16872*x^2*y^3 - 60480*x^2*y^2*z^3 + 90720*x^2*y^2*z^2 - 50616*x^2*y^2*z + 11088*x^2*y^2 + 30240*x^2*y*z^4 - 60480*x^2*y*z^3 + 50616*x^2*y*z^2 - 22122*x^2*y*z + 4071*x^2*y - 6048*x^2*z^5 + 15120*x^2*z^4 - 16872*x^2*z^3 + 11034*x^2*z^2 - 4044*x^2*z + 652*x^2 - 1344*x*y^6 + 8064*x*y^5*z - 4032*x*y^5 - 20160*x*y^4*z^2 + 20160*x*y^4*z - 5632*x*y^4 + 26880*x*y^3*z^3 - 40320*x*y^3*z^2 + 22528*x*y^3*z - 4960*x*y^3 - 20160*x*y^2*z^4 + 40320*x*y^2*z^3 - 33792*x*y^2*z^2 + 14844*x*y^2*z - 2752*x*y^2 + 8064*x*y*z^5 - 20160*x*y*z^4 + 22528*x*y*z^3 - 14808*x*y*z^2 + 5468*x*y*z - 893*x*y - 1344*x*z^6 + 4032*x*z^5 - 5632*x*z^4 + 4924*x*z^3 - 2716*x*z^2 + 880*x*z - 130*x + 128*y^7 - 896*y^6*z + 448*y^6 + 2688*y^5*z^2 - 2688*y^5*z + 752*y^5 - 4480*y^4*z^3 + 6720*y^4*z^2 - 3760*y^4*z + 832*y^4 + 4480*y^3*z^4 - 8960*y^3*z^3 + 7520*y^3*z^2 - 3320*y^3*z + 620*y^3 - 2688*y^2*z^5 + 6720*y^2*z^4 - 7520*y^2*z^3 + 4968*y^2*z^2 - 1848*y^2*z + 306*y^2 + 896*y*z^6 - 2688*y*z^5 + 3760*y*z^4 - 3304*y*z^3 + 1836*y*z^2 - 603*y*z + 95*y - 128*z^7 + 448*z^6 - 752*z^5 + 824*z^4 - 608*z^3 + 297*z^2 - 89*z + 1, -2187*x^7 + 10206*x^6*y - 10206*x^6*z + 5103*x^6 - 20412*x^5*y^2 + 40824*x^5*y*z - 20412*x^5*y - 20412*x^5*z^2 + 20412*x^5*z - 5670*x^5 + 22680*x^4*y^3 - 68040*x^4*y^2*z + 34020*x^4*y^2 + 68040*x^4*y*z^2 - 68040*x^4*y*z + 18927*x^4*y - 22680*x^4*z^3 + 34020*x^4*z^2 - 18927*x^4*z + 4185*x^4 - 15120*x^3*y^4 + 60480*x^3*y^3*z - 30240*x^3*y^3 - 90720*x^3*y^2*z^2 + 90720*x^3*y^2*z - 25272*x^3*y^2 + 60480*x^3*y*z^3 - 90720*x^3*y*z^2 + 50544*x^3*y*z - 11232*x^3*y - 15120*x^3*z^4 + 30240*x^3*z^3 - 25272*x^3*z^2 + 11205*x^3*z - 2115*x^3 + 6048*x^2*y^5 - 30240*x^2*y^4*z + 15120*x^2*y^4 + 60480*x^2*y^3*z^2 - 60480*x^2*y^3*z + 16872*x^2*y^3 - 60480*x^2*y^2*z^3 + 90720*x^2*y^2*z^2 - 50616*x^2*y^2*z + 11304*x^2*y^2 + 30240*x^2*y*z^4 - 60480*x^2*y*z^3 + 50616*x^2*y*z^2 - 22554*x^2*y*z + 4287*x^2*y - 6048*x^2*z^5 + 15120*x^2*z^4 - 16872*x^2*z^3 + 11250*x^2*z^2 - 4260*x^2*z + 715*x^2 - 1344*x*y^6 + 8064*x*y^5*z - 4032*x*y^5 - 20160*x*y^4*z^2 + 20160*x*y^4*z - 5632*x*y^4 + 26880*x*y^3*z^3 - 40320*x*y^3*z^2 + 22528*x*y^3*z - 5056*x*y^3 - 20160*x*y^2*z^4 + 40320*x*y^2*z^3 - 33792*x*y^2*z^2 + 15132*x*y^2*z - 2896*x*y^2 + 8064*x*y*z^5 - 20160*x*y*z^4 + 22528*x*y*z^3 - 15096*x*y*z^2 + 5756*x*y*z - 977*x*y - 1344*x*z^6 + 4032*x*z^5 - 5632*x*z^4 + 5020*x*z^3 - 2860*x*z^2 + 964*x*z - 149*x + 128*y^7 - 896*y^6*z + 448*y^6 + 2688*y^5*z^2 - 2688*y^5*z + 752*y^5 - 4480*y^4*z^3 + 6720*y^4*z^2 - 3760*y^4*z + 848*y^4 + 4480*y^3*z^4 - 8960*y^3*z^3 + 7520*y^3*z^2 - 3384*y^3*z + 652*y^3 - 2688*y^2*z^5 + 6720*y^2*z^4 - 7520*y^2*z^3 + 5064*y^2*z^2 - 1944*y^2*z + 334*y^2 + 896*y*z^6 - 2688*y*z^5 + 3760*y*z^4 - 3368*y*z^3 + 1932*y*z^2 - 659*y*z + 109*y - 128*z^7 + 448*z^6 - 752*z^5 + 840*z^4 - 640*z^3 + 325*z^2 - 102*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -3, 2, -2 ], [ -1, 2, -1 ], [ -1, 1, -1 ] ], "source_shift": [ 1, 1, 1 ], "target_matrix": [ [ 1, 1, 2 ], [ 1, 1, 1 ], [ 1, 2, 1 ] ], "target_shift": [ -6, -12, -15 ], "inverse_recipe": { "p": 4, "q": 3, "f": "x**4 + x**2", "g": "x**3 + x", "h": "y*z + 2*y", "polynomials": { "f": [ [ [ 4, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "g": [ [ [ 3, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 1, 0 ], 2 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 0.9677560819982318 ], "surface_features": [ 4.26224200057749, 4.9577030415488315, 3.90655051910145, 0.0, 4.351150074434779, 5147 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C100", "pair": 100, "family": "counterexample", "mapping": "F(x,y,z)=(-2*x^7 + 19*x^6*y + 13*x^6*z - 29*x^6 - 75*x^5*y^2 - 105*x^5*y*z + 234*x^5*y - 36*x^5*z^2 + 162*x^5*z - 177*x^5 + 159*x^4*y^3 + 342*x^4*y^2*z - 762*x^4*y^2 + 240*x^4*y*z^2 - 1080*x^4*y*z + 1173*x^4*y + 55*x^4*z^3 - 375*x^4*z^2 + 825*x^4*z - 590*x^4 - 195*x^3*y^4 - 573*x^3*y^3*z + 1278*x^3*y^3 - 618*x^3*y^2*z^2 + 2784*x^3*y^2*z - 3006*x^3*y^2 - 290*x^3*y*z^3 + 1980*x^3*y*z^2 - 4335*x^3*y*z + 3064*x^3*y - 50*x^3*z^4 + 460*x^3*z^3 - 1530*x^3*z^2 + 2199*x^3*z - 1161*x^3 + 138*x^2*y^5 + 519*x^2*y^4*z - 1161*x^2*y^4 + 765*x^2*y^3*z^2 - 3456*x^2*y^3*z + 3714*x^2*y^3 + 552*x^2*y^2*z^3 - 3780*x^2*y^2*z^2 + 8241*x^2*y^2*z - 5754*x^2*y^2 + 195*x^2*y*z^4 - 1800*x^2*y*z^3 + 5967*x^2*y*z^2 - 8484*x^2*y*z + 4398*x^2*y + 27*x^2*z^5 - 315*x^2*z^4 + 1410*x^2*z^3 - 3057*x^2*z^2 + 3237*x^2*z - 1348*x^2 - 52*x*y^6 - 240*x*y^5*z + 540*x*y^5 - 453*x*y^4*z^2 + 2058*x*y^4*z - 2208*x*y^4 - 447*x*y^3*z^3 + 3078*x*y^3*z^2 - 6696*x*y^3*z + 4625*x*y^3 - 243*x*y^2*z^4 + 2256*x*y^2*z^3 - 7464*x*y^2*z^2 + 10500*x*y^2*z - 5340*x*y^2 - 69*x*y*z^5 + 810*x*y*z^4 - 3621*x*y*z^3 + 7776*x*y*z^2 - 8091*x*y*z + 3281*x*y - 8*x*z^6 + 114*x*z^5 - 645*x*z^4 + 1877*x*z^3 - 2991*x*z^2 + 2492*x*z - 854*x + 8*y^7 + 44*y^6*z - 100*y^6 + 102*y^5*z^2 - 468*y^5*z + 504*y^5 + 129*y^4*z^3 - 897*y^4*z^2 + 1956*y^4*z - 1342*y^4 + 96*y^3*z^4 - 900*y^3*z^3 + 2982*y^3*z^2 - 4161*y^3*z + 2082*y^3 + 42*y^2*z^5 - 498*y^2*z^4 + 2229*y^2*z^3 - 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240*x*y^5*z + 672*x*y^5 - 453*x*y^4*z^2 + 2538*x*y^4*z - 3096*x*y^4 - 447*x*y^3*z^3 + 3759*x*y^3*z^2 - 9294*x*y^3*z + 6953*x*y^3 - 243*x*y^2*z^4 + 2727*x*y^2*z^3 - 10245*x*y^2*z^2 + 15618*x*y^2*z - 8424*x*y^2 - 69*x*y*z^5 + 969*x*y*z^4 - 4911*x*y*z^3 + 11433*x*y*z^2 - 12609*x*y*z + 5372*x*y - 8*x*z^6 + 135*x*z^5 - 864*x*z^4 + 2726*x*z^3 - 4602*x*z^2 + 4019*x*z - 1437*x + 8*y^7 + 44*y^6*z - 124*y^6 + 102*y^5*z^2 - 576*y^5*z + 708*y^5 + 129*y^4*z^3 - 1095*y^4*z^2 + 2724*y^4*z - 2026*y^4 + 96*y^3*z^4 - 1089*y^3*z^3 + 4113*y^3*z^2 - 6225*y^3*z + 3306*y^3 + 42*y^2*z^5 - 597*y^2*z^4 + 3042*y^2*z^3 - 7029*y^2*z^2 + 7626*y^2*z - 3172*y^2 + 10*y*z^6 - 171*y*z^5 + 1101*y*z^4 - 3451*y*z^3 + 5736*y*z^2 - 4892*y*z + 1697*y + z^7 - 20*z^6 + 156*z^5 - 621*z^4 + 1404*z^3 - 1840*z^2 + 1310*z - 1, 2*x^7 - 19*x^6*y - 13*x^6*z + 29*x^6 + 75*x^5*y^2 + 105*x^5*y*z - 234*x^5*y + 36*x^5*z^2 - 162*x^5*z + 177*x^5 - 159*x^4*y^3 - 342*x^4*y^2*z + 762*x^4*y^2 - 240*x^4*y*z^2 + 1080*x^4*y*z - 1173*x^4*y - 55*x^4*z^3 + 375*x^4*z^2 - 825*x^4*z + 588*x^4 + 195*x^3*y^4 + 573*x^3*y^3*z - 1278*x^3*y^3 + 618*x^3*y^2*z^2 - 2784*x^3*y^2*z + 3006*x^3*y^2 + 290*x^3*y*z^3 - 1980*x^3*y*z^2 + 4335*x^3*y*z - 3051*x^3*y + 50*x^3*z^4 - 460*x^3*z^3 + 1530*x^3*z^2 - 2192*x^3*z + 1144*x^3 - 138*x^2*y^5 - 519*x^2*y^4*z + 1161*x^2*y^4 - 765*x^2*y^3*z^2 + 3456*x^2*y^3*z - 3714*x^2*y^3 - 552*x^2*y^2*z^3 + 3780*x^2*y^2*z^2 - 8241*x^2*y^2*z + 5724*x^2*y^2 - 195*x^2*y*z^4 + 1800*x^2*y*z^3 - 5967*x^2*y*z^2 + 8451*x^2*y*z - 4317*x^2*y - 27*x^2*z^5 + 315*x^2*z^4 - 1410*x^2*z^3 + 3048*x^2*z^2 - 3192*x^2*z + 1297*x^2 + 52*x*y^6 + 240*x*y^5*z - 540*x*y^5 + 453*x*y^4*z^2 - 2058*x*y^4*z + 2208*x*y^4 + 447*x*y^3*z^3 - 3078*x*y^3*z^2 + 6696*x*y^3*z - 4597*x*y^3 + 243*x*y^2*z^4 - 2256*x*y^2*z^3 + 7464*x*y^2*z^2 - 10452*x*y^2*z + 5220*x*y^2 + 69*x*y*z^5 - 810*x*y*z^4 + 3621*x*y*z^3 - 7749*x*y*z^2 + 7953*x*y*z - 3125*x*y + 8*x*z^6 - 114*x*z^5 + 645*x*z^4 - 1872*x*z^3 + 2952*x*z^2 - 2402*x*z + 792*x - 8*y^7 - 44*y^6*z + 100*y^6 - 102*y^5*z^2 + 468*y^5*z - 504*y^5 - 129*y^4*z^3 + 897*y^4*z^2 - 1956*y^4*z + 1334*y^4 - 96*y^3*z^4 + 900*y^3*z^3 - 2982*y^3*z^2 + 4141*y^3*z - 2030*y^3 - 42*y^2*z^5 + 498*y^2*z^4 - 2229*y^2*z^3 + 4728*y^2*z^2 - 4758*y^2*z + 1810*y^2 - 10*y*z^6 + 144*y*z^5 - 816*y*z^4 + 2349*y*z^3 - 3636*y*z^2 + 2867*y*z - 906*y - z^7 + 17*z^6 - 117*z^5 + 428*z^4 - 904*z^3 + 1105*z^2 - 727*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ 1, -2, -1 ], [ 1, -1, -1 ], [ 2, -1, -1 ] ], "source_shift": [ 2, 1, 2 ], "target_matrix": [ [ -1, -1, -1 ], [ -1, -2, -1 ], [ 1, 1, 2 ] ], "target_shift": [ 228, 397, -204 ], "collision_preimages": [ [ "-5/4", "1", "-5/4" ], [ "7", "-3/2", "11" ], [ "4", "7/2", "0" ] ], "collision_image": [ "913/4", "1589/4", "-817/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0375032409974665 ], "surface_features": [ 3.3024301454846094, 4.193625418492879, 2.944729360303296, 0.0, 3.4137895128703493, 4785 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C117", "pair": 117, "family": "counterexample", "mapping": "F(x,y,z)=(-x^7 - 10*x^6*y - 10*x^6*z - 16*x^6 - 42*x^5*y^2 - 87*x^5*y*z - 138*x^5*y - 42*x^5*z^2 - 138*x^5*z - 108*x^5 - 96*x^4*y^3 - 309*x^4*y^2*z - 486*x^4*y^2 - 309*x^4*y*z^2 - 1005*x^4*y*z - 780*x^4*y - 96*x^4*z^3 - 489*x^4*z^2 - 783*x^4*z - 401*x^4 - 129*x^3*y^4 - 573*x^3*y^3*z - 894*x^3*y^3 - 891*x^3*y^2*z^2 - 2868*x^3*y^2*z - 2205*x^3*y^2 - 573*x^3*y*z^3 - 2886*x^3*y*z^2 - 4587*x^3*y*z - 2331*x^3*y - 129*x^3*z^4 - 912*x^3*z^3 - 2244*x^3*z^2 - 2339*x^3*z - 887*x^3 - 102*x^2*y^5 - 585*x^2*y^4*z - 906*x^2*y^4 - 1257*x^2*y^3*z^2 - 4005*x^2*y^3*z - 3048*x^2*y^3 - 1257*x^2*y^2*z^3 - 6255*x^2*y^2*z^2 - 9852*x^2*y^2*z - 4959*x^2*y^2 - 585*x^2*y*z^4 - 4089*x^2*y*z^3 - 10005*x^2*y*z^2 - 10368*x^2*y*z - 3897*x^2*y - 102*x^2*z^5 - 945*x^2*z^4 - 3177*x^2*z^3 - 5049*x^2*z^2 - 3885*x^2*z - 1177*x^2 - 44*x*y^6 - 312*x*y^5*z - 480*x*y^5 - 867*x*y^4*z^2 - 2736*x*y^4*z - 2061*x*y^4 - 1199*x*y^3*z^3 - 5892*x*y^3*z^2 - 9183*x*y^3*z - 4571*x*y^3 - 867*x*y^2*z^4 - 5982*x*y^2*z^3 - 14526*x*y^2*z^2 - 14937*x*y^2*z - 5553*x*y^2 - 312*x*y*z^5 - 2862*x*y*z^4 - 9594*x*y*z^3 - 15192*x*y*z^2 - 11607*x*y*z - 3476*x*y - 44*x*z^6 - 516*x*z^5 - 2220*x*z^4 - 4774*x*z^3 - 5583*x*z^2 - 3430*x*z - 879*x - 8*y^7 - 68*y^6*z - 104*y^6 - 234*y^5*z^2 - 732*y^5*z - 546*y^5 - 419*y^4*z^3 - 2034*y^4*z^2 - 3135*y^4*z - 1542*y^4 - 419*y^3*z^4 - 2849*y^3*z^3 - 6849*y^3*z^2 - 6974*y^3*z - 2561*y^3 - 234*y^2*z^5 - 2121*y^2*z^4 - 7077*y^2*z^3 - 11145*y^2*z^2 - 8430*y^2*z - 2485*y^2 - 68*y*z^6 - 792*y*z^5 - 3408*y*z^4 - 7320*y*z^3 - 8523*y*z^2 - 5191*y*z - 1314*y - 8*z^7 - 116*z^6 - 612*z^5 - 1666*z^4 - 2626*z^3 - 2449*z^2 - 1271*z - 1, x^7 + 10*x^6*y + 10*x^6*z + 16*x^6 + 42*x^5*y^2 + 87*x^5*y*z + 138*x^5*y + 42*x^5*z^2 + 138*x^5*z + 108*x^5 + 96*x^4*y^3 + 309*x^4*y^2*z + 486*x^4*y^2 + 309*x^4*y*z^2 + 1005*x^4*y*z + 780*x^4*y + 96*x^4*z^3 + 489*x^4*z^2 + 783*x^4*z + 402*x^4 + 129*x^3*y^4 + 573*x^3*y^3*z + 894*x^3*y^3 + 891*x^3*y^2*z^2 + 2868*x^3*y^2*z + 2205*x^3*y^2 + 573*x^3*y*z^3 + 2886*x^3*y*z^2 + 4587*x^3*y*z + 2338*x^3*y + 129*x^3*z^4 + 912*x^3*z^3 + 2244*x^3*z^2 + 2343*x^3*z + 894*x^3 + 102*x^2*y^5 + 585*x^2*y^4*z + 906*x^2*y^4 + 1257*x^2*y^3*z^2 + 4005*x^2*y^3*z + 3048*x^2*y^3 + 1257*x^2*y^2*z^3 + 6255*x^2*y^2*z^2 + 9852*x^2*y^2*z + 4977*x^2*y^2 + 585*x^2*y*z^4 + 4089*x^2*y*z^3 + 10005*x^2*y*z^2 + 10389*x^2*y*z + 3933*x^2*y + 102*x^2*z^5 + 945*x^2*z^4 + 3177*x^2*z^3 + 5055*x^2*z^2 + 3909*x^2*z + 1195*x^2 + 44*x*y^6 + 312*x*y^5*z + 480*x*y^5 + 867*x*y^4*z^2 + 2736*x*y^4*z + 2061*x*y^4 + 1199*x*y^3*z^3 + 5892*x*y^3*z^2 + 9183*x*y^3*z + 4591*x*y^3 + 867*x*y^2*z^4 + 5982*x*y^2*z^3 + 14526*x*y^2*z^2 + 14973*x*y^2*z + 5613*x*y^2 + 312*x*y*z^5 + 2862*x*y*z^4 + 9594*x*y*z^3 + 15213*x*y*z^2 + 11691*x*y*z + 3539*x*y + 44*x*z^6 + 516*x*z^5 + 2220*x*z^4 + 4778*x*z^3 + 5610*x*z^2 + 3472*x*z + 896*x + 8*y^7 + 68*y^6*z + 104*y^6 + 234*y^5*z^2 + 732*y^5*z + 546*y^5 + 419*y^4*z^3 + 2034*y^4*z^2 + 3135*y^4*z + 1550*y^4 + 419*y^3*z^4 + 2849*y^3*z^3 + 6849*y^3*z^2 + 6994*y^3*z + 2593*y^3 + 234*y^2*z^5 + 2121*y^2*z^4 + 7077*y^2*z^3 + 11163*y^2*z^2 + 8502*y^2*z + 2539*y^2 + 68*y*z^6 + 792*y*z^5 + 3408*y*z^4 + 7327*y*z^3 + 8571*y*z^2 + 5266*y*z + 1344*y + 8*z^7 + 116*z^6 + 612*z^5 + 1667*z^4 + 2636*z^3 + 2473*z^2 + 1291*z + 1, -x^7 - 10*x^6*y - 10*x^6*z - 19*x^6 - 42*x^5*y^2 - 87*x^5*y*z - 165*x^5*y - 42*x^5*z^2 - 162*x^5*z - 141*x^5 - 96*x^4*y^3 - 309*x^4*y^2*z - 585*x^4*y^2 - 309*x^4*y*z^2 - 1188*x^4*y*z - 1026*x^4*y - 96*x^4*z^3 - 567*x^4*z^2 - 1011*x^4*z - 558*x^4 - 129*x^3*y^4 - 573*x^3*y^3*z - 1083*x^3*y^3 - 891*x^3*y^2*z^2 - 3414*x^3*y^2*z - 2922*x^3*y^2 - 573*x^3*y*z^3 - 3369*x^3*y*z^2 - 5967*x^3*y*z - 3271*x^3*y - 129*x^3*z^4 - 1044*x^3*z^3 - 2865*x^3*z^2 - 3219*x^3*z - 1293*x^3 - 102*x^2*y^5 - 585*x^2*y^4*z - 1104*x^2*y^4 - 1257*x^2*y^3*z^2 - 4800*x^2*y^3*z - 4068*x^2*y^3 - 1257*x^2*y^2*z^3 - 7353*x^2*y^2*z^2 - 12912*x^2*y^2*z - 7017*x^2*y^2 - 585*x^2*y*z^4 - 4710*x^2*y*z^3 - 12867*x^2*y*z^2 - 14388*x^2*y*z - 5739*x^2*y - 102*x^2*z^5 - 1068*x^2*z^4 - 4011*x^2*z^3 - 6873*x^2*z^2 - 5595*x^2*z - 1777*x^2 - 44*x*y^6 - 312*x*y^5*z - 588*x*y^5 - 867*x*y^4*z^2 - 3300*x*y^4*z - 2769*x*y^4 - 1199*x*y^3*z^3 - 6975*x*y^3*z^2 - 12123*x*y^3*z - 6520*x*y^3 - 867*x*y^2*z^4 - 6936*x*y^2*z^3 - 18819*x*y^2*z^2 - 20901*x*y^2*z - 8262*x*y^2 - 312*x*y*z^5 - 3252*x*y*z^4 - 12198*x*y*z^3 - 20853*x*y*z^2 - 16887*x*y*z - 5318*x*y - 44*x*z^6 - 576*x*z^5 - 2772*x*z^4 - 6428*x*z^3 - 7944*x*z^2 - 5104*x*z - 1353*x - 8*y^7 - 68*y^6*z - 128*y^6 - 234*y^5*z^2 - 888*y^5*z - 738*y^5 - 419*y^4*z^3 - 2424*y^4*z^2 - 4167*y^4*z - 2216*y^4 - 419*y^3*z^4 - 3326*y^3*z^3 - 8937*y^3*z^2 - 9835*y^3*z - 3847*y^3 - 234*y^2*z^5 - 2424*y^2*z^4 - 9063*y^2*z^3 - 15429*y^2*z^2 - 12396*y^2*z - 3856*y^2 - 68*y*z^6 - 888*y*z^5 - 4284*y*z^4 - 9937*y*z^3 - 12249*y*z^2 - 7825*y*z - 2056*y - 8*z^7 - 128*z^6 - 756*z^5 - 2219*z^4 - 3692*z^3 - 3592*z^2 - 1923*z - 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ 1, 2, 1 ], [ 1, 1, 2 ], [ 1, 1, 1 ] ], "source_shift": [ 1, 2, 1 ], "target_matrix": [ [ -1, -1, -2 ], [ 1, 1, 1 ], [ -1, -2, -1 ] ], "target_shift": [ 285, -290, 439 ], "collision_preimages": [ [ "-3/4", "1/4", "-3/4" ], [ "20", "-11/2", "-9" ], [ "19", "-15/2", "-6" ] ], "collision_image": [ "1141/4", "-1161/4", "1757/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0866266829980304 ], "surface_features": [ 3.4855376719006035, 4.320167065254232, 3.0993352776859577, 0.0, 3.594037692001176, 4845 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C203", "pair": 203, "family": "counterexample", "mapping": "F(x,y,z)=(2*x^7 - 19*x^6*y - 13*x^6*z + 28*x^6 + 75*x^5*y^2 + 105*x^5*y*z - 231*x^5*y + 36*x^5*z^2 - 156*x^5*z + 144*x^5 - 159*x^4*y^3 - 342*x^4*y^2*z + 768*x^4*y^2 - 240*x^4*y*z^2 + 1062*x^4*y*z - 987*x^4*y - 55*x^4*z^3 + 360*x^4*z^2 - 672*x^4*z + 396*x^4 + 195*x^3*y^4 + 573*x^3*y^3*z - 1311*x^3*y^3 + 618*x^3*y^2*z^2 - 2790*x^3*y^2*z + 2610*x^3*y^2 + 290*x^3*y*z^3 - 1938*x^3*y*z^2 + 3648*x^3*y*z - 2154*x^3*y + 50*x^3*z^4 - 440*x^3*z^3 + 1248*x^3*z^2 - 1481*x^3*z + 650*x^3 - 138*x^2*y^5 - 519*x^2*y^4*z + 1206*x^2*y^4 - 765*x^2*y^3*z^2 + 3516*x^2*y^3*z - 3312*x^2*y^3 - 552*x^2*y^2*z^3 + 3762*x^2*y^2*z^2 - 7143*x^2*y^2*z + 4218*x^2*y^2 - 195*x^2*y*z^4 + 1752*x^2*y*z^3 - 5022*x^2*y*z^2 + 5976*x^2*y*z - 2613*x^2*y - 27*x^2*z^5 + 300*x^2*z^4 - 1152*x^2*z^3 + 2067*x^2*z^2 - 1821*x^2*z + 649*x^2 + 52*x*y^6 + 240*x*y^5*z - 564*x*y^5 + 453*x*y^4*z^2 - 2112*x*y^4*z + 2007*x*y^4 + 447*x*y^3*z^3 - 3099*x*y^3*z^2 + 5943*x*y^3*z - 3504*x*y^3 + 243*x*y^2*z^4 - 2226*x*y^2*z^3 + 6456*x*y^2*z^2 - 7692*x*y^2*z + 3336*x*y^2 + 69*x*y*z^5 - 783*x*y*z^4 + 3048*x*y*z^3 - 5490*x*y*z^2 + 4818*x*y*z - 1700*x*y + 8*x*z^6 - 108*x*z^5 + 528*x*z^4 - 1275*x*z^3 + 1692*x*z^2 - 1205*x*z + 366*x - 8*y^7 - 44*y^6*z + 104*y^6 - 102*y^5*z^2 + 480*y^5*z - 462*y^5 - 129*y^4*z^3 + 906*y^4*z^2 - 1761*y^4*z + 1038*y^4 - 96*y^3*z^4 + 894*y^3*z^3 - 2631*y^3*z^2 + 3137*y^3*z - 1346*y^3 - 42*y^2*z^5 + 486*y^2*z^4 - 1923*y^2*z^3 + 3474*y^2*z^2 - 3024*y^2*z + 1045*y^2 - 10*y*z^6 + 138*y*z^5 - 687*y*z^4 + 1668*y*z^3 - 2205*y*z^2 + 1550*y*z - 461*y - z^7 + 16*z^6 - 96*z^5 + 293*z^4 - 521*z^3 + 556*z^2 - 335*z + 1, 2*x^7 - 19*x^6*y - 13*x^6*z + 22*x^6 + 75*x^5*y^2 + 105*x^5*y*z - 180*x^5*y + 36*x^5*z^2 - 123*x^5*z + 102*x^5 - 159*x^4*y^3 - 342*x^4*y^2*z + 594*x^4*y^2 - 240*x^4*y*z^2 + 831*x^4*y*z - 693*x^4*y - 55*x^4*z^3 + 285*x^4*z^2 - 477*x^4*z + 264*x^4 + 195*x^3*y^4 + 573*x^3*y^3*z - 1008*x^3*y^3 + 618*x^3*y^2*z^2 - 2169*x^3*y^2*z + 1818*x^3*y^2 + 290*x^3*y*z^3 - 1524*x^3*y*z^2 + 2568*x^3*y*z - 1422*x^3*y + 50*x^3*z^4 - 350*x^3*z^3 + 888*x^3*z^2 - 989*x^3*z + 416*x^3 - 138*x^2*y^5 - 519*x^2*y^4*z + 924*x^2*y^4 - 765*x^2*y^3*z^2 + 2721*x^2*y^3*z - 2292*x^2*y^3 - 552*x^2*y^2*z^3 + 2943*x^2*y^2*z^2 - 4992*x^2*y^2*z + 2760*x^2*y^2 - 195*x^2*y*z^4 + 1386*x^2*y*z^3 - 3546*x^2*y*z^2 + 3954*x^2*y*z - 1653*x^2*y - 27*x^2*z^5 + 240*x^2*z^4 - 822*x^2*z^3 + 1383*x^2*z^2 - 1167*x^2*z + 403*x^2 + 52*x*y^6 + 240*x*y^5*z - 432*x*y^5 + 453*x*y^4*z^2 - 1632*x*y^4*z + 1383*x*y^4 + 447*x*y^3*z^3 - 2418*x*y^3*z^2 + 4131*x*y^3*z - 2277*x*y^3 + 243*x*y^2*z^4 - 1755*x*y^2*z^3 + 4530*x*y^2*z^2 - 5049*x*y^2*z + 2088*x*y^2 + 69*x*y*z^5 - 624*x*y*z^4 + 2160*x*y*z^3 - 3642*x*y*z^2 + 3054*x*y*z - 1040*x*y + 8*x*z^6 - 87*x*z^5 + 378*x*z^4 - 855*x*z^3 + 1086*x*z^2 - 749*x*z + 224*x - 8*y^7 - 44*y^6*z + 80*y^6 - 102*y^5*z^2 + 372*y^5*z - 318*y^5 - 129*y^4*z^3 + 708*y^4*z^2 - 1221*y^4*z + 672*y^4 - 96*y^3*z^4 + 705*y^3*z^3 - 1839*y^3*z^2 + 2048*y^3*z - 836*y^3 - 42*y^2*z^5 + 387*y^2*z^4 - 1356*y^2*z^3 + 2289*y^2*z^2 - 1899*y^2*z + 631*y^2 - 10*y*z^6 + 111*y*z^5 - 489*y*z^4 + 1110*y*z^3 - 1401*y*z^2 + 950*y*z - 277*y - z^7 + 13*z^6 - 69*z^5 + 197*z^4 - 335*z^3 + 346*z^2 - 205*z + 1, 4*x^7 - 38*x^6*y - 26*x^6*z + 38*x^6 + 150*x^5*y^2 + 210*x^5*y*z - 309*x^5*y + 72*x^5*z^2 - 213*x^5*z + 162*x^5 - 318*x^4*y^3 - 684*x^4*y^2*z + 1014*x^4*y^2 - 480*x^4*y*z^2 + 1431*x^4*y*z - 1092*x^4*y - 110*x^4*z^3 + 495*x^4*z^2 - 759*x^4*z + 398*x^4 + 390*x^3*y^4 + 1146*x^3*y^3*z - 1713*x^3*y^3 + 1236*x^3*y^2*z^2 - 3717*x^3*y^2*z + 2844*x^3*y^2 + 580*x^3*y*z^3 - 2634*x^3*y*z^2 + 4056*x^3*y*z - 2125*x^3*y + 100*x^3*z^4 - 610*x^3*z^3 + 1416*x^3*z^2 - 1493*x^3*z + 608*x^3 - 276*x^2*y^5 - 1038*x^2*y^4*z + 1566*x^2*y^4 - 1530*x^2*y^3*z^2 + 4647*x^2*y^3*z - 3564*x^2*y^3 - 1104*x^2*y^2*z^3 + 5067*x^2*y^2*z^2 - 7833*x^2*y^2*z + 4092*x^2*y^2 - 390*x^2*y*z^4 + 2406*x^2*y*z^3 - 5616*x^2*y*z^2 + 5919*x^2*y*z - 2394*x^2*y - 54*x^2*z^5 + 420*x^2*z^4 - 1314*x^2*z^3 + 2091*x^2*z^2 - 1707*x^2*z + 578*x^2 + 104*x*y^6 + 480*x*y^5*z - 732*x*y^5 + 906*x*y^4*z^2 - 2784*x*y^4*z + 2142*x*y^4 + 894*x*y^3*z^3 - 4155*x*y^3*z^2 + 6450*x*y^3*z - 3355*x*y^3 + 486*x*y^2*z^4 - 3039*x*y^2*z^3 + 7134*x*y^2*z^2 - 7503*x*y^2*z + 3000*x*y^2 + 138*x*y*z^5 - 1089*x*y*z^4 + 3432*x*y*z^3 - 5463*x*y*z^2 + 4428*x*y*z - 1477*x*y + 16*x*z^6 - 153*x*z^5 + 606*x*z^4 - 1295*x*z^3 + 1590*x*z^2 - 1075*x*z + 318*x - 16*y^7 - 88*y^6*z + 136*y^6 - 204*y^5*z^2 + 636*y^5*z - 492*y^5 - 258*y^4*z^3 + 1218*y^4*z^2 - 1902*y^4*z + 986*y^4 - 192*y^3*z^4 + 1221*y^3*z^3 - 2886*y^3*z^2 + 3027*y^3*z - 1194*y^3 - 84*y^2*z^5 + 675*y^2*z^4 - 2145*y^2*z^3 + 3411*y^2*z^2 - 2733*y^2*z + 890*y^2 - 20*y*z^6 + 195*y*z^5 - 780*y*z^4 + 1669*y*z^3 - 2034*y*z^2 + 1351*y*z - 388*y - 2*z^7 + 23*z^6 - 111*z^5 + 299*z^4 - 491*z^3 + 497*z^2 - 291*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ 1, -2, -1 ], [ 1, -1, -1 ], [ 2, -1, -1 ] ], "source_shift": [ 1, 1, 1 ], "target_matrix": [ [ 1, 2, 1 ], [ 1, 1, 1 ], [ 2, 1, 1 ] ], "target_shift": [ -87, -53, -75 ], "collision_preimages": [ [ "-1/4", "0", "3/4" ], [ "8", "-5/2", "13" ], [ "5", "5/2", "2" ] ], "collision_image": [ "-349/4", "-213/4", "-151/2" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0395617780013708 ], "surface_features": [ 3.1083695547501957, 3.893928126542607, 2.807873132003332, 0.0, 3.191738207620116, 4758 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B051", "pair": 51, "family": "bijection", "mapping": "F(x,y,z)=(-3*x^7 - 21*x^6*y - 42*x^6*z - 30*x^6 - 63*x^5*y^2 - 252*x^5*y*z - 180*x^5*y - 252*x^5*z^2 - 354*x^5*z - 138*x^5 - 105*x^4*y^3 - 630*x^4*y^2*z - 450*x^4*y^2 - 1260*x^4*y*z^2 - 1770*x^4*y*z - 693*x^4*y - 840*x^4*z^3 - 1740*x^4*z^2 - 1329*x^4*z - 372*x^4 - 105*x^3*y^4 - 840*x^3*y^3*z - 600*x^3*y^3 - 2520*x^3*y^2*z^2 - 3540*x^3*y^2*z - 1392*x^3*y^2 - 3360*x^3*y*z^3 - 6960*x^3*y*z^2 - 5340*x^3*y*z - 1506*x^3*y - 1680*x^3*z^4 - 4560*x^3*z^3 - 5115*x^3*z^2 - 2784*x^3*z - 614*x^3 - 63*x^2*y^5 - 630*x^2*y^4*z - 450*x^2*y^4 - 2520*x^2*y^3*z^2 - 3540*x^2*y^3*z - 1398*x^2*y^3 - 5040*x^2*y^2*z^3 - 10440*x^2*y^2*z^2 - 8046*x^2*y^2*z - 2286*x^2*y^2 - 5040*x^2*y*z^4 - 13680*x^2*y*z^3 - 15417*x^2*y*z^2 - 8454*x^2*y*z - 1881*x^2*y - 2016*x^2*z^5 - 6720*x^2*z^4 - 9834*x^2*z^3 - 7806*x^2*z^2 - 3321*x^2*z - 605*x^2 - 21*x*y^6 - 252*x*y^5*z - 180*x*y^5 - 1260*x*y^4*z^2 - 1770*x*y^4*z - 702*x*y^4 - 3360*x*y^3*z^3 - 6960*x*y^3*z^2 - 5388*x*y^3*z - 1542*x*y^3 - 5040*x*y^2*z^4 - 13680*x*y^2*z^3 - 15489*x*y^2*z^2 - 8556*x*y^2*z - 1920*x*y^2 - 4032*x*y*z^5 - 13440*x*y*z^4 - 19764*x*y*z^3 - 15804*x*y*z^2 - 6780*x*y*z - 1246*x*y - 1344*x*z^6 - 5280*x*z^5 - 9444*x*z^4 - 9720*x*z^3 - 5997*x*z^2 - 2096*x*z - 327*x - 3*y^7 - 42*y^6*z - 30*y^6 - 252*y^5*z^2 - 354*y^5*z - 141*y^5 - 840*y^4*z^3 - 1740*y^4*z^2 - 1353*y^4*z - 390*y^4 - 1680*y^3*z^4 - 4560*y^3*z^3 - 5187*y^3*z^2 - 2886*y^3*z - 653*y^3 - 2016*y^2*z^5 - 6720*y^2*z^4 - 9930*y^2*z^3 - 7998*y^2*z^2 - 3459*y^2*z - 641*y^2 - 1344*y*z^6 - 5280*y*z^5 - 9492*y*z^4 - 9840*y*z^3 - 6120*y*z^2 - 2156*y*z - 338*y - 384*z^7 - 1728*z^6 - 3624*z^5 - 4536*z^4 - 3613*z^3 - 1826*z^2 - 546*z - 1, -x^7 - 7*x^6*y - 14*x^6*z - 10*x^6 - 21*x^5*y^2 - 84*x^5*y*z - 60*x^5*y - 84*x^5*z^2 - 118*x^5*z - 46*x^5 - 35*x^4*y^3 - 210*x^4*y^2*z - 150*x^4*y^2 - 420*x^4*y*z^2 - 590*x^4*y*z - 231*x^4*y - 280*x^4*z^3 - 580*x^4*z^2 - 443*x^4*z - 124*x^4 - 35*x^3*y^4 - 280*x^3*y^3*z - 200*x^3*y^3 - 840*x^3*y^2*z^2 - 1180*x^3*y^2*z - 464*x^3*y^2 - 1120*x^3*y*z^3 - 2320*x^3*y*z^2 - 1780*x^3*y*z - 502*x^3*y - 560*x^3*z^4 - 1520*x^3*z^3 - 1705*x^3*z^2 - 928*x^3*z - 204*x^3 - 21*x^2*y^5 - 210*x^2*y^4*z - 150*x^2*y^4 - 840*x^2*y^3*z^2 - 1180*x^2*y^3*z - 466*x^2*y^3 - 1680*x^2*y^2*z^3 - 3480*x^2*y^2*z^2 - 2682*x^2*y^2*z - 762*x^2*y^2 - 1680*x^2*y*z^4 - 4560*x^2*y*z^3 - 5139*x^2*y*z^2 - 2818*x^2*y*z - 625*x^2*y - 672*x^2*z^5 - 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1967*x*z^2 + 668*x*z + 94*x + y^7 + 14*y^6*z + 10*y^6 + 84*y^5*z^2 + 118*y^5*z + 47*y^5 + 280*y^4*z^3 + 580*y^4*z^2 + 451*y^4*z + 130*y^4 + 560*y^3*z^4 + 1520*y^3*z^3 + 1729*y^3*z^2 + 962*y^3*z + 215*y^3 + 672*y^2*z^5 + 2240*y^2*z^4 + 3310*y^2*z^3 + 2666*y^2*z^2 + 1137*y^2*z + 206*y^2 + 448*y*z^6 + 1760*y*z^5 + 3164*y*z^4 + 3280*y*z^3 + 2008*y*z^2 + 688*y*z + 95*y + 128*z^7 + 576*z^6 + 1208*z^5 + 1512*z^4 + 1183*z^3 + 578*z^2 + 157*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ 1, 1, 2 ], [ 1, 1, 1 ], [ 2, 3, 2 ] ], "source_shift": [ 1, 1, 1 ], "target_matrix": [ [ -3, -2, -2 ], [ -1, -1, -1 ], [ 1, 1, 2 ] ], "target_shift": [ 75, 23, -17 ], "inverse_recipe": { "p": 2, "q": 3, "f": "x**2", "g": "x**3 + x", "h": "y**3 + y**2*z + y**2 + z", "polynomials": { "f": [ [ [ 2, 0, 0 ], 1 ] ], "g": [ [ [ 3, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "h": [ [ [ 0, 2, 1 ], 1 ], [ [ 0, 2, 0 ], 1 ], [ [ 0, 0, 1 ], 1 ], [ [ 0, 3, 0 ], 1 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0926015250006458 ], "surface_features": [ 3.2923643852702895, 4.295874845217379, 2.9242792860618816, 0.0, 3.4542933533080937, 4791 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B234", "pair": 234, "family": "bijection", "mapping": "F(x,y,z)=(-256*x^7 + 2688*x^6*y - 1792*x^6*z + 1920*x^6 - 12096*x^5*y^2 + 16128*x^5*y*z - 17280*x^5*y - 5376*x^5*z^2 + 11520*x^5*z - 6304*x^5 + 30240*x^4*y^3 - 60480*x^4*y^2*z + 64800*x^4*y^2 + 40320*x^4*y*z^2 - 86400*x^4*y*z + 47264*x^4*y - 8960*x^4*z^3 + 28800*x^4*z^2 - 31552*x^4*z + 11648*x^4 - 45360*x^3*y^4 + 120960*x^3*y^3*z - 129600*x^3*y^3 - 120960*x^3*y^2*z^2 + 259200*x^3*y^2*z - 141744*x^3*y^2 + 53760*x^3*y*z^3 - 172800*x^3*y*z^2 + 189248*x^3*y*z - 69808*x^3*y - 8960*x^3*z^4 + 38400*x^3*z^3 - 63168*x^3*z^2 + 46736*x^3*z - 13000*x^3 + 40824*x^2*y^5 - 136080*x^2*y^4*z + 145800*x^2*y^4 + 181440*x^2*y^3*z^2 - 388800*x^2*y^3*z + 212544*x^2*y^3 - 120960*x^2*y^2*z^3 + 388800*x^2*y^2*z^2 - 425664*x^2*y^2*z + 156888*x^2*y^2 + 40320*x^2*y*z^4 - 172800*x^2*y*z^3 + 284160*x^2*y*z^2 - 210072*x^2*y*z + 58352*x^2*y - 5376*x^2*z^5 + 28800*x^2*z^4 - 63232*x^2*z^3 + 70320*x^2*z^2 - 39248*x^2*z + 8710*x^2 - 20412*x*y^6 + 81648*x*y^5*z - 87480*x*y^5 - 136080*x*y^4*z^2 + 291600*x*y^4*z - 159354*x*y^4 + 120960*x*y^3*z^3 - 388800*x*y^3*z^2 + 425520*x*y^3*z - 156708*x*y^3 - 60480*x*y^2*z^4 + 259200*x*y^2*z^3 - 426096*x*y^2*z^2 + 314748*x*y^2*z - 87306*x*y^2 + 16128*x*y*z^5 - 86400*x*y*z^4 + 189632*x*y*z^3 - 210720*x*y*z^2 + 117448*x*y*z - 26006*x*y - 1792*x*z^6 + 11520*x*z^5 - 31648*x*z^4 + 47024*x*z^3 - 39496*x*z^2 + 17614*x*z - 3230*x + 4374*y^7 - 20412*y^6*z + 21870*y^6 + 40824*y^5*z^2 - 87480*y^5*z + 47790*y^5 - 45360*y^4*z^3 + 145800*y^4*z^2 - 159516*y^4*z + 58698*y^4 + 30240*y^3*z^4 - 129600*y^3*z^3 + 212976*y^3*z^2 - 157194*y^3*z + 43542*y^3 - 12096*y^2*z^5 + 64800*y^2*z^4 - 142176*y^2*z^3 + 157860*y^2*z^2 - 87864*y^2*z + 19412*y^2 + 2688*y*z^6 - 17280*y*z^5 + 47456*y*z^4 - 70456*y*z^3 + 59096*y*z^2 - 26296*y*z + 4806*y - 256*z^7 + 1920*z^6 - 6336*z^5 + 11792*z^4 - 13248*z^3 + 8904*z^2 - 3292*z + 1, 128*x^7 - 1344*x^6*y + 896*x^6*z - 960*x^6 + 6048*x^5*y^2 - 8064*x^5*y*z + 8640*x^5*y + 2688*x^5*z^2 - 5760*x^5*z + 3152*x^5 - 15120*x^4*y^3 + 30240*x^4*y^2*z - 32400*x^4*y^2 - 20160*x^4*y*z^2 + 43200*x^4*y*z - 23632*x^4*y + 4480*x^4*z^3 - 14400*x^4*z^2 + 15776*x^4*z - 5840*x^4 + 22680*x^3*y^4 - 60480*x^3*y^3*z + 64800*x^3*y^3 + 60480*x^3*y^2*z^2 - 129600*x^3*y^2*z + 70872*x^3*y^2 - 26880*x^3*y*z^3 + 86400*x^3*y*z^2 - 94624*x^3*y*z + 35000*x^3*y + 4480*x^3*z^4 - 19200*x^3*z^3 + 31584*x^3*z^2 - 23432*x^3*z + 6572*x^3 - 20412*x^2*y^5 + 68040*x^2*y^4*z - 72900*x^2*y^4 - 90720*x^2*y^3*z^2 + 194400*x^2*y^3*z - 106272*x^2*y^3 + 60480*x^2*y^2*z^3 - 194400*x^2*y^2*z^2 + 212832*x^2*y^2*z - 78660*x^2*y^2 - 20160*x^2*y*z^4 + 86400*x^2*y*z^3 - 142080*x^2*y*z^2 + 105324*x^2*y*z - 29500*x^2*y + 2688*x^2*z^5 - 14400*x^2*z^4 + 31616*x^2*z^3 - 35256*x^2*z^2 + 19840*x^2*z - 4479*x^2 + 10206*x*y^6 - 40824*x*y^5*z + 43740*x*y^5 + 68040*x*y^4*z^2 - 145800*x*y^4*z + 79677*x*y^4 - 60480*x*y^3*z^3 + 194400*x*y^3*z^2 - 212760*x*y^3*z + 78570*x*y^3 + 30240*x*y^2*z^4 - 129600*x*y^2*z^3 + 213048*x*y^2*z^2 - 157806*x*y^2*z + 44139*x*y^2 - 8064*x*y*z^5 + 43200*x*y*z^4 - 94816*x*y*z^3 + 105648*x*y*z^2 - 59372*x*y*z + 13375*x*y + 896*x*z^6 - 5760*x*z^5 + 15824*x*z^4 - 23576*x*z^3 + 19964*x*z^2 - 9055*x*z + 1712*x - 2187*y^7 + 10206*y^6*z - 10935*y^6 - 20412*y^5*z^2 + 43740*y^5*z - 23895*y^5 + 22680*y^4*z^3 - 72900*y^4*z^2 + 79758*y^4*z - 29430*y^4 - 15120*y^3*z^4 + 64800*y^3*z^3 - 106488*y^3*z^2 + 78813*y^3*z - 22014*y^3 + 6048*y^2*z^5 - 32400*y^2*z^4 + 71088*y^2*z^3 - 79146*y^2*z^2 + 44418*y^2*z - 9985*y^2 - 1344*y*z^6 + 8640*y*z^5 - 23728*y*z^4 + 35324*y*z^3 - 29872*y*z^2 + 13520*y*z - 2548*y + 128*z^7 - 960*z^6 + 3168*z^5 - 5912*z^4 + 6696*z^3 - 4576*z^2 + 1743*z - 1, 128*x^7 - 1344*x^6*y + 896*x^6*z - 960*x^6 + 6048*x^5*y^2 - 8064*x^5*y*z + 8640*x^5*y + 2688*x^5*z^2 - 5760*x^5*z + 3152*x^5 - 15120*x^4*y^3 + 30240*x^4*y^2*z - 32400*x^4*y^2 - 20160*x^4*y*z^2 + 43200*x^4*y*z - 23632*x^4*y + 4480*x^4*z^3 - 14400*x^4*z^2 + 15776*x^4*z - 5840*x^4 + 22680*x^3*y^4 - 60480*x^3*y^3*z + 64800*x^3*y^3 + 60480*x^3*y^2*z^2 - 129600*x^3*y^2*z + 70872*x^3*y^2 - 26880*x^3*y*z^3 + 86400*x^3*y*z^2 - 94624*x^3*y*z + 35000*x^3*y + 4480*x^3*z^4 - 19200*x^3*z^3 + 31584*x^3*z^2 - 23432*x^3*z + 6580*x^3 - 20412*x^2*y^5 + 68040*x^2*y^4*z - 72900*x^2*y^4 - 90720*x^2*y^3*z^2 + 194400*x^2*y^3*z - 106272*x^2*y^3 + 60480*x^2*y^2*z^3 - 194400*x^2*y^2*z^2 + 212832*x^2*y^2*z - 78660*x^2*y^2 - 20160*x^2*y*z^4 + 86400*x^2*y*z^3 - 142080*x^2*y*z^2 + 105324*x^2*y*z - 29536*x^2*y + 2688*x^2*z^5 - 14400*x^2*z^4 + 31616*x^2*z^3 - 35256*x^2*z^2 + 19864*x^2*z - 4503*x^2 + 10206*x*y^6 - 40824*x*y^5*z + 43740*x*y^5 + 68040*x*y^4*z^2 - 145800*x*y^4*z + 79677*x*y^4 - 60480*x*y^3*z^3 + 194400*x*y^3*z^2 - 212760*x*y^3*z + 78570*x*y^3 + 30240*x*y^2*z^4 - 129600*x*y^2*z^3 + 213048*x*y^2*z^2 - 157806*x*y^2*z + 44193*x*y^2 - 8064*x*y*z^5 + 43200*x*y*z^4 - 94816*x*y*z^3 + 105648*x*y*z^2 - 59444*x*y*z + 13447*x*y + 896*x*z^6 - 5760*x*z^5 + 15824*x*z^4 - 23576*x*z^3 + 19988*x*z^2 - 9103*x*z + 1739*x - 2187*y^7 + 10206*y^6*z - 10935*y^6 - 20412*y^5*z^2 + 43740*y^5*z - 23895*y^5 + 22680*y^4*z^3 - 72900*y^4*z^2 + 79758*y^4*z - 29430*y^4 - 15120*y^3*z^4 + 64800*y^3*z^3 - 106488*y^3*z^2 + 78813*y^3*z - 22041*y^3 + 6048*y^2*z^5 - 32400*y^2*z^4 + 71088*y^2*z^3 - 79146*y^2*z^2 + 44472*y^2*z - 10039*y^2 - 1344*y*z^6 + 8640*y*z^5 - 23728*y*z^4 + 35324*y*z^3 - 29908*y*z^2 + 13592*y*z - 2588*y + 128*z^7 - 960*z^6 + 3168*z^5 - 5912*z^4 + 6704*z^3 - 4600*z^2 + 1771*z - 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -2, 3, -2 ], [ -1, 1, -2 ], [ -1, 1, -1 ] ], "source_shift": [ 2, 1, 2 ], "target_matrix": [ [ 2, 2, 3 ], [ -1, -1, -1 ], [ -1, -2, -1 ] ], "target_shift": [ -505, 282, 293 ], "inverse_recipe": { "p": 3, "q": 4, "f": "x**3 + x", "g": "x**4 + x**3 + x**2", "h": "y*z", "polynomials": { "f": [ [ [ 3, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "g": [ [ [ 4, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ], [ [ 3, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 0.9990523929991468 ], "surface_features": [ 4.78315655304848, 5.629507457255882, 4.499467130989543, 0.0, 4.8792401514437564, 5363 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B082", "pair": 82, "family": "bijection", "mapping": "F(x,y,z)=(4*x^7 + 28*x^6*y - 27*x^6*z - 56*x^6 + 84*x^5*y^2 - 162*x^5*y*z - 336*x^5*y + 78*x^5*z^2 + 323*x^5*z + 332*x^5 + 140*x^4*y^3 - 405*x^4*y^2*z - 840*x^4*y^2 + 390*x^4*y*z^2 + 1615*x^4*y*z + 1660*x^4*y - 125*x^4*z^3 - 775*x^4*z^2 - 1590*x^4*z - 1084*x^4 + 140*x^3*y^4 - 540*x^3*y^3*z - 1120*x^3*y^3 + 780*x^3*y^2*z^2 + 3230*x^3*y^2*z + 3320*x^3*y^2 - 500*x^3*y*z^3 - 3100*x^3*y*z^2 - 6360*x^3*y*z - 4336*x^3*y + 120*x^3*z^4 + 990*x^3*z^3 + 3040*x^3*z^2 + 4135*x^3*z + 2172*x^3 + 84*x^2*y^5 - 405*x^2*y^4*z - 840*x^2*y^4 + 780*x^2*y^3*z^2 + 3230*x^2*y^3*z + 3320*x^2*y^3 - 750*x^2*y^2*z^3 - 4650*x^2*y^2*z^2 - 9540*x^2*y^2*z - 6504*x^2*y^2 + 360*x^2*y*z^4 + 2970*x^2*y*z^3 + 9120*x^2*y*z^2 + 12405*x^2*y*z + 6516*x^2*y - 69*x^2*z^5 - 710*x^2*z^4 - 2900*x^2*z^3 - 5901*x^2*z^2 - 6142*x^2*z - 2628*x^2 + 28*x*y^6 - 162*x*y^5*z - 336*x*y^5 + 390*x*y^4*z^2 + 1615*x*y^4*z + 1660*x*y^4 - 500*x*y^3*z^3 - 3100*x*y^3*z^2 - 6360*x*y^3*z - 4336*x*y^3 + 360*x*y^2*z^4 + 2970*x*y^2*z^3 + 9120*x*y^2*z^2 + 12405*x*y^2*z + 6516*x*y^2 - 138*x*y*z^5 - 1420*x*y*z^4 - 5800*x*y*z^3 - 11802*x*y*z^2 - 12284*x*y*z - 5252*x*y + 22*x*z^6 + 271*x*z^5 + 1380*x*z^4 + 3733*x*z^3 + 5780*x*z^2 + 4877*x*z + 1725*x + 4*y^7 - 27*y^6*z - 56*y^6 + 78*y^5*z^2 + 323*y^5*z + 332*y^5 - 125*y^4*z^3 - 775*y^4*z^2 - 1590*y^4*z - 1084*y^4 + 120*y^3*z^4 + 990*y^3*z^3 + 3040*y^3*z^2 + 4135*y^3*z + 2172*y^3 - 69*y^2*z^5 - 710*y^2*z^4 - 2900*y^2*z^3 - 5901*y^2*z^2 - 6142*y^2*z - 2624*y^2 + 22*y*z^6 + 271*y*z^5 + 1380*y*z^4 + 3733*y*z^3 + 5780*y*z^2 + 4874*y*z + 1721*y - 3*z^7 - 43*z^6 - 262*z^5 - 883*z^4 - 1809*z^3 - 2261*z^2 - 1577*z - 1, 4*x^7 + 28*x^6*y - 27*x^6*z - 56*x^6 + 84*x^5*y^2 - 162*x^5*y*z - 336*x^5*y + 78*x^5*z^2 + 323*x^5*z + 332*x^5 + 140*x^4*y^3 - 405*x^4*y^2*z - 840*x^4*y^2 + 390*x^4*y*z^2 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4877*x*z + 1717*x + 4*y^7 - 27*y^6*z - 56*y^6 + 78*y^5*z^2 + 323*y^5*z + 332*y^5 - 125*y^4*z^3 - 775*y^4*z^2 - 1590*y^4*z - 1084*y^4 + 120*y^3*z^4 + 990*y^3*z^3 + 3040*y^3*z^2 + 4135*y^3*z + 2172*y^3 - 69*y^2*z^5 - 710*y^2*z^4 - 2900*y^2*z^3 - 5901*y^2*z^2 - 6142*y^2*z - 2624*y^2 + 22*y*z^6 + 271*y*z^5 + 1380*y*z^4 + 3733*y*z^3 + 5780*y*z^2 + 4874*y*z + 1713*y - 3*z^7 - 43*z^6 - 262*z^5 - 883*z^4 - 1809*z^3 - 2261*z^2 - 1571*z - 1, -8*x^7 - 56*x^6*y + 54*x^6*z + 111*x^6 - 168*x^5*y^2 + 324*x^5*y*z + 666*x^5*y - 156*x^5*z^2 - 640*x^5*z - 651*x^5 - 280*x^4*y^3 + 810*x^4*y^2*z + 1665*x^4*y^2 - 780*x^4*y*z^2 - 3200*x^4*y*z - 3255*x^4*y + 250*x^4*z^3 + 1535*x^4*z^2 + 3115*x^4*z + 2098*x^4 - 280*x^3*y^4 + 1080*x^3*y^3*z + 2220*x^3*y^3 - 1560*x^3*y^2*z^2 - 6400*x^3*y^2*z - 6510*x^3*y^2 + 1000*x^3*y*z^3 + 6140*x^3*y*z^2 + 12460*x^3*y*z + 8392*x^3*y - 240*x^3*z^4 - 1960*x^3*z^3 - 5950*x^3*z^2 - 7990*x^3*z - 4143*x^3 - 168*x^2*y^5 + 810*x^2*y^4*z + 1665*x^2*y^4 - 1560*x^2*y^3*z^2 - 6400*x^2*y^3*z - 6510*x^2*y^3 + 1500*x^2*y^2*z^3 + 9210*x^2*y^2*z^2 + 18690*x^2*y^2*z + 12588*x^2*y^2 - 720*x^2*y*z^4 - 5880*x^2*y*z^3 - 17850*x^2*y*z^2 - 23970*x^2*y*z - 12429*x^2*y + 138*x^2*z^5 + 1405*x^2*z^4 + 5670*x^2*z^3 + 11382*x^2*z^2 + 11681*x^2*z + 4930*x^2 - 56*x*y^6 + 324*x*y^5*z + 666*x*y^5 - 780*x*y^4*z^2 - 3200*x*y^4*z - 3255*x*y^4 + 1000*x*y^3*z^3 + 6140*x*y^3*z^2 + 12460*x*y^3*z + 8392*x*y^3 - 720*x*y^2*z^4 - 5880*x*y^2*z^3 - 17850*x*y^2*z^2 - 23970*x*y^2*z - 12429*x*y^2 + 276*x*y*z^5 + 2810*x*y*z^4 + 11340*x*y*z^3 + 22764*x*y*z^2 + 23362*x*y*z + 9852*x*y - 44*x*z^6 - 536*x*z^5 - 2695*x*z^4 - 7186*x*z^3 - 10957*x*z^2 - 9102*x*z - 3171*x - 8*y^7 + 54*y^6*z + 111*y^6 - 156*y^5*z^2 - 640*y^5*z - 651*y^5 + 250*y^4*z^3 + 1535*y^4*z^2 + 3115*y^4*z + 2098*y^4 - 240*y^3*z^4 - 1960*y^3*z^3 - 5950*y^3*z^2 - 7990*y^3*z - 4143*y^3 + 138*y^2*z^5 + 1405*y^2*z^4 + 5670*y^2*z^3 + 11382*y^2*z^2 + 11681*y^2*z + 4922*y^2 - 44*y*z^6 - 536*y*z^5 - 2695*y*z^4 - 7186*y*z^3 - 10957*y*z^2 - 9096*y*z - 3164*y + 6*z^7 + 85*z^6 + 511*z^5 + 1696*z^4 + 3417*z^3 + 4196*z^2 + 2874*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -1, -1, 1 ], [ -2, -1, 1 ], [ -3, -3, 2 ] ], "source_shift": [ 2, 1, 1 ], "target_matrix": [ [ -1, -1, -1 ], [ -1, -1, -2 ], [ 2, 1, 1 ] ], "target_shift": [ 468, 462, -836 ], "inverse_recipe": { "p": 6, "q": 1, "f": "x**6 + x**5 + x**3 + x", "g": "x", "h": "y*z + z**3 + 2*z**2", "polynomials": { "f": [ [ [ 6, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ], [ [ 3, 0, 0 ], 1 ], [ [ 5, 0, 0 ], 1 ] ], "g": [ [ [ 1, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 0, 2 ], 2 ], [ [ 0, 0, 3 ], 1 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0211868419974053 ], "surface_features": [ 3.4948118020839996, 4.379668034033654, 3.1931245983544616, 0.0, 3.618242155781335, 4854 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B222", "pair": 222, "family": "bijection", "mapping": "F(x,y,z)=(8*x^7 - 112*x^6*y + 56*x^6*z - 72*x^6 + 672*x^5*y^2 - 672*x^5*y*z + 864*x^5*y + 168*x^5*z^2 - 432*x^5*z + 288*x^5 - 2240*x^4*y^3 + 3360*x^4*y^2*z - 4320*x^4*y^2 - 1680*x^4*y*z^2 + 4320*x^4*y*z - 2856*x^4*y + 280*x^4*z^3 - 1080*x^4*z^2 + 1432*x^4*z - 689*x^4 + 4480*x^3*y^4 - 8960*x^3*y^3*z + 11520*x^3*y^3 + 6720*x^3*y^2*z^2 - 17280*x^3*y^2*z + 11328*x^3*y^2 - 2240*x^3*y*z^3 + 8640*x^3*y*z^2 - 11360*x^3*y*z + 5360*x^3*y + 280*x^3*z^4 - 1440*x^3*z^3 + 2848*x^3*z^2 - 2708*x^3*z + 1051*x^3 - 5376*x^2*y^5 + 13440*x^2*y^4*z - 17280*x^2*y^4 - 13440*x^2*y^3*z^2 + 34560*x^2*y^3*z - 22464*x^2*y^3 + 6720*x^2*y^2*z^3 - 25920*x^2*y^2*z^2 + 33792*x^2*y^2*z - 15624*x^2*y^2 - 1680*x^2*y*z^4 + 8640*x^2*y*z^3 - 16944*x^2*y*z^2 + 15792*x^2*y*z - 5962*x^2*y + 168*x^2*z^5 - 1080*x^2*z^4 + 2832*x^2*z^3 - 3990*x^2*z^2 + 3049*x^2*z - 1033*x^2 + 3584*x*y^6 - 10752*x*y^5*z + 13824*x*y^5 + 13440*x*y^4*z^2 - 34560*x*y^4*z + 22272*x*y^4 - 8960*x*y^3*z^3 + 34560*x*y^3*z^2 - 44672*x*y^3*z + 20224*x*y^3 + 3360*x*y^2*z^4 - 17280*x*y^2*z^3 + 33600*x*y^2*z^2 - 30672*x*y^2*z + 11236*x*y^2 - 672*x*y*z^5 + 4320*x*y*z^4 - 11232*x*y*z^3 + 15504*x*y*z^2 - 11508*x*y*z + 3644*x*y + 56*x*z^6 - 432*x*z^5 + 1408*x*z^4 - 2612*x*z^3 + 2945*x*z^2 - 1922*x*z + 574*x - 1024*y^7 + 3584*y^6*z - 4608*y^6 - 5376*y^5*z^2 + 13824*y^5*z - 8832*y^5 + 4480*y^4*z^3 - 17280*y^4*z^2 + 22144*y^4*z - 9808*y^4 - 2240*y^3*z^4 + 11520*y^3*z^3 - 22208*y^3*z^2 + 19840*y^3*z - 7032*y^3 + 672*y^2*z^5 - 4320*y^2*z^4 + 11136*y^2*z^3 - 15048*y^2*z^2 + 10820*y^2*z - 3252*y^2 - 112*y*z^6 + 864*y*z^5 - 2792*y*z^4 + 5072*y*z^3 - 5546*y*z^2 + 3412*y*z - 926*y + 8*z^7 - 72*z^6 + 280*z^5 - 641*z^4 + 947*z^3 - 897*z^2 + 511*z - 1, 8*x^7 - 112*x^6*y + 56*x^6*z - 72*x^6 + 672*x^5*y^2 - 672*x^5*y*z + 864*x^5*y + 168*x^5*z^2 - 432*x^5*z + 288*x^5 - 2240*x^4*y^3 + 3360*x^4*y^2*z - 4320*x^4*y^2 - 1680*x^4*y*z^2 + 4320*x^4*y*z - 2856*x^4*y + 280*x^4*z^3 - 1080*x^4*z^2 + 1432*x^4*z - 689*x^4 + 4480*x^3*y^4 - 8960*x^3*y^3*z + 11520*x^3*y^3 + 6720*x^3*y^2*z^2 - 17280*x^3*y^2*z + 11328*x^3*y^2 - 2240*x^3*y*z^3 + 8640*x^3*y*z^2 - 11360*x^3*y*z + 5360*x^3*y + 280*x^3*z^4 - 1440*x^3*z^3 + 2848*x^3*z^2 - 2708*x^3*z + 1049*x^3 - 5376*x^2*y^5 + 13440*x^2*y^4*z - 17280*x^2*y^4 - 13440*x^2*y^3*z^2 + 34560*x^2*y^3*z - 22464*x^2*y^3 + 6720*x^2*y^2*z^3 - 25920*x^2*y^2*z^2 + 33792*x^2*y^2*z - 15624*x^2*y^2 - 1680*x^2*y*z^4 + 8640*x^2*y*z^3 - 16944*x^2*y*z^2 + 15792*x^2*y*z - 5950*x^2*y + 168*x^2*z^5 - 1080*x^2*z^4 + 2832*x^2*z^3 - 3990*x^2*z^2 + 3043*x^2*z - 1025*x^2 + 3584*x*y^6 - 10752*x*y^5*z + 13824*x*y^5 + 13440*x*y^4*z^2 - 34560*x*y^4*z + 22272*x*y^4 - 8960*x*y^3*z^3 + 34560*x*y^3*z^2 - 44672*x*y^3*z + 20224*x*y^3 + 3360*x*y^2*z^4 - 17280*x*y^2*z^3 + 33600*x*y^2*z^2 - 30672*x*y^2*z + 11212*x*y^2 - 672*x*y*z^5 + 4320*x*y*z^4 - 11232*x*y*z^3 + 15504*x*y*z^2 - 11484*x*y*z + 3612*x*y + 56*x*z^6 - 432*x*z^5 + 1408*x*z^4 - 2612*x*z^3 + 2939*x*z^2 - 1906*x*z + 560*x - 1024*y^7 + 3584*y^6*z - 4608*y^6 - 5376*y^5*z^2 + 13824*y^5*z - 8832*y^5 + 4480*y^4*z^3 - 17280*y^4*z^2 + 22144*y^4*z - 9808*y^4 - 2240*y^3*z^4 + 11520*y^3*z^3 - 22208*y^3*z^2 + 19840*y^3*z - 7016*y^3 + 672*y^2*z^5 - 4320*y^2*z^4 + 11136*y^2*z^3 - 15048*y^2*z^2 + 10796*y^2*z - 3220*y^2 - 112*y*z^6 + 864*y*z^5 - 2792*y*z^4 + 5072*y*z^3 - 5534*y*z^2 + 3380*y*z - 904*y + 8*z^7 - 72*z^6 + 280*z^5 - 641*z^4 + 945*z^3 - 889*z^2 + 499*z - 1, -16*x^7 + 224*x^6*y - 112*x^6*z + 144*x^6 - 1344*x^5*y^2 + 1344*x^5*y*z - 1728*x^5*y - 336*x^5*z^2 + 864*x^5*z - 576*x^5 + 4480*x^4*y^3 - 6720*x^4*y^2*z + 8640*x^4*y^2 + 3360*x^4*y*z^2 - 8640*x^4*y*z + 5712*x^4*y - 560*x^4*z^3 + 2160*x^4*z^2 - 2864*x^4*z + 1377*x^4 - 8960*x^3*y^4 + 17920*x^3*y^3*z - 23040*x^3*y^3 - 13440*x^3*y^2*z^2 + 34560*x^3*y^2*z - 22656*x^3*y^2 + 4480*x^3*y*z^3 - 17280*x^3*y*z^2 + 22720*x^3*y*z - 10712*x^3*y - 560*x^3*z^4 + 2880*x^3*z^3 - 5696*x^3*z^2 + 5412*x^3*z - 2099*x^3 + 10752*x^2*y^5 - 26880*x^2*y^4*z + 34560*x^2*y^4 + 26880*x^2*y^3*z^2 - 69120*x^2*y^3*z + 44928*x^2*y^3 - 13440*x^2*y^2*z^3 + 51840*x^2*y^2*z^2 - 67584*x^2*y^2*z + 31224*x^2*y^2 + 3360*x^2*y*z^4 - 17280*x^2*y*z^3 + 33888*x^2*y*z^2 - 31560*x^2*y*z + 11906*x^2*y - 336*x^2*z^5 + 2160*x^2*z^4 - 5664*x^2*z^3 + 7974*x^2*z^2 - 6089*x^2*z + 2065*x^2 - 7168*x*y^6 + 21504*x*y^5*z - 27648*x*y^5 - 26880*x*y^4*z^2 + 69120*x*y^4*z - 44544*x*y^4 + 17920*x*y^3*z^3 - 69120*x*y^3*z^2 + 89344*x*y^3*z - 40416*x*y^3 - 6720*x*y^2*z^4 + 34560*x*y^2*z^3 - 67200*x*y^2*z^2 + 61296*x*y^2*z - 22436*x*y^2 + 1344*x*y*z^5 - 8640*x*y*z^4 + 22464*x*y*z^3 - 30984*x*y*z^2 + 22980*x*y*z - 7284*x*y - 112*x*z^6 + 864*x*z^5 - 2816*x*z^4 + 5220*x*z^3 - 5881*x*z^2 + 3842*x*z - 1151*x + 2048*y^7 - 7168*y^6*z + 9216*y^6 + 10752*y^5*z^2 - 27648*y^5*z + 17664*y^5 - 8960*y^4*z^3 + 34560*y^4*z^2 - 44288*y^4*z + 19600*y^4 + 4480*y^3*z^4 - 23040*y^3*z^3 + 44416*y^3*z^2 - 39648*y^3*z + 14040*y^3 - 1344*y^2*z^5 + 8640*y^2*z^4 - 22272*y^2*z^3 + 30072*y^2*z^2 - 21604*y^2*z + 6500*y^2 + 224*y*z^6 - 1728*y*z^5 + 5584*y*z^4 - 10136*y*z^3 + 11074*y*z^2 - 6820*y*z + 1856*y - 16*z^7 + 144*z^6 - 560*z^5 + 1281*z^4 - 1891*z^3 + 1793*z^2 - 1024*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -1, 2, -1 ], [ -3, 2, -2 ], [ -2, 1, -1 ] ], "source_shift": [ 1, 1, 2 ], "target_matrix": [ [ -1, -1, -1 ], [ -1, -1, -2 ], [ 2, 1, 1 ] ], "target_shift": [ 124, 116, -253 ], "inverse_recipe": { "p": 4, "q": 3, "f": "x**4 + x**3 + x", "g": "x**3 + x**2", "h": "y*z", "polynomials": { "f": [ [ [ 4, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ], [ [ 3, 0, 0 ], 1 ] ], "g": [ [ [ 3, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ] ] }, "h_multiplier": 8 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0280979350027337 ], "surface_features": [ 4.024561373013846, 4.95106539227103, 3.7114697818743276, 0.0, 4.137546309672313, 5075 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B095", "pair": 95, "family": "bijection", "mapping": "F(x,y,z)=(-2187*x^7 - 10206*x^6*y - 10206*x^6*z - 10206*x^6 - 20412*x^5*y^2 - 40824*x^5*y*z - 40824*x^5*y - 20412*x^5*z^2 - 40824*x^5*z - 20736*x^5 - 22680*x^4*y^3 - 68040*x^4*y^2*z - 68040*x^4*y^2 - 68040*x^4*y*z^2 - 136080*x^4*y*z - 69228*x^4*y - 22680*x^4*z^3 - 68040*x^4*z^2 - 69147*x^4*z - 24165*x^4 - 15120*x^3*y^4 - 60480*x^3*y^3*z - 60480*x^3*y^3 - 90720*x^3*y^2*z^2 - 181440*x^3*y^2*z - 92448*x^3*y^2 - 60480*x^3*y*z^3 - 181440*x^3*y*z^2 - 184680*x^3*y*z - 64719*x^3*y - 15120*x^3*z^4 - 60480*x^3*z^3 - 92232*x^3*z^2 - 64503*x^3*z - 17577*x^3 - 6048*x^2*y^5 - 30240*x^2*y^4*z - 30240*x^2*y^4 - 60480*x^2*y^3*z^2 - 120960*x^2*y^3*z - 61728*x^2*y^3 - 60480*x^2*y^2*z^3 - 181440*x^2*y^2*z^2 - 184968*x^2*y^2*z - 64998*x^2*y^2 - 30240*x^2*y*z^4 - 120960*x^2*y*z^3 - 184752*x^2*y*z^2 - 129564*x^2*y*z - 35424*x^2*y - 6048*x^2*z^5 - 30240*x^2*z^4 - 61512*x^2*z^3 - 64566*x^2*z^2 - 35208*x^2*z - 7970*x^2 - 1344*x*y^6 - 8064*x*y^5*z - 8064*x*y^5 - 20160*x*y^4*z^2 - 40320*x*y^4*z - 20608*x*y^4 - 26880*x*y^3*z^3 - 80640*x*y^3*z^2 - 82336*x*y^3*z - 29012*x*y^3 - 20160*x*y^2*z^4 - 80640*x*y^2*z^3 - 123360*x*y^2*z^2 - 86748*x*y^2*z - 23796*x*y^2 - 8064*x*y*z^5 - 40320*x*y*z^4 - 82144*x*y*z^3 - 86460*x*y*z^2 - 47304*x*y*z - 10744*x*y - 1344*x*z^6 - 8064*x*z^5 - 20512*x*z^4 - 28724*x*z^3 - 23508*x*z^2 - 10646*x*z - 2073*x - 128*y^7 - 896*y^6*z - 896*y^6 - 2688*y^5*z^2 - 5376*y^5*z - 2752*y^5 - 4480*y^4*z^3 - 13440*y^4*z^2 - 13744*y^4*z - 4856*y^4 - 4480*y^3*z^4 - 17920*y^3*z^3 - 27456*y^3*z^2 - 19360*y^3*z - 5328*y^3 - 2688*y^2*z^5 - 13440*y^2*z^4 - 27424*y^2*z^3 - 28944*y^2*z^2 - 15888*y^2*z - 3620*y^2 - 896*y*z^6 - 5376*y*z^5 - 13696*y*z^4 - 19232*y*z^3 - 15792*y*z^2 - 7175*y*z - 1397*y - 128*z^7 - 896*z^6 - 2736*z^5 - 4792*z^4 - 5232*z^3 - 3555*z^2 - 1382*z - 1, 4374*x^7 + 20412*x^6*y + 20412*x^6*z + 20412*x^6 + 40824*x^5*y^2 + 81648*x^5*y*z + 81648*x^5*y + 40824*x^5*z^2 + 81648*x^5*z + 41472*x^5 + 45360*x^4*y^3 + 136080*x^4*y^2*z + 136080*x^4*y^2 + 136080*x^4*y*z^2 + 272160*x^4*y*z + 138456*x^4*y + 45360*x^4*z^3 + 136080*x^4*z^2 + 138294*x^4*z + 48168*x^4 + 30240*x^3*y^4 + 120960*x^3*y^3*z + 120960*x^3*y^3 + 181440*x^3*y^2*z^2 + 362880*x^3*y^2*z + 184896*x^3*y^2 + 120960*x^3*y*z^3 + 362880*x^3*y*z^2 + 369360*x^3*y*z + 129006*x^3*y + 30240*x^3*z^4 + 120960*x^3*z^3 + 184464*x^3*z^2 + 128574*x^3*z + 34668*x^3 + 12096*x^2*y^5 + 60480*x^2*y^4*z + 60480*x^2*y^4 + 120960*x^2*y^3*z^2 + 241920*x^2*y^3*z + 123456*x^2*y^3 + 120960*x^2*y^2*z^3 + 362880*x^2*y^2*z^2 + 369936*x^2*y^2*z + 129564*x^2*y^2 + 60480*x^2*y*z^4 + 241920*x^2*y*z^3 + 369504*x^2*y*z^2 + 258264*x^2*y*z + 69876*x^2*y + 12096*x^2*z^5 + 60480*x^2*z^4 + 123024*x^2*z^3 + 128700*x^2*z^2 + 69444*x^2*z + 15400*x^2 + 2688*x*y^6 + 16128*x*y^5*z + 16128*x*y^5 + 40320*x*y^4*z^2 + 80640*x*y^4*z + 41216*x*y^4 + 53760*x*y^3*z^3 + 161280*x*y^3*z^2 + 164672*x*y^3*z + 57832*x*y^3 + 40320*x*y^2*z^4 + 161280*x*y^2*z^3 + 246720*x*y^2*z^2 + 172920*x*y^2*z + 46944*x*y^2 + 16128*x*y*z^5 + 80640*x*y*z^4 + 164288*x*y*z^3 + 172344*x*y*z^2 + 93312*x*y*z + 20768*x*y + 2688*x*z^6 + 16128*x*z^5 + 41024*x*z^4 + 57256*x*z^3 + 46368*x*z^2 + 20572*x*z + 3870*x + 256*y^7 + 1792*y^6*z + 1792*y^6 + 5376*y^5*z^2 + 10752*y^5*z + 5504*y^5 + 8960*y^4*z^3 + 26880*y^4*z^2 + 27488*y^4*z + 9680*y^4 + 8960*y^3*z^4 + 35840*y^3*z^3 + 54912*y^3*z^2 + 38592*y^3*z + 10512*y^3 + 5376*y^2*z^5 + 26880*y^2*z^4 + 54848*y^2*z^3 + 57696*y^2*z^2 + 31344*y^2*z + 7000*y^2 + 1792*y*z^6 + 10752*y*z^5 + 27392*y*z^4 + 38336*y*z^3 + 31152*y*z^2 + 13870*y*z + 2608*y + 256*z^7 + 1792*z^6 + 5472*z^5 + 9552*z^4 + 10320*z^3 + 6870*z^2 + 2580*z + 1, -2187*x^7 - 10206*x^6*y - 10206*x^6*z - 10206*x^6 - 20412*x^5*y^2 - 40824*x^5*y*z - 40824*x^5*y - 20412*x^5*z^2 - 40824*x^5*z - 20736*x^5 - 22680*x^4*y^3 - 68040*x^4*y^2*z - 68040*x^4*y^2 - 68040*x^4*y*z^2 - 136080*x^4*y*z - 69228*x^4*y - 22680*x^4*z^3 - 68040*x^4*z^2 - 69147*x^4*z - 24084*x^4 - 15120*x^3*y^4 - 60480*x^3*y^3*z - 60480*x^3*y^3 - 90720*x^3*y^2*z^2 - 181440*x^3*y^2*z - 92448*x^3*y^2 - 60480*x^3*y*z^3 - 181440*x^3*y*z^2 - 184680*x^3*y*z - 64503*x^3*y - 15120*x^3*z^4 - 60480*x^3*z^3 - 92232*x^3*z^2 - 64287*x^3*z - 17361*x^3 - 6048*x^2*y^5 - 30240*x^2*y^4*z - 30240*x^2*y^4 - 60480*x^2*y^3*z^2 - 120960*x^2*y^3*z - 61728*x^2*y^3 - 60480*x^2*y^2*z^3 - 181440*x^2*y^2*z^2 - 184968*x^2*y^2*z - 64782*x^2*y^2 - 30240*x^2*y*z^4 - 120960*x^2*y*z^3 - 184752*x^2*y*z^2 - 129132*x^2*y*z - 34992*x^2*y - 6048*x^2*z^5 - 30240*x^2*z^4 - 61512*x^2*z^3 - 64350*x^2*z^2 - 34776*x^2*z - 7754*x^2 - 1344*x*y^6 - 8064*x*y^5*z - 8064*x*y^5 - 20160*x*y^4*z^2 - 40320*x*y^4*z - 20608*x*y^4 - 26880*x*y^3*z^3 - 80640*x*y^3*z^2 - 82336*x*y^3*z - 28916*x*y^3 - 20160*x*y^2*z^4 - 80640*x*y^2*z^3 - 123360*x*y^2*z^2 - 86460*x*y^2*z - 23508*x*y^2 - 8064*x*y*z^5 - 40320*x*y*z^4 - 82144*x*y*z^3 - 86172*x*y*z^2 - 46728*x*y*z - 10456*x*y - 1344*x*z^6 - 8064*x*z^5 - 20512*x*z^4 - 28628*x*z^3 - 23220*x*z^2 - 10358*x*z - 1975*x - 128*y^7 - 896*y^6*z - 896*y^6 - 2688*y^5*z^2 - 5376*y^5*z - 2752*y^5 - 4480*y^4*z^3 - 13440*y^4*z^2 - 13744*y^4*z - 4840*y^4 - 4480*y^3*z^4 - 17920*y^3*z^3 - 27456*y^3*z^2 - 19296*y^3*z - 5264*y^3 - 2688*y^2*z^5 - 13440*y^2*z^4 - 27424*y^2*z^3 - 28848*y^2*z^2 - 15696*y^2*z - 3524*y^2 - 896*y*z^6 - 5376*y*z^5 - 13696*y*z^4 - 19168*y*z^3 - 15600*y*z^2 - 6983*y*z - 1332*y - 128*z^7 - 896*z^6 - 2736*z^5 - 4776*z^4 - 5168*z^3 - 3459*z^2 - 1317*z 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256*x*y^5 + 420*x*y^4*z^2 - 1240*x*y^4*z + 924*x*y^4 + 560*x*y^3*z^3 - 2400*x*y^3*z^2 + 3496*x*y^3*z - 1768*x*y^3 + 420*x*y^2*z^4 - 2320*x*y^2*z^3 + 4956*x*y^2*z^2 - 4960*x*y^2*z + 1986*x*y^2 + 168*x*y*z^5 - 1120*x*y*z^4 + 3120*x*y*z^3 - 4624*x*y*z^2 + 3660*x*y*z - 1206*x*y + 28*x*z^6 - 216*x*z^5 + 736*x*z^4 - 1432*x*z^3 + 1670*x*z^2 - 1078*x*z + 306*x - 4*y^7 - 28*y^6*z + 44*y^6 - 84*y^5*z^2 + 256*y^5*z - 196*y^5 - 140*y^4*z^3 + 620*y^4*z^2 - 928*y^4*z + 480*y^4 - 140*y^3*z^4 + 800*y^3*z^3 - 1756*y^3*z^2 + 1800*y^3*z - 742*y^3 - 84*y^2*z^5 + 580*y^2*z^4 - 1660*y^2*z^3 + 2524*y^2*z^2 - 2066*y^2*z + 707*y^2 - 28*y*z^6 + 224*y*z^5 - 784*y*z^4 + 1568*y*z^3 - 1894*y*z^2 + 1270*y*z - 371*y - 4*z^7 + 36*z^6 - 148*z^5 + 364*z^4 - 574*z^3 + 563*z^2 - 328*z + 1, 8*x^7 - 56*x^6*y - 56*x^6*z + 72*x^6 + 168*x^5*y^2 + 336*x^5*y*z - 448*x^5*y + 168*x^5*z^2 - 432*x^5*z + 288*x^5 - 280*x^4*y^3 - 840*x^4*y^2*z + 1160*x^4*y^2 - 840*x^4*y*z^2 + 2240*x^4*y*z - 1536*x^4*y - 280*x^4*z^3 + 1080*x^4*z^2 - 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4932*x^2*y^2*z + 2436*x^2*y^2 - 420*x^2*y*z^4 + 2240*x^2*y*z^3 - 4656*x^2*y*z^2 + 4544*x^2*y*z - 1766*x^2*y - 84*x^2*z^5 + 540*x^2*z^4 - 1464*x^2*z^3 + 2112*x^2*z^2 - 1618*x^2*z + 516*x^2 + 28*x*y^6 + 168*x*y^5*z - 256*x*y^5 + 420*x*y^4*z^2 - 1240*x*y^4*z + 924*x*y^4 + 560*x*y^3*z^3 - 2400*x*y^3*z^2 + 3496*x*y^3*z - 1768*x*y^3 + 420*x*y^2*z^4 - 2320*x*y^2*z^3 + 4956*x*y^2*z^2 - 4960*x*y^2*z + 1986*x*y^2 + 168*x*y*z^5 - 1120*x*y*z^4 + 3120*x*y*z^3 - 4624*x*y*z^2 + 3660*x*y*z - 1208*x*y + 28*x*z^6 - 216*x*z^5 + 736*x*z^4 - 1432*x*z^3 + 1670*x*z^2 - 1080*x*z + 309*x - 4*y^7 - 28*y^6*z + 44*y^6 - 84*y^5*z^2 + 256*y^5*z - 196*y^5 - 140*y^4*z^3 + 620*y^4*z^2 - 928*y^4*z + 480*y^4 - 140*y^3*z^4 + 800*y^3*z^3 - 1756*y^3*z^2 + 1800*y^3*z - 742*y^3 - 84*y^2*z^5 + 580*y^2*z^4 - 1660*y^2*z^3 + 2524*y^2*z^2 - 2066*y^2*z + 708*y^2 - 28*y*z^6 + 224*y*z^5 - 784*y*z^4 + 1568*y*z^3 - 1894*y*z^2 + 1272*y*z - 375*y - 4*z^7 + 36*z^6 - 148*z^5 + 364*z^4 - 574*z^3 + 564*z^2 - 331*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ 1, -1, -1 ], [ 1, -2, -1 ], [ 1, -1, -2 ] ], "source_shift": [ 1, 1, 2 ], "target_matrix": [ [ 1, 1, 1 ], [ 2, 1, 1 ], [ 1, 2, 1 ] ], "target_shift": [ -90, -187, -92 ], "inverse_recipe": { "p": 2, "q": 3, "f": "x**2", "g": "x**3 + x", "h": "y**2*z + 2*z", "polynomials": { "f": [ [ [ 2, 0, 0 ], 1 ] ], "g": [ [ [ 3, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "h": [ [ [ 0, 2, 1 ], 1 ], [ [ 0, 0, 1 ], 2 ] ] }, "h_multiplier": 4 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0742534889977833 ], "surface_features": [ 3.1628621641984123, 3.9965116721541785, 2.8987251815894934, 0.0, 3.2521623744659136, 4782 ], "component_degrees": [ 7, 7, 7 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1950*x*y^4*z^2 - 5334*x*y^4*z + 2988*x*y^4 - 2875*x*y^3*z^3 + 11874*x*y^3*z^2 - 13863*x*y^3*z + 4929*x*y^3 + 2211*x*y^2*z^4 - 12312*x*y^2*z^3 + 22545*x*y^2*z^2 - 16815*x*y^2*z + 4488*x*y^2 - 840*x*y*z^5 + 5934*x*y*z^4 - 15093*x*y*z^3 + 17706*x*y*z^2 - 9984*x*y*z + 2201*x*y + 124*x*z^6 - 1068*x*z^5 + 3507*x*z^4 - 5721*x*z^3 + 5085*x*z^2 - 2374*x*z + 459*x - 16*y^7 + 144*y^6*z - 196*y^6 - 528*y^5*z^2 + 1440*y^5*z - 792*y^5 + 1012*y^4*z^3 - 4161*y^4*z^2 + 4758*y^4*z - 1651*y^4 - 1083*y^3*z^4 + 5995*y^3*z^3 - 10749*y^3*z^2 + 7816*y^3*z - 2025*y^3 + 642*y^2*z^5 - 4509*y^2*z^4 + 11286*y^2*z^3 - 12933*y^2*z^2 + 7077*y^2*z - 1507*y^2 - 196*y*z^6 + 1680*y*z^5 - 5457*y*z^4 + 8747*y*z^3 - 7599*y*z^2 + 3439*y*z - 642*y + 24*z^7 - 244*z^6 + 978*z^5 - 2036*z^4 + 2469*z^3 - 1777*z^2 + 704*z - 1, 2*x^7 - 20*x^6*y + 21*x^6*z - 23*x^6 + 84*x^5*y^2 - 183*x^5*y*z + 198*x^5*y + 93*x^5*z^2 - 207*x^5*z + 111*x^5 - 192*x^4*y^3 + 651*x^4*y^2*z - 696*x^4*y^2 - 687*x^4*y*z^2 + 1509*x^4*y*z - 795*x^4*y + 225*x^4*z^3 - 765*x^4*z^2 + 837*x^4*z - 297*x^4 + 258*x^3*y^4 - 1209*x^3*y^3*z + 1278*x^3*y^3 + 1989*x^3*y^2*z^2 - 4311*x^3*y^2*z + 2229*x^3*y^2 - 1353*x^3*y*z^3 + 4536*x^3*y*z^2 - 4878*x^3*y*z + 1693*x^3*y + 321*x^3*z^4 - 1485*x^3*z^3 + 2487*x^3*z^2 - 1804*x^3*z + 483*x^3 - 204*x^2*y^5 + 1236*x^2*y^4*z - 1293*x^2*y^4 - 2817*x^2*y^3*z^2 + 6027*x^2*y^3*z - 3057*x^2*y^3 + 2991*x^2*y^2*z^3 - 9882*x^2*y^2*z^2 + 10428*x^2*y^2*z - 3534*x^2*y^2 - 1473*x^2*y*z^4 + 6717*x^2*y*z^3 - 11067*x^2*y*z^2 + 7857*x^2*y*z - 2046*x^2*y + 270*x^2*z^5 - 1596*x^2*z^4 + 3636*x^2*z^3 - 4044*x^2*z^2 + 2220*x^2*z - 484*x^2 + 88*x*y^6 - 660*x*y^5*z + 684*x*y^5 + 1950*x*y^4*z^2 - 4122*x*y^4*z + 2052*x*y^4 - 2875*x*y^3*z^3 + 9360*x*y^3*z^2 - 9681*x*y^3*z + 3201*x*y^3 + 2211*x*y^2*z^4 - 9927*x*y^2*z^3 + 16056*x*y^2*z^2 - 11130*x*y^2*z + 2811*x*y^2 - 840*x*y*z^5 + 4896*x*y*z^4 - 10989*x*y*z^3 + 11976*x*y*z^2 - 6402*x*y*z + 1349*x*y + 124*x*z^6 - 900*x*z^5 + 2613*x*z^4 - 3960*x*z^3 + 3342*x*z^2 - 1498*x*z + 281*x - 16*y^7 + 144*y^6*z - 148*y^6 - 528*y^5*z^2 + 1104*y^5*z - 540*y^5 + 1012*y^4*z^3 - 3249*y^4*z^2 + 3294*y^4*z - 1063*y^4 - 1083*y^3*z^4 + 4783*y^3*z^3 - 7578*y^3*z^2 + 5122*y^3*z - 1254*y^3 + 642*y^2*z^5 - 3684*y^2*z^4 + 8127*y^2*z^3 - 8649*y^2*z^2 + 4482*y^2*z - 910*y^2 - 196*y*z^6 + 1404*y*z^5 - 4023*y*z^4 + 5987*y*z^3 - 4932*y*z^2 + 2137*y*z - 386*y + 24*z^7 - 208*z^6 + 738*z^5 - 1427*z^4 + 1644*z^3 - 1138*z^2 + 438*z - 1, -4*x^7 + 40*x^6*y - 42*x^6*z + 40*x^6 - 168*x^5*y^2 + 366*x^5*y*z - 342*x^5*y - 186*x^5*z^2 + 363*x^5*z - 177*x^5 + 384*x^4*y^3 - 1302*x^4*y^2*z + 1194*x^4*y^2 + 1374*x^4*y*z^2 - 2628*x^4*y*z + 1257*x^4*y - 450*x^4*z^3 + 1353*x^4*z^2 - 1347*x^4*z + 449*x^4 - 516*x^3*y^4 + 2418*x^3*y^3*z - 2178*x^3*y^3 - 3978*x^3*y^2*z^2 + 7455*x^3*y^2*z - 3495*x^3*y^2 + 2706*x^3*y*z^3 - 7968*x^3*y*z^2 + 7782*x^3*y*z - 2536*x^3*y - 642*x^3*z^4 + 2649*x^3*z^3 - 4041*x^3*z^2 + 2753*x^3*z - 707*x^3 + 408*x^2*y^5 - 2472*x^2*y^4*z + 2190*x^2*y^4 + 5634*x^2*y^3*z^2 - 10350*x^2*y^3*z + 4755*x^2*y^3 - 5982*x^2*y^2*z^3 + 17235*x^2*y^2*z^2 - 16491*x^2*y^2*z + 5247*x^2*y^2 + 2946*x^2*y*z^4 - 11904*x^2*y*z^3 + 17823*x^2*y*z^2 - 11874*x^2*y*z + 2967*x^2*y - 540*x^2*z^5 + 2871*x^2*z^4 - 5967*x^2*z^3 + 6234*x^2*z^2 - 3279*x^2*z + 695*x^2 - 176*x*y^6 + 1320*x*y^5*z - 1152*x*y^5 - 3900*x*y^4*z^2 + 7032*x*y^4*z - 3168*x*y^4 + 5750*x*y^3*z^3 - 16206*x*y^3*z^2 + 15180*x*y^3*z - 4714*x*y^3 - 4422*x*y^2*z^4 + 17469*x*y^2*z^3 - 25623*x*y^2*z^2 + 16659*x*y^2*z - 4041*x*y^2 + 1680*x*y*z^5 - 8754*x*y*z^4 + 17874*x*y*z^3 - 18276*x*y*z^2 + 9360*x*y*z - 1918*x*y - 248*x*z^6 + 1632*x*z^5 - 4332*x*z^4 + 6170*x*z^3 - 4986*x*z^2 + 2171*x*z - 399*x + 32*y^7 - 288*y^6*z + 248*y^6 + 1056*y^5*z^2 - 1872*y^5*z + 828*y^5 - 2024*y^4*z^3 + 5586*y^4*z^2 - 5124*y^4*z + 1554*y^4 + 2166*y^3*z^4 - 8354*y^3*z^3 + 11985*y^3*z^2 - 7598*y^3*z + 1789*y^3 - 1284*y^2*z^5 + 6543*y^2*z^4 - 13095*y^2*z^3 + 13062*y^2*z^2 - 6489*y^2*z + 1283*y^2 + 392*y*z^6 - 2532*y*z^5 + 6612*y*z^4 - 9234*y*z^3 + 7278*y*z^2 - 3062*y*z + 541*y - 48*z^7 + 380*z^6 - 1236*z^5 + 2248*z^4 - 2480*z^3 + 1667*z^2 - 629*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -1, 2, -1 ], [ -1, 1, -2 ], [ -2, 2, -3 ] ], "source_shift": [ 1, 1, 2 ], "target_matrix": [ [ -1, -2, -1 ], [ -1, -1, -1 ], [ 2, 1, 1 ] ], "target_shift": [ 118, 72, -102 ], "collision_preimages": [ [ "7/4", "1/4", "-1/4" ], [ "-47/2", "-7", "19/2" ], [ "-27/2", "-6", "7/2" ] ], "collision_image": [ "473/4", "289/4", "-205/2" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0942810069973348 ], "surface_features": [ 3.563261509460975, 4.40862997658628, 3.286456469746983, 0.0, 3.6474157869202055, 4930 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B149", "pair": 149, "family": "bijection", "mapping": "F(x,y,z)=(2187*x^7 + 15309*x^6*y - 10206*x^6*z - 5346*x^6 + 45927*x^5*y^2 - 61236*x^5*y*z - 32076*x^5*y + 20412*x^5*z^2 + 21465*x^5*z + 5751*x^5 + 76545*x^4*y^3 - 153090*x^4*y^2*z - 80190*x^4*y^2 + 102060*x^4*y*z^2 + 107325*x^4*y*z + 28755*x^4*y - 22680*x^4*z^3 - 35910*x^4*z^2 - 19305*x^4*z - 3618*x^4 + 76545*x^3*y^4 - 204120*x^3*y^3*z - 106920*x^3*y^3 + 204120*x^3*y^2*z^2 + 214650*x^3*y^2*z + 57510*x^3*y^2 - 90720*x^3*y*z^3 - 143640*x^3*y*z^2 - 77220*x^3*y*z - 14472*x^3*y + 15120*x^3*z^4 + 32040*x^3*z^3 + 25920*x^3*z^2 + 9747*x^3*z + 1503*x^3 + 45927*x^2*y^5 - 153090*x^2*y^4*z - 80190*x^2*y^4 + 204120*x^2*y^3*z^2 + 214650*x^2*y^3*z + 57510*x^2*y^3 - 136080*x^2*y^2*z^3 - 215460*x^2*y^2*z^2 - 115830*x^2*y^2*z - 21708*x^2*y^2 + 45360*x^2*y*z^4 + 96120*x^2*y*z^3 + 77760*x^2*y*z^2 + 29241*x^2*y*z + 4500*x^2*y - 6048*x^2*z^5 - 16080*x^2*z^4 - 17400*x^2*z^3 - 9846*x^2*z^2 - 3048*x^2*z - 495*x^2 + 15309*x*y^6 - 61236*x*y^5*z - 32076*x*y^5 + 102060*x*y^4*z^2 + 107325*x*y^4*z + 28755*x*y^4 - 90720*x*y^3*z^3 - 143640*x*y^3*z^2 - 77220*x*y^3*z - 14472*x*y^3 + 45360*x*y^2*z^4 + 96120*x*y^2*z^3 + 77760*x*y^2*z^2 + 29241*x*y^2*z + 4491*x*y^2 - 12096*x*y*z^5 - 32160*x*y*z^4 - 34800*x*y*z^3 - 19692*x*y*z^2 - 6084*x*y*z - 983*x*y + 1344*x*z^6 + 4304*x*z^5 + 5840*x*z^4 + 4420*x*z^3 + 2060*x*z^2 + 672*x*z + 130*x + 2187*y^7 - 10206*y^6*z - 5346*y^6 + 20412*y^5*z^2 + 21465*y^5*z + 5751*y^5 - 22680*y^4*z^3 - 35910*y^4*z^2 - 19305*y^4*z - 3618*y^4 + 15120*y^3*z^4 + 32040*y^3*z^3 + 25920*y^3*z^2 + 9747*y^3*z + 1494*y^3 - 6048*y^2*z^5 - 16080*y^2*z^4 - 17400*y^2*z^3 - 9846*y^2*z^2 - 3036*y^2*z - 488*y^2 + 1344*y*z^6 + 4304*y*z^5 + 5840*y*z^4 + 4420*y*z^3 + 2056*y*z^2 + 667*y*z + 129*y - 128*z^7 - 480*z^6 - 784*z^5 - 744*z^4 - 464*z^3 - 228*z^2 - 91*z - 1, 2187*x^7 + 15309*x^6*y - 10206*x^6*z - 5346*x^6 + 45927*x^5*y^2 - 61236*x^5*y*z - 32076*x^5*y + 20412*x^5*z^2 + 21465*x^5*z + 5265*x^5 + 76545*x^4*y^3 - 153090*x^4*y^2*z - 80190*x^4*y^2 + 102060*x^4*y*z^2 + 107325*x^4*y*z + 26325*x^4*y - 22680*x^4*z^3 - 35910*x^4*z^2 - 17685*x^4*z - 2808*x^4 + 76545*x^3*y^4 - 204120*x^3*y^3*z - 106920*x^3*y^3 + 204120*x^3*y^2*z^2 + 214650*x^3*y^2*z + 52650*x^3*y^2 - 90720*x^3*y*z^3 - 143640*x^3*y*z^2 - 70740*x^3*y*z - 11232*x^3*y + 15120*x^3*z^4 + 32040*x^3*z^3 + 23760*x^3*z^2 + 7587*x^3*z + 909*x^3 + 45927*x^2*y^5 - 153090*x^2*y^4*z - 80190*x^2*y^4 + 204120*x^2*y^3*z^2 + 214650*x^2*y^3*z + 52650*x^2*y^3 - 136080*x^2*y^2*z^3 - 215460*x^2*y^2*z^2 - 106110*x^2*y^2*z - 16848*x^2*y^2 + 45360*x^2*y*z^4 + 96120*x^2*y*z^3 + 71280*x^2*y*z^2 + 22761*x^2*y*z + 2718*x^2*y - 6048*x^2*z^5 - 16080*x^2*z^4 - 15960*x^2*z^3 - 7686*x^2*z^2 - 1860*x^2*z - 252*x^2 + 15309*x*y^6 - 61236*x*y^5*z - 32076*x*y^5 + 102060*x*y^4*z^2 + 107325*x*y^4*z + 26325*x*y^4 - 90720*x*y^3*z^3 - 143640*x*y^3*z^2 - 70740*x*y^3*z - 11232*x*y^3 + 45360*x*y^2*z^4 + 96120*x*y^2*z^3 + 71280*x*y^2*z^2 + 22761*x*y^2*z + 2709*x*y^2 - 12096*x*y*z^5 - 32160*x*y*z^4 - 31920*x*y*z^3 - 15372*x*y*z^2 - 3708*x*y*z - 497*x*y + 1344*x*z^6 + 4304*x*z^5 + 5360*x*z^4 + 3460*x*z^3 + 1268*x*z^2 + 348*x*z + 65*x + 2187*y^7 - 10206*y^6*z - 5346*y^6 + 20412*y^5*z^2 + 21465*y^5*z + 5265*y^5 - 22680*y^4*z^3 - 35910*y^4*z^2 - 17685*y^4*z - 2808*y^4 + 15120*y^3*z^4 + 32040*y^3*z^3 + 23760*y^3*z^2 + 7587*y^3*z + 900*y^3 - 6048*y^2*z^5 - 16080*y^2*z^4 - 15960*y^2*z^3 - 7686*y^2*z^2 - 1848*y^2*z - 245*y^2 + 1344*y*z^6 + 4304*y*z^5 + 5360*y*z^4 + 3460*y*z^3 + 1264*y*z^2 + 343*y*z + 66*y - 128*z^7 - 480*z^6 - 720*z^5 - 584*z^4 - 288*z^3 - 120*z^2 - 48*z - 1, -2187*x^7 - 15309*x^6*y + 10206*x^6*z + 5346*x^6 - 45927*x^5*y^2 + 61236*x^5*y*z + 32076*x^5*y - 20412*x^5*z^2 - 21465*x^5*z - 5751*x^5 - 76545*x^4*y^3 + 153090*x^4*y^2*z + 80190*x^4*y^2 - 102060*x^4*y*z^2 - 107325*x^4*y*z - 28755*x^4*y + 22680*x^4*z^3 + 35910*x^4*z^2 + 19305*x^4*z + 3618*x^4 - 76545*x^3*y^4 + 204120*x^3*y^3*z + 106920*x^3*y^3 - 204120*x^3*y^2*z^2 - 214650*x^3*y^2*z - 57510*x^3*y^2 + 90720*x^3*y*z^3 + 143640*x^3*y*z^2 + 77220*x^3*y*z + 14472*x^3*y - 15120*x^3*z^4 - 32040*x^3*z^3 - 25920*x^3*z^2 - 9747*x^3*z - 1503*x^3 - 45927*x^2*y^5 + 153090*x^2*y^4*z + 80190*x^2*y^4 - 204120*x^2*y^3*z^2 - 214650*x^2*y^3*z - 57510*x^2*y^3 + 136080*x^2*y^2*z^3 + 215460*x^2*y^2*z^2 + 115830*x^2*y^2*z + 21708*x^2*y^2 - 45360*x^2*y*z^4 - 96120*x^2*y*z^3 - 77760*x^2*y*z^2 - 29241*x^2*y*z - 4500*x^2*y + 6048*x^2*z^5 + 16080*x^2*z^4 + 17400*x^2*z^3 + 9846*x^2*z^2 + 3048*x^2*z + 486*x^2 - 15309*x*y^6 + 61236*x*y^5*z + 32076*x*y^5 - 102060*x*y^4*z^2 - 107325*x*y^4*z - 28755*x*y^4 + 90720*x*y^3*z^3 + 143640*x*y^3*z^2 + 77220*x*y^3*z + 14472*x*y^3 - 45360*x*y^2*z^4 - 96120*x*y^2*z^3 - 77760*x*y^2*z^2 - 29241*x*y^2*z - 4491*x*y^2 + 12096*x*y*z^5 + 32160*x*y*z^4 + 34800*x*y*z^3 + 19692*x*y*z^2 + 6084*x*y*z + 965*x*y - 1344*x*z^6 - 4304*x*z^5 - 5840*x*z^4 - 4420*x*z^3 - 2060*x*z^2 - 660*x*z - 123*x - 2187*y^7 + 10206*y^6*z + 5346*y^6 - 20412*y^5*z^2 - 21465*y^5*z - 5751*y^5 + 22680*y^4*z^3 + 35910*y^4*z^2 + 19305*y^4*z + 3618*y^4 - 15120*y^3*z^4 - 32040*y^3*z^3 - 25920*y^3*z^2 - 9747*y^3*z - 1494*y^3 + 6048*y^2*z^5 + 16080*y^2*z^4 + 17400*y^2*z^3 + 9846*y^2*z^2 + 3036*y^2*z + 479*y^2 - 1344*y*z^6 - 4304*y*z^5 - 5840*y*z^4 - 4420*y*z^3 - 2056*y*z^2 - 655*y*z - 122*y + 128*z^7 + 480*z^6 + 784*z^5 + 744*z^4 + 464*z^3 + 224*z^2 + 86*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -3, -3, 2 ], [ -1, -1, 1 ], [ -2, -1, 1 ] ], "source_shift": [ 1, 2, 2 ], "target_matrix": [ [ -1, -2, -1 ], [ -1, -1, -2 ], [ 1, 1, 1 ] ], "target_shift": [ 23, 10, -20 ], "inverse_recipe": { "p": 2, "q": 5, "f": "x**2", "g": "x**5 + x**3 + x", "h": "y**2 + y*z + y", "polynomials": { "f": [ [ [ 2, 0, 0 ], 1 ] ], "g": [ [ [ 5, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ], [ [ 3, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 1, 0 ], 1 ], [ [ 0, 2, 0 ], 1 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0920170760000474 ], "surface_features": [ 4.618948715720777, 5.333366655509717, 4.24760506415077, 0.0, 4.736135180996655, 5262 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B217", "pair": 217, "family": "bijection", "mapping": "F(x,y,z)=(22*x^7 - 154*x^6*y - 308*x^6*z + 176*x^6 + 462*x^5*y^2 + 1848*x^5*y*z - 1056*x^5*y + 1848*x^5*z^2 - 2112*x^5*z + 638*x^5 - 770*x^4*y^3 - 4620*x^4*y^2*z + 2640*x^4*y^2 - 9240*x^4*y*z^2 + 10560*x^4*y*z - 3212*x^4*y - 6160*x^4*z^3 + 10560*x^4*z^2 - 6336*x^4*z + 1338*x^4 + 770*x^3*y^4 + 6160*x^3*y^3*z - 3520*x^3*y^3 + 18480*x^3*y^2*z^2 - 21120*x^3*y^2*z + 6468*x^3*y^2 + 24640*x^3*y*z^3 - 42240*x^3*y*z^2 + 25520*x^3*y*z - 5462*x^3*y + 12320*x^3*z^4 - 28160*x^3*z^3 + 25168*x^3*z^2 - 10506*x^3*z + 1747*x^3 - 462*x^2*y^5 - 4620*x^2*y^4*z + 2640*x^2*y^4 - 18480*x^2*y^3*z^2 + 21120*x^2*y^3*z - 6512*x^2*y^3 - 36960*x^2*y^2*z^3 + 63360*x^2*y^2*z^2 - 38544*x^2*y^2*z + 8358*x^2*y^2 - 36960*x^2*y*z^4 + 84480*x^2*y*z^3 - 76032*x^2*y*z^2 + 32178*x^2*y*z - 5461*x^2*y - 14784*x^2*z^5 + 42240*x^2*z^4 - 49984*x^2*z^3 + 30924*x^2*z^2 - 10130*x^2*z + 1462*x^2 + 154*x*y^6 + 1848*x*y^5*z - 1056*x*y^5 + 9240*x*y^4*z^2 - 10560*x*y^4*z + 3278*x*y^4 + 24640*x*y^3*z^3 - 42240*x*y^3*z^2 + 25872*x*y^3*z - 5682*x*y^3 + 36960*x*y^2*z^4 - 84480*x*y^2*z^3 + 76560*x*y^2*z^2 - 32838*x*y^2*z + 5681*x*y^2 + 29568*x*y*z^5 - 84480*x*y*z^4 + 100672*x*y*z^3 - 63168*x*y*z^2 + 21140*x*y*z - 3188*x*y + 9856*x*z^6 - 33792*x*z^5 + 49632*x*z^4 - 40440*x*z^3 + 19556*x*z^2 - 5474*x*z + 751*x - 22*y^7 - 308*y^6*z + 176*y^6 - 1848*y^5*z^2 + 2112*y^5*z - 660*y^5 - 6160*y^4*z^3 + 10560*y^4*z^2 - 6512*y^4*z + 1448*y^4 - 12320*y^3*z^4 + 28160*y^3*z^3 - 25696*y^3*z^2 + 11166*y^3*z - 1967*y^3 - 14784*y^2*z^5 + 42240*y^2*z^4 - 50688*y^2*z^3 + 32244*y^2*z^2 - 11010*y^2*z + 1748*y^2 - 9856*y*z^6 + 33792*y*z^5 - 49984*y*z^4 + 41320*y*z^3 - 20436*y*z^2 + 5936*y*z - 882*y - 2816*z^7 + 11264*z^6 - 19712*z^5 + 19824*z^4 - 12568*z^3 + 5144*z^2 - 1324*z + 1, 11*x^7 - 77*x^6*y - 154*x^6*z + 88*x^6 + 231*x^5*y^2 + 924*x^5*y*z - 528*x^5*y + 924*x^5*z^2 - 1056*x^5*z + 319*x^5 - 385*x^4*y^3 - 2310*x^4*y^2*z + 1320*x^4*y^2 - 4620*x^4*y*z^2 + 5280*x^4*y*z - 1606*x^4*y - 3080*x^4*z^3 + 5280*x^4*z^2 - 3168*x^4*z + 669*x^4 + 385*x^3*y^4 + 3080*x^3*y^3*z - 1760*x^3*y^3 + 9240*x^3*y^2*z^2 - 10560*x^3*y^2*z + 3234*x^3*y^2 + 12320*x^3*y*z^3 - 21120*x^3*y*z^2 + 12760*x^3*y*z - 2731*x^3*y + 6160*x^3*z^4 - 14080*x^3*z^3 + 12584*x^3*z^2 - 5253*x^3*z + 874*x^3 - 231*x^2*y^5 - 2310*x^2*y^4*z + 1320*x^2*y^4 - 9240*x^2*y^3*z^2 + 10560*x^2*y^3*z - 3256*x^2*y^3 - 18480*x^2*y^2*z^3 + 31680*x^2*y^2*z^2 - 19272*x^2*y^2*z + 4179*x^2*y^2 - 18480*x^2*y*z^4 + 42240*x^2*y*z^3 - 38016*x^2*y*z^2 + 16089*x^2*y*z - 2732*x^2*y - 7392*x^2*z^5 + 21120*x^2*z^4 - 24992*x^2*z^3 + 15462*x^2*z^2 - 5068*x^2*z + 733*x^2 + 77*x*y^6 + 924*x*y^5*z - 528*x*y^5 + 4620*x*y^4*z^2 - 5280*x*y^4*z + 1639*x*y^4 + 12320*x*y^3*z^3 - 21120*x*y^3*z^2 + 12936*x*y^3*z - 2841*x*y^3 + 18480*x*y^2*z^4 - 42240*x*y^2*z^3 + 38280*x*y^2*z^2 - 16419*x*y^2*z + 2842*x*y^2 + 14784*x*y*z^5 - 42240*x*y*z^4 + 50336*x*y*z^3 - 31584*x*y*z^2 + 10576*x*y*z - 1598*x*y + 4928*x*z^6 - 16896*x*z^5 + 24816*x*z^4 - 20220*x*z^3 + 9784*x*z^2 - 2745*x*z + 379*x - 11*y^7 - 154*y^6*z + 88*y^6 - 924*y^5*z^2 + 1056*y^5*z - 330*y^5 - 3080*y^4*z^3 + 5280*y^4*z^2 - 3256*y^4*z + 724*y^4 - 6160*y^3*z^4 + 14080*y^3*z^3 - 12848*y^3*z^2 + 5583*y^3*z - 984*y^3 - 7392*y^2*z^5 + 21120*y^2*z^4 - 25344*y^2*z^3 + 16122*y^2*z^2 - 5508*y^2*z + 876*y^2 - 4928*y*z^6 + 16896*y*z^5 - 24992*y*z^4 + 20660*y*z^3 - 10224*y*z^2 + 2976*y*z - 445*y - 1408*z^7 + 5632*z^6 - 9856*z^5 + 9912*z^4 - 6288*z^3 + 2580*z^2 - 668*z + 1, 11*x^7 - 77*x^6*y - 154*x^6*z + 88*x^6 + 231*x^5*y^2 + 924*x^5*y*z - 528*x^5*y + 924*x^5*z^2 - 1056*x^5*z + 319*x^5 - 385*x^4*y^3 - 2310*x^4*y^2*z + 1320*x^4*y^2 - 4620*x^4*y*z^2 + 5280*x^4*y*z - 1606*x^4*y - 3080*x^4*z^3 + 5280*x^4*z^2 - 3168*x^4*z + 667*x^4 + 385*x^3*y^4 + 3080*x^3*y^3*z - 1760*x^3*y^3 + 9240*x^3*y^2*z^2 - 10560*x^3*y^2*z + 3234*x^3*y^2 + 12320*x^3*y*z^3 - 21120*x^3*y*z^2 + 12760*x^3*y*z - 2723*x^3*y + 6160*x^3*z^4 - 14080*x^3*z^3 + 12584*x^3*z^2 - 5237*x^3*z + 865*x^3 - 231*x^2*y^5 - 2310*x^2*y^4*z + 1320*x^2*y^4 - 9240*x^2*y^3*z^2 + 10560*x^2*y^3*z - 3256*x^2*y^3 - 18480*x^2*y^2*z^3 + 31680*x^2*y^2*z^2 - 19272*x^2*y^2*z + 4167*x^2*y^2 - 18480*x^2*y*z^4 + 42240*x^2*y*z^3 - 38016*x^2*y*z^2 + 16041*x^2*y*z - 2705*x^2*y - 7392*x^2*z^5 + 21120*x^2*z^4 - 24992*x^2*z^3 + 15414*x^2*z^2 - 5014*x^2*z + 717*x^2 + 77*x*y^6 + 924*x*y^5*z - 528*x*y^5 + 4620*x*y^4*z^2 - 5280*x*y^4*z + 1639*x*y^4 + 12320*x*y^3*z^3 - 21120*x*y^3*z^2 + 12936*x*y^3*z - 2833*x*y^3 + 18480*x*y^2*z^4 - 42240*x*y^2*z^3 + 38280*x*y^2*z^2 - 16371*x*y^2*z + 2815*x*y^2 + 14784*x*y*z^5 - 42240*x*y*z^4 + 50336*x*y*z^3 - 31488*x*y*z^2 + 10468*x*y*z - 1566*x*y + 4928*x*z^6 - 16896*x*z^5 + 24816*x*z^4 - 20156*x*z^3 + 9676*x*z^2 - 2681*x*z + 362*x - 11*y^7 - 154*y^6*z + 88*y^6 - 924*y^5*z^2 + 1056*y^5*z - 330*y^5 - 3080*y^4*z^3 + 5280*y^4*z^2 - 3256*y^4*z + 722*y^4 - 6160*y^3*z^4 + 14080*y^3*z^3 - 12848*y^3*z^2 + 5567*y^3*z - 975*y^3 - 7392*y^2*z^5 + 21120*y^2*z^4 - 25344*y^2*z^3 + 16074*y^2*z^2 - 5454*y^2*z + 860*y^2 - 4928*y*z^6 + 16896*y*z^5 - 24992*y*z^4 + 20596*y*z^3 - 10116*y*z^2 + 2912*y*z - 425*y - 1408*z^7 + 5632*z^6 - 9856*z^5 + 9880*z^4 - 6216*z^3 + 2516*z^2 - 638*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ 1, -1, -2 ], [ 2, -3, -2 ], [ 1, -2, -1 ] ], "source_shift": [ 1, 1, 1 ], "target_matrix": [ [ 2, 3, 2 ], [ 1, 2, 1 ], [ 1, 1, 2 ] ], "target_shift": [ -178, -90, -83 ], "inverse_recipe": { "p": 3, "q": 4, "f": "x**3 + x**2", "g": "x**4", "h": "y*z + z", "polynomials": { "f": [ [ [ 3, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "g": [ [ [ 4, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 0, 1 ], 1 ] ] }, "h_multiplier": 11 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0517072989969165 ], "surface_features": [ 4.087327509323577, 5.002908696607174, 3.746244871720198, 0.0, 4.2014748730893094, 5092 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B200", "pair": 200, "family": "bijection", "mapping": "F(x,y,z)=(-2*x^7 + 14*x^6*y - 28*x^6*z + 34*x^6 - 42*x^5*y^2 + 168*x^5*y*z - 202*x^5*y - 168*x^5*z^2 + 402*x^5*z - 240*x^5 + 70*x^4*y^3 - 420*x^4*y^2*z + 500*x^4*y^2 + 840*x^4*y*z^2 - 1990*x^4*y*z + 1178*x^4*y - 560*x^4*z^3 + 1980*x^4*z^2 - 2334*x^4*z + 920*x^4 - 70*x^3*y^4 + 560*x^3*y^3*z - 660*x^3*y^3 - 1680*x^3*y^2*z^2 + 3940*x^3*y^2*z - 2312*x^3*y^2 + 2240*x^3*y*z^3 - 7840*x^3*y*z^2 + 9160*x^3*y*z - 3582*x^3*y - 1120*x^3*z^4 + 5200*x^3*z^3 - 9072*x^3*z^2 + 7066*x^3*z - 2080*x^3 + 42*x^2*y^5 - 420*x^2*y^4*z + 490*x^2*y^4 + 1680*x^2*y^3*z^2 - 3900*x^2*y^3*z + 2268*x^2*y^3 - 3360*x^2*y^2*z^3 + 11640*x^2*y^2*z^2 - 13476*x^2*y^2*z + 5226*x^2*y^2 + 3360*x^2*y*z^4 - 15440*x^2*y*z^3 + 26688*x^2*y*z^2 - 20610*x^2*y*z + 6018*x^2*y - 1344*x^2*z^5 + 7680*x^2*z^4 - 17616*x^2*z^3 + 20316*x^2*z^2 - 11812*x^2*z + 2784*x^2 - 14*x*y^6 + 168*x*y^5*z - 194*x*y^5 - 840*x*y^4*z^2 + 1930*x*y^4*z - 1112*x*y^4 + 2240*x*y^3*z^3 - 7680*x*y^3*z^2 + 8808*x*y^3*z - 3386*x*y^3 - 3360*x*y^2*z^4 + 15280*x*y^2*z^3 - 26160*x*y^2*z^2 + 20022*x*y^2*z - 5796*x*y^2 + 2688*x*y*z^5 - 15200*x*y*z^4 + 34528*x*y*z^3 - 39456*x*y*z^2 + 22736*x*y*z - 5310*x*y - 896*x*z^6 + 6048*x*z^5 - 17088*x*z^4 + 25912*x*z^3 - 22288*x*z^2 + 10350*x*z - 2047*x + 2*y^7 - 28*y^6*z + 32*y^6 + 168*y^5*z^2 - 382*y^5*z + 218*y^5 - 560*y^4*z^3 + 1900*y^4*z^2 - 2158*y^4*z + 822*y^4 + 1120*y^3*z^4 - 5040*y^3*z^3 + 8544*y^3*z^2 - 6478*y^3*z + 1858*y^3 - 1344*y^2*z^5 + 7520*y^2*z^4 - 16912*y^2*z^3 + 19140*y^2*z^2 - 10924*y^2*z + 2526*y^2 + 896*y*z^6 - 5984*y*z^5 + 16736*y*z^4 - 25128*y*z^3 + 21400*y*z^2 - 9836*y*z + 1921*y - 256*z^7 + 1984*z^6 - 6624*z^5 + 12368*z^4 - 13968*z^3 + 9570*z^2 - 3706*z + 1, 2*x^7 - 14*x^6*y + 28*x^6*z - 34*x^6 + 42*x^5*y^2 - 168*x^5*y*z + 202*x^5*y + 168*x^5*z^2 - 402*x^5*z + 238*x^5 - 70*x^4*y^3 + 420*x^4*y^2*z - 500*x^4*y^2 - 840*x^4*y*z^2 + 1990*x^4*y*z - 1168*x^4*y + 560*x^4*z^3 - 1980*x^4*z^2 + 2314*x^4*z - 898*x^4 + 70*x^3*y^4 - 560*x^3*y^3*z + 660*x^3*y^3 + 1680*x^3*y^2*z^2 - 3940*x^3*y^2*z + 2292*x^3*y^2 - 2240*x^3*y*z^3 + 7840*x^3*y*z^2 - 9080*x^3*y*z + 3494*x^3*y + 1120*x^3*z^4 - 5200*x^3*z^3 + 8992*x^3*z^2 - 6890*x^3*z + 1982*x^3 - 42*x^2*y^5 + 420*x^2*y^4*z - 490*x^2*y^4 - 1680*x^2*y^3*z^2 + 3900*x^2*y^3*z - 2248*x^2*y^3 + 3360*x^2*y^2*z^3 - 11640*x^2*y^2*z^2 + 13356*x^2*y^2*z - 5094*x^2*y^2 - 3360*x^2*y*z^4 + 15440*x^2*y*z^3 - 26448*x^2*y*z^2 + 20082*x^2*y*z - 5724*x^2*y + 1344*x^2*z^5 - 7680*x^2*z^4 + 17456*x^2*z^3 - 19788*x^2*z^2 + 11224*x^2*z - 2561*x^2 + 14*x*y^6 - 168*x*y^5*z + 194*x*y^5 + 840*x*y^4*z^2 - 1930*x*y^4*z + 1102*x*y^4 - 2240*x*y^3*z^3 + 7680*x*y^3*z^2 - 8728*x*y^3*z + 3298*x*y^3 + 3360*x*y^2*z^4 - 15280*x*y^2*z^3 + 25920*x*y^2*z^2 - 19494*x*y^2*z + 5502*x*y^2 - 2688*x*y*z^5 + 15200*x*y*z^4 - 34208*x*y*z^3 + 38400*x*y*z^2 - 21560*x*y*z + 4864*x*y + 896*x*z^6 - 6048*x*z^5 + 16928*x*z^4 - 25208*x*z^3 + 21112*x*z^2 - 9458*x*z + 1783*x - 2*y^7 + 28*y^6*z - 32*y^6 - 168*y^5*z^2 + 382*y^5*z - 216*y^5 + 560*y^4*z^3 - 1900*y^4*z^2 + 2138*y^4*z - 800*y^4 - 1120*y^3*z^4 + 5040*y^3*z^3 - 8464*y^3*z^2 + 6302*y^3*z - 1760*y^3 + 1344*y^2*z^5 - 7520*y^2*z^4 + 16752*y^2*z^3 - 18612*y^2*z^2 + 10336*y^2*z - 2303*y^2 - 896*y*z^6 + 5984*y*z^5 - 16576*y*z^4 + 24424*y*z^3 - 20224*y*z^2 + 8944*y*z - 1658*y + 256*z^7 - 1984*z^6 + 6560*z^5 - 12016*z^4 + 13184*z^3 - 8678*z^2 + 3183*z - 1, 2*x^7 - 14*x^6*y + 28*x^6*z - 34*x^6 + 42*x^5*y^2 - 168*x^5*y*z + 202*x^5*y + 168*x^5*z^2 - 402*x^5*z + 240*x^5 - 70*x^4*y^3 + 420*x^4*y^2*z - 500*x^4*y^2 - 840*x^4*y*z^2 + 1990*x^4*y*z - 1178*x^4*y + 560*x^4*z^3 - 1980*x^4*z^2 + 2334*x^4*z - 920*x^4 + 70*x^3*y^4 - 560*x^3*y^3*z + 660*x^3*y^3 + 1680*x^3*y^2*z^2 - 3940*x^3*y^2*z + 2312*x^3*y^2 - 2240*x^3*y*z^3 + 7840*x^3*y*z^2 - 9160*x^3*y*z + 3582*x^3*y + 1120*x^3*z^4 - 5200*x^3*z^3 + 9072*x^3*z^2 - 7066*x^3*z + 2080*x^3 - 42*x^2*y^5 + 420*x^2*y^4*z - 490*x^2*y^4 - 1680*x^2*y^3*z^2 + 3900*x^2*y^3*z - 2268*x^2*y^3 + 3360*x^2*y^2*z^3 - 11640*x^2*y^2*z^2 + 13476*x^2*y^2*z - 5226*x^2*y^2 - 3360*x^2*y*z^4 + 15440*x^2*y*z^3 - 26688*x^2*y*z^2 + 20610*x^2*y*z - 6018*x^2*y + 1344*x^2*z^5 - 7680*x^2*z^4 + 17616*x^2*z^3 - 20316*x^2*z^2 + 11812*x^2*z - 2783*x^2 + 14*x*y^6 - 168*x*y^5*z + 194*x*y^5 + 840*x*y^4*z^2 - 1930*x*y^4*z + 1112*x*y^4 - 2240*x*y^3*z^3 + 7680*x*y^3*z^2 - 8808*x*y^3*z + 3386*x*y^3 + 3360*x*y^2*z^4 - 15280*x*y^2*z^3 + 26160*x*y^2*z^2 - 20022*x*y^2*z + 5796*x*y^2 - 2688*x*y*z^5 + 15200*x*y*z^4 - 34528*x*y*z^3 + 39456*x*y*z^2 - 22736*x*y*z + 5308*x*y + 896*x*z^6 - 6048*x*z^5 + 17088*x*z^4 - 25912*x*z^3 + 22288*x*z^2 - 10346*x*z + 2041*x - 2*y^7 + 28*y^6*z - 32*y^6 - 168*y^5*z^2 + 382*y^5*z - 218*y^5 + 560*y^4*z^3 - 1900*y^4*z^2 + 2158*y^4*z - 822*y^4 - 1120*y^3*z^4 + 5040*y^3*z^3 - 8544*y^3*z^2 + 6478*y^3*z - 1858*y^3 + 1344*y^2*z^5 - 7520*y^2*z^4 + 16912*y^2*z^3 - 19140*y^2*z^2 + 10924*y^2*z - 2525*y^2 - 896*y*z^6 + 5984*y*z^5 - 16736*y*z^4 + 25128*y*z^3 - 21400*y*z^2 + 9832*y*z - 1916*y + 256*z^7 - 1984*z^6 + 6624*z^5 - 12368*z^4 + 13968*z^3 - 9566*z^2 + 3697*z - 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -1, 1, -2 ], [ -2, 1, -1 ], [ -1, 1, -1 ] ], "source_shift": [ 2, 2, 2 ], "target_matrix": [ [ 1, 2, 1 ], [ -1, -1, -2 ], [ -1, -1, -1 ] ], "target_shift": [ -633, 503, 627 ], "inverse_recipe": { "p": 2, "q": 5, "f": "x**2", "g": "x**5 + x**4 + x**3 + x**2", "h": "y*z", "polynomials": { "f": [ [ [ 2, 0, 0 ], 1 ] ], "g": [ [ [ 5, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ], [ [ 3, 0, 0 ], 1 ], [ [ 4, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ] ] }, "h_multiplier": 2 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0816922259982675 ], "surface_features": [ 3.800621435923593, 4.596113054915637, 3.3639878297484915, 0.0, 3.905782591174688, 4965 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C039", "pair": 39, "family": "counterexample", "mapping": "F(x,y,z)=(128*x^7 - 704*x^6*y + 1024*x^6*z - 1312*x^6 + 1632*x^5*y^2 - 4704*x^5*y*z + 6192*x^5*y + 3360*x^5*z^2 - 8784*x^5*z + 5040*x^5 - 2064*x^4*y^3 + 8832*x^4*y^2*z - 11976*x^4*y^2 - 12480*x^4*y*z^2 + 33648*x^4*y*z - 19632*x^4*y + 5824*x^4*z^3 - 23472*x^4*z^2 + 27408*x^4*z - 10080*x^4 + 1536*x^3*y^4 - 8664*x^3*y^3*z + 12132*x^3*y^3 + 18144*x^3*y^2*z^2 - 50580*x^3*y^2*z + 30036*x^3*y^2 - 16720*x^3*y*z^3 + 69768*x^3*y*z^2 - 82992*x^3*y*z + 31016*x^3*y + 5720*x^3*z^4 - 31832*x^3*z^3 + 56952*x^3*z^2 - 42768*x^3*z + 11712*x^3 - 672*x^2*y^5 + 4680*x^2*y^4*z - 6780*x^2*y^4 - 12900*x^2*y^3*z^2 + 37248*x^2*y^3*z - 22524*x^2*y^3 + 17592*x^2*y^2*z^3 - 76122*x^2*y^2*z^2 + 92280*x^2*y^2*z - 35070*x^2*y^2 - 11868*x^2*y*z^4 + 68568*x^2*y*z^3 - 125112*x^2*y*z^2 + 95616*x^2*y*z - 26664*x^2*y + 3168*x^2*z^5 - 22962*x^2*z^4 + 56112*x^2*z^3 - 64728*x^2*z^2 + 36384*x^2*z - 8072*x^2 + 160*x*y^6 - 1320*x*y^5*z + 1980*x*y^5 + 4488*x*y^4*z^2 - 13428*x*y^4*z + 8268*x*y^4 - 8050*x*y^3*z^3 + 36117*x*y^3*z^2 - 44604*x*y^3*z + 17241*x*y^3 + 8034*x*y^2*z^4 - 48147*x*y^2*z^3 + 89529*x*y^2*z^2 - 69627*x*y^2*z + 19779*x*y^2 - 4230*x*y*z^5 + 31803*x*y*z^4 - 79212*x*y*z^3 + 93018*x*y*z^2 - 53292*x*y*z + 12076*x*y + 918*x*z^6 - 8325*x*z^5 + 26055*x*z^4 - 41088*x*z^3 + 35628*x*z^2 - 16320*x*z + 3104*x - 16*y^7 + 152*y^6*z - 236*y^6 - 612*y^5*z^2 + 1896*y^5*z - 1188*y^5 + 1354*y^4*z^3 - 6291*y^4*z^2 + 7908*y^4*z - 3107*y^4 - 1778*y^3*z^4 + 11033*y^3*z^3 - 20883*y^3*z^2 + 16509*y^3*z - 4773*y^3 + 1386*y^2*z^5 - 10785*y^2*z^4 + 27339*y^2*z^3 - 32628*y^2*z^2 + 19023*y^2*z - 4400*y^2 - 594*y*z^6 + 5571*y*z^5 - 17739*y*z^4 + 28416*y*z^3 - 25062*y*z^2 + 11718*y*z - 2282*y + 108*z^7 - 1188*z^6 + 4563*z^5 - 9198*z^4 + 10908*z^3 - 7734*z^2 + 3052*z - 1, -64*x^7 + 352*x^6*y - 512*x^6*z + 704*x^6 - 816*x^5*y^2 + 2352*x^5*y*z - 3312*x^5*y - 1680*x^5*z^2 + 4704*x^5*z - 2928*x^5 + 1032*x^4*y^3 - 4416*x^4*y^2*z + 6384*x^4*y^2 + 6240*x^4*y*z^2 - 17952*x^4*y*z + 11352*x^4*y - 2912*x^4*z^3 + 12528*x^4*z^2 - 15840*x^4*z + 6200*x^4 - 768*x^3*y^4 + 4332*x^3*y^3*z - 6444*x^3*y^3 - 9072*x^3*y^2*z^2 + 26880*x^3*y^2*z - 17280*x^3*y^2 + 8360*x^3*y*z^3 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889*y^3*z^4 - 5788*y^3*z^3 + 11760*y^3*z^2 - 9845*y^3*z + 2975*y^3 - 693*y^2*z^5 + 5652*y^2*z^4 - 15354*y^2*z^3 + 19386*y^2*z^2 - 11808*y^2*z + 2824*y^2 + 297*y*z^6 - 2916*y*z^5 + 9933*y*z^4 - 16816*y*z^3 + 15486*y*z^2 - 7485*y*z + 1495*y - 54*z^7 + 621*z^6 - 2547*z^5 + 5420*z^4 - 6707*z^3 + 4914*z^2 - 1988*z + 1, 64*x^7 - 352*x^6*y + 512*x^6*z - 704*x^6 + 816*x^5*y^2 - 2352*x^5*y*z + 3312*x^5*y + 1680*x^5*z^2 - 4704*x^5*z + 2928*x^5 - 1032*x^4*y^3 + 4416*x^4*y^2*z - 6384*x^4*y^2 - 6240*x^4*y*z^2 + 17952*x^4*y*z - 11352*x^4*y + 2912*x^4*z^3 - 12528*x^4*z^2 + 15840*x^4*z - 6192*x^4 + 768*x^3*y^4 - 4332*x^3*y^3*z + 6444*x^3*y^3 + 9072*x^3*y^2*z^2 - 26880*x^3*y^2*z + 17280*x^3*y^2 - 8360*x^3*y*z^3 + 37080*x^3*y*z^2 - 47700*x^3*y*z + 18940*x^3*y + 2860*x^3*z^4 - 16912*x^3*z^3 + 32688*x^3*z^2 - 26064*x^3*z + 7428*x^3 - 336*x^2*y^5 + 2340*x^2*y^4*z - 3588*x^2*y^4 - 6450*x^2*y^3*z^2 + 19716*x^2*y^3*z - 12888*x^2*y^3 + 8796*x^2*y^2*z^3 - 40284*x^2*y^2*z^2 + 52728*x^2*y^2*z - 21276*x^2*y^2 - 5934*x^2*y*z^4 + 36264*x^2*y*z^3 - 71358*x^2*y*z^2 + 57864*x^2*y*z - 16806*x^2*y + 1584*x^2*z^5 - 12132*x^2*z^4 + 31932*x^2*z^3 - 39060*x^2*z^2 + 22872*x^2*z - 5200*x^2 + 80*x*y^6 - 660*x*y^5*z + 1044*x*y^5 + 2244*x*y^4*z^2 - 7080*x*y^4*z + 4704*x*y^4 - 4025*x*y^3*z^3 + 19035*x*y^3*z^2 - 25332*x*y^3*z + 10386*x*y^3 + 4017*x*y^2*z^4 - 25356*x*y^2*z^3 + 50736*x*y^2*z^2 - 41820*x*y^2*z + 12378*x*y^2 - 2115*x*y*z^5 + 16731*x*y*z^4 - 44775*x*y*z^3 + 55683*x*y*z^2 - 33243*x*y*z + 7733*x*y + 459*x*z^6 - 4374*x*z^5 + 14685*x*z^4 - 24504*x*z^3 + 22143*x*z^2 - 10422*x*z + 2018*x - 8*y^7 + 76*y^6*z - 124*y^6 - 306*y^5*z^2 + 996*y^5*z - 672*y^5 + 677*y^4*z^3 - 3303*y^4*z^2 + 4464*y^4*z - 1858*y^4 - 889*y^3*z^4 + 5788*y^3*z^3 - 11760*y^3*z^2 + 9840*y^3*z - 2964*y^3 + 693*y^2*z^5 - 5652*y^2*z^4 + 15354*y^2*z^3 - 19377*y^2*z^2 + 11769*y^2*z - 2797*y^2 - 297*y*z^6 + 2916*y*z^5 - 9933*y*z^4 + 16809*y*z^3 - 15441*y*z^2 + 7422*y*z - 1472*y + 54*z^7 - 621*z^6 + 2547*z^5 - 5418*z^4 + 6690*z^3 - 4878*z^2 + 1961*z - 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -2, 1, -1 ], [ -2, 2, -3 ], [ -1, 1, -2 ] ], "source_shift": [ 1, 2, 2 ], "target_matrix": [ [ -2, -1, -1 ], [ 1, 1, 1 ], [ -1, -1, -2 ] ], "target_shift": [ 517, -343, 337 ], "collision_preimages": [ [ "5/4", "4", "5/2" ], [ "-8", "-57/2", "-25/2" ], [ "-3", "-35/2", "-19/2" ] ], "collision_image": [ "1035/2", "-1373/4", "1349/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.063323300997581 ], "surface_features": [ 4.182388098722474, 5.0972989666386255, 3.874046725512229, 0.0, 4.288702244984499, 5127 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B207", "pair": 207, "family": "bijection", "mapping": "F(x,y,z)=(-14*x^7 - 98*x^6*y - 98*x^6*z - 112*x^6 - 294*x^5*y^2 - 588*x^5*y*z - 672*x^5*y - 294*x^5*z^2 - 672*x^5*z - 420*x^5 - 490*x^4*y^3 - 1470*x^4*y^2*z - 1680*x^4*y^2 - 1470*x^4*y*z^2 - 3360*x^4*y*z - 2086*x^4*y - 490*x^4*z^3 - 1680*x^4*z^2 - 2086*x^4*z - 950*x^4 - 490*x^3*y^4 - 1960*x^3*y^3*z - 2240*x^3*y^3 - 2940*x^3*y^2*z^2 - 6720*x^3*y^2*z - 4144*x^3*y^2 - 1960*x^3*y*z^3 - 6720*x^3*y*z^2 - 8288*x^3*y*z - 3744*x^3*y - 490*x^3*z^4 - 2240*x^3*z^3 - 4144*x^3*z^2 - 3730*x^3*z - 1379*x^3 - 294*x^2*y^5 - 1470*x^2*y^4*z - 1680*x^2*y^4 - 2940*x^2*y^3*z^2 - 6720*x^2*y^3*z - 4116*x^2*y^3 - 2940*x^2*y^2*z^3 - 10080*x^2*y^2*z^2 - 12348*x^2*y^2*z - 5532*x^2*y^2 - 1470*x^2*y*z^4 - 6720*x^2*y*z^3 - 12348*x^2*y*z^2 - 11022*x^2*y*z - 4053*x^2*y - 294*x^2*z^5 - 1680*x^2*z^4 - 4116*x^2*z^3 - 5490*x^2*z^2 - 3997*x^2*z - 1392*x^2 - 98*x*y^6 - 588*x*y^5*z - 672*x*y^5 - 1470*x*y^4*z^2 - 3360*x*y^4*z - 2044*x*y^4 - 1960*x*y^3*z^3 - 6720*x*y^3*z^2 - 8176*x*y^3*z - 3632*x*y^3 - 1470*x*y^2*z^4 - 6720*x*y^2*z^3 - 12264*x*y^2*z^2 - 10854*x*y^2*z - 3969*x*y^2 - 588*x*y*z^5 - 3360*x*y*z^4 - 8176*x*y*z^3 - 10812*x*y*z^2 - 7826*x*y*z - 2672*x*y - 98*x*z^6 - 672*x*z^5 - 2044*x*z^4 - 3590*x*z^3 - 3857*x*z^2 - 2560*x*z - 778*x - 14*y^7 - 98*y^6*z - 112*y^6 - 294*y^5*z^2 - 672*y^5*z - 406*y^5 - 490*y^4*z^3 - 1680*y^4*z^2 - 2030*y^4*z - 894*y^4 - 490*y^3*z^4 - 2240*y^3*z^3 - 4060*y^3*z^2 - 3562*y^3*z - 1295*y^3 - 294*y^2*z^5 - 1680*y^2*z^4 - 4060*y^2*z^3 - 5322*y^2*z^2 - 3829*y^2*z - 1280*y^2 - 98*y*z^6 - 672*y*z^5 - 2030*y*z^4 - 3534*y*z^3 - 3773*y*z^2 - 2462*y*z - 735*y - 14*z^7 - 112*z^6 - 406*z^5 - 880*z^4 - 1239*z^3 - 1182*z^2 - 681*z - 1, 7*x^7 + 49*x^6*y + 49*x^6*z + 56*x^6 + 147*x^5*y^2 + 294*x^5*y*z + 336*x^5*y + 147*x^5*z^2 + 336*x^5*z + 210*x^5 + 245*x^4*y^3 + 735*x^4*y^2*z + 840*x^4*y^2 + 735*x^4*y*z^2 + 1680*x^4*y*z + 1043*x^4*y + 245*x^4*z^3 + 840*x^4*z^2 + 1043*x^4*z + 474*x^4 + 245*x^3*y^4 + 980*x^3*y^3*z + 1120*x^3*y^3 + 1470*x^3*y^2*z^2 + 3360*x^3*y^2*z + 2072*x^3*y^2 + 980*x^3*y*z^3 + 3360*x^3*y*z^2 + 4144*x^3*y*z + 1868*x^3*y + 245*x^3*z^4 + 1120*x^3*z^3 + 2072*x^3*z^2 + 1861*x^3*z + 686*x^3 + 147*x^2*y^5 + 735*x^2*y^4*z + 840*x^2*y^4 + 1470*x^2*y^3*z^2 + 3360*x^2*y^3*z + 2058*x^2*y^3 + 1470*x^2*y^2*z^3 + 5040*x^2*y^2*z^2 + 6174*x^2*y^2*z + 2760*x^2*y^2 + 735*x^2*y*z^4 + 3360*x^2*y*z^3 + 6174*x^2*y*z^2 + 5499*x^2*y*z + 2016*x^2*y + 147*x^2*z^5 + 840*x^2*z^4 + 2058*x^2*z^3 + 2739*x^2*z^2 + 1988*x^2*z + 692*x^2 + 49*x*y^6 + 294*x*y^5*z + 336*x*y^5 + 735*x*y^4*z^2 + 1680*x*y^4*z + 1022*x*y^4 + 980*x*y^3*z^3 + 3360*x*y^3*z^2 + 4088*x*y^3*z + 1812*x*y^3 + 735*x*y^2*z^4 + 3360*x*y^2*z^3 + 6132*x*y^2*z^2 + 5415*x*y^2*z + 1974*x*y^2 + 294*x*y*z^5 + 1680*x*y*z^4 + 4088*x*y*z^3 + 5394*x*y*z^2 + 3892*x*y*z + 1328*x*y + 49*x*z^6 + 336*x*z^5 + 1022*x*z^4 + 1791*x*z^3 + 1918*x*z^2 + 1272*x*z + 385*x + 7*y^7 + 49*y^6*z + 56*y^6 + 147*y^5*z^2 + 336*y^5*z + 203*y^5 + 245*y^4*z^3 + 840*y^4*z^2 + 1015*y^4*z + 446*y^4 + 245*y^3*z^4 + 1120*y^3*z^3 + 2030*y^3*z^2 + 1777*y^3*z + 644*y^3 + 147*y^2*z^5 + 840*y^2*z^4 + 2030*y^2*z^3 + 2655*y^2*z^2 + 1904*y^2*z + 636*y^2 + 49*y*z^6 + 336*y*z^5 + 1015*y*z^4 + 1763*y*z^3 + 1876*y*z^2 + 1223*y*z + 363*y + 7*z^7 + 56*z^6 + 203*z^5 + 439*z^4 + 616*z^3 + 587*z^2 + 337*z + 1, -7*x^7 - 49*x^6*y - 49*x^6*z - 56*x^6 - 147*x^5*y^2 - 294*x^5*y*z - 336*x^5*y - 147*x^5*z^2 - 336*x^5*z - 210*x^5 - 245*x^4*y^3 - 735*x^4*y^2*z - 840*x^4*y^2 - 735*x^4*y*z^2 - 1680*x^4*y*z - 1043*x^4*y - 245*x^4*z^3 - 840*x^4*z^2 - 1043*x^4*z - 472*x^4 - 245*x^3*y^4 - 980*x^3*y^3*z - 1120*x^3*y^3 - 1470*x^3*y^2*z^2 - 3360*x^3*y^2*z - 2072*x^3*y^2 - 980*x^3*y*z^3 - 3360*x^3*y*z^2 - 4144*x^3*y*z - 1860*x^3*y - 245*x^3*z^4 - 1120*x^3*z^3 - 2072*x^3*z^2 - 1853*x^3*z - 678*x^3 - 147*x^2*y^5 - 735*x^2*y^4*z - 840*x^2*y^4 - 1470*x^2*y^3*z^2 - 3360*x^2*y^3*z - 2058*x^2*y^3 - 1470*x^2*y^2*z^3 - 5040*x^2*y^2*z^2 - 6174*x^2*y^2*z - 2748*x^2*y^2 - 735*x^2*y*z^4 - 3360*x^2*y*z^3 - 6174*x^2*y*z^2 - 5475*x^2*y*z - 1992*x^2*y - 147*x^2*z^5 - 840*x^2*z^4 - 2058*x^2*z^3 - 2727*x^2*z^2 - 1964*x^2*z - 680*x^2 - 49*x*y^6 - 294*x*y^5*z - 336*x*y^5 - 735*x*y^4*z^2 - 1680*x*y^4*z - 1022*x*y^4 - 980*x*y^3*z^3 - 3360*x*y^3*z^2 - 4088*x*y^3*z - 1804*x*y^3 - 735*x*y^2*z^4 - 3360*x*y^2*z^3 - 6132*x*y^2*z^2 - 5391*x*y^2*z - 1950*x*y^2 - 294*x*y*z^5 - 1680*x*y*z^4 - 4088*x*y*z^3 - 5370*x*y*z^2 - 3844*x*y*z - 1304*x*y - 49*x*z^6 - 336*x*z^5 - 1022*x*z^4 - 1783*x*z^3 - 1894*x*z^2 - 1248*x*z - 369*x - 7*y^7 - 49*y^6*z - 56*y^6 - 147*y^5*z^2 - 336*y^5*z - 203*y^5 - 245*y^4*z^3 - 840*y^4*z^2 - 1015*y^4*z - 444*y^4 - 245*y^3*z^4 - 1120*y^3*z^3 - 2030*y^3*z^2 - 1769*y^3*z - 636*y^3 - 147*y^2*z^5 - 840*y^2*z^4 - 2030*y^2*z^3 - 2643*y^2*z^2 - 1880*y^2*z - 624*y^2 - 49*y*z^6 - 336*y*z^5 - 1015*y*z^4 - 1755*y*z^3 - 1852*y*z^2 - 1199*y*z - 347*y - 7*z^7 - 56*z^6 - 203*z^5 - 437*z^4 - 608*z^3 - 575*z^2 - 323*z - 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ 1, 1, 1 ], [ 2, 1, 1 ], [ 3, 3, 2 ] ], "source_shift": [ 1, 1, 1 ], "target_matrix": [ [ -2, -1, -1 ], [ 1, 1, 1 ], [ -1, -1, -2 ] ], "target_shift": [ 167, -82, 76 ], "inverse_recipe": { "p": 3, "q": 4, "f": "x**3 + x**2 + x", "g": "x**4 + x", "h": "y*z", "polynomials": { "f": [ [ [ 3, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "g": [ [ [ 4, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ] ] }, "h_multiplier": 7 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.002667985001608 ], "surface_features": [ 3.268574071950804, 4.091596620810058, 3.018284308426531, 0.0, 3.342539230220441, 4839 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C143", "pair": 143, "family": "counterexample", "mapping": "F(x,y,z)=(-8*x^7 + 52*x^6*y + 44*x^6*z - 140*x^6 - 138*x^5*y^2 - 240*x^5*y*z + 708*x^5*y - 102*x^5*z^2 + 624*x^5*z - 912*x^5 + 195*x^4*y^3 + 519*x^4*y^2*z - 1455*x^4*y^2 + 453*x^4*y*z^2 - 2598*x^4*y*z + 3648*x^4*y + 129*x^4*z^3 - 1143*x^4*z^2 + 3276*x^4*z - 3067*x^4 - 159*x^3*y^4 - 573*x^3*y^3*z + 1554*x^3*y^3 - 765*x^3*y^2*z^2 + 4209*x^3*y^2*z - 5736*x^3*y^2 - 447*x^3*y*z^3 + 3756*x^3*y*z^2 - 10380*x^3*y*z + 9465*x^3*y - 96*x^3*z^4 + 1101*x^3*z^3 - 4644*x^3*z^2 + 8578*x^3*z - 5889*x^3 + 75*x^2*y^5 + 342*x^2*y^4*z - 909*x^2*y^4 + 618*x^2*y^3*z^2 - 3315*x^2*y^3*z + 4422*x^2*y^3 + 552*x^2*y^2*z^3 - 4491*x^2*y^2*z^2 + 12093*x^2*y^2*z - 10800*x^2*y^2 + 243*x^2*y*z^4 - 2673*x^2*y*z^3 + 10920*x^2*y*z^2 - 19677*x^2*y*z + 13245*x^2*y + 42*x^2*z^5 - 588*x^2*z^4 + 3249*x^2*z^3 - 8877*x^2*z^2 + 12045*x^2*z - 6523*x^2 - 19*x*y^6 - 105*x*y^5*z + 276*x*y^5 - 240*x*y^4*z^2 + 1269*x*y^4*z - 1668*x*y^4 - 290*x*y^3*z^3 + 2316*x*y^3*z^2 - 6126*x*y^3*z + 5386*x*y^3 - 195*x*y^2*z^4 + 2094*x*y^2*z^3 - 8370*x*y^2*z^2 + 14805*x*y^2*z - 9810*x*y^2 - 69*x*y*z^5 + 936*x*y*z^4 - 5034*x*y*z^3 + 13452*x*y*z^2 - 17919*x*y*z + 9550*x*y - 10*x*z^6 + 165*x*z^5 - 1122*x*z^4 + 4033*x*z^3 - 8109*x*z^2 + 8677*x*z - 3872*x + 2*y^7 + 13*y^6*z - 34*y^6 + 36*y^5*z^2 - 189*y^5*z + 246*y^5 + 55*y^4*z^3 - 435*y^4*z^2 + 1137*y^4*z - 988*y^4 + 50*y^3*z^4 - 530*y^3*z^3 + 2088*y^3*z^2 - 3643*y^3*z + 2384*y^3 + 27*y^2*z^5 - 360*y^2*z^4 + 1902*y^2*z^3 - 5001*y^2*z^2 + 6567*y^2*z - 3457*y^2 + 8*y*z^6 - 129*y*z^5 + 858*y*z^4 - 3025*y*z^3 + 5982*y*z^2 - 6311*y*z + 2784*y + z^7 - 19*z^6 + 153*z^5 - 679*z^4 + 1799*z^3 - 2854*z^2 + 2517*z - 1, -8*x^7 + 52*x^6*y + 44*x^6*z - 140*x^6 - 138*x^5*y^2 - 240*x^5*y*z + 708*x^5*y - 102*x^5*z^2 + 624*x^5*z - 912*x^5 + 195*x^4*y^3 + 519*x^4*y^2*z - 1455*x^4*y^2 + 453*x^4*y*z^2 - 2598*x^4*y*z + 3648*x^4*y + 129*x^4*z^3 - 1143*x^4*z^2 + 3276*x^4*z - 3066*x^4 - 159*x^3*y^4 - 573*x^3*y^3*z + 1554*x^3*y^3 - 765*x^3*y^2*z^2 + 4209*x^3*y^2*z - 5736*x^3*y^2 - 447*x^3*y*z^3 + 3756*x^3*y*z^2 - 10380*x^3*y*z + 9460*x^3*y - 96*x^3*z^4 + 1101*x^3*z^3 - 4644*x^3*z^2 + 8574*x^3*z - 5876*x^3 + 75*x^2*y^5 + 342*x^2*y^4*z - 909*x^2*y^4 + 618*x^2*y^3*z^2 - 3315*x^2*y^3*z + 4422*x^2*y^3 + 552*x^2*y^2*z^3 - 4491*x^2*y^2*z^2 + 12093*x^2*y^2*z - 10791*x^2*y^2 + 243*x^2*y*z^4 - 2673*x^2*y*z^3 + 10920*x^2*y*z^2 - 19662*x^2*y*z + 13203*x^2*y + 42*x^2*z^5 - 588*x^2*z^4 + 3249*x^2*z^3 - 8871*x^2*z^2 + 12009*x^2*z - 6475*x^2 - 19*x*y^6 - 105*x*y^5*z + 276*x*y^5 - 240*x*y^4*z^2 + 1269*x*y^4*z - 1668*x*y^4 - 290*x*y^3*z^3 + 2316*x*y^3*z^2 - 6126*x*y^3*z + 5379*x*y^3 - 195*x*y^2*z^4 + 2094*x*y^2*z^3 - 8370*x*y^2*z^2 + 14787*x*y^2*z - 9765*x*y^2 - 69*x*y*z^5 + 936*x*y*z^4 - 5034*x*y*z^3 + 13437*x*y*z^2 - 17841*x*y*z + 9454*x*y - 10*x*z^6 + 165*x*z^5 - 1122*x*z^4 + 4029*x*z^3 - 8076*x*z^2 + 8593*x*z - 3806*x + 2*y^7 + 13*y^6*z - 34*y^6 + 36*y^5*z^2 - 189*y^5*z + 246*y^5 + 55*y^4*z^3 - 435*y^4*z^2 + 1137*y^4*z - 986*y^4 + 50*y^3*z^4 - 530*y^3*z^3 + 2088*y^3*z^2 - 3636*y^3*z + 2368*y^3 + 27*y^2*z^5 - 360*y^2*z^4 + 1902*y^2*z^3 - 4992*y^2*z^2 + 6525*y^2*z - 3409*y^2 + 8*y*z^6 - 129*y*z^5 + 858*y*z^4 - 3020*y*z^3 + 5946*y*z^2 - 6227*y*z + 2722*y + z^7 - 19*z^6 + 153*z^5 - 678*z^4 + 1789*z^3 - 2818*z^2 + 2463*z - 1, 16*x^7 - 104*x^6*y - 88*x^6*z + 268*x^6 + 276*x^5*y^2 + 480*x^5*y*z - 1344*x^5*y + 204*x^5*z^2 - 1188*x^5*z + 1644*x^5 - 390*x^4*y^3 - 1038*x^4*y^2*z + 2739*x^4*y^2 - 906*x^4*y*z^2 + 4902*x^4*y*z - 6504*x^4*y - 258*x^4*z^3 + 2163*x^4*z^2 - 5856*x^4*z + 5223*x^4 + 318*x^3*y^4 + 1146*x^3*y^3*z - 2901*x^3*y^3 + 1530*x^3*y^2*z^2 - 7872*x^3*y^2*z + 10113*x^3*y^2 + 894*x^3*y*z^3 - 7041*x^3*y*z^2 + 18342*x^3*y*z - 15929*x^3*y + 192*x^3*z^4 - 2070*x^3*z^3 + 8229*x^3*z^2 - 14466*x^3*z + 9547*x^3 - 150*x^2*y^5 - 684*x^2*y^4*z + 1683*x^2*y^4 - 1236*x^2*y^3*z^2 + 6147*x^2*y^3*z - 7710*x^2*y^3 - 1104*x^2*y^2*z^3 + 8343*x^2*y^2*z^2 - 21126*x^2*y^2*z + 17961*x^2*y^2 - 486*x^2*y*z^4 + 4977*x^2*y*z^3 - 19122*x^2*y*z^2 + 32787*x^2*y*z - 21222*x^2*y - 84*x^2*z^5 + 1098*x^2*z^4 - 5706*x^2*z^3 + 14826*x^2*z^2 - 19332*x^2*z + 10151*x^2 + 38*x*y^6 + 210*x*y^5*z - 507*x*y^5 + 480*x*y^4*z^2 - 2334*x*y^4*z + 2877*x*y^4 + 580*x*y^3*z^3 - 4266*x*y^3*z^2 + 10584*x*y^3*z - 8853*x*y^3 + 390*x*y^2*z^4 - 3864*x*y^2*z^3 + 14490*x*y^2*z^2 - 24378*x*y^2*z + 15537*x*y^2 + 138*x*y*z^5 - 1731*x*y*z^4 + 8736*x*y*z^3 - 22197*x*y*z^2 + 28428*x*y*z - 14699*x*y + 20*x*z^6 - 306*x*z^5 + 1953*x*z^4 - 6672*x*z^3 + 12891*x*z^2 - 13376*x*z + 5831*x - 4*y^7 - 26*y^6*z + 62*y^6 - 72*y^5*z^2 + 345*y^5*z - 420*y^5 - 110*y^4*z^3 + 795*y^4*z^2 - 1944*y^4*z + 1606*y^4 - 100*y^3*z^4 + 970*y^3*z^3 - 3576*y^3*z^2 + 5931*y^3*z - 3734*y^3 - 54*y^2*z^5 + 660*y^2*z^4 - 3264*y^2*z^3 + 8157*y^2*z^2 - 10302*y^2*z + 5264*y^2 - 16*y*z^6 + 237*y*z^5 - 1476*y*z^4 + 4945*y*z^3 - 9402*y*z^2 + 9625*y*z - 4150*y - 2*z^7 + 35*z^6 - 264*z^5 + 1113*z^4 - 2834*z^3 + 4361*z^2 - 3758*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ 1, -1, -1 ], [ 2, -1, -1 ], [ 1, -2, -1 ] ], "source_shift": [ 2, 2, 1 ], "target_matrix": [ [ -1, -1, -1 ], [ -1, -1, -2 ], [ 2, 1, 1 ] ], "target_shift": [ 952, 924, -1397 ], "collision_preimages": [ [ "0", "-3/4", "11/4" ], [ "-5/2", "-13/2", "5" ], [ "5/2", "-17/2", "14" ] ], "collision_image": [ "3809/4", "3697/4", "-2795/2" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0331053949994384 ], "surface_features": [ 3.5824503668650416, 4.515701680681241, 3.1659858227744544, 0.0, 3.72006452520477, 4859 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C082", "pair": 82, "family": "counterexample", "mapping": "F(x,y,z)=(24*x^7 + 132*x^6*y - 124*x^6*z - 272*x^6 + 306*x^5*y^2 - 576*x^5*y*z - 1236*x^5*y + 270*x^5*z^2 + 1164*x^5*z + 1242*x^5 + 387*x^4*y^3 - 1095*x^4*y^2*z - 2304*x^4*y^2 + 1029*x^4*y*z^2 + 4344*x^4*y*z + 4551*x^4*y - 321*x^4*z^3 - 2040*x^4*z^2 - 4281*x^4*z - 2975*x^4 + 288*x^3*y^4 - 1089*x^3*y^3*z - 2253*x^3*y^3 + 1539*x^3*y^2*z^2 + 6381*x^3*y^2*z + 6573*x^3*y^2 - 963*x^3*y*z^3 - 6003*x^3*y*z^2 - 12375*x^3*y*z - 8441*x^3*y + 225*x^3*z^4 + 1875*x^3*z^3 + 5802*x^3*z^2 + 7906*x^3*z + 3998*x^3 + 126*x^2*y^5 - 597*x^2*y^4*z - 1218*x^2*y^4 + 1128*x^2*y^3*z^2 + 4608*x^2*y^3*z + 4674*x^2*y^3 - 1062*x^2*y^2*z^3 - 6516*x^2*y^2*z^2 - 13218*x^2*y^2*z - 8859*x^2*y^2 + 498*x^2*y*z^4 + 4080*x^2*y*z^3 + 12414*x^2*y*z^2 + 16611*x^2*y*z + 8235*x^2*y - 93*x^2*z^5 - 954*x^2*z^4 - 3870*x^2*z^3 - 7752*x^2*z^2 - 7650*x^2*z - 2962*x^2 + 30*x*y^6 - 171*x*y^5*z - 345*x*y^5 + 405*x*y^4*z^2 + 1635*x*y^4*z + 1635*x*y^4 - 510*x*y^3*z^3 - 3090*x*y^3*z^2 - 6177*x*y^3*z - 4072*x*y^3 + 360*x*y^2*z^4 + 2910*x*y^2*z^3 + 8721*x*y^2*z^2 + 11475*x*y^2*z + 5589*x*y^2 - 135*x*y*z^5 - 1365*x*y*z^4 - 5451*x*y*z^3 - 10734*x*y*z^2 - 10407*x*y*z - 3958*x*y + 21*x*z^6 + 255*x*z^5 + 1272*x*z^4 + 3331*x*z^3 + 4818*x*z^2 + 3628*x*z + 1092*x + 3*y^7 - 20*y^6*z - 40*y^6 + 57*y^5*z^2 + 228*y^5*z + 225*y^5 - 90*y^4*z^3 - 540*y^4*z^2 - 1065*y^4*z - 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+ 2454*x*y*z^4 + 9000*x*y*z^3 + 16614*x*y*z^2 + 15354*x*y*z + 5642*x*y - 42*x*z^6 - 459*x*z^5 - 2103*x*z^4 - 5162*x*z^3 - 7116*x*z^2 - 5177*x*z - 1523*x - 6*y^7 + 40*y^6*z + 71*y^6 - 114*y^5*z^2 - 405*y^5*z - 366*y^5 + 180*y^4*z^3 + 960*y^4*z^2 + 1734*y^4*z + 1055*y^4 - 170*y^3*z^4 - 1210*y^3*z^3 - 3276*y^3*z^2 - 3978*y^3*z - 1818*y^3 + 96*y^2*z^5 + 855*y^2*z^4 + 3084*y^2*z^3 + 5604*y^2*z^2 + 5100*y^2*z + 1847*y^2 - 30*y*z^6 - 321*y*z^5 - 1446*y*z^4 - 3494*y*z^3 - 4746*y*z^2 - 3409*y*z - 1002*y + 4*z^7 + 50*z^6 + 270*z^5 + 813*z^4 + 1464*z^3 + 1562*z^2 + 902*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -1, -1, 1 ], [ -2, -1, 1 ], [ -3, -3, 2 ] ], "source_shift": [ 2, 1, 1 ], "target_matrix": [ [ -1, -1, -1 ], [ -1, -1, -2 ], [ 2, 1, 1 ] ], "target_shift": [ 155, 139, -212 ], "collision_preimages": [ [ "-1", "-7/4", "-19/4" ], [ "3/2", "-9", "-17/2" ], [ "-7/2", "-8", "-29/2" ] ], "collision_image": [ "621/4", "557/4", "-425/2" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0568399639996642 ], "surface_features": [ 3.520825476867103, 4.415040157367443, 3.1742052269401473, 0.0, 3.6276026356374307, 4891 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C050", "pair": 50, "family": "counterexample", "mapping": "F(x,y,z)=(-16*x^7 - 144*x^6*y - 88*x^6*z - 148*x^6 - 528*x^5*y^2 - 660*x^5*y*z - 1104*x^5*y - 204*x^5*z^2 - 684*x^5*z - 540*x^5 - 1012*x^4*y^3 - 1950*x^4*y^2*z - 3249*x^4*y^2 - 1236*x^4*y*z^2 - 4122*x^4*y*z - 3294*x^4*y - 258*x^4*z^3 - 1293*x^4*z^2 - 2052*x^4*z - 1047*x^4 - 1083*x^3*y^4 - 2875*x^3*y^3*z - 4783*x^3*y^3 - 2817*x^3*y^2*z^2 - 9360*x^3*y^2*z - 7578*x^3*y^2 - 1209*x^3*y*z^3 - 6027*x^3*y*z^2 - 9681*x^3*y*z - 5074*x^3*y - 192*x^3*z^4 - 1278*x^3*z^3 - 3057*x^3*z^2 - 3161*x^3*z - 1202*x^3 - 642*x^2*y^5 - 2211*x^2*y^4*z - 3684*x^2*y^4 - 2991*x^2*y^3*z^2 - 9927*x^2*y^3*z - 8127*x^2*y^3 - 1989*x^2*y^2*z^3 - 9882*x^2*y^2*z^2 - 16056*x^2*y^2*z - 8601*x^2*y^2 - 651*x^2*y*z^4 - 4311*x^2*y*z^3 - 10428*x^2*y*z^2 - 11046*x^2*y*z - 4362*x^2*y - 84*x^2*z^5 - 696*x^2*z^4 - 2229*x^2*z^3 - 3498*x^2*z^2 - 2715*x^2*z - 850*x^2 - 196*x*y^6 - 840*x*y^5*z - 1404*x*y^5 - 1473*x*y^4*z^2 - 4896*x*y^4*z - 4023*x*y^4 - 1353*x*y^3*z^3 - 6717*x*y^3*z^2 - 10989*x*y^3*z - 5967*x*y^3 - 687*x*y^2*z^4 - 4536*x*y^2*z^3 - 11067*x*y^2*z^2 - 11922*x*y^2*z - 4851*x*y^2 - 183*x*y*z^5 - 1509*x*y*z^4 - 4878*x*y*z^3 - 7809*x*y*z^2 - 6264*x*y*z - 2047*x*y - 20*x*z^6 - 198*x*z^5 - 795*x*z^4 - 1679*x*z^3 - 1989*x*z^2 - 1277*x*z - 361*x - 24*y^7 - 124*y^6*z - 208*y^6 - 270*y^5*z^2 - 900*y^5*z - 738*y^5 - 321*y^4*z^3 - 1596*y^4*z^2 - 2613*y^4*z - 1424*y^4 - 225*y^3*z^4 - 1485*y^3*z^3 - 3636*y^3*z^2 - 3949*y^3*z - 1627*y^3 - 93*y^2*z^5 - 765*y^2*z^4 - 2487*y^2*z^3 - 4029*y^2*z^2 - 3297*y^2*z - 1108*y^2 - 21*y*z^6 - 207*y*z^5 - 837*y*z^4 - 1795*y*z^3 - 2181*y*z^2 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124*y^6*z - 208*y^6 - 270*y^5*z^2 - 900*y^5*z - 738*y^5 - 321*y^4*z^3 - 1596*y^4*z^2 - 2613*y^4*z - 1427*y^4 - 225*y^3*z^4 - 1485*y^3*z^3 - 3636*y^3*z^2 - 3960*y^3*z - 1644*y^3 - 93*y^2*z^5 - 765*y^2*z^4 - 2487*y^2*z^3 - 4044*y^2*z^2 - 3342*y^2*z - 1138*y^2 - 21*y*z^6 - 207*y*z^5 - 837*y*z^4 - 1804*y*z^3 - 2220*y*z^2 - 1498*y*z - 438*y - 2*z^7 - 23*z^6 - 111*z^5 - 297*z^4 - 483*z^3 - 484*z^2 - 281*z - 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ 2, 1, 1 ], [ 1, 2, 1 ], [ 2, 3, 2 ] ], "source_shift": [ 1, 1, 2 ], "target_matrix": [ [ -1, -1, -2 ], [ 3, 2, 3 ], [ -1, -1, -1 ] ], "target_shift": [ 69, -172, 72 ], "collision_preimages": [ [ "1/4", "1/4", "-7/4" ], [ "-7", "-19/2", "47/2" ], [ "-6", "-7/2", "27/2" ] ], "collision_image": [ "277/4", "-691/4", "289/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 0.9817973539975355 ], "surface_features": [ 3.626105588691867, 4.619917290433218, 3.304813543394108, 0.0, 3.7699426396095177, 4938 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B225", "pair": 225, "family": "bijection", "mapping": "F(x,y,z)=(4*x^7 - 56*x^6*y - 28*x^6*z + 52*x^6 + 336*x^5*y^2 + 336*x^5*y*z - 600*x^5*y + 84*x^5*z^2 - 304*x^5*z + 256*x^5 - 1120*x^4*y^3 - 1680*x^4*y^2*z + 2880*x^4*y^2 - 840*x^4*y*z^2 + 2920*x^4*y*z - 2364*x^4*y - 140*x^4*z^3 + 740*x^4*z^2 - 1216*x^4*z + 644*x^4 + 2240*x^3*y^4 + 4480*x^3*y^3*z - 7360*x^3*y^3 + 3360*x^3*y^2*z^2 - 11200*x^3*y^2*z + 8708*x^3*y^2 + 1120*x^3*y*z^3 - 5680*x^3*y*z^2 + 8968*x^3*y*z - 4584*x^3*y + 140*x^3*z^4 - 960*x^3*z^3 + 2308*x^3*z^2 - 2400*x^3*z + 1001*x^3 - 2688*x^2*y^5 - 6720*x^2*y^4*z + 10560*x^2*y^4 - 6720*x^2*y^3*z^2 + 21440*x^2*y^3*z - 15992*x^2*y^3 - 3360*x^2*y^2*z^3 + 16320*x^2*y^2*z^2 - 24732*x^2*y^2*z + 12180*x^2*y^2 - 840*x^2*y*z^4 + 5520*x^2*y*z^3 - 12744*x^2*y*z^2 + 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2840*x^3*y*z^2 + 4484*x^3*y*z - 2292*x^3*y + 70*x^3*z^4 - 480*x^3*z^3 + 1154*x^3*z^2 - 1200*x^3*z + 501*x^3 - 1344*x^2*y^5 - 3360*x^2*y^4*z + 5280*x^2*y^4 - 3360*x^2*y^3*z^2 + 10720*x^2*y^3*z - 7996*x^2*y^3 - 1680*x^2*y^2*z^3 + 8160*x^2*y^2*z^2 - 12366*x^2*y^2*z + 6090*x^2*y^2 - 420*x^2*y*z^4 + 2760*x^2*y*z^3 - 6372*x^2*y*z^2 + 6388*x^2*y*z - 2528*x^2*y - 42*x^2*z^5 + 350*x^2*z^4 - 1094*x^2*z^3 + 1674*x^2*z^2 - 1363*x^2*z + 465*x^2 + 896*x*y^6 + 2688*x*y^5*z - 4032*x*y^5 + 3360*x*y^4*z^2 - 10240*x*y^4*z + 7320*x*y^4 + 2240*x*y^3*z^3 - 10400*x*y^3*z^2 + 15112*x*y^3*z - 7160*x*y^3 + 840*x*y^2*z^4 - 5280*x*y^2*z^3 + 11694*x*y^2*z^2 - 11284*x*y^2*z + 4250*x*y^2 + 168*x*y*z^5 - 1340*x*y*z^4 + 4020*x*y*z^3 - 5924*x*y*z^2 + 4576*x*y*z - 1460*x*y + 14*x*z^6 - 136*x*z^5 + 518*x*z^4 - 1036*x*z^3 + 1233*x*z^2 - 810*x*z + 216*x - 256*y^7 - 896*y^6*z + 1280*y^6 - 1344*y^5*z^2 + 3904*y^5*z - 2672*y^5 - 1120*y^4*z^3 + 4960*y^4*z^2 - 6904*y^4*z + 3144*y^4 - 560*y^3*z^4 + 3360*y^3*z^3 - 7132*y^3*z^2 + 6616*y^3*z - 2378*y^3 - 168*y^2*z^5 + 1280*y^2*z^4 - 3682*y^2*z^3 + 5218*y^2*z^2 - 3838*y^2*z + 1156*y^2 - 28*y*z^6 + 260*y*z^5 - 950*y*z^4 + 1828*y*z^3 - 2066*y*z^2 + 1276*y*z - 325*y - 2*z^7 + 22*z^6 - 98*z^5 + 240*z^4 - 371*z^3 + 353*z^2 - 184*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ 1, -2, -1 ], [ 1, -1, -1 ], [ 3, -3, -2 ] ], "source_shift": [ 1, 2, 1 ], "target_matrix": [ [ 2, 1, 1 ], [ 3, 3, 2 ], [ 1, 1, 1 ] ], "target_shift": [ -89, -138, -44 ], "inverse_recipe": { "p": 3, "q": 2, "f": "x**3 + x", "g": "x**2", "h": "y*z**2 + y + z", "polynomials": { "f": [ [ [ 3, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "g": [ [ [ 2, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 2 ], 1 ], [ [ 0, 0, 1 ], 1 ], [ [ 0, 1, 0 ], 1 ] ] }, "h_multiplier": 2 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "Maybe" ], "seconds": [ 0.9838506379965111 ], "surface_features": [ 3.7166779246609956, 4.65644319963344, 3.401400540781544, 0.0, 3.848720695131469, 4954 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B054", "pair": 54, "family": "bijection", "mapping": "F(x,y,z)=(-384*x^7 + 1408*x^6*y + 1536*x^6*z - 1792*x^6 - 2208*x^5*y^2 - 4800*x^5*y*z + 5600*x^5*y - 2592*x^5*z^2 + 6048*x^5*z - 3488*x^5 + 1920*x^4*y^3 + 6240*x^4*y^2*z - 7280*x^4*y^2 + 6720*x^4*y*z^2 - 15680*x^4*y*z + 9040*x^4*y + 2400*x^4*z^3 - 8400*x^4*z^2 + 9680*x^4*z - 3680*x^4 - 1000*x^3*y^4 - 4320*x^3*y^3*z + 5040*x^3*y^3 - 6960*x^3*y^2*z^2 + 16240*x^3*y^2*z - 9360*x^3*y^2 - 4960*x^3*y*z^3 + 17360*x^3*y*z^2 - 20000*x^3*y*z + 7600*x^3*y - 1320*x^3*z^4 + 6160*x^3*z^3 - 10640*x^3*z^2 + 8080*x^3*z - 2280*x^3 + 312*x^2*y^5 + 1680*x^2*y^4*z - 1960*x^2*y^4 + 3600*x^2*y^3*z^2 - 8400*x^2*y^3*z + 4840*x^2*y^3 + 3840*x^2*y^2*z^3 - 13440*x^2*y^2*z^2 + 15480*x^2*y^2*z - 5880*x^2*y^2 + 2040*x^2*y*z^4 - 9520*x^2*y*z^3 + 16440*x^2*y*z^2 - 12480*x^2*y*z + 3520*x^2*y + 432*x^2*z^5 - 2520*x^2*z^4 + 5800*x^2*z^3 - 6600*x^2*z^2 + 3720*x^2*z - 838*x^2 - 54*x*y^6 - 348*x*y^5*z + 406*x*y^5 - 930*x*y^4*z^2 + 2170*x*y^4*z - 1250*x*y^4 - 1320*x*y^3*z^3 + 4620*x*y^3*z^2 - 5320*x*y^3*z + 2020*x*y^3 - 1050*x*y^2*z^4 + 4900*x*y^2*z^3 - 8460*x*y^2*z^2 + 6420*x*y^2*z - 1810*x*y^2 - 444*x*y*z^5 + 2590*x*y*z^4 - 5960*x*y*z^3 + 6780*x*y*z^2 - 3820*x*y*z + 864*x*y - 78*x*z^6 + 546*x*z^5 - 1570*x*z^4 + 2380*x*z^3 - 2010*x*z^2 + 910*x*z - 189*x + 4*y^7 + 30*y^6*z - 35*y^6 + 96*y^5*z^2 - 224*y^5*z + 129*y^5 + 170*y^4*z^3 - 595*y^4*z^2 + 685*y^4*z - 260*y^4 + 180*y^3*z^4 - 840*y^3*z^3 + 1450*y^3*z^2 - 1100*y^3*z + 310*y^3 + 114*y^2*z^5 - 665*y^2*z^4 + 1530*y^2*z^3 - 1740*y^2*z^2 + 980*y^2*z - 223*y^2 + 40*y*z^6 - 280*y*z^5 + 805*y*z^4 - 1220*y*z^3 + 1030*y*z^2 - 470*y*z + 102*y + 6*z^7 - 49*z^6 + 169*z^5 - 320*z^4 + 360*z^3 - 247*z^2 + 110*z - 1, -192*x^7 + 704*x^6*y + 768*x^6*z - 928*x^6 - 1104*x^5*y^2 - 2400*x^5*y*z + 2896*x^5*y - 1296*x^5*z^2 + 3120*x^5*z - 1856*x^5 + 960*x^4*y^3 + 3120*x^4*y^2*z - 3760*x^4*y^2 + 3360*x^4*y*z^2 - 8080*x^4*y*z + 4800*x^4*y + 1200*x^4*z^3 - 4320*x^4*z^2 + 5120*x^4*z - 2000*x^4 - 500*x^3*y^4 - 2160*x^3*y^3*z + 2600*x^3*y^3 - 3480*x^3*y^2*z^2 + 8360*x^3*y^2*z - 4960*x^3*y^2 - 2480*x^3*y*z^3 + 8920*x^3*y*z^2 - 10560*x^3*y*z + 4120*x^3*y - 660*x^3*z^4 + 3160*x^3*z^3 - 5600*x^3*z^2 + 4360*x^3*z - 1260*x^3 + 156*x^2*y^5 + 840*x^2*y^4*z - 1010*x^2*y^4 + 1800*x^2*y^3*z^2 - 4320*x^2*y^3*z + 2560*x^2*y^3 + 1920*x^2*y^2*z^3 - 6900*x^2*y^2*z^2 + 8160*x^2*y^2*z - 3180*x^2*y^2 + 1020*x^2*y*z^4 - 4880*x^2*y*z^3 + 8640*x^2*y*z^2 - 6720*x^2*y*z + 1940*x^2*y + 216*x^2*z^5 - 1290*x^2*z^4 + 3040*x^2*z^3 - 3540*x^2*z^2 + 2040*x^2*z - 469*x^2 - 27*x*y^6 - 174*x*y^5*z + 209*x*y^5 - 465*x*y^4*z^2 + 1115*x*y^4*z - 660*x*y^4 - 660*x*y^3*z^3 + 2370*x*y^3*z^2 - 2800*x*y^3*z + 1090*x*y^3 - 525*x*y^2*z^4 + 2510*x*y^2*z^3 - 4440*x*y^2*z^2 + 3450*x*y^2*z - 995*x*y^2 - 222*x*y*z^5 + 1325*x*y*z^4 - 3120*x*y*z^3 + 3630*x*y*z^2 - 2090*x*y*z + 482*x*y - 39*x*z^6 + 279*x*z^5 - 820*x*z^4 + 1270*x*z^3 - 1095*x*z^2 + 505*x*z - 103*x + 2*y^7 + 15*y^6*z - 18*y^6 + 48*y^5*z^2 - 115*y^5*z + 68*y^5 + 85*y^4*z^3 - 305*y^4*z^2 + 360*y^4*z - 140*y^4 + 90*y^3*z^4 - 430*y^3*z^3 + 760*y^3*z^2 - 590*y^3*z + 170*y^3 + 57*y^2*z^5 - 340*y^2*z^4 + 800*y^2*z^3 - 930*y^2*z^2 + 535*y^2*z - 124*y^2 + 20*y*z^6 - 143*y*z^5 + 420*y*z^4 - 650*y*z^3 + 560*y*z^2 - 260*y*z + 55*y + 3*z^7 - 25*z^6 + 88*z^5 - 170*z^4 + 195*z^3 - 136*z^2 + 58*z - 1, 384*x^7 - 1408*x^6*y - 1536*x^6*z + 1856*x^6 + 2208*x^5*y^2 + 4800*x^5*y*z - 5792*x^5*y + 2592*x^5*z^2 - 6240*x^5*z + 3712*x^5 - 1920*x^4*y^3 - 6240*x^4*y^2*z + 7520*x^4*y^2 - 6720*x^4*y*z^2 + 16160*x^4*y*z - 9600*x^4*y - 2400*x^4*z^3 + 8640*x^4*z^2 - 10240*x^4*z + 4000*x^4 + 1000*x^3*y^4 + 4320*x^3*y^3*z - 5200*x^3*y^3 + 6960*x^3*y^2*z^2 - 16720*x^3*y^2*z + 9920*x^3*y^2 + 4960*x^3*y*z^3 - 17840*x^3*y*z^2 + 21120*x^3*y*z - 8240*x^3*y + 1320*x^3*z^4 - 6320*x^3*z^3 + 11200*x^3*z^2 - 8720*x^3*z + 2520*x^3 - 312*x^2*y^5 - 1680*x^2*y^4*z + 2020*x^2*y^4 - 3600*x^2*y^3*z^2 + 8640*x^2*y^3*z - 5120*x^2*y^3 - 3840*x^2*y^2*z^3 + 13800*x^2*y^2*z^2 - 16320*x^2*y^2*z + 6360*x^2*y^2 - 2040*x^2*y*z^4 + 9760*x^2*y*z^3 - 17280*x^2*y*z^2 + 13440*x^2*y*z - 3880*x^2*y - 432*x^2*z^5 + 2580*x^2*z^4 - 6080*x^2*z^3 + 7080*x^2*z^2 - 4080*x^2*z + 938*x^2 + 54*x*y^6 + 348*x*y^5*z - 418*x*y^5 + 930*x*y^4*z^2 - 2230*x*y^4*z + 1320*x*y^4 + 1320*x*y^3*z^3 - 4740*x*y^3*z^2 + 5600*x*y^3*z - 2180*x*y^3 + 1050*x*y^2*z^4 - 5020*x*y^2*z^3 + 8880*x*y^2*z^2 - 6900*x*y^2*z + 1990*x*y^2 + 444*x*y*z^5 - 2650*x*y*z^4 + 6240*x*y*z^3 - 7260*x*y*z^2 + 4180*x*y*z - 964*x*y + 78*x*z^6 - 558*x*z^5 + 1640*x*z^4 - 2540*x*z^3 + 2190*x*z^2 - 1010*x*z + 200*x - 4*y^7 - 30*y^6*z + 36*y^6 - 96*y^5*z^2 + 230*y^5*z - 136*y^5 - 170*y^4*z^3 + 610*y^4*z^2 - 720*y^4*z + 280*y^4 - 180*y^3*z^4 + 860*y^3*z^3 - 1520*y^3*z^2 + 1180*y^3*z - 340*y^3 - 114*y^2*z^5 + 680*y^2*z^4 - 1600*y^2*z^3 + 1860*y^2*z^2 - 1070*y^2*z + 248*y^2 - 40*y*z^6 + 286*y*z^5 - 840*y*z^4 + 1300*y*z^3 - 1120*y*z^2 + 520*y*z - 106*y - 6*z^7 + 50*z^6 - 176*z^5 + 340*z^4 - 390*z^3 + 272*z^2 - 110*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ 2, -1, -1 ], [ 1, -1, -1 ], [ 1, -1, -2 ] ], "source_shift": [ 1, 2, 2 ], "target_matrix": [ [ -2, -1, -1 ], [ -1, -1, -1 ], [ 2, 2, 3 ] ], "target_shift": [ 29, 13, -21 ], "inverse_recipe": { "p": 6, "q": 1, "f": "x**6 + x**5", "g": "x", "h": "y*z + z", "polynomials": { "f": [ [ [ 6, 0, 0 ], 1 ], [ [ 5, 0, 0 ], 1 ] ], "g": [ [ [ 1, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 0, 1 ], 1 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 0.9918379900009313 ], "surface_features": [ 3.4876300773809445, 4.324693913861775, 3.1550322287909704, 0.0, 3.5932421061406403, 4874 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C098", "pair": 98, "family": "counterexample", "mapping": "F(x,y,z)=(32*x^7 - 352*x^6*y + 208*x^6*z - 328*x^6 + 1632*x^5*y^2 - 1944*x^5*y*z + 3132*x^5*y + 576*x^5*z^2 - 1860*x^5*z + 1404*x^5 - 4120*x^4*y^3 + 7428*x^4*y^2*z - 12294*x^4*y^2 - 4440*x^4*y*z^2 + 14712*x^4*y*z - 11244*x^4*y + 880*x^4*z^3 - 4380*x^4*z^2 + 6696*x^4*z - 3330*x^4 + 6086*x^3*y^4 - 14786*x^3*y^3*z + 25337*x^3*y^3 + 13392*x^3*y^2*z^2 - 45861*x^3*y^2*z + 35583*x^3*y^2 - 5360*x^3*y*z^3 + 27528*x^3*y*z^2 - 42696*x^3*y*z + 21387*x^3*y + 800*x^3*z^4 - 5480*x^3*z^3 + 12744*x^3*z^2 - 12773*x^3*z + 4775*x^3 - 5220*x^2*y^5 + 16062*x^2*y^4*z - 28830*x^2*y^4 - 19638*x^2*y^3*z^2 + 70239*x^2*y^3*z - 55530*x^2*y^3 + 11928*x^2*y^2*z^3 - 63819*x^2*y^2*z^2 + 100758*x^2*y^2*z - 50907*x^2*y^2 - 3600*x^2*y*z^4 + 25632*x^2*y*z^3 - 60624*x^2*y*z^2 + 61269*x^2*y*z - 22980*x^2*y + 432*x^2*z^5 - 3840*x^2*z^4 + 12096*x^2*z^3 - 18339*x^2*z^2 + 13776*x^2*z - 4163*x^2 + 2376*x*y^6 - 8928*x*y^5*z + 17100*x*y^5 + 13866*x*y^4*z^2 - 52656*x*y^4*z + 42642*x*y^4 - 11398*x*y^3*z^3 + 64467*x*y^3*z^2 - 104094*x*y^3*z + 53151*x*y^3 + 5232*x*y^2*z^4 - 39231*x*y^2*z^3 + 94767*x*y^2*z^2 - 96741*x*y^2*z + 36423*x*y^2 - 1272*x*y*z^5 + 11868*x*y*z^4 - 38136*x*y*z^3 + 58377*x*y*z^2 - 44022*x*y*z + 13296*x*y + 128*x*z^6 - 1428*x*z^5 + 5724*x*z^4 - 11679*x*z^3 + 13227*x*z^2 - 8014*x*z + 2050*x - 432*y^7 + 1944*y^6*z - 4104*y^6 - 3708*y^5*z^2 + 15372*y^5*z - 12852*y^5 + 3890*y^4*z^3 - 23826*y^4*z^2 + 39618*y^4*z - 20502*y^4 - 2426*y^3*z^4 + 19565*y^3*z^3 - 48564*y^3*z^2 + 50199*y^3*z - 18990*y^3 + 900*y^2*z^5 - 8979*y^2*z^4 + 29592*y^2*z^3 - 45834*y^2*z^2 + 34722*y^2*z - 10479*y^2 - 184*y*z^6 + 2184*y*z^5 - 8964*y*z^4 + 18495*y*z^3 - 21042*y*z^2 + 12744*y*z - 3242*y + 16*z^7 - 220*z^6 + 1080*z^5 - 2783*z^4 + 4226*z^3 - 3851*z^2 + 1970*z - 1, -16*x^7 + 176*x^6*y - 104*x^6*z + 176*x^6 - 816*x^5*y^2 + 972*x^5*y*z - 1680*x^5*y - 288*x^5*z^2 + 996*x^5*z - 804*x^5 + 2060*x^4*y^3 - 3714*x^4*y^2*z + 6588*x^4*y^2 + 2220*x^4*y*z^2 - 7872*x^4*y*z + 6444*x^4*y - 440*x^4*z^3 + 2340*x^4*z^2 - 3828*x^4*z + 2006*x^4 - 3043*x^3*y^4 + 7393*x^3*y^3*z - 13552*x^3*y^3 - 6696*x^3*y^2*z^2 + 24501*x^3*y^2*z - 20397*x^3*y^2 + 2680*x^3*y*z^3 - 14688*x^3*y*z^2 + 24420*x^3*y*z - 12907*x^3*y - 400*x^3*z^4 + 2920*x^3*z^3 - 7272*x^3*z^2 + 7685*x^3*z - 2987*x^3 + 2610*x^2*y^5 - 8031*x^2*y^4*z + 15372*x^2*y^4 + 9819*x^2*y^3*z^2 - 37425*x^2*y^3*z + 31812*x^2*y^3 - 5964*x^2*y^2*z^3 + 33975*x^2*y^2*z^2 - 57615*x^2*y^2*z + 30762*x^2*y^2 + 1800*x^2*y*z^4 - 13632*x^2*y*z^3 + 34596*x^2*y*z^2 - 36921*x^2*y*z + 14412*x^2*y - 216*x^2*z^5 + 2040*x^2*z^4 - 6888*x^2*z^3 + 11019*x^2*z^2 - 8610*x^2*z + 2677*x^2 - 1188*x*y^6 + 4464*x*y^5*z - 9072*x*y^5 - 6933*x*y^4*z^2 + 27936*x*y^4*z - 24387*x*y^4 + 5699*x*y^3*z^3 - 34194*x*y^3*z^2 + 59451*x*y^3*z - 32137*x*y^3 - 2616*x*y^2*z^4 + 20799*x*y^2*z^3 - 54039*x*y^2*z^2 + 58353*x*y^2*z - 22890*x*y^2 + 636*x*y*z^5 - 6288*x*y*z^4 + 21708*x*y*z^3 - 35121*x*y*z^2 + 27579*x*y*z - 8577*x*y - 64*x*z^6 + 756*x*z^5 - 3252*x*z^4 + 7007*x*z^3 - 8259*x*z^2 + 5150*x*z - 1344*x + 216*y^7 - 972*y^6*z + 2160*y^6 + 1854*y^5*z^2 - 8100*y^5*z + 7326*y^5 - 1945*y^4*z^3 + 12564*y^4*z^2 - 22569*y^4*z + 12392*y^4 + 1213*y^3*z^4 - 10321*y^3*z^3 + 27639*y^3*z^2 - 30284*y^3*z + 11951*y^3 - 450*y^2*z^5 + 4737*y^2*z^4 - 16821*y^2*z^3 + 27591*y^2*z^2 - 21792*y^2*z + 6777*y^2 + 92*y*z^6 - 1152*y*z^5 + 5088*y*z^4 - 11107*y*z^3 + 13167*y*z^2 - 8214*y*z + 2134*y - 8*z^7 + 116*z^6 - 612*z^5 + 1667*z^4 - 2636*z^3 + 2473*z^2 - 1291*z + 1, 16*x^7 - 176*x^6*y + 104*x^6*z - 176*x^6 + 816*x^5*y^2 - 972*x^5*y*z + 1680*x^5*y + 288*x^5*z^2 - 996*x^5*z + 804*x^5 - 2060*x^4*y^3 + 3714*x^4*y^2*z - 6588*x^4*y^2 - 2220*x^4*y*z^2 + 7872*x^4*y*z - 6444*x^4*y + 440*x^4*z^3 - 2340*x^4*z^2 + 3828*x^4*z - 2004*x^4 + 3043*x^3*y^4 - 7393*x^3*y^3*z + 13552*x^3*y^3 + 6696*x^3*y^2*z^2 - 24501*x^3*y^2*z + 20397*x^3*y^2 - 2680*x^3*y*z^3 + 14688*x^3*y*z^2 - 24420*x^3*y*z + 12894*x^3*y + 400*x^3*z^4 - 2920*x^3*z^3 + 7272*x^3*z^2 - 7678*x^3*z + 2974*x^3 - 2610*x^2*y^5 + 8031*x^2*y^4*z - 15372*x^2*y^4 - 9819*x^2*y^3*z^2 + 37425*x^2*y^3*z - 31812*x^2*y^3 + 5964*x^2*y^2*z^3 - 33975*x^2*y^2*z^2 + 57615*x^2*y^2*z - 30732*x^2*y^2 - 1800*x^2*y*z^4 + 13632*x^2*y*z^3 - 34596*x^2*y*z^2 + 36888*x^2*y*z - 14346*x^2*y + 216*x^2*z^5 - 2040*x^2*z^4 + 6888*x^2*z^3 - 11010*x^2*z^2 + 8574*x^2*z - 2650*x^2 + 1188*x*y^6 - 4464*x*y^5*z + 9072*x*y^5 + 6933*x*y^4*z^2 - 27936*x*y^4*z + 24387*x*y^4 - 5699*x*y^3*z^3 + 34194*x*y^3*z^2 - 59451*x*y^3*z + 32109*x*y^3 + 2616*x*y^2*z^4 - 20799*x*y^2*z^3 + 54039*x*y^2*z^2 - 58305*x*y^2*z + 22782*x*y^2 - 636*x*y*z^5 + 6288*x*y*z^4 - 21708*x*y*z^3 + 35094*x*y*z^2 - 27459*x*y*z + 8484*x*y + 64*x*z^6 - 756*x*z^5 + 3252*x*z^4 - 7002*x*z^3 + 8226*x*z^2 - 5099*x*z + 1323*x - 216*y^7 + 972*y^6*z - 2160*y^6 - 1854*y^5*z^2 + 8100*y^5*z - 7326*y^5 + 1945*y^4*z^3 - 12564*y^4*z^2 + 22569*y^4*z - 12384*y^4 - 1213*y^3*z^4 + 10321*y^3*z^3 - 27639*y^3*z^2 + 30264*y^3*z - 11895*y^3 + 450*y^2*z^5 - 4737*y^2*z^4 + 16821*y^2*z^3 - 27573*y^2*z^2 + 21696*y^2*z - 6699*y^2 - 92*y*z^6 + 1152*y*z^5 - 5088*y*z^4 + 11100*y*z^3 - 13113*y*z^2 + 8127*y*z - 2098*y + 8*z^7 - 116*z^6 + 612*z^5 - 1666*z^4 + 2626*z^3 - 2449*z^2 + 1271*z - 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -1, 2, -1 ], [ -2, 3, -2 ], [ -2, 1, -1 ] ], "source_shift": [ 1, 2, 1 ], "target_matrix": [ [ -2, -1, -1 ], [ 1, 1, 1 ], [ -1, -1, -2 ] ], "target_shift": [ 437, -290, 285 ], "collision_preimages": [ [ "1/4", "0", "3/4" ], [ "-9", "7/2", "16" ], [ "-4", "-7/2", "-1" ] ], "collision_image": [ "875/2", "-1161/4", "1141/4" ] }, "probe_target": "No", "jev_answers": [ "Maybe" ], "seconds": [ 1.0210119980001764 ], "surface_features": [ 4.142231774551217, 5.0174256974069005, 3.8409212001987716, 0.0, 4.249636359111385, 5116 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B224", "pair": 224, "family": "bijection", "mapping": "F(x,y,z)=(5*x^7 + 35*x^6*y - 35*x^6*z - 60*x^6 + 105*x^5*y^2 - 210*x^5*y*z - 360*x^5*y + 105*x^5*z^2 + 370*x^5*z + 315*x^5 + 175*x^4*y^3 - 525*x^4*y^2*z - 900*x^4*y^2 + 525*x^4*y*z^2 + 1850*x^4*y*z + 1570*x^4*y - 175*x^4*z^3 - 950*x^4*z^2 - 1650*x^4*z - 940*x^4 + 175*x^3*y^4 - 700*x^3*y^3*z - 1200*x^3*y^3 + 1050*x^3*y^2*z^2 + 3700*x^3*y^2*z + 3130*x^3*y^2 - 700*x^3*y*z^3 - 3800*x^3*y*z^2 - 6580*x^3*y*z - 3720*x^3*y + 175*x^3*z^4 + 1300*x^3*z^3 + 3455*x^3*z^2 + 3990*x^3*z + 1718*x^3 + 105*x^2*y^5 - 525*x^2*y^4*z - 900*x^2*y^4 + 1050*x^2*y^3*z^2 + 3700*x^2*y^3*z + 3120*x^2*y^3 - 1050*x^2*y^2*z^3 - 5700*x^2*y^2*z^2 - 9840*x^2*y^2*z - 5520*x^2*y^2 + 525*x^2*y*z^4 + 3900*x^2*y*z^3 + 10335*x^2*y*z^2 + 11840*x^2*y*z + 5034*x^2*y - 105*x^2*z^5 - 1000*x^2*z^4 - 3615*x^2*z^3 - 6340*x^2*z^2 - 5514*x^2*z - 1913*x^2 + 35*x*y^6 - 210*x*y^5*z - 360*x*y^5 + 525*x*y^4*z^2 + 1850*x*y^4*z + 1555*x*y^4 - 700*x*y^3*z^3 - 3800*x*y^3*z^2 - 6540*x*y^3*z - 3640*x*y^3 + 525*x*y^2*z^4 + 3900*x*y^2*z^3 + 10305*x*y^2*z^2 + 11710*x*y^2*z + 4914*x*y^2 - 210*x*y*z^5 - 2000*x*y*z^4 - 7210*x*y*z^3 - 12540*x*y*z^2 - 10748*x*y*z - 3666*x*y + 35*x*z^6 + 410*x*z^5 + 1890*x*z^4 + 4470*x*z^3 + 5869*x*z^2 + 4106*x*z + 1191*x + 5*y^7 - 35*y^6*z - 60*y^6 + 105*y^5*z^2 + 370*y^5*z + 310*y^5 - 175*y^4*z^3 - 950*y^4*z^2 - 1630*y^4*z - 900*y^4 + 175*y^3*z^4 + 1300*y^3*z^3 + 3425*y^3*z^2 + 3860*y^3*z + 1598*y^3 - 105*y^2*z^5 - 1000*y^2*z^4 - 3595*y^2*z^3 - 6200*y^2*z^2 - 5234*y^2*z - 1753*y^2 + 35*y*z^6 + 410*y*z^5 + 1885*y*z^4 + 4420*y*z^3 + 5704*y*z^2 + 3906*y*z + 1113*y - 5*z^7 - 70*z^6 - 395*z^5 - 1180*z^4 - 2068*z^3 - 2173*z^2 - 1274*z - 1, 5*x^7 + 35*x^6*y - 35*x^6*z - 60*x^6 + 105*x^5*y^2 - 210*x^5*y*z - 360*x^5*y + 105*x^5*z^2 + 370*x^5*z + 315*x^5 + 175*x^4*y^3 - 525*x^4*y^2*z - 900*x^4*y^2 + 525*x^4*y*z^2 + 1850*x^4*y*z + 1570*x^4*y - 175*x^4*z^3 - 950*x^4*z^2 - 1650*x^4*z - 940*x^4 + 175*x^3*y^4 - 700*x^3*y^3*z - 1200*x^3*y^3 + 1050*x^3*y^2*z^2 + 3700*x^3*y^2*z + 3130*x^3*y^2 - 700*x^3*y*z^3 - 3800*x^3*y*z^2 - 6580*x^3*y*z - 3720*x^3*y + 175*x^3*z^4 + 1300*x^3*z^3 + 3455*x^3*z^2 + 3990*x^3*z + 1718*x^3 + 105*x^2*y^5 - 525*x^2*y^4*z - 900*x^2*y^4 + 1050*x^2*y^3*z^2 + 3700*x^2*y^3*z + 3120*x^2*y^3 - 1050*x^2*y^2*z^3 - 5700*x^2*y^2*z^2 - 9840*x^2*y^2*z - 5520*x^2*y^2 + 525*x^2*y*z^4 + 3900*x^2*y*z^3 + 10335*x^2*y*z^2 + 11840*x^2*y*z + 5034*x^2*y - 105*x^2*z^5 - 1000*x^2*z^4 - 3615*x^2*z^3 - 6340*x^2*z^2 - 5514*x^2*z - 1914*x^2 + 35*x*y^6 - 210*x*y^5*z - 360*x*y^5 + 525*x*y^4*z^2 + 1850*x*y^4*z + 1555*x*y^4 - 700*x*y^3*z^3 - 3800*x*y^3*z^2 - 6540*x*y^3*z - 3640*x*y^3 + 525*x*y^2*z^4 + 3900*x*y^2*z^3 + 10305*x*y^2*z^2 + 11710*x*y^2*z + 4914*x*y^2 - 210*x*y*z^5 - 2000*x*y*z^4 - 7210*x*y*z^3 - 12540*x*y*z^2 - 10748*x*y*z - 3668*x*y + 35*x*z^6 + 410*x*z^5 + 1890*x*z^4 + 4470*x*z^3 + 5869*x*z^2 + 4108*x*z + 1195*x + 5*y^7 - 35*y^6*z - 60*y^6 + 105*y^5*z^2 + 370*y^5*z + 310*y^5 - 175*y^4*z^3 - 950*y^4*z^2 - 1630*y^4*z - 900*y^4 + 175*y^3*z^4 + 1300*y^3*z^3 + 3425*y^3*z^2 + 3860*y^3*z + 1598*y^3 - 105*y^2*z^5 - 1000*y^2*z^4 - 3595*y^2*z^3 - 6200*y^2*z^2 - 5234*y^2*z - 1754*y^2 + 35*y*z^6 + 410*y*z^5 + 1885*y*z^4 + 4420*y*z^3 + 5704*y*z^2 + 3908*y*z + 1117*y - 5*z^7 - 70*z^6 - 395*z^5 - 1180*z^4 - 2068*z^3 - 2174*z^2 - 1279*z - 1, -10*x^7 - 70*x^6*y + 70*x^6*z + 120*x^6 - 210*x^5*y^2 + 420*x^5*y*z + 720*x^5*y - 210*x^5*z^2 - 740*x^5*z - 630*x^5 - 350*x^4*y^3 + 1050*x^4*y^2*z + 1800*x^4*y^2 - 1050*x^4*y*z^2 - 3700*x^4*y*z - 3140*x^4*y + 350*x^4*z^3 + 1900*x^4*z^2 + 3300*x^4*z + 1880*x^4 - 350*x^3*y^4 + 1400*x^3*y^3*z + 2400*x^3*y^3 - 2100*x^3*y^2*z^2 - 7400*x^3*y^2*z - 6260*x^3*y^2 + 1400*x^3*y*z^3 + 7600*x^3*y*z^2 + 13160*x^3*y*z + 7440*x^3*y - 350*x^3*z^4 - 2600*x^3*z^3 - 6910*x^3*z^2 - 7980*x^3*z - 3434*x^3 - 210*x^2*y^5 + 1050*x^2*y^4*z + 1800*x^2*y^4 - 2100*x^2*y^3*z^2 - 7400*x^2*y^3*z - 6240*x^2*y^3 + 2100*x^2*y^2*z^3 + 11400*x^2*y^2*z^2 + 19680*x^2*y^2*z + 11040*x^2*y^2 - 1050*x^2*y*z^4 - 7800*x^2*y*z^3 - 20670*x^2*y*z^2 - 23680*x^2*y*z - 10062*x^2*y + 210*x^2*z^5 + 2000*x^2*z^4 + 7230*x^2*z^3 + 12680*x^2*z^2 + 11022*x^2*z + 3819*x^2 - 70*x*y^6 + 420*x*y^5*z + 720*x*y^5 - 1050*x*y^4*z^2 - 3700*x*y^4*z - 3110*x*y^4 + 1400*x*y^3*z^3 + 7600*x*y^3*z^2 + 13080*x*y^3*z + 7280*x*y^3 - 1050*x*y^2*z^4 - 7800*x*y^2*z^3 - 20610*x*y^2*z^2 - 23420*x*y^2*z - 9822*x*y^2 + 420*x*y*z^5 + 4000*x*y*z^4 + 14420*x*y*z^3 + 25080*x*y*z^2 + 21484*x*y*z + 7318*x*y - 70*x*z^6 - 820*x*z^5 - 3780*x*z^4 - 8940*x*z^3 - 11732*x*z^2 - 8198*x*z - 2372*x - 10*y^7 + 70*y^6*z + 120*y^6 - 210*y^5*z^2 - 740*y^5*z - 620*y^5 + 350*y^4*z^3 + 1900*y^4*z^2 + 3260*y^4*z + 1800*y^4 - 350*y^3*z^4 - 2600*y^3*z^3 - 6850*y^3*z^2 - 7720*y^3*z - 3194*y^3 + 210*y^2*z^5 + 2000*y^2*z^4 + 7190*y^2*z^3 + 12400*y^2*z^2 + 10462*y^2*z + 3499*y^2 - 70*y*z^6 - 820*y*z^5 - 3770*y*z^4 - 8840*y*z^3 - 11402*y*z^2 - 7798*y*z - 2218*y + 10*z^7 + 140*z^6 + 790*z^5 + 2360*z^4 + 4134*z^3 + 4339*z^2 + 2541*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -1, -1, 1 ], [ -1, -1, 2 ], [ -2, -1, 1 ] ], "source_shift": [ 1, 2, 2 ], "target_matrix": [ [ -1, -1, -1 ], [ -1, -2, -1 ], [ 2, 3, 3 ] ], "target_shift": [ 316, 320, -629 ], "inverse_recipe": { "p": 2, "q": 3, "f": "x**2 + x", "g": "x**3 + x**2", "h": "y**2*z", "polynomials": { "f": [ [ [ 2, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "g": [ [ [ 3, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "h": [ [ [ 0, 2, 1 ], 1 ] ] }, "h_multiplier": 5 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 0.9776652290020138 ], "surface_features": [ 3.508442552072435, 4.3993275321586784, 3.2079035303860515, 0.0, 3.6244296236251414, 4892 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C134", "pair": 134, "family": "counterexample", "mapping": "F(x,y,z)=(54*x^7 + 324*x^6*y + 432*x^6*z + 648*x^6 + 828*x^5*y^2 + 2214*x^5*y*z + 3276*x^5*y + 1458*x^5*z^2 + 4428*x^5*z + 3024*x^5 + 1168*x^4*y^3 + 4698*x^4*y^2*z + 6852*x^4*y^2 + 6210*x^4*y*z^2 + 18558*x^4*y*z + 12528*x^4*y + 2700*x^4*z^3 + 12420*x^4*z^2 + 16974*x^4*z + 7385*x^4 + 982*x^3*y^4 + 5282*x^3*y^3*z + 7588*x^3*y^3 + 10512*x^3*y^2*z^2 + 30888*x^3*y^2*z + 20598*x^3*y^2 + 9180*x^3*y*z^3 + 41472*x^3*y*z^2 + 55866*x^3*y*z + 24075*x^3*y + 2970*x^3*z^4 + 18360*x^3*z^3 + 37656*x^3*z^2 + 32359*x^3*z + 10447*x^3 + 492*x^2*y^5 + 3318*x^2*y^4*z + 4692*x^2*y^4 + 8838*x^2*y^3*z^2 + 25518*x^2*y^3*z + 16800*x^2*y^3 + 11628*x^2*y^2*z^3 + 51552*x^2*y^2*z^2 + 68424*x^2*y^2*z + 29172*x^2*y^2 + 7560*x^2*y*z^4 + 45828*x^2*y*z^3 + 92430*x^2*y*z^2 + 78537*x^2*y*z + 25014*x^2*y + 1944*x^2*z^5 + 15120*x^2*z^4 + 41364*x^2*z^3 + 52767*x^2*z^2 + 33462*x^2*z + 8438*x^2 + 136*x*y^6 + 1104*x*y^5*z + 1536*x*y^5 + 3690*x*y^4*z^2 + 10464*x*y^4*z + 6798*x*y^4 + 6502*x*y^3*z^3 + 28272*x*y^3*z^2 + 36966*x*y^3*z + 15574*x*y^3 + 6372*x*y^2*z^4 + 37848*x*y^2*z^3 + 75054*x*y^2*z^2 + 63024*x*y^2*z + 19758*x*y^2 + 3294*x*y*z^5 + 25092*x*y*z^4 + 67374*x*y*z^3 + 84849*x*y*z^2 + 53094*x*y*z + 13081*x*y + 702*x*z^6 + 6588*x*z^5 + 22536*x*z^4 + 37997*x*z^3 + 35583*x*z^2 + 17641*x*z + 3541*x + 16*y^7 + 152*y^6*z + 208*y^6 + 612*y^5*z^2 + 1704*y^5*z + 1092*y^5 + 1354*y^4*z^3 + 5772*y^4*z^2 + 7434*y^4*z + 3092*y^4 + 1778*y^3*z^4 + 10342*y^3*z^3 + 20166*y^3*z^2 + 16724*y^3*z + 5154*y^3 + 1386*y^2*z^5 + 10332*y^2*z^4 + 27228*y^2*z^3 + 33852*y^2*z^2 + 20862*y^2*z + 5024*y^2 + 594*y*z^6 + 5454*y*z^5 + 18282*y*z^4 + 30387*y*z^3 + 28080*y*z^2 + 13621*y*z + 2658*y + 108*z^7 + 1188*z^6 + 4878*z^5 + 10204*z^4 + 12568*z^3 + 9203*z^2 + 3646*z + 1, -27*x^7 - 162*x^6*y - 216*x^6*z - 324*x^6 - 414*x^5*y^2 - 1107*x^5*y*z - 1638*x^5*y - 729*x^5*z^2 - 2214*x^5*z - 1512*x^5 - 584*x^4*y^3 - 2349*x^4*y^2*z - 3426*x^4*y^2 - 3105*x^4*y*z^2 - 9279*x^4*y*z - 6264*x^4*y - 1350*x^4*z^3 - 6210*x^4*z^2 - 8487*x^4*z - 3679*x^4 - 491*x^3*y^4 - 2641*x^3*y^3*z - 3794*x^3*y^3 - 5256*x^3*y^2*z^2 - 15444*x^3*y^2*z - 10299*x^3*y^2 - 4590*x^3*y*z^3 - 20736*x^3*y*z^2 - 27933*x^3*y*z - 11997*x^3*y - 1485*x^3*z^4 - 9180*x^3*z^3 - 18828*x^3*z^2 - 16112*x^3*z - 5183*x^3 - 246*x^2*y^5 - 1659*x^2*y^4*z - 2346*x^2*y^4 - 4419*x^2*y^3*z^2 - 12759*x^2*y^3*z - 8400*x^2*y^3 - 5814*x^2*y^2*z^3 - 25776*x^2*y^2*z^2 - 34212*x^2*y^2*z - 14541*x^2*y^2 - 3780*x^2*y*z^4 - 22914*x^2*y*z^3 - 46215*x^2*y*z^2 - 39120*x^2*y*z - 12417*x^2*y - 972*x^2*z^5 - 7560*x^2*z^4 - 20682*x^2*z^3 - 26262*x^2*z^2 - 16596*x^2*z - 4165*x^2 - 68*x*y^6 - 552*x*y^5*z - 768*x*y^5 - 1845*x*y^4*z^2 - 5232*x*y^4*z - 3399*x*y^4 - 3251*x*y^3*z^3 - 14136*x*y^3*z^2 - 18483*x*y^3*z - 7765*x*y^3 - 3186*x*y^2*z^4 - 18924*x*y^2*z^3 - 37527*x*y^2*z^2 - 31404*x*y^2*z - 9813*x*y^2 - 1647*x*y*z^5 - 12546*x*y*z^4 - 33687*x*y*z^3 - 42249*x*y*z^2 - 26349*x*y*z - 6464*x*y - 351*x*z^6 - 3294*x*z^5 - 11268*x*z^4 - 18904*x*z^3 - 17643*x*z^2 - 8708*x*z - 1749*x - 8*y^7 - 76*y^6*z - 104*y^6 - 306*y^5*z^2 - 852*y^5*z - 546*y^5 - 677*y^4*z^3 - 2886*y^4*z^2 - 3717*y^4*z - 1542*y^4 - 889*y^3*z^4 - 5171*y^3*z^3 - 10083*y^3*z^2 - 8336*y^3*z - 2561*y^3 - 693*y^2*z^5 - 5166*y^2*z^4 - 13614*y^2*z^3 - 16863*y^2*z^2 - 10359*y^2*z - 2485*y^2 - 297*y*z^6 - 2727*y*z^5 - 9141*y*z^4 - 15126*y*z^3 - 13932*y*z^2 - 6731*y*z - 1314*y - 54*z^7 - 594*z^6 - 2439*z^5 - 5075*z^4 - 6230*z^3 - 4543*z^2 - 1801*z - 1, -54*x^7 - 324*x^6*y - 432*x^6*z - 567*x^6 - 828*x^5*y^2 - 2214*x^5*y*z - 2871*x^5*y - 1458*x^5*z^2 - 3861*x^5*z - 2457*x^5 - 1168*x^4*y^3 - 4698*x^4*y^2*z - 6015*x^4*y^2 - 6210*x^4*y*z^2 - 16209*x^4*y*z - 10206*x^4*y - 2700*x^4*z^3 - 10800*x^4*z^2 - 13734*x^4*z - 5765*x^4 - 982*x^3*y^4 - 5282*x^3*y^3*z - 6673*x^3*y^3 - 10512*x^3*y^2*z^2 - 27027*x^3*y^2*z - 16827*x^3*y^2 - 9180*x^3*y*z^3 - 36126*x^3*y*z^2 - 45336*x^3*y*z - 18846*x^3*y - 2970*x^3*z^4 - 15930*x^3*z^3 - 30366*x^3*z^2 - 25204*x^3*z - 7960*x^3 - 492*x^2*y^5 - 3318*x^2*y^4*z - 4134*x^2*y^4 - 8838*x^2*y^3*z^2 - 22371*x^2*y^3*z - 13764*x^2*y^3 - 11628*x^2*y^2*z^3 - 44991*x^2*y^2*z^2 - 55698*x^2*y^2*z - 22902*x^2*y^2 - 7560*x^2*y*z^4 - 39834*x^2*y*z^3 - 74772*x^2*y*z^2 - 61356*x^2*y*z - 19116*x^2*y - 1944*x^2*z^5 - 13095*x^2*z^4 - 33264*x^2*z^3 - 41022*x^2*z^2 - 25476*x^2*z - 6362*x^2 - 136*x*y^6 - 1104*x*y^5*z - 1356*x*y^5 - 3690*x*y^4*z^2 - 9192*x*y^4*z - 5586*x*y^4 - 6502*x*y^3*z^3 - 24723*x*y^3*z^2 - 30186*x*y^3*z - 12263*x*y^3 - 6372*x*y^2*z^4 - 32961*x*y^2*z^3 - 60915*x*y^2*z^2 - 49389*x*y^2*z - 15147*x*y^2 - 3294*x*y*z^5 - 21771*x*y*z^4 - 54360*x*y*z^3 - 66174*x*y*z^2 - 40548*x*y*z - 9898*x*y - 702*x*z^6 - 5697*x*z^5 - 18081*x*z^4 - 29492*x*z^3 - 27072*x*z^2 - 13300*x*z - 2660*x - 16*y^7 - 152*y^6*z - 184*y^6 - 612*y^5*z^2 - 1500*y^5*z - 900*y^5 - 1354*y^4*z^3 - 5058*y^4*z^2 - 6090*y^4*z - 2442*y^4 - 1778*y^3*z^4 - 9025*y^3*z^3 - 16422*y^3*z^2 - 13147*y^3*z - 3964*y^3 - 1386*y^2*z^5 - 8982*y^2*z^4 - 22044*y^2*z^3 - 26487*y^2*z^2 - 15984*y^2*z - 3815*y^2 - 594*y*z^6 - 4725*y*z^5 - 14718*y*z^4 - 23664*y*z^3 - 21432*y*z^2 - 10306*y*z - 2006*y - 108*z^7 - 1026*z^6 - 3906*z^5 - 7909*z^4 - 9556*z^3 - 6938*z^2 - 2740*z - 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ 3, 2, 3 ], [ 1, 1, 1 ], [ 1, 1, 2 ] ], "source_shift": [ 1, 2, 1 ], "target_matrix": [ [ 2, 2, 3 ], [ -1, -1, -2 ], [ -2, -1, -1 ] ], "target_shift": [ -576, 285, 437 ], "collision_preimages": [ [ "9/4", "-5", "3/4" ], [ "-2", "-21/2", "9" ], [ "-7", "1/2", "6" ] ], "collision_image": [ "-1153/2", "1141/4", "875/2" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 0.9927310620005301 ], "surface_features": [ 4.11766342020905, 4.965812953036603, 3.8130469651601078, 0.0, 4.21290416029156, 5117 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C157", "pair": 157, "family": "counterexample", "mapping": "F(x,y,z)=(-2*x^7 - 13*x^6*y + 19*x^6*z + 35*x^6 - 36*x^5*y^2 + 105*x^5*y*z + 195*x^5*y - 75*x^5*z^2 - 279*x^5*z - 234*x^5 - 55*x^4*y^3 + 240*x^4*y^2*z + 450*x^4*y^2 - 342*x^4*y*z^2 - 1287*x^4*y*z - 1092*x^4*y + 159*x^4*z^3 + 903*x^4*z^2 + 1554*x^4*z + 842*x^4 - 50*x^3*y^4 + 290*x^3*y^3*z + 550*x^3*y^3 - 618*x^3*y^2*z^2 - 2358*x^3*y^2*z - 2028*x^3*y^2 + 573*x^3*y*z^3 + 3309*x^3*y*z^2 + 5781*x^3*y*z + 3162*x^3*y - 195*x^3*z^4 - 1521*x^3*z^3 - 4053*x^3*z^2 - 4491*x^3*z - 1802*x^3 - 27*x^2*y^5 + 195*x^2*y^4*z + 375*x^2*y^4 - 552*x^2*y^3*z^2 - 2142*x^2*y^3*z - 1872*x^2*y^3 + 765*x^2*y^2*z^3 + 4509*x^2*y^2*z^2 + 8019*x^2*y^2*z + 4434*x^2*y^2 - 519*x^2*y*z^4 - 4149*x^2*y*z^3 - 11274*x^2*y*z^2 - 12636*x^2*y*z - 5103*x^2*y + 138*x^2*z^5 + 1410*x^2*z^4 + 5205*x^2*z^3 + 8859*x^2*z^2 + 7227*x^2*z + 2320*x^2 - 8*x*y^6 + 69*x*y^5*z + 135*x*y^5 - 243*x*y^4*z^2 - 963*x*y^4*z - 858*x*y^4 + 447*x*y^3*z^3 + 2703*x*y^3*z^2 + 4911*x*y^3*z + 2750*x*y^3 - 453*x*y^2*z^4 - 3735*x*y^2*z^3 - 10389*x*y^2*z^2 - 11799*x*y^2*z - 4800*x*y^2 + 240*x*y*z^5 + 2544*x*y*z^4 + 9630*x*y*z^3 + 16617*x*y*z^2 + 13659*x*y*z + 4400*x*y - 52*x*z^6 - 684*x*z^5 - 3300*x*z^4 - 7679*x*z^3 - 9552*x*z^2 - 6202*x*z - 1673*x - y^7 + 10*y^6*z + 20*y^6 - 42*y^5*z^2 - 171*y^5*z - 156*y^5 + 96*y^4*z^3 + 600*y^4*z^2 + 1119*y^4*z + 636*y^4 - 129*y^3*z^4 - 1107*y^3*z^3 - 3168*y^3*z^2 - 3654*y^3*z - 1499*y^3 + 102*y^2*z^5 + 1134*y^2*z^4 + 4425*y^2*z^3 + 7758*y^2*z^2 + 6432*y^2*z + 2080*y^2 - 44*y*z^6 - 612*y*z^5 - 3048*y*z^4 - 7208*y*z^3 - 9045*y*z^2 - 5899*y*z - 1593*y + 8*z^7 + 136*z^6 + 828*z^5 + 2471*z^4 + 4163*z^3 + 4099*z^2 + 2226*z + 1, -2*x^7 - 13*x^6*y + 19*x^6*z + 29*x^6 - 36*x^5*y^2 + 105*x^5*y*z + 162*x^5*y - 75*x^5*z^2 - 234*x^5*z - 177*x^5 - 55*x^4*y^3 + 240*x^4*y^2*z + 375*x^4*y^2 - 342*x^4*y*z^2 - 1083*x^4*y*z - 828*x^4*y + 159*x^4*z^3 + 768*x^4*z^2 + 1191*x^4*z + 602*x^4 - 50*x^3*y^4 + 290*x^3*y^3*z + 460*x^3*y^3 - 618*x^3*y^2*z^2 - 1992*x^3*y^2*z - 1542*x^3*y^2 + 573*x^3*y*z^3 + 2826*x^3*y*z^2 + 4443*x^3*y*z + 2265*x^3*y - 195*x^3*z^4 - 1314*x^3*z^3 - 3150*x^3*z^2 - 3252*x^3*z - 1241*x^3 - 27*x^2*y^5 + 195*x^2*y^4*z + 315*x^2*y^4 - 552*x^2*y^3*z^2 - 1818*x^2*y^3*z - 1428*x^2*y^3 + 765*x^2*y^2*z^3 + 3870*x^2*y^2*z^2 + 6183*x^2*y^2*z + 3183*x^2*y^2 - 519*x^2*y*z^4 - 3603*x^2*y*z^3 - 8790*x^2*y*z^2 - 9168*x^2*y*z - 3519*x^2*y + 138*x^2*z^5 + 1239*x^2*z^4 + 4104*x^2*z^3 + 6498*x^2*z^2 + 5040*x^2*z + 1558*x^2 - 8*x*y^6 + 69*x*y^5*z + 114*x*y^5 - 243*x*y^4*z^2 - 822*x*y^4*z - 657*x*y^4 + 447*x*y^3*z^3 + 2334*x*y^3*z^2 + 3801*x*y^3*z + 1979*x*y^3 - 453*x*y^2*z^4 - 3264*x*y^2*z^3 - 8130*x*y^2*z^2 - 8580*x*y^2*z - 3315*x*y^2 + 240*x*y*z^5 + 2250*x*y*z^4 + 7620*x*y*z^3 + 12213*x*y*z^2 + 9537*x*y*z + 2957*x*y - 52*x*z^6 - 612*x*z^5 - 2640*x*z^4 - 5705*x*z^3 - 6744*x*z^2 - 4219*x*z - 1108*x - y^7 + 10*y^6*z + 17*y^6 - 42*y^5*z^2 - 147*y^5*z - 120*y^5 + 96*y^4*z^3 + 522*y^4*z^2 + 870*y^4*z + 459*y^4 - 129*y^3*z^4 - 975*y^3*z^3 - 2490*y^3*z^2 - 2664*y^3*z - 1037*y^3 + 102*y^2*z^5 + 1011*y^2*z^4 + 3516*y^2*z^3 + 5715*y^2*z^2 + 4497*y^2*z + 1399*y^2 - 44*y*z^6 - 552*y*z^5 - 2448*y*z^4 - 5366*y*z^3 - 6393*y*z^2 - 4015*y*z - 1055*y + 8*z^7 + 124*z^6 + 672*z^5 + 1859*z^4 + 2975*z^3 + 2824*z^2 + 1495*z + 1, -4*x^7 - 26*x^6*y + 38*x^6*z + 52*x^6 - 72*x^5*y^2 + 210*x^5*y*z + 291*x^5*y - 150*x^5*z^2 - 423*x^5*z - 297*x^5 - 110*x^4*y^3 + 480*x^4*y^2*z + 675*x^4*y^2 - 684*x^4*y*z^2 - 1962*x^4*y*z - 1392*x^4*y + 318*x^4*z^3 + 1401*x^4*z^2 + 2019*x^4*z + 966*x^4 - 100*x^3*y^4 + 580*x^3*y^3*z + 830*x^3*y^3 - 1236*x^3*y^2*z^2 - 3618*x^3*y^2*z - 2598*x^3*y^2 + 1146*x^3*y*z^3 + 5169*x^3*y*z^2 + 7548*x^3*y*z + 3640*x^3*y - 390*x^3*z^4 - 2421*x^3*z^3 - 5397*x^3*z^2 - 5272*x^3*z - 1932*x^3 - 54*x^2*y^5 + 390*x^2*y^4*z + 570*x^2*y^4 - 1104*x^2*y^3*z^2 - 3312*x^2*y^3*z - 2412*x^2*y^3 + 1530*x^2*y^2*z^3 + 7101*x^2*y^2*z^2 + 10530*x^2*y^2*z + 5124*x^2*y^2 - 1038*x^2*y*z^4 - 6660*x^2*y*z^3 - 15096*x^2*y*z^2 - 14886*x^2*y*z - 5484*x^2*y + 276*x^2*z^5 + 2307*x^2*z^4 + 7107*x^2*z^3 + 10644*x^2*z^2 + 7926*x^2*z + 2378*x^2 - 16*x*y^6 + 138*x*y^5*z + 207*x*y^5 - 486*x*y^4*z^2 - 1503*x*y^4*z - 1113*x*y^4 + 894*x*y^3*z^3 + 4299*x*y^3*z^2 + 6492*x*y^3*z + 3192*x*y^3 - 906*x*y^2*z^4 - 6057*x*y^2*z^3 - 14001*x*y^2*z^2 - 13956*x*y^2*z - 5172*x*y^2 + 480*x*y*z^5 + 4206*x*y*z^4 + 13230*x*y*z^3 + 20037*x*y*z^2 + 15012*x*y*z + 4516*x*y - 104*x*z^6 - 1152*x*z^5 - 4620*x*z^4 - 9441*x*z^3 - 10713*x*z^2 - 6506*x*z - 1672*x - 2*y^7 + 20*y^6*z + 31*y^6 - 84*y^5*z^2 - 270*y^5*z - 204*y^5 + 192*y^4*z^3 + 966*y^4*z^2 + 1491*y^4*z + 742*y^4 - 258*y^3*z^4 - 1818*y^3*z^3 - 4302*y^3*z^2 - 4342*y^3*z - 1620*y^3 + 204*y^2*z^5 + 1899*y^2*z^4 + 6123*y^2*z^3 + 9393*y^2*z^2 + 7086*y^2*z + 2138*y^2 - 88*y*z^6 - 1044*y*z^5 - 4296*y*z^4 - 8894*y*z^3 - 10164*y*z^2 - 6194*y*z - 1592*y + 16*z^7 + 236*z^6 + 1188*z^5 + 3107*z^4 + 4773*z^3 + 4400*z^2 + 2282*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -1, -1, 1 ], [ -1, -1, 2 ], [ -2, -1, 1 ] ], "source_shift": [ 1, 2, 2 ], "target_matrix": [ [ 1, 2, 1 ], [ 1, 1, 1 ], [ 2, 1, 1 ] ], "target_shift": [ -519, -343, -517 ], "collision_preimages": [ [ "5/4", "-5/4", "-1" ], [ "-9/2", "1", "-7/2" ], [ "-13/2", "10", "3/2" ] ], "collision_image": [ "-2077/4", "-1373/4", "-1035/2" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 0.9875584850015002 ], "surface_features": [ 3.42441924972109, 4.301832698184397, 3.060320028688285, 0.0, 3.5356406525377446, 4830 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C164", "pair": 164, "family": "counterexample", "mapping": "F(x,y,z)=(16*x^7 - 152*x^6*y - 136*x^6*z + 176*x^6 + 612*x^5*y^2 + 1104*x^5*y*z - 1416*x^5*y + 492*x^5*z^2 - 1296*x^5*z + 756*x^5 - 1354*x^4*y^3 - 3690*x^4*y^2*z + 4716*x^4*y^2 - 3318*x^4*y*z^2 + 8664*x^4*y*z - 5010*x^4*y - 982*x^4*z^3 + 3948*x^4*z^2 - 4686*x^4*z + 1660*x^4 + 1778*x^3*y^4 + 6502*x^3*y^3*z - 8318*x^3*y^3 + 8838*x^3*y^2*z^2 - 23004*x^3*y^2*z + 13218*x^3*y^2 + 5282*x^3*y*z^3 - 21054*x^3*y*z^2 + 24750*x^3*y*z - 8748*x^3*y + 1168*x^3*z^4 - 6368*x^3*z^3 + 11532*x^3*z^2 - 8298*x^3*z + 2090*x^3 - 1386*x^2*y^5 - 6372*x^2*y^4*z + 8190*x^2*y^4 - 11628*x^2*y^3*z^2 + 30306*x^2*y^3*z - 17346*x^2*y^3 - 10512*x^2*y^2*z^3 + 41778*x^2*y^2*z^2 - 48774*x^2*y^2*z + 17202*x^2*y^2 - 4698*x^2*y*z^4 + 25398*x^2*y*z^3 - 45510*x^2*y*z^2 + 32634*x^2*y*z - 8238*x^2*y - 828*x^2*z^5 + 5736*x^2*z^4 - 14082*x^2*z^3 + 15432*x^2*z^2 - 7872*x^2*z + 1586*x^2 + 594*x*y^6 + 3294*x*y^5*z - 4266*x*y^5 + 7560*x*y^4*z^2 - 19800*x*y^4*z + 11316*x*y^4 + 9180*x*y^3*z^3 - 36540*x*y^3*z^2 + 42480*x*y^3*z - 14949*x*y^3 + 6210*x*y^2*z^4 - 33480*x*y^2*z^3 + 59544*x*y^2*z^2 - 42549*x*y^2*z + 10764*x*y^2 + 2214*x*y*z^5 - 15210*x*y*z^4 + 36912*x*y*z^3 - 40251*x*y*z^2 + 20556*x*y*z - 4177*x*y + 324*x*z^6 - 2736*x*z^5 + 8532*x*z^4 - 12651*x*z^3 + 9792*x*z^2 - 4003*x*z + 710*x - 108*y^7 - 702*y^6*z + 918*y^6 - 1944*y^5*z^2 + 5130*y^5*z - 2934*y^5 - 2970*y^4*z^3 + 11880*y^4*z^2 - 13788*y^4*z + 4840*y^4 - 2700*y^3*z^4 + 14580*y^3*z^3 - 25812*y^3*z^2 + 18377*y^3*z - 4659*y^3 - 1458*y^2*z^5 + 9990*y^2*z^4 - 24048*y^2*z^3 + 26091*y^2*z^2 - 13338*y^2*z + 2735*y^2 - 432*y*z^6 + 3618*y*z^5 - 11142*y*z^4 + 16411*y*z^3 - 12699*y*z^2 + 5233*y*z - 947*y - 54*z^7 + 540*z^6 - 2052*z^5 + 3857*z^4 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7001*x^3*y^3 - 8838*x^3*y^2*z^2 + 19455*x^3*y^2*z - 10140*x^3*y^2 - 5282*x^3*y*z^3 + 17907*x^3*y*z^2 - 19038*x^3*y*z + 6431*x^3*y - 1168*x^3*z^4 + 5453*x^3*z^3 - 8898*x^3*z^2 + 6103*x^3*z - 1522*x^3 + 1386*x^2*y^5 + 6372*x^2*y^4*z - 6840*x^2*y^4 + 11628*x^2*y^3*z^2 - 25419*x^2*y^3*z + 13215*x^2*y^3 + 10512*x^2*y^2*z^3 - 35217*x^2*y^2*z^2 + 37254*x^2*y^2*z - 12591*x^2*y^2 + 4698*x^2*y*z^4 - 21537*x^2*y*z^3 + 34863*x^2*y*z^2 - 23895*x^2*y*z + 5985*x^2*y + 828*x^2*z^5 - 4899*x^2*z^4 + 10824*x^2*z^3 - 11304*x^2*z^2 + 5718*x^2*z - 1157*x^2 - 594*x*y^6 - 3294*x*y^5*z + 3537*x*y^5 - 7560*x*y^4*z^2 + 16479*x*y^4*z - 8562*x*y^4 - 9180*x*y^3*z^3 + 30546*x*y^3*z^2 - 32220*x*y^3*z + 10899*x*y^3 - 6210*x*y^2*z^4 + 28134*x*y^2*z^3 - 45288*x*y^2*z^2 + 31029*x*y^2*z - 7806*x*y^2 - 2214*x*y*z^5 + 12861*x*y*z^4 - 28164*x*y*z^3 + 29361*x*y*z^2 - 14904*x*y*z + 3043*x*y - 324*x*z^6 + 2331*x*z^5 - 6534*x*z^4 + 9231*x*z^3 - 7098*x*z^2 + 2917*x*z - 514*x + 108*y^7 + 702*y^6*z - 756*y^6 + 1944*y^5*z^2 - 4239*y^5*z + 2205*y^5 + 2970*y^4*z^3 - 9855*y^4*z^2 + 10386*y^4*z - 3517*y^4 + 2700*y^3*z^4 - 12150*y^3*z^3 + 19494*y^3*z^2 - 13355*y^3*z + 3375*y^3 + 1458*y^2*z^5 - 8370*y^2*z^4 + 18216*y^2*z^3 - 18963*y^2*z^2 + 9660*y^2*z - 1991*y^2 + 432*y*z^6 - 3051*y*z^5 + 8469*y*z^4 - 11929*y*z^3 + 9195*y*z^2 - 3811*y*z + 683*y + 54*z^7 - 459*z^6 + 1566*z^5 - 2804*z^4 + 2910*z^3 - 1820*z^2 + 656*z - 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ 2, -3, -3 ], [ 1, -1, -1 ], [ 1, -2, -1 ] ], "source_shift": [ 1, 1, 2 ], "target_matrix": [ [ 2, 2, 3 ], [ -1, -1, -2 ], [ -2, -1, -1 ] ], "target_shift": [ -142, 69, 102 ], "collision_preimages": [ [ "-2", "5/4", "-9/4" ], [ "-15/2", "-7", "2" ], [ "7/2", "-4", "7" ] ], "collision_image": [ "-285/2", "277/4", "205/2" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 0.8123321730017778 ], "surface_features": [ 3.906019563837884, 4.774838005983815, 3.63002085111341, 0.0, 3.994484606953852, 5053 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B138", "pair": 138, "family": "bijection", "mapping": "F(x,y,z)=(-256*x^7 - 896*x^6*y + 896*x^6*z + 1856*x^6 - 1344*x^5*y^2 + 2688*x^5*y*z + 5664*x^5*y - 1344*x^5*z^2 - 5600*x^5*z - 5696*x^5 - 1120*x^4*y^3 + 3360*x^4*y^2*z + 7200*x^4*y^2 - 3360*x^4*y*z^2 - 14240*x^4*y*z - 14720*x^4*y + 1120*x^4*z^3 + 7040*x^4*z^2 + 14400*x^4*z + 9632*x^4 - 560*x^3*y^4 + 2240*x^3*y^3*z + 4880*x^3*y^3 - 3360*x^3*y^2*z^2 - 14480*x^3*y^2*z - 15200*x^3*y^2 + 2240*x^3*y*z^3 + 14320*x^3*y*z^2 + 29760*x^3*y*z + 20224*x^3*y - 560*x^3*z^4 - 4720*x^3*z^3 - 14560*x^3*z^2 - 19584*x^3*z - 9760*x^3 - 168*x^2*y^5 + 840*x^2*y^4*z + 1860*x^2*y^4 - 1680*x^2*y^3*z^2 - 7360*x^2*y^3*z - 7840*x^2*y^3 + 1680*x^2*y^2*z^3 + 10920*x^2*y^2*z^2 + 23040*x^2*y^2*z + 15888*x^2*y^2 - 840*x^2*y*z^4 - 7200*x^2*y*z^3 - 22560*x^2*y*z^2 - 30816*x^2*y*z - 15616*x^2*y + 168*x^2*z^5 + 1780*x^2*z^4 + 7360*x^2*z^3 + 14928*x^2*z^2 + 14968*x^2*z + 6010*x^2 - 28*x*y^6 + 168*x*y^5*z + 378*x*y^5 - 420*x*y^4*z^2 - 1870*x*y^4*z - 2020*x*y^4 + 560*x*y^3*z^3 + 3700*x*y^3*z^2 + 7920*x*y^3*z + 5536*x*y^3 - 420*x*y^2*z^4 - 3660*x*y^2*z^3 - 11640*x*y^2*z^2 - 16128*x*y^2*z - 8296*x*y^2 + 168*x*y*z^5 + 1810*x*y*z^4 + 7600*x*y*z^3 + 15648*x*y*z^2 + 15944*x*y*z + 6532*x*y - 28*x*z^6 - 358*x*z^5 - 1860*x*z^4 - 5056*x*z^3 - 7648*x*z^2 - 6190*x*z - 2137*x - 2*y^7 + 14*y^6*z + 32*y^6 - 42*y^5*z^2 - 190*y^5*z - 208*y^5 + 70*y^4*z^3 + 470*y^4*z^2 + 1020*y^4*z + 722*y^4 - 70*y^3*z^4 - 620*y^3*z^3 - 2000*y^3*z^2 - 2808*y^3*z - 1464*y^3 + 42*y^2*z^5 + 460*y^2*z^4 + 1960*y^2*z^3 + 4092*y^2*z^2 + 4230*y^2*z + 1765*y^2 - 14*y*z^6 - 182*y*z^5 - 960*y*z^4 - 2648*y*z^3 - 4068*y*z^2 - 3358*y*z - 1200*y + 2*z^7 + 30*z^6 + 188*z^5 + 642*z^4 + 1302*z^3 + 1593*z^2 + 1115*z + 1, -128*x^7 - 448*x^6*y + 448*x^6*z + 928*x^6 - 672*x^5*y^2 + 1344*x^5*y*z + 2832*x^5*y 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7584*x*y*z^2 + 7492*x*y*z + 2948*x*y - 14*x*z^6 - 179*x*z^5 - 920*x*z^4 - 2448*x*z^3 - 3584*x*z^2 - 2777*x*z - 911*x - y^7 + 7*y^6*z + 16*y^6 - 21*y^5*z^2 - 95*y^5*z - 103*y^5 + 35*y^4*z^3 + 235*y^4*z^2 + 505*y^4*z + 351*y^4 - 35*y^3*z^4 - 310*y^3*z^3 - 990*y^3*z^2 - 1364*y^3*z - 692*y^3 + 21*y^2*z^5 + 230*y^2*z^4 + 970*y^2*z^3 + 1986*y^2*z^2 + 1995*y^2*z + 803*y^2 - 7*y*z^6 - 91*y*z^5 - 475*y*z^4 - 1284*y*z^3 - 1914*y*z^2 - 1520*y*z - 523*y + z^7 + 15*z^6 + 93*z^5 + 311*z^4 + 611*z^3 + 717*z^2 + 479*z + 1, -128*x^7 - 448*x^6*y + 448*x^6*z + 928*x^6 - 672*x^5*y^2 + 1344*x^5*y*z + 2832*x^5*y - 672*x^5*z^2 - 2800*x^5*z - 2816*x^5 - 560*x^4*y^3 + 1680*x^4*y^2*z + 3600*x^4*y^2 - 1680*x^4*y*z^2 - 7120*x^4*y*z - 7280*x^4*y + 560*x^4*z^3 + 3520*x^4*z^2 + 7120*x^4*z + 4656*x^4 - 280*x^3*y^4 + 1120*x^3*y^3*z + 2440*x^3*y^3 - 1680*x^3*y^2*z^2 - 7240*x^3*y^2*z - 7520*x^3*y^2 + 1120*x^3*y*z^3 + 7160*x^3*y*z^2 + 14720*x^3*y*z + 9792*x^3*y - 280*x^3*z^4 - 2360*x^3*z^3 - 7200*x^3*z^2 - 9472*x^3*z - 4560*x^3 - 84*x^2*y^5 + 420*x^2*y^4*z + 930*x^2*y^4 - 840*x^2*y^3*z^2 - 3680*x^2*y^3*z - 3880*x^2*y^3 + 840*x^2*y^2*z^3 + 5460*x^2*y^2*z^2 + 11400*x^2*y^2*z + 7704*x^2*y^2 - 420*x^2*y*z^4 - 3600*x^2*y*z^3 - 11160*x^2*y*z^2 - 14928*x^2*y*z - 7328*x^2*y + 84*x^2*z^5 + 890*x^2*z^4 + 3640*x^2*z^3 + 7224*x^2*z^2 + 7004*x^2*z + 2683*x^2 - 14*x*y^6 + 84*x*y^5*z + 189*x*y^5 - 210*x*y^4*z^2 - 935*x*y^4*z - 1000*x*y^4 + 280*x*y^3*z^3 + 1850*x*y^3*z^2 + 3920*x*y^3*z + 2688*x*y^3 - 210*x*y^2*z^4 - 1830*x*y^2*z^3 - 5760*x*y^2*z^2 - 7824*x*y^2*z - 3908*x*y^2 + 84*x*y*z^5 + 905*x*y*z^4 + 3760*x*y*z^3 + 7584*x*y*z^2 + 7492*x*y*z + 2944*x*y - 14*x*z^6 - 179*x*z^5 - 920*x*z^4 - 2448*x*z^3 - 3584*x*z^2 - 2773*x*z - 902*x - y^7 + 7*y^6*z + 16*y^6 - 21*y^5*z^2 - 95*y^5*z - 103*y^5 + 35*y^4*z^3 + 235*y^4*z^2 + 505*y^4*z + 351*y^4 - 35*y^3*z^4 - 310*y^3*z^3 - 990*y^3*z^2 - 1364*y^3*z - 692*y^3 + 21*y^2*z^5 + 230*y^2*z^4 + 970*y^2*z^3 + 1986*y^2*z^2 + 1995*y^2*z + 802*y^2 - 7*y*z^6 - 91*y*z^5 - 475*y*z^4 - 1284*y*z^3 - 1914*y*z^2 - 1518*y*z - 517*y + z^7 + 15*z^6 + 93*z^5 + 311*z^4 + 611*z^3 + 716*z^2 + 474*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -2, -1, 1 ], [ -1, -2, 1 ], [ -1, -1, 1 ] ], "source_shift": [ 2, 1, 2 ], "target_matrix": [ [ 2, 1, 1 ], [ 1, 2, 1 ], [ 1, 1, 1 ] ], "target_shift": [ -348, -141, -136 ], "inverse_recipe": { "p": 2, "q": 5, "f": "x**2", "g": "x**5 + x**2 + x", "h": "y*z + 2*y", "polynomials": { "f": [ [ [ 2, 0, 0 ], 1 ] ], "g": [ [ [ 5, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 1, 0 ], 2 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 0.9873169339989545 ], "surface_features": [ 3.535689616365434, 4.48877626544444, 3.13481437032046, 0.0, 3.6781367929589637, 4844 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B059", "pair": 59, "family": "bijection", "mapping": "F(x,y,z)=(2*x^7 - 28*x^6*y - 14*x^6*z + 18*x^6 + 168*x^5*y^2 + 168*x^5*y*z - 214*x^5*y + 42*x^5*z^2 - 108*x^5*z + 68*x^5 - 560*x^4*y^3 - 840*x^4*y^2*z + 1060*x^4*y^2 - 420*x^4*y*z^2 + 1070*x^4*y*z - 668*x^4*y - 70*x^4*z^3 + 270*x^4*z^2 - 340*x^4*z + 144*x^4 + 1120*x^3*y^4 + 2240*x^3*y^3*z - 2800*x^3*y^3 + 1680*x^3*y^2*z^2 - 4240*x^3*y^2*z + 2624*x^3*y^2 + 560*x^3*y*z^3 - 2140*x^3*y*z^2 + 2672*x^3*y*z - 1124*x^3*y + 70*x^3*z^4 - 360*x^3*z^3 + 680*x^3*z^2 - 576*x^3*z + 196*x^3 - 1344*x^2*y^5 - 3360*x^2*y^4*z + 4160*x^2*y^4 - 3360*x^2*y^3*z^2 + 8400*x^2*y^3*z - 5152*x^2*y^3 - 1680*x^2*y^2*z^3 + 6360*x^2*y^2*z^2 - 7872*x^2*y^2*z + 3288*x^2*y^2 - 420*x^2*y*z^4 + 2140*x^2*y*z^3 - 4008*x^2*y*z^2 + 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7*x*z^6 + 54*x*z^5 - 165*x*z^4 + 264*x*z^3 - 250*x*z^2 + 147*x*z - 44*x + 128*y^7 + 448*y^6*z - 544*y^6 + 672*y^5*z^2 - 1648*y^5*z + 960*y^5 + 560*y^4*z^3 - 2080*y^4*z^2 + 2448*y^4*z - 944*y^4 + 280*y^3*z^4 - 1400*y^3*z^3 + 2496*y^3*z^2 - 1944*y^3*z + 584*y^3 + 84*y^2*z^5 - 530*y^2*z^4 + 1272*y^2*z^3 - 1500*y^2*z^2 + 916*y^2*z - 248*y^2 + 14*y*z^6 - 107*y*z^5 + 324*y*z^4 - 514*y*z^3 + 478*y*z^2 - 267*y*z + 75*y + z^7 - 9*z^6 + 33*z^5 - 66*z^4 + 83*z^3 - 72*z^2 + 43*z - 1, -3*x^7 + 42*x^6*y + 21*x^6*z - 27*x^6 - 252*x^5*y^2 - 252*x^5*y*z + 321*x^5*y - 63*x^5*z^2 + 162*x^5*z - 101*x^5 + 840*x^4*y^3 + 1260*x^4*y^2*z - 1590*x^4*y^2 + 630*x^4*y*z^2 - 1605*x^4*y*z + 992*x^4*y + 105*x^4*z^3 - 405*x^4*z^2 + 505*x^4*z - 210*x^4 - 1680*x^3*y^4 - 3360*x^3*y^3*z + 4200*x^3*y^3 - 2520*x^3*y^2*z^2 + 6360*x^3*y^2*z - 3896*x^3*y^2 - 840*x^3*y*z^3 + 3210*x^3*y*z^2 - 3968*x^3*y*z + 1638*x^3*y - 105*x^3*z^4 + 540*x^3*z^3 - 1010*x^3*z^2 + 840*x^3*z - 280*x^3 + 2016*x^2*y^5 + 5040*x^2*y^4*z - 6240*x^2*y^4 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1602*y*z^2 - 921*y*z + 251*y + 3*z^7 - 27*z^6 + 101*z^5 - 210*z^4 + 277*z^3 - 246*z^2 + 141*z - 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ 1, -2, -1 ], [ 1, -1, -1 ], [ 2, -1, -1 ] ], "source_shift": [ 1, 1, 1 ], "target_matrix": [ [ 2, 1, 1 ], [ -1, -2, -1 ], [ -3, -2, -2 ] ], "target_shift": [ -27, 12, 38 ], "inverse_recipe": { "p": 2, "q": 5, "f": "x**2", "g": "x**5 + x**4 + x**2", "h": "y**2 + y*z + z", "polynomials": { "f": [ [ [ 2, 0, 0 ], 1 ] ], "g": [ [ [ 5, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ], [ [ 4, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 0, 1 ], 1 ], [ [ 0, 2, 0 ], 1 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": 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36528*x^2*y^2 - 68040*x^2*y*z^4 + 189000*x^2*y*z^3 - 210960*x^2*y*z^2 + 110076*x^2*y*z - 22360*x^2*y + 20412*x^2*z^5 - 71010*x^2*z^4 + 105840*x^2*z^3 - 82926*x^2*z^2 + 33704*x^2*z - 5635*x^2 + 896*x*y^6 - 8064*x*y^5*z + 5568*x*y^5 + 30240*x*y^4*z^2 - 41840*x*y^4*z + 15520*x*y^4 - 60480*x*y^3*z^3 + 125760*x*y^3*z^2 - 93440*x*y^3*z + 24352*x*y^3 + 68040*x*y^2*z^4 - 189000*x*y^2*z^3 + 210960*x*y^2*z^2 - 110076*x*y^2*z + 22364*x*y^2 - 40824*x*y*z^5 + 142020*x*y*z^4 - 211680*x*y*z^3 + 165852*x*y*z^2 - 67420*x*y*z + 11279*x*y + 10206*x*z^6 - 42687*x*z^5 + 79650*x*z^4 - 83295*x*z^3 + 50811*x*z^2 - 16994*x*z + 2437*x - 128*y^7 + 1344*y^6*z - 928*y^6 - 6048*y^5*z^2 + 8368*y^5*z - 3104*y^5 + 15120*y^4*z^3 - 31440*y^4*z^2 + 23360*y^4*z - 6088*y^4 - 22680*y^3*z^4 + 63000*y^3*z^3 - 70320*y^3*z^2 + 36692*y^3*z - 7456*y^3 + 20412*y^2*z^5 - 71010*y^2*z^4 + 105840*y^2*z^3 - 82926*y^2*z^2 + 33716*y^2*z - 5644*y^2 - 10206*y*z^6 + 42687*y*z^5 - 79650*y*z^4 + 83295*y*z^3 - 50820*y*z^2 + 17008*y*z - 2448*y + 2187*z^7 - 10692*z^6 + 23976*z^5 - 31374*z^4 + 25533*z^3 - 12812*z^2 + 3671*z - 1, 128*x^7 - 896*x^6*y + 1344*x^6*z - 928*x^6 + 2688*x^5*y^2 - 8064*x^5*y*z + 5568*x^5*y + 6048*x^5*z^2 - 8368*x^5*z + 3072*x^5 - 4480*x^4*y^3 + 20160*x^4*y^2*z - 13920*x^4*y^2 - 30240*x^4*y*z^2 + 41840*x^4*y*z - 15360*x^4*y + 15120*x^4*z^3 - 31440*x^4*z^2 + 23120*x^4*z - 5928*x^4 + 4480*x^3*y^4 - 26880*x^3*y^3*z + 18560*x^3*y^3 + 60480*x^3*y^2*z^2 - 83680*x^3*y^2*z + 30720*x^3*y^2 - 60480*x^3*y*z^3 + 125760*x^3*y*z^2 - 92480*x^3*y*z + 23712*x^3*y + 22680*x^3*z^4 - 63000*x^3*z^3 + 69600*x^3*z^2 - 35732*x^3*z + 7124*x^3 - 2688*x^2*y^5 + 20160*x^2*y^4*z - 13920*x^2*y^4 - 60480*x^2*y^3*z^2 + 83680*x^2*y^3*z - 30720*x^2*y^3 + 90720*x^2*y^2*z^3 - 188640*x^2*y^2*z^2 + 138720*x^2*y^2*z - 35568*x^2*y^2 - 68040*x^2*y*z^4 + 189000*x^2*y*z^3 - 208800*x^2*y*z^2 + 107196*x^2*y*z - 21376*x^2*y + 20412*x^2*z^5 - 71010*x^2*z^4 + 104760*x^2*z^3 - 80766*x^2*z^2 + 32228*x^2*z - 5283*x^2 + 896*x*y^6 - 8064*x*y^5*z + 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4480*x^4*y^3 - 20160*x^4*y^2*z + 13920*x^4*y^2 + 30240*x^4*y*z^2 - 41840*x^4*y*z + 15360*x^4*y - 15120*x^4*z^3 + 31440*x^4*z^2 - 23120*x^4*z + 5928*x^4 - 4480*x^3*y^4 + 26880*x^3*y^3*z - 18560*x^3*y^3 - 60480*x^3*y^2*z^2 + 83680*x^3*y^2*z - 30720*x^3*y^2 + 60480*x^3*y*z^3 - 125760*x^3*y*z^2 + 92480*x^3*y*z - 23712*x^3*y - 22680*x^3*z^4 + 63000*x^3*z^3 - 69600*x^3*z^2 + 35732*x^3*z - 7124*x^3 + 2688*x^2*y^5 - 20160*x^2*y^4*z + 13920*x^2*y^4 + 60480*x^2*y^3*z^2 - 83680*x^2*y^3*z + 30720*x^2*y^3 - 90720*x^2*y^2*z^3 + 188640*x^2*y^2*z^2 - 138720*x^2*y^2*z + 35568*x^2*y^2 + 68040*x^2*y*z^4 - 189000*x^2*y*z^3 + 208800*x^2*y*z^2 - 107196*x^2*y*z + 21376*x^2*y - 20412*x^2*z^5 + 71010*x^2*z^4 - 104760*x^2*z^3 + 80766*x^2*z^2 - 32228*x^2*z + 5291*x^2 - 896*x*y^6 + 8064*x*y^5*z - 5568*x*y^5 - 30240*x*y^4*z^2 + 41840*x*y^4*z - 15360*x*y^4 + 60480*x*y^3*z^3 - 125760*x*y^3*z^2 + 92480*x*y^3*z - 23712*x*y^3 - 68040*x*y^2*z^4 + 189000*x*y^2*z^3 - 208800*x*y^2*z^2 + 107196*x*y^2*z - 21380*x*y^2 + 40824*x*y*z^5 - 142020*x*y*z^4 + 209520*x*y*z^3 - 161532*x*y*z^2 + 64468*x*y*z - 10591*x*y - 10206*x*z^6 + 42687*x*z^5 - 78840*x*z^4 + 81135*x*z^3 - 48597*x*z^2 + 15962*x*z - 2250*x + 128*y^7 - 1344*y^6*z + 928*y^6 + 6048*y^5*z^2 - 8368*y^5*z + 3072*y^5 - 15120*y^4*z^3 + 31440*y^4*z^2 - 23120*y^4*z + 5928*y^4 + 22680*y^3*z^4 - 63000*y^3*z^3 + 69600*y^3*z^2 - 35732*y^3*z + 7128*y^3 - 20412*y^2*z^5 + 71010*y^2*z^4 - 104760*y^2*z^3 + 80766*y^2*z^2 - 32240*y^2*z + 5300*y^2 + 10206*y*z^6 - 42687*y*z^5 + 78840*y*z^4 - 81135*y*z^3 + 48606*y*z^2 - 15976*y*z + 2260*y - 2187*z^7 + 10692*z^6 - 23733*z^5 + 30564*z^4 - 24426*z^3 + 12038*z^2 - 3391*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -2, 2, -3 ], [ -1, 2, -1 ], [ -1, 1, -2 ] ], "source_shift": [ 2, 2, 2 ], "target_matrix": [ [ -1, -2, -1 ], [ -1, -1, -2 ], [ 1, 1, 1 ] ], "target_shift": [ 473, 417, -429 ], "inverse_recipe": { "p": 5, "q": 2, "f": "x**5 + x**3 + x", "g": "x**2", "h": "y*z + 3*y", "polynomials": { "f": [ [ [ 5, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ], [ [ 3, 0, 0 ], 1 ] ], "g": [ [ [ 2, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 1, 0 ], 3 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 0.9593154309986858 ], "surface_features": [ 4.658045704035806, 5.325679826843426, 4.387193993696869, 0.0, 4.7033262992999445, 5326 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C070", "pair": 70, "family": "counterexample", "mapping": "F(x,y,z)=(-16*x^7 - 88*x^6*y + 144*x^6*z + 176*x^6 - 204*x^5*y^2 + 660*x^5*y*z + 804*x^5*y - 528*x^5*z^2 - 1296*x^5*z - 744*x^5 - 258*x^4*y^3 + 1236*x^4*y^2*z + 1512*x^4*y^2 - 1950*x^4*y*z^2 - 4824*x^4*y*z - 2754*x^4*y + 1012*x^4*z^3 + 3804*x^4*z^2 + 4368*x^4*z + 1590*x^4 - 192*x^3*y^4 + 1209*x^3*y^3*z + 1497*x^3*y^3 - 2817*x^3*y^2*z^2 - 7086*x^3*y^2*z - 4023*x^3*y^2 + 2875*x^3*y*z^3 + 11049*x^3*y*z^2 + 12597*x^3*y*z + 4587*x^3*y - 1083*x^3*z^4 - 5672*x^3*z^3 - 9720*x^3*z^2 - 7129*x^3*z - 1948*x^3 - 84*x^2*y^5 + 651*x^2*y^4*z + 822*x^2*y^4 - 1989*x^2*y^3*z^2 - 5127*x^2*y^3*z - 2898*x^2*y^3 + 2991*x^2*y^2*z^3 + 11844*x^2*y^2*z^2 + 13434*x^2*y^2*z + 4893*x^2*y^2 - 2211*x^2*y*z^4 - 12003*x^2*y*z^3 - 20457*x^2*y*z^2 - 14988*x^2*y*z - 4110*x^2*y + 642*x^2*z^5 + 4500*x^2*z^4 + 10227*x^2*z^3 + 11283*x^2*z^2 + 6237*x^2*z + 1399*x^2 - 20*x*y^6 + 183*x*y^5*z + 237*x*y^5 - 687*x*y^4*z^2 - 1824*x*y^4*z - 1029*x*y^4 + 1353*x*y^3*z^3 + 5541*x*y^3*z^2 + 6276*x*y^3*z + 2289*x*y^3 - 1473*x*y^2*z^4 - 8298*x*y^2*z^3 - 14145*x*y^2*z^2 - 10374*x*y^2*z - 2850*x*y^2 + 840*x*y*z^5 + 6120*x*y*z^4 + 13959*x*y*z^3 + 15420*x*y*z^2 + 8514*x*y*z + 1915*x*y - 196*x*z^6 - 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3167*y*z^3 - 2496*y*z^2 - 1091*y*z - 212*y + 24*z^7 + 200*z^6 + 642*z^5 + 1106*z^4 + 1158*z^3 + 757*z^2 + 295*z + 1, -16*x^7 - 88*x^6*y + 144*x^6*z + 152*x^6 - 204*x^5*y^2 + 660*x^5*y*z + 684*x^5*y - 528*x^5*z^2 - 1092*x^5*z - 564*x^5 - 258*x^4*y^3 + 1236*x^4*y^2*z + 1266*x^4*y^2 - 1950*x^4*y*z^2 - 3996*x^4*y*z - 2058*x^4*y + 1012*x^4*z^3 + 3114*x^4*z^2 + 3228*x^4*z + 1116*x^4 - 192*x^3*y^4 + 1209*x^3*y^3*z + 1233*x^3*y^3 - 2817*x^3*y^2*z^2 - 5769*x^3*y^2*z - 2964*x^3*y^2 + 2875*x^3*y*z^3 + 8883*x^3*y*z^2 + 9177*x^3*y*z + 3177*x^3*y - 1083*x^3*z^4 - 4499*x^3*z^3 - 6999*x^3*z^2 - 4882*x^3*z - 1297*x^3 - 84*x^2*y^5 + 651*x^2*y^4*z + 666*x^2*y^4 - 1989*x^2*y^3*z^2 - 4104*x^2*y^3*z - 2106*x^2*y^3 + 2991*x^2*y^2*z^3 + 9360*x^2*y^2*z^2 + 9654*x^2*y^2*z + 3345*x^2*y^2 - 2211*x^2*y*z^4 - 9360*x^2*y*z^3 - 14535*x^2*y*z^2 - 10131*x^2*y*z - 2700*x^2*y + 642*x^2*z^5 + 3462*x^2*z^4 + 7185*x^2*z^3 + 7539*x^2*z^2 + 4047*x^2*z + 895*x^2 - 20*x*y^6 + 183*x*y^5*z + 189*x*y^5 - 687*x*y^4*z^2 - 1437*x*y^4*z - 738*x*y^4 + 1353*x*y^3*z^3 + 4311*x*y^3*z^2 + 4452*x*y^3*z + 1545*x*y^3 - 1473*x*y^2*z^4 - 6375*x*y^2*z^3 - 9924*x*y^2*z^2 - 6924*x*y^2*z - 1848*x*y^2 + 840*x*y*z^5 + 4644*x*y*z^4 + 9687*x*y*z^3 + 10176*x*y*z^2 + 5454*x*y*z + 1207*x*y - 196*x*z^6 - 1332*x*z^5 - 3495*x*z^4 - 4908*x*z^3 - 3945*x*z^2 - 1750*x*z - 338*x - 2*y^7 + 21*y^6*z + 22*y^6 - 93*y^5*z^2 - 198*y^5*z - 102*y^5 + 225*y^4*z^3 + 732*y^4*z^2 + 759*y^4*z + 264*y^4 - 321*y^3*z^4 - 1422*y^3*z^3 - 2226*y^3*z^2 - 1556*y^3*z - 416*y^3 + 270*y^2*z^5 + 1530*y^2*z^4 + 3216*y^2*z^3 + 3384*y^2*z^2 + 1815*y^2*z + 403*y^2 - 124*y*z^6 - 864*y*z^5 - 2289*y*z^4 - 3219*y*z^3 - 2592*y*z^2 - 1154*y*z - 224*y + 24*z^7 + 200*z^6 + 642*z^5 + 1130*z^4 + 1214*z^3 + 811*z^2 + 315*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -1, -1, 2 ], [ -2, -1, 1 ], [ -2, -2, 3 ] ], "source_shift": [ 1, 1, 1 ], "target_matrix": [ [ 1, 2, 1 ], [ 1, 1, 2 ], [ 1, 1, 1 ] ], "target_shift": [ -87, -51, -53 ], "collision_preimages": [ [ "3/4", "-5/4", "-3/4" ], [ "8", "-19", "-11/2" ], [ "7", "-24", "-19/2" ] ], "collision_image": [ "-349/4", "-205/4", "-213/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 0.9986582910023571 ], "surface_features": [ 3.449402538878376, 4.310841945164296, 3.157456768134226, 0.0, 3.5302186724308253, 4896 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B135", "pair": 135, "family": "bijection", "mapping": "F(x,y,z)=(-384*x^7 - 2688*x^6*y - 4032*x^6*z - 2688*x^6 - 8064*x^5*y^2 - 24192*x^5*y*z - 16128*x^5*y - 18144*x^5*z^2 - 24192*x^5*z - 8208*x^5 - 13440*x^4*y^3 - 60480*x^4*y^2*z - 40320*x^4*y^2 - 90720*x^4*y*z^2 - 120960*x^4*y*z - 41040*x^4*y - 45360*x^4*z^3 - 90720*x^4*z^2 - 61584*x^4*z - 14288*x^4 - 13440*x^3*y^4 - 80640*x^3*y^3*z - 53760*x^3*y^3 - 181440*x^3*y^2*z^2 - 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7286*z^2 - 1846*z - 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ 2, 2, 3 ], [ 1, 1, 2 ], [ 2, 1, 1 ] ], "source_shift": [ 2, 2, 2 ], "target_matrix": [ [ -3, -2, -2 ], [ 1, 1, 1 ], [ -1, -1, -2 ] ], "target_shift": [ 801, -257, 217 ], "inverse_recipe": { "p": 3, "q": 4, "f": "x**3 + x", "g": "x**4 + x", "h": "y*z + 2*y + z", "polynomials": { "f": [ [ [ 3, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "g": [ [ [ 4, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 0, 1 ], 1 ], [ [ 0, 1, 0 ], 2 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 0.9782697230002668 ], "surface_features": [ 4.831781392650426, 5.744048770547747, 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15282*x^3*y^2*z^2 - 62736*x^3*y^2*z - 50676*x^3*y^2 - 8637*x^3*y*z^3 - 57852*x^3*y*z^2 - 98508*x^3*y*z - 49690*x^3*y - 1752*x^3*z^4 - 16992*x^3*z^3 - 45837*x^3*z^2 - 48460*x^3*z - 17786*x^3 - 2448*x^2*y^5 - 11664*x^2*y^4*z - 21312*x^2*y^4 - 21312*x^2*y^3*z^2 - 84216*x^2*y^3*z - 66180*x^2*y^3 - 18612*x^2*y^2*z^3 - 120060*x^2*y^2*z^2 - 198432*x^2*y^2*z - 97404*x^2*y^2 - 7767*x^2*y*z^4 - 72810*x^2*y*z^3 - 190557*x^2*y*z^2 - 195678*x^2*y*z - 69780*x^2*y - 1242*x^2*z^5 - 15801*x^2*z^4 - 58281*x^2*z^3 - 94200*x^2*z^2 - 70281*x^2*z - 19764*x^2 - 1056*x*y^6 - 6192*x*y^5*z - 10944*x*y^5 - 14544*x*y^4*z^2 - 55392*x*y^4*z - 42432*x*y^4 - 17448*x*y^3*z^3 - 108288*x*y^3*z^2 - 173976*x*y^3*z - 83264*x*y^3 - 11250*x*y^2*z^4 - 101664*x*y^2*z^3 - 258084*x*y^2*z^2 - 257652*x*y^2*z - 89442*x*y^2 - 3699*x*y*z^5 - 45588*x*y*z^4 - 163146*x*y*z^3 - 255948*x*y*z^2 - 185331*x*y*z - 50604*x*y - 486*x*z^6 - 7794*x*z^5 - 36867*x*z^4 - 81025*x*z^3 - 92226*x*z^2 - 52656*x*z - 11899*x - 192*y^7 - 1344*y^6*z - 2304*y^6 - 3888*y^5*z^2 - 14304*y^5*z - 10704*y^5 - 6000*y^4*z^3 - 35856*y^4*z^2 - 56112*y^4*z - 26240*y^4 - 5316*y^3*z^4 - 46224*y^3*z^3 - 113976*y^3*z^2 - 110840*y^3*z - 37540*y^3 - 2700*y^2*z^5 - 32136*y^2*z^4 - 111480*y^2*z^3 - 169860*y^2*z^2 - 119604*y^2*z - 31788*y^2 - 729*y*z^6 - 11358*y*z^5 - 52149*y*z^4 - 111178*y*z^3 - 122667*y*z^2 - 67914*y*z - 14903*y - 81*z^7 - 1593*z^6 - 9279*z^5 - 26011*z^4 - 40181*z^3 - 34956*z^2 - 16017*z - 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ 1, 2, 1 ], [ 2, 2, 3 ], [ 1, 1, 1 ] ], "source_shift": [ 2, 2, 2 ], "target_matrix": [ [ -1, -2, -1 ], [ 1, 1, 1 ], [ -3, -2, -2 ] ], "target_shift": [ 1905, -1219, 3009 ], "collision_preimages": [ [ "-5", "1/4", "5/2" ], [ "45/2", "-11/2", "-25/2" ], [ "43/2", "-15/2", "-19/2" ] ], "collision_image": [ "7621/4", "-4877/4", "12039/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 0.985523536997789 ], "surface_features": [ 4.461322036908564, 5.411761081153323, 4.089481202687437, 0.0, 4.62339434119316, 5188 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B228", "pair": 228, "family": "bijection", "mapping": "F(x,y,z)=(-128*x^7 - 448*x^6*y + 448*x^6*z + 992*x^6 - 672*x^5*y^2 + 1344*x^5*y*z + 2992*x^5*y - 672*x^5*z^2 - 2992*x^5*z - 3376*x^5 - 560*x^4*y^3 + 1680*x^4*y^2*z + 3760*x^4*y^2 - 1680*x^4*y*z^2 - 7520*x^4*y*z - 8544*x^4*y + 560*x^4*z^3 + 3760*x^4*z^2 + 8544*x^4*z + 6588*x^4 - 280*x^3*y^4 + 1120*x^3*y^3*z + 2520*x^3*y^3 - 1680*x^3*y^2*z^2 - 7560*x^3*y^2*z - 8648*x^3*y^2 + 1120*x^3*y*z^3 + 7560*x^3*y*z^2 + 17296*x^3*y*z + 13444*x^3*y - 280*x^3*z^4 - 2520*x^3*z^3 - 8648*x^3*z^2 - 13444*x^3*z - 8001*x^3 - 84*x^2*y^5 + 420*x^2*y^4*z + 950*x^2*y^4 - 840*x^2*y^3*z^2 - 3800*x^2*y^3*z - 4376*x^2*y^3 + 840*x^2*y^2*z^3 + 5700*x^2*y^2*z^2 + 13128*x^2*y^2*z + 10287*x^2*y^2 - 420*x^2*y*z^4 - 3800*x^2*y*z^3 - 13128*x^2*y*z^2 - 20574*x^2*y*z - 12371*x^2*y + 84*x^2*z^5 + 950*x^2*z^4 + 4376*x^2*z^3 + 10287*x^2*z^2 + 12367*x^2*z + 6077*x^2 - 14*x*y^6 + 84*x*y^5*z + 191*x*y^5 - 210*x*y^4*z^2 - 955*x*y^4*z - 1107*x*y^4 + 280*x*y^3*z^3 + 1910*x*y^3*z^2 + 4428*x*y^3*z + 3498*x*y^3 - 210*x*y^2*z^4 - 1910*x*y^2*z^3 - 6642*x*y^2*z^2 - 10494*x*y^2*z - 6377*x*y^2 + 84*x*y*z^5 + 955*x*y*z^4 + 4428*x*y*z^3 + 10494*x*y*z^2 + 12750*x*y*z + 6355*x*y - 14*x*z^6 - 191*x*z^5 - 1107*x*z^4 - 3498*x*z^3 - 6373*x*z^2 - 6346*x*z - 2688*x - y^7 + 7*y^6*z + 16*y^6 - 21*y^5*z^2 - 96*y^5*z - 112*y^5 + 35*y^4*z^3 + 240*y^4*z^2 + 560*y^4*z + 446*y^4 - 35*y^3*z^4 - 320*y^3*z^3 - 1120*y^3*z^2 - 1784*y^3*z - 1096*y^3 + 21*y^2*z^5 + 240*y^2*z^4 + 1120*y^2*z^3 + 2676*y^2*z^2 + 3287*y^2*z + 1664*y^2 - 7*y*z^6 - 96*y*z^5 - 560*y*z^4 - 1784*y*z^3 - 3286*y*z^2 - 3323*y*z - 1438*y + z^7 + 16*z^6 + 112*z^5 + 446*z^4 + 1095*z^3 + 1659*z^2 + 1431*z + 1, -128*x^7 - 448*x^6*y + 448*x^6*z + 992*x^6 - 672*x^5*y^2 + 1344*x^5*y*z + 2992*x^5*y - 672*x^5*z^2 - 2992*x^5*z - 3408*x^5 - 560*x^4*y^3 + 1680*x^4*y^2*z + 3760*x^4*y^2 - 1680*x^4*y*z^2 - 7520*x^4*y*z - 8624*x^4*y + 560*x^4*z^3 + 3760*x^4*z^2 + 8624*x^4*z + 6748*x^4 - 280*x^3*y^4 + 1120*x^3*y^3*z + 2520*x^3*y^3 - 1680*x^3*y^2*z^2 - 7560*x^3*y^2*z - 8728*x^3*y^2 + 1120*x^3*y*z^3 + 7560*x^3*y*z^2 + 17456*x^3*y*z + 13764*x^3*y - 280*x^3*z^4 - 2520*x^3*z^3 - 8728*x^3*z^2 - 13764*x^3*z - 8321*x^3 - 84*x^2*y^5 + 420*x^2*y^4*z + 950*x^2*y^4 - 840*x^2*y^3*z^2 - 3800*x^2*y^3*z - 4416*x^2*y^3 + 840*x^2*y^2*z^3 + 5700*x^2*y^2*z^2 + 13248*x^2*y^2*z + 10527*x^2*y^2 - 420*x^2*y*z^4 - 3800*x^2*y*z^3 - 13248*x^2*y*z^2 - 21054*x^2*y*z - 12851*x^2*y + 84*x^2*z^5 + 950*x^2*z^4 + 4416*x^2*z^3 + 10527*x^2*z^2 + 12847*x^2*z + 6401*x^2 - 14*x*y^6 + 84*x*y^5*z + 191*x*y^5 - 210*x*y^4*z^2 - 955*x*y^4*z - 1117*x*y^4 + 280*x*y^3*z^3 + 1910*x*y^3*z^2 + 4468*x*y^3*z + 3578*x*y^3 - 210*x*y^2*z^4 - 1910*x*y^2*z^3 - 6702*x*y^2*z^2 - 10734*x*y^2*z - 6617*x*y^2 + 84*x*y*z^5 + 955*x*y*z^4 + 4468*x*y*z^3 + 10734*x*y*z^2 + 13230*x*y*z + 6679*x*y - 14*x*z^6 - 191*x*z^5 - 1117*x*z^4 - 3578*x*z^3 - 6613*x*z^2 - 6670*x*z - 2857*x - y^7 + 7*y^6*z + 16*y^6 - 21*y^5*z^2 - 96*y^5*z - 113*y^5 + 35*y^4*z^3 + 240*y^4*z^2 + 565*y^4*z + 456*y^4 - 35*y^3*z^4 - 320*y^3*z^3 - 1130*y^3*z^2 - 1824*y^3*z - 1136*y^3 + 21*y^2*z^5 + 240*y^2*z^4 + 1130*y^2*z^3 + 2736*y^2*z^2 + 3407*y^2*z + 1745*y^2 - 7*y*z^6 - 96*y*z^5 - 565*y*z^4 - 1824*y*z^3 - 3406*y*z^2 - 3485*y*z - 1524*y + z^7 + 16*z^6 + 113*z^5 + 456*z^4 + 1135*z^3 + 1740*z^2 + 1516*z + 1, -128*x^7 - 448*x^6*y + 448*x^6*z + 992*x^6 - 672*x^5*y^2 + 1344*x^5*y*z + 2992*x^5*y - 672*x^5*z^2 - 2992*x^5*z - 3376*x^5 - 560*x^4*y^3 + 1680*x^4*y^2*z + 3760*x^4*y^2 - 1680*x^4*y*z^2 - 7520*x^4*y*z - 8544*x^4*y + 560*x^4*z^3 + 3760*x^4*z^2 + 8544*x^4*z + 6588*x^4 - 280*x^3*y^4 + 1120*x^3*y^3*z + 2520*x^3*y^3 - 1680*x^3*y^2*z^2 - 7560*x^3*y^2*z - 8648*x^3*y^2 + 1120*x^3*y*z^3 + 7560*x^3*y*z^2 + 17296*x^3*y*z + 13444*x^3*y - 280*x^3*z^4 - 2520*x^3*z^3 - 8648*x^3*z^2 - 13444*x^3*z - 8001*x^3 - 84*x^2*y^5 + 420*x^2*y^4*z + 950*x^2*y^4 - 840*x^2*y^3*z^2 - 3800*x^2*y^3*z - 4376*x^2*y^3 + 840*x^2*y^2*z^3 + 5700*x^2*y^2*z^2 + 13128*x^2*y^2*z + 10287*x^2*y^2 - 420*x^2*y*z^4 - 3800*x^2*y*z^3 - 13128*x^2*y*z^2 - 20574*x^2*y*z - 12371*x^2*y + 84*x^2*z^5 + 950*x^2*z^4 + 4376*x^2*z^3 + 10287*x^2*z^2 + 12367*x^2*z + 6069*x^2 - 14*x*y^6 + 84*x*y^5*z + 191*x*y^5 - 210*x*y^4*z^2 - 955*x*y^4*z - 1107*x*y^4 + 280*x*y^3*z^3 + 1910*x*y^3*z^2 + 4428*x*y^3*z + 3498*x*y^3 - 210*x*y^2*z^4 - 1910*x*y^2*z^3 - 6642*x*y^2*z^2 - 10494*x*y^2*z - 6377*x*y^2 + 84*x*y*z^5 + 955*x*y*z^4 + 4428*x*y*z^3 + 10494*x*y*z^2 + 12750*x*y*z + 6347*x*y - 14*x*z^6 - 191*x*z^5 - 1107*x*z^4 - 3498*x*z^3 - 6373*x*z^2 - 6338*x*z - 2670*x - y^7 + 7*y^6*z + 16*y^6 - 21*y^5*z^2 - 96*y^5*z - 112*y^5 + 35*y^4*z^3 + 240*y^4*z^2 + 560*y^4*z + 446*y^4 - 35*y^3*z^4 - 320*y^3*z^3 - 1120*y^3*z^2 - 1784*y^3*z - 1096*y^3 + 21*y^2*z^5 + 240*y^2*z^4 + 1120*y^2*z^3 + 2676*y^2*z^2 + 3287*y^2*z + 1662*y^2 - 7*y*z^6 - 96*y*z^5 - 560*y*z^4 - 1784*y*z^3 - 3286*y*z^2 - 3319*y*z - 1428*y + z^7 + 16*z^6 + 112*z^5 + 446*z^4 + 1095*z^3 + 1657*z^2 + 1421*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -2, -1, 1 ], [ -1, -2, 1 ], [ -1, -1, 1 ] ], "source_shift": [ 2, 1, 2 ], "target_matrix": [ [ 1, 1, 1 ], [ 1, 2, 1 ], [ 1, 1, 2 ] ], "target_shift": [ -536, -573, -524 ], "inverse_recipe": { "p": 5, "q": 2, "f": "x**5 + x**2", "g": "x**2", "h": "y*z + z**3 + 2*z**2", "polynomials": { "f": [ [ [ 5, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "g": [ [ [ 2, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 0, 2 ], 2 ], [ [ 0, 0, 3 ], 1 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.093914181998116 ], "surface_features": [ 3.525643979189622, 4.32333461859905, 3.140665139976736, 0.0, 3.6295822987141615, 4851 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C092", "pair": 92, "family": "counterexample", "mapping": "F(x,y,z)=(16*x^7 + 104*x^6*y + 128*x^6*z + 200*x^6 + 288*x^5*y^2 + 708*x^5*y*z + 1128*x^5*y + 432*x^5*z^2 + 1428*x^5*z + 972*x^5 + 440*x^4*y^3 + 1620*x^4*y^2*z + 2640*x^4*y^2 + 1974*x^4*y*z^2 + 6684*x^4*y*z + 4590*x^4*y + 796*x^4*z^3 + 4206*x^4*z^2 + 5940*x^4*z + 2438*x^4 + 400*x^3*y^4 + 1960*x^3*y^3*z + 3280*x^3*y^3 + 3576*x^3*y^2*z^2 + 12456*x^3*y^2*z + 8640*x^3*y^2 + 2879*x^3*y*z^3 + 15678*x^3*y*z^2 + 22371*x^3*y*z + 9218*x^3*y + 863*x^3*z^4 + 6539*x^3*z^3 + 14409*x^3*z^2 + 12179*x^3*z + 3394*x^3 + 216*x^2*y^5 + 1320*x^2*y^4*z + 2280*x^2*y^4 + 3204*x^2*y^3*z^2 + 11544*x^2*y^3*z + 8100*x^2*y^3 + 3861*x^2*y^2*z^3 + 21798*x^2*y^2*z^2 + 31473*x^2*y^2*z + 13026*x^2*y^2 + 2310*x^2*y*z^4 + 18189*x^2*y*z^3 + 40569*x^2*y*z^2 + 34437*x^2*y*z + 9603*x^2*y + 549*x^2*z^5 + 5658*x^2*z^4 + 17343*x^2*z^3 + 22665*x^2*z^2 + 12960*x^2*z + 2575*x^2 + 64*x*y^6 + 468*x*y^5*z + 840*x*y^5 + 1416*x*y^4*z^2 + 5316*x*y^4*z + 3780*x*y^4 + 2269*x*y^3*z^3 + 13386*x*y^3*z^2 + 19593*x*y^3*z + 8150*x*y^3 + 2031*x*y^2*z^4 + 16761*x*y^2*z^3 + 37911*x*y^2*z^2 + 32337*x*y^2*z + 9024*x*y^2 + 963*x*y*z^5 + 10434*x*y*z^4 + 32445*x*y*z^3 + 42600*x*y*z^2 + 24363*x*y*z + 4832*x*y + 189*x*z^6 + 2583*x*z^5 + 10359*x*z^4 + 18628*x*z^3 + 16392*x*z^2 + 6660*x*z + 957*x + 8*y^7 + 68*y^6*z + 128*y^6 + 246*y^5*z^2 + 972*y^5*z + 702*y^5 + 491*y^4*z^3 + 3060*y^4*z^2 + 4551*y^4*z + 1904*y^4 + 584*y^3*z^4 + 5111*y^3*z^3 + 11751*y^3*z^2 + 10079*y^3*z + 2815*y^3 + 414*y^2*z^5 + 4776*y^2*z^4 + 15102*y^2*z^3 + 19935*y^2*z^2 + 11403*y^2*z + 2257*y^2 + 162*y*z^6 + 2367*y*z^5 + 9657*y*z^4 + 17456*y*z^3 + 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648*y^3*z^4 - 4146*y^3*z^3 - 9096*y^3*z^2 - 8464*y^3*z - 2876*y^3 - 480*y^2*z^5 - 3978*y^2*z^4 - 12072*y^2*z^3 - 17466*y^2*z^2 - 12327*y^2*z - 3407*y^2 - 192*y*z^6 - 1968*y*z^5 - 7716*y*z^4 - 15404*y*z^3 - 16944*y*z^2 - 9760*y*z - 2300*y - 32*z^7 - 392*z^6 - 1896*z^5 - 4878*z^4 - 7408*z^3 - 6668*z^2 - 3303*z - 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ 1, 1, 2 ], [ 1, 1, 1 ], [ 2, 3, 2 ] ], "source_shift": [ 1, 2, 2 ], "target_matrix": [ [ 2, 1, 1 ], [ -1, -1, -1 ], [ -2, -2, -3 ] ], "target_shift": [ -517, 343, 681 ], "collision_preimages": [ [ "-19/4", "7/4", "1" ], [ "-37/2", "23/2", "7/2" ], [ "-9/2", "11/2", "-3/2" ] ], "collision_image": [ "-1035/2", "1373/4", "1363/2" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0723835780008812 ], "surface_features": [ 3.6261803436728144, 4.51470709754628, 3.2873537727147464, 0.0, 3.735008443876838, 4913 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C055", "pair": 55, "family": "counterexample", "mapping": "F(x,y,z)=(-x^7 - 10*x^6*y + 10*x^6*z + 20*x^6 - 42*x^5*y^2 + 87*x^5*y*z + 171*x^5*y - 42*x^5*z^2 - 174*x^5*z - 168*x^5 - 96*x^4*y^3 + 309*x^4*y^2*z + 597*x^4*y^2 - 309*x^4*y*z^2 - 1257*x^4*y*z - 1194*x^4*y + 96*x^4*z^3 + 621*x^4*z^2 + 1233*x^4*z + 771*x^4 - 129*x^3*y^4 + 573*x^3*y^3*z + 1089*x^3*y^3 - 891*x^3*y^2*z^2 - 3558*x^3*y^2*z - 3324*x^3*y^2 + 573*x^3*y*z^3 + 3639*x^3*y*z^2 + 7110*x^3*y*z + 4369*x^3*y - 129*x^3*z^4 - 1164*x^3*z^3 - 3570*x^3*z^2 - 4575*x^3*z - 2097*x^3 - 102*x^2*y^5 + 585*x^2*y^4*z + 1095*x^2*y^4 - 1257*x^2*y^3*z^2 - 4929*x^2*y^3*z - 4530*x^2*y^3 + 1257*x^2*y^2*z^3 + 7830*x^2*y^2*z^2 + 15045*x^2*y^2*z + 9081*x^2*y^2 - 585*x^2*y*z^4 - 5187*x^2*y*z^3 - 15666*x^2*y*z^2 - 19737*x^2*y*z - 8868*x^2*y + 102*x^2*z^5 + 1209*x^2*z^4 + 5097*x^2*z^3 + 10044*x^2*z^2 + 9426*x^2*z + 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618*x^4*y*z^2 - 2331*x^4*y*z - 2055*x^4*y + 192*x^4*z^3 + 1164*x^4*z^2 + 2154*x^4*z + 1269*x^4 - 258*x^3*y^4 + 1146*x^3*y^3*z + 1989*x^3*y^3 - 1782*x^3*y^2*z^2 - 6570*x^3*y^2*z - 5685*x^3*y^2 + 1146*x^3*y*z^3 + 6795*x^3*y*z^2 + 12342*x^3*y*z + 7139*x^3*y - 258*x^3*z^4 - 2196*x^3*z^3 - 6291*x^3*z^2 - 7596*x^3*z - 3327*x^3 - 204*x^2*y^5 + 1170*x^2*y^4*z + 1992*x^2*y^4 - 2514*x^2*y^3*z^2 - 9063*x^2*y^3*z - 7701*x^2*y^3 + 2514*x^2*y^2*z^3 + 14562*x^2*y^2*z^2 + 25950*x^2*y^2*z + 14736*x^2*y^2 - 1170*x^2*y*z^4 - 9753*x^2*y*z^3 - 27435*x^2*y*z^2 - 32523*x^2*y*z - 13971*x^2*y + 204*x^2*z^5 + 2295*x^2*z^4 + 9060*x^2*z^3 + 16830*x^2*z^2 + 15072*x^2*z + 5261*x^2 - 88*x*y^6 + 624*x*y^5*z + 1044*x*y^5 - 1734*x*y^4*z^2 - 6120*x*y^4*z - 5112*x*y^4 + 2398*x*y^3*z^3 + 13563*x*y^3*z^2 + 23718*x*y^3*z + 13230*x*y^3 - 1734*x*y^2*z^4 - 14130*x*y^2*z^3 - 38997*x*y^2*z^2 - 45366*x*y^2*z - 19110*x*y^2 + 624*x*y*z^5 + 6894*x*y*z^4 + 26748*x*y*z^3 + 48795*x*y*z^2 + 42852*x*y*z + 14641*x*y - 88*x*z^6 - 1260*x*z^5 - 6432*x*z^4 - 16350*x*z^3 - 22467*x*z^2 - 16022*x*z - 4661*x - 16*y^7 + 136*y^6*z + 224*y^6 - 468*y^5*z^2 - 1620*y^5*z - 1332*y^5 + 838*y^4*z^3 + 4632*y^4*z^2 + 7956*y^4*z + 4364*y^4 - 838*y^3*z^4 - 6665*y^3*z^3 - 18045*y^3*z^2 - 20610*y^3*z - 8520*y^3 + 468*y^2*z^5 + 5067*y^2*z^4 + 19296*y^2*z^3 + 34530*y^2*z^2 + 29718*y^2*z + 9938*y^2 - 136*y*z^6 - 1920*y*z^5 - 9648*y*z^4 - 24103*y*z^3 - 32481*y*z^2 - 22661*y*z - 6437*y + 16*z^7 + 284*z^6 + 1800*z^5 + 5871*z^4 + 11010*z^3 + 12041*z^2 + 7154*z + 1, -2*x^7 - 20*x^6*y + 20*x^6*z + 40*x^6 - 84*x^5*y^2 + 174*x^5*y*z + 342*x^5*y - 84*x^5*z^2 - 348*x^5*z - 336*x^5 - 192*x^4*y^3 + 618*x^4*y^2*z + 1194*x^4*y^2 - 618*x^4*y*z^2 - 2514*x^4*y*z - 2388*x^4*y + 192*x^4*z^3 + 1242*x^4*z^2 + 2466*x^4*z + 1543*x^4 - 258*x^3*y^4 + 1146*x^3*y^3*z + 2178*x^3*y^3 - 1782*x^3*y^2*z^2 - 7116*x^3*y^2*z - 6648*x^3*y^2 + 1146*x^3*y*z^3 + 7278*x^3*y*z^2 + 14220*x^3*y*z + 8745*x^3*y - 258*x^3*z^4 - 2328*x^3*z^3 - 7140*x^3*z^2 - 9154*x^3*z - 4205*x^3 - 204*x^2*y^5 + 1170*x^2*y^4*z + 2190*x^2*y^4 - 2514*x^2*y^3*z^2 - 9858*x^2*y^3*z - 9060*x^2*y^3 + 2514*x^2*y^2*z^3 + 15660*x^2*y^2*z^2 + 30090*x^2*y^2*z + 18180*x^2*y^2 - 1170*x^2*y*z^4 - 10374*x^2*y*z^3 - 31332*x^2*y*z^2 - 39495*x^2*y*z - 17793*x^2*y + 204*x^2*z^5 + 2418*x^2*z^4 + 10194*x^2*z^3 + 20094*x^2*z^2 + 18888*x^2*z + 6836*x^2 - 88*x*y^6 + 624*x*y^5*z + 1152*x*y^5 - 1734*x*y^4*z^2 - 6684*x*y^4*z - 6048*x*y^4 + 2398*x*y^3*z^3 + 14646*x*y^3*z^2 + 27672*x*y^3*z + 16430*x*y^3 - 1734*x*y^2*z^4 - 15084*x*y^2*z^3 - 44820*x*y^2*z^2 - 55506*x*y^2*z - 24510*x*y^2 + 624*x*y*z^5 + 7284*x*y*z^4 + 30276*x*y*z^3 + 58713*x*y*z^2 + 54144*x*y*z + 19162*x*y - 88*x*z^6 - 1320*x*z^5 - 7176*x*z^4 - 19336*x*z^3 - 27897*x*z^2 - 20660*x*z - 6174*x - 16*y^7 + 136*y^6*z + 248*y^6 - 468*y^5*z^2 - 1776*y^5*z - 1584*y^5 + 838*y^4*z^3 + 5022*y^4*z^2 + 9336*y^4*z + 5452*y^4 - 838*y^3*z^4 - 7142*y^3*z^3 - 20868*y^3*z^2 - 25394*y^3*z - 10996*y^3 + 468*y^2*z^5 + 5370*y^2*z^4 + 21978*y^2*z^3 + 41874*y^2*z^2 + 37842*y^2*z + 13088*y^2 - 136*y*z^6 - 2016*y*z^5 - 10824*y*z^4 - 28727*y*z^3 - 40683*y*z^2 - 29462*y*z - 8584*y + 16*z^7 + 296*z^6 + 1992*z^5 + 6877*z^4 + 13538*z^3 + 15392*z^2 + 9412*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -1, -2, 1 ], [ -1, -1, 2 ], [ -1, -1, 1 ] ], "source_shift": [ 2, 2, 2 ], "target_matrix": [ [ 1, 1, 2 ], [ 2, 1, 1 ], [ 2, 2, 3 ] ], "target_shift": [ -1183, -1789, -2403 ], "collision_preimages": [ [ "11/4", "-1/4", "1/4" ], [ "-18", "11/2", "-8" ], [ "-17", "15/2", "-5" ] ], "collision_image": [ "-4733/4", "-3579/2", "-4807/2" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0174798340012785 ], "surface_features": [ 3.887701988252947, 4.768734271659541, 3.4319263739116437, 0.0, 4.027509337320764, 4979 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { 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15360*x*y^4 + 5816*x*y^3*z^3 + 37176*x*y^3*z^2 + 62568*x*y^3*z + 31200*x*y^3 - 3750*x*y^2*z^4 - 34752*x*y^2*z^3 - 92172*x*y^2*z^2 - 95844*x*y^2*z - 34326*x*y^2 + 1233*x*y*z^5 + 15525*x*y*z^4 + 57846*x*y*z^3 + 94428*x*y*z^2 + 70641*x*y*z + 19703*x*y - 162*x*z^6 - 2646*x*z^5 - 12981*x*z^4 - 29625*x*z^3 - 34854*x*z^2 - 20394*x*z - 4667*x - 64*y^7 + 448*y^6*z + 800*y^6 - 1296*y^5*z^2 - 4944*y^5*z - 3888*y^5 + 2000*y^4*z^3 + 12336*y^4*z^2 + 20256*y^4*z + 9872*y^4 - 1772*y^3*z^4 - 15832*y^3*z^3 - 40872*y^3*z^2 - 41424*y^3*z - 14460*y^3 + 900*y^2*z^5 + 10962*y^2*z^4 + 39696*y^2*z^3 + 63000*y^2*z^2 + 45804*y^2*z + 12406*y^2 - 243*y*z^6 - 3861*y*z^5 - 18435*y*z^4 - 40884*y*z^3 - 46629*y*z^2 - 26409*y*z - 5849*y + 27*z^7 + 540*z^6 + 3258*z^5 + 9477*z^4 + 15135*z^3 + 13509*z^2 + 6280*z + 1, -16*x^7 - 160*x^6*y + 136*x^6*z + 308*x^6 - 672*x^5*y^2 + 1176*x^5*y*z + 2580*x^5*y - 492*x^5*z^2 - 2316*x^5*z - 2184*x^5 - 1536*x^4*y^3 + 4152*x^4*y^2*z + 8820*x^4*y^2 - 3576*x^4*y*z^2 - 16308*x^4*y*z - 15000*x^4*y + 982*x^4*z^3 + 7221*x^4*z^2 + 13920*x^4*z + 7886*x^4 - 2064*x^3*y^4 + 7656*x^3*y^3*z + 15756*x^3*y^3 - 10188*x^3*y^2*z^2 - 44964*x^3*y^2*z - 40344*x^3*y^2 + 5758*x^3*y*z^3 + 41043*x^3*y*z^2 + 77172*x^3*y*z + 42582*x^3*y - 1168*x^3*z^4 - 11948*x^3*z^3 - 35328*x^3*z^2 - 40754*x^3*z - 16108*x^3 - 1632*x^2*y^5 + 7776*x^2*y^4*z + 15528*x^2*y^4 - 14208*x^2*y^3*z^2 - 60684*x^2*y^3*z - 53160*x^2*y^3 + 12408*x^2*y^2*z^3 + 85590*x^2*y^2*z^2 + 156948*x^2*y^2*z + 84384*x^2*y^2 - 5178*x^2*y*z^4 - 51405*x^2*y*z^3 - 148278*x^2*y*z^2 - 166548*x^2*y*z - 63972*x^2*y + 828*x^2*z^5 + 11064*x^2*z^4 + 44619*x^2*z^3 + 78636*x^2*z^2 + 63204*x^2*z + 18905*x^2 - 704*x*y^6 + 4128*x*y^5*z + 8016*x*y^5 - 9696*x*y^4*z^2 - 40128*x*y^4*z - 34368*x*y^4 + 11632*x*y^3*z^3 + 77592*x*y^3*z^2 + 138864*x*y^3*z + 72856*x*y^3 - 7500*x*y^2*z^4 - 72096*x*y^2*z^3 - 202776*x*y^2*z^2 - 221784*x*y^2*z - 82908*x*y^2 + 2466*x*y*z^5 + 32037*x*y*z^4 + 126084*x*y*z^3 + 216318*x*y*z^2 + 168810*x*y*z + 48991*x*y - 324*x*z^6 - 5436*x*z^5 - 28038*x*z^4 - 67124*x*z^3 - 82332*x*z^2 - 50034*x*z - 11904*x - 128*y^7 + 896*y^6*z + 1696*y^6 - 2592*y^5*z^2 - 10416*y^5*z - 8736*y^5 + 4000*y^4*z^3 + 25824*y^4*z^2 + 45168*y^4*z + 23168*y^4 - 3544*y^3*z^4 - 32936*y^3*z^3 - 90384*y^3*z^2 - 96400*y^3*z - 35144*y^3 + 1800*y^2*z^5 + 22674*y^2*z^4 + 87000*y^2*z^3 + 145248*y^2*z^2 + 110208*y^2*z + 31106*y^2 - 486*y*z^6 - 7947*y*z^5 - 40026*y*z^4 - 93284*y*z^3 - 110982*y*z^2 - 65355*y*z - 15086*y + 54*z^7 + 1107*z^6 + 7011*z^5 + 21380*z^4 + 35596*z^3 + 32985*z^2 + 15923*z + 1, -8*x^7 - 80*x^6*y + 68*x^6*z + 160*x^6 - 336*x^5*y^2 + 588*x^5*y*z + 1344*x^5*y - 246*x^5*z^2 - 1200*x^5*z - 1188*x^5 - 768*x^4*y^3 + 2076*x^4*y^2*z + 4608*x^4*y^2 - 1788*x^4*y*z^2 - 8472*x^4*y*z - 8196*x^4*y + 491*x^4*z^3 + 3732*x^4*z^2 + 7548*x^4*z + 4447*x^4 - 1032*x^3*y^4 + 3828*x^3*y^3*z + 8256*x^3*y^3 - 5094*x^3*y^2*z^2 - 23424*x^3*y^2*z - 22140*x^3*y^2 + 2879*x^3*y*z^3 + 21264*x^3*y*z^2 + 42036*x^3*y*z + 24129*x^3*y - 584*x^3*z^4 - 6160*x^3*z^3 - 19095*x^3*z^2 - 22912*x^3*z - 9332*x^3 - 816*x^2*y^5 + 3888*x^2*y^4*z + 8160*x^2*y^4 - 7104*x^2*y^3*z^2 - 31704*x^2*y^3*z - 29292*x^2*y^3 + 6204*x^2*y^2*z^3 + 44460*x^2*y^2*z^2 + 85872*x^2*y^2*z + 48030*x^2*y^2 - 2589*x^2*y*z^4 - 26562*x^2*y*z^3 - 80511*x^2*y*z^2 - 94107*x^2*y*z - 37236*x^2*y + 414*x^2*z^5 + 5691*x^2*z^4 + 24039*x^2*z^3 + 44070*x^2*z^2 + 36516*x^2*z + 11182*x^2 - 352*x*y^6 + 2064*x*y^5*z + 4224*x*y^5 - 4848*x*y^4*z^2 - 21024*x*y^4*z - 19008*x*y^4 + 5816*x*y^3*z^3 + 40416*x*y^3*z^2 + 76296*x*y^3*z + 41636*x*y^3 - 3750*x*y^2*z^4 - 37344*x*y^2*z^3 - 110604*x*y^2*z^2 - 125904*x*y^2*z - 48462*x*y^2 + 1233*x*y*z^5 + 16512*x*y*z^4 + 68238*x*y*z^3 + 121869*x*y*z^2 + 98013*x*y*z + 29108*x*y - 162*x*z^6 - 2790*x*z^5 - 15057*x*z^4 - 37495*x*z^3 - 47430*x*z^2 - 29520*x*z - 7159*x - 64*y^7 + 448*y^6*z + 896*y^6 - 1296*y^5*z^2 - 5472*y^5*z - 4848*y^5 + 2000*y^4*z^3 + 13488*y^4*z^2 + 24912*y^4*z + 13288*y^4 - 1772*y^3*z^4 - 17104*y^3*z^3 - 49512*y^3*z^2 - 54956*y^3*z - 20620*y^3 + 900*y^2*z^5 + 11712*y^2*z^4 + 47304*y^2*z^3 + 82230*y^2*z^2 + 64272*y^2*z + 18556*y^2 - 243*y*z^6 - 4086*y*z^5 - 21591*y*z^4 - 52393*y*z^3 - 64269*y*z^2 - 38742*y*z - 9113*y + 27*z^7 + 567*z^6 + 3753*z^5 + 11902*z^4 + 20444*z^3 + 19410*z^2 + 9553*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -1, -2, 1 ], [ -2, -2, 3 ], [ -1, -1, 1 ] ], "source_shift": [ 2, 2, 2 ], "target_matrix": [ [ 1, 1, 2 ], [ 2, 3, 2 ], [ 1, 2, 1 ] ], "target_shift": [ -1183, -3125, -1905 ], "collision_preimages": [ [ "5", "-1/4", "5/2" ], [ "-45/2", "11/2", "-25/2" ], [ "-43/2", "15/2", "-19/2" ] ], "collision_image": [ "-4733/4", "-6251/2", "-7621/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0275823740012129 ], "surface_features": [ 4.414737994967702, 5.345930211955827, 4.046221937221636, 0.0, 4.5588936620903695, 5168 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B216", "pair": 216, "family": "bijection", "mapping": "F(x,y,z)=(3*x^7 - 42*x^6*y - 21*x^6*z + 33*x^6 + 252*x^5*y^2 + 252*x^5*y*z - 390*x^5*y + 63*x^5*z^2 - 204*x^5*z + 159*x^5 - 840*x^4*y^3 - 1260*x^4*y^2*z + 1920*x^4*y^2 - 630*x^4*y*z^2 + 2010*x^4*y*z - 1536*x^4*y - 105*x^4*z^3 + 525*x^4*z^2 - 846*x^4*z + 429*x^4 + 1680*x^3*y^4 + 3360*x^3*y^3*z - 5040*x^3*y^3 + 2520*x^3*y^2*z^2 - 7920*x^3*y^2*z + 5931*x^3*y^2 + 840*x^3*y*z^3 - 4140*x^3*y*z^2 + 6546*x^3*y*z - 3234*x^3*y + 105*x^3*z^4 - 720*x^3*z^3 + 1797*x^3*z^2 - 1896*x^3*z + 686*x^3 - 2016*x^2*y^5 - 5040*x^2*y^4*z + 7440*x^2*y^4 - 5040*x^2*y^3*z^2 + 15600*x^2*y^3*z - 11442*x^2*y^3 - 2520*x^2*y^2*z^3 + 12240*x^2*y^2*z^2 - 18981*x^2*y^2*z + 9135*x^2*y^2 - 630*x^2*y*z^4 + 4260*x^2*y*z^3 - 10440*x^2*y*z^2 + 10734*x^2*y*z - 3762*x^2*y - 63*x^2*z^5 + 555*x^2*z^4 - 1905*x^2*z^3 + 3135*x^2*z^2 - 2373*x^2*z + 640*x^2 + 1344*x*y^6 + 4032*x*y^5*z - 5856*x*y^5 + 5040*x*y^4*z^2 - 15360*x*y^4*z + 11028*x*y^4 + 3360*x*y^3*z^3 - 16080*x*y^3*z^2 + 24444*x*y^3*z - 11460*x*y^3 + 1260*x*y^2*z^4 - 8400*x*y^2*z^3 + 20205*x*y^2*z^2 - 20238*x*y^2*z + 6882*x*y^2 + 252*x*y*z^5 - 2190*x*y*z^4 + 7386*x*y*z^3 - 11850*x*y*z^2 + 8670*x*y*z - 2260*x*y + 21*x*z^6 - 228*x*z^5 + 1008*x*z^4 - 2298*x*z^3 + 2736*x*z^2 - 1544*x*z + 317*x - 384*y^7 - 1344*y^6*z + 1920*y^6 - 2016*y^5*z^2 + 6048*y^5*z - 4248*y^5 - 1680*y^4*z^3 + 7920*y^4*z^2 - 11796*y^4*z + 5388*y^4 - 840*y^3*z^4 + 5520*y^3*z^3 - 13026*y^3*z^2 + 12708*y^3*z - 4198*y^3 - 252*y^2*z^5 + 2160*y^2*z^4 - 7155*y^2*z^3 + 11187*y^2*z^2 - 7929*y^2*z + 2002*y^2 - 42*y*z^6 + 450*y*z^5 - 1956*y*z^4 + 4350*y*z^3 - 4992*y*z^2 + 2716*y*z - 540*y - 3*z^7 + 39*z^6 - 213*z^5 + 630*z^4 - 1052*z^3 + 934*z^2 - 403*z + 1, x^7 - 14*x^6*y - 7*x^6*z + 11*x^6 + 84*x^5*y^2 + 84*x^5*y*z - 130*x^5*y + 21*x^5*z^2 - 68*x^5*z + 53*x^5 - 280*x^4*y^3 - 420*x^4*y^2*z + 640*x^4*y^2 - 210*x^4*y*z^2 + 670*x^4*y*z - 512*x^4*y - 35*x^4*z^3 + 175*x^4*z^2 - 282*x^4*z + 143*x^4 + 560*x^3*y^4 + 1120*x^3*y^3*z - 1680*x^3*y^3 + 840*x^3*y^2*z^2 - 2640*x^3*y^2*z + 1977*x^3*y^2 + 280*x^3*y*z^3 - 1380*x^3*y*z^2 + 2182*x^3*y*z - 1078*x^3*y + 35*x^3*z^4 - 240*x^3*z^3 + 599*x^3*z^2 - 632*x^3*z + 226*x^3 - 672*x^2*y^5 - 1680*x^2*y^4*z + 2480*x^2*y^4 - 1680*x^2*y^3*z^2 + 5200*x^2*y^3*z - 3814*x^2*y^3 - 840*x^2*y^2*z^3 + 4080*x^2*y^2*z^2 - 6327*x^2*y^2*z + 3045*x^2*y^2 - 210*x^2*y*z^4 + 1420*x^2*y*z^3 - 3480*x^2*y*z^2 + 3578*x^2*y*z - 1238*x^2*y - 21*x^2*z^5 + 185*x^2*z^4 - 635*x^2*z^3 + 1045*x^2*z^2 - 783*x^2*z + 203*x^2 + 448*x*y^6 + 1344*x*y^5*z - 1952*x*y^5 + 1680*x*y^4*z^2 - 5120*x*y^4*z + 3676*x*y^4 + 1120*x*y^3*z^3 - 5360*x*y^3*z^2 + 8148*x*y^3*z - 3820*x*y^3 + 420*x*y^2*z^4 - 2800*x*y^2*z^3 + 6735*x*y^2*z^2 - 6746*x*y^2*z + 2262*x*y^2 + 84*x*y*z^5 - 730*x*y*z^4 + 2462*x*y*z^3 - 3950*x*y*z^2 + 2858*x*y*z - 712*x*y + 7*x*z^6 - 76*x*z^5 + 336*x*z^4 - 766*x*z^3 + 904*x*z^2 - 494*x*z + 88*x - 128*y^7 - 448*y^6*z + 640*y^6 - 672*y^5*z^2 + 2016*y^5*z - 1416*y^5 - 560*y^4*z^3 + 2640*y^4*z^2 - 3932*y^4*z + 1796*y^4 - 280*y^3*z^4 + 1840*y^3*z^3 - 4342*y^3*z^2 + 4236*y^3*z - 1378*y^3 - 84*y^2*z^5 + 720*y^2*z^4 - 2385*y^2*z^3 + 3729*y^2*z^2 - 2611*y^2*z + 626*y^2 - 14*y*z^6 + 150*y*z^5 - 652*y*z^4 + 1450*y*z^3 - 1648*y*z^2 + 864*y*z - 147*y - z^7 + 13*z^6 - 71*z^5 + 210*z^4 - 348*z^3 + 301*z^2 - 117*z + 1, 2*x^7 - 28*x^6*y - 14*x^6*z + 22*x^6 + 168*x^5*y^2 + 168*x^5*y*z - 260*x^5*y + 42*x^5*z^2 - 136*x^5*z + 106*x^5 - 560*x^4*y^3 - 840*x^4*y^2*z + 1280*x^4*y^2 - 420*x^4*y*z^2 + 1340*x^4*y*z - 1024*x^4*y - 70*x^4*z^3 + 350*x^4*z^2 - 564*x^4*z + 286*x^4 + 1120*x^3*y^4 + 2240*x^3*y^3*z - 3360*x^3*y^3 + 1680*x^3*y^2*z^2 - 5280*x^3*y^2*z + 3954*x^3*y^2 + 560*x^3*y*z^3 - 2760*x^3*y*z^2 + 4364*x^3*y*z - 2156*x^3*y + 70*x^3*z^4 - 480*x^3*z^3 + 1198*x^3*z^2 - 1264*x^3*z + 458*x^3 - 1344*x^2*y^5 - 3360*x^2*y^4*z + 4960*x^2*y^4 - 3360*x^2*y^3*z^2 + 10400*x^2*y^3*z - 7628*x^2*y^3 - 1680*x^2*y^2*z^3 + 8160*x^2*y^2*z^2 - 12654*x^2*y^2*z + 6090*x^2*y^2 - 420*x^2*y*z^4 + 2840*x^2*y*z^3 - 6960*x^2*y*z^2 + 7156*x^2*y*z - 2512*x^2*y - 42*x^2*z^5 + 370*x^2*z^4 - 1270*x^2*z^3 + 2090*x^2*z^2 - 1584*x^2*z + 429*x^2 + 896*x*y^6 + 2688*x*y^5*z - 3904*x*y^5 + 3360*x*y^4*z^2 - 10240*x*y^4*z + 7352*x*y^4 + 2240*x*y^3*z^3 - 10720*x*y^3*z^2 + 16296*x*y^3*z - 7640*x*y^3 + 840*x*y^2*z^4 - 5600*x*y^2*z^3 + 13470*x*y^2*z^2 - 13492*x*y^2*z + 4596*x*y^2 + 168*x*y*z^5 - 1460*x*y*z^4 + 4924*x*y*z^3 - 7900*x*y*z^2 + 5788*x*y*z - 1516*x*y + 14*x*z^6 - 152*x*z^5 + 672*x*z^4 - 1532*x*z^3 + 1826*x*z^2 - 1034*x*z + 215*x - 256*y^7 - 896*y^6*z + 1280*y^6 - 1344*y^5*z^2 + 4032*y^5*z - 2832*y^5 - 1120*y^4*z^3 + 5280*y^4*z^2 - 7864*y^4*z + 3592*y^4 - 560*y^3*z^4 + 3680*y^3*z^3 - 8684*y^3*z^2 + 8472*y^3*z - 2804*y^3 - 168*y^2*z^5 + 1440*y^2*z^4 - 4770*y^2*z^3 + 7458*y^2*z^2 - 5294*y^2*z + 1344*y^2 - 28*y*z^6 + 300*y*z^5 - 1304*y*z^4 + 2900*y*z^3 - 3332*y*z^2 + 1820*y*z - 367*y - 2*z^7 + 26*z^6 - 142*z^5 + 420*z^4 - 702*z^3 + 625*z^2 - 272*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ 1, -2, -1 ], [ 1, -1, -2 ], [ 1, -1, -1 ] ], "source_shift": [ 1, 1, 1 ], "target_matrix": [ [ 3, 2, 2 ], [ 1, 1, 2 ], [ 2, 1, 1 ] ], "target_shift": [ -62, -10, -43 ], "inverse_recipe": { "p": 2, "q": 3, "f": "x**2", "g": "x**3 + x**2 + x", "h": "y**3 + y**2*z", "polynomials": { "f": [ [ [ 2, 0, 0 ], 1 ] ], "g": [ [ [ 3, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "h": [ [ [ 0, 2, 1 ], 1 ], [ [ 0, 3, 0 ], 1 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0358775829990918 ], "surface_features": [ 3.4518908616691433, 4.388172275047789, 3.1283992687178066, 0.0, 3.58334744275859, 4852 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C003", "pair": 3, "family": "counterexample", "mapping": "F(x,y,z)=(4*x^7 + 26*x^6*y - 38*x^6*z - 50*x^6 + 72*x^5*y^2 - 210*x^5*y*z - 279*x^5*y + 150*x^5*z^2 + 417*x^5*z + 270*x^5 + 110*x^4*y^3 - 480*x^4*y^2*z - 645*x^4*y^2 + 684*x^4*y*z^2 + 1926*x^4*y*z + 1254*x^4*y - 318*x^4*z^3 - 1413*x^4*z^2 - 1911*x^4*z - 813*x^4 + 100*x^3*y^4 - 580*x^3*y^3*z - 790*x^3*y^3 + 1236*x^3*y^2*z^2 + 3534*x^3*y^2*z + 2316*x^3*y^2 - 1146*x^3*y*z^3 - 5181*x^3*y*z^2 - 7056*x^3*y*z - 3010*x^3*y + 390*x^3*z^4 + 2487*x^3*z^3 + 5292*x^3*z^2 + 4678*x^3*z + 1464*x^3 + 54*x^2*y^5 - 390*x^2*y^4*z - 540*x^2*y^4 + 1104*x^2*y^3*z^2 + 3216*x^2*y^3*z + 2124*x^2*y^3 - 1530*x^2*y^2*z^3 - 7065*x^2*y^2*z^2 - 9702*x^2*y^2*z - 4152*x^2*y^2 + 1038*x^2*y*z^4 + 6780*x^2*y*z^3 + 14556*x^2*y*z^2 + 12906*x^2*y*z + 4038*x^2*y - 276*x^2*z^5 - 2397*x^2*z^4 - 7167*x^2*z^3 - 9891*x^2*z^2 - 6408*x^2*z - 1562*x^2 + 16*x*y^6 - 138*x*y^5*z - 195*x*y^5 + 486*x*y^4*z^2 + 1449*x*y^4*z + 966*x*y^4 - 894*x*y^3*z^3 - 4239*x*y^3*z^2 - 5880*x*y^3*z - 2526*x*y^3 + 906*x*y^2*z^4 + 6099*x*y^2*z^3 + 13236*x*y^2*z^2 + 11778*x*y^2*z + 3684*x*y^2 - 480*x*y*z^5 - 4314*x*y*z^4 - 13050*x*y*z^3 - 18072*x*y*z^2 - 11694*x*y*z - 2839*x*y + 104*x*z^6 + 1200*x*z^5 + 4752*x*z^4 + 9118*x*z^3 + 9168*x*z^2 + 4607*x*z + 902*x + 2*y^7 - 20*y^6*z - 29*y^6 + 84*y^5*z^2 + 258*y^5*z + 174*y^5 - 192*y^4*z^3 - 942*y^4*z^2 - 1323*y^4*z - 571*y^4 + 258*y^3*z^4 + 1806*y^3*z^3 + 3972*y^3*z^2 + 3550*y^3*z + 1110*y^3 - 204*y^2*z^5 - 1917*y^2*z^4 - 5883*y^2*z^3 - 8181*y^2*z^2 - 5286*y^2*z - 1277*y^2 + 88*y*z^6 + 1068*y*z^5 + 4296*y*z^4 + 8278*y*z^3 + 8304*y*z^2 + 4142*y*z + 802*y - 16*z^7 - 244*z^6 - 1236*z^5 - 3100*z^4 - 4304*z^3 - 3329*z^2 - 1323*z - 1, 2*x^7 + 13*x^6*y - 19*x^6*z - 34*x^6 + 36*x^5*y^2 - 105*x^5*y*z - 189*x^5*y + 75*x^5*z^2 + 276*x^5*z + 225*x^5 + 55*x^4*y^3 - 240*x^4*y^2*z - 435*x^4*y^2 + 342*x^4*y*z^2 + 1269*x^4*y*z + 1041*x^4*y - 159*x^4*z^3 - 909*x^4*z^2 - 1545*x^4*z - 779*x^4 + 50*x^3*y^4 - 290*x^3*y^3*z - 530*x^3*y^3 + 618*x^3*y^2*z^2 + 2316*x^3*y^2*z + 1914*x^3*y^2 - 573*x^3*y*z^3 - 3315*x^3*y*z^2 - 5679*x^3*y*z - 2874*x^3*y + 195*x^3*z^4 + 1554*x^3*z^3 + 4140*x^3*z^2 + 4356*x^3*z + 1552*x^3 + 27*x^2*y^5 - 195*x^2*y^4*z - 360*x^2*y^4 + 552*x^2*y^3*z^2 + 2094*x^2*y^3*z + 1746*x^2*y^3 - 765*x^2*y^2*z^3 - 4491*x^2*y^2*z^2 - 7767*x^2*y^2*z - 3948*x^2*y^2 + 519*x^2*y*z^4 + 4209*x^2*y*z^3 + 11328*x^2*y*z^2 + 11970*x^2*y*z + 4266*x^2*y - 138*x^2*z^5 - 1455*x^2*z^4 - 5415*x^2*z^3 - 8928*x^2*z^2 - 6615*x^2*z - 1789*x^2 + 8*x*y^6 - 69*x*y^5*z - 129*x*y^5 + 243*x*y^4*z^2 + 936*x*y^4*z + 789*x*y^4 - 447*x*y^3*z^3 - 2673*x*y^3*z^2 - 4677*x*y^3*z - 2390*x*y^3 + 453*x*y^2*z^4 + 3756*x*y^2*z^3 + 10236*x*y^2*z^2 + 10872*x*y^2*z + 3876*x*y^2 - 240*x*y*z^5 - 2598*x*y*z^4 - 9801*x*y*z^3 - 16242*x*y*z^2 - 12027*x*y*z - 3239*x*y + 52*x*z^6 + 708*x*z^5 + 3465*x*z^4 + 7963*x*z^3 + 9198*x*z^2 + 5143*x*z + 1098*x + y^7 - 10*y^6*z - 19*y^6 + 42*y^5*z^2 + 165*y^5*z + 141*y^5 - 96*y^4*z^3 - 588*y^4*z^2 - 1044*y^4*z - 537*y^4 + 129*y^3*z^4 + 1101*y^3*z^3 + 3048*y^3*z^2 + 3258*y^3*z + 1162*y^3 - 102*y^2*z^5 - 1143*y^2*z^4 - 4386*y^2*z^3 - 7314*y^2*z^2 - 5412*y^2*z - 1450*y^2 + 44*y*z^6 + 624*y*z^5 + 3111*y*z^4 + 7197*y*z^3 + 8301*y*z^2 + 4606*y*z + 971*y - 8*z^7 - 140*z^6 - 870*z^5 - 2616*z^4 - 4193*z^3 - 3622*z^2 - 1569*z - 1, -2*x^7 - 13*x^6*y + 19*x^6*z + 28*x^6 - 36*x^5*y^2 + 105*x^5*y*z + 156*x^5*y - 75*x^5*z^2 - 231*x^5*z - 165*x^5 - 55*x^4*y^3 + 240*x^4*y^2*z + 360*x^4*y^2 - 342*x^4*y*z^2 - 1065*x^4*y*z - 765*x^4*y + 159*x^4*z^3 + 774*x^4*z^2 + 1152*x^4*z + 530*x^4 - 50*x^3*y^4 + 290*x^3*y^3*z + 440*x^3*y^3 - 618*x^3*y^2*z^2 - 1950*x^3*y^2*z - 1410*x^3*y^2 + 573*x^3*y*z^3 + 2832*x^3*y*z^2 + 4245*x^3*y*z + 1959*x^3*y - 195*x^3*z^4 - 1347*x^3*z^3 - 3144*x^3*z^2 - 3009*x^3*z - 1000*x^3 - 27*x^2*y^5 + 195*x^2*y^4*z + 300*x^2*y^4 - 552*x^2*y^3*z^2 - 1770*x^2*y^3*z - 1290*x^2*y^3 + 765*x^2*y^2*z^3 + 3852*x^2*y^2*z^2 + 5823*x^2*y^2*z + 2697*x^2*y^2 - 519*x^2*y*z^4 - 3663*x^2*y*z^3 - 8628*x^2*y*z^2 - 8286*x^2*y*z - 2754*x^2*y + 138*x^2*z^5 + 1284*x^2*z^4 + 4194*x^2*z^3 + 6270*x^2*z^2 + 4326*x^2*z + 1102*x^2 - 8*x*y^6 + 69*x*y^5*z + 108*x*y^5 - 243*x*y^4*z^2 - 795*x*y^4*z - 585*x*y^4 + 447*x*y^3*z^3 + 2304*x*y^3*z^2 + 3519*x*y^3*z + 1637*x*y^3 - 453*x*y^2*z^4 - 3285*x*y^2*z^3 - 7824*x*y^2*z^2 - 7545*x*y^2*z - 2508*x*y^2 + 240*x*y*z^5 + 2304*x*y*z^4 + 7617*x*y*z^3 + 11433*x*y*z^2 + 7881*x*y*z + 2000*x*y - 52*x*z^6 - 636*x*z^5 - 2739*x*z^4 - 5692*x*z^3 - 6108*x*z^2 - 3220*x*z - 650*x - y^7 + 10*y^6*z + 16*y^6 - 42*y^5*z^2 - 141*y^5*z - 105*y^5 + 96*y^4*z^3 + 510*y^4*z^2 + 789*y^4*z + 369*y^4 - 129*y^3*z^4 - 969*y^3*z^3 - 2340*y^3*z^2 - 2268*y^3*z - 754*y^3 + 102*y^2*z^5 + 1020*y^2*z^4 + 3423*y^2*z^3 + 5163*y^2*z^2 + 3555*y^2*z + 898*y^2 - 44*y*z^6 - 564*y*z^5 - 2469*y*z^4 - 5157*y*z^3 - 5523*y*z^2 - 2890*y*z - 577*y + 8*z^7 + 128*z^6 + 702*z^5 + 1905*z^4 + 2828*z^3 + 2302*z^2 + 946*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -1, -1, 1 ], [ -1, -1, 2 ], [ -2, -1, 1 ] ], "source_shift": [ 2, 1, 1 ], "target_matrix": [ [ -2, -1, -1 ], [ -1, -2, -1 ], [ 1, 1, 1 ] ], "target_shift": [ 212, 270, -155 ], "collision_preimages": [ [ "-3/4", "15/4", "1" ], [ "-13/2", "6", "-3/2" ], [ "-17/2", "15", "7/2" ] ], "collision_image": [ "425/2", "1081/4", "-621/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0635290419995727 ], "surface_features": [ 3.3855233585296034, 4.2570062179123065, 3.023046584075505, 0.0, 3.4999756502641306, 4815 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C120", "pair": 120, "family": "counterexample", "mapping": "F(x,y,z)=(-2*x^7 + 20*x^6*y + 16*x^6*z - 31*x^6 - 84*x^5*y^2 - 138*x^5*y*z + 270*x^5*y - 54*x^5*z^2 + 207*x^5*z - 186*x^5 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573*x*y^3*z^3 - 3603*x*y^3*z^2 + 7041*x*y^3*z - 4231*x*y^3 + 342*x*y^2*z^4 - 2826*x*y^2*z^3 + 8025*x*y^2*z^2 - 9453*x*y^2*z + 4104*x*y^2 + 105*x*y*z^5 - 1083*x*y*z^4 + 3996*x*y*z^3 - 6945*x*y*z^2 + 5958*x*y*z - 2059*x*y + 13*x*z^6 - 162*x*z^5 + 732*x*z^4 - 1673*x*z^3 + 2130*x*z^2 - 1463*x*z + 428*x - 8*y^7 - 52*y^6*z + 124*y^6 - 138*y^5*z^2 + 612*y^5*z - 630*y^5 - 195*y^4*z^3 + 1239*y^4*z^2 - 2457*y^4*z + 1492*y^4 - 159*y^3*z^4 + 1314*y^3*z^3 - 3783*y^3*z^2 + 4503*y^3*z - 1971*y^3 - 75*y^2*z^5 + 768*y^2*z^4 - 2868*y^2*z^3 + 5031*y^2*z^2 - 4353*y^2*z + 1519*y^2 - 19*y*z^6 + 234*y*z^5 - 1068*y*z^4 + 2462*y*z^3 - 3162*y*z^2 + 2188*y*z - 642*y - 2*z^7 + 29*z^6 - 156*z^5 + 444*z^4 - 753*z^3 + 778*z^2 - 459*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ 1, -1, -1 ], [ 1, -2, -1 ], [ 1, -1, -2 ] ], "source_shift": [ 1, 1, 2 ], "target_matrix": [ [ -2, -3, -2 ], [ -2, -1, -1 ], [ 1, 2, 1 ] ], "target_shift": [ 191, 102, -118 ], "collision_preimages": [ [ "1/4", "0", "5/4" ], [ "-2", "5/2", "-9/2" ], [ "-11", "-5/2", "-13/2" ] ], "collision_image": [ "383/2", "205/2", "-473/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0295833829986805 ], "surface_features": [ 3.318003592163001, 4.203277720892418, 3.0136796972911926, 0.0, 3.420917672590198, 4818 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C060", "pair": 60, "family": "counterexample", "mapping": "F(x,y,z)=(-16*x^7 + 88*x^6*y - 104*x^6*z + 272*x^6 - 204*x^5*y^2 + 480*x^5*y*z - 1260*x^5*y - 276*x^5*z^2 + 1476*x^5*z - 1836*x^5 + 258*x^4*y^3 - 906*x^4*y^2*z + 2388*x^4*y^2 + 1038*x^4*y*z^2 - 5556*x^4*y*z + 6984*x^4*y - 390*x^4*z^3 + 3168*x^4*z^2 - 8040*x^4*z + 6564*x^4 - 192*x^3*y^4 + 894*x^3*y^3*z - 2367*x^3*y^3 - 1530*x^3*y^2*z^2 + 8199*x^3*y^2*z - 10419*x^3*y^2 + 1146*x^3*y*z^3 - 9297*x^3*y*z^2 + 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30696*x*y*z - 14641*x*y - 38*x*z^6 + 519*x*z^5 - 3003*x*z^4 + 9395*x*z^3 - 16725*x*z^2 + 16027*x*z - 6437*x + 2*y^7 - 16*y^6*z + 37*y^6 + 54*y^5*z^2 - 249*y^5*z + 291*y^5 - 100*y^4*z^3 + 690*y^4*z^2 - 1611*y^4*z + 1269*y^4 + 110*y^3*z^4 - 1010*y^3*z^3 + 3534*y^3*z^2 - 5569*y^3*z + 3327*y^3 - 72*y^2*z^5 + 825*y^2*z^4 - 3846*y^2*z^3 + 9093*y^2*z^2 - 10881*y^2*z + 5261*y^2 + 26*y*z^6 - 357*y*z^5 + 2079*y*z^4 - 6555*y*z^3 + 11781*y*z^2 - 11425*y*z + 4661*y - 4*z^7 + 64*z^6 - 447*z^5 + 1762*z^4 - 4227*z^3 + 6164*z^2 - 5053*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -2, 1, -1 ], [ -1, 1, -1 ], [ -1, 1, -2 ] ], "source_shift": [ 2, 2, 2 ], "target_matrix": [ [ 2, 3, 2 ], [ 1, 1, 1 ], [ 2, 1, 1 ] ], "target_shift": [ -3125, -1219, -1789 ], "collision_preimages": [ [ "0", "-7/4", "1/4" ], [ "-5/2", "-14", "-8" ], [ "5/2", "-3", "-5" ] ], "collision_image": [ "-6251/2", "-4877/4", "-3579/2" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0349089829978766 ], "surface_features": [ 3.7392681175159836, 4.686636269262293, 3.305136318943639, 0.0, 3.898095244717543, 4913 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B199", "pair": 199, "family": "bijection", "mapping": "F(x,y,z)=(-128*x^7 - 448*x^6*y - 448*x^6*z - 544*x^6 - 672*x^5*y^2 - 1344*x^5*y*z - 1648*x^5*y - 672*x^5*z^2 - 1680*x^5*z - 1072*x^5 - 560*x^4*y^3 - 1680*x^4*y^2*z - 2080*x^4*y^2 - 1680*x^4*y*z^2 - 4240*x^4*y*z - 2720*x^4*y - 560*x^4*z^3 - 2160*x^4*z^2 - 2800*x^4*z - 1256*x^4 - 280*x^3*y^4 - 1120*x^3*y^3*z - 1400*x^3*y^3 - 1680*x^3*y^2*z^2 - 4280*x^3*y^2*z - 2760*x^3*y^2 - 1120*x^3*y*z^3 - 4360*x^3*y*z^2 - 5680*x^3*y*z - 2556*x^3*y - 280*x^3*z^4 - 1480*x^3*z^3 - 2920*x^3*z^2 - 2644*x^3*z - 972*x^3 - 84*x^2*y^5 - 420*x^2*y^4*z - 530*x^2*y^4 - 840*x^2*y^3*z^2 - 2160*x^2*y^3*z - 1400*x^2*y^3 - 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42*y^2*z^5 + 290*y^2*z^4 + 750*y^2*z^3 + 984*y^2*z^2 + 713*y^2*z + 236*y^2 + 14*y*z^6 + 118*y*z^5 + 385*y*z^4 + 678*y*z^3 + 745*y*z^2 + 510*y*z + 151*y + 2*z^7 + 20*z^6 + 79*z^5 + 175*z^4 + 259*z^3 + 276*z^2 + 169*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ 2, 1, 1 ], [ 1, 1, 1 ], [ 1, 1, 2 ] ], "source_shift": [ 1, 1, 1 ], "target_matrix": [ [ -1, -2, -1 ], [ -2, -3, -2 ], [ 2, 1, 1 ] ], "target_shift": [ 23, 43, -41 ], "inverse_recipe": { "p": 5, "q": 2, "f": "x**5 + x**3 + x**2", "g": "x**2 + x", "h": "y*z + z**2", "polynomials": { "f": [ [ [ 5, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ], [ [ 3, 0, 0 ], 1 ] ], "g": [ [ [ 2, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 0, 2 ], 1 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.031460213001992 ], "surface_features": [ 3.221314238611918, 4.049218022670182, 2.9242792860618816, 0.0, 3.315025525092439, 4786 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B114", "pair": 114, "family": "bijection", "mapping": "F(x,y,z)=(-384*x^7 + 1344*x^6*y - 1344*x^6*z + 2016*x^6 - 2016*x^5*y^2 + 4032*x^5*y*z - 6192*x^5*y - 2016*x^5*z^2 + 6096*x^5*z - 4448*x^5 + 1680*x^4*y^3 - 5040*x^4*y^2*z + 7920*x^4*y^2 + 5040*x^4*y*z^2 - 15600*x^4*y*z + 11984*x^4*y - 1680*x^4*z^3 + 7680*x^4*z^2 - 11408*x^4*z + 5832*x^4 - 840*x^3*y^4 + 3360*x^3*y^3*z - 5400*x^3*y^3 - 5040*x^3*y^2*z^2 + 15960*x^3*y^2*z - 12848*x^3*y^2 + 3360*x^3*y*z^3 - 15720*x^3*y*z^2 + 24544*x^3*y*z - 13248*x^3*y - 840*x^3*z^4 + 5160*x^3*z^3 - 11696*x^3*z^2 + 12192*x^3*z - 5026*x^3 + 252*x^2*y^5 - 1260*x^2*y^4*z + 2070*x^2*y^4 + 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3272*x*z + 1099*x - 3*y^7 + 21*y^6*z - 36*y^6 - 63*y^5*z^2 + 213*y^5*z - 193*y^5 + 105*y^4*z^3 - 525*y^4*z^2 + 929*y^4*z - 603*y^4 - 105*y^3*z^4 + 690*y^3*z^3 - 1786*y^3*z^2 + 2226*y^3*z - 1163*y^3 + 63*y^2*z^5 - 510*y^2*z^4 + 1714*y^2*z^3 - 3078*y^2*z^2 + 3039*y^2*z - 1346*y^2 - 21*y*z^6 + 201*y*z^5 - 821*y*z^4 + 1890*y*z^3 - 2661*y*z^2 + 2203*y*z - 838*y + 3*z^7 - 33*z^6 + 157*z^5 - 435*z^4 + 779*z^3 - 914*z^2 + 648*z - 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -2, 1, -1 ], [ -1, 2, -1 ], [ -3, 2, -2 ] ], "source_shift": [ 1, 2, 1 ], "target_matrix": [ [ 1, 2, 1 ], [ -2, -2, -3 ], [ -1, -1, -1 ] ], "target_shift": [ -216, 421, 213 ], "inverse_recipe": { "p": 2, "q": 5, "f": "x**2", "g": "x**5 + x**4", "h": "2*y**3 + y**2 + y*z", "polynomials": { "f": [ [ [ 2, 0, 0 ], 1 ] ], "g": [ [ [ 5, 0, 0 ], 1 ], [ [ 4, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 2, 0 ], 1 ], [ [ 0, 3, 0 ], 2 ] ] }, "h_multiplier": 3 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0644590760020947 ], "surface_features": [ 3.7630611427027096, 4.688134945323488, 3.469822015978163, 0.0, 3.8728947507480815, 4977 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B208", "pair": 208, "family": "bijection", "mapping": "F(x,y,z)=(2*x^7 - 14*x^6*y - 28*x^6*z + 30*x^6 + 42*x^5*y^2 + 168*x^5*y*z - 180*x^5*y + 168*x^5*z^2 - 360*x^5*z + 198*x^5 - 70*x^4*y^3 - 420*x^4*y^2*z + 450*x^4*y^2 - 840*x^4*y*z^2 + 1800*x^4*y*z - 992*x^4*y - 560*x^4*z^3 + 1800*x^4*z^2 - 1978*x^4*z + 743*x^4 + 70*x^3*y^4 + 560*x^3*y^3*z - 600*x^3*y^3 + 1680*x^3*y^2*z^2 - 3600*x^3*y^2*z + 1988*x^3*y^2 + 2240*x^3*y*z^3 - 7200*x^3*y*z^2 + 7928*x^3*y*z - 2988*x^3*y + 1120*x^3*z^4 - 4800*x^3*z^3 + 7904*x^3*z^2 - 5926*x^3*z + 1708*x^3 - 42*x^2*y^5 - 420*x^2*y^4*z + 450*x^2*y^4 - 1680*x^2*y^3*z^2 + 3600*x^2*y^3*z - 1992*x^2*y^3 - 3360*x^2*y^2*z^3 + 10800*x^2*y^2*z^2 - 11916*x^2*y^2*z + 4506*x^2*y^2 - 3360*x^2*y*z^4 + 14400*x^2*y*z^3 - 23760*x^2*y*z^2 + 17874*x^2*y*z - 5174*x^2*y - 1344*x^2*z^5 + 7200*x^2*z^4 - 15792*x^2*z^3 + 17724*x^2*z^2 - 10184*x^2*z + 2401*x^2 + 14*x*y^6 + 168*x*y^5*z - 180*x*y^5 + 840*x*y^4*z^2 - 1800*x*y^4*z + 998*x*y^4 + 2240*x*y^3*z^3 - 7200*x*y^3*z^2 + 7960*x*y^3*z - 3020*x*y^3 + 3360*x*y^2*z^4 - 14400*x*y^2*z^3 + 23808*x*y^2*z^2 - 17970*x*y^2*z + 5224*x*y^2 + 2688*x*y*z^5 - 14400*x*y*z^4 + 31648*x*y*z^3 - 35640*x*y*z^2 + 20568*x*y*z - 4876*x*y + 896*x*z^6 - 5760*x*z^5 + 15776*x*z^4 - 23560*x*z^3 + 20240*x*z^2 - 9494*x*z + 1905*x - 2*y^7 - 28*y^6*z + 30*y^6 - 168*y^5*z^2 + 360*y^5*z - 200*y^5 - 560*y^4*z^3 + 1800*y^4*z^2 - 1994*y^4*z + 759*y^4 - 1120*y^3*z^4 + 4800*y^3*z^3 - 7952*y^3*z^2 + 6022*y^3*z - 1758*y^3 - 1344*y^2*z^5 + 7200*y^2*z^4 - 15856*y^2*z^3 + 17916*y^2*z^2 - 10384*y^2*z + 2475*y^2 - 896*y*z^6 + 5760*y*z^5 - 15808*y*z^4 + 23688*y*z^3 - 20440*y*z^2 + 9640*y*z - 1947*y - 256*z^7 + 1920*z^6 - 6304*z^5 + 11744*z^4 - 13408*z^3 + 9386*z^2 - 3737*z + 1, x^7 - 7*x^6*y - 14*x^6*z + 15*x^6 + 21*x^5*y^2 + 84*x^5*y*z - 90*x^5*y + 84*x^5*z^2 - 180*x^5*z + 99*x^5 - 35*x^4*y^3 - 210*x^4*y^2*z + 225*x^4*y^2 - 420*x^4*y*z^2 + 900*x^4*y*z - 496*x^4*y - 280*x^4*z^3 + 900*x^4*z^2 - 989*x^4*z + 372*x^4 + 35*x^3*y^4 + 280*x^3*y^3*z - 300*x^3*y^3 + 840*x^3*y^2*z^2 - 1800*x^3*y^2*z + 994*x^3*y^2 + 1120*x^3*y*z^3 - 3600*x^3*y*z^2 + 3964*x^3*y*z - 1496*x^3*y + 560*x^3*z^4 - 2400*x^3*z^3 + 3952*x^3*z^2 - 2967*x^3*z + 857*x^3 - 21*x^2*y^5 - 210*x^2*y^4*z + 225*x^2*y^4 - 840*x^2*y^3*z^2 + 1800*x^2*y^3*z - 996*x^2*y^3 - 1680*x^2*y^2*z^3 + 5400*x^2*y^2*z^2 - 5958*x^2*y^2*z + 2256*x^2*y^2 - 1680*x^2*y*z^4 + 7200*x^2*y*z^3 - 11880*x^2*y*z^2 + 8949*x^2*y*z - 2596*x^2*y - 672*x^2*z^5 + 3600*x^2*z^4 - 7896*x^2*z^3 + 8874*x^2*z^2 - 5110*x^2*z + 1206*x^2 + 7*x*y^6 + 84*x*y^5*z - 90*x*y^5 + 420*x*y^4*z^2 - 900*x*y^4*z + 499*x*y^4 + 1120*x*y^3*z^3 - 3600*x*y^3*z^2 + 3980*x*y^3*z - 1512*x*y^3 + 1680*x*y^2*z^4 - 7200*x*y^2*z^3 + 11904*x*y^2*z^2 - 8997*x*y^2*z + 2621*x*y^2 + 1344*x*y*z^5 - 7200*x*y*z^4 + 15824*x*y*z^3 - 17844*x*y*z^2 + 10320*x*y*z - 2449*x*y + 448*x*z^6 - 2880*x*z^5 + 7888*x*z^4 - 11796*x*z^3 + 10156*x*z^2 - 4769*x*z + 953*x - y^7 - 14*y^6*z + 15*y^6 - 84*y^5*z^2 + 180*y^5*z - 100*y^5 - 280*y^4*z^3 + 900*y^4*z^2 - 997*y^4*z + 380*y^4 - 560*y^3*z^4 + 2400*y^3*z^3 - 3976*y^3*z^2 + 3015*y^3*z - 882*y^3 - 672*y^2*z^5 + 3600*y^2*z^4 - 7928*y^2*z^3 + 8970*y^2*z^2 - 5210*y^2*z + 1243*y^2 - 448*y*z^6 + 2880*y*z^5 - 7904*y*z^4 + 11860*y*z^3 - 10256*y*z^2 + 4842*y*z - 973*y - 128*z^7 + 960*z^6 - 3152*z^5 + 5880*z^4 - 6728*z^3 + 4715*z^2 - 1870*z + 1, 2*x^7 - 14*x^6*y - 28*x^6*z + 30*x^6 + 42*x^5*y^2 + 168*x^5*y*z - 180*x^5*y + 168*x^5*z^2 - 360*x^5*z + 198*x^5 - 70*x^4*y^3 - 420*x^4*y^2*z + 450*x^4*y^2 - 840*x^4*y*z^2 + 1800*x^4*y*z - 992*x^4*y - 560*x^4*z^3 + 1800*x^4*z^2 - 1978*x^4*z + 744*x^4 + 70*x^3*y^4 + 560*x^3*y^3*z - 600*x^3*y^3 + 1680*x^3*y^2*z^2 - 3600*x^3*y^2*z + 1988*x^3*y^2 + 2240*x^3*y*z^3 - 7200*x^3*y*z^2 + 7928*x^3*y*z - 2992*x^3*y + 1120*x^3*z^4 - 4800*x^3*z^3 + 7904*x^3*z^2 - 5934*x^3*z + 1712*x^3 - 42*x^2*y^5 - 420*x^2*y^4*z + 450*x^2*y^4 - 1680*x^2*y^3*z^2 + 3600*x^2*y^3*z - 1992*x^2*y^3 - 3360*x^2*y^2*z^3 + 10800*x^2*y^2*z^2 - 11916*x^2*y^2*z + 4512*x^2*y^2 - 3360*x^2*y*z^4 + 14400*x^2*y*z^3 - 23760*x^2*y*z^2 + 17898*x^2*y*z - 5186*x^2*y - 1344*x^2*z^5 + 7200*x^2*z^4 - 15792*x^2*z^3 + 17748*x^2*z^2 - 10208*x^2*z + 2398*x^2 + 14*x*y^6 + 168*x*y^5*z - 180*x*y^5 + 840*x*y^4*z^2 - 1800*x*y^4*z + 998*x*y^4 + 2240*x*y^3*z^3 - 7200*x*y^3*z^2 + 7960*x*y^3*z - 3024*x*y^3 + 3360*x*y^2*z^4 - 14400*x*y^2*z^3 + 23808*x*y^2*z^2 - 17994*x*y^2*z + 5236*x*y^2 + 2688*x*y*z^5 - 14400*x*y*z^4 + 31648*x*y*z^3 - 35688*x*y*z^2 + 20616*x*y*z - 4870*x*y + 896*x*z^6 - 5760*x*z^5 + 15776*x*z^4 - 23592*x*z^3 + 20288*x*z^2 - 9482*x*z + 1870*x - 2*y^7 - 28*y^6*z + 30*y^6 - 168*y^5*z^2 + 360*y^5*z - 200*y^5 - 560*y^4*z^3 + 1800*y^4*z^2 - 1994*y^4*z + 760*y^4 - 1120*y^3*z^4 + 4800*y^3*z^3 - 7952*y^3*z^2 + 6030*y^3*z - 1762*y^3 - 1344*y^2*z^5 + 7200*y^2*z^4 - 15856*y^2*z^3 + 17940*y^2*z^2 - 10408*y^2*z + 2472*y^2 - 896*y*z^6 + 5760*y*z^5 - 15808*y*z^4 + 23720*y*z^3 - 20488*y*z^2 + 9628*y*z - 1908*y - 256*z^7 + 1920*z^6 - 6304*z^5 + 11760*z^4 - 13440*z^3 + 9374*z^2 - 3670*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ 1, -1, -2 ], [ 1, -1, -1 ], [ 1, -2, -1 ] ], "source_shift": [ 2, 2, 1 ], "target_matrix": [ [ 2, 1, 1 ], [ 1, 1, 1 ], [ 2, 2, 3 ] ], "target_shift": [ -655, -323, -617 ], "inverse_recipe": { "p": 4, "q": 3, "f": "x**4 + x**2", "g": "x**3 + x**2 + x", "h": "y*z", "polynomials": { "f": [ [ [ 4, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "g": [ [ [ 3, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.1513135330023943 ], "surface_features": [ 3.6890393767748257, 4.55252221023356, 3.2833012287035497, 0.0, 3.8169779361021483, 4907 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C061", "pair": 61, "family": "counterexample", "mapping": "F(x,y,z)=(-3*x^7 + 30*x^6*y - 24*x^6*z + 57*x^6 - 126*x^5*y^2 + 207*x^5*y*z - 495*x^5*y - 81*x^5*z^2 + 387*x^5*z - 459*x^5 + 288*x^4*y^3 - 729*x^4*y^2*z + 1764*x^4*y^2 + 585*x^4*y*z^2 - 2808*x^4*y*z + 3366*x^4*y - 150*x^4*z^3 + 1080*x^4*z^2 - 2574*x^4*z + 2038*x^4 - 387*x^3*y^4 + 1341*x^3*y^3*z - 3303*x^3*y^3 - 1656*x^3*y^2*z^2 + 8019*x^3*y^2*z - 9747*x^3*y^2 + 870*x^3*y*z^3 - 6282*x^3*y*z^2 + 15102*x^3*y*z - 12103*x^3*y - 165*x^3*z^4 + 1590*x^3*z^3 - 5706*x^3*z^2 + 9080*x^3*z - 5407*x^3 + 306*x^2*y^5 - 1359*x^2*y^4*z + 3429*x^2*y^4 + 2295*x^2*y^3*z^2 - 11268*x^2*y^3*z + 13932*x^2*y^3 - 1854*x^2*y^2*z^3 + 13473*x^2*y^2*z^2 - 32760*x^2*y^2*z + 26637*x^2*y^2 + 720*x^2*y*z^4 - 6948*x^2*y*z^3 + 25110*x^2*y*z^2 - 40371*x^2*y*z + 24360*x^2*y - 108*x^2*z^5 + 1305*x^2*z^4 - 6264*x^2*z^3 + 15012*x^2*z^2 - 17982*x^2*z + 8598*x^2 - 132*x*y^6 + 720*x*y^5*z - 1872*x*y^5 - 1557*x*y^4*z^2 + 7794*x*y^4*z - 9828*x*y^4 + 1719*x*y^3*z^3 - 12627*x*y^3*z^2 + 31140*x*y^3*z - 25738*x*y^3 - 1026*x*y^2*z^4 + 9945*x*y^2*z^3 - 36279*x*y^2*z^2 + 59046*x*y^2*z - 36171*x*y^2 + 315*x*y*z^5 - 3807*x*y*z^4 + 18378*x*y*z^3 - 44433*x*y*z^2 + 53835*x*y*z - 26115*x*y - 39*x*z^6 + 567*x*z^5 - 3411*x*z^4 + 10936*x*z^3 - 19743*x*z^2 + 19017*x*z - 7605*x + 24*y^7 - 156*y^6*z + 420*y^6 + 414*y^5*z^2 - 2124*y^5*z + 2736*y^5 - 585*y^4*z^3 + 4365*y^4*z^2 - 10944*y^4*z + 9202*y^4 + 477*y^3*z^4 - 4662*y^3*z^3 + 17208*y^3*z^2 - 28400*y^3*z + 17680*y^3 - 225*y^2*z^5 + 2727*y^2*z^4 - 13266*y^2*z^3 + 32409*y^2*z^2 - 39777*y^2*z + 19617*y^2 + 57*y*z^6 - 828*y*z^5 + 5004*y*z^4 - 16165*y*z^3 + 29475*y*z^2 - 28746*y*z + 11682*y - 6*z^7 + 102*z^6 - 738*z^5 + 2966*z^4 - 7168*z^3 + 10419*z^2 - 8414*z + 1, x^7 - 10*x^6*y + 8*x^6*z - 22*x^6 + 42*x^5*y^2 - 69*x^5*y*z + 189*x^5*y + 27*x^5*z^2 - 150*x^5*z + 195*x^5 - 96*x^4*y^3 + 243*x^4*y^2*z - 666*x^4*y^2 - 195*x^4*y*z^2 + 1077*x^4*y*z - 1413*x^4*y + 50*x^4*z^3 - 420*x^4*z^2 + 1098*x^4*z - 925*x^4 + 129*x^3*y^4 - 447*x^3*y^3*z + 1233*x^3*y^3 + 552*x^3*y^2*z^2 - 3042*x^3*y^2*z + 4041*x^3*y^2 - 290*x^3*y*z^3 + 2418*x^3*y*z^2 - 6366*x^3*y*z + 5431*x^3*y + 55*x^3*z^4 - 620*x^3*z^3 + 2442*x^3*z^2 - 4135*x^3*z + 2573*x^3 - 102*x^2*y^5 + 453*x^2*y^4*z - 1266*x^2*y^4 - 765*x^2*y^3*z^2 + 4227*x^2*y^3*z - 5703*x^2*y^3 + 618*x^2*y^2*z^3 - 5130*x^2*y^2*z^2 + 13638*x^2*y^2*z - 11811*x^2*y^2 - 240*x^2*y*z^4 + 2682*x^2*y*z^3 - 10620*x^2*y*z^2 + 18174*x^2*y*z - 11469*x^2*y + 36*x^2*z^5 - 510*x^2*z^4 + 2688*x^2*z^3 - 6855*x^2*z^2 + 8580*x^2*z - 4237*x^2 + 44*x*y^6 - 240*x*y^5*z + 684*x*y^5 + 519*x*y^4*z^2 - 2892*x*y^4*z + 3972*x*y^4 - 573*x*y^3*z^3 + 4755*x*y^3*z^2 - 12798*x*y^3*z + 11272*x*y^3 + 342*x*y^2*z^4 - 3798*x*y^2*z^3 + 15153*x*y^2*z^2 - 26259*x*y^2*z + 16839*x*y^2 - 105*x*y*z^5 + 1473*x*y*z^4 - 7794*x*y*z^3 + 20055*x*y*z^2 - 25407*x*y*z + 12739*x*y + 13*x*z^6 - 222*x*z^5 + 1467*x*z^4 - 5005*x*z^3 + 9441*x*z^2 - 9392*x*z + 3847*x - 8*y^7 + 52*y^6*z - 152*y^6 - 138*y^5*z^2 + 780*y^5*z - 1092*y^5 + 195*y^4*z^3 - 1626*y^4*z^2 + 4440*y^4*z - 3979*y^4 - 159*y^3*z^4 + 1761*y^3*z^3 - 7095*y^3*z^2 + 12471*y^3*z - 8133*y^3 + 75*y^2*z^5 - 1044*y^2*z^4 + 5556*y^2*z^3 - 14448*y^2*z^2 + 18555*y^2*z - 9466*y^2 - 19*y*z^6 + 321*y*z^5 - 2127*y*z^4 + 7312*y*z^3 - 13938*y*z^2 + 14047*y*z - 5851*y + 2*z^7 - 40*z^6 + 318*z^5 - 1360*z^4 + 3434*z^3 - 5155*z^2 + 4264*z - 1, x^7 - 10*x^6*y + 8*x^6*z - 19*x^6 + 42*x^5*y^2 - 69*x^5*y*z + 165*x^5*y + 27*x^5*z^2 - 129*x^5*z + 153*x^5 - 96*x^4*y^3 + 243*x^4*y^2*z - 588*x^4*y^2 - 195*x^4*y*z^2 + 936*x^4*y*z - 1122*x^4*y + 50*x^4*z^3 - 360*x^4*z^2 + 858*x^4*z - 679*x^4 + 129*x^3*y^4 - 447*x^3*y^3*z + 1101*x^3*y^3 + 552*x^3*y^2*z^2 - 2673*x^3*y^2*z + 3249*x^3*y^2 - 290*x^3*y*z^3 + 2094*x^3*y*z^2 - 5034*x^3*y*z + 4033*x^3*y + 55*x^3*z^4 - 530*x^3*z^3 + 1902*x^3*z^2 - 3025*x^3*z + 1799*x^3 - 102*x^2*y^5 + 453*x^2*y^4*z - 1143*x^2*y^4 - 765*x^2*y^3*z^2 + 3756*x^2*y^3*z - 4644*x^2*y^3 + 618*x^2*y^2*z^3 - 4491*x^2*y^2*z^2 + 10920*x^2*y^2*z - 8877*x^2*y^2 - 240*x^2*y*z^4 + 2316*x^2*y*z^3 - 8370*x^2*y*z^2 + 13452*x^2*y*z - 8109*x^2*y + 36*x^2*z^5 - 435*x^2*z^4 + 2088*x^2*z^3 - 5001*x^2*z^2 + 5982*x^2*z - 2854*x^2 + 44*x*y^6 - 240*x*y^5*z + 624*x*y^5 + 519*x*y^4*z^2 - 2598*x*y^4*z + 3276*x*y^4 - 573*x*y^3*z^3 + 4209*x*y^3*z^2 - 10380*x*y^3*z + 8578*x*y^3 + 342*x*y^2*z^4 - 3315*x*y^2*z^3 + 12093*x*y^2*z^2 - 19677*x*y^2*z + 12045*x*y^2 - 105*x*y*z^5 + 1269*x*y*z^4 - 6126*x*y*z^3 + 14805*x*y*z^2 - 17919*x*y*z + 8677*x*y + 13*x*z^6 - 189*x*z^5 + 1137*x*z^4 - 3643*x*z^3 + 6567*x*z^2 - 6311*x*z + 2517*x - 8*y^7 + 52*y^6*z - 140*y^6 - 138*y^5*z^2 + 708*y^5*z - 912*y^5 + 195*y^4*z^3 - 1455*y^4*z^2 + 3648*y^4*z - 3067*y^4 - 159*y^3*z^4 + 1554*y^3*z^3 - 5736*y^3*z^2 + 9465*y^3*z - 5889*y^3 + 75*y^2*z^5 - 909*y^2*z^4 + 4422*y^2*z^3 - 10800*y^2*z^2 + 13245*y^2*z - 6523*y^2 - 19*y*z^6 + 276*y*z^5 - 1668*y*z^4 + 5386*y*z^3 - 9810*y*z^2 + 9550*y*z - 3872*y + 2*z^7 - 34*z^6 + 246*z^5 - 988*z^4 + 2384*z^3 - 3457*z^2 + 2784*z - 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -1, 1, -1 ], [ -1, 2, -1 ], [ -1, 1, -2 ] ], "source_shift": [ 2, 2, 1 ], "target_matrix": [ [ 3, 3, 2 ], [ -1, -2, -1 ], [ -1, -1, -1 ] ], "target_shift": [ -2886, 1488, 952 ], "collision_preimages": [ [ "11/4", "0", "-3/4" ], [ "5", "-5/2", "-13/2" ], [ "14", "5/2", "-17/2" ] ], "collision_image": [ "-11547/4", "5953/4", "3809/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0931804519968864 ], "surface_features": [ 3.744022279839258, 4.771190482190553, 3.2616196765479355, 0.0, 3.9366087148814017, 4897 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C130", "pair": 130, "family": "counterexample", "mapping": "F(x,y,z)=(-16*x^7 - 96*x^6*y + 88*x^6*z + 176*x^6 - 240*x^5*y^2 + 444*x^5*y*z + 864*x^5*y - 204*x^5*z^2 - 804*x^5*z - 804*x^5 - 324*x^4*y^3 + 906*x^4*y^2*z + 1740*x^4*y^2 - 840*x^4*y*z^2 - 3252*x^4*y*z - 3192*x^4*y + 258*x^4*z^3 + 1512*x^4*z^2 + 3000*x^4*z + 2006*x^4 - 255*x^3*y^4 + 957*x^3*y^3*z + 1836*x^3*y^3 - 1341*x^3*y^2*z^2 - 5169*x^3*y^2*z - 5013*x^3*y^2 + 831*x^3*y*z^3 + 4830*x^3*y*z^2 + 9438*x^3*y*z + 6239*x^3*y - 192*x^3*z^4 - 1497*x^3*z^3 - 4425*x^3*z^2 - 5900*x^3*z - 2987*x^3 - 117*x^2*y^5 + 552*x^2*y^4*z + 1068*x^2*y^4 - 1038*x^2*y^3*z^2 - 4026*x^2*y^3*z - 3885*x^2*y^3 + 972*x^2*y^2*z^3 + 5670*x^2*y^2*z^2 + 10995*x^2*y^2*z + 7209*x^2*y^2 - 453*x^2*y*z^4 - 3534*x^2*y*z^3 - 10335*x^2*y*z^2 - 13635*x^2*y*z - 6855*x^2*y + 84*x^2*z^5 + 822*x^2*z^4 + 3225*x^2*z^3 + 6426*x^2*z^2 + 6504*x^2*z + 2677*x^2 - 29*x*y^6 + 165*x*y^5*z + 324*x*y^5 - 390*x*y^4*z^2 - 1533*x*y^4*z - 1482*x*y^4 + 490*x*y^3*z^3 + 2892*x*y^3*z^2 + 5607*x*y^3*z + 3662*x*y^3 - 345*x*y^2*z^4 - 2718*x*y^2*z^3 - 7929*x*y^2*z^2 - 10398*x*y^2*z - 5199*x*y^2 + 129*x*y*z^5 + 1272*x*y*z^4 + 4965*x*y*z^3 + 9810*x*y*z^2 + 9855*x*y*z + 4039*x*y - 20*x*z^6 - 237*x*z^5 - 1161*x*z^4 - 3074*x*z^3 - 4656*x*z^2 - 3835*x*z - 1344*x - 3*y^7 + 20*y^6*z + 40*y^6 - 57*y^5*z^2 - 228*y^5*z - 222*y^5 + 90*y^4*z^3 + 540*y^4*z^2 + 1053*y^4*z + 688*y^4 - 85*y^3*z^4 - 680*y^3*z^3 - 1992*y^3*z^2 - 2609*y^3*z - 1302*y^3 + 48*y^2*z^5 + 480*y^2*z^4 + 1878*y^2*z^3 + 3699*y^2*z^2 + 3702*y^2*z + 1513*y^2 - 15*y*z^6 - 180*y*z^5 - 882*y*z^4 - 2323*y*z^3 - 3498*y*z^2 - 2867*y*z - 1002*y + 2*z^7 + 28*z^6 + 165*z^5 + 545*z^4 + 1098*z^3 + 1354*z^2 + 949*z + 1, 16*x^7 + 96*x^6*y - 88*x^6*z - 176*x^6 + 240*x^5*y^2 - 444*x^5*y*z - 864*x^5*y + 204*x^5*z^2 + 804*x^5*z + 804*x^5 + 324*x^4*y^3 - 906*x^4*y^2*z - 1740*x^4*y^2 + 840*x^4*y*z^2 + 3252*x^4*y*z + 3192*x^4*y - 258*x^4*z^3 - 1512*x^4*z^2 - 3000*x^4*z - 2004*x^4 + 255*x^3*y^4 - 957*x^3*y^3*z - 1836*x^3*y^3 + 1341*x^3*y^2*z^2 + 5169*x^3*y^2*z + 5013*x^3*y^2 - 831*x^3*y*z^3 - 4830*x^3*y*z^2 - 9438*x^3*y*z - 6230*x^3*y + 192*x^3*z^4 + 1497*x^3*z^3 + 4425*x^3*z^2 + 5892*x^3*z + 2974*x^3 + 117*x^2*y^5 - 552*x^2*y^4*z - 1068*x^2*y^4 + 1038*x^2*y^3*z^2 + 4026*x^2*y^3*z + 3885*x^2*y^3 - 972*x^2*y^2*z^3 - 5670*x^2*y^2*z^2 - 10995*x^2*y^2*z - 7194*x^2*y^2 + 453*x^2*y*z^4 + 3534*x^2*y*z^3 + 10335*x^2*y*z^2 + 13608*x^2*y*z + 6816*x^2*y - 84*x^2*z^5 - 822*x^2*z^4 - 3225*x^2*z^3 - 6414*x^2*z^2 - 6468*x^2*z - 2650*x^2 + 29*x*y^6 - 165*x*y^5*z - 324*x*y^5 + 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5196*x^4*z - 3330*x^4 + 510*x^3*y^4 - 1914*x^3*y^3*z - 3333*x^3*y^3 + 2682*x^3*y^2*z^2 + 9396*x^3*y^2*z + 8583*x^3*y^2 - 1662*x^3*y*z^3 - 8793*x^3*y*z^2 - 16182*x^3*y*z - 10255*x^3*y + 384*x^3*z^4 + 2730*x^3*z^3 + 7599*x^3*z^2 + 9708*x^3*z + 4775*x^3 + 234*x^2*y^5 - 1104*x^2*y^4*z - 1923*x^2*y^4 + 2076*x^2*y^3*z^2 + 7257*x^2*y^3*z + 6585*x^2*y^3 - 1944*x^2*y^2*z^3 - 10233*x^2*y^2*z^2 - 18660*x^2*y^2*z - 11730*x^2*y^2 + 906*x^2*y*z^4 + 6387*x^2*y*z^3 + 17565*x^2*y*z^2 + 22209*x^2*y*z + 10851*x^2*y - 168*x^2*z^5 - 1488*x^2*z^4 - 5490*x^2*z^3 - 10479*x^2*z^2 - 10302*x^2*z - 4163*x^2 + 58*x*y^6 - 330*x*y^5*z - 579*x*y^5 + 780*x*y^4*z^2 + 2742*x*y^4*z + 2487*x*y^4 - 980*x*y^3*z^3 - 5178*x*y^3*z^2 - 9420*x*y^3*z - 5898*x*y^3 + 690*x*y^2*z^4 + 4872*x*y^2*z^3 + 13338*x*y^2*z^2 + 16764*x*y^2*z + 8148*x*y^2 - 258*x*y*z^5 - 2283*x*y*z^4 - 8364*x*y*z^3 - 15834*x*y*z^2 - 15456*x*y*z - 6218*x*y + 40*x*z^6 + 426*x*z^5 + 1959*x*z^4 + 4968*x*z^3 + 7308*x*z^2 + 5906*x*z + 2050*x + 6*y^7 - 40*y^6*z - 71*y^6 + 114*y^5*z^2 + 405*y^5*z + 369*y^5 - 180*y^4*z^3 - 960*y^4*z^2 - 1752*y^4*z - 1097*y^4 + 170*y^3*z^4 + 1210*y^3*z^3 + 3318*y^3*z^2 + 4164*y^3*z + 2020*y^3 - 96*y^2*z^5 - 855*y^2*z^4 - 3132*y^2*z^3 - 5910*y^2*z^2 - 5748*y^2*z - 2306*y^2 + 30*y*z^6 + 321*y*z^5 + 1473*y*z^4 + 3716*y*z^3 + 5436*y*z^2 + 4372*y*z + 1512*y - 4*z^7 - 50*z^6 - 276*z^5 - 873*z^4 - 1708*z^3 - 2066*z^2 - 1432*z - 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -1, -1, 1 ], [ -2, -1, 1 ], [ -2, -3, 2 ] ], "source_shift": [ 1, 2, 1 ], "target_matrix": [ [ 1, 1, 1 ], [ -1, -1, -2 ], [ -2, -1, -1 ] ], "target_shift": [ -290, 285, 437 ], "collision_preimages": [ [ "1", "-3/4", "-3/4" ], [ "7/2", "-11/2", "-2" ], [ "-3/2", "-19/2", "-13" ] ], "collision_image": [ "-1161/4", "1141/4", "875/2" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0444342760019936 ], 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15172*x*y*z^2 - 8642*x*y*z - 2065*x*y - 448*x*z^6 - 2560*x*z^5 - 6352*x*z^4 - 8752*x*z^3 - 7060*x*z^2 - 3153*x*z - 608*x - y^7 - 14*y^6*z - 17*y^6 - 84*y^5*z^2 - 196*y^5*z - 118*y^5 - 280*y^4*z^3 - 940*y^4*z^2 - 1087*y^4*z - 443*y^4 - 560*y^3*z^4 - 2400*y^3*z^3 - 3992*y^3*z^2 - 3104*y^3*z - 972*y^3 - 672*y^2*z^5 - 3440*y^2*z^4 - 7304*y^2*z^3 - 8112*y^2*z^2 - 4739*y^2*z - 1166*y^2 - 448*y*z^6 - 2624*y*z^5 - 6656*y*z^4 - 9376*y*z^3 - 7740*y*z^2 - 3542*y*z - 700*y - 128*z^7 - 832*z^6 - 2416*z^5 - 4048*z^4 - 4220*z^3 - 2728*z^2 - 1011*z - 1, -2*x^7 - 14*x^6*y - 28*x^6*z - 30*x^6 - 42*x^5*y^2 - 168*x^5*y*z - 184*x^5*y - 168*x^5*z^2 - 352*x^5*z - 192*x^5 - 70*x^4*y^3 - 420*x^4*y^2*z - 470*x^4*y^2 - 840*x^4*y*z^2 - 1800*x^4*y*z - 1002*x^4*y - 560*x^4*z^3 - 1720*x^4*z^2 - 1834*x^4*z - 686*x^4 - 70*x^3*y^4 - 560*x^3*y^3*z - 640*x^3*y^3 - 1680*x^3*y^2*z^2 - 3680*x^3*y^2*z - 2090*x^3*y^2 - 2240*x^3*y*z^3 - 7040*x^3*y*z^2 - 7664*x^3*y*z - 2930*x^3*y - 1120*x^3*z^4 - 4480*x^3*z^3 - 7000*x^3*z^2 - 5096*x^3*z - 1468*x^3 - 42*x^2*y^5 - 420*x^2*y^4*z - 490*x^2*y^4 - 1680*x^2*y^3*z^2 - 3760*x^2*y^3*z - 2178*x^2*y^3 - 3360*x^2*y^2*z^3 - 10800*x^2*y^2*z^2 - 12000*x^2*y^2*z - 4688*x^2*y^2 - 3360*x^2*y*z^4 - 13760*x^2*y*z^3 - 21960*x^2*y*z^2 - 16344*x^2*y*z - 4834*x^2*y - 1344*x^2*z^5 - 6560*x^2*z^4 - 13344*x^2*z^3 - 14176*x^2*z^2 - 7880*x^2*z - 1830*x^2 - 14*x*y^6 - 168*x*y^5*z - 200*x*y^5 - 840*x*y^4*z^2 - 1920*x*y^4*z - 1134*x*y^4 - 2240*x*y^3*z^3 - 7360*x*y^3*z^2 - 8344*x*y^3*z - 3330*x*y^3 - 3360*x*y^2*z^4 - 14080*x*y^2*z^3 - 22944*x*y^2*z^2 - 17456*x*y^2*z - 5308*x*y^2 - 2688*x*y*z^5 - 13440*x*y*z^4 - 27936*x*y*z^3 - 30344*x*y*z^2 - 17284*x*y*z - 4126*x*y - 896*x*z^6 - 5120*x*z^5 - 12704*x*z^4 - 17504*x*z^3 - 14120*x*z^2 - 6298*x*z - 1206*x - 2*y^7 - 28*y^6*z - 34*y^6 - 168*y^5*z^2 - 392*y^5*z - 236*y^5 - 560*y^4*z^3 - 1880*y^4*z^2 - 2174*y^4*z - 886*y^4 - 1120*y^3*z^4 - 4800*y^3*z^3 - 7984*y^3*z^2 - 6208*y^3*z - 1944*y^3 - 1344*y^2*z^5 - 6880*y^2*z^4 - 14608*y^2*z^3 - 16224*y^2*z^2 - 9478*y^2*z - 2330*y^2 - 896*y*z^6 - 5248*y*z^5 - 13312*y*z^4 - 18752*y*z^3 - 15480*y*z^2 - 7076*y*z - 1388*y - 256*z^7 - 1664*z^6 - 4832*z^5 - 8096*z^4 - 8440*z^3 - 5448*z^2 - 2006*z - 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ 1, 1, 2 ], [ 1, 1, 1 ], [ 2, 3, 2 ] ], "source_shift": [ 1, 2, 2 ], "target_matrix": [ [ 2, 1, 1 ], [ -1, -1, -1 ], [ -2, -2, -3 ] ], "target_shift": [ -334, 165, 323 ], "inverse_recipe": { "p": 3, "q": 2, "f": "x**3 + x**2 + x", "g": "x**2 + x", "h": "y*z**2 + y*z + z**3 + z", "polynomials": { "f": [ [ [ 3, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "g": [ [ [ 2, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 2 ], 1 ], [ [ 0, 1, 1 ], 1 ], [ [ 0, 0, 1 ], 1 ], [ [ 0, 0, 3 ], 1 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0778271799972572 ], "surface_features": [ 3.632413140605071, 4.482072829694734, 3.2741578492636796, 0.0, 3.749647550449925, 4903 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B168", "pair": 168, "family": "bijection", "mapping": "F(x,y,z)=(-3*x^7 - 21*x^6*y + 21*x^6*z + 30*x^6 - 63*x^5*y^2 + 126*x^5*y*z + 186*x^5*y - 63*x^5*z^2 - 180*x^5*z - 129*x^5 - 105*x^4*y^3 + 315*x^4*y^2*z + 480*x^4*y^2 - 315*x^4*y*z^2 - 930*x^4*y*z - 687*x^4*y + 105*x^4*z^3 + 450*x^4*z^2 + 648*x^4*z + 309*x^4 - 105*x^3*y^4 + 420*x^3*y^3*z + 660*x^3*y^3 - 630*x^3*y^2*z^2 - 1920*x^3*y^2*z - 1461*x^3*y^2 + 420*x^3*y*z^3 + 1860*x^3*y*z^2 + 2760*x^3*y*z + 1356*x^3*y - 105*x^3*z^4 - 600*x^3*z^3 - 1302*x^3*z^2 - 1254*x^3*z - 442*x^3 - 63*x^2*y^5 + 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3840*x^3*y^2*z - 2922*x^3*y^2 + 840*x^3*y*z^3 + 3720*x^3*y*z^2 + 5520*x^3*y*z + 2712*x^3*y - 210*x^3*z^4 - 1200*x^3*z^3 - 2604*x^3*z^2 - 2508*x^3*z - 886*x^3 - 126*x^2*y^5 + 630*x^2*y^4*z + 1020*x^2*y^4 - 1260*x^2*y^3*z^2 - 3960*x^2*y^3*z - 3102*x^2*y^3 + 1260*x^2*y^2*z^3 + 5760*x^2*y^2*z^2 + 8802*x^2*y^2*z + 4452*x^2*y^2 - 630*x^2*y*z^4 - 3720*x^2*y*z^3 - 8316*x^2*y*z^2 - 8256*x^2*y*z - 3006*x^2*y + 126*x^2*z^5 + 900*x^2*z^4 + 2616*x^2*z^3 + 3816*x^2*z^2 + 2736*x^2*z + 755*x^2 - 42*x*y^6 + 252*x*y^5*z + 420*x*y^5 - 630*x*y^4*z^2 - 2040*x*y^4*z - 1644*x*y^4 + 840*x*y^3*z^3 + 3960*x*y^3*z^2 + 6228*x*y^3*z + 3240*x*y^3 - 630*x*y^2*z^4 - 3840*x*y^2*z^3 - 8838*x*y^2*z^2 - 9036*x*y^2*z - 3390*x*y^2 + 252*x*y*z^5 + 1860*x*y*z^4 + 5568*x*y*z^3 + 8376*x*y*z^2 + 6204*x*y*z + 1762*x*y - 42*x*z^6 - 360*x*z^5 - 1314*x*z^4 - 2580*x*z^3 - 2814*x*z^2 - 1582*x*z - 353*x - 6*y^7 + 42*y^6*z + 72*y^6 - 126*y^5*z^2 - 420*y^5*z - 348*y^5 + 210*y^4*z^3 + 1020*y^4*z^2 + 1650*y^4*z + 882*y^4 - 210*y^3*z^4 - 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774*y^5 - 491*y^4*z^3 - 3246*y^4*z^2 - 5007*y^4*z - 2175*y^4 - 584*y^3*z^4 - 5416*y^3*z^3 - 12897*y^3*z^2 - 11472*y^3*z - 3363*y^3 - 414*y^2*z^5 - 5055*y^2*z^4 - 16530*y^2*z^3 - 22599*y^2*z^2 - 13548*y^2*z - 2887*y^2 - 162*y*z^6 - 2502*y*z^5 - 10539*y*z^4 - 19701*y*z^3 - 18132*y*z^2 - 7887*y*z - 1276*y - 27*z^7 - 513*z^6 - 2673*z^5 - 6411*z^4 - 8061*z^3 - 5370*z^2 - 1771*z - 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ 1, 1, 1 ], [ 2, 2, 3 ], [ 2, 1, 1 ] ], "source_shift": [ 2, 1, 2 ], "target_matrix": [ [ 3, 2, 2 ], [ -1, -2, -1 ], [ -1, -1, -1 ] ], "target_shift": [ -541, 397, 228 ], "collision_preimages": [ [ "-1/4", "-19/4", "3" ], [ "11/2", "-6", "-1/2" ], [ "15/2", "-17", "13/2" ] ], "collision_image": [ "-2167/4", "1589/4", "913/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0395035729998199 ], "surface_features": [ 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1452*y^2*z^4 - 6687*y^2*z^3 + 15411*y^2*z^2 - 17775*y^2*z + 8197*y^2 - 30*y*z^6 + 417*y*z^5 - 2400*y*z^4 + 7345*y*z^3 - 12612*y*z^2 + 11497*y*z - 4324*y - 3*z^7 + 49*z^6 - 339*z^5 + 1294*z^4 - 2947*z^3 + 4000*z^2 - 2981*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ 1, -1, -1 ], [ 1, -2, -1 ], [ 2, -3, -3 ] ], "source_shift": [ 2, 2, 1 ], "target_matrix": [ [ 2, 3, 2 ], [ 1, 2, 1 ], [ 1, 1, 2 ] ], "target_shift": [ -2441, -1488, -924 ], "collision_preimages": [ [ "-19/4", "0", "-11/4" ], [ "-17/2", "5/2", "-10" ], [ "-29/2", "-5/2", "-9" ] ], "collision_image": [ "-4883/2", "-5953/4", "-3697/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.1613198820014077 ], "surface_features": [ 3.8523478146230006, 4.8324578162320435, 3.4316853446860116, 0.0, 4.008013008427356, 4965 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B173", "pair": 173, "family": "bijection", "mapping": "F(x,y,z)=(-4*x^7 + 54*x^6*y + 28*x^6*z - 35*x^6 - 312*x^5*y^2 - 324*x^5*y*z + 406*x^5*y - 84*x^5*z^2 + 210*x^5*z - 129*x^5 + 1000*x^4*y^3 + 1560*x^4*y^2*z - 1960*x^4*y^2 + 810*x^4*y*z^2 - 2030*x^4*y*z + 1250*x^4*y + 140*x^4*z^3 - 525*x^4*z^2 + 645*x^4*z - 264*x^4 - 1920*x^3*y^4 - 4000*x^3*y^3*z + 5040*x^3*y^3 - 3120*x^3*y^2*z^2 + 7840*x^3*y^2*z - 4840*x^3*y^2 - 1080*x^3*y*z^3 + 4060*x^3*y*z^2 - 5000*x^3*y*z + 2050*x^3*y - 140*x^3*z^4 + 700*x^3*z^3 - 1290*x^3*z^2 + 1056*x^3*z - 349*x^3 + 2208*x^2*y^5 + 5760*x^2*y^4*z - 7280*x^2*y^4 + 6000*x^2*y^3*z^2 - 15120*x^2*y^3*z + 9360*x^2*y^3 + 3120*x^2*y^2*z^3 - 11760*x^2*y^2*z^2 + 14520*x^2*y^2*z - 5964*x^2*y^2 + 810*x^2*y*z^4 - 4060*x^2*y*z^3 + 7500*x^2*y*z^2 - 6150*x^2*y*z + 2012*x^2*y + 84*x^2*z^5 - 525*x^2*z^4 + 1290*x^2*z^3 - 1584*x^2*z^2 + 1047*x^2*z - 343*x^2 - 1408*x*y^6 - 4416*x*y^5*z + 5600*x*y^5 - 5760*x*y^4*z^2 + 14560*x*y^4*z 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960*y^4*z^3 - 4000*y^4*z^2 + 5360*y^4*z - 2344*y^4 + 500*y^3*z^4 - 2760*y^3*z^3 + 5520*y^3*z^2 - 4812*y^3*z + 1615*y^3 + 156*y^2*z^5 - 1070*y^2*z^4 + 2840*y^2*z^3 - 3702*y^2*z^2 + 2494*y^2*z - 760*y^2 + 27*y*z^6 - 221*y*z^5 + 730*y*z^4 - 1265*y*z^3 + 1285*y*z^2 - 808*y*z + 260*y + 2*z^7 - 19*z^6 + 75*z^5 - 162*z^4 + 221*z^3 - 217*z^2 + 153*z - 1, 2*x^7 - 27*x^6*y - 14*x^6*z + 18*x^6 + 156*x^5*y^2 + 162*x^5*y*z - 209*x^5*y + 42*x^5*z^2 - 108*x^5*z + 68*x^5 - 500*x^4*y^3 - 780*x^4*y^2*z + 1010*x^4*y^2 - 405*x^4*y*z^2 + 1045*x^4*y*z - 660*x^4*y - 70*x^4*z^3 + 270*x^4*z^2 - 340*x^4*z + 142*x^4 + 960*x^3*y^4 + 2000*x^3*y^3*z - 2600*x^3*y^3 + 1560*x^3*y^2*z^2 - 4040*x^3*y^2*z + 2560*x^3*y^2 + 540*x^3*y*z^3 - 2090*x^3*y*z^2 + 2640*x^3*y*z - 1105*x^3*y + 70*x^3*z^4 - 360*x^3*z^3 + 680*x^3*z^2 - 568*x^3*z + 190*x^3 - 1104*x^2*y^5 - 2880*x^2*y^4*z + 3760*x^2*y^4 - 3000*x^2*y^3*z^2 + 7800*x^2*y^3*z - 4960*x^2*y^3 - 1560*x^2*y^2*z^3 + 6060*x^2*y^2*z^2 - 7680*x^2*y^2*z + 3222*x^2*y^2 - 405*x^2*y*z^4 + 2090*x^2*y*z^3 - 3960*x^2*y*z^2 + 3315*x^2*y*z - 1099*x^2*y - 42*x^2*z^5 + 270*x^2*z^4 - 680*x^2*z^3 + 852*x^2*z^2 - 570*x^2*z + 186*x^2 + 704*x*y^6 + 2208*x*y^5*z - 2896*x*y^5 + 2880*x*y^4*z^2 - 7520*x*y^4*z + 4800*x*y^4 + 2000*x*y^3*z^3 - 7800*x*y^3*z^2 + 9920*x*y^3*z - 4172*x*y^3 + 780*x*y^2*z^4 - 4040*x*y^2*z^3 + 7680*x*y^2*z^2 - 6444*x*y^2*z + 2122*x*y^2 + 162*x*y*z^5 - 1045*x*y*z^4 + 2640*x*y*z^3 - 3315*x*y*z^2 + 2198*x*y*z - 689*x*y + 14*x*z^6 - 108*x*z^5 + 340*x*z^4 - 568*x*z^3 + 570*x*z^2 - 374*x*z + 131*x - 192*y^7 - 704*y^6*z + 928*y^6 - 1104*y^5*z^2 + 2896*y^5*z - 1856*y^5 - 960*y^4*z^3 + 3760*y^4*z^2 - 4800*y^4*z + 2024*y^4 - 500*y^3*z^4 + 2600*y^3*z^3 - 4960*y^3*z^2 + 4172*y^3*z - 1367*y^3 - 156*y^2*z^5 + 1010*y^2*z^4 - 2560*y^2*z^3 + 3222*y^2*z^2 - 2122*y^2*z + 644*y^2 - 27*y*z^6 + 209*y*z^5 - 660*y*z^4 + 1105*y*z^3 - 1099*y*z^2 + 692*y*z - 227*y - 2*z^7 + 18*z^6 - 68*z^5 + 142*z^4 - 190*z^3 + 188*z^2 - 135*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ 1, -2, -1 ], [ 1, -1, -2 ], [ 1, -1, -1 ] ], "source_shift": [ 1, 1, 2 ], "target_matrix": [ [ -2, -1, -1 ], [ -1, -2, -1 ], [ 1, 1, 1 ] ], "target_shift": [ 90, 49, -44 ], "inverse_recipe": { "p": 6, "q": 1, "f": "x**6 + x**5 + x**3 + x**2", "g": "x", "h": "y*z + z**3 + z", "polynomials": { "f": [ [ [ 6, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ], [ [ 3, 0, 0 ], 1 ], [ [ 5, 0, 0 ], 1 ] ], "g": [ [ [ 1, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 0, 1 ], 1 ], [ [ 0, 0, 3 ], 1 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0387851859995862 ], "surface_features": [ 3.3657151604836346, 4.272305844402086, 3.0372272345822746, 0.0, 3.4736767690713974, 4840 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C054", "pair": 54, "family": "counterexample", "mapping": "F(x,y,z)=(-16*x^7 + 88*x^6*y + 104*x^6*z - 200*x^6 - 204*x^5*y^2 - 480*x^5*y*z + 924*x^5*y - 276*x^5*z^2 + 1068*x^5*z - 1044*x^5 + 258*x^4*y^3 + 906*x^4*y^2*z - 1746*x^4*y^2 + 1038*x^4*y*z^2 - 4008*x^4*y*z + 3930*x^4*y + 390*x^4*z^3 - 2262*x^4*z^2 + 4434*x^4*z - 2946*x^4 - 192*x^3*y^4 - 894*x^3*y^3*z + 1725*x^3*y^3 - 1530*x^3*y^2*z^2 + 5895*x^3*y^2*z - 5796*x^3*y^2 - 1146*x^3*y*z^3 + 6615*x^3*y*z^2 - 12966*x^3*y*z + 8647*x^3*y - 318*x^3*z^4 + 2445*x^3*z^3 - 7170*x^3*z^2 + 9531*x^3*z - 4848*x^3 + 84*x^2*y^5 + 486*x^2*y^4*z - 939*x^2*y^4 + 1104*x^2*y^3*z^2 - 4245*x^2*y^3*z + 4182*x^2*y^3 + 1236*x^2*y^2*z^3 - 7101*x^2*y^2*z^2 + 13911*x^2*y^2*z - 9297*x^2*y^2 + 684*x^2*y*z^4 - 5223*x^2*y*z^3 + 15276*x^2*y*z^2 - 20313*x^2*y*z + 10356*x^2*y + 150*x^2*z^5 - 1428*x^2*z^4 + 5547*x^2*z^3 - 11016*x^2*z^2 + 11193*x^2*z - 4652*x^2 - 20*x*y^6 - 138*x*y^5*z + 267*x*y^5 - 390*x*y^4*z^2 + 1497*x*y^4*z - 1476*x*y^4 - 580*x*y^3*z^3 + 3318*x*y^3*z^2 - 6492*x*y^3*z + 4338*x*y^3 - 480*x*y^2*z^4 + 3642*x*y^2*z^3 - 10620*x*y^2*z^2 + 14100*x*y^2*z - 7182*x*y^2 - 210*x*y*z^5 + 1983*x*y*z^4 - 7668*x*y*z^3 + 15186*x*y*z^2 - 15396*x*y*z + 6394*x*y - 38*x*z^6 + 429*x*z^5 - 2064*x*z^4 + 5424*x*z^3 - 8214*x*z^2 + 6802*x*z - 2414*x + 2*y^7 + 16*y^6*z - 31*y^6 + 54*y^5*z^2 - 207*y^5*z + 204*y^5 + 100*y^4*z^3 - 570*y^4*z^2 + 1113*y^4*z - 742*y^4 + 110*y^3*z^4 - 830*y^3*z^3 + 2412*y^3*z^2 - 3192*y^3*z + 1620*y^3 + 72*y^2*z^5 - 675*y^2*z^4 + 2598*y^2*z^3 - 5124*y^2*z^2 + 5172*y^2*z - 2138*y^2 + 26*y*z^6 - 291*y*z^5 + 1392*y*z^4 - 3640*y*z^3 + 5484*y*z^2 - 4516*y*z + 1592*y + 4*z^7 - 52*z^6 + 297*z^5 - 966*z^4 + 1932*z^3 - 2378*z^2 + 1672*z - 1, -8*x^7 + 44*x^6*y + 52*x^6*z - 112*x^6 - 102*x^5*y^2 - 240*x^5*y*z + 516*x^5*y - 138*x^5*z^2 + 600*x^5*z - 630*x^5 + 129*x^4*y^3 + 453*x^4*y^2*z - 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3400*x*z^3 - 5313*x*z^2 + 4493*x*z - 1617*x + y^7 + 8*y^6*z - 17*y^6 + 27*y^5*z^2 - 114*y^5*z + 120*y^5 + 50*y^4*z^3 - 315*y^4*z^2 + 657*y^4*z - 459*y^4 + 55*y^3*z^4 - 460*y^3*z^3 + 1428*y^3*z^2 - 1979*y^3*z + 1037*y^3 + 36*y^2*z^5 - 375*y^2*z^4 + 1542*y^2*z^3 - 3183*y^2*z^2 + 3315*y^2*z - 1399*y^2 + 13*y*z^6 - 162*y*z^5 + 828*y*z^4 - 2265*y*z^3 + 3519*y*z^2 - 2957*y*z + 1055*y + 2*z^7 - 29*z^6 + 177*z^5 - 602*z^4 + 1241*z^3 - 1558*z^2 + 1108*z - 1, 16*x^7 - 88*x^6*y - 104*x^6*z + 224*x^6 + 204*x^5*y^2 + 480*x^5*y*z - 1032*x^5*y + 276*x^5*z^2 - 1200*x^5*z + 1260*x^5 - 258*x^4*y^3 - 906*x^4*y^2*z + 1944*x^4*y^2 - 1038*x^4*y*z^2 + 4488*x^4*y*z - 4722*x^4*y - 390*x^4*z^3 + 2544*x^4*z^2 - 5358*x^4*z + 3728*x^4 + 192*x^3*y^4 + 894*x^3*y^3*z - 1914*x^3*y^3 + 1530*x^3*y^2*z^2 - 6576*x^3*y^2*z + 6930*x^3*y^2 + 1146*x^3*y*z^3 - 7410*x^3*y*z^2 + 15582*x^3*y*z - 10884*x^3*y + 318*x^3*z^4 - 2748*x^3*z^3 + 8652*x^3*z^2 - 12034*x^3*z + 6304*x^3 - 84*x^2*y^5 - 486*x^2*y^4*z + 1038*x^2*y^4 - 1104*x^2*y^3*z^2 + 4716*x^2*y^3*z - 4974*x^2*y^3 - 1236*x^2*y^2*z^3 + 7920*x^2*y^2*z^2 - 16620*x^2*y^2*z + 11634*x^2*y^2 - 684*x^2*y*z^4 + 5844*x^2*y*z^3 - 18318*x^2*y*z^2 + 25488*x^2*y*z - 13386*x^2*y - 150*x^2*z^5 + 1602*x^2*z^4 - 6672*x^2*z^3 + 13854*x^2*z^2 - 14490*x^2*z + 6128*x^2 + 20*x*y^6 + 138*x*y^5*z - 294*x*y^5 + 390*x*y^4*z^2 - 1656*x*y^4*z + 1746*x*y^4 + 580*x*y^3*z^3 - 3684*x*y^3*z^2 + 7710*x*y^3*z - 5395*x*y^3 + 480*x*y^2*z^4 - 4056*x*y^2*z^3 + 12654*x*y^2*z^2 - 17577*x*y^2*z + 9222*x*y^2 + 210*x*y*z^5 - 2214*x*y*z^4 + 9162*x*y*z^3 - 18969*x*y*z^2 + 19794*x*y*z - 8365*x*y + 38*x*z^6 - 480*x*z^5 + 2472*x*z^4 - 6787*x*z^3 + 10572*x*z^2 - 8905*x*z + 3198*x - 2*y^7 - 16*y^6*z + 34*y^6 - 54*y^5*z^2 + 228*y^5*z - 240*y^5 - 100*y^4*z^3 + 630*y^4*z^2 - 1314*y^4*z + 917*y^4 - 110*y^3*z^4 + 920*y^3*z^3 - 2856*y^3*z^2 + 3953*y^3*z - 2066*y^3 - 72*y^2*z^5 + 750*y^2*z^4 - 3084*y^2*z^3 + 6357*y^2*z^2 - 6603*y^2*z + 2777*y^2 - 26*y*z^6 + 324*y*z^5 - 1656*y*z^4 + 4523*y*z^3 - 7008*y*z^2 + 5869*y*z - 2090*y - 4*z^7 + 58*z^6 - 354*z^5 + 1202*z^4 - 2471*z^3 + 3092*z^2 - 2195*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ 2, -1, -1 ], [ 1, -1, -1 ], [ 1, -1, -2 ] ], "source_shift": [ 1, 2, 2 ], "target_matrix": [ [ -2, -1, -1 ], [ -1, -1, -1 ], [ 2, 2, 3 ] ], "target_shift": [ 517, 343, -681 ], "collision_preimages": [ [ "1", "11/4", "1/4" ], [ "7/2", "15", "-8" ], [ "-3/2", "4", "-5" ] ], "collision_image": [ "1035/2", "1373/4", "-1363/2" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0052169609989505 ], "surface_features": [ 3.5001426695325724, 4.406335758457057, 3.1408221801093106, 0.0, 3.6222105639410525, 4859 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B080", "pair": 80, "family": "bijection", "mapping": "F(x,y,z)=(6*x^7 + 42*x^6*y + 82*x^6*z + 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19185*x*y^2*z^2 + 26772*x*y^2*z - 13350*x*y^2 + 312*x*y*z^5 - 3906*x*y*z^4 + 17790*x*y*z^3 - 37983*x*y*z^2 + 38679*x*y*z - 15208*x*y - 68*x*z^6 + 1044*x*z^5 - 6048*x*z^4 + 17531*x*z^3 - 27315*x*z^2 + 21952*x*z - 7159*x + y^7 - 10*y^6*z + 23*y^6 + 42*y^5*z^2 - 198*y^5*z + 213*y^5 - 96*y^4*z^3 + 696*y^4*z^2 - 1527*y^4*z + 1048*y^4 + 129*y^3*z^4 - 1278*y^3*z^3 + 4287*y^3*z^2 - 5996*y^3*z + 3008*y^3 - 102*y^2*z^5 + 1293*y^2*z^4 - 5889*y^2*z^3 + 12579*y^2*z^2 - 12861*y^2*z + 5098*y^2 + 44*y*z^6 - 684*y*z^5 + 3960*y*z^4 - 11465*y*z^3 + 17895*y*z^2 - 14462*y*z + 4765*y - 8*z^7 + 148*z^6 - 1044*z^5 + 3834*z^4 - 8104*z^3 + 9994*z^2 - 6719*z + 1, 8*x^7 - 44*x^6*y + 68*x^6*z - 148*x^6 + 102*x^5*y^2 - 312*x^5*y*z + 660*x^5*y + 234*x^5*z^2 - 1008*x^5*z + 996*x^5 - 129*x^4*y^3 + 585*x^4*y^2*z - 1209*x^4*y^2 - 867*x^4*y*z^2 + 3642*x^4*y*z - 3588*x^4*y + 419*x^4*z^3 - 2685*x^4*z^2 + 5412*x^4*z - 3438*x^4 + 96*x^3*y^4 - 573*x^3*y^3*z + 1164*x^3*y^3 + 1257*x^3*y^2*z^2 - 5187*x^3*y^2*z + 5097*x^3*y^2 - 1199*x^3*y*z^3 + 7542*x^3*y*z^2 - 15138*x^3*y*z + 9666*x^3*y + 419*x^3*z^4 - 3571*x^3*z^3 + 10989*x^3*z^2 - 14360*x^3*z + 6762*x^3 - 42*x^2*y^5 + 309*x^2*y^4*z - 621*x^2*y^4 - 891*x^2*y^3*z^2 + 3639*x^2*y^3*z - 3570*x^2*y^3 + 1257*x^2*y^2*z^3 - 7830*x^2*y^2*z^2 + 15666*x^2*y^2*z - 10044*x^2*y^2 - 867*x^2*y*z^4 + 7323*x^2*y*z^3 - 22410*x^2*y*z^2 + 29346*x^2*y*z - 13929*x^2*y + 234*x^2*z^5 - 2511*x^2*z^4 + 10434*x^2*z^3 - 20928*x^2*z^2 + 20307*x^2*z - 7669*x^2 + 10*x*y^6 - 87*x*y^5*z + 174*x*y^5 + 309*x*y^4*z^2 - 1257*x*y^4*z + 1233*x*y^4 - 573*x*y^3*z^3 + 3558*x*y^3*z^2 - 7110*x*y^3*z + 4575*x*y^3 + 585*x*y^2*z^4 - 4929*x*y^2*z^3 + 15045*x*y^2*z^2 - 19737*x*y^2*z + 9426*x*y^2 - 312*x*y*z^5 + 3342*x*y*z^4 - 13836*x*y*z^3 + 27735*x*y*z^2 - 27009*x*y*z + 10282*x*y + 68*x*z^6 - 888*x*z^5 + 4668*x*z^4 - 12687*x*z^3 + 18867*x*z^2 - 14647*x*z + 4667*x - y^7 + 10*y^6*z - 20*y^6 - 42*y^5*z^2 + 171*y^5*z - 168*y^5 + 96*y^4*z^3 - 597*y^4*z^2 + 1194*y^4*z - 771*y^4 - 129*y^3*z^4 + 1089*y^3*z^3 - 3324*y^3*z^2 + 4369*y^3*z - 2097*y^3 + 102*y^2*z^5 - 1095*y^2*z^4 + 4530*y^2*z^3 - 9081*y^2*z^2 + 8868*y^2*z - 3397*y^2 - 44*y*z^6 + 576*y*z^5 - 3024*y*z^4 + 8205*y*z^3 - 12207*y*z^2 + 9509*y*z - 3054*y + 8*z^7 - 124*z^6 + 792*z^5 - 2722*z^4 + 5472*z^3 - 6484*z^2 + 4236*z - 1, 8*x^7 - 44*x^6*y + 68*x^6*z - 148*x^6 + 102*x^5*y^2 - 312*x^5*y*z + 660*x^5*y + 234*x^5*z^2 - 1008*x^5*z + 996*x^5 - 129*x^4*y^3 + 585*x^4*y^2*z - 1209*x^4*y^2 - 867*x^4*y*z^2 + 3642*x^4*y*z - 3588*x^4*y + 419*x^4*z^3 - 2685*x^4*z^2 + 5412*x^4*z - 3439*x^4 + 96*x^3*y^4 - 573*x^3*y^3*z + 1164*x^3*y^3 + 1257*x^3*y^2*z^2 - 5187*x^3*y^2*z + 5097*x^3*y^2 - 1199*x^3*y*z^3 + 7542*x^3*y*z^2 - 15138*x^3*y*z + 9670*x^3*y + 419*x^3*z^4 - 3571*x^3*z^3 + 10989*x^3*z^2 - 14367*x^3*z + 6776*x^3 - 42*x^2*y^5 + 309*x^2*y^4*z - 621*x^2*y^4 - 891*x^2*y^3*z^2 + 3639*x^2*y^3*z - 3570*x^2*y^3 + 1257*x^2*y^2*z^3 - 7830*x^2*y^2*z^2 + 15666*x^2*y^2*z - 10050*x^2*y^2 - 867*x^2*y*z^4 + 7323*x^2*y*z^3 - 22410*x^2*y*z^2 + 29367*x^2*y*z - 13968*x^2*y + 234*x^2*z^5 - 2511*x^2*z^4 + 10434*x^2*z^3 - 20946*x^2*z^2 + 20376*x^2*z - 7723*x^2 + 10*x*y^6 - 87*x*y^5*z + 174*x*y^5 + 309*x*y^4*z^2 - 1257*x*y^4*z + 1233*x*y^4 - 573*x*y^3*z^3 + 3558*x*y^3*z^2 - 7110*x*y^3*z + 4579*x*y^3 + 585*x*y^2*z^4 - 4929*x*y^2*z^3 + 15045*x*y^2*z^2 - 19758*x*y^2*z + 9462*x*y^2 - 312*x*y*z^5 + 3342*x*y*z^4 - 13836*x*y*z^3 + 27771*x*y*z^2 - 27135*x*y*z + 10378*x*y + 68*x*z^6 - 888*x*z^5 + 4668*x*z^4 - 12707*x*z^3 + 18975*x*z^2 - 14815*x*z + 4745*x - y^7 + 10*y^6*z - 20*y^6 - 42*y^5*z^2 + 171*y^5*z - 168*y^5 + 96*y^4*z^3 - 597*y^4*z^2 + 1194*y^4*z - 772*y^4 - 129*y^3*z^4 + 1089*y^3*z^3 - 3324*y^3*z^2 + 4376*y^3*z - 2108*y^3 + 102*y^2*z^5 - 1095*y^2*z^4 + 4530*y^2*z^3 - 9099*y^2*z^2 + 8925*y^2*z - 3439*y^2 - 44*y*z^6 + 576*y*z^5 - 3024*y*z^4 + 8225*y*z^3 - 12303*y*z^2 + 9653*y*z - 3120*y + 8*z^7 - 124*z^6 + 792*z^5 - 2730*z^4 + 5524*z^3 - 6604*z^2 + 4348*z - 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -1, 1, -2 ], [ -2, 1, -1 ], [ -1, 1, -1 ] ], "source_shift": [ 2, 2, 2 ], "target_matrix": [ [ 1, 2, 1 ], [ -1, -1, -2 ], [ -1, -1, -1 ] ], "target_shift": [ -1905, 1183, 1219 ], "collision_preimages": [ [ "-1/4", "-11/4", "-1/4" ], [ "8", "18", "11/2" ], [ "5", "17", "15/2" ] ], "collision_image": [ "-7621/4", "4733/4", "4877/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.1048065289978695 ], "surface_features": [ 3.740218778188363, 4.59469081818349, 3.3998467127129226, 0.0, 3.8724040679259666, 4911 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B126", "pair": 126, "family": "bijection", "mapping": "F(x,y,z)=(3*x^7 - 21*x^6*y - 21*x^6*z + 33*x^6 + 63*x^5*y^2 + 126*x^5*y*z - 204*x^5*y + 63*x^5*z^2 - 198*x^5*z + 150*x^5 - 105*x^4*y^3 - 315*x^4*y^2*z + 525*x^4*y^2 - 315*x^4*y*z^2 + 1020*x^4*y*z - 792*x^4*y - 105*x^4*z^3 + 495*x^4*z^2 - 753*x^4*z + 375*x^4 + 105*x^3*y^4 + 420*x^3*y^3*z - 720*x^3*y^3 + 630*x^3*y^2*z^2 - 2100*x^3*y^2*z + 1671*x^3*y^2 + 420*x^3*y*z^3 - 2040*x^3*y*z^2 + 3180*x^3*y*z - 1614*x^3*y + 105*x^3*z^4 - 660*x^3*z^3 + 1512*x^3*z^2 - 1524*x^3*z + 580*x^3 - 63*x^2*y^5 - 315*x^2*y^4*z + 555*x^2*y^4 - 630*x^2*y^3*z^2 + 2160*x^2*y^3*z - 1761*x^2*y^3 - 630*x^2*y^2*z^3 + 3150*x^2*y^2*z^2 - 5031*x^2*y^2*z + 2601*x^2*y^2 - 315*x^2*y*z^4 + 2040*x^2*y*z^3 - 4788*x^2*y*z^2 + 4920*x^2*y*z - 1908*x^2*y - 63*x^2*z^5 + 495*x^2*z^4 - 1518*x^2*z^3 + 2322*x^2*z^2 - 1806*x^2*z + 575*x^2 + 21*x*y^6 + 126*x*y^5*z - 228*x*y^5 + 315*x*y^4*z^2 - 1110*x*y^4*z + 927*x*y^4 + 420*x*y^3*z^3 - 2160*x*y^3*z^2 + 3534*x*y^3*z - 1860*x*y^3 + 315*x*y^2*z^4 - 2100*x*y^2*z^3 + 5049*x*y^2*z^2 - 5286*x*y^2*z + 2088*x*y^2 + 126*x*y*z^5 - 1020*x*y*z^4 + 3204*x*y*z^3 - 4998*x*y*z^2 + 3972*x*y*z - 1294*x*y + 21*x*z^6 - 198*x*z^5 + 762*x*z^4 - 1572*x*z^3 + 1872*x*z^2 - 1222*x*z + 349*x - 3*y^7 - 21*y^6*z + 39*y^6 - 63*y^5*z^2 + 228*y^5*z - 195*y^5 - 105*y^4*z^3 + 555*y^4*z^2 - 930*y^4*z + 498*y^4 - 105*y^3*z^4 + 720*y^3*z^3 - 1773*y^3*z^2 + 1890*y^3*z - 760*y^3 - 63*y^2*z^5 + 525*y^2*z^4 - 1689*y^2*z^3 + 2685*y^2*z^2 - 2181*y^2*z + 728*y^2 - 21*y*z^6 + 204*y*z^5 - 804*y*z^4 + 1692*y*z^3 - 2064*y*z^2 + 1384*y*z - 408*y - 3*z^7 + 33*z^6 - 153*z^5 + 399*z^4 - 646*z^3 + 647*z^2 - 377*z + 1, 6*x^7 - 42*x^6*y - 42*x^6*z + 66*x^6 + 126*x^5*y^2 + 252*x^5*y*z - 408*x^5*y + 126*x^5*z^2 - 396*x^5*z + 300*x^5 - 210*x^4*y^3 - 630*x^4*y^2*z + 1050*x^4*y^2 - 630*x^4*y*z^2 + 2040*x^4*y*z - 1584*x^4*y - 210*x^4*z^3 + 990*x^4*z^2 - 1506*x^4*z + 750*x^4 + 210*x^3*y^4 + 840*x^3*y^3*z - 1440*x^3*y^3 + 1260*x^3*y^2*z^2 - 4200*x^3*y^2*z + 3342*x^3*y^2 + 840*x^3*y*z^3 - 4080*x^3*y*z^2 + 6360*x^3*y*z - 3228*x^3*y + 210*x^3*z^4 - 1320*x^3*z^3 + 3024*x^3*z^2 - 3048*x^3*z + 1158*x^3 - 126*x^2*y^5 - 630*x^2*y^4*z + 1110*x^2*y^4 - 1260*x^2*y^3*z^2 + 4320*x^2*y^3*z - 3522*x^2*y^3 - 1260*x^2*y^2*z^3 + 6300*x^2*y^2*z^2 - 10062*x^2*y^2*z + 5202*x^2*y^2 - 630*x^2*y*z^4 + 4080*x^2*y*z^3 - 9576*x^2*y*z^2 + 9840*x^2*y*z - 3810*x^2*y - 126*x^2*z^5 + 990*x^2*z^4 - 3036*x^2*z^3 + 4644*x^2*z^2 - 3606*x^2*z + 1142*x^2 + 42*x*y^6 + 252*x*y^5*z - 456*x*y^5 + 630*x*y^4*z^2 - 2220*x*y^4*z + 1854*x*y^4 + 840*x*y^3*z^3 - 4320*x*y^3*z^2 + 7068*x*y^3*z - 3720*x*y^3 + 630*x*y^2*z^4 - 4200*x*y^2*z^3 + 10098*x*y^2*z^2 - 10572*x*y^2*z + 4170*x*y^2 + 252*x*y*z^5 - 2040*x*y*z^4 + 6408*x*y*z^3 - 9996*x*y*z^2 + 7932*x*y*z - 2572*x*y + 42*x*z^6 - 396*x*z^5 + 1524*x*z^4 - 3144*x*z^3 + 3738*x*z^2 - 2428*x*z + 682*x - 6*y^7 - 42*y^6*z + 78*y^6 - 126*y^5*z^2 + 456*y^5*z - 390*y^5 - 210*y^4*z^3 + 1110*y^4*z^2 - 1860*y^4*z + 996*y^4 - 210*y^3*z^4 + 1440*y^3*z^3 - 3546*y^3*z^2 + 3780*y^3*z - 1518*y^3 - 126*y^2*z^5 + 1050*y^2*z^4 - 3378*y^2*z^3 + 5370*y^2*z^2 - 4356*y^2*z + 1448*y^2 - 42*y*z^6 + 408*y*z^5 - 1608*y*z^4 + 3384*y*z^3 - 4122*y*z^2 + 2752*y*z - 798*y - 6*z^7 + 66*z^6 - 306*z^5 + 798*z^4 - 1290*z^3 + 1286*z^2 - 736*z + 1, 3*x^7 - 21*x^6*y - 21*x^6*z + 33*x^6 + 63*x^5*y^2 + 126*x^5*y*z - 204*x^5*y + 63*x^5*z^2 - 198*x^5*z + 150*x^5 - 105*x^4*y^3 - 315*x^4*y^2*z + 525*x^4*y^2 - 315*x^4*y*z^2 + 1020*x^4*y*z - 792*x^4*y - 105*x^4*z^3 + 495*x^4*z^2 - 753*x^4*z + 375*x^4 + 105*x^3*y^4 + 420*x^3*y^3*z - 720*x^3*y^3 + 630*x^3*y^2*z^2 - 2100*x^3*y^2*z + 1671*x^3*y^2 + 420*x^3*y*z^3 - 2040*x^3*y*z^2 + 3180*x^3*y*z - 1614*x^3*y + 105*x^3*z^4 - 660*x^3*z^3 + 1512*x^3*z^2 - 1524*x^3*z + 580*x^3 - 63*x^2*y^5 - 315*x^2*y^4*z + 555*x^2*y^4 - 630*x^2*y^3*z^2 + 2160*x^2*y^3*z - 1761*x^2*y^3 - 630*x^2*y^2*z^3 + 3150*x^2*y^2*z^2 - 5031*x^2*y^2*z + 2601*x^2*y^2 - 315*x^2*y*z^4 + 2040*x^2*y*z^3 - 4788*x^2*y*z^2 + 4920*x^2*y*z - 1908*x^2*y - 63*x^2*z^5 + 495*x^2*z^4 - 1518*x^2*z^3 + 2322*x^2*z^2 - 1806*x^2*z + 574*x^2 + 21*x*y^6 + 126*x*y^5*z - 228*x*y^5 + 315*x*y^4*z^2 - 1110*x*y^4*z + 927*x*y^4 + 420*x*y^3*z^3 - 2160*x*y^3*z^2 + 3534*x*y^3*z - 1860*x*y^3 + 315*x*y^2*z^4 - 2100*x*y^2*z^3 + 5049*x*y^2*z^2 - 5286*x*y^2*z + 2088*x*y^2 + 126*x*y*z^5 - 1020*x*y*z^4 + 3204*x*y*z^3 - 4998*x*y*z^2 + 3972*x*y*z - 1292*x*y + 21*x*z^6 - 198*x*z^5 + 762*x*z^4 - 1572*x*z^3 + 1872*x*z^2 - 1220*x*z + 345*x - 3*y^7 - 21*y^6*z + 39*y^6 - 63*y^5*z^2 + 228*y^5*z - 195*y^5 - 105*y^4*z^3 + 555*y^4*z^2 - 930*y^4*z + 498*y^4 - 105*y^3*z^4 + 720*y^3*z^3 - 1773*y^3*z^2 + 1890*y^3*z - 760*y^3 - 63*y^2*z^5 + 525*y^2*z^4 - 1689*y^2*z^3 + 2685*y^2*z^2 - 2181*y^2*z + 727*y^2 - 21*y*z^6 + 204*y*z^5 - 804*y*z^4 + 1692*y*z^3 - 2064*y*z^2 + 1382*y*z - 403*y - 3*z^7 + 33*z^6 - 153*z^5 + 399*z^4 - 646*z^3 + 646*z^2 - 373*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ 1, -1, -1 ], [ 1, -2, -1 ], [ 1, -1, -2 ] ], "source_shift": [ 1, 1, 1 ], "target_matrix": [ [ 1, 2, 1 ], [ 2, 2, 3 ], [ 1, 1, 1 ] ], "target_shift": [ -98, -187, -95 ], "inverse_recipe": { "p": 2, "q": 3, "f": "x**2 + x", "g": "x**3", "h": "y**2*z + y**2 + y + z", "polynomials": { "f": [ [ [ 2, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "g": [ [ [ 3, 0, 0 ], 1 ] ], "h": [ [ [ 0, 2, 1 ], 1 ], [ [ 0, 2, 0 ], 1 ], [ [ 0, 1, 0 ], 1 ], [ [ 0, 0, 1 ], 1 ] ] }, "h_multiplier": 3 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0836302660027286 ], "surface_features": [ 3.164021389736797, 4.024157154459672, 2.859438535455056, 0.0, 3.269181233235913, 4780 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C012", "pair": 12, "family": "counterexample", "mapping": "F(x,y,z)=(-x^7 + 10*x^6*y + 8*x^6*z - 16*x^6 - 42*x^5*y^2 - 69*x^5*y*z + 138*x^5*y - 27*x^5*z^2 + 108*x^5*z - 108*x^5 + 96*x^4*y^3 + 243*x^4*y^2*z - 489*x^4*y^2 + 195*x^4*y*z^2 - 777*x^4*y*z + 783*x^4*y + 50*x^4*z^3 - 300*x^4*z^2 + 597*x^4*z - 401*x^4 - 129*x^3*y^4 - 447*x^3*y^3*z + 912*x^3*y^3 - 552*x^3*y^2*z^2 + 2202*x^3*y^2*z - 2244*x^3*y^2 - 290*x^3*y*z^3 + 1728*x^3*y*z^2 - 3453*x^3*y*z + 2339*x^3*y - 55*x^3*z^4 + 440*x^3*z^3 - 1308*x^3*z^2 + 1746*x^3*z - 887*x^3 + 102*x^2*y^5 + 453*x^2*y^4*z - 945*x^2*y^4 + 765*x^2*y^3*z^2 - 3075*x^2*y^3*z + 3177*x^2*y^3 + 618*x^2*y^2*z^3 - 3672*x^2*y^2*z^2 + 7392*x^2*y^2*z - 5049*x^2*y^2 + 240*x^2*y*z^4 - 1902*x^2*y*z^3 + 5661*x^2*y*z^2 - 7602*x^2*y*z + 3885*x^2*y + 36*x^2*z^5 - 360*x^2*z^4 + 1422*x^2*z^3 - 2832*x^2*z^2 + 2862*x^2*z - 1177*x^2 - 44*x*y^6 - 240*x*y^5*z + 516*x*y^5 - 519*x*y^4*z^2 + 2118*x*y^4*z - 2220*x*y^4 - 573*x*y^3*z^3 + 3414*x*y^3*z^2 - 6942*x*y^3*z + 4774*x*y^3 - 342*x*y^2*z^4 + 2694*x*y^2*z^3 - 8052*x*y^2*z^2 + 10878*x*y^2*z - 5583*x*y^2 - 105*x*y*z^5 + 1038*x*y*z^4 - 4095*x*y*z^3 + 8187*x*y*z^2 - 8307*x*y*z + 3430*x*y - 13*x*z^6 + 156*x*z^5 - 768*x*z^4 + 2030*x*z^3 - 3063*x*z^2 + 2510*x*z - 879*x + 8*y^7 + 52*y^6*z - 116*y^6 + 138*y^5*z^2 - 576*y^5*z + 612*y^5 + 195*y^4*z^3 - 1173*y^4*z^2 + 2412*y^4*z - 1666*y^4 + 159*y^3*z^4 - 1251*y^3*z^3 + 3765*y^3*z^2 - 5112*y^3*z + 2626*y^3 + 75*y^2*z^5 - 735*y^2*z^4 + 2904*y^2*z^3 - 5829*y^2*z^2 + 5931*y^2*z - 2449*y^2 + 19*y*z^6 - 225*y*z^5 + 1104*y*z^4 - 2924*y*z^3 + 4422*y*z^2 - 3631*y*z + 1271*y + 2*z^7 - 28*z^6 + 165*z^5 - 543*z^4 + 1088*z^3 - 1333*z^2 + 931*z - 1, -2*x^7 + 20*x^6*y + 16*x^6*z - 35*x^6 - 84*x^5*y^2 - 138*x^5*y*z + 300*x^5*y - 54*x^5*z^2 + 237*x^5*z - 249*x^5 + 192*x^4*y^3 + 486*x^4*y^2*z - 1056*x^4*y^2 + 390*x^4*y*z^2 - 1695*x^4*y*z + 1794*x^4*y + 100*x^4*z^3 - 660*x^4*z^2 + 1380*x^4*z - 960*x^4 - 258*x^3*y^4 - 894*x^3*y^3*z + 1956*x^3*y^3 - 1104*x^3*y^2*z^2 + 4773*x^3*y^2*z - 5109*x^3*y^2 - 580*x^3*y*z^3 + 3780*x^3*y*z^2 - 7932*x^3*y*z + 5562*x^3*y - 110*x^3*z^4 + 970*x^3*z^3 - 3030*x^3*z^2 + 4189*x^3*z - 2187*x^3 + 204*x^2*y^5 + 906*x^2*y^4*z - 2013*x^2*y^4 + 1530*x^2*y^3*z^2 - 6621*x^2*y^3*z + 7188*x^2*y^3 + 1236*x^2*y^2*z^3 - 7983*x^2*y^2*z^2 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3371*z^2 + 2382*z - 1, x^7 - 10*x^6*y - 8*x^6*z + 19*x^6 + 42*x^5*y^2 + 69*x^5*y*z - 162*x^5*y + 27*x^5*z^2 - 129*x^5*z + 141*x^5 - 96*x^4*y^3 - 243*x^4*y^2*z + 567*x^4*y^2 - 195*x^4*y*z^2 + 918*x^4*y*z - 1011*x^4*y - 50*x^4*z^3 + 360*x^4*z^2 - 783*x^4*z + 558*x^4 + 129*x^3*y^4 + 447*x^3*y^3*z - 1044*x^3*y^3 + 552*x^3*y^2*z^2 - 2571*x^3*y^2*z + 2865*x^3*y^2 + 290*x^3*y*z^3 - 2052*x^3*y*z^2 + 4479*x^3*y*z - 3219*x^3*y + 55*x^3*z^4 - 530*x^3*z^3 + 1722*x^3*z^2 - 2438*x^3*z + 1293*x^3 - 102*x^2*y^5 - 453*x^2*y^4*z + 1068*x^2*y^4 - 765*x^2*y^3*z^2 + 3546*x^2*y^3*z - 4011*x^2*y^3 - 618*x^2*y^2*z^3 + 4311*x^2*y^2*z^2 - 9480*x^2*y^2*z + 6873*x^2*y^2 - 240*x^2*y*z^4 + 2268*x^2*y*z^3 - 7371*x^2*y*z^2 + 10494*x^2*y*z - 5595*x^2*y - 36*x^2*z^5 + 435*x^2*z^4 - 1878*x^2*z^3 + 3966*x^2*z^2 - 4182*x^2*z + 1777*x^2 + 44*x*y^6 + 240*x*y^5*z - 576*x*y^5 + 519*x*y^4*z^2 - 2412*x*y^4*z + 2772*x*y^4 + 573*x*y^3*z^3 - 3960*x*y^3*z^2 + 8802*x*y^3*z - 6428*x*y^3 + 342*x*y^2*z^4 - 3177*x*y^2*z^3 + 10365*x*y^2*z^2 - 14847*x*y^2*z + 7944*x*y^2 + 105*x*y*z^5 - 1242*x*y*z^4 + 5349*x*y*z^3 - 11331*x*y*z^2 + 11985*x*y*z - 5104*x*y + 13*x*z^6 - 189*x*z^5 + 1017*x*z^4 - 2850*x*z^3 + 4485*x*z^2 - 3794*x*z + 1353*x - 8*y^7 - 52*y^6*z + 128*y^6 - 138*y^5*z^2 + 648*y^5*z - 756*y^5 - 195*y^4*z^3 + 1344*y^4*z^2 - 3024*y^4*z + 2219*y^4 - 159*y^3*z^4 + 1458*y^3*z^3 - 4791*y^3*z^2 + 6899*y^3*z - 3692*y^3 - 75*y^2*z^5 + 870*y^2*z^4 - 3750*y^2*z^3 + 7974*y^2*z^2 - 8451*y^2*z + 3592*y^2 - 19*y*z^6 + 270*y*z^5 - 1446*y*z^4 + 4056*y*z^3 - 6390*y*z^2 + 5407*y*z - 1923*y - 2*z^7 + 34*z^6 - 219*z^5 + 764*z^4 - 1596*z^3 + 2017*z^2 - 1433*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ 1, -1, -1 ], [ 1, -2, -1 ], [ 1, -1, -2 ] ], "source_shift": [ 1, 2, 1 ], "target_matrix": [ [ -1, -1, -2 ], [ -2, -3, -2 ], [ 1, 2, 1 ] ], "target_shift": [ 285, 730, -439 ], "collision_preimages": [ [ "1/4", "1", "1/4" ], [ "-2", "7/2", "-11/2" ], [ "-11", "-3/2", "-15/2" ] ], "collision_image": [ "1141/4", "1461/2", "-1757/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.080486868002481 ], "surface_features": [ 3.4452901732053833, 4.410608542568368, 3.0642707529740063, 0.0, 3.5870501395055543, 4830 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B013", "pair": 13, "family": "bijection", "mapping": "F(x,y,z)=(-9*x^7 - 57*x^6*y - 57*x^6*z - 72*x^6 - 154*x^5*y^2 - 308*x^5*y*z - 390*x^5*y - 154*x^5*z^2 - 390*x^5*z - 245*x^5 - 230*x^4*y^3 - 690*x^4*y^2*z - 876*x^4*y^2 - 690*x^4*y*z^2 - 1752*x^4*y*z - 1105*x^4*y - 230*x^4*z^3 - 876*x^4*z^2 - 1105*x^4*z - 471*x^4 - 205*x^3*y^4 - 820*x^3*y^3*z - 1044*x^3*y^3 - 1230*x^3*y^2*z^2 - 3132*x^3*y^2*z - 1984*x^3*y^2 - 820*x^3*y*z^3 - 3132*x^3*y*z^2 - 3968*x^3*y*z - 1698*x^3*y - 205*x^3*z^4 - 1044*x^3*z^3 - 1984*x^3*z^2 - 1698*x^3*z - 568*x^3 - 109*x^2*y^5 - 545*x^2*y^4*z - 696*x^2*y^4 - 1090*x^2*y^3*z^2 - 2784*x^2*y^3*z - 1772*x^2*y^3 - 1090*x^2*y^2*z^3 - 4176*x^2*y^2*z^2 - 5316*x^2*y^2*z - 2285*x^2*y^2 - 545*x^2*y*z^4 - 2784*x^2*y*z^3 - 5316*x^2*y*z^2 - 4570*x^2*y*z - 1530*x^2*y - 109*x^2*z^5 - 696*x^2*z^4 - 1772*x^2*z^3 - 2285*x^2*z^2 - 1539*x^2*z - 439*x^2 - 32*x*y^6 - 192*x*y^5*z - 246*x*y^5 - 480*x*y^4*z^2 - 1230*x*y^4*z - 787*x*y^4 - 640*x*y^3*z^3 - 2460*x*y^3*z^2 - 3148*x*y^3*z - 1360*x*y^3 - 480*x*y^2*z^4 - 2460*x*y^2*z^3 - 4722*x*y^2*z^2 - 4080*x*y^2*z - 1368*x*y^2 - 192*x*y*z^5 - 1230*x*y*z^4 - 3148*x*y*z^3 - 4080*x*y*z^2 - 2748*x*y*z - 782*x*y - 32*x*z^6 - 246*x*z^5 - 787*x*z^4 - 1360*x*z^3 - 1380*x*z^2 - 800*x*z - 199*x - 4*y^7 - 28*y^6*z - 36*y^6 - 84*y^5*z^2 - 216*y^5*z - 139*y^5 - 140*y^4*z^3 - 540*y^4*z^2 - 695*y^4*z - 302*y^4 - 140*y^3*z^4 - 720*y^3*z^3 - 1390*y^3*z^2 - 1208*y^3*z - 406*y^3 - 84*y^2*z^5 - 540*y^2*z^4 - 1390*y^2*z^3 - 1812*y^2*z^2 - 1222*y^2*z - 347*y^2 - 28*y*z^6 - 216*y*z^5 - 695*y*z^4 - 1208*y*z^3 - 1226*y*z^2 - 706*y*z - 177*y - 4*z^7 - 36*z^6 - 139*z^5 - 302*z^4 - 410*z^3 - 359*z^2 - 188*z - 1, -9*x^7 - 57*x^6*y - 57*x^6*z - 72*x^6 - 154*x^5*y^2 - 308*x^5*y*z - 390*x^5*y - 154*x^5*z^2 - 390*x^5*z - 244*x^5 - 230*x^4*y^3 - 690*x^4*y^2*z - 876*x^4*y^2 - 690*x^4*y*z^2 - 1752*x^4*y*z - 1100*x^4*y - 230*x^4*z^3 - 876*x^4*z^2 - 1100*x^4*z - 465*x^4 - 205*x^3*y^4 - 820*x^3*y^3*z - 1044*x^3*y^3 - 1230*x^3*y^2*z^2 - 3132*x^3*y^2*z - 1974*x^3*y^2 - 820*x^3*y*z^3 - 3132*x^3*y*z^2 - 3948*x^3*y*z - 1674*x^3*y - 205*x^3*z^4 - 1044*x^3*z^3 - 1974*x^3*z^2 - 1674*x^3*z - 554*x^3 - 109*x^2*y^5 - 545*x^2*y^4*z - 696*x^2*y^4 - 1090*x^2*y^3*z^2 - 2784*x^2*y^3*z - 1762*x^2*y^3 - 1090*x^2*y^2*z^3 - 4176*x^2*y^2*z^2 - 5286*x^2*y^2*z - 2249*x^2*y^2 - 545*x^2*y*z^4 - 2784*x^2*y*z^3 - 5286*x^2*y*z^2 - 4498*x^2*y*z - 1488*x^2*y - 109*x^2*z^5 - 696*x^2*z^4 - 1762*x^2*z^3 - 2249*x^2*z^2 - 1497*x^2*z - 422*x^2 - 32*x*y^6 - 192*x*y^5*z - 246*x*y^5 - 480*x*y^4*z^2 - 1230*x*y^4*z - 782*x*y^4 - 640*x*y^3*z^3 - 2460*x*y^3*z^2 - 3128*x*y^3*z - 1336*x*y^3 - 480*x*y^2*z^4 - 2460*x*y^2*z^3 - 4692*x*y^2*z^2 - 4008*x*y^2*z - 1326*x*y^2 - 192*x*y*z^5 - 1230*x*y*z^4 - 3128*x*y*z^3 - 4008*x*y*z^2 - 2664*x*y*z - 748*x*y - 32*x*z^6 - 246*x*z^5 - 782*x*z^4 - 1336*x*z^3 - 1338*x*z^2 - 766*x*z - 187*x - 4*y^7 - 28*y^6*z - 36*y^6 - 84*y^5*z^2 - 216*y^5*z - 138*y^5 - 140*y^4*z^3 - 540*y^4*z^2 - 690*y^4*z - 296*y^4 - 140*y^3*z^4 - 720*y^3*z^3 - 1380*y^3*z^2 - 1184*y^3*z - 392*y^3 - 84*y^2*z^5 - 540*y^2*z^4 - 1380*y^2*z^3 - 1776*y^2*z^2 - 1180*y^2*z - 330*y^2 - 28*y*z^6 - 216*y*z^5 - 690*y*z^4 - 1184*y*z^3 - 1184*y*z^2 - 672*y*z - 165*y - 4*z^7 - 36*z^6 - 138*z^5 - 296*z^4 - 396*z^3 - 342*z^2 - 175*z - 1, 18*x^7 + 114*x^6*y + 114*x^6*z + 144*x^6 + 308*x^5*y^2 + 616*x^5*y*z + 780*x^5*y + 308*x^5*z^2 + 780*x^5*z + 487*x^5 + 460*x^4*y^3 + 1380*x^4*y^2*z + 1752*x^4*y^2 + 1380*x^4*y*z^2 + 3504*x^4*y*z + 2195*x^4*y + 460*x^4*z^3 + 1752*x^4*z^2 + 2195*x^4*z + 924*x^4 + 410*x^3*y^4 + 1640*x^3*y^3*z + 2088*x^3*y^3 + 2460*x^3*y^2*z^2 + 6264*x^3*y^2*z + 3938*x^3*y^2 + 1640*x^3*y*z^3 + 6264*x^3*y*z^2 + 7876*x^3*y*z + 3324*x^3*y + 410*x^3*z^4 + 2088*x^3*z^3 + 3938*x^3*z^2 + 3324*x^3*z + 1094*x^3 + 218*x^2*y^5 + 1090*x^2*y^4*z + 1392*x^2*y^4 + 2180*x^2*y^3*z^2 + 5568*x^2*y^3*z + 3514*x^2*y^3 + 2180*x^2*y^2*z^3 + 8352*x^2*y^2*z^2 + 10542*x^2*y^2*z + 4462*x^2*y^2 + 1090*x^2*y*z^4 + 5568*x^2*y*z^3 + 10542*x^2*y*z^2 + 8924*x^2*y*z + 2934*x^2*y + 218*x^2*z^5 + 1392*x^2*z^4 + 3514*x^2*z^3 + 4462*x^2*z^2 + 2952*x^2*z + 827*x^2 + 64*x*y^6 + 384*x*y^5*z + 492*x*y^5 + 960*x*y^4*z^2 + 2460*x*y^4*z + 1559*x*y^4 + 1280*x*y^3*z^3 + 4920*x*y^3*z^2 + 6236*x*y^3*z + 2648*x*y^3 + 960*x*y^2*z^4 + 4920*x*y^2*z^3 + 9354*x*y^2*z^2 + 7944*x*y^2*z + 2610*x*y^2 + 384*x*y*z^5 + 2460*x*y*z^4 + 6236*x*y*z^3 + 7944*x*y*z^2 + 5244*x*y*z + 1462*x*y + 64*x*z^6 + 492*x*z^5 + 1559*x*z^4 + 2648*x*z^3 + 2634*x*z^2 + 1498*x*z + 368*x + 8*y^7 + 56*y^6*z + 72*y^6 + 168*y^5*z^2 + 432*y^5*z + 275*y^5 + 280*y^4*z^3 + 1080*y^4*z^2 + 1375*y^4*z + 586*y^4 + 280*y^3*z^4 + 1440*y^3*z^3 + 2750*y^3*z^2 + 2344*y^3*z + 770*y^3 + 168*y^2*z^5 + 1080*y^2*z^4 + 2750*y^2*z^3 + 3516*y^2*z^2 + 2318*y^2*z + 643*y^2 + 56*y*z^6 + 432*y*z^5 + 1375*y*z^4 + 2344*y*z^3 + 2326*y*z^2 + 1310*y*z + 322*y + 8*z^7 + 72*z^6 + 275*z^5 + 586*z^4 + 778*z^3 + 667*z^2 + 341*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ 1, 1, 1 ], [ 1, 1, 2 ], [ 2, 1, 1 ] ], "source_shift": [ 1, 1, 2 ], "target_matrix": [ [ -1, -2, -1 ], [ -1, -1, -1 ], [ 2, 1, 1 ] ], "target_shift": [ 38, 34, -71 ], "inverse_recipe": { "p": 5, "q": 1, "f": "x**5 + x**4 + x**2", "g": "x", "h": "y*z**2", "polynomials": { "f": [ [ [ 5, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ], [ [ 4, 0, 0 ], 1 ] ], "g": [ [ [ 1, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 2 ], 1 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.004819554000278 ], "surface_features": [ 3.1642016196179594, 4.022923011878938, 2.8959747323590648, 0.0, 3.251291632606556, 4805 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B215", "pair": 215, "family": "bijection", "mapping": "F(x,y,z)=(25*x^7 - 175*x^6*y - 175*x^6*z + 250*x^6 + 525*x^5*y^2 + 1050*x^5*y*z - 1500*x^5*y + 525*x^5*z^2 - 1525*x^5*z + 1151*x^5 - 875*x^4*y^3 - 2625*x^4*y^2*z + 3750*x^4*y^2 - 2625*x^4*y*z^2 + 7625*x^4*y*z - 5755*x^4*y - 875*x^4*z^3 + 3875*x^4*z^2 - 6030*x^4*z + 3055*x^4 + 875*x^3*y^4 + 3500*x^3*y^3*z - 5000*x^3*y^3 + 5250*x^3*y^2*z^2 - 15250*x^3*y^2*z + 11510*x^3*y^2 + 3500*x^3*y*z^3 - 15500*x^3*y*z^2 + 24120*x^3*y*z - 12220*x^3*y + 875*x^3*z^4 - 5250*x^3*z^3 + 12610*x^3*z^2 - 13370*x^3*z + 4985*x^3 - 525*x^2*y^5 - 2625*x^2*y^4*z + 3750*x^2*y^4 - 5250*x^2*y^3*z^2 + 15250*x^2*y^3*z - 11510*x^2*y^3 - 5250*x^2*y^2*z^3 + 23250*x^2*y^2*z^2 - 36180*x^2*y^2*z + 18330*x^2*y^2 - 2625*x^2*y*z^4 + 15750*x^2*y*z^3 - 37830*x^2*y*z^2 + 40110*x^2*y*z - 14980*x^2*y - 525*x^2*z^5 + 4000*x^2*z^4 - 13160*x^2*z^3 + 21930*x^2*z^2 - 17205*x^2*z + 5009*x^2 + 175*x*y^6 + 1050*x*y^5*z - 1500*x*y^5 + 2625*x*y^4*z^2 - 7625*x*y^4*z + 5755*x*y^4 + 3500*x*y^3*z^3 - 15500*x*y^3*z^2 + 24120*x*y^3*z - 12220*x*y^3 + 2625*x*y^2*z^4 - 15750*x*y^2*z^3 + 37830*x*y^2*z^2 - 40110*x*y^2*z + 15005*x*y^2 + 1050*x*y*z^5 - 8000*x*y*z^4 + 26320*x*y*z^3 - 43860*x*y*z^2 + 34460*x*y*z - 10093*x*y + 175*x*z^6 - 1625*x*z^5 + 6855*x*z^4 - 15970*x*z^3 + 19905*x*z^2 - 12143*x*z + 2930*x - 25*y^7 - 175*y^6*z + 250*y^6 - 525*y^5*z^2 + 1525*y^5*z - 1151*y^5 - 875*y^4*z^3 + 3875*y^4*z^2 - 6030*y^4*z + 3055*y^4 - 875*y^3*z^4 + 5250*y^3*z^3 - 12610*y^3*z^2 + 13370*y^3*z - 5010*y^3 - 525*y^2*z^5 + 4000*y^2*z^4 - 13160*y^2*z^3 + 21930*y^2*z^2 - 17255*y^2*z + 5084*y^2 - 175*y*z^6 + 1625*y*z^5 - 6855*y*z^4 + 15970*y*z^3 - 19930*y*z^2 + 12243*y*z - 3006*y - 25*z^7 + 275*z^6 - 1426*z^5 + 4355*z^4 - 7735*z^3 + 7459*z^2 - 3729*z + 1, -25*x^7 + 175*x^6*y + 175*x^6*z - 250*x^6 - 525*x^5*y^2 - 1050*x^5*y*z + 1500*x^5*y - 525*x^5*z^2 + 1525*x^5*z - 1151*x^5 + 875*x^4*y^3 + 2625*x^4*y^2*z - 3750*x^4*y^2 + 2625*x^4*y*z^2 - 7625*x^4*y*z + 5755*x^4*y + 875*x^4*z^3 - 3875*x^4*z^2 + 6030*x^4*z - 3055*x^4 - 875*x^3*y^4 - 3500*x^3*y^3*z + 5000*x^3*y^3 - 5250*x^3*y^2*z^2 + 15250*x^3*y^2*z - 11510*x^3*y^2 - 3500*x^3*y*z^3 + 15500*x^3*y*z^2 - 24120*x^3*y*z + 12220*x^3*y - 875*x^3*z^4 + 5250*x^3*z^3 - 12610*x^3*z^2 + 13370*x^3*z - 4985*x^3 + 525*x^2*y^5 + 2625*x^2*y^4*z - 3750*x^2*y^4 + 5250*x^2*y^3*z^2 - 15250*x^2*y^3*z + 11510*x^2*y^3 + 5250*x^2*y^2*z^3 - 23250*x^2*y^2*z^2 + 36180*x^2*y^2*z - 18330*x^2*y^2 + 2625*x^2*y*z^4 - 15750*x^2*y*z^3 + 37830*x^2*y*z^2 - 40110*x^2*y*z + 14980*x^2*y + 525*x^2*z^5 - 4000*x^2*z^4 + 13160*x^2*z^3 - 21930*x^2*z^2 + 17205*x^2*z - 5007*x^2 - 175*x*y^6 - 1050*x*y^5*z + 1500*x*y^5 - 2625*x*y^4*z^2 + 7625*x*y^4*z - 5755*x*y^4 - 3500*x*y^3*z^3 + 15500*x*y^3*z^2 - 24120*x*y^3*z + 12220*x*y^3 - 2625*x*y^2*z^4 + 15750*x*y^2*z^3 - 37830*x*y^2*z^2 + 40110*x*y^2*z - 15005*x*y^2 - 1050*x*y*z^5 + 8000*x*y*z^4 - 26320*x*y*z^3 + 43860*x*y*z^2 - 34460*x*y*z + 10089*x*y - 175*x*z^6 + 1625*x*z^5 - 6855*x*z^4 + 15970*x*z^3 - 19905*x*z^2 + 12139*x*z - 2924*x + 25*y^7 + 175*y^6*z - 250*y^6 + 525*y^5*z^2 - 1525*y^5*z + 1151*y^5 + 875*y^4*z^3 - 3875*y^4*z^2 + 6030*y^4*z - 3055*y^4 + 875*y^3*z^4 - 5250*y^3*z^3 + 12610*y^3*z^2 - 13370*y^3*z + 5010*y^3 + 525*y^2*z^5 - 4000*y^2*z^4 + 13160*y^2*z^3 - 21930*y^2*z^2 + 17255*y^2*z - 5082*y^2 + 175*y*z^6 - 1625*y*z^5 + 6855*y*z^4 - 15970*y*z^3 + 19930*y*z^2 - 12239*y*z + 3000*y + 25*z^7 - 275*z^6 + 1426*z^5 - 4355*z^4 + 7735*z^3 - 7457*z^2 + 3721*z - 1, -75*x^7 + 525*x^6*y + 525*x^6*z - 750*x^6 - 1575*x^5*y^2 - 3150*x^5*y*z + 4500*x^5*y - 1575*x^5*z^2 + 4575*x^5*z - 3452*x^5 + 2625*x^4*y^3 + 7875*x^4*y^2*z - 11250*x^4*y^2 + 7875*x^4*y*z^2 - 22875*x^4*y*z + 17260*x^4*y + 2625*x^4*z^3 - 11625*x^4*z^2 + 18085*x^4*z - 9160*x^4 - 2625*x^3*y^4 - 10500*x^3*y^3*z + 15000*x^3*y^3 - 15750*x^3*y^2*z^2 + 45750*x^3*y^2*z - 34520*x^3*y^2 - 10500*x^3*y*z^3 + 46500*x^3*y*z^2 - 72340*x^3*y*z + 36640*x^3*y - 2625*x^3*z^4 + 15750*x^3*z^3 - 37820*x^3*z^2 + 40090*x^3*z - 14945*x^3 + 1575*x^2*y^5 + 7875*x^2*y^4*z - 11250*x^2*y^4 + 15750*x^2*y^3*z^2 - 45750*x^2*y^3*z + 34520*x^2*y^3 + 15750*x^2*y^2*z^3 - 69750*x^2*y^2*z^2 + 108510*x^2*y^2*z - 54960*x^2*y^2 + 7875*x^2*y*z^4 - 47250*x^2*y*z^3 + 113460*x^2*y*z^2 - 120270*x^2*y*z + 44910*x^2*y + 1575*x^2*z^5 - 12000*x^2*z^4 + 39470*x^2*z^3 - 65760*x^2*z^2 + 51585*x^2*z - 15016*x^2 - 525*x*y^6 - 3150*x*y^5*z + 4500*x*y^5 - 7875*x*y^4*z^2 + 22875*x*y^4*z - 17260*x*y^4 - 10500*x*y^3*z^3 + 46500*x*y^3*z^2 - 72340*x*y^3*z + 36640*x*y^3 - 7875*x*y^2*z^4 + 47250*x*y^2*z^3 - 113460*x*y^2*z^2 + 120270*x*y^2*z - 44985*x*y^2 - 3150*x*y*z^5 + 24000*x*y*z^4 - 78940*x*y*z^3 + 131520*x*y*z^2 - 103320*x*y*z + 30257*x*y - 525*x*z^6 + 4875*x*z^5 - 20560*x*z^4 + 47890*x*z^3 - 59685*x*z^2 + 36407*x*z - 8780*x + 75*y^7 + 525*y^6*z - 750*y^6 + 1575*y^5*z^2 - 4575*y^5*z + 3452*y^5 + 2625*y^4*z^3 - 11625*y^4*z^2 + 18085*y^4*z - 9160*y^4 + 2625*y^3*z^4 - 15750*y^3*z^3 + 37820*y^3*z^2 - 40090*y^3*z + 15020*y^3 + 1575*y^2*z^5 - 12000*y^2*z^4 + 39470*y^2*z^3 - 65760*y^2*z^2 + 51735*y^2*z - 15241*y^2 + 525*y*z^6 - 4875*y*z^5 + 20560*y*z^4 - 47890*y*z^3 + 59760*y*z^2 - 36707*y*z + 9007*y + 75*z^7 - 825*z^6 + 4277*z^5 - 13060*z^4 + 23195*z^3 - 22366*z^2 + 11176*z - 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ 1, -1, -1 ], [ 2, -3, -3 ], [ 1, -1, -2 ] ], "source_shift": [ 1, 2, 1 ], "target_matrix": [ [ 1, 1, 1 ], [ -1, -1, -2 ], [ -3, -2, -3 ] ], "target_shift": [ -776, 772, 2325 ], "inverse_recipe": { "p": 5, "q": 2, "f": "x**5 + x**2 + x", "g": "x**2", "h": "y*z + y + 2*z**3", "polynomials": { "f": [ [ [ 5, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "g": [ [ [ 2, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 0, 3 ], 2 ], [ [ 0, 1, 0 ], 1 ] ] }, "h_multiplier": 25 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0295274029995198 ], "surface_features": [ 4.160856948056265, 5.1189918001959756, 3.780317312140151, 0.0, 4.317935120327827, 5132 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B231", "pair": 231, "family": "bijection", "mapping": "F(x,y,z)=(-576*x^7 - 2304*x^6*y - 2112*x^6*z - 4800*x^6 - 3888*x^5*y^2 - 7200*x^5*y*z - 16416*x^5*y - 3312*x^5*z^2 - 15072*x^5*z - 17472*x^5 - 3600*x^4*y^3 - 10080*x^4*y^2*z - 23040*x^4*y^2 - 9360*x^4*y*z^2 - 42720*x^4*y*z - 49800*x^4*y - 2880*x^4*z^3 - 19680*x^4*z^2 - 45768*x^4*z - 35640*x^4 - 1980*x^3*y^4 - 7440*x^3*y^3*z - 17040*x^3*y^3 - 10440*x^3*y^2*z^2 - 47760*x^3*y^2*z - 55956*x^3*y^2 - 6480*x^3*y*z^3 - 44400*x^3*y*z^2 - 103824*x^3*y*z - 81384*x^3*y - 1500*x^3*z^4 - 13680*x^3*z^3 - 47868*x^3*z^2 - 74880*x^3*z - 43561*x^3 - 648*x^2*y^5 - 3060*x^2*y^4*z - 7020*x^2*y^4 - 5760*x^2*y^3*z^2 - 26400*x^2*y^3*z - 31074*x^2*y^3 - 5400*x^2*y^2*z^3 - 37080*x^2*y^2*z^2 - 87138*x^2*y^2*z - 68742*x^2*y^2 - 2520*x^2*y*z^4 - 23040*x^2*y*z^3 - 81054*x^2*y*z^2 - 127644*x^2*y*z - 74781*x^2*y - 468*x^2*z^5 - 5340*x^2*z^4 - 24990*x^2*z^3 - 58902*x^2*z^2 - 68922*x^2*z - 31514*x^2 - 117*x*y^6 - 666*x*y^5*z - 1530*x*y^5 - 1575*x*y^4*z^2 - 7230*x*y^4*z - 8547*x*y^4 - 1980*x*y^3*z^3 - 13620*x*y^3*z^2 - 32154*x*y^3*z - 25524*x*y^3 - 1395*x*y^2*z^4 - 12780*x*y^2*z^3 - 45180*x*y^2*z^2 - 71610*x*y^2*z - 42270*x*y^2 - 522*x*y*z^5 - 5970*x*y*z^4 - 28086*x*y*z^3 - 66648*x*y*z^2 - 78576*x*y*z - 36146*x*y - 81*x*z^6 - 1110*x*z^5 - 6513*x*z^4 - 20562*x*z^3 - 36306*x*z^2 - 33416*x*z - 12313*x - 9*y^7 - 60*y^6*z - 138*y^6 - 171*y^5*z^2 - 786*y^5*z - 933*y^5 - 270*y^4*z^3 - 1860*y^4*z^2 - 4410*y^4*z - 3522*y^4 - 255*y^3*z^4 - 2340*y^3*z^3 - 8310*y^3*z^2 - 13254*y^3*z - 7886*y^3 - 144*y^2*z^5 - 1650*y^2*z^4 - 7800*y^2*z^3 - 18630*y^2*z^2 - 22140*y^2*z - 10256*y^2 - 45*y*z^6 - 618*y*z^5 - 3645*y*z^4 - 11586*y*z^3 - 20622*y*z^2 - 19108*y*z - 7074*y - 6*z^7 - 96*z^6 - 678*z^5 - 2688*z^4 - 6368*z^3 - 8852*z^2 - 6565*z - 1, 192*x^7 + 768*x^6*y + 704*x^6*z + 1600*x^6 + 1296*x^5*y^2 + 2400*x^5*y*z + 5472*x^5*y + 1104*x^5*z^2 + 5024*x^5*z + 5824*x^5 + 1200*x^4*y^3 + 3360*x^4*y^2*z + 7680*x^4*y^2 + 3120*x^4*y*z^2 + 14240*x^4*y*z + 16600*x^4*y + 960*x^4*z^3 + 6560*x^4*z^2 + 15256*x^4*z + 11880*x^4 + 660*x^3*y^4 + 2480*x^3*y^3*z + 5680*x^3*y^3 + 3480*x^3*y^2*z^2 + 15920*x^3*y^2*z + 18652*x^3*y^2 + 2160*x^3*y*z^3 + 14800*x^3*y*z^2 + 34608*x^3*y*z + 27128*x^3*y + 500*x^3*z^4 + 4560*x^3*z^3 + 15956*x^3*z^2 + 24960*x^3*z + 14515*x^3 + 216*x^2*y^5 + 1020*x^2*y^4*z + 2340*x^2*y^4 + 1920*x^2*y^3*z^2 + 8800*x^2*y^3*z + 10358*x^2*y^3 + 1800*x^2*y^2*z^3 + 12360*x^2*y^2*z^2 + 29046*x^2*y^2*z + 22914*x^2*y^2 + 840*x^2*y*z^4 + 7680*x^2*y*z^3 + 27018*x^2*y*z^2 + 42548*x^2*y*z + 24919*x^2*y + 156*x^2*z^5 + 1780*x^2*z^4 + 8330*x^2*z^3 + 19634*x^2*z^2 + 22966*x^2*z + 10486*x^2 + 39*x*y^6 + 222*x*y^5*z + 510*x*y^5 + 525*x*y^4*z^2 + 2410*x*y^4*z + 2849*x*y^4 + 660*x*y^3*z^3 + 4540*x*y^3*z^2 + 10718*x*y^3*z + 8508*x*y^3 + 465*x*y^2*z^4 + 4260*x*y^2*z^3 + 15060*x*y^2*z^2 + 23870*x*y^2*z + 14086*x*y^2 + 174*x*y*z^5 + 1990*x*y*z^4 + 9362*x*y*z^3 + 22216*x*y*z^2 + 26184*x*y*z + 12030*x*y + 27*x*z^6 + 370*x*z^5 + 2171*x*z^4 + 6854*x*z^3 + 12098*x*z^2 + 11120*x*z + 4082*x + 3*y^7 + 20*y^6*z + 46*y^6 + 57*y^5*z^2 + 262*y^5*z + 311*y^5 + 90*y^4*z^3 + 620*y^4*z^2 + 1470*y^4*z + 1174*y^4 + 85*y^3*z^4 + 780*y^3*z^3 + 2770*y^3*z^2 + 4418*y^3*z + 2628*y^3 + 48*y^2*z^5 + 550*y^2*z^4 + 2600*y^2*z^3 + 6210*y^2*z^2 + 7378*y^2*z + 3414*y^2 + 15*y*z^6 + 206*y*z^5 + 1215*y*z^4 + 3862*y*z^3 + 6872*y*z^2 + 6360*y*z + 2347*y + 2*z^7 + 32*z^6 + 226*z^5 + 896*z^4 + 2122*z^3 + 2946*z^2 + 2177*z + 1, -192*x^7 - 768*x^6*y - 704*x^6*z - 1600*x^6 - 1296*x^5*y^2 - 2400*x^5*y*z - 5472*x^5*y - 1104*x^5*z^2 - 5024*x^5*z - 5824*x^5 - 1200*x^4*y^3 - 3360*x^4*y^2*z - 7680*x^4*y^2 - 3120*x^4*y*z^2 - 14240*x^4*y*z - 16600*x^4*y - 960*x^4*z^3 - 6560*x^4*z^2 - 15256*x^4*z - 11880*x^4 - 660*x^3*y^4 - 2480*x^3*y^3*z - 5680*x^3*y^3 - 3480*x^3*y^2*z^2 - 15920*x^3*y^2*z - 18652*x^3*y^2 - 2160*x^3*y*z^3 - 14800*x^3*y*z^2 - 34608*x^3*y*z - 27128*x^3*y - 500*x^3*z^4 - 4560*x^3*z^3 - 15956*x^3*z^2 - 24960*x^3*z - 14499*x^3 - 216*x^2*y^5 - 1020*x^2*y^4*z - 2340*x^2*y^4 - 1920*x^2*y^3*z^2 - 8800*x^2*y^3*z - 10358*x^2*y^3 - 1800*x^2*y^2*z^3 - 12360*x^2*y^2*z^2 - 29046*x^2*y^2*z - 22914*x^2*y^2 - 840*x^2*y*z^4 - 7680*x^2*y*z^3 - 27018*x^2*y*z^2 - 42548*x^2*y*z - 24895*x^2*y - 156*x^2*z^5 - 1780*x^2*z^4 - 8330*x^2*z^3 - 19634*x^2*z^2 - 22942*x^2*z - 10430*x^2 - 39*x*y^6 - 222*x*y^5*z - 510*x*y^5 - 525*x*y^4*z^2 - 2410*x*y^4*z - 2849*x*y^4 - 660*x*y^3*z^3 - 4540*x*y^3*z^2 - 10718*x*y^3*z - 8508*x*y^3 - 465*x*y^2*z^4 - 4260*x*y^2*z^3 - 15060*x*y^2*z^2 - 23870*x*y^2*z - 14074*x*y^2 - 174*x*y*z^5 - 1990*x*y*z^4 - 9362*x*y*z^3 - 22216*x*y*z^2 - 26160*x*y*z - 11974*x*y - 27*x*z^6 - 370*x*z^5 - 2171*x*z^4 - 6854*x*z^3 - 12086*x*z^2 - 11064*x*z - 4012*x - 3*y^7 - 20*y^6*z - 46*y^6 - 57*y^5*z^2 - 262*y^5*z - 311*y^5 - 90*y^4*z^3 - 620*y^4*z^2 - 1470*y^4*z - 1174*y^4 - 85*y^3*z^4 - 780*y^3*z^3 - 2770*y^3*z^2 - 4418*y^3*z - 2626*y^3 - 48*y^2*z^5 - 550*y^2*z^4 - 2600*y^2*z^3 - 6210*y^2*z^2 - 7372*y^2*z - 3400*y^2 - 15*y*z^6 - 206*y*z^5 - 1215*y*z^4 - 3862*y*z^3 - 6866*y*z^2 - 6332*y*z - 2311*y - 2*z^7 - 32*z^6 - 226*z^5 - 896*z^4 - 2120*z^3 - 2932*z^2 - 2141*z - 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ 2, 1, 1 ], [ 1, 2, 1 ], [ 3, 2, 2 ] ], "source_shift": [ 2, 2, 1 ], "target_matrix": [ [ -3, -2, -2 ], [ 1, 1, 1 ], [ -1, -1, -2 ] ], "target_shift": [ 1989, -655, 629 ], "inverse_recipe": { "p": 1, "q": 3, "f": "x", "g": "x**3 + x**2", "h": "y*z**2", "polynomials": { "f": [ [ [ 1, 0, 0 ], 1 ] ], "g": [ [ [ 3, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 2 ], 1 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0014511230001517 ], "surface_features": [ 4.051174504698154, 5.106000405288851, 3.6071332043915665, 0.0, 4.246072997628687, 5030 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B100", "pair": 100, "family": "bijection", "mapping": "F(x,y,z)=(-x^7 + 14*x^6*y + 7*x^6*z - 16*x^6 - 84*x^5*y^2 - 84*x^5*y*z + 189*x^5*y - 21*x^5*z^2 + 95*x^5*z - 107*x^5 + 280*x^4*y^3 + 420*x^4*y^2*z - 930*x^4*y^2 + 210*x^4*y*z^2 - 935*x^4*y*z + 1040*x^4*y + 35*x^4*z^3 - 235*x^4*z^2 + 525*x^4*z - 390*x^4 - 560*x^3*y^4 - 1120*x^3*y^3*z + 2440*x^3*y^3 - 840*x^3*y^2*z^2 + 3680*x^3*y^2*z - 4040*x^3*y^2 - 280*x^3*y*z^3 + 1850*x^3*y*z^2 - 4080*x^3*y*z + 3000*x^3*y - 35*x^3*z^4 + 310*x^3*z^3 - 1030*x^3*z^2 + 1520*x^3*z - 841*x^3 + 672*x^2*y^5 + 1680*x^2*y^4*z - 3600*x^2*y^4 + 1680*x^2*y^3*z^2 - 7240*x^2*y^3*z + 7840*x^2*y^3 + 840*x^2*y^2*z^3 - 5460*x^2*y^2*z^2 + 11880*x^2*y^2*z - 8640*x^2*y^2 + 210*x^2*y*z^4 - 1830*x^2*y*z^3 + 6000*x^2*y*z^2 - 8760*x^2*y*z + 4805*x^2*y + 21*x^2*z^5 - 230*x^2*z^4 + 1010*x^2*z^3 - 2220*x^2*z^2 + 2443*x^2*z - 1078*x^2 - 448*x*y^6 - 1344*x*y^5*z + 2832*x*y^5 - 1680*x*y^4*z^2 + 7120*x*y^4*z - 7600*x*y^4 - 1120*x*y^3*z^3 + 7160*x*y^3*z^2 - 15360*x*y^3*z + 11040*x*y^3 - 420*x*y^2*z^4 + 3600*x*y^2*z^3 - 11640*x*y^2*z^2 + 16800*x*y^2*z - 9128*x*y^2 - 84*x*y*z^5 + 905*x*y*z^4 - 3920*x*y*z^3 + 8520*x*y*z^2 - 9290*x*y*z + 4063*x*y - 7*x*z^6 + 91*x*z^5 - 495*x*z^4 + 1440*x*z^3 - 2363*x*z^2 + 2075*x*z - 760*x + 128*y^7 + 448*y^6*z - 928*y^6 + 672*y^5*z^2 - 2800*y^5*z + 2944*y^5 + 560*y^4*z^3 - 3520*y^4*z^2 + 7440*y^4*z - 5280*y^4 + 280*y^3*z^4 - 2360*y^3*z^3 + 7520*y^3*z^2 - 10720*y^3*z + 5764*y^3 + 84*y^2*z^5 - 890*y^2*z^4 + 3800*y^2*z^3 - 8160*y^2*z^2 + 8808*y^2*z - 3817*y^2 + 14*y*z^6 - 179*y*z^5 + 960*y*z^4 - 2760*y*z^3 + 4485*y*z^2 - 3902*y*z + 1417*y + z^7 - 15*z^6 + 97*z^5 - 350*z^4 + 761*z^3 - 997*z^2 + 728*z - 1, -x^7 + 14*x^6*y + 7*x^6*z - 16*x^6 - 84*x^5*y^2 - 84*x^5*y*z + 189*x^5*y - 21*x^5*z^2 + 95*x^5*z - 108*x^5 + 280*x^4*y^3 + 420*x^4*y^2*z - 930*x^4*y^2 + 210*x^4*y*z^2 - 935*x^4*y*z + 1050*x^4*y + 35*x^4*z^3 - 235*x^4*z^2 + 530*x^4*z - 400*x^4 - 560*x^3*y^4 - 1120*x^3*y^3*z + 2440*x^3*y^3 - 840*x^3*y^2*z^2 + 3680*x^3*y^2*z - 4080*x^3*y^2 - 280*x^3*y*z^3 + 1850*x^3*y*z^2 - 4120*x^3*y*z + 3080*x^3*y - 35*x^3*z^4 + 310*x^3*z^3 - 1040*x^3*z^2 + 1560*x^3*z - 881*x^3 + 672*x^2*y^5 + 1680*x^2*y^4*z - 3600*x^2*y^4 + 1680*x^2*y^3*z^2 - 7240*x^2*y^3*z + 7920*x^2*y^3 + 840*x^2*y^2*z^3 - 5460*x^2*y^2*z^2 + 12000*x^2*y^2*z - 8880*x^2*y^2 + 210*x^2*y*z^4 - 1830*x^2*y*z^3 + 6060*x^2*y*z^2 - 9000*x^2*y*z + 5045*x^2*y + 21*x^2*z^5 - 230*x^2*z^4 + 1020*x^2*z^3 - 2280*x^2*z^2 + 2563*x^2*z - 1158*x^2 - 448*x*y^6 - 1344*x*y^5*z + 2832*x*y^5 - 1680*x*y^4*z^2 + 7120*x*y^4*z - 7680*x*y^4 - 1120*x*y^3*z^3 + 7160*x*y^3*z^2 - 15520*x*y^3*z + 11360*x*y^3 - 420*x*y^2*z^4 + 3600*x*y^2*z^3 - 11760*x*y^2*z^2 + 17280*x*y^2*z - 9608*x*y^2 - 84*x*y*z^5 + 905*x*y*z^4 - 3960*x*y*z^3 + 8760*x*y*z^2 - 9770*x*y*z + 4383*x*y - 7*x*z^6 + 91*x*z^5 - 500*x*z^4 + 1480*x*z^3 - 2483*x*z^2 + 2235*x*z - 841*x + 128*y^7 + 448*y^6*z - 928*y^6 + 672*y^5*z^2 - 2800*y^5*z + 2976*y^5 + 560*y^4*z^3 - 3520*y^4*z^2 + 7520*y^4*z - 5440*y^4 + 280*y^3*z^4 - 2360*y^3*z^3 + 7600*y^3*z^2 - 11040*y^3*z + 6084*y^3 + 84*y^2*z^5 - 890*y^2*z^4 + 3840*y^2*z^3 - 8400*y^2*z^2 + 9288*y^2*z - 4137*y^2 + 14*y*z^6 - 179*y*z^5 + 970*y*z^4 - 2840*y*z^3 + 4725*y*z^2 - 4222*y*z + 1578*y + z^7 - 15*z^6 + 98*z^5 - 360*z^4 + 801*z^3 - 1077*z^2 + 809*z - 1, x^7 - 14*x^6*y - 7*x^6*z + 16*x^6 + 84*x^5*y^2 + 84*x^5*y*z - 189*x^5*y + 21*x^5*z^2 - 95*x^5*z + 107*x^5 - 280*x^4*y^3 - 420*x^4*y^2*z + 930*x^4*y^2 - 210*x^4*y*z^2 + 935*x^4*y*z - 1040*x^4*y - 35*x^4*z^3 + 235*x^4*z^2 - 525*x^4*z + 390*x^4 + 560*x^3*y^4 + 1120*x^3*y^3*z - 2440*x^3*y^3 + 840*x^3*y^2*z^2 - 3680*x^3*y^2*z + 4040*x^3*y^2 + 280*x^3*y*z^3 - 1850*x^3*y*z^2 + 4080*x^3*y*z - 3000*x^3*y + 35*x^3*z^4 - 310*x^3*z^3 + 1030*x^3*z^2 - 1520*x^3*z + 841*x^3 - 672*x^2*y^5 - 1680*x^2*y^4*z + 3600*x^2*y^4 - 1680*x^2*y^3*z^2 + 7240*x^2*y^3*z - 7840*x^2*y^3 - 840*x^2*y^2*z^3 + 5460*x^2*y^2*z^2 - 11880*x^2*y^2*z + 8640*x^2*y^2 - 210*x^2*y*z^4 + 1830*x^2*y*z^3 - 6000*x^2*y*z^2 + 8760*x^2*y*z - 4805*x^2*y - 21*x^2*z^5 + 230*x^2*z^4 - 1010*x^2*z^3 + 2220*x^2*z^2 - 2443*x^2*z + 1076*x^2 + 448*x*y^6 + 1344*x*y^5*z - 2832*x*y^5 + 1680*x*y^4*z^2 - 7120*x*y^4*z + 7600*x*y^4 + 1120*x*y^3*z^3 - 7160*x*y^3*z^2 + 15360*x*y^3*z - 11040*x*y^3 + 420*x*y^2*z^4 - 3600*x*y^2*z^3 + 11640*x*y^2*z^2 - 16800*x*y^2*z + 9128*x*y^2 + 84*x*y*z^5 - 905*x*y*z^4 + 3920*x*y*z^3 - 8520*x*y*z^2 + 9290*x*y*z - 4055*x*y + 7*x*z^6 - 91*x*z^5 + 495*x*z^4 - 1440*x*z^3 + 2363*x*z^2 - 2071*x*z + 748*x - 128*y^7 - 448*y^6*z + 928*y^6 - 672*y^5*z^2 + 2800*y^5*z - 2944*y^5 - 560*y^4*z^3 + 3520*y^4*z^2 - 7440*y^4*z + 5280*y^4 - 280*y^3*z^4 + 2360*y^3*z^3 - 7520*y^3*z^2 + 10720*y^3*z - 5764*y^3 - 84*y^2*z^5 + 890*y^2*z^4 - 3800*y^2*z^3 + 8160*y^2*z^2 - 8808*y^2*z + 3809*y^2 - 14*y*z^6 + 179*y*z^5 - 960*y*z^4 + 2760*y*z^3 - 4485*y*z^2 + 3894*y*z - 1399*y - z^7 + 15*z^6 - 97*z^5 + 350*z^4 - 761*z^3 + 995*z^2 - 718*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ 1, -2, -1 ], [ 1, -1, -1 ], [ 2, -1, -1 ] ], "source_shift": [ 2, 1, 2 ], "target_matrix": [ [ -1, -1, -1 ], [ -1, -2, -1 ], [ 1, 1, 2 ] ], "target_shift": [ 226, 259, -214 ], "inverse_recipe": { "p": 5, "q": 2, "f": "x**5", "g": "x**2", "h": "y*z + z", "polynomials": { "f": [ [ [ 5, 0, 0 ], 1 ] ], "g": [ [ [ 2, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 0, 1 ], 1 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.04262071100311 ], "surface_features": [ 3.44823768510277, 4.237543738142874, 3.032014034159506, 0.0, 3.547969322550791, 4809 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B189", "pair": 189, "family": "bijection", "mapping": "F(x,y,z)=(512*x^7 + 1792*x^6*y - 1792*x^6*z - 2432*x^6 + 2688*x^5*y^2 - 5376*x^5*y*z - 7360*x^5*y + 2688*x^5*z^2 + 7360*x^5*z + 5056*x^5 + 2240*x^4*y^3 - 6720*x^4*y^2*z - 9280*x^4*y^2 + 6720*x^4*y*z^2 + 18560*x^4*y*z + 12832*x^4*y - 2240*x^4*z^3 - 9280*x^4*z^2 - 12832*x^4*z - 6048*x^4 + 1120*x^3*y^4 - 4480*x^3*y^3*z - 6240*x^3*y^3 + 6720*x^3*y^2*z^2 + 18720*x^3*y^2*z + 13024*x^3*y^2 - 4480*x^3*y*z^3 - 18720*x^3*y*z^2 - 26048*x^3*y*z - 12336*x^3*y + 1120*x^3*z^4 + 6240*x^3*z^3 + 13024*x^3*z^2 + 12336*x^3*z + 4688*x^3 + 336*x^2*y^5 - 1680*x^2*y^4*z - 2360*x^2*y^4 + 3360*x^2*y^3*z^2 + 9440*x^2*y^3*z + 6608*x^2*y^3 - 3360*x^2*y^2*z^3 - 14160*x^2*y^2*z^2 - 19824*x^2*y^2*z - 9432*x^2*y^2 + 1680*x^2*y*z^4 + 9440*x^2*y*z^3 + 19824*x^2*y*z^2 + 18864*x^2*y*z + 7224*x^2*y - 336*x^2*z^5 - 2360*x^2*z^4 - 6608*x^2*z^3 - 9432*x^2*z^2 - 7240*x^2*z - 2584*x^2 + 56*x*y^6 - 336*x*y^5*z - 476*x*y^5 + 840*x*y^4*z^2 + 2380*x*y^4*z + 1676*x*y^4 - 1120*x*y^3*z^3 - 4760*x*y^3*z^2 - 6704*x*y^3*z - 3204*x*y^3 + 840*x*y^2*z^4 + 4760*x*y^2*z^3 + 10056*x*y^2*z^2 + 9612*x*y^2*z + 3708*x*y^2 - 336*x*y*z^5 - 2380*x*y*z^4 - 6704*x*y*z^3 - 9612*x*y*z^2 - 7432*x*y*z - 2708*x*y + 56*x*z^6 + 476*x*z^5 + 1676*x*z^4 + 3204*x*z^3 + 3724*x*z^2 + 2736*x*z + 1033*x + 4*y^7 - 28*y^6*z - 40*y^6 + 84*y^5*z^2 + 240*y^5*z + 170*y^5 - 140*y^4*z^3 - 600*y^4*z^2 - 850*y^4*z - 408*y^4 + 140*y^3*z^4 + 800*y^3*z^3 + 1700*y^3*z^2 + 1632*y^3*z + 634*y^3 - 84*y^2*z^5 - 600*y^2*z^4 - 1700*y^2*z^3 - 2448*y^2*z^2 - 1906*y^2*z - 711*y^2 + 28*y*z^6 + 240*y*z^5 + 850*y*z^4 + 1632*y*z^3 + 1910*y*z^2 + 1438*y*z + 570*y - 4*z^7 - 40*z^6 - 170*z^5 - 408*z^4 - 638*z^3 - 727*z^2 - 584*z - 1, -256*x^7 - 896*x^6*y + 896*x^6*z + 1216*x^6 - 1344*x^5*y^2 + 2688*x^5*y*z + 3680*x^5*y - 1344*x^5*z^2 - 3680*x^5*z - 2496*x^5 - 1120*x^4*y^3 + 3360*x^4*y^2*z + 4640*x^4*y^2 - 3360*x^4*y*z^2 - 9280*x^4*y*z - 6336*x^4*y + 1120*x^4*z^3 + 4640*x^4*z^2 + 6336*x^4*z + 2928*x^4 - 560*x^3*y^4 + 2240*x^3*y^3*z + 3120*x^3*y^3 - 3360*x^3*y^2*z^2 - 9360*x^3*y^2*z - 6432*x^3*y^2 + 2240*x^3*y*z^3 + 9360*x^3*y*z^2 + 12864*x^3*y*z + 5976*x^3*y - 560*x^3*z^4 - 3120*x^3*z^3 - 6432*x^3*z^2 - 5976*x^3*z - 2224*x^3 - 168*x^2*y^5 + 840*x^2*y^4*z + 1180*x^2*y^4 - 1680*x^2*y^3*z^2 - 4720*x^2*y^3*z - 3264*x^2*y^3 + 1680*x^2*y^2*z^3 + 7080*x^2*y^2*z^2 + 9792*x^2*y^2*z + 4572*x^2*y^2 - 840*x^2*y*z^4 - 4720*x^2*y*z^3 - 9792*x^2*y*z^2 - 9144*x^2*y*z - 3432*x^2*y + 168*x^2*z^5 + 1180*x^2*z^4 + 3264*x^2*z^3 + 4572*x^2*z^2 + 3440*x^2*z + 1222*x^2 - 28*x*y^6 + 168*x*y^5*z + 238*x*y^5 - 420*x*y^4*z^2 - 1190*x*y^4*z - 828*x*y^4 + 560*x*y^3*z^3 + 2380*x*y^3*z^2 + 3312*x*y^3*z + 1554*x*y^3 - 420*x*y^2*z^4 - 2380*x*y^2*z^3 - 4968*x*y^2*z^2 - 4662*x*y^2*z - 1764*x*y^2 + 168*x*y*z^5 + 1190*x*y*z^4 + 3312*x*y*z^3 + 4662*x*y*z^2 + 3536*x*y*z + 1284*x*y - 28*x*z^6 - 238*x*z^5 - 828*x*z^4 - 1554*x*z^3 - 1772*x*z^2 - 1298*x*z - 500*x - 2*y^7 + 14*y^6*z + 20*y^6 - 42*y^5*z^2 - 120*y^5*z - 84*y^5 + 70*y^4*z^3 + 300*y^4*z^2 + 420*y^4*z + 198*y^4 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9440*x^2*y*z^3 + 19584*x^2*y*z^2 + 18288*x^2*y*z + 6864*x^2*y - 336*x^2*z^5 - 2360*x^2*z^4 - 6528*x^2*z^3 - 9144*x^2*z^2 - 6880*x^2*z - 2440*x^2 + 56*x*y^6 - 336*x*y^5*z - 476*x*y^5 + 840*x*y^4*z^2 + 2380*x*y^4*z + 1656*x*y^4 - 1120*x*y^3*z^3 - 4760*x*y^3*z^2 - 6624*x*y^3*z - 3108*x*y^3 + 840*x*y^2*z^4 + 4760*x*y^2*z^3 + 9936*x*y^2*z^2 + 9324*x*y^2*z + 3528*x*y^2 - 336*x*y*z^5 - 2380*x*y*z^4 - 6624*x*y*z^3 - 9324*x*y*z^2 - 7072*x*y*z - 2564*x*y + 56*x*z^6 + 476*x*z^5 + 1656*x*z^4 + 3108*x*z^3 + 3544*x*z^2 + 2592*x*z + 993*x + 4*y^7 - 28*y^6*z - 40*y^6 + 84*y^5*z^2 + 240*y^5*z + 168*y^5 - 140*y^4*z^3 - 600*y^4*z^2 - 840*y^4*z - 396*y^4 + 140*y^3*z^4 + 800*y^3*z^3 + 1680*y^3*z^2 + 1584*y^3*z + 604*y^3 - 84*y^2*z^5 - 600*y^2*z^4 - 1680*y^2*z^3 - 2376*y^2*z^2 - 1816*y^2*z - 675*y^2 + 28*y*z^6 + 240*y*z^5 + 840*y*z^4 + 1584*y*z^3 + 1820*y*z^2 + 1366*y*z + 550*y - 4*z^7 - 40*z^6 - 168*z^5 - 396*z^4 - 608*z^3 - 691*z^2 - 562*z - 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -2, -1, 1 ], [ -1, -1, 1 ], [ -1, -1, 2 ] ], "source_shift": [ 1, 2, 2 ], "target_matrix": [ [ -2, -1, -1 ], [ 1, 2, 1 ], [ -2, -3, -2 ] ], "target_shift": [ 233, -116, 229 ], "inverse_recipe": { "p": 2, "q": 5, "f": "x**2 + x", "g": "x**5 + x**4 + x**3 + x", "h": "2*y**2 + y*z + y", "polynomials": { "f": [ [ [ 2, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "g": [ [ [ 5, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ], [ [ 3, 0, 0 ], 1 ], [ [ 4, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 2, 0 ], 2 ], [ [ 0, 1, 0 ], 1 ] ] }, "h_multiplier": 2 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0777049339994846 ], "surface_features": [ 3.5751618557834495, 4.415774383209107, 3.2533380053261065, 0.0, 3.6742845994144813, 4896 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C194", "pair": 194, "family": "counterexample", "mapping": "F(x,y,z)=(-32*x^7 - 208*x^6*y - 352*x^6*z - 296*x^6 - 576*x^5*y^2 - 1944*x^5*y*z - 1656*x^5*y - 1632*x^5*z^2 - 2808*x^5*z - 1188*x^5 - 880*x^4*y^3 - 4440*x^4*y^2*z - 3840*x^4*y^2 - 7428*x^4*y*z^2 - 12996*x^4*y*z - 5556*x^4*y - 4120*x^4*z^3 - 10950*x^4*z^2 - 9444*x^4*z - 2680*x^4 - 800*x^3*y^4 - 5360*x^3*y^3*z - 4720*x^3*y^3 - 13392*x^3*y^2*z^2 - 23904*x^3*y^2*z - 10344*x^3*y^2 - 14786*x^3*y*z^3 - 40182*x^3*y*z^2 - 35112*x^3*y*z - 10048*x^3*y - 6086*x^3*z^4 - 22420*x^3*z^3 - 29685*x^3*z^2 - 17130*x^3*z - 3676*x^3 - 432*x^2*y^5 - 3600*x^2*y^4*z - 3240*x^2*y^4 - 11928*x^2*y^3*z^2 - 21816*x^2*y^3*z - 9576*x^2*y^3 - 19638*x^2*y^2*z^3 - 54846*x^2*y^2*z^2 - 48672*x^2*y^2*z - 14064*x^2*y^2 - 16062*x^2*y*z^4 - 61017*x^2*y*z^3 - 82155*x^2*y*z^2 - 47898*x^2*y*z - 10344*x^2*y 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288*x^5*y^2 + 972*x^5*y*z + 960*x^5*y + 816*x^5*z^2 + 1620*x^5*z + 756*x^5 + 440*x^4*y^3 + 2220*x^4*y^2*z + 2220*x^4*y^2 + 3714*x^4*y*z^2 + 7476*x^4*y*z + 3528*x^4*y + 2060*x^4*z^3 + 6267*x^4*z^2 + 5964*x^4*z + 1806*x^4 + 400*x^3*y^4 + 2680*x^3*y^3*z + 2720*x^3*y^3 + 6696*x^3*y^2*z^2 + 13704*x^3*y^2*z + 6552*x^3*y^2 + 7393*x^3*y*z^3 + 22914*x^3*y*z^2 + 22116*x^3*y*z + 6760*x^3*y + 3043*x^3*z^4 + 12716*x^3*z^3 + 18591*x^3*z^2 + 11465*x^3*z + 2568*x^3 + 216*x^2*y^5 + 1800*x^2*y^4*z + 1860*x^2*y^4 + 5964*x^2*y^3*z^2 + 12456*x^2*y^3*z + 6048*x^2*y^3 + 9819*x^2*y^2*z^3 + 31140*x^2*y^2*z^2 + 30564*x^2*y^2*z + 9444*x^2*y^2 + 8031*x^2*y*z^4 + 34446*x^2*y*z^3 + 51288*x^2*y*z^2 + 31995*x^2*y*z + 7218*x^2*y + 2610*x^2*z^5 + 14226*x^2*z^4 + 28581*x^2*z^3 + 26997*x^2*z^2 + 12291*x^2*z + 2200*x^2 + 64*x*y^6 + 636*x*y^5*z + 672*x*y^5 + 2616*x*y^4*z^2 + 5604*x*y^4*z + 2772*x*y^4 + 5699*x*y^3*z^3 + 18606*x*y^3*z^2 + 18636*x*y^3*z + 5832*x*y^3 + 6933*x*y^2*z^4 + 30744*x*y^2*z^3 + 46803*x*y^2*z^2 + 29595*x*y^2*z + 6732*x*y^2 + 4464*x*y*z^5 + 25284*x*y*z^4 + 52047*x*y*z^3 + 49878*x*y*z^2 + 22905*x*y*z + 4118*x*y + 1188*x*z^6 + 8280*x*z^5 + 21627*x*z^4 + 27924*x*z^3 + 19407*x*z^2 + 7039*x*z + 1067*x + 8*y^7 + 92*y^6*z + 100*y^6 + 450*y^5*z^2 + 996*y^5*z + 504*y^5 + 1213*y^4*z^3 + 4113*y^4*z^2 + 4224*y^4*z + 1342*y^4 + 1945*y^3*z^4 + 9014*y^3*z^3 + 14106*y^3*z^2 + 9065*y^3*z + 2082*y^3 + 1854*y^2*z^5 + 11058*y^2*z^4 + 23466*y^2*z^3 + 22881*y^2*z^2 + 10614*y^2*z + 1918*y^2 + 972*y*z^6 + 7200*y*z^5 + 19449*y*z^4 + 25583*y*z^3 + 17970*y*z^2 + 6553*y*z + 994*y + 216*z^7 + 1944*z^6 + 6426*z^5 + 10693*z^4 + 10107*z^3 + 5575*z^2 + 1702*z + 1, -48*x^7 - 312*x^6*y - 528*x^6*z - 468*x^6 - 864*x^5*y^2 - 2916*x^5*y*z - 2616*x^5*y - 2448*x^5*z^2 - 4428*x^5*z - 1944*x^5 - 1320*x^4*y^3 - 6660*x^4*y^2*z - 6060*x^4*y^2 - 11142*x^4*y*z^2 - 20472*x^4*y*z - 9084*x^4*y - 6180*x^4*z^3 - 17217*x^4*z^2 - 15408*x^4*z - 4470*x^4 - 1200*x^3*y^4 - 8040*x^3*y^3*z - 7440*x^3*y^3 - 20088*x^3*y^2*z^2 - 37608*x^3*y^2*z - 16896*x^3*y^2 - 22179*x^3*y*z^3 - 63096*x^3*y*z^2 - 57228*x^3*y*z - 16752*x^3*y - 9129*x^3*z^4 - 35136*x^3*z^3 - 48276*x^3*z^2 - 28515*x^3*z - 6168*x^3 - 648*x^2*y^5 - 5400*x^2*y^4*z - 5100*x^2*y^4 - 17892*x^2*y^3*z^2 - 34272*x^2*y^3*z - 15624*x^2*y^3 - 29457*x^2*y^2*z^3 - 85986*x^2*y^2*z^2 - 79236*x^2*y^2*z - 23436*x^2*y^2 - 24093*x^2*y*z^4 - 95463*x^2*y*z^3 - 133443*x^2*y*z^2 - 79689*x^2*y*z - 17358*x^2*y - 7830*x^2*z^5 - 39573*x^2*z^4 - 74637*x^2*z^3 - 67488*x^2*z^2 - 29685*x^2*z - 5148*x^2 - 192*x*y^6 - 1908*x*y^5*z - 1848*x*y^5 - 7848*x*y^4*z^2 - 15468*x*y^4*z - 7176*x*y^4 - 17097*x*y^3*z^3 - 51552*x*y^3*z^2 - 48420*x*y^3*z - 14496*x*y^3 - 20799*x*y^2*z^4 - 85518*x*y^2*z^3 - 122058*x*y^2*z^2 - 73833*x*y^2*z - 16212*x*y^2 - 13392*x*y*z^5 - 70614*x*y*z^4 - 136251*x*y*z^3 - 124896*x*y*z^2 - 55395*x*y*z - 9648*x*y - 3564*x*z^6 - 23220*x*z^5 - 56835*x*z^4 - 70182*x*z^3 - 47130*x*z^2 - 16581*x*z - 2444*x - 24*y^7 - 276*y^6*z - 276*y^6 - 1350*y^5*z^2 - 2760*y^5*z - 1308*y^5 - 3639*y^4*z^3 - 11445*y^4*z^2 - 11004*y^4*z - 3342*y^4 - 5835*y^3*z^4 - 25191*y^3*z^3 - 36891*y^3*z^2 - 22659*y^3*z - 5022*y^3 - 5562*y^2*z^5 - 31041*y^2*z^4 - 61614*y^2*z^3 - 57408*y^2*z^2 - 25710*y^2*z - 4500*y^2 - 2916*y*z^6 - 20304*y*z^5 - 51273*y*z^4 - 64428*y*z^3 - 43707*y*z^2 - 15456*y*z - 2279*y - 648*z^7 - 5508*z^6 - 17010*z^5 - 27030*z^4 - 24681*z^3 - 13215*z^2 - 3928*z - 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ 2, 2, 3 ], [ 1, 1, 2 ], [ 2, 1, 1 ] ], "source_shift": [ 2, 1, 2 ], "target_matrix": [ [ -2, -1, -1 ], [ 1, 1, 1 ], [ -3, -2, -3 ] ], "target_shift": [ 312, -228, 517 ], "collision_preimages": [ [ "-5/4", "1/4", "0" ], [ "3", "5/2", "-4" ], [ "8", "-31/2", "4" ] ], "collision_image": [ "625/2", "-913/4", "2071/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.1736118640001223 ], "surface_features": [ 4.23474022260642, 5.134339698419419, 3.9007493580610797, 0.0, 4.355805890596741, 5146 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C122", "pair": 122, "family": "counterexample", "mapping": "F(x,y,z)=(-x^7 - 10*x^6*y + 10*x^6*z + 17*x^6 - 42*x^5*y^2 + 87*x^5*y*z + 144*x^5*y - 42*x^5*z^2 - 150*x^5*z - 117*x^5 - 96*x^4*y^3 + 309*x^4*y^2*z + 498*x^4*y^2 - 309*x^4*y*z^2 - 1074*x^4*y*z - 816*x^4*y + 96*x^4*z^3 + 543*x^4*z^2 + 879*x^4*z + 429*x^4 - 129*x^3*y^4 + 573*x^3*y^3*z + 900*x^3*y^3 - 891*x^3*y^2*z^2 - 3012*x^3*y^2*z - 2229*x^3*y^2 + 573*x^3*y*z^3 + 3156*x^3*y*z^2 + 4977*x^3*y*z + 2356*x^3*y - 129*x^3*z^4 - 1032*x^3*z^3 - 2604*x^3*z^2 - 2631*x^3*z - 915*x^3 - 102*x^2*y^5 + 585*x^2*y^4*z + 897*x^2*y^4 - 1257*x^2*y^3*z^2 - 4134*x^2*y^3*z - 2982*x^2*y^3 + 1257*x^2*y^2*z^3 + 6732*x^2*y^2*z^2 + 10338*x^2*y^2*z + 4746*x^2*y^2 - 585*x^2*y*z^4 - 4566*x^2*y*z^3 - 11229*x^2*y*z^2 - 11004*x^2*y*z - 3693*x^2*y + 102*x^2*z^5 + 1086*x^2*z^4 + 3801*x^2*z^3 + 5967*x^2*z^2 + 4296*x^2*z + 1144*x^2 - 44*x*y^6 + 312*x*y^5*z + 468*x*y^5 - 867*x*y^4*z^2 - 2778*x*y^4*z - 1956*x*y^4 + 1199*x*y^3*z^3 + 6240*x*y^3*z^2 + 9336*x*y^3*z + 4161*x*y^3 - 867*x*y^2*z^4 - 6588*x*y^2*z^3 - 15777*x*y^2*z^2 - 14985*x*y^2*z - 4854*x*y^2 + 312*x*y*z^5 + 3252*x*y*z^4 + 11103*x*y*z^3 + 16911*x*y*z^2 + 11745*x*y*z + 2999*x*y - 44*x*z^6 - 600*x*z^5 - 2733*x*z^4 - 5928*x*z^3 - 6630*x*z^2 - 3661*x*z - 781*x - 8*y^7 + 68*y^6*z + 100*y^6 - 234*y^5*z^2 - 732*y^5*z - 504*y^5 + 419*y^4*z^3 + 2121*y^4*z^2 + 3096*y^4*z + 1342*y^4 - 419*y^3*z^4 - 3094*y^3*z^3 - 7218*y^3*z^2 - 6651*y^3*z - 2082*y^3 + 234*y^2*z^5 + 2382*y^2*z^4 + 7920*y^2*z^3 + 11685*y^2*z^2 + 7830*y^2*z + 1918*y^2 - 68*y*z^6 - 912*y*z^5 - 4059*y*z^4 - 8549*y*z^3 - 9216*y*z^2 - 4879*y*z - 994*y + 8*z^7 + 136*z^6 + 774*z^5 + 2175*z^4 + 3363*z^3 + 2887*z^2 + 1276*z + 1, 3*x^7 + 30*x^6*y - 30*x^6*z - 48*x^6 + 126*x^5*y^2 - 261*x^5*y*z - 405*x^5*y + 126*x^5*z^2 + 426*x^5*z + 312*x^5 + 288*x^4*y^3 - 927*x^4*y^2*z - 1395*x^4*y^2 + 927*x^4*y*z^2 + 3039*x^4*y*z + 2163*x^4*y - 288*x^4*z^3 - 1551*x^4*z^2 - 2361*x^4*z - 1095*x^4 + 387*x^3*y^4 - 1719*x^3*y^3*z - 2511*x^3*y^3 + 2673*x^3*y^2*z^2 + 8490*x^3*y^2*z + 5874*x^3*y^2 - 1719*x^3*y*z^3 - 8985*x^3*y*z^2 - 13287*x^3*y*z - 5973*x^3*y + 387*x^3*z^4 + 2964*x^3*z^3 + 7047*x^3*z^2 + 6765*x^3*z + 2256*x^3 + 306*x^2*y^5 - 1755*x^2*y^4*z - 2493*x^2*y^4 + 3771*x^2*y^3*z^2 + 11607*x^2*y^3*z + 7815*x^2*y^3 - 3771*x^2*y^2*z^3 - 19098*x^2*y^2*z^2 - 27432*x^2*y^2*z - 11955*x^2*y^2 + 1755*x^2*y*z^4 + 13077*x^2*y*z^3 + 30204*x^2*y*z^2 + 28095*x^2*y*z + 9036*x^2*y - 306*x^2*z^5 - 3135*x^2*z^4 - 10365*x^2*z^3 - 15462*x^2*z^2 - 10674*x^2*z - 2736*x^2 + 132*x*y^6 - 936*x*y^5*z - 1296*x*y^5 + 2601*x*y^4*z^2 + 7770*x*y^4*z + 5100*x*y^4 - 3597*x*y^3*z^3 - 17637*x*y^3*z^2 - 24630*x*y^3*z - 10419*x*y^3 + 2601*x*y^2*z^4 + 18810*x*y^2*z^3 + 42174*x*y^2*z^2 + 38001*x*y^2*z + 11790*x*y^2 - 936*x*y*z^5 - 9366*x*y*z^4 - 30099*x*y*z^3 - 43500*x*y*z^2 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4443*x^3*z^2 + 4138*x^3*z + 1352*x^3 + 204*x^2*y^5 - 1170*x^2*y^4*z - 1596*x^2*y^4 + 2514*x^2*y^3*z^2 + 7473*x^2*y^3*z + 4833*x^2*y^3 - 2514*x^2*y^2*z^3 - 12366*x^2*y^2*z^2 - 17094*x^2*y^2*z - 7227*x^2*y^2 + 1170*x^2*y*z^4 + 8511*x^2*y*z^3 + 18975*x^2*y*z^2 + 17112*x^2*y*z + 5400*x^2*y - 204*x^2*z^5 - 2049*x^2*z^4 - 6564*x^2*z^3 - 9501*x^2*z^2 - 6414*x^2*z - 1631*x^2 + 88*x*y^6 - 624*x*y^5*z - 828*x*y^5 + 1734*x*y^4*z^2 + 4992*x*y^4*z + 3144*x*y^4 - 2398*x*y^3*z^3 - 11397*x*y^3*z^2 - 15294*x*y^3*z - 6278*x*y^3 + 1734*x*y^2*z^4 + 12222*x*y^2*z^3 + 26397*x*y^2*z^2 + 23052*x*y^2*z + 7032*x*y^2 - 624*x*y*z^5 - 6114*x*y*z^4 - 18996*x*y*z^3 - 26610*x*y*z^2 - 17334*x*y*z - 4237*x*y + 88*x*z^6 + 1140*x*z^5 + 4776*x*z^4 + 9562*x*z^3 + 10014*x*z^2 + 5264*x*z + 1087*x + 16*y^7 - 136*y^6*z - 176*y^6 + 468*y^5*z^2 + 1308*y^5*z + 804*y^5 - 838*y^4*z^3 - 3852*y^4*z^2 - 5028*y^4*z - 2008*y^4 + 838*y^3*z^4 + 5711*y^3*z^3 + 11961*y^3*z^2 + 10130*y^3*z + 2992*y^3 - 468*y^2*z^5 - 4461*y^2*z^4 - 13416*y^2*z^3 - 18177*y^2*z^2 - 11436*y^2*z - 2690*y^2 + 136*y*z^6 + 1728*y*z^5 + 7032*y*z^4 + 13622*y*z^3 + 13746*y*z^2 + 6941*y*z + 1373*y - 16*z^7 - 260*z^6 - 1368*z^5 - 3556*z^4 - 5144*z^3 - 4193*z^2 - 1788*z - 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -1, -2, 1 ], [ -1, -1, 2 ], [ -1, -1, 1 ] ], "source_shift": [ 2, 1, 2 ], "target_matrix": [ [ 1, 1, 1 ], [ -3, -2, -3 ], [ -2, -1, -1 ] ], "target_shift": [ -228, 517, 312 ], "collision_preimages": [ [ "15/4", "-1/4", "5/4" ], [ "-17", "11/2", "-7" ], [ "-16", "15/2", "-4" ] ], "collision_image": [ "-913/4", "2071/4", "625/2" ] }, "probe_target": "No", "jev_answers": [ "Maybe" ], "seconds": [ 1.1388414409993857 ], "surface_features": [ 3.7077860869404446, 4.638489256954637, 3.3507324517161297, 0.0, 3.849271752214216, 4940 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C193", "pair": 193, "family": "counterexample", "mapping": "F(x,y,z)=(8*x^7 - 80*x^6*y + 76*x^6*z - 148*x^6 + 336*x^5*y^2 - 660*x^5*y*z + 1248*x^5*y + 306*x^5*z^2 - 1212*x^5*z + 1044*x^5 - 768*x^4*y^3 + 2340*x^4*y^2*z - 4308*x^4*y^2 - 2244*x^4*y*z^2 + 8604*x^4*y*z - 7308*x^4*y + 677*x^4*z^3 - 4101*x^4*z^2 + 7092*x^4*z - 3834*x^4 + 1032*x^3*y^4 - 4332*x^3*y^3*z + 7800*x^3*y^3 + 6450*x^3*y^2*z^2 - 23976*x^3*y^2*z + 20148*x^3*y^2 - 4025*x^3*y*z^3 + 23550*x^3*y*z^2 - 40020*x^3*y*z + 21402*x^3*y + 889*x^3*z^4 - 7336*x^3*z^3 + 19161*x^3*z^2 - 20735*x^3*z + 8104*x^3 - 816*x^2*y^5 + 4416*x^2*y^4*z - 7824*x^2*y^4 - 9072*x^2*y^3*z^2 + 32808*x^2*y^3*z - 27372*x^2*y^3 + 8796*x^2*y^2*z^3 - 49716*x^2*y^2*z^2 + 83292*x^2*y^2*z - 44124*x^2*y^2 - 4017*x^2*y*z^4 + 31974*x^2*y*z^3 - 81795*x^2*y*z^2 + 87384*x^2*y*z - 33732*x^2*y + 693*x^2*z^5 - 7314*x^2*z^4 + 25722*x^2*z^3 - 41877*x^2*z^2 + 32787*x^2*z - 9994*x^2 + 352*x*y^6 - 2352*x*y^5*z + 4128*x*y^5 + 6240*x*y^4*z^2 - 22080*x*y^4*z + 18336*x*y^4 - 8360*x*y^3*z^3 + 45744*x*y^3*z^2 - 75864*x*y^3*z + 39832*x*y^3 + 5934*x*y^2*z^4 - 45504*x*y^2*z^3 + 114324*x*y^2*z^2 - 120816*x*y^2*z + 46050*x*y^2 - 2115*x*y*z^5 + 21522*x*y*z^4 - 73914*x*y*z^3 + 118518*x*y*z^2 - 91575*x*y*z + 27478*x*y + 297*x*z^6 - 3852*x*z^5 + 17145*x*z^4 - 37414*x*z^3 + 44064*x*z^2 - 26960*x*z + 6719*x - 64*y^7 + 512*y^6*z - 896*y^6 - 1680*y^5*z^2 + 5856*y^5*z - 4848*y^5 + 2912*y^4*z^3 - 15504*y^4*z^2 + 25536*y^4*z - 13288*y^4 - 2860*y^3*z^4 + 21136*y^3*z^3 - 52392*y^3*z^2 + 54828*y^3*z - 20620*y^3 + 1584*y^2*z^5 - 15504*y^2*z^4 + 52080*y^2*z^3 - 82458*y^2*z^2 + 62892*y^2*z - 18556*y^2 - 459*y*z^6 + 5742*y*z^5 - 24903*y*z^4 + 53377*y*z^3 - 61965*y*z^2 + 37318*y*z - 9113*y + 54*z^7 - 837*z^6 + 4536*z^5 - 12469*z^4 + 19665*z^3 - 18124*z^2 + 9090*z - 1, 8*x^7 - 80*x^6*y + 76*x^6*z - 124*x^6 + 336*x^5*y^2 - 660*x^5*y*z + 1056*x^5*y + 306*x^5*z^2 - 1008*x^5*z + 792*x^5 - 768*x^4*y^3 + 2340*x^4*y^2*z - 3684*x^4*y^2 - 2244*x^4*y*z^2 + 7224*x^4*y*z - 5604*x^4*y + 677*x^4*z^3 - 3387*x^4*z^2 + 5340*x^4*z - 2722*x^4 + 1032*x^3*y^4 - 4332*x^3*y^3*z + 6744*x^3*y^3 + 6450*x^3*y^2*z^2 - 20340*x^3*y^2*z + 15624*x^3*y^2 - 4025*x^3*y*z^3 + 19626*x^3*y*z^2 - 30456*x^3*y*z + 15370*x^3*y + 889*x^3*z^4 - 6019*x^3*z^3 + 14322*x^3*z^2 - 14611*x^3*z + 5472*x^3 - 816*x^2*y^5 + 4416*x^2*y^4*z - 6840*x^2*y^4 - 9072*x^2*y^3*z^2 + 28140*x^2*y^3*z - 21468*x^2*y^3 + 8796*x^2*y^2*z^3 - 41850*x^2*y^2*z^2 + 64104*x^2*y^2*z - 32052*x^2*y^2 - 4017*x^2*y*z^4 + 26457*x^2*y*z^3 - 61779*x^2*y*z^2 + 62304*x^2*y*z - 23076*x^2*y + 693*x^2*z^5 - 5964*x^2*z^4 + 19089*x^2*z^3 - 29283*x^2*z^2 + 21963*x^2*z - 6484*x^2 + 352*x*y^6 - 2352*x*y^5*z + 3648*x*y^5 + 6240*x*y^4*z^2 - 19152*x*y^4*z + 14544*x*y^4 - 8360*x*y^3*z^3 + 38928*x*y^3*z^2 - 59064*x*y^3*z + 29264*x*y^3 + 5934*x*y^2*z^4 - 38016*x*y^2*z^3 + 87324*x*y^2*z^2 - 87156*x*y^2*z + 31902*x*y^2 - 2115*x*y*z^5 + 17688*x*y*z^4 - 55410*x*y*z^3 + 83880*x*y*z^2 - 62187*x*y*z + 18106*x*y + 297*x*z^6 - 3123*x*z^5 + 12636*x*z^4 - 25957*x*z^3 + 29271*x*z^2 - 17339*x*z + 4236*x - 64*y^7 + 512*y^6*z - 800*y^6 - 1680*y^5*z^2 + 5136*y^5*z - 3888*y^5 + 2912*y^4*z^3 - 13344*y^4*z^2 + 20112*y^4*z - 9872*y^4 - 2860*y^3*z^4 + 17848*y^3*z^3 - 40488*y^3*z^2 + 40016*y^3*z - 14460*y^3 + 1584*y^2*z^5 - 12858*y^2*z^4 + 39480*y^2*z^3 - 59064*y^2*z^2 + 43272*y^2*z - 12406*y^2 - 459*y*z^6 + 4689*y*z^5 - 18531*y*z^4 + 37492*y*z^3 - 41757*y*z^2 + 24397*y*z - 5849*y + 54*z^7 - 675*z^6 + 3321*z^5 - 8581*z^4 + 12948*z^3 - 11548*z^2 + 5677*z - 1, -8*x^7 + 80*x^6*y - 76*x^6*z + 124*x^6 - 336*x^5*y^2 + 660*x^5*y*z - 1056*x^5*y - 306*x^5*z^2 + 1008*x^5*z - 792*x^5 + 768*x^4*y^3 - 2340*x^4*y^2*z + 3684*x^4*y^2 + 2244*x^4*y*z^2 - 7224*x^4*y*z + 5604*x^4*y - 677*x^4*z^3 + 3387*x^4*z^2 - 5340*x^4*z + 2730*x^4 - 1032*x^3*y^4 + 4332*x^3*y^3*z - 6744*x^3*y^3 - 6450*x^3*y^2*z^2 + 20340*x^3*y^2*z - 15624*x^3*y^2 + 4025*x^3*y*z^3 - 19626*x^3*y*z^2 + 30456*x^3*y*z - 15402*x^3*y - 889*x^3*z^4 + 6019*x^3*z^3 - 14322*x^3*z^2 + 14663*x^3*z - 5524*x^3 + 816*x^2*y^5 - 4416*x^2*y^4*z + 6840*x^2*y^4 + 9072*x^2*y^3*z^2 - 28140*x^2*y^3*z + 21468*x^2*y^3 - 8796*x^2*y^2*z^3 + 41850*x^2*y^2*z^2 - 64104*x^2*y^2*z + 32100*x^2*y^2 + 4017*x^2*y*z^4 - 26457*x^2*y*z^3 + 61779*x^2*y*z^2 - 62460*x^2*y*z + 23244*x^2*y - 693*x^2*z^5 + 5964*x^2*z^4 - 19089*x^2*z^3 + 29409*x^2*z^2 - 22203*x^2*z + 6604*x^2 - 352*x*y^6 + 2352*x*y^5*z - 3648*x*y^5 - 6240*x*y^4*z^2 + 19152*x*y^4*z - 14544*x*y^4 + 8360*x*y^3*z^3 - 38928*x*y^3*z^2 + 59064*x*y^3*z - 29296*x*y^3 - 5934*x*y^2*z^4 + 38016*x*y^2*z^3 - 87324*x*y^2*z^2 + 87312*x*y^2*z - 32082*x*y^2 + 2115*x*y*z^5 - 17688*x*y*z^4 + 55410*x*y*z^3 - 84132*x*y*z^2 + 62703*x*y*z - 18370*x*y - 297*x*z^6 + 3123*x*z^5 - 12636*x*z^4 + 26092*x*z^3 - 29640*x*z^2 + 17699*x*z - 4348*x + 64*y^7 - 512*y^6*z + 800*y^6 + 1680*y^5*z^2 - 5136*y^5*z + 3888*y^5 - 2912*y^4*z^3 + 13344*y^4*z^2 - 20112*y^4*z + 9880*y^4 + 2860*y^3*z^4 - 17848*y^3*z^3 + 40488*y^3*z^2 - 40068*y^3*z + 14524*y^3 - 1584*y^2*z^5 + 12858*y^2*z^4 - 39480*y^2*z^3 + 59190*y^2*z^2 - 43548*y^2*z + 12550*y^2 + 459*y*z^6 - 4689*y*z^5 + 18531*y*z^4 - 37627*y*z^3 + 42153*y*z^2 - 24793*y*z + 5973*y - 54*z^7 + 675*z^6 - 3321*z^5 + 8635*z^4 - 13137*z^3 + 11818*z^2 - 5843*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -2, 2, -3 ], [ -1, 2, -1 ], [ -1, 1, -2 ] ], "source_shift": [ 2, 2, 2 ], "target_matrix": [ [ -1, -2, -1 ], [ -1, -1, -2 ], [ 1, 1, 1 ] ], "target_shift": [ 1905, 1183, -1219 ], "collision_preimages": [ [ "-5", "-9/4", "5/2" ], [ "35/2", "2", "-10" ], [ "53/2", "7", "-12" ] ], "collision_image": [ "7621/4", "4733/4", "-4877/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.032448406000185 ], "surface_features": [ 4.260525793927986, 5.08212445292395, 3.9222062774390163, 0.0, 4.3559312412190145, 5125 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B111", "pair": 111, "family": "bijection", "mapping": "F(x,y,z)=(5*x^7 + 70*x^6*y - 35*x^6*z - 70*x^6 + 420*x^5*y^2 - 420*x^5*y*z - 840*x^5*y + 105*x^5*z^2 + 420*x^5*z + 435*x^5 + 1400*x^4*y^3 - 2100*x^4*y^2*z - 4200*x^4*y^2 + 1050*x^4*y*z^2 + 4200*x^4*y*z + 4335*x^4*y - 175*x^4*z^3 - 1050*x^4*z^2 - 2170*x^4*z - 1551*x^4 + 2800*x^3*y^4 - 5600*x^3*y^3*z - 11200*x^3*y^3 + 4200*x^3*y^2*z^2 + 16800*x^3*y^2*z + 17280*x^3*y^2 - 1400*x^3*y*z^3 - 8400*x^3*y*z^2 - 17300*x^3*y*z - 12283*x^3*y + 175*x^3*z^4 + 1400*x^3*z^3 + 4330*x^3*z^2 + 6164*x^3*z + 3391*x^3 + 3360*x^2*y^5 - 8400*x^2*y^4*z - 16800*x^2*y^4 + 8400*x^2*y^3*z^2 + 33600*x^2*y^3*z + 34440*x^2*y^3 - 4200*x^2*y^2*z^3 - 25200*x^2*y^2*z^2 - 51720*x^2*y^2*z - 36474*x^2*y^2 + 1050*x^2*y*z^4 + 8400*x^2*y*z^3 + 25890*x^2*y*z^2 + 36609*x^2*y*z + 19956*x^2*y - 105*x^2*z^5 - 1050*x^2*z^4 - 4320*x^2*z^3 - 9186*x^2*z^2 - 10053*x^2*z - 4512*x^2 + 2240*x*y^6 - 6720*x*y^5*z - 13440*x*y^5 + 8400*x*y^4*z^2 + 33600*x*y^4*z + 34320*x*y^4 - 5600*x*y^3*z^3 - 33600*x*y^3*z^2 - 68720*x*y^3*z - 48132*x*y^3 + 2100*x*y^2*z^4 + 16800*x*y^2*z^3 + 51600*x*y^2*z^2 + 72468*x*y^2*z + 39132*x*y^2 - 420*x*y*z^5 - 4200*x*y*z^4 - 17220*x*y*z^3 - 36369*x*y*z^2 - 39432*x*y*z - 17463*x*y + 35*x*z^6 + 420*x*z^5 + 2155*x*z^4 + 6084*x*z^3 + 9933*x*z^2 + 8854*x*z + 3320*x + 640*y^7 - 2240*y^6*z - 4480*y^6 + 3360*y^5*z^2 + 13440*y^5*z + 13680*y^5 - 2800*y^4*z^3 - 16800*y^4*z^2 - 34240*y^4*z - 23816*y^4 + 1400*y^3*z^4 + 11200*y^3*z^3 + 34280*y^3*z^2 + 47812*y^3*z + 25568*y^3 - 420*y^2*z^5 - 4200*y^2*z^4 - 17160*y^2*z^3 - 35994*y^2*z^2 - 38652*y^2*z - 16893*y^2 + 70*y*z^6 + 840*y*z^5 + 4295*y*z^4 + 12043*y*z^3 + 19476*y*z^2 + 17128*y*z + 6330*y - 5*z^7 - 70*z^6 - 430*z^5 - 1511*z^4 - 3271*z^3 - 4342*z^2 - 3237*z - 1, 5*x^7 + 70*x^6*y - 35*x^6*z - 70*x^6 + 420*x^5*y^2 - 420*x^5*y*z - 840*x^5*y + 105*x^5*z^2 + 420*x^5*z + 435*x^5 + 1400*x^4*y^3 - 2100*x^4*y^2*z - 4200*x^4*y^2 + 1050*x^4*y*z^2 + 4200*x^4*y*z + 4335*x^4*y - 175*x^4*z^3 - 1050*x^4*z^2 - 2170*x^4*z - 1551*x^4 + 2800*x^3*y^4 - 5600*x^3*y^3*z - 11200*x^3*y^3 + 4200*x^3*y^2*z^2 + 16800*x^3*y^2*z + 17280*x^3*y^2 - 1400*x^3*y*z^3 - 8400*x^3*y*z^2 - 17300*x^3*y*z - 12283*x^3*y + 175*x^3*z^4 + 1400*x^3*z^3 + 4330*x^3*z^2 + 6164*x^3*z + 3389*x^3 + 3360*x^2*y^5 - 8400*x^2*y^4*z - 16800*x^2*y^4 + 8400*x^2*y^3*z^2 + 33600*x^2*y^3*z + 34440*x^2*y^3 - 4200*x^2*y^2*z^3 - 25200*x^2*y^2*z^2 - 51720*x^2*y^2*z - 36474*x^2*y^2 + 1050*x^2*y*z^4 + 8400*x^2*y*z^3 + 25890*x^2*y*z^2 + 36609*x^2*y*z + 19944*x^2*y - 105*x^2*z^5 - 1050*x^2*z^4 - 4320*x^2*z^3 - 9186*x^2*z^2 - 10047*x^2*z - 4500*x^2 + 2240*x*y^6 - 6720*x*y^5*z - 13440*x*y^5 + 8400*x*y^4*z^2 + 33600*x*y^4*z + 34320*x*y^4 - 5600*x*y^3*z^3 - 33600*x*y^3*z^2 - 68720*x*y^3*z - 48132*x*y^3 + 2100*x*y^2*z^4 + 16800*x*y^2*z^3 + 51600*x*y^2*z^2 + 72468*x*y^2*z + 39108*x*y^2 - 420*x*y*z^5 - 4200*x*y*z^4 - 17220*x*y*z^3 - 36369*x*y*z^2 - 39408*x*y*z - 17415*x*y + 35*x*z^6 + 420*x*z^5 + 2155*x*z^4 + 6084*x*z^3 + 9927*x*z^2 + 8830*x*z + 3290*x + 640*y^7 - 2240*y^6*z - 4480*y^6 + 3360*y^5*z^2 + 13440*y^5*z + 13680*y^5 - 2800*y^4*z^3 - 16800*y^4*z^2 - 34240*y^4*z - 23816*y^4 + 1400*y^3*z^4 + 11200*y^3*z^3 + 34280*y^3*z^2 + 47812*y^3*z + 25552*y^3 - 420*y^2*z^5 - 4200*y^2*z^4 - 17160*y^2*z^3 - 35994*y^2*z^2 - 38628*y^2*z - 16845*y^2 + 70*y*z^6 + 840*y*z^5 + 4295*y*z^4 + 12043*y*z^3 + 19464*y*z^2 + 17080*y*z + 6276*y - 5*z^7 - 70*z^6 - 430*z^5 - 1511*z^4 - 3269*z^3 - 4330*z^2 - 3209*z - 1, -10*x^7 - 140*x^6*y + 70*x^6*z + 140*x^6 - 840*x^5*y^2 + 840*x^5*y*z + 1680*x^5*y - 210*x^5*z^2 - 840*x^5*z - 870*x^5 - 2800*x^4*y^3 + 4200*x^4*y^2*z + 8400*x^4*y^2 - 2100*x^4*y*z^2 - 8400*x^4*y*z - 8670*x^4*y + 350*x^4*z^3 + 2100*x^4*z^2 + 4340*x^4*z + 3101*x^4 - 5600*x^3*y^4 + 11200*x^3*y^3*z + 22400*x^3*y^3 - 8400*x^3*y^2*z^2 - 33600*x^3*y^2*z - 34560*x^3*y^2 + 2800*x^3*y*z^3 + 16800*x^3*y*z^2 + 34600*x^3*y*z + 24558*x^3*y - 350*x^3*z^4 - 2800*x^3*z^3 - 8660*x^3*z^2 - 12324*x^3*z - 6776*x^3 - 6720*x^2*y^5 + 16800*x^2*y^4*z + 33600*x^2*y^4 - 16800*x^2*y^3*z^2 - 67200*x^2*y^3*z - 68880*x^2*y^3 + 8400*x^2*y^2*z^3 + 50400*x^2*y^2*z^2 + 103440*x^2*y^2*z + 72924*x^2*y^2 - 2100*x^2*y*z^4 - 16800*x^2*y*z^3 - 51780*x^2*y*z^2 - 73194*x^2*y*z - 39876*x^2*y + 210*x^2*z^5 + 2100*x^2*z^4 + 8640*x^2*z^3 + 18366*x^2*z^2 + 20088*x^2*z + 9012*x^2 - 4480*x*y^6 + 13440*x*y^5*z + 26880*x*y^5 - 16800*x*y^4*z^2 - 67200*x*y^4*z - 68640*x*y^4 + 11200*x*y^3*z^3 + 67200*x*y^3*z^2 + 137440*x*y^3*z + 96232*x*y^3 - 4200*x*y^2*z^4 - 33600*x*y^2*z^3 - 103200*x*y^2*z^2 - 144888*x*y^2*z - 78192*x*y^2 + 840*x*y*z^5 + 8400*x*y*z^4 + 34440*x*y*z^3 + 72714*x*y*z^2 + 78792*x*y*z + 34878*x*y - 70*x*z^6 - 840*x*z^5 - 4310*x*z^4 - 12164*x*z^3 - 19848*x*z^2 - 17684*x*z - 6636*x - 1280*y^7 + 4480*y^6*z + 8960*y^6 - 6720*y^5*z^2 - 26880*y^5*z - 27360*y^5 + 5600*y^4*z^3 + 33600*y^4*z^2 + 68480*y^4*z + 47616*y^4 - 2800*y^3*z^4 - 22400*y^3*z^3 - 68560*y^3*z^2 - 95592*y^3*z - 51088*y^3 + 840*y^2*z^5 + 8400*y^2*z^4 + 34320*y^2*z^3 + 71964*y^2*z^2 + 77232*y^2*z + 33738*y^2 - 140*y*z^6 - 1680*y*z^5 - 8590*y*z^4 - 24078*y*z^3 - 38916*y*z^2 - 34208*y*z - 12647*y + 10*z^7 + 140*z^6 + 860*z^5 + 3021*z^4 + 6536*z^3 + 8672*z^2 + 6468*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -1, -2, 1 ], [ -2, -3, 2 ], [ -2, -1, 1 ] ], "source_shift": [ 2, 1, 2 ], "target_matrix": [ [ -1, -1, -1 ], [ -1, -1, -2 ], [ 2, 1, 1 ] ], "target_shift": [ 1014, 990, -2036 ], "inverse_recipe": { "p": 4, "q": 3, "f": "x**4", "g": "x**3 + x", "h": "y*z", "polynomials": { "f": [ [ [ 4, 0, 0 ], 1 ] ], "g": [ [ [ 3, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ] ] }, "h_multiplier": 5 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0215128140007437 ], "surface_features": [ 4.21779909546341, 5.161032417565079, 3.8273692730538253, 0.0, 4.3524719890866574, 5101 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C218", "pair": 218, "family": "counterexample", "mapping": "F(x,y,z)=(2*x^7 - 20*x^6*y - 20*x^6*z + 37*x^6 + 84*x^5*y^2 + 174*x^5*y*z - 315*x^5*y + 84*x^5*z^2 - 324*x^5*z + 291*x^5 - 192*x^4*y^3 - 618*x^4*y^2*z + 1095*x^4*y^2 - 618*x^4*y*z^2 + 2331*x^4*y*z - 2055*x^4*y - 192*x^4*z^3 + 1164*x^4*z^2 - 2154*x^4*z + 1269*x^4 + 258*x^3*y^4 + 1146*x^3*y^3*z - 1989*x^3*y^3 + 1782*x^3*y^2*z^2 - 6570*x^3*y^2*z + 5685*x^3*y^2 + 1146*x^3*y*z^3 - 6795*x^3*y*z^2 + 12342*x^3*y*z - 7139*x^3*y + 258*x^3*z^4 - 2196*x^3*z^3 + 6291*x^3*z^2 - 7596*x^3*z + 3327*x^3 - 204*x^2*y^5 - 1170*x^2*y^4*z + 1992*x^2*y^4 - 2514*x^2*y^3*z^2 + 9063*x^2*y^3*z - 7701*x^2*y^3 - 2514*x^2*y^2*z^3 + 14562*x^2*y^2*z^2 - 25950*x^2*y^2*z + 14736*x^2*y^2 - 1170*x^2*y*z^4 + 9753*x^2*y*z^3 - 27435*x^2*y*z^2 + 32523*x^2*y*z - 13971*x^2*y - 204*x^2*z^5 + 2295*x^2*z^4 - 9060*x^2*z^3 + 16830*x^2*z^2 - 15072*x^2*z + 5261*x^2 + 88*x*y^6 + 624*x*y^5*z - 1044*x*y^5 + 1734*x*y^4*z^2 - 6120*x*y^4*z + 5112*x*y^4 + 2398*x*y^3*z^3 - 13563*x*y^3*z^2 + 23718*x*y^3*z - 13230*x*y^3 + 1734*x*y^2*z^4 - 14130*x*y^2*z^3 + 38997*x*y^2*z^2 - 45366*x*y^2*z + 19110*x*y^2 + 624*x*y*z^5 - 6894*x*y*z^4 + 26748*x*y*z^3 - 48795*x*y*z^2 + 42852*x*y*z - 14641*x*y + 88*x*z^6 - 1260*x*z^5 + 6432*x*z^4 - 16350*x*z^3 + 22467*x*z^2 - 16022*x*z + 4661*x - 16*y^7 - 136*y^6*z + 224*y^6 - 468*y^5*z^2 + 1620*y^5*z - 1332*y^5 - 838*y^4*z^3 + 4632*y^4*z^2 - 7956*y^4*z + 4364*y^4 - 838*y^3*z^4 + 6665*y^3*z^3 - 18045*y^3*z^2 + 20610*y^3*z - 8520*y^3 - 468*y^2*z^5 + 5067*y^2*z^4 - 19296*y^2*z^3 + 34530*y^2*z^2 - 29718*y^2*z + 9938*y^2 - 136*y*z^6 + 1920*y*z^5 - 9648*y*z^4 + 24103*y*z^3 - 32481*y*z^2 + 22661*y*z - 6437*y - 16*z^7 + 284*z^6 - 1800*z^5 + 5871*z^4 - 11010*z^3 + 12041*z^2 - 7154*z + 1, 3*x^7 - 30*x^6*y - 30*x^6*z + 60*x^6 + 126*x^5*y^2 + 261*x^5*y*z - 513*x^5*y + 126*x^5*z^2 - 522*x^5*z + 504*x^5 - 288*x^4*y^3 - 927*x^4*y^2*z + 1791*x^4*y^2 - 927*x^4*y*z^2 + 3771*x^4*y*z - 3582*x^4*y - 288*x^4*z^3 + 1863*x^4*z^2 - 3699*x^4*z + 2317*x^4 + 387*x^3*y^4 + 1719*x^3*y^3*z - 3267*x^3*y^3 + 2673*x^3*y^2*z^2 - 10674*x^3*y^2*z + 9972*x^3*y^2 + 1719*x^3*y*z^3 - 10917*x^3*y*z^2 + 21330*x^3*y*z - 13135*x^3*y + 387*x^3*z^4 - 3492*x^3*z^3 + 10710*x^3*z^2 - 13741*x^3*z + 6335*x^3 - 306*x^2*y^5 - 1755*x^2*y^4*z + 3285*x^2*y^4 - 3771*x^2*y^3*z^2 + 14787*x^2*y^3*z - 13590*x^2*y^3 - 3771*x^2*y^2*z^3 + 23490*x^2*y^2*z^2 - 45135*x^2*y^2*z + 27315*x^2*y^2 - 1755*x^2*y*z^4 + 15561*x^2*y*z^3 - 46998*x^2*y*z^2 + 59295*x^2*y*z - 26832*x^2*y - 306*x^2*z^5 + 3627*x^2*z^4 - 15291*x^2*z^3 + 30156*x^2*z^2 - 28422*x^2*z + 10359*x^2 + 132*x*y^6 + 936*x*y^5*z - 1728*x*y^5 + 2601*x*y^4*z^2 - 10026*x*y^4*z + 9072*x*y^4 + 3597*x*y^3*z^3 - 21969*x*y^3*z^2 + 41508*x*y^3*z - 24695*x*y^3 + 2601*x*y^2*z^4 - 22626*x*y^2*z^3 + 67230*x*y^2*z^2 - 83349*x*y^2*z + 37005*x*y^2 + 936*x*y*z^5 - 10926*x*y*z^4 + 45414*x*y*z^3 - 88122*x*y*z^2 + 81531*x*y*z - 29103*x*y + 132*x*z^6 - 1980*x*z^5 + 10764*x*z^4 - 29014*x*z^3 + 41943*x*z^2 - 31230*x*z + 9426*x - 24*y^7 - 204*y^6*z + 372*y^6 - 702*y^5*z^2 + 2664*y^5*z - 2376*y^5 - 1257*y^4*z^3 + 7533*y^4*z^2 - 14004*y^4*z + 8198*y^4 - 1257*y^3*z^4 + 10713*y^3*z^3 - 31302*y^3*z^2 + 38141*y^3*z - 16624*y^3 - 702*y^2*z^5 + 8055*y^2*z^4 - 32967*y^2*z^3 + 62856*y^2*z^2 - 57033*y^2*z + 19932*y^2 - 204*y*z^6 + 3024*y*z^5 - 16236*y*z^4 + 43108*y*z^3 - 61197*y*z^2 + 44613*y*z - 13156*y - 24*z^7 + 444*z^6 - 2988*z^5 + 10318*z^4 - 20342*z^3 + 23223*z^2 - 14313*z + 1, x^7 - 10*x^6*y - 10*x^6*z + 20*x^6 + 42*x^5*y^2 + 87*x^5*y*z - 171*x^5*y + 42*x^5*z^2 - 174*x^5*z + 168*x^5 - 96*x^4*y^3 - 309*x^4*y^2*z + 597*x^4*y^2 - 309*x^4*y*z^2 + 1257*x^4*y*z - 1194*x^4*y - 96*x^4*z^3 + 621*x^4*z^2 - 1233*x^4*z + 772*x^4 + 129*x^3*y^4 + 573*x^3*y^3*z - 1089*x^3*y^3 + 891*x^3*y^2*z^2 - 3558*x^3*y^2*z + 3324*x^3*y^2 + 573*x^3*y*z^3 - 3639*x^3*y*z^2 + 7110*x^3*y*z - 4376*x^3*y + 129*x^3*z^4 - 1164*x^3*z^3 + 3570*x^3*z^2 - 4579*x^3*z + 2108*x^3 - 102*x^2*y^5 - 585*x^2*y^4*z + 1095*x^2*y^4 - 1257*x^2*y^3*z^2 + 4929*x^2*y^3*z - 4530*x^2*y^3 - 1257*x^2*y^2*z^3 + 7830*x^2*y^2*z^2 - 15045*x^2*y^2*z + 9099*x^2*y^2 - 585*x^2*y*z^4 + 5187*x^2*y*z^3 - 15666*x^2*y*z^2 + 19758*x^2*y*z - 8925*x^2*y - 102*x^2*z^5 + 1209*x^2*z^4 - 5097*x^2*z^3 + 10050*x^2*z^2 - 9462*x^2*z + 3439*x^2 + 44*x*y^6 + 312*x*y^5*z - 576*x*y^5 + 867*x*y^4*z^2 - 3342*x*y^4*z + 3024*x*y^4 + 1199*x*y^3*z^3 - 7323*x*y^3*z^2 + 13836*x*y^3*z - 8225*x*y^3 + 867*x*y^2*z^4 - 7542*x*y^2*z^3 + 22410*x*y^2*z^2 - 27771*x*y^2*z + 12303*x*y^2 + 312*x*y*z^5 - 3642*x*y*z^4 + 15138*x*y*z^3 - 29367*x*y*z^2 + 27135*x*y*z - 9653*x*y + 44*x*z^6 - 660*x*z^5 + 3588*x*z^4 - 9670*x*z^3 + 13968*x*z^2 - 10378*x*z + 3120*x - 8*y^7 - 68*y^6*z + 124*y^6 - 234*y^5*z^2 + 888*y^5*z - 792*y^5 - 419*y^4*z^3 + 2511*y^4*z^2 - 4668*y^4*z + 2730*y^4 - 419*y^3*z^4 + 3571*y^3*z^3 - 10434*y^3*z^2 + 12707*y^3*z - 5524*y^3 - 234*y^2*z^5 + 2685*y^2*z^4 - 10989*y^2*z^3 + 20946*y^2*z^2 - 18975*y^2*z + 6604*y^2 - 68*y*z^6 + 1008*y*z^5 - 5412*y*z^4 + 14367*y*z^3 - 20376*y*z^2 + 14815*y*z - 4348*y - 8*z^7 + 148*z^6 - 996*z^5 + 3439*z^4 - 6776*z^3 + 7723*z^2 - 4745*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ 1, -2, -1 ], [ 1, -1, -2 ], [ 1, -1, -1 ] ], "source_shift": [ 2, 2, 2 ], "target_matrix": [ [ 2, 1, 1 ], [ 3, 3, 2 ], [ 1, 1, 1 ] ], "target_shift": [ -1789, -3695, -1219 ], "collision_preimages": [ [ "-11/4", "-1/4", "-1/4" ], [ "18", "11/2", "8" ], [ "17", "15/2", "5" ] ], "collision_image": [ "-3579/2", "-14783/4", "-4877/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0314362250028353 ], "surface_features": [ 3.9726253779522804, 4.945084345254798, 3.5397659031831123, 0.0, 4.14807981746707, 4998 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B248", "pair": 248, "family": "bijection", "mapping": "F(x,y,z)=(2187*x^7 + 10206*x^6*y - 10206*x^6*z - 10206*x^6 + 20412*x^5*y^2 - 40824*x^5*y*z - 40824*x^5*y + 20412*x^5*z^2 + 40824*x^5*z + 20736*x^5 + 22680*x^4*y^3 - 68040*x^4*y^2*z - 68040*x^4*y^2 + 68040*x^4*y*z^2 + 136080*x^4*y*z + 69147*x^4*y - 22680*x^4*z^3 - 68040*x^4*z^2 - 69228*x^4*z - 23976*x^4 + 15120*x^3*y^4 - 60480*x^3*y^3*z - 60480*x^3*y^3 + 90720*x^3*y^2*z^2 + 181440*x^3*y^2*z + 92232*x^3*y^2 - 60480*x^3*y*z^3 - 181440*x^3*y*z^2 - 184680*x^3*y*z - 64017*x^3*y + 15120*x^3*z^4 + 60480*x^3*z^3 + 92448*x^3*z^2 + 64233*x^3*z + 17154*x^3 + 6048*x^2*y^5 - 30240*x^2*y^4*z - 30240*x^2*y^4 + 60480*x^2*y^3*z^2 + 120960*x^2*y^3*z + 61512*x^2*y^3 - 60480*x^2*y^2*z^3 - 181440*x^2*y^2*z^2 - 184752*x^2*y^2*z - 64098*x^2*y^2 + 30240*x^2*y*z^4 + 120960*x^2*y*z^3 + 184968*x^2*y*z^2 + 128628*x^2*y*z + 34401*x^2*y - 6048*x^2*z^5 - 30240*x^2*z^4 - 61728*x^2*z^3 - 64530*x^2*z^2 - 34626*x^2*z - 7642*x^2 + 1344*x*y^6 - 8064*x*y^5*z - 8064*x*y^5 + 20160*x*y^4*z^2 + 40320*x*y^4*z + 20512*x*y^4 - 26880*x*y^3*z^3 - 80640*x*y^3*z^2 - 82144*x*y^3*z - 28524*x*y^3 + 20160*x*y^2*z^4 + 80640*x*y^2*z^3 + 123360*x*y^2*z^2 + 85860*x*y^2*z + 22996*x*y^2 - 8064*x*y*z^5 - 40320*x*y*z^4 - 82336*x*y*z^3 - 86148*x*y*z^2 - 46292*x*y*z - 10238*x*y + 1344*x*z^6 + 8064*x*z^5 + 20608*x*z^4 + 28812*x*z^3 + 23296*x*z^2 + 10347*x*z + 1979*x + 128*y^7 - 896*y^6*z - 896*y^6 + 2688*y^5*z^2 + 5376*y^5*z + 2736*y^5 - 4480*y^4*z^3 - 13440*y^4*z^2 - 13696*y^4*z - 4760*y^4 + 4480*y^3*z^4 + 17920*y^3*z^3 + 27424*y^3*z^2 + 19104*y^3*z + 5124*y^3 - 2688*y^2*z^5 - 13440*y^2*z^4 - 27456*y^2*z^3 - 28752*y^2*z^2 - 15472*y^2*z - 3429*y^2 + 896*y*z^6 + 5376*y*z^5 + 13744*y*z^4 + 19232*y*z^3 + 15572*y*z^2 + 6931*y*z + 1330*y - 128*z^7 - 896*z^6 - 2752*z^5 - 4824*z^4 - 5224*z^3 - 3502*z^2 - 1351*z - 1, -6561*x^7 - 30618*x^6*y + 30618*x^6*z + 30618*x^6 - 61236*x^5*y^2 + 122472*x^5*y*z + 122472*x^5*y - 61236*x^5*z^2 - 122472*x^5*z - 62208*x^5 - 68040*x^4*y^3 + 204120*x^4*y^2*z + 204120*x^4*y^2 - 204120*x^4*y*z^2 - 408240*x^4*y*z - 207441*x^4*y + 68040*x^4*z^3 + 204120*x^4*z^2 + 207684*x^4*z + 71847*x^4 - 45360*x^3*y^4 + 181440*x^3*y^3*z + 181440*x^3*y^3 - 272160*x^3*y^2*z^2 - 544320*x^3*y^2*z - 276696*x^3*y^2 + 181440*x^3*y*z^3 + 544320*x^3*y*z^2 + 554040*x^3*y*z + 191835*x^3*y - 45360*x^3*z^4 - 181440*x^3*z^3 - 277344*x^3*z^2 - 192483*x^3*z - 51300*x^3 - 18144*x^2*y^5 + 90720*x^2*y^4*z + 90720*x^2*y^4 - 181440*x^2*y^3*z^2 - 362880*x^2*y^3*z - 184536*x^2*y^3 + 181440*x^2*y^2*z^3 + 544320*x^2*y^2*z^2 + 554256*x^2*y^2*z + 192078*x^2*y^2 - 90720*x^2*y*z^4 - 362880*x^2*y*z^3 - 554904*x^2*y*z^2 - 385452*x^2*y*z - 102879*x^2*y + 18144*x^2*z^5 + 90720*x^2*z^4 + 185184*x^2*z^3 + 193374*x^2*z^2 + 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40824*x^5*y*z - 40824*x^5*y + 20412*x^5*z^2 + 40824*x^5*z + 20736*x^5 + 22680*x^4*y^3 - 68040*x^4*y^2*z - 68040*x^4*y^2 + 68040*x^4*y*z^2 + 136080*x^4*y*z + 69147*x^4*y - 22680*x^4*z^3 - 68040*x^4*z^2 - 69228*x^4*z - 24057*x^4 + 15120*x^3*y^4 - 60480*x^3*y^3*z - 60480*x^3*y^3 + 90720*x^3*y^2*z^2 + 181440*x^3*y^2*z + 92232*x^3*y^2 - 60480*x^3*y*z^3 - 181440*x^3*y*z^2 - 184680*x^3*y*z - 64233*x^3*y + 15120*x^3*z^4 + 60480*x^3*z^3 + 92448*x^3*z^2 + 64449*x^3*z + 17370*x^3 + 6048*x^2*y^5 - 30240*x^2*y^4*z - 30240*x^2*y^4 + 60480*x^2*y^3*z^2 + 120960*x^2*y^3*z + 61512*x^2*y^3 - 60480*x^2*y^2*z^3 - 181440*x^2*y^2*z^2 - 184752*x^2*y^2*z - 64314*x^2*y^2 + 30240*x^2*y*z^4 + 120960*x^2*y*z^3 + 184968*x^2*y*z^2 + 129060*x^2*y*z + 34833*x^2*y - 6048*x^2*z^5 - 30240*x^2*z^4 - 61728*x^2*z^3 - 64746*x^2*z^2 - 35058*x^2*z - 7867*x^2 + 1344*x*y^6 - 8064*x*y^5*z - 8064*x*y^5 + 20160*x*y^4*z^2 + 40320*x*y^4*z + 20512*x*y^4 - 26880*x*y^3*z^3 - 80640*x*y^3*z^2 - 82144*x*y^3*z - 28620*x*y^3 + 20160*x*y^2*z^4 + 80640*x*y^2*z^3 + 123360*x*y^2*z^2 + 86148*x*y^2*z + 23284*x*y^2 - 8064*x*y*z^5 - 40320*x*y*z^4 - 82336*x*y*z^3 - 86436*x*y*z^2 - 46868*x*y*z - 10538*x*y + 1344*x*z^6 + 8064*x*z^5 + 20608*x*z^4 + 28908*x*z^3 + 23584*x*z^2 + 10647*x*z + 2088*x + 128*y^7 - 896*y^6*z - 896*y^6 + 2688*y^5*z^2 + 5376*y^5*z + 2736*y^5 - 4480*y^4*z^3 - 13440*y^4*z^2 - 13696*y^4*z - 4776*y^4 + 4480*y^3*z^4 + 17920*y^3*z^3 + 27424*y^3*z^2 + 19168*y^3*z + 5188*y^3 - 2688*y^2*z^5 - 13440*y^2*z^4 - 27456*y^2*z^3 - 28848*y^2*z^2 - 15664*y^2*z - 3529*y^2 + 896*y*z^6 + 5376*y*z^5 + 13744*y*z^4 + 19296*y*z^3 + 15764*y*z^2 + 7131*y*z + 1403*y - 128*z^7 - 896*z^6 - 2752*z^5 - 4840*z^4 - 5288*z^3 - 3602*z^2 - 1424*z - 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -3, -2, 2 ], [ -1, -1, 1 ], [ -1, -1, 2 ] ], "source_shift": [ 2, 2, 2 ], "target_matrix": [ [ -1, -1, -1 ], [ 3, 2, 2 ], [ -1, -2, -1 ] ], "target_shift": [ 233, -699, 255 ], "inverse_recipe": { "p": 4, "q": 3, "f": "x**4 + x**2", "g": "x**3", "h": "y*z + z", "polynomials": { "f": [ [ [ 4, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "g": [ [ [ 3, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 0, 1 ], 1 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0441111259970057 ], "surface_features": [ 4.832111567978998, 5.744217855417277, 4.480581786829169, 0.0, 4.982086316280405, 5365 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C168", "pair": 168, "family": "counterexample", "mapping": "F(x,y,z)=(-x^7 - 10*x^6*y + 8*x^6*z + 13*x^6 - 42*x^5*y^2 + 69*x^5*y*z + 114*x^5*y - 27*x^5*z^2 - 87*x^5*z - 69*x^5 - 96*x^4*y^3 + 243*x^4*y^2*z + 411*x^4*y^2 - 195*x^4*y*z^2 - 636*x^4*y*z - 513*x^4*y + 50*x^4*z^3 + 240*x^4*z^2 + 378*x^4*z + 197*x^4 - 129*x^3*y^4 + 447*x^3*y^3*z + 780*x^3*y^3 - 552*x^3*y^2*z^2 - 1833*x^3*y^2*z - 1506*x^3*y^2 + 290*x^3*y*z^3 + 1404*x^3*y*z^2 + 2244*x^3*y*z + 1187*x^3*y - 55*x^3*z^4 - 350*x^3*z^3 - 822*x^3*z^2 - 855*x^3*z - 335*x^3 - 102*x^2*y^5 + 453*x^2*y^4*z + 822*x^2*y^4 - 765*x^2*y^3*z^2 - 2604*x^2*y^3*z - 2181*x^2*y^3 + 618*x^2*y^2*z^3 + 3033*x^2*y^2*z^2 + 4926*x^2*y^2*z + 2646*x^2*y^2 - 240*x^2*y*z^4 - 1536*x^2*y*z^3 - 3654*x^2*y*z^2 - 3852*x^2*y*z - 1533*x^2*y + 36*x^2*z^5 + 285*x^2*z^4 + 888*x^2*z^3 + 1383*x^2*z^2 + 1086*x^2*z + 346*x^2 - 44*x*y^6 + 240*x*y^5*z + 456*x*y^5 - 519*x*y^4*z^2 - 1824*x*y^4*z - 1557*x*y^4 + 573*x*y^3*z^3 + 2868*x*y^3*z^2 + 4737*x*y^3*z + 2584*x*y^3 - 342*x*y^2*z^4 - 2211*x*y^2*z^3 - 5334*x*y^2*z^2 - 5700*x*y^2*z - 2304*x*y^2 + 105*x*y*z^5 + 834*x*y*z^4 + 2628*x*y*z^3 + 4143*x*y*z^2 + 3300*x*y*z + 1069*x*y - 13*x*z^6 - 123*x*z^5 - 477*x*z^4 - 989*x*z^3 - 1167*x*z^2 - 749*x*z - 205*x - 8*y^7 + 52*y^6*z + 104*y^6 - 138*y^5*z^2 - 504*y^5*z - 438*y^5 + 195*y^4*z^3 + 1002*y^4*z^2 + 1683*y^4*z + 931*y^4 - 159*y^3*z^4 - 1044*y^3*z^3 - 2556*y^3*z^2 - 2769*y^3*z - 1135*y^3 + 75*y^2*z^5 + 600*y^2*z^4 + 1914*y^2*z^3 + 3054*y^2*z^2 + 2466*y^2*z + 814*y^2 - 19*y*z^6 - 180*y*z^5 - 705*y*z^4 - 1478*y*z^3 - 1767*y*z^2 - 1150*y*z - 319*y + 2*z^7 + 22*z^6 + 102*z^5 + 264*z^4 + 416*z^3 + 403*z^2 + 224*z + 1, -x^7 - 10*x^6*y + 8*x^6*z + 13*x^6 - 42*x^5*y^2 + 69*x^5*y*z + 114*x^5*y - 27*x^5*z^2 - 87*x^5*z - 69*x^5 - 96*x^4*y^3 + 243*x^4*y^2*z + 411*x^4*y^2 - 195*x^4*y*z^2 - 636*x^4*y*z - 513*x^4*y + 50*x^4*z^3 + 240*x^4*z^2 + 378*x^4*z + 196*x^4 - 129*x^3*y^4 + 447*x^3*y^3*z + 780*x^3*y^3 - 552*x^3*y^2*z^2 - 1833*x^3*y^2*z - 1506*x^3*y^2 + 290*x^3*y*z^3 + 1404*x^3*y*z^2 + 2244*x^3*y*z + 1183*x^3*y - 55*x^3*z^4 - 350*x^3*z^3 - 822*x^3*z^2 - 850*x^3*z - 328*x^3 - 102*x^2*y^5 + 453*x^2*y^4*z + 822*x^2*y^4 - 765*x^2*y^3*z^2 - 2604*x^2*y^3*z - 2181*x^2*y^3 + 618*x^2*y^2*z^3 + 3033*x^2*y^2*z^2 + 4926*x^2*y^2*z + 2640*x^2*y^2 - 240*x^2*y*z^4 - 1536*x^2*y*z^3 - 3654*x^2*y*z^2 - 3837*x^2*y*z - 1509*x^2*y + 36*x^2*z^5 + 285*x^2*z^4 + 888*x^2*z^3 + 1374*x^2*z^2 + 1062*x^2*z + 331*x^2 - 44*x*y^6 + 240*x*y^5*z + 456*x*y^5 - 519*x*y^4*z^2 - 1824*x*y^4*z - 1557*x*y^4 + 573*x*y^3*z^3 + 2868*x*y^3*z^2 + 4737*x*y^3*z + 2580*x*y^3 - 342*x*y^2*z^4 - 2211*x*y^2*z^3 - 5334*x*y^2*z^2 - 5685*x*y^2*z - 2277*x*y^2 + 105*x*y*z^5 + 834*x*y*z^4 + 2628*x*y*z^3 + 4125*x*y*z^2 + 3246*x*y*z + 1033*x*y - 13*x*z^6 - 123*x*z^5 - 477*x*z^4 - 982*x*z^3 - 1140*x*z^2 - 716*x*z - 194*x - 8*y^7 + 52*y^6*z + 104*y^6 - 138*y^5*z^2 - 504*y^5*z - 438*y^5 + 195*y^4*z^3 + 1002*y^4*z^2 + 1683*y^4*z + 930*y^4 - 159*y^3*z^4 - 1044*y^3*z^3 - 2556*y^3*z^2 - 2764*y^3*z - 1125*y^3 + 75*y^2*z^5 + 600*y^2*z^4 + 1914*y^2*z^3 + 3045*y^2*z^2 + 2436*y^2*z + 793*y^2 - 19*y*z^6 - 180*y*z^5 - 705*y*z^4 - 1471*y*z^3 - 1737*y*z^2 - 1111*y*z - 305*y + 2*z^7 + 22*z^6 + 102*z^5 + 262*z^4 + 406*z^3 + 385*z^2 + 212*z + 1, -2*x^7 - 20*x^6*y + 16*x^6*z + 23*x^6 - 84*x^5*y^2 + 138*x^5*y*z + 204*x^5*y - 54*x^5*z^2 - 153*x^5*z - 111*x^5 - 192*x^4*y^3 + 486*x^4*y^2*z + 744*x^4*y^2 - 390*x^4*y*z^2 - 1131*x^4*y*z - 834*x^4*y + 100*x^4*z^3 + 420*x^4*z^2 + 606*x^4*z + 299*x^4 - 258*x^3*y^4 + 894*x^3*y^3*z + 1428*x^3*y^3 - 1104*x^3*y^2*z^2 - 3297*x^3*y^2*z - 2475*x^3*y^2 + 580*x^3*y*z^3 + 2484*x^3*y*z^2 + 3636*x^3*y*z + 1820*x^3*y - 110*x^3*z^4 - 610*x^3*z^3 - 1314*x^3*z^2 - 1295*x^3*z - 491*x^3 - 204*x^2*y^5 + 906*x^2*y^4*z + 1521*x^2*y^4 - 1530*x^2*y^3*z^2 - 4737*x^2*y^3*z - 3624*x^2*y^3 + 1236*x^2*y^2*z^3 + 5427*x^2*y^2*z^2 + 8070*x^2*y^2*z + 4101*x^2*y^2 - 480*x^2*y*z^4 - 2706*x^2*y*z^3 - 5904*x^2*y*z^2 - 5895*x^2*y*z - 2268*x^2*y + 72*x^2*z^5 + 495*x^2*z^4 + 1416*x^2*z^3 + 2091*x^2*z^2 + 1590*x^2*z + 497*x^2 - 88*x*y^6 + 480*x*y^5*z + 852*x*y^5 - 1038*x*y^4*z^2 - 3354*x*y^4*z - 2616*x*y^4 + 1146*x*y^3*z^3 + 5190*x*y^3*z^2 + 7848*x*y^3*z + 4050*x*y^3 - 684*x*y^2*z^4 - 3939*x*y^2*z^3 - 8715*x*y^2*z^2 - 8820*x*y^2*z - 3444*x*y^2 + 210*x*y*z^5 + 1464*x*y*z^4 + 4236*x*y*z^3 + 6330*x*y*z^2 + 4878*x*y*z + 1550*x*y - 26*x*z^6 - 213*x*z^5 - 759*x*z^4 - 1493*x*z^3 - 1707*x*z^2 - 1075*x*z - 291*x - 16*y^7 + 104*y^6*z + 196*y^6 - 276*y^5*z^2 - 936*y^5*z - 744*y^5 + 390*y^4*z^3 + 1833*y^4*z^2 + 2820*y^4*z + 1476*y^4 - 318*y^3*z^4 - 1881*y^3*z^3 - 4224*y^3*z^2 - 4334*y^3*z - 1716*y^3 + 150*y^2*z^5 + 1065*y^2*z^4 + 3120*y^2*z^3 + 4719*y^2*z^2 + 3684*y^2*z + 1193*y^2 - 38*y*z^6 - 315*y*z^5 - 1134*y*z^4 - 2255*y*z^3 - 2610*y*z^2 - 1667*y*z - 458*y + 4*z^7 + 38*z^6 + 162*z^5 + 398*z^4 + 608*z^3 + 578*z^2 + 318*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -1, -1, 1 ], [ -1, -2, 1 ], [ -1, -1, 2 ] ], "source_shift": [ 1, 1, 1 ], "target_matrix": [ [ 1, 1, 1 ], [ 1, 1, 2 ], [ 2, 1, 1 ] ], "target_shift": [ -53, -51, -75 ], "collision_preimages": [ [ "3/4", "0", "-1/4" ], [ "3", "5/2", "11/2" ], [ "12", "-5/2", "15/2" ] ], "collision_image": [ "-213/4", "-205/4", "-151/2" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0845484290002787 ], "surface_features": [ 3.1009714011787106, 3.94546858513182, 2.790988475088816, 0.0, 3.1932695877896013, 4758 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B047", "pair": 47, "family": "bijection", "mapping": "F(x,y,z)=(2*x^7 + 28*x^6*y + 14*x^6*z + 34*x^6 + 168*x^5*y^2 + 168*x^5*y*z + 408*x^5*y + 42*x^5*z^2 + 204*x^5*z + 254*x^5 + 560*x^4*y^3 + 840*x^4*y^2*z + 2040*x^4*y^2 + 420*x^4*y*z^2 + 2040*x^4*y*z + 2532*x^4*y + 70*x^4*z^3 + 510*x^4*z^2 + 1268*x^4*z + 1077*x^4 + 1120*x^3*y^4 + 2240*x^3*y^3*z + 5440*x^3*y^3 + 1680*x^3*y^2*z^2 + 8160*x^3*y^2*z + 10096*x^3*y^2 + 560*x^3*y*z^3 + 4080*x^3*y*z^2 + 10112*x^3*y*z + 8526*x^3*y + 70*x^3*z^4 + 680*x^3*z^3 + 2532*x^3*z^2 + 4286*x^3*z + 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4077*x*z^2 + 4211*x*z + 1800*x + 128*y^7 + 448*y^6*z + 1088*y^6 + 672*y^5*z^2 + 3264*y^5*z + 4000*y^5 + 560*y^4*z^3 + 4080*y^4*z^2 + 10016*y^4*z + 8264*y^4 + 280*y^3*z^4 + 2720*y^3*z^3 + 10032*y^3*z^2 + 16620*y^3*z + 10364*y^3 + 84*y^2*z^5 + 1020*y^2*z^4 + 5024*y^2*z^3 + 12534*y^2*z^2 + 15740*y^2*z + 7882*y^2 + 14*y*z^6 + 204*y*z^5 + 1258*y*z^4 + 4201*y*z^3 + 7967*y*z^2 + 8064*y*z + 3361*y + z^7 + 17*z^6 + 126*z^5 + 528*z^4 + 1344*z^3 + 2063*z^2 + 1744*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ 1, 2, 1 ], [ 1, 1, 1 ], [ 3, 2, 2 ] ], "source_shift": [ 2, 2, 2 ], "target_matrix": [ [ 2, 1, 1 ], [ 1, 2, 1 ], [ 1, 1, 1 ] ], "target_shift": [ -1237, -649, -619 ], "inverse_recipe": { "p": 4, "q": 3, "f": "x**4 + x**3 + x**2", "g": "x**3 + x**2", "h": "y*z + z**2", "polynomials": { "f": [ [ [ 4, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ], [ [ 3, 0, 0 ], 1 ] ], "g": [ [ [ 3, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 0, 2 ], 1 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.1048573719999695 ], "surface_features": [ 3.7215314801058037, 4.702016725569145, 3.3096301674258988, 0.0, 3.8753565016946263, 4919 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B235", "pair": 235, "family": "bijection", "mapping": "F(x,y,z)=(4*x^7 - 28*x^6*y - 28*x^6*z + 56*x^6 + 84*x^5*y^2 + 168*x^5*y*z - 328*x^5*y + 84*x^5*z^2 - 328*x^5*z + 296*x^5 - 140*x^4*y^3 - 420*x^4*y^2*z + 800*x^4*y^2 - 420*x^4*y*z^2 + 1600*x^4*y*z - 1420*x^4*y - 140*x^4*z^3 + 800*x^4*z^2 - 1416*x^4*z + 824*x^4 + 140*x^3*y^4 + 560*x^3*y^3*z - 1040*x^3*y^3 + 840*x^3*y^2*z^2 - 3120*x^3*y^2*z + 2724*x^3*y^2 + 560*x^3*y*z^3 - 3120*x^3*y*z^2 + 5432*x^3*y*z - 3156*x^3*y + 140*x^3*z^4 - 1040*x^3*z^3 + 2708*x^3*z^2 - 3108*x^3*z + 1354*x^3 - 84*x^2*y^5 - 420*x^2*y^4*z + 760*x^2*y^4 - 840*x^2*y^3*z^2 + 3040*x^2*y^3*z - 2612*x^2*y^3 - 840*x^2*y^2*z^3 + 4560*x^2*y^2*z^2 - 7812*x^2*y^2*z + 4528*x^2*y^2 - 420*x^2*y*z^4 + 3040*x^2*y*z^3 - 7788*x^2*y*z^2 + 8920*x^2*y*z - 3938*x^2*y - 84*x^2*z^5 + 760*x^2*z^4 - 2588*x^2*z^3 + 4392*x^2*z^2 - 3778*x^2*z + 1340*x^2 + 28*x*y^6 + 168*x*y^5*z - 296*x*y^5 + 420*x*y^4*z^2 - 1480*x*y^4*z + 1252*x*y^4 + 560*x*y^3*z^3 - 2960*x*y^3*z^2 + 4992*x*y^3*z - 2884*x*y^3 + 420*x*y^2*z^4 - 2960*x*y^2*z^3 + 7464*x*y^2*z^2 - 8524*x*y^2*z + 3790*x*y^2 + 168*x*y*z^5 - 1480*x*y*z^4 + 4960*x*y*z^3 - 8396*x*y*z^2 + 7300*x*y*z - 2668*x*y + 28*x*z^6 - 296*x*z^5 + 1236*x*z^4 - 2756*x*z^3 + 3510*x*z^2 - 2456*x*z + 747*x - 4*y^7 - 28*y^6*z + 48*y^6 - 84*y^5*z^2 + 288*y^5*z - 240*y^5 - 140*y^4*z^3 + 720*y^4*z^2 - 1196*y^4*z + 688*y^4 - 140*y^3*z^4 + 960*y^3*z^3 - 2384*y^3*z^2 + 2712*y^3*z - 1210*y^3 - 84*y^2*z^5 + 720*y^2*z^4 - 2376*y^2*z^3 + 4008*y^2*z^2 - 3506*y^2*z + 1304*y^2 - 28*y*z^6 + 288*y*z^5 - 1184*y*z^4 + 2632*y*z^3 - 3382*y*z^2 + 2424*y*z - 767*y - 4*z^7 + 48*z^6 - 236*z^5 + 648*z^4 - 1086*z^3 + 1124*z^2 - 673*z + 1, -4*x^7 + 28*x^6*y + 28*x^6*z - 56*x^6 - 84*x^5*y^2 - 168*x^5*y*z + 328*x^5*y - 84*x^5*z^2 + 328*x^5*z - 296*x^5 + 140*x^4*y^3 + 420*x^4*y^2*z - 800*x^4*y^2 + 420*x^4*y*z^2 - 1600*x^4*y*z + 1420*x^4*y + 140*x^4*z^3 - 800*x^4*z^2 + 1416*x^4*z - 824*x^4 - 140*x^3*y^4 - 560*x^3*y^3*z + 1040*x^3*y^3 - 840*x^3*y^2*z^2 + 3120*x^3*y^2*z - 2724*x^3*y^2 - 560*x^3*y*z^3 + 3120*x^3*y*z^2 - 5432*x^3*y*z + 3156*x^3*y - 140*x^3*z^4 + 1040*x^3*z^3 - 2708*x^3*z^2 + 3108*x^3*z - 1354*x^3 + 84*x^2*y^5 + 420*x^2*y^4*z - 760*x^2*y^4 + 840*x^2*y^3*z^2 - 3040*x^2*y^3*z + 2612*x^2*y^3 + 840*x^2*y^2*z^3 - 4560*x^2*y^2*z^2 + 7812*x^2*y^2*z - 4528*x^2*y^2 + 420*x^2*y*z^4 - 3040*x^2*y*z^3 + 7788*x^2*y*z^2 - 8920*x^2*y*z + 3938*x^2*y + 84*x^2*z^5 - 760*x^2*z^4 + 2588*x^2*z^3 - 4392*x^2*z^2 + 3778*x^2*z - 1339*x^2 - 28*x*y^6 - 168*x*y^5*z + 296*x*y^5 - 420*x*y^4*z^2 + 1480*x*y^4*z - 1252*x*y^4 - 560*x*y^3*z^3 + 2960*x*y^3*z^2 - 4992*x*y^3*z + 2884*x*y^3 - 420*x*y^2*z^4 + 2960*x*y^2*z^3 - 7464*x*y^2*z^2 + 8524*x*y^2*z - 3790*x*y^2 - 168*x*y*z^5 + 1480*x*y*z^4 - 4960*x*y*z^3 + 8396*x*y*z^2 - 7300*x*y*z + 2666*x*y - 28*x*z^6 + 296*x*z^5 - 1236*x*z^4 + 2756*x*z^3 - 3510*x*z^2 + 2454*x*z - 742*x + 4*y^7 + 28*y^6*z - 48*y^6 + 84*y^5*z^2 - 288*y^5*z + 240*y^5 + 140*y^4*z^3 - 720*y^4*z^2 + 1196*y^4*z - 688*y^4 + 140*y^3*z^4 - 960*y^3*z^3 + 2384*y^3*z^2 - 2712*y^3*z + 1210*y^3 + 84*y^2*z^5 - 720*y^2*z^4 + 2376*y^2*z^3 - 4008*y^2*z^2 + 3506*y^2*z - 1303*y^2 + 28*y*z^6 - 288*y*z^5 + 1184*y*z^4 - 2632*y*z^3 + 3382*y*z^2 - 2422*y*z + 763*y + 4*z^7 - 48*z^6 + 236*z^5 - 648*z^4 + 1086*z^3 - 1123*z^2 + 669*z - 1, -8*x^7 + 56*x^6*y + 56*x^6*z - 112*x^6 - 168*x^5*y^2 - 336*x^5*y*z + 656*x^5*y - 168*x^5*z^2 + 656*x^5*z - 592*x^5 + 280*x^4*y^3 + 840*x^4*y^2*z - 1600*x^4*y^2 + 840*x^4*y*z^2 - 3200*x^4*y*z + 2840*x^4*y + 280*x^4*z^3 - 1600*x^4*z^2 + 2832*x^4*z - 1648*x^4 - 280*x^3*y^4 - 1120*x^3*y^3*z + 2080*x^3*y^3 - 1680*x^3*y^2*z^2 + 6240*x^3*y^2*z - 5448*x^3*y^2 - 1120*x^3*y*z^3 + 6240*x^3*y*z^2 - 10864*x^3*y*z + 6312*x^3*y - 280*x^3*z^4 + 2080*x^3*z^3 - 5416*x^3*z^2 + 6216*x^3*z - 2710*x^3 + 168*x^2*y^5 + 840*x^2*y^4*z - 1520*x^2*y^4 + 1680*x^2*y^3*z^2 - 6080*x^2*y^3*z + 5224*x^2*y^3 + 1680*x^2*y^2*z^3 - 9120*x^2*y^2*z^2 + 15624*x^2*y^2*z - 9056*x^2*y^2 + 840*x^2*y*z^4 - 6080*x^2*y*z^3 + 15576*x^2*y*z^2 - 17840*x^2*y*z + 7882*x^2*y + 168*x^2*z^5 - 1520*x^2*z^4 + 5176*x^2*z^3 - 8784*x^2*z^2 + 7562*x^2*z - 2683*x^2 - 56*x*y^6 - 336*x*y^5*z + 592*x*y^5 - 840*x*y^4*z^2 + 2960*x*y^4*z - 2504*x*y^4 - 1120*x*y^3*z^3 + 5920*x*y^3*z^2 - 9984*x*y^3*z + 5768*x*y^3 - 840*x*y^2*z^4 + 5920*x*y^2*z^3 - 14928*x*y^2*z^2 + 17048*x*y^2*z - 7586*x*y^2 - 336*x*y*z^5 + 2960*x*y*z^4 - 9920*x*y*z^3 + 16792*x*y*z^2 - 14612*x*y*z + 5342*x*y - 56*x*z^6 + 592*x*z^5 - 2472*x*z^4 + 5512*x*z^3 - 7026*x*z^2 + 4918*x*z - 1487*x + 8*y^7 + 56*y^6*z - 96*y^6 + 168*y^5*z^2 - 576*y^5*z + 480*y^5 + 280*y^4*z^3 - 1440*y^4*z^2 + 2392*y^4*z - 1376*y^4 + 280*y^3*z^4 - 1920*y^3*z^3 + 4768*y^3*z^2 - 5424*y^3*z + 2422*y^3 + 168*y^2*z^5 - 1440*y^2*z^4 + 4752*y^2*z^3 - 8016*y^2*z^2 + 7018*y^2*z - 2611*y^2 + 56*y*z^6 - 576*y*z^5 + 2368*y*z^4 - 5264*y*z^3 + 6770*y*z^2 - 4854*y*z + 1532*y + 8*z^7 - 96*z^6 + 472*z^5 - 1296*z^4 + 2174*z^3 - 2251*z^2 + 1342*z - 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ 1, -1, -1 ], [ 2, -1, -1 ], [ 1, -2, -1 ] ], "source_shift": [ 1, 2, 1 ], "target_matrix": [ [ 1, 2, 1 ], [ -1, -1, -1 ], [ -2, -1, -1 ] ], "target_shift": [ -180, 176, 353 ], "inverse_recipe": { "p": 2, "q": 3, "f": "x**2 + x", "g": "x**3", "h": "y**2*z + y*z + z**2", "polynomials": { "f": [ [ [ 2, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "g": [ [ [ 3, 0, 0 ], 1 ] ], "h": [ [ [ 0, 2, 1 ], 1 ], [ [ 0, 1, 1 ], 1 ], [ [ 0, 0, 2 ], 1 ] ] }, "h_multiplier": 4 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0443248589981522 ], "surface_features": [ 3.3732088942800016, 4.251394850040104, 3.0755469613925306, 0.0, 3.485050544437248, 4827 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C129", "pair": 129, "family": "counterexample", "mapping": "F(x,y,z)=(8*x^7 - 76*x^6*y + 68*x^6*z - 104*x^6 + 306*x^5*y^2 - 552*x^5*y*z + 852*x^5*y + 246*x^5*z^2 - 768*x^5*z + 546*x^5 - 677*x^4*y^3 + 1845*x^4*y^2*z - 2886*x^4*y^2 - 1659*x^4*y*z^2 + 5232*x^4*y*z - 3717*x^4*y + 491*x^4*z^3 - 2346*x^4*z^2 + 3399*x^4*z - 1550*x^4 + 889*x^3*y^4 - 3251*x^3*y^3*z + 5171*x^3*y^3 + 4419*x^3*y^2*z^2 - 14136*x^3*y^2*z + 10083*x^3*y^2 - 2641*x^3*y*z^3 + 12759*x^3*y*z^2 - 18483*x^3*y*z + 8388*x^3*y + 584*x^3*z^4 - 3794*x^3*z^3 + 8400*x^3*z^2 - 7809*x^3*z + 2593*x^3 - 693*x^2*y^5 + 3186*x^2*y^4*z - 5166*x^2*y^4 - 5814*x^2*y^3*z^2 + 18924*x^2*y^3*z - 13614*x^2*y^3 + 5256*x^2*y^2*z^3 - 25776*x^2*y^2*z^2 + 37527*x^2*y^2*z - 16989*x^2*y^2 - 2349*x^2*y*z^4 + 15444*x^2*y*z^3 - 34212*x^2*y*z^2 + 31620*x^2*y*z - 10503*x^2*y + 414*x^2*z^5 - 3426*x^2*z^4 + 10299*x^2*z^3 - 14631*x^2*z^2 + 9945*x^2*z - 2539*x^2 + 297*x*y^6 - 1647*x*y^5*z + 2727*x*y^5 + 3780*x*y^4*z^2 - 12546*x*y^4*z + 9141*x*y^4 - 4590*x*y^3*z^3 + 22914*x*y^3*z^2 - 33687*x*y^3*z + 15261*x*y^3 + 3105*x*y^2*z^4 - 20736*x*y^2*z^3 + 46215*x*y^2*z^2 - 42600*x*y^2*z + 14148*x*y^2 - 1107*x*y*z^5 + 9279*x*y*z^4 - 27933*x*y*z^3 + 39417*x*y*z^2 - 26745*x*y*z + 6890*x*y + 162*x*z^6 - 1638*x*z^5 + 6264*x*z^4 - 12078*x*z^3 + 12597*x*z^2 - 6617*x*z + 1344*x - 54*y^7 + 351*y^6*z - 594*y^6 - 972*y^5*z^2 + 3294*y^5*z - 2439*y^5 + 1485*y^4*z^3 - 7560*y^4*z^2 + 11268*y^4*z - 5129*y^4 - 1350*y^3*z^4 + 9180*y^3*z^3 - 20682*y^3*z^2 + 19093*y^3*z - 6338*y^3 + 729*y^2*z^5 - 6210*y^2*z^4 + 18828*y^2*z^3 - 26505*y^2*z^2 + 17940*y^2*z - 4660*y^2 - 216*y*z^6 + 2214*y*z^5 - 8487*y*z^4 + 16247*y*z^3 - 16866*y*z^2 + 8933*y*z - 1845*y + 27*z^7 - 324*z^6 + 1512*z^5 - 3706*z^4 + 5264*z^3 - 4273*z^2 + 1792*z - 1, 8*x^7 - 76*x^6*y + 68*x^6*z - 104*x^6 + 306*x^5*y^2 - 552*x^5*y*z + 852*x^5*y + 246*x^5*z^2 - 768*x^5*z + 546*x^5 - 677*x^4*y^3 + 1845*x^4*y^2*z - 2886*x^4*y^2 - 1659*x^4*y*z^2 + 5232*x^4*y*z - 3717*x^4*y + 491*x^4*z^3 - 2346*x^4*z^2 + 3399*x^4*z - 1542*x^4 + 889*x^3*y^4 - 3251*x^3*y^3*z + 5171*x^3*y^3 + 4419*x^3*y^2*z^2 - 14136*x^3*y^2*z + 10083*x^3*y^2 - 2641*x^3*y*z^3 + 12759*x^3*y*z^2 - 18483*x^3*y*z + 8336*x^3*y + 584*x^3*z^4 - 3794*x^3*z^3 + 8400*x^3*z^2 - 7765*x^3*z + 2561*x^3 - 693*x^2*y^5 + 3186*x^2*y^4*z - 5166*x^2*y^4 - 5814*x^2*y^3*z^2 + 18924*x^2*y^3*z - 13614*x^2*y^3 + 5256*x^2*y^2*z^3 - 25776*x^2*y^2*z^2 + 37527*x^2*y^2*z - 16863*x^2*y^2 - 2349*x^2*y*z^4 + 15444*x^2*y*z^3 - 34212*x^2*y*z^2 + 31404*x^2*y*z - 10359*x^2*y + 414*x^2*z^5 - 3426*x^2*z^4 + 10299*x^2*z^3 - 14541*x^2*z^2 + 9813*x^2*z - 2485*x^2 + 297*x*y^6 - 1647*x*y^5*z + 2727*x*y^5 + 3780*x*y^4*z^2 - 12546*x*y^4*z + 9141*x*y^4 - 4590*x*y^3*z^3 + 22914*x*y^3*z^2 - 33687*x*y^3*z + 15126*x*y^3 + 3105*x*y^2*z^4 - 20736*x*y^2*z^3 + 46215*x*y^2*z^2 - 42249*x*y^2*z + 13932*x*y^2 - 1107*x*y*z^5 + 9279*x*y*z^4 - 27933*x*y*z^3 + 39120*x*y*z^2 - 26349*x*y*z + 6731*x*y + 162*x*z^6 - 1638*x*z^5 + 6264*x*z^4 - 11997*x*z^3 + 12417*x*z^2 - 6464*x*z + 1314*x - 54*y^7 + 351*y^6*z - 594*y^6 - 972*y^5*z^2 + 3294*y^5*z - 2439*y^5 + 1485*y^4*z^3 - 7560*y^4*z^2 + 11268*y^4*z - 5075*y^4 - 1350*y^3*z^4 + 9180*y^3*z^3 - 20682*y^3*z^2 + 18904*y^3*z - 6230*y^3 + 729*y^2*z^5 - 6210*y^2*z^4 + 18828*y^2*z^3 - 26262*y^2*z^2 + 17643*y^2*z - 4543*y^2 - 216*y*z^6 + 2214*y*z^5 - 8487*y*z^4 + 16112*y*z^3 - 16596*y*z^2 + 8708*y*z - 1801*y + 27*z^7 - 324*z^6 + 1512*z^5 - 3679*z^4 + 5183*z^3 - 4165*z^2 + 1749*z - 1, -24*x^7 + 228*x^6*y - 204*x^6*z + 288*x^6 - 918*x^5*y^2 + 1656*x^5*y*z - 2352*x^5*y - 738*x^5*z^2 + 2124*x^5*z - 1446*x^5 + 2031*x^4*y^3 - 5535*x^4*y^2*z + 7944*x^4*y^2 + 4977*x^4*y*z^2 - 14424*x^4*y*z + 9807*x^4*y - 1473*x^4*z^3 + 6480*x^4*z^2 - 8985*x^4*z + 3984*x^4 - 2667*x^3*y^4 + 9753*x^3*y^3*z - 14196*x^3*y^3 - 13257*x^3*y^2*z^2 + 38859*x^3*y^2*z - 26505*x^3*y^2 + 7923*x^3*y*z^3 - 35130*x^3*y*z^2 + 48669*x^3*y*z - 21483*x^3*y - 1752*x^3*z^4 + 10467*x^3*z^3 - 22164*x^3*z^2 + 20028*x^3*z - 6525*x^3 + 2079*x^2*y^5 - 9558*x^2*y^4*z + 14148*x^2*y^4 + 17442*x^2*y^3*z^2 - 51885*x^2*y^3*z + 35658*x^2*y^3 - 15768*x^2*y^2*z^3 + 70767*x^2*y^2*z^2 - 98442*x^2*y^2*z + 43350*x^2*y^2 + 7047*x^2*y*z^4 - 42471*x^2*y*z^3 + 89910*x^2*y*z^2 - 80793*x^2*y*z + 26343*x^2*y - 1242*x^2*z^5 + 9441*x^2*z^4 - 27126*x^2*z^3 + 37443*x^2*z^2 - 24960*x^2*z + 6300*x^2 - 891*x*y^6 + 4941*x*y^5*z - 7452*x*y^5 - 11340*x*y^4*z^2 + 34317*x*y^4*z - 23859*x*y^4 + 13770*x*y^3*z^3 - 62748*x*y^3*z^2 + 88047*x*y^3*z - 38790*x*y^3 - 9315*x*y^2*z^4 + 56862*x*y^2*z^3 - 120987*x*y^2*z^2 + 108423*x*y^2*z - 35364*x*y^2 + 3321*x*y*z^5 - 25488*x*y*z^4 + 73269*x*y*z^3 - 100476*x*y*z^2 + 66897*x*y*z - 17037*x*y - 486*x*z^6 + 4509*x*z^5 - 16470*x*z^4 + 30843*x*z^3 - 31533*x*z^2 + 16362*x*z - 3320*x + 162*y^7 - 1053*y^6*z + 1620*y^6 + 2916*y^5*z^2 - 8991*y^5*z + 6345*y^5 - 4455*y^4*z^3 + 20655*y^4*z^2 - 29349*y^4*z + 12984*y^4 + 4050*y^3*z^4 - 25110*y^3*z^3 + 53946*y^3*z^2 - 48396*y^3*z + 15786*y^3 - 2187*y^2*z^5 + 17010*y^2*z^4 - 49194*y^2*z^3 + 67284*y^2*z^2 - 44715*y^2*z + 11481*y^2 + 648*y*z^6 - 6075*y*z^5 + 22221*y*z^4 - 41316*y*z^3 + 42072*y*z^2 - 22008*y*z + 4541*y - 81*z^7 + 891*z^6 - 3969*z^5 + 9444*z^4 - 13143*z^3 + 10527*z^2 - 4409*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -2, 3, -3 ], [ -1, 1, -1 ], [ -1, 2, -1 ] ], "source_shift": [ 1, 2, 1 ], "target_matrix": [ [ -1, -1, -1 ], [ -1, -1, -2 ], [ 3, 2, 3 ] ], "target_shift": [ 290, 285, -723 ], "collision_preimages": [ [ "5", "3/4", "-9/4" ], [ "21/2", "9", "2" ], [ "-1/2", "6", "7" ] ], "collision_image": [ "1161/4", "1141/4", "-2895/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0008442349972029 ], "surface_features": [ 4.11766342020905, 5.08273870807203, 3.7937903846908188, 0.0, 4.263764426143052, 5099 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B028", "pair": 28, "family": "bijection", "mapping": "F(x,y,z)=(x^7 + 14*x^6*y - 7*x^6*z - 15*x^6 + 84*x^5*y^2 - 84*x^5*y*z - 180*x^5*y + 21*x^5*z^2 + 90*x^5*z + 98*x^5 + 280*x^4*y^3 - 420*x^4*y^2*z - 900*x^4*y^2 + 210*x^4*y*z^2 + 900*x^4*y*z + 979*x^4*y - 35*x^4*z^3 - 225*x^4*z^2 - 490*x^4*z - 359*x^4 + 560*x^3*y^4 - 1120*x^3*y^3*z - 2400*x^3*y^3 + 840*x^3*y^2*z^2 + 3600*x^3*y^2*z + 3912*x^3*y^2 - 280*x^3*y*z^3 - 1800*x^3*y*z^2 - 3916*x^3*y*z - 2861*x^3*y + 35*x^3*z^4 + 300*x^3*z^3 + 980*x^3*z^2 + 1435*x^3*z + 793*x^3 + 672*x^2*y^5 - 1680*x^2*y^4*z - 3600*x^2*y^4 + 1680*x^2*y^3*z^2 + 7200*x^2*y^3*z + 7816*x^2*y^3 - 840*x^2*y^2*z^3 - 5400*x^2*y^2*z^2 - 11736*x^2*y^2*z - 8550*x^2*y^2 + 210*x^2*y*z^4 + 1800*x^2*y*z^3 + 5874*x^2*y*z^2 + 8577*x^2*y*z + 4713*x^2*y - 21*x^2*z^5 - 225*x^2*z^4 - 980*x^2*z^3 - 2151*x^2*z^2 - 2372*x^2*z - 1058*x^2 + 448*x*y^6 - 1344*x*y^5*z - 2880*x*y^5 + 1680*x*y^4*z^2 + 7200*x*y^4*z + 7808*x*y^4 - 1120*x*y^3*z^3 - 7200*x*y^3*z^2 - 15632*x*y^3*z - 11356*x*y^3 + 420*x*y^2*z^4 + 3600*x*y^2*z^3 + 11736*x*y^2*z^2 + 17088*x*y^2*z + 9336*x*y^2 - 84*x*y*z^5 - 900*x*y*z^4 - 3916*x*y*z^3 - 8571*x*y*z^2 - 9398*x*y*z - 4143*x*y + 7*x*z^6 + 90*x*z^5 + 490*x*z^4 + 1433*x*z^3 + 2365*x*z^2 + 2098*x*z + 781*x + 128*y^7 - 448*y^6*z - 960*y^6 + 672*y^5*z^2 + 2880*y^5*z + 3120*y^5 - 560*y^4*z^3 - 3600*y^4*z^2 - 7808*y^4*z - 5656*y^4 + 280*y^3*z^4 + 2400*y^3*z^3 + 7816*y^3*z^2 + 11348*y^3*z + 6164*y^3 - 84*y^2*z^5 - 900*y^2*z^4 - 3912*y^2*z^3 - 8538*y^2*z^2 - 9308*y^2*z - 4057*y^2 + 14*y*z^6 + 180*y*z^5 + 979*y*z^4 + 2855*y*z^3 + 4685*y*z^2 + 4108*y*z + 1506*y - z^7 - 15*z^6 - 98*z^5 - 358*z^4 - 786*z^3 - 1040*z^2 - 769*z - 1, 2*x^7 + 28*x^6*y - 14*x^6*z - 30*x^6 + 168*x^5*y^2 - 168*x^5*y*z - 360*x^5*y + 42*x^5*z^2 + 180*x^5*z + 196*x^5 + 560*x^4*y^3 - 840*x^4*y^2*z - 1800*x^4*y^2 + 420*x^4*y*z^2 + 1800*x^4*y*z + 1958*x^4*y - 70*x^4*z^3 - 450*x^4*z^2 - 980*x^4*z - 724*x^4 + 1120*x^3*y^4 - 2240*x^3*y^3*z - 4800*x^3*y^3 + 1680*x^3*y^2*z^2 + 7200*x^3*y^2*z + 7824*x^3*y^2 - 560*x^3*y*z^3 - 3600*x^3*y*z^2 - 7832*x^3*y*z - 5770*x^3*y + 70*x^3*z^4 + 600*x^3*z^3 + 1960*x^3*z^2 + 2894*x^3*z + 1633*x^3 + 1344*x^2*y^5 - 3360*x^2*y^4*z - 7200*x^2*y^4 + 3360*x^2*y^3*z^2 + 14400*x^2*y^3*z + 15632*x^2*y^3 - 1680*x^2*y^2*z^3 - 10800*x^2*y^2*z^2 - 23472*x^2*y^2*z - 17244*x^2*y^2 + 420*x^2*y*z^4 + 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[ 245, 589, -529 ], "inverse_recipe": { "p": 3, "q": 4, "f": "x**3 + x**2 + x", "g": "x**4 + x", "h": "y*z", "polynomials": { "f": [ [ [ 3, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "g": [ [ [ 4, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 0.9874611330014886 ], "surface_features": [ 3.6748458376323323, 4.537365681617888, 3.269512944217916, 0.0, 3.8032554196275967, 4900 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B150", "pair": 150, "family": "bijection", "mapping": "F(x,y,z)=(-512*x^7 - 2048*x^6*y + 2304*x^6*z + 3840*x^6 - 3488*x^5*y^2 + 7808*x^5*y*z + 13184*x^5*y - 4352*x^5*z^2 - 14848*x^5*z - 12224*x^5 - 3280*x^4*y^3 + 10960*x^4*y^2*z + 18736*x^4*y^2 - 12160*x^4*y*z^2 - 41984*x^4*y*z - 35040*x^4*y + 4480*x^4*z^3 + 23424*x^4*z^2 + 39520*x^4*z + 21408*x^4 - 1840*x^3*y^4 + 8160*x^3*y^3*z + 14112*x^3*y^3 - 13520*x^3*y^2*z^2 - 47200*x^3*y^2*z - 39904*x^3*y^2 + 9920*x^3*y*z^3 + 52416*x^3*y*z^2 + 89536*x^3*y*z + 49216*x^3*y - 2720*x^3*z^4 - 19328*x^3*z^3 - 50016*x^3*z^2 - 55616*x^3*z - 22384*x^3 - 616*x^2*y^5 + 3400*x^2*y^4*z + 5944*x^2*y^4 - 7480*x^2*y^3*z^2 - 26384*x^2*y^3*z - 22576*x^2*y^3 + 8200*x^2*y^2*z^3 + 43752*x^2*y^2*z^2 + 75600*x^2*y^2*z + 42128*x^2*y^2 - 4480*x^2*y*z^4 - 32128*x^2*y*z^3 - 84048*x^2*y*z^2 - 94688*x^2*y*z - 38776*x^2*y + 976*x^2*z^5 + 8816*x^2*z^4 + 31024*x^2*z^3 + 52976*x^2*z^2 + 43984*x^2*z + 14096*x^2 - 114*x*y^6 + 752*x*y^5*z + 1328*x*y^5 - 2060*x*y^4*z^2 - 7336*x*y^4*z - 6348*x*y^4 + 3000*x*y^3*z^3 + 16152*x*y^3*z^2 + 28208*x*y^3*z + 15920*x*y^3 - 2450*x*y^2*z^4 - 17720*x*y^2*z^3 - 46824*x*y^2*z^2 - 53392*x*y^2*z - 22231*x*y^2 + 1064*x*y*z^5 + 9688*x*y*z^4 + 34416*x*y*z^3 + 59440*x*y*z^2 + 50160*x*y*z + 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19328*x^3*z^3 - 50016*x^3*z^2 - 55616*x^3*z - 22384*x^3 - 616*x^2*y^5 + 3400*x^2*y^4*z + 5944*x^2*y^4 - 7480*x^2*y^3*z^2 - 26384*x^2*y^3*z - 22576*x^2*y^3 + 8200*x^2*y^2*z^3 + 43752*x^2*y^2*z^2 + 75600*x^2*y^2*z + 42128*x^2*y^2 - 4480*x^2*y*z^4 - 32128*x^2*y*z^3 - 84048*x^2*y*z^2 - 94688*x^2*y*z - 38776*x^2*y + 976*x^2*z^5 + 8816*x^2*z^4 + 31024*x^2*z^3 + 52976*x^2*z^2 + 43984*x^2*z + 14096*x^2 - 114*x*y^6 + 752*x*y^5*z + 1328*x*y^5 - 2060*x*y^4*z^2 - 7336*x*y^4*z - 6348*x*y^4 + 3000*x*y^3*z^3 + 16152*x*y^3*z^2 + 28208*x*y^3*z + 15920*x*y^3 - 2450*x*y^2*z^4 - 17720*x*y^2*z^3 - 46824*x*y^2*z^2 - 53392*x*y^2*z - 22231*x*y^2 + 1064*x*y*z^5 + 9688*x*y*z^4 + 34416*x*y*z^3 + 59440*x*y*z^2 + 50160*x*y*z + 16432*x*y - 192*x*z^6 - 2112*x*z^5 - 9452*x*z^4 - 21968*x*z^3 - 28176*x*z^2 - 18784*x*z - 5008*x - 9*y^7 + 69*y^6*z + 123*y^6 - 226*y^5*z^2 - 812*y^5*z - 710*y^5 + 410*y^4*z^3 + 2226*y^4*z^2 + 3926*y^4*z + 2242*y^4 - 445*y^3*z^4 - 3244*y^3*z^3 - 8652*y^3*z^2 - 9976*y^3*z - 4221*y^3 + 289*y^2*z^5 + 2651*y^2*z^4 + 9500*y^2*z^3 + 16580*y^2*z^2 + 14215*y^2*z + 4759*y^2 - 104*y*z^6 - 1152*y*z^5 - 5198*y*z^4 - 12200*y*z^3 - 15896*y*z^2 - 10832*y*z - 2968*y + 16*z^7 + 208*z^6 + 1134*z^5 + 3354*z^4 + 5904*z^3 + 6144*z^2 + 3439*z + 1, -1024*x^7 - 4096*x^6*y + 4608*x^6*z + 7680*x^6 - 6976*x^5*y^2 + 15616*x^5*y*z + 26368*x^5*y - 8704*x^5*z^2 - 29696*x^5*z - 24512*x^5 - 6560*x^4*y^3 + 21920*x^4*y^2*z + 37472*x^4*y^2 - 24320*x^4*y*z^2 - 83968*x^4*y*z - 70240*x^4*y + 8960*x^4*z^3 + 46848*x^4*z^2 + 79200*x^4*z + 43168*x^4 - 3680*x^3*y^4 + 16320*x^3*y^3*z + 28224*x^3*y^3 - 27040*x^3*y^2*z^2 - 94400*x^3*y^2*z - 79968*x^3*y^2 + 19840*x^3*y*z^3 + 104832*x^3*y*z^2 + 179392*x^3*y*z + 99136*x^3*y - 5440*x^3*z^4 - 38656*x^3*z^3 - 100192*x^3*z^2 - 111936*x^3*z - 45536*x^3 - 1232*x^2*y^5 + 6800*x^2*y^4*z + 11888*x^2*y^4 - 14960*x^2*y^3*z^2 - 52768*x^2*y^3*z - 45232*x^2*y^3 + 16400*x^2*y^2*z^3 + 87504*x^2*y^2*z^2 + 151440*x^2*y^2*z + 84784*x^2*y^2 - 8960*x^2*y*z^4 - 64256*x^2*y*z^3 - 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"determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -2, -1, 1 ], [ -2, -2, 3 ], [ -1, -1, 2 ] ], "source_shift": [ 2, 2, 1 ], "target_matrix": [ [ 1, 2, 1 ], [ 1, 1, 1 ], [ 2, 1, 1 ] ], "target_shift": [ -787, -783, -1665 ], "inverse_recipe": { "p": 1, "q": 5, "f": "x", "g": "x**5 + x**4 + x", "h": "y**3 + y**2*z", "polynomials": { "f": [ [ [ 1, 0, 0 ], 1 ] ], "g": [ [ [ 5, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ], [ [ 4, 0, 0 ], 1 ] ], "h": [ [ [ 0, 2, 1 ], 1 ], [ [ 0, 3, 0 ], 1 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0554768519978097 ], "surface_features": [ 4.353817477563669, 5.2797399285847, 4.008216801589691, 0.0, 4.477936817824286, 5173 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C135", "pair": 135, "family": "counterexample", "mapping": "F(x,y,z)=(-48*x^7 - 312*x^6*y - 528*x^6*z - 612*x^6 - 864*x^5*y^2 - 2916*x^5*y*z - 3408*x^5*y - 2448*x^5*z^2 - 5724*x^5*z - 3288*x^5 - 1320*x^4*y^3 - 6660*x^4*y^2*z - 7860*x^4*y^2 - 11142*x^4*y*z^2 - 26340*x^4*y*z - 15276*x^4*y - 6180*x^4*z^3 - 21969*x^4*z^2 - 25536*x^4*z - 9766*x^4 - 1200*x^3*y^4 - 8040*x^3*y^3*z - 9600*x^3*y^3 - 20088*x^3*y^2*z^2 - 48120*x^3*y^2*z - 28224*x^3*y^2 - 22179*x^3*y*z^3 - 80034*x^3*y*z^2 - 94200*x^3*y*z - 36392*x^3*y - 9129*x^3*z^4 - 44172*x^3*z^3 - 78312*x^3*z^2 - 60683*x^3*z - 17458*x^3 - 648*x^2*y^5 - 5400*x^2*y^4*z - 6540*x^2*y^4 - 17892*x^2*y^3*z^2 - 43560*x^2*y^3*z - 25896*x^2*y^3 - 29457*x^2*y^2*z^3 - 108288*x^2*y^2*z^2 - 129384*x^2*y^2*z - 50580*x^2*y^2 - 24093*x^2*y*z^4 - 119088*x^2*y*z^3 - 214695*x^2*y*z^2 - 168525*x^2*y*z - 48960*x^2*y - 7830*x^2*z^5 - 48888*x^2*z^4 - 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17208*x^3*y^2*z - 10584*x^3*y^2 - 7393*x^3*y*z^3 - 28560*x^3*y*z^2 - 35232*x^3*y*z - 14008*x^3*y - 3043*x^3*z^4 - 15728*x^3*z^3 - 29211*x^3*z^2 - 23309*x^3*z - 6810*x^3 - 216*x^2*y^5 - 1800*x^2*y^4*z - 2340*x^2*y^4 - 5964*x^2*y^3*z^2 - 15552*x^2*y^3*z - 9696*x^2*y^3 - 9819*x^2*y^2*z^3 - 38574*x^2*y^2*z^2 - 48312*x^2*y^2*z - 19452*x^2*y^2 - 8031*x^2*y*z^4 - 42321*x^2*y*z^3 - 79944*x^2*y*z^2 - 64671*x^2*y*z - 19092*x^2*y - 2610*x^2*z^5 - 17331*x^2*z^4 - 43941*x^2*z^3 - 53586*x^2*z^2 - 31746*x^2*z - 7396*x^2 - 64*x*y^6 - 636*x*y^5*z - 840*x*y^5 - 2616*x*y^4*z^2 - 6948*x*y^4*z - 4404*x*y^4 - 5699*x*y^3*z^3 - 22872*x*y^3*z^2 - 29184*x*y^3*z - 11928*x*y^3 - 6933*x*y^2*z^4 - 37458*x*y^2*z^3 - 72255*x*y^2*z^2 - 59415*x*y^2*z - 17754*x*y^2 - 4464*x*y*z^5 - 30522*x*y*z^4 - 79230*x*y*z^3 - 98358*x*y*z^2 - 59031*x*y*z - 13880*x*y - 1188*x*z^6 - 9900*x*z^5 - 32472*x*z^4 - 54126*x*z^3 - 48939*x*z^2 - 23068*x*z - 4495*x - 8*y^7 - 92*y^6*z - 124*y^6 - 450*y^5*z^2 - 1224*y^5*z - 792*y^5 - 1213*y^4*z^3 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3240*y^4*z^2 + 3942*y^4*z - 1811*y^4 + 945*y^3*z^4 - 4320*y^3*z^3 + 7938*y^3*z^2 - 7406*y^3*z + 2868*y^3 + 567*y^2*z^5 - 3240*y^2*z^4 + 7992*y^2*z^3 - 11352*y^2*z^2 + 8928*y^2*z - 3057*y^2 + 189*y*z^6 - 1296*y*z^5 + 4023*y*z^4 - 7730*y*z^3 + 9252*y*z^2 - 6519*y*z + 2163*y + 27*z^7 - 216*z^6 + 810*z^5 - 1973*z^4 + 3192*z^3 - 3489*z^2 + 2458*z - 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ 2, -1, -1 ], [ 2, -2, -3 ], [ 1, -1, -2 ] ], "source_shift": [ 1, 1, 2 ], "target_matrix": [ [ -1, -1, -1 ], [ 1, 1, 2 ], [ -3, -2, -2 ] ], "target_shift": [ 248, -240, 750 ], "inverse_recipe": { "p": 4, "q": 3, "f": "x**4 + x**3 + x", "g": "x**3 + x", "h": "y*z + y + 2*z", "polynomials": { "f": [ [ [ 4, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ], [ [ 3, 0, 0 ], 1 ] ], "g": [ [ [ 3, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 1, 0 ], 1 ], [ [ 0, 0, 1 ], 2 ] ] }, "h_multiplier": 9 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0469845619991247 ], "surface_features": [ 4.117327834980339, 5.09638158957146, 3.767044494243667, 0.0, 4.271910737036982, 5105 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B230", "pair": 230, "family": "bijection", "mapping": "F(x,y,z)=(128*x^7 + 896*x^6*y - 1344*x^6*z - 960*x^6 + 2688*x^5*y^2 - 8064*x^5*y*z - 5760*x^5*y + 6048*x^5*z^2 + 8640*x^5*z + 3120*x^5 + 4480*x^4*y^3 - 20160*x^4*y^2*z - 14400*x^4*y^2 + 30240*x^4*y*z^2 + 43200*x^4*y*z + 15600*x^4*y - 15120*x^4*z^3 - 32400*x^4*z^2 - 23408*x^4*z - 5720*x^4 + 4480*x^3*y^4 - 26880*x^3*y^3*z - 19200*x^3*y^3 + 60480*x^3*y^2*z^2 + 86400*x^3*y^2*z + 31200*x^3*y^2 - 60480*x^3*y*z^3 - 129600*x^3*y*z^2 - 93632*x^3*y*z - 22872*x^3*y + 22680*x^3*z^4 + 64800*x^3*z^3 + 70248*x^3*z^2 + 34340*x^3*z + 6400*x^3 + 2688*x^2*y^5 - 20160*x^2*y^4*z - 14400*x^2*y^4 + 60480*x^2*y^3*z^2 + 86400*x^2*y^3*z + 31200*x^2*y^3 - 90720*x^2*y^2*z^3 - 194400*x^2*y^2*z^2 - 140448*x^2*y^2*z - 34296*x^2*y^2 + 68040*x^2*y*z^4 + 194400*x^2*y*z^3 + 210744*x^2*y*z^2 + 102984*x^2*y*z + 19176*x^2*y - 20412*x^2*z^5 - 72900*x^2*z^4 - 105408*x^2*z^3 - 77310*x^2*z^2 - 28812*x^2*z - 4380*x^2 + 896*x*y^6 - 8064*x*y^5*z - 5760*x*y^5 + 30240*x*y^4*z^2 + 43200*x*y^4*z + 15600*x*y^4 - 60480*x*y^3*z^3 - 129600*x*y^3*z^2 - 93632*x*y^3*z - 22856*x*y^3 + 68040*x*y^2*z^4 + 194400*x*y^2*z^3 + 210744*x*y^2*z^2 + 102948*x*y^2*z + 19152*x*y^2 - 40824*x*y*z^5 - 145800*x*y*z^4 - 210816*x*y*z^3 - 154566*x*y*z^2 - 57552*x*y*z - 8733*x*y + 10206*x*z^6 + 43740*x*z^5 + 79083*x*z^4 + 77355*x*z^3 + 43236*x*z^2 + 13132*x*z + 1705*x + 128*y^7 - 1344*y^6*z - 960*y^6 + 6048*y^5*z^2 + 8640*y^5*z + 3120*y^5 - 15120*y^4*z^3 - 32400*y^4*z^2 - 23408*y^4*z - 5712*y^4 + 22680*y^3*z^4 + 64800*y^3*z^3 + 70248*y^3*z^2 + 34304*y^3*z + 6376*y^3 - 20412*y^2*z^5 - 72900*y^2*z^4 - 105408*y^2*z^3 - 77256*y^2*z^2 - 28740*y^2*z - 4353*y^2 + 10206*y*z^6 + 43740*y*z^5 + 79083*y*z^4 + 77328*y*z^3 + 43182*y*z^2 + 13091*y*z + 1695*y - 2187*z^7 - 10935*z^6 - 23733*z^5 - 29025*z^4 - 21627*z^3 - 9842*z^2 - 2552*z - 1, 384*x^7 + 2688*x^6*y - 4032*x^6*z - 2880*x^6 + 8064*x^5*y^2 - 24192*x^5*y*z - 17280*x^5*y + 18144*x^5*z^2 + 25920*x^5*z + 9360*x^5 + 13440*x^4*y^3 - 60480*x^4*y^2*z - 43200*x^4*y^2 + 90720*x^4*y*z^2 + 129600*x^4*y*z + 46800*x^4*y - 45360*x^4*z^3 - 97200*x^4*z^2 - 70224*x^4*z - 17160*x^4 + 13440*x^3*y^4 - 80640*x^3*y^3*z - 57600*x^3*y^3 + 181440*x^3*y^2*z^2 + 259200*x^3*y^2*z + 93600*x^3*y^2 - 181440*x^3*y*z^3 - 388800*x^3*y*z^2 - 280896*x^3*y*z - 68616*x^3*y + 68040*x^3*z^4 + 194400*x^3*z^3 + 210744*x^3*z^2 + 103020*x^3*z + 19192*x^3 + 8064*x^2*y^5 - 60480*x^2*y^4*z - 43200*x^2*y^4 + 181440*x^2*y^3*z^2 + 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206544*x^2*y*z - 38760*x^2*y + 40824*x^2*z^5 + 145800*x^2*z^4 + 210816*x^2*z^3 + 155052*x^2*z^2 + 58236*x^2*z + 8976*x^2 - 1792*x*y^6 + 16128*x*y^5*z + 11520*x*y^5 - 60480*x*y^4*z^2 - 86400*x*y^4*z - 31200*x*y^4 + 120960*x*y^3*z^3 + 259200*x*y^3*z^2 + 187264*x*y^3*z + 45840*x*y^3 - 136080*x*y^2*z^4 - 388800*x*y^2*z^3 - 421488*x*y^2*z^2 - 206472*x*y^2*z - 38712*x*y^2 + 81648*x*y*z^5 + 291600*x*y*z^4 + 421632*x*y*z^3 + 309996*x*y*z^2 + 116328*x*y*z + 17898*x*y - 20412*x*z^6 - 87480*x*z^5 - 158166*x*z^4 - 155142*x*z^3 - 87390*x*z^2 - 26912*x*z - 3567*x - 256*y^7 + 2688*y^6*z + 1920*y^6 - 12096*y^5*z^2 - 17280*y^5*z - 6240*y^5 + 30240*y^4*z^3 + 64800*y^4*z^2 + 46816*y^4*z + 11456*y^4 - 45360*y^3*z^4 - 129600*y^3*z^3 - 140496*y^3*z^2 - 68800*y^3*z - 12888*y^3 + 40824*y^2*z^5 + 145800*y^2*z^4 + 210816*y^2*z^3 + 154944*y^2*z^2 + 58092*y^2*z + 8922*y^2 - 20412*y*z^6 - 87480*y*z^5 - 158166*y*z^4 - 155088*y*z^3 - 87282*y*z^2 - 26830*y*z - 3545*y + 4374*z^7 + 21870*z^6 + 47466*z^5 + 58212*z^4 + 43713*z^3 + 20170*z^2 + 5335*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -2, -2, 3 ], [ -1, -1, 2 ], [ -2, -1, 1 ] ], "source_shift": [ 2, 2, 1 ], "target_matrix": [ [ -1, -1, -1 ], [ -3, -2, -3 ], [ 2, 1, 1 ] ], "target_shift": [ 295, 875, -633 ], "inverse_recipe": { "p": 3, "q": 4, "f": "x**3 + x", "g": "x**4 + x**3 + x", "h": "y*z + y", "polynomials": { "f": [ [ [ 3, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "g": [ [ [ 4, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ], [ [ 3, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 1, 0 ], 1 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0060442649992183 ], "surface_features": [ 4.956084443274992, 5.801024823537285, 4.65667304588485, 0.0, 5.065275922349004, 5418 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B088", "pair": 88, "family": "bijection", "mapping": "F(x,y,z)=(-6*x^7 - 40*x^6*y - 40*x^6*z - 84*x^6 - 114*x^5*y^2 - 228*x^5*y*z - 478*x^5*y - 114*x^5*z^2 - 478*x^5*z - 984*x^5 - 180*x^4*y^3 - 540*x^4*y^2*z - 1130*x^4*y^2 - 540*x^4*y*z^2 - 2260*x^4*y*z - 3970*x^4*y - 180*x^4*z^3 - 1130*x^4*z^2 - 3970*x^4*z - 4866*x^4 - 170*x^3*y^4 - 680*x^3*y^3*z - 1420*x^3*y^3 - 1020*x^3*y^2*z^2 - 4260*x^3*y^2*z - 6580*x^3*y^2 - 680*x^3*y*z^3 - 4260*x^3*y*z^2 - 13160*x^3*y*z - 14742*x^3*y - 170*x^3*z^4 - 1420*x^3*z^3 - 6580*x^3*z^2 - 14742*x^3*z - 11856*x^3 - 96*x^2*y^5 - 480*x^2*y^4*z - 1000*x^2*y^4 - 960*x^2*y^3*z^2 - 4000*x^2*y^3*z - 5580*x^2*y^3 - 960*x^2*y^2*z^3 - 6000*x^2*y^2*z^2 - 16740*x^2*y^2*z - 17190*x^2*y^2 - 480*x^2*y*z^4 - 4000*x^2*y*z^3 - 16740*x^2*y*z^2 - 34380*x^2*y*z - 26222*x^2*y - 96*x^2*z^5 - 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28*y*z^6 - 336*y*z^5 - 1980*y*z^4 - 6896*y*z^3 - 14052*y*z^2 - 15332*y*z - 6887*y - 4*z^7 - 56*z^6 - 396*z^5 - 1724*z^4 - 4684*z^3 - 7664*z^2 - 6883*z - 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ 1, 1, 1 ], [ 2, 1, 1 ], [ 3, 3, 2 ] ], "source_shift": [ 2, 2, 2 ], "target_matrix": [ [ -1, -1, -1 ], [ 1, 2, 1 ], [ -1, -1, -2 ] ], "target_shift": [ 2817, -2821, 2605 ], "inverse_recipe": { "p": 1, "q": 6, "f": "x", "g": "x**6 + x**5 + x**3", "h": "y**5 + y**3 + y*z", "polynomials": { "f": [ [ [ 1, 0, 0 ], 1 ] ], "g": [ [ [ 6, 0, 0 ], 1 ], [ [ 3, 0, 0 ], 1 ], [ [ 5, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 5, 0 ], 1 ], [ [ 0, 3, 0 ], 1 ] ] }, "h_multiplier": 2 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, 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104*z^6 + 438*z^5 - 931*z^4 + 1135*z^3 - 814*z^2 + 319*z - 1, 16*x^7 - 176*x^6*y + 104*x^6*z - 152*x^6 + 816*x^5*y^2 - 972*x^5*y*z + 1452*x^5*y + 288*x^5*z^2 - 864*x^5*z + 564*x^5 - 2060*x^4*y^3 + 3714*x^4*y^2*z - 5706*x^4*y^2 - 2220*x^4*y*z^2 + 6840*x^4*y*z - 4500*x^4*y + 440*x^4*z^3 - 2040*x^4*z^2 + 2694*x^4*z - 1114*x^4 + 3043*x^3*y^4 - 7393*x^3*y^3*z + 11785*x^3*y^3 + 6696*x^3*y^2*z^2 - 21360*x^3*y^2*z + 14199*x^3*y^2 - 2680*x^3*y*z^3 + 12840*x^3*y*z^2 - 17121*x^3*y*z + 7073*x^3*y + 400*x^3*z^4 - 2560*x^3*z^3 + 5136*x^3*z^2 - 4272*x^3*z + 1284*x^3 - 2610*x^2*y^5 + 8031*x^2*y^4*z - 13458*x^2*y^4 - 9819*x^2*y^3*z^2 + 32814*x^2*y^3*z - 22119*x^2*y^3 + 5964*x^2*y^2*z^3 - 29844*x^2*y^2*z^2 + 40308*x^2*y^2*z - 16653*x^2*y^2 - 1800*x^2*y*z^4 + 12000*x^2*y*z^3 - 24363*x^2*y*z^2 + 20262*x^2*y*z - 6039*x^2*y + 216*x^2*z^5 - 1800*x^2*z^4 + 4884*x^2*z^3 - 6132*x^2*z^2 + 3693*x^2*z - 871*x^2 + 1188*x*y^6 - 4464*x*y^5*z + 8028*x*y^5 + 6933*x*y^4*z^2 - 24720*x*y^4*z + 16983*x*y^4 - 5699*x*y^3*z^3 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1760*x^4 - 6086*x^3*y^4 + 14786*x^3*y^3*z - 21803*x^3*y^3 - 13392*x^3*y^2*z^2 + 39579*x^3*y^2*z - 23781*x^3*y^2 + 5360*x^3*y*z^3 - 23832*x^3*y*z^2 + 28776*x^3*y*z - 11140*x^3*y - 800*x^3*z^4 + 4760*x^3*z^3 - 8664*x^3*z^2 + 6756*x^3*z - 1956*x^3 + 5220*x^2*y^5 - 16062*x^2*y^4*z + 25002*x^2*y^4 + 19638*x^2*y^3*z^2 - 61017*x^2*y^3*z + 37020*x^2*y^3 - 11928*x^2*y^2*z^3 + 55557*x^2*y^2*z^2 - 67674*x^2*y^2*z + 26160*x^2*y^2 + 3600*x^2*y*z^4 - 22368*x^2*y*z^3 + 41040*x^2*y*z^2 - 31950*x^2*y*z + 9174*x^2*y - 432*x^2*z^5 + 3360*x^2*z^4 - 8256*x^2*z^3 + 9708*x^2*z^2 - 5628*x^2*z + 1310*x^2 - 2376*x*y^6 + 8928*x*y^5*z - 15012*x*y^5 - 13866*x*y^4*z^2 + 46224*x*y^4*z - 28434*x*y^4 + 11398*x*y^3*z^3 - 56625*x*y^3*z^2 + 69834*x*y^3*z - 26989*x*y^3 - 5232*x*y^2*z^4 + 34497*x*y^2*z^3 - 64005*x*y^2*z^2 + 49797*x*y^2*z - 14181*x*y^2 + 1272*x*y*z^5 - 10452*x*y*z^4 + 25944*x*y*z^3 - 30480*x*y*z^2 + 17526*x*y*z - 4017*x*y - 128*x*z^6 + 1260*x*z^5 - 3924*x*z^4 + 6188*x*z^3 - 5388*x*z^2 + 2503*x*z - 485*x + 432*y^7 - 1944*y^6*z + 3672*y^6 + 3708*y^5*z^2 - 13716*y^5*z + 8604*y^5 - 3890*y^4*z^3 + 21222*y^4*z^2 - 26634*y^4*z + 10310*y^4 + 2426*y^3*z^4 - 17411*y^3*z^3 + 32814*y^3*z^2 - 25553*y^3*z + 7220*y^3 - 900*y^2*z^5 + 7989*y^2*z^4 - 20112*y^2*z^3 + 23637*y^2*z^2 - 13482*y^2*z + 3042*y^2 + 184*y*z^6 - 1944*y*z^5 + 6132*y*z^4 - 9670*y*z^3 + 8352*y*z^2 - 3819*y*z + 722*y - 16*z^7 + 196*z^6 - 744*z^5 + 1476*z^4 - 1716*z^3 + 1193*z^2 - 458*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -1, 2, -1 ], [ -2, 3, -2 ], [ -2, 1, -1 ] ], "source_shift": [ 1, 1, 1 ], "target_matrix": [ [ -1, -1, -1 ], [ -1, -1, -2 ], [ 2, 1, 1 ] ], "target_shift": [ 53, 51, -75 ], "collision_preimages": [ [ "5/4", "-1", "-9/4" ], [ "-8", "5/2", "13" ], [ "-3", "-9/2", "-4" ] ], "collision_image": [ "213/4", "205/4", "-151/2" ] }, "probe_target": "No", "jev_answers": [ "No" ], 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13350*y^2*z - 5098*y^2 + 44*y*z^6 + 720*y*z^5 + 4332*y*z^4 + 12664*y*z^3 + 19476*y*z^2 + 15208*y*z + 4765*y - 8*z^7 - 160*z^6 - 1188*z^5 - 4447*z^4 - 9332*z^3 - 11182*z^2 - 7159*z - 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -1, -1, 1 ], [ -1, -1, 2 ], [ -2, -1, 1 ] ], "source_shift": [ 2, 2, 2 ], "target_matrix": [ [ -1, -1, -2 ], [ 1, 1, 1 ], [ -1, -2, -1 ] ], "target_shift": [ 1183, -1219, 1905 ], "collision_preimages": [ [ "1/4", "7/4", "0" ], [ "-11/2", "4", "-5/2" ], [ "-15/2", "13", "5/2" ] ], "collision_image": [ "4733/4", "-4877/4", "7621/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0607545279999613 ], "surface_features": [ 3.58699559203061, 4.466704544030668, 3.0909630765957314, 0.0, 3.7249194658773637, 4851 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B238", "pair": 238, "family": "bijection", "mapping": "F(x,y,z)=(10*x^7 - 70*x^6*y - 70*x^6*z + 100*x^6 + 210*x^5*y^2 + 420*x^5*y*z - 600*x^5*y + 210*x^5*z^2 - 620*x^5*z + 460*x^5 - 350*x^4*y^3 - 1050*x^4*y^2*z + 1500*x^4*y^2 - 1050*x^4*y*z^2 + 3100*x^4*y*z - 2290*x^4*y - 350*x^4*z^3 + 1600*x^4*z^2 - 2430*x^4*z + 1240*x^4 + 350*x^3*y^4 + 1400*x^3*y^3*z - 2000*x^3*y^3 + 2100*x^3*y^2*z^2 - 6200*x^3*y^2*z + 4560*x^3*y^2 + 1400*x^3*y*z^3 - 6400*x^3*y*z^2 + 9680*x^3*y*z - 4900*x^3*y + 350*x^3*z^4 - 2200*x^3*z^3 + 5130*x^3*z^2 - 5340*x^3*z + 2118*x^3 - 210*x^2*y^5 - 1050*x^2*y^4*z + 1500*x^2*y^4 - 2100*x^2*y^3*z^2 + 6200*x^2*y^3*z - 4540*x^2*y^3 - 2100*x^2*y^2*z^3 + 9600*x^2*y^2*z^2 - 14460*x^2*y^2*z + 7260*x^2*y^2 - 1050*x^2*y*z^4 + 6600*x^2*y*z^3 - 15330*x^2*y*z^2 + 15820*x^2*y*z - 6204*x^2*y - 210*x^2*z^5 + 1700*x^2*z^4 - 5410*x^2*z^3 + 8600*x^2*z^2 - 6944*x^2*z + 2293*x^2 + 70*x*y^6 + 420*x*y^5*z - 600*x*y^5 + 1050*x*y^4*z^2 - 3100*x*y^4*z + 2260*x*y^4 + 1400*x*y^3*z^3 - 6400*x*y^3*z^2 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1000*x^3*y^3 + 1050*x^3*y^2*z^2 - 3100*x^3*y^2*z + 2280*x^3*y^2 + 700*x^3*y*z^3 - 3200*x^3*y*z^2 + 4840*x^3*y*z - 2450*x^3*y + 175*x^3*z^4 - 1100*x^3*z^3 + 2565*x^3*z^2 - 2670*x^3*z + 1058*x^3 - 105*x^2*y^5 - 525*x^2*y^4*z + 750*x^2*y^4 - 1050*x^2*y^3*z^2 + 3100*x^2*y^3*z - 2270*x^2*y^3 - 1050*x^2*y^2*z^3 + 4800*x^2*y^2*z^2 - 7230*x^2*y^2*z + 3630*x^2*y^2 - 525*x^2*y*z^4 + 3300*x^2*y*z^3 - 7665*x^2*y*z^2 + 7910*x^2*y*z - 3099*x^2*y - 105*x^2*z^5 + 850*x^2*z^4 - 2705*x^2*z^3 + 4300*x^2*z^2 - 3469*x^2*z + 1143*x^2 + 35*x*y^6 + 210*x*y^5*z - 300*x*y^5 + 525*x*y^4*z^2 - 1550*x*y^4*z + 1130*x*y^4 + 700*x*y^3*z^3 - 3200*x*y^3*z^2 + 4800*x*y^3*z - 2390*x*y^3 + 525*x*y^2*z^4 - 3300*x*y^2*z^3 + 7635*x*y^2*z^2 - 7810*x*y^2*z + 3024*x*y^2 + 210*x*y*z^5 - 1700*x*y*z^4 + 5390*x*y*z^3 - 8490*x*y*z^2 + 6758*x*y*z - 2196*x*y + 35*x*z^6 - 350*x*z^5 + 1425*x*z^4 - 3070*x*z^3 + 3769*x*z^2 - 2526*x*z + 735*x - 5*y^7 - 35*y^6*z + 50*y^6 - 105*y^5*z^2 + 310*y^5*z - 225*y^5 - 175*y^4*z^3 + 800*y^4*z^2 - 1195*y^4*z + 590*y^4 - 175*y^3*z^4 + 1100*y^3*z^3 - 2535*y^3*z^2 + 2570*y^3*z - 983*y^3 - 105*y^2*z^5 + 850*y^2*z^4 - 2685*y^2*z^3 + 4190*y^2*z^2 - 3289*y^2*z + 1053*y^2 - 35*y*z^6 + 350*y*z^5 - 1420*y*z^4 + 3030*y*z^3 - 3659*y*z^2 + 2406*y*z - 682*y - 5*z^7 + 60*z^6 - 300*z^5 + 820*z^4 - 1353*z^3 + 1373*z^2 - 803*z + 1, 5*x^7 - 35*x^6*y - 35*x^6*z + 50*x^6 + 105*x^5*y^2 + 210*x^5*y*z - 300*x^5*y + 105*x^5*z^2 - 310*x^5*z + 230*x^5 - 175*x^4*y^3 - 525*x^4*y^2*z + 750*x^4*y^2 - 525*x^4*y*z^2 + 1550*x^4*y*z - 1145*x^4*y - 175*x^4*z^3 + 800*x^4*z^2 - 1215*x^4*z + 620*x^4 + 175*x^3*y^4 + 700*x^3*y^3*z - 1000*x^3*y^3 + 1050*x^3*y^2*z^2 - 3100*x^3*y^2*z + 2280*x^3*y^2 + 700*x^3*y*z^3 - 3200*x^3*y*z^2 + 4840*x^3*y*z - 2450*x^3*y + 175*x^3*z^4 - 1100*x^3*z^3 + 2565*x^3*z^2 - 2670*x^3*z + 1056*x^3 - 105*x^2*y^5 - 525*x^2*y^4*z + 750*x^2*y^4 - 1050*x^2*y^3*z^2 + 3100*x^2*y^3*z - 2270*x^2*y^3 - 1050*x^2*y^2*z^3 + 4800*x^2*y^2*z^2 - 7230*x^2*y^2*z + 3630*x^2*y^2 - 525*x^2*y*z^4 + 3300*x^2*y*z^3 - 7665*x^2*y*z^2 + 7910*x^2*y*z - 3093*x^2*y - 105*x^2*z^5 + 850*x^2*z^4 - 2705*x^2*z^3 + 4300*x^2*z^2 - 3463*x^2*z + 1135*x^2 + 35*x*y^6 + 210*x*y^5*z - 300*x*y^5 + 525*x*y^4*z^2 - 1550*x*y^4*z + 1130*x*y^4 + 700*x*y^3*z^3 - 3200*x*y^3*z^2 + 4800*x*y^3*z - 2390*x*y^3 + 525*x*y^2*z^4 - 3300*x*y^2*z^3 + 7635*x*y^2*z^2 - 7810*x*y^2*z + 3018*x*y^2 + 210*x*y*z^5 - 1700*x*y*z^4 + 5390*x*y*z^3 - 8490*x*y*z^2 + 6746*x*y*z - 2180*x*y + 35*x*z^6 - 350*x*z^5 + 1425*x*z^4 - 3070*x*z^3 + 3763*x*z^2 - 2510*x*z + 721*x - 5*y^7 - 35*y^6*z + 50*y^6 - 105*y^5*z^2 + 310*y^5*z - 225*y^5 - 175*y^4*z^3 + 800*y^4*z^2 - 1195*y^4*z + 590*y^4 - 175*y^3*z^4 + 1100*y^3*z^3 - 2535*y^3*z^2 + 2570*y^3*z - 981*y^3 - 105*y^2*z^5 + 850*y^2*z^4 - 2685*y^2*z^3 + 4190*y^2*z^2 - 3283*y^2*z + 1045*y^2 - 35*y*z^6 + 350*y*z^5 - 1420*y*z^4 + 3030*y*z^3 - 3653*y*z^2 + 2390*y*z - 670*y - 5*z^7 + 60*z^6 - 300*z^5 + 820*z^4 - 1351*z^3 + 1365*z^2 - 791*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ 1, -1, -1 ], [ 1, -1, -2 ], [ 2, -1, -1 ] ], "source_shift": [ 1, 2, 2 ], "target_matrix": [ [ 2, 1, 1 ], [ 1, 1, 1 ], [ 1, 1, 2 ] ], "target_shift": [ -436, -215, -207 ], "inverse_recipe": { "p": 2, "q": 3, "f": "x**2", "g": "x**3 + x**2", "h": "y**2*z + 2*z", "polynomials": { "f": [ [ [ 2, 0, 0 ], 1 ] ], "g": [ [ [ 3, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "h": [ [ [ 0, 2, 1 ], 1 ], [ [ 0, 0, 1 ], 2 ] ] }, "h_multiplier": 5 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0327982280032302 ], "surface_features": [ 3.365100940953799, 4.229937685907934, 3.0565237240791006, 0.0, 3.4645066449695774, 4854 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B236", "pair": 236, "family": "bijection", "mapping": "F(x,y,z)=(6561*x^7 - 30618*x^6*y + 45927*x^6*z - 33534*x^6 + 61236*x^5*y^2 - 183708*x^5*y*z + 134379*x^5*y + 137781*x^5*z^2 - 201204*x^5*z + 73629*x^5 - 68040*x^4*y^3 + 306180*x^4*y^2*z - 224370*x^4*y^2 - 459270*x^4*y*z^2 + 671895*x^4*y*z - 246240*x^4*y + 229635*x^4*z^3 - 503010*x^4*z^2 + 368145*x^4*z - 89910*x^4 + 45360*x^3*y^4 - 272160*x^3*y^3*z + 199800*x^3*y^3 + 612360*x^3*y^2*z^2 - 897480*x^3*y^2*z + 329400*x^3*y^2 - 612360*x^3*y*z^3 + 1343790*x^3*y*z^2 - 984960*x^3*y*z + 240840*x^3*y + 229635*x^3*z^4 - 670680*x^3*z^3 + 736290*x^3*z^2 - 359640*x^3*z + 65934*x^3 - 18144*x^2*y^5 + 136080*x^2*y^4*z - 100080*x^2*y^4 - 408240*x^2*y^3*z^2 + 599400*x^2*y^3*z - 220320*x^2*y^3 + 612360*x^2*y^2*z^3 - 1346220*x^2*y^2*z^2 + 988200*x^2*y^2*z - 241920*x^2*y^2 - 459270*x^2*y*z^4 + 1343790*x^2*y*z^3 - 1477440*x^2*y*z^2 + 722520*x^2*y*z - 132579*x^2*y + 137781*x^2*z^5 - 503010*x^2*z^4 + 736290*x^2*z^3 - 539460*x^2*z^2 + 197775*x^2*z - 29031*x^2 + 4032*x*y^6 - 36288*x*y^5*z + 26736*x*y^5 + 136080*x*y^4*z^2 - 200160*x*y^4*z + 73680*x*y^4 - 272160*x*y^3*z^3 + 599400*x*y^3*z^2 - 440640*x*y^3*z + 108000*x*y^3 + 306180*x*y^2*z^4 - 897480*x*y^2*z^3 + 988200*x*y^2*z^2 - 483840*x*y^2*z + 88860*x*y^2 - 183708*x*y*z^5 + 671895*x*y*z^4 - 984960*x*y*z^3 + 722520*x*y*z^2 - 265122*x*y*z + 38934*x*y + 45927*x*z^6 - 201204*x*z^5 + 368145*x*z^4 - 359640*x*z^3 + 197748*x*z^2 - 58014*x*z + 7095*x - 384*y^7 + 4032*y^6*z - 2976*y^6 - 18144*y^5*z^2 + 26736*y^5*z - 9856*y^5 + 45360*y^4*z^3 - 100080*y^4*z^2 + 73680*y^4*z - 18080*y^4 - 68040*y^3*z^4 + 199800*y^3*z^3 - 220320*y^3*z^2 + 108000*y^3*z - 19852*y^3 + 61236*y^2*z^5 - 224370*y^2*z^4 + 329400*y^2*z^3 - 241920*y^2*z^2 + 88848*y^2*z - 13053*y^2 - 30618*y*z^6 + 134379*y*z^5 - 246240*y*z^4 + 240840*y*z^3 - 132543*y*z^2 + 38901*y*z - 4754*y + 6561*z^7 - 33534*z^6 + 73629*z^5 - 89910*z^4 + 65907*z^3 - 28983*z^2 + 7072*z - 1, 2187*x^7 - 10206*x^6*y + 15309*x^6*z - 11178*x^6 + 20412*x^5*y^2 - 61236*x^5*y*z + 44793*x^5*y + 45927*x^5*z^2 - 67068*x^5*z + 24867*x^5 - 22680*x^4*y^3 + 102060*x^4*y^2*z - 74790*x^4*y^2 - 153090*x^4*y*z^2 + 223965*x^4*y*z - 83160*x^4*y + 76545*x^4*z^3 - 167670*x^4*z^2 + 124335*x^4*z - 31050*x^4 + 15120*x^3*y^4 - 90720*x^3*y^3*z + 66600*x^3*y^3 + 204120*x^3*y^2*z^2 - 299160*x^3*y^2*z + 111240*x^3*y^2 - 204120*x^3*y*z^3 + 447930*x^3*y*z^2 - 332640*x^3*y*z + 83160*x^3*y + 76545*x^3*z^4 - 223560*x^3*z^3 + 248670*x^3*z^2 - 124200*x^3*z + 23418*x^3 - 6048*x^2*y^5 + 45360*x^2*y^4*z - 33360*x^2*y^4 - 136080*x^2*y^3*z^2 + 199800*x^2*y^3*z - 74400*x^2*y^3 + 204120*x^2*y^2*z^3 - 448740*x^2*y^2*z^2 + 333720*x^2*y^2*z - 83520*x^2*y^2 - 153090*x^2*y*z^4 + 447930*x^2*y*z^3 - 498960*x^2*y*z^2 + 249480*x^2*y*z - 47073*x^2*y + 45927*x^2*z^5 - 167670*x^2*z^4 + 248670*x^2*z^3 - 186300*x^2*z^2 + 70245*x^2*z - 10631*x^2 + 1344*x*y^6 - 12096*x*y^5*z + 8912*x*y^5 + 45360*x*y^4*z^2 - 66720*x*y^4*z + 24880*x*y^4 - 90720*x*y^3*z^3 + 199800*x*y^3*z^2 - 148800*x*y^3*z + 37280*x*y^3 + 102060*x*y^2*z^4 - 299160*x*y^2*z^3 + 333720*x*y^2*z^2 - 167040*x*y^2*z + 31540*x*y^2 - 61236*x*y*z^5 + 223965*x*y*z^4 - 332640*x*y*z^3 + 249480*x*y*z^2 - 94134*x*y*z + 14250*x*y + 15309*x*z^6 - 67068*x*z^5 + 124335*x*z^4 - 124200*x*z^3 + 70236*x*z^2 - 21246*x*z + 2677*x - 128*y^7 + 1344*y^6*z - 992*y^6 - 6048*y^5*z^2 + 8912*y^5*z - 3328*y^5 + 15120*y^4*z^3 - 33360*y^4*z^2 + 24880*y^4*z - 6240*y^4 - 22680*y^3*z^4 + 66600*y^3*z^3 - 74400*y^3*z^2 + 37280*y^3*z - 7044*y^3 + 20412*y^2*z^5 - 74790*y^2*z^4 + 111240*y^2*z^3 - 83520*y^2*z^2 + 31536*y^2*z - 4775*y^2 - 10206*y*z^6 + 44793*y*z^5 - 83160*y*z^4 + 83160*y*z^3 - 47061*y*z^2 + 14239*y*z - 1792*y + 2187*z^7 - 11178*z^6 + 24867*z^5 - 31050*z^4 + 23409*z^3 - 10615*z^2 + 2668*z - 1, -2187*x^7 + 10206*x^6*y - 15309*x^6*z + 11178*x^6 - 20412*x^5*y^2 + 61236*x^5*y*z - 44793*x^5*y - 45927*x^5*z^2 + 67068*x^5*z - 24624*x^5 + 22680*x^4*y^3 - 102060*x^4*y^2*z + 74790*x^4*y^2 + 153090*x^4*y*z^2 - 223965*x^4*y*z + 82350*x^4*y - 76545*x^4*z^3 + 167670*x^4*z^2 - 123120*x^4*z + 30240*x^4 - 15120*x^3*y^4 + 90720*x^3*y^3*z - 66600*x^3*y^3 - 204120*x^3*y^2*z^2 + 299160*x^3*y^2*z - 110160*x^3*y^2 + 204120*x^3*y*z^3 - 447930*x^3*y*z^2 + 329400*x^3*y*z - 81000*x^3*y - 76545*x^3*z^4 + 223560*x^3*z^3 - 246240*x^3*z^2 + 120960*x^3*z - 22338*x^3 + 6048*x^2*y^5 - 45360*x^2*y^4*z + 33360*x^2*y^4 + 136080*x^2*y^3*z^2 - 199800*x^2*y^3*z + 73680*x^2*y^3 - 204120*x^2*y^2*z^3 + 448740*x^2*y^2*z^2 - 330480*x^2*y^2*z + 81360*x^2*y^2 + 153090*x^2*y*z^4 - 447930*x^2*y*z^3 + 494100*x^2*y*z^2 - 243000*x^2*y*z + 44913*x^2*y - 45927*x^2*z^5 + 167670*x^2*z^4 - 246240*x^2*z^3 + 181440*x^2*z^2 - 67005*x^2*z + 9911*x^2 - 1344*x*y^6 + 12096*x*y^5*z - 8912*x*y^5 - 45360*x*y^4*z^2 + 66720*x*y^4*z - 24640*x*y^4 + 90720*x*y^3*z^3 - 199800*x*y^3*z^2 + 147360*x*y^3*z - 36320*x*y^3 - 102060*x*y^2*z^4 + 299160*x*y^2*z^3 - 330480*x*y^2*z^2 + 162720*x*y^2*z - 30100*x*y^2 + 61236*x*y*z^5 - 223965*x*y*z^4 + 329400*x*y*z^3 - 243000*x*y*z^2 + 89814*x*y*z - 13290*x*y - 15309*x*z^6 + 67068*x*z^5 - 123120*x*z^4 + 120960*x*z^3 - 66996*x*z^2 + 19806*x*z - 2435*x + 128*y^7 - 1344*y^6*z + 992*y^6 + 6048*y^5*z^2 - 8912*y^5*z + 3296*y^5 - 15120*y^4*z^3 + 33360*y^4*z^2 - 24640*y^4*z + 6080*y^4 + 22680*y^3*z^4 - 66600*y^3*z^3 + 73680*y^3*z^2 - 36320*y^3*z + 6724*y^3 - 20412*y^2*z^5 + 74790*y^2*z^4 - 110160*y^2*z^3 + 81360*y^2*z^2 - 30096*y^2*z + 4455*y^2 + 10206*y*z^6 - 44793*y*z^5 + 82350*y*z^4 - 81000*y*z^3 + 44901*y*z^2 - 13279*y*z + 1631*y - 2187*z^7 + 11178*z^6 - 24624*z^5 + 30240*z^4 - 22329*z^3 + 9895*z^2 - 2427*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -3, 2, -3 ], [ -2, 1, -1 ], [ -1, 1, -1 ] ], "source_shift": [ 2, 1, 1 ], "target_matrix": [ [ -3, -2, -2 ], [ -1, -2, -1 ], [ 1, 1, 1 ] ], "target_shift": [ 736, 284, -251 ], "inverse_recipe": { "p": 5, "q": 2, "f": "x**5", "g": "x**2 + x", "h": "y*z", "polynomials": { "f": [ [ [ 5, 0, 0 ], 1 ] ], "g": [ [ [ 2, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0525055949983653 ], "surface_features": [ 5.203370948173237, 6.169509852871047, 4.872709718287906, 0.0, 5.356786431491006, 5498 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C212", "pair": 212, "family": "counterexample", "mapping": "F(x,y,z)=(-8*x^7 + 44*x^6*y + 68*x^6*z - 124*x^6 - 102*x^5*y^2 - 312*x^5*y*z + 552*x^5*y - 234*x^5*z^2 + 852*x^5*z - 630*x^5 + 129*x^4*y^3 + 585*x^4*y^2*z - 1011*x^4*y^2 + 867*x^4*y*z^2 - 3078*x^4*y*z + 2283*x^4*y + 419*x^4*z^3 - 2295*x^4*z^2 + 3525*x^4*z - 1492*x^4 - 96*x^3*y^4 - 573*x^3*y^3*z + 975*x^3*y^3 - 1257*x^3*y^2*z^2 + 4392*x^3*y^2*z - 3258*x^3*y^2 - 1199*x^3*y*z^3 + 6459*x^3*y*z^2 - 9891*x^3*y*z + 4231*x^3*y - 419*x^3*z^4 + 3094*x^3*z^3 - 7347*x^3*z^2 + 6453*x^3*z - 1971*x^3 + 42*x^2*y^5 + 309*x^2*y^4*z - 522*x^2*y^4 + 891*x^2*y^3*z^2 - 3093*x^2*y^3*z + 2289*x^2*y^3 + 1257*x^2*y^2*z^3 - 6732*x^2*y^2*z^2 + 10248*x^2*y^2*z - 4422*x^2*y^2 + 867*x^2*y*z^4 - 6369*x^2*y*z^3 + 14958*x^2*y*z^2 - 13230*x^2*y*z + 4104*x^2*y + 234*x^2*z^5 - 2208*x^2*z^4 + 7101*x^2*z^3 - 9660*x^2*z^2 + 6150*x^2*z - 1519*x^2 - 10*x*y^6 - 87*x*y^5*z + 147*x*y^5 - 309*x*y^4*z^2 + 1074*x*y^4*z - 792*x*y^4 - 573*x*y^3*z^3 + 3075*x*y^3*z^2 - 4650*x*y^3*z + 2021*x*y^3 - 585*x*y^2*z^4 + 4308*x*y^2*z^3 - 10014*x*y^2*z^2 + 8904*x*y^2*z - 2796*x*y^2 - 312*x*y*z^5 + 2952*x*y*z^4 - 9363*x*y*z^3 + 12768*x*y*z^2 - 8202*x*y*z + 2059*x*y - 68*x*z^6 + 792*x*z^5 - 3207*x*z^4 + 5951*x*z^3 - 5853*x*z^2 + 3010*x*z - 642*x + y^7 + 10*y^6*z - 17*y^6 + 42*y^5*z^2 - 147*y^5*z + 108*y^5 + 96*y^4*z^3 - 519*y^4*z^2 + 780*y^4*z - 341*y^4 + 129*y^3*z^4 - 957*y^3*z^3 + 2205*y^3*z^2 - 1968*y^3*z + 624*y^3 + 102*y^2*z^5 - 972*y^2*z^4 + 3048*y^2*z^3 - 4161*y^2*z^2 + 2691*y^2*z - 685*y^2 + 44*y*z^6 - 516*y*z^5 + 2061*y*z^4 - 3818*y*z^3 + 3768*y*z^2 - 1958*y*z + 428*y + 8*z^7 - 112*z^6 + 546*z^5 - 1284*z^4 + 1713*z^3 - 1357*z^2 + 611*z - 1, -8*x^7 + 44*x^6*y + 68*x^6*z - 112*x^6 - 102*x^5*y^2 - 312*x^5*y*z + 492*x^5*y - 234*x^5*z^2 + 756*x^5*z - 486*x^5 + 129*x^4*y^3 + 585*x^4*y^2*z - 888*x^4*y^2 + 867*x^4*y*z^2 - 2688*x^4*y*z + 1737*x^4*y + 419*x^4*z^3 - 1992*x^4*z^2 + 2655*x^4*z - 1056*x^4 - 96*x^3*y^4 - 573*x^3*y^3*z + 843*x^3*y^3 - 1257*x^3*y^2*z^2 + 3771*x^3*y^2*z - 2445*x^3*y^2 - 1199*x^3*y*z^3 + 5505*x^3*y*z^2 - 7347*x^3*y*z + 2955*x^3*y - 419*x^3*z^4 + 2617*x^3*z^3 - 5400*x^3*z^2 + 4457*x^3*z - 1309*x^3 + 42*x^2*y^5 + 309*x^2*y^4*z - 444*x^2*y^4 + 891*x^2*y^3*z^2 - 2610*x^2*y^3*z + 1695*x^2*y^3 + 1257*x^2*y^2*z^3 - 5634*x^2*y^2*z^2 + 7512*x^2*y^2*z - 3048*x^2*y^2 + 867*x^2*y*z^4 - 5286*x^2*y*z^3 + 10854*x^2*y*z^2 - 9018*x^2*y*z + 2682*x^2*y + 234*x^2*z^5 - 1818*x^2*z^4 + 5103*x^2*z^3 - 6510*x^2*z^2 + 3960*x^2*z - 952*x^2 - 10*x*y^6 - 87*x*y^5*z + 123*x*y^5 - 309*x*y^4*z^2 + 891*x*y^4*z - 579*x*y^4 - 573*x*y^3*z^3 + 2529*x*y^3*z^2 - 3366*x*y^3*z + 1375*x*y^3 - 585*x*y^2*z^4 + 3513*x*y^2*z^3 - 7179*x*y^2*z^2 + 5991*x*y^2*z - 1797*x*y^2 - 312*x*y*z^5 + 2388*x*y*z^4 - 6651*x*y*z^3 + 8496*x*y*z^2 - 5190*x*y*z + 1261*x*y - 68*x*z^6 + 636*x*z^5 - 2259*x*z^4 + 3918*x*z^3 - 3642*x*z^2 + 1801*x*z - 383*x + y^7 + 10*y^6*z - 14*y^6 + 42*y^5*z^2 - 120*y^5*z + 78*y^5 + 96*y^4*z^3 - 420*y^4*z^2 + 558*y^4*z - 229*y^4 + 129*y^3*z^4 - 768*y^3*z^3 + 1563*y^3*z^2 - 1307*y^3*z + 394*y^3 + 102*y^2*z^5 - 774*y^2*z^4 + 2142*y^2*z^3 - 2733*y^2*z^2 + 1671*y^2*z - 409*y^2 + 44*y*z^6 - 408*y*z^5 + 1437*y*z^4 - 2481*y*z^3 + 2298*y*z^2 - 1139*y*z + 248*y + 8*z^7 - 88*z^6 + 378*z^5 - 826*z^4 + 1025*z^3 - 766*z^2 + 343*z - 1, 8*x^7 - 44*x^6*y - 68*x^6*z + 112*x^6 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2388*x*y*z^4 + 6651*x*y*z^3 - 8532*x*y*z^2 + 5286*x*y*z - 1303*x*y + 68*x*z^6 - 636*x*z^5 + 2259*x*z^4 - 3938*x*z^3 + 3726*x*z^2 - 1876*x*z + 400*x - y^7 - 10*y^6*z + 14*y^6 - 42*y^5*z^2 + 120*y^5*z - 78*y^5 - 96*y^4*z^3 + 420*y^4*z^2 - 558*y^4*z + 230*y^4 - 129*y^3*z^4 + 768*y^3*z^3 - 1563*y^3*z^2 + 1314*y^3*z - 402*y^3 - 102*y^2*z^5 + 774*y^2*z^4 - 2142*y^2*z^3 + 2751*y^2*z^2 - 1713*y^2*z + 427*y^2 - 44*y*z^6 + 408*y*z^5 - 1437*y*z^4 + 2501*y*z^3 - 2370*y*z^2 + 1202*y*z - 262*y - 8*z^7 + 88*z^6 - 378*z^5 + 834*z^4 - 1065*z^3 + 820*z^2 - 367*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ 1, -1, -2 ], [ 2, -1, -1 ], [ 1, -1, -1 ] ], "source_shift": [ 1, 1, 2 ], "target_matrix": [ [ -1, -2, -1 ], [ -1, -1, -2 ], [ 1, 1, 1 ] ], "target_shift": [ 118, 69, -72 ], "collision_preimages": [ [ "5/4", "19/4", "-5/4" ], [ "-7", "-16", "9/2" ], [ "-4", "-15", "13/2" ] ], "collision_image": [ "473/4", "277/4", "-289/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0294860399990284 ], "surface_features": [ 3.3098613274948527, 4.174873528887417, 2.949877704036875, 0.0, 3.4125662167763293, 4798 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C188", "pair": 188, "family": "counterexample", "mapping": "F(x,y,z)=(-2*x^7 + 20*x^6*y - 21*x^6*z + 23*x^6 - 84*x^5*y^2 + 183*x^5*y*z - 198*x^5*y - 93*x^5*z^2 + 207*x^5*z - 111*x^5 + 192*x^4*y^3 - 651*x^4*y^2*z + 696*x^4*y^2 + 687*x^4*y*z^2 - 1509*x^4*y*z + 795*x^4*y - 225*x^4*z^3 + 765*x^4*z^2 - 837*x^4*z + 297*x^4 - 258*x^3*y^4 + 1209*x^3*y^3*z - 1278*x^3*y^3 - 1989*x^3*y^2*z^2 + 4311*x^3*y^2*z - 2229*x^3*y^2 + 1353*x^3*y*z^3 - 4536*x^3*y*z^2 + 4878*x^3*y*z - 1693*x^3*y - 321*x^3*z^4 + 1485*x^3*z^3 - 2487*x^3*z^2 + 1804*x^3*z - 483*x^3 + 204*x^2*y^5 - 1236*x^2*y^4*z + 1293*x^2*y^4 + 2817*x^2*y^3*z^2 - 6027*x^2*y^3*z + 3057*x^2*y^3 - 2991*x^2*y^2*z^3 + 9882*x^2*y^2*z^2 - 10428*x^2*y^2*z + 3534*x^2*y^2 + 1473*x^2*y*z^4 - 6717*x^2*y*z^3 + 11067*x^2*y*z^2 - 7857*x^2*y*z + 2046*x^2*y - 270*x^2*z^5 + 1596*x^2*z^4 - 3636*x^2*z^3 + 4044*x^2*z^2 - 2220*x^2*z + 484*x^2 - 88*x*y^6 + 660*x*y^5*z - 684*x*y^5 - 1950*x*y^4*z^2 + 4122*x*y^4*z - 2052*x*y^4 + 2875*x*y^3*z^3 - 9360*x*y^3*z^2 + 9681*x*y^3*z - 3201*x*y^3 - 2211*x*y^2*z^4 + 9927*x*y^2*z^3 - 16056*x*y^2*z^2 + 11130*x*y^2*z - 2811*x*y^2 + 840*x*y*z^5 - 4896*x*y*z^4 + 10989*x*y*z^3 - 11976*x*y*z^2 + 6402*x*y*z - 1349*x*y - 124*x*z^6 + 900*x*z^5 - 2613*x*z^4 + 3960*x*z^3 - 3342*x*z^2 + 1498*x*z - 281*x + 16*y^7 - 144*y^6*z + 148*y^6 + 528*y^5*z^2 - 1104*y^5*z + 540*y^5 - 1012*y^4*z^3 + 3249*y^4*z^2 - 3294*y^4*z + 1063*y^4 + 1083*y^3*z^4 - 4783*y^3*z^3 + 7578*y^3*z^2 - 5122*y^3*z + 1254*y^3 - 642*y^2*z^5 + 3684*y^2*z^4 - 8127*y^2*z^3 + 8649*y^2*z^2 - 4482*y^2*z + 910*y^2 + 196*y*z^6 - 1404*y*z^5 + 4023*y*z^4 - 5987*y*z^3 + 4932*y*z^2 - 2137*y*z + 386*y - 24*z^7 + 208*z^6 - 738*z^5 + 1427*z^4 - 1644*z^3 + 1138*z^2 - 438*z + 1, -6*x^7 + 60*x^6*y - 63*x^6*z + 63*x^6 - 252*x^5*y^2 + 549*x^5*y*z - 540*x^5*y - 279*x^5*z^2 + 570*x^5*z - 288*x^5 + 576*x^4*y^3 - 1953*x^4*y^2*z + 1890*x^4*y^2 + 2061*x^4*y*z^2 - 4137*x^4*y*z + 2052*x^4*y - 675*x^4*z^3 + 2118*x^4*z^2 - 2184*x^4*z + 744*x^4 - 774*x^3*y^4 + 3627*x^3*y^3*z - 3456*x^3*y^3 - 5967*x^3*y^2*z^2 + 11766*x^3*y^2*z - 5724*x^3*y^2 + 4059*x^3*y*z^3 - 12504*x^3*y*z^2 + 12660*x^3*y*z - 4215*x^3*y - 963*x^3*z^4 + 4134*x^3*z^3 - 6528*x^3*z^2 + 4548*x^3*z - 1179*x^3 + 612*x^2*y^5 - 3708*x^2*y^4*z + 3483*x^2*y^4 + 8451*x^2*y^3*z^2 - 16377*x^2*y^3*z + 7812*x^2*y^3 - 8973*x^2*y^2*z^3 + 27117*x^2*y^2*z^2 - 26919*x^2*y^2*z + 8745*x^2*y^2 + 4419*x^2*y*z^4 - 18621*x^2*y*z^3 + 28890*x^2*y*z^2 - 19683*x^2*y*z + 4956*x^2*y - 810*x^2*z^5 + 4467*x^2*z^4 - 9603*x^2*z^3 + 10263*x^2*z^2 - 5460*x^2*z + 1158*x^2 - 264*x*y^6 + 1980*x*y^5*z - 1836*x*y^5 - 5850*x*y^4*z^2 + 11154*x*y^4*z - 5220*x*y^4 + 8625*x*y^3*z^3 - 25566*x*y^3*z^2 + 24861*x*y^3*z - 7875*x*y^3 - 6633*x*y^2*z^4 + 27396*x*y^2*z^3 - 41679*x*y^2*z^2 + 27705*x*y^2*z - 6756*x*y^2 + 2520*x*y*z^5 - 13650*x*y*z^4 + 28863*x*y*z^3 - 30198*x*y*z^2 + 15624*x*y*z - 3195*x*y - 372*x*z^6 + 2532*x*z^5 - 6945*x*z^4 + 10119*x*z^3 - 8283*x*z^2 + 3618*x*z - 665*x + 48*y^7 - 432*y^6*z + 396*y^6 + 1584*y^5*z^2 - 2976*y^5*z + 1368*y^5 - 3036*y^4*z^3 + 8835*y^4*z^2 - 8418*y^4*z + 2601*y^4 + 3249*y^3*z^4 - 13137*y^3*z^3 + 19563*y^3*z^2 - 12672*y^3*z + 2991*y^3 - 1926*y^2*z^5 + 10227*y^2*z^4 - 21222*y^2*z^3 + 21663*y^2*z^2 - 10851*y^2*z + 2133*y^2 + 588*y*z^6 - 3936*y*z^5 + 10635*y*z^4 - 15201*y*z^3 + 12129*y*z^2 - 5109*y*z + 902*y - 72*z^7 + 588*z^6 - 1974*z^5 + 3672*z^4 - 4107*z^3 + 2775*z^2 - 1048*z + 1, -4*x^7 + 40*x^6*y - 42*x^6*z + 40*x^6 - 168*x^5*y^2 + 366*x^5*y*z - 342*x^5*y - 186*x^5*z^2 + 363*x^5*z - 177*x^5 + 384*x^4*y^3 - 1302*x^4*y^2*z + 1194*x^4*y^2 + 1374*x^4*y*z^2 - 2628*x^4*y*z + 1257*x^4*y - 450*x^4*z^3 + 1353*x^4*z^2 - 1347*x^4*z + 449*x^4 - 516*x^3*y^4 + 2418*x^3*y^3*z - 2178*x^3*y^3 - 3978*x^3*y^2*z^2 + 7455*x^3*y^2*z - 3495*x^3*y^2 + 2706*x^3*y*z^3 - 7968*x^3*y*z^2 + 7782*x^3*y*z - 2536*x^3*y - 642*x^3*z^4 + 2649*x^3*z^3 - 4041*x^3*z^2 + 2753*x^3*z - 707*x^3 + 408*x^2*y^5 - 2472*x^2*y^4*z + 2190*x^2*y^4 + 5634*x^2*y^3*z^2 - 10350*x^2*y^3*z + 4755*x^2*y^3 - 5982*x^2*y^2*z^3 + 17235*x^2*y^2*z^2 - 16491*x^2*y^2*z + 5247*x^2*y^2 + 2946*x^2*y*z^4 - 11904*x^2*y*z^3 + 17823*x^2*y*z^2 - 11874*x^2*y*z + 2967*x^2*y - 540*x^2*z^5 + 2871*x^2*z^4 - 5967*x^2*z^3 + 6234*x^2*z^2 - 3279*x^2*z + 695*x^2 - 176*x*y^6 + 1320*x*y^5*z - 1152*x*y^5 - 3900*x*y^4*z^2 + 7032*x*y^4*z - 3168*x*y^4 + 5750*x*y^3*z^3 - 16206*x*y^3*z^2 + 15180*x*y^3*z - 4714*x*y^3 - 4422*x*y^2*z^4 + 17469*x*y^2*z^3 - 25623*x*y^2*z^2 + 16659*x*y^2*z - 4041*x*y^2 + 1680*x*y*z^5 - 8754*x*y*z^4 + 17874*x*y*z^3 - 18276*x*y*z^2 + 9360*x*y*z - 1918*x*y - 248*x*z^6 + 1632*x*z^5 - 4332*x*z^4 + 6170*x*z^3 - 4986*x*z^2 + 2171*x*z - 399*x + 32*y^7 - 288*y^6*z + 248*y^6 + 1056*y^5*z^2 - 1872*y^5*z + 828*y^5 - 2024*y^4*z^3 + 5586*y^4*z^2 - 5124*y^4*z + 1554*y^4 + 2166*y^3*z^4 - 8354*y^3*z^3 + 11985*y^3*z^2 - 7598*y^3*z + 1789*y^3 - 1284*y^2*z^5 + 6543*y^2*z^4 - 13095*y^2*z^3 + 13062*y^2*z^2 - 6489*y^2*z + 1283*y^2 + 392*y*z^6 - 2532*y*z^5 + 6612*y*z^4 - 9234*y*z^3 + 7278*y*z^2 - 3062*y*z + 541*y - 48*z^7 + 380*z^6 - 1236*z^5 + 2248*z^4 - 2480*z^3 + 1667*z^2 - 629*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -1, 2, -1 ], [ -1, 1, -2 ], [ -2, 2, -3 ] ], "source_shift": [ 1, 1, 2 ], "target_matrix": [ [ 1, 1, 1 ], [ 3, 2, 3 ], [ 2, 1, 1 ] ], "target_shift": [ -72, -172, -102 ], "collision_preimages": [ [ "7/4", "1/4", "-1/4" ], [ "-47/2", "-7", "19/2" ], [ "-27/2", "-6", "7/2" ] ], "collision_image": [ "-289/4", "-691/4", "-205/2" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 0.9873189559984894 ], "surface_features": [ 3.682488042586139, 4.619917290433218, 3.395064164331242, 0.0, 3.7956591175424648, 4966 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C121", "pair": 121, "family": "counterexample", "mapping": "F(x,y,z)=(24*x^7 - 132*x^6*y + 156*x^6*z - 288*x^6 + 306*x^5*y^2 - 720*x^5*y*z + 1332*x^5*y + 414*x^5*z^2 - 1560*x^5*z + 1446*x^5 - 387*x^4*y^3 + 1359*x^4*y^2*z - 2520*x^4*y^2 - 1557*x^4*y*z^2 + 5856*x^4*y*z - 5475*x^4*y + 585*x^4*z^3 - 3336*x^4*z^2 + 6297*x^4*z - 3992*x^4 + 288*x^3*y^4 - 1341*x^3*y^3*z + 2493*x^3*y^3 + 2295*x^3*y^2*z^2 - 8616*x^3*y^2*z + 8124*x^3*y^2 - 1719*x^3*y*z^3 + 9753*x^3*y*z^2 - 18483*x^3*y*z + 11864*x^3*y + 477*x^3*z^4 - 3630*x^3*z^3 + 10359*x^3*z^2 - 13327*x^3*z + 6557*x^3 - 126*x^2*y^5 + 729*x^2*y^4*z - 1359*x^2*y^4 - 1656*x^2*y^3*z^2 + 6207*x^2*y^3*z - 5898*x^2*y^3 + 1854*x^2*y^2*z^3 - 10467*x^2*y^2*z^2 + 19908*x^2*y^2*z - 12909*x^2*y^2 - 1026*x^2*y*z^4 + 7749*x^2*y*z^3 - 22122*x^2*y*z^2 + 28653*x^2*y*z - 14250*x^2*y + 225*x^2*z^5 - 2130*x^2*z^4 + 8112*x^2*z^3 - 15744*x^2*z^2 + 15645*x^2*z - 6354*x^2 + 30*x*y^6 - 207*x*y^5*z + 387*x*y^5 + 585*x*y^4*z^2 - 2190*x*y^4*z + 2094*x*y^4 - 870*x*y^3*z^3 + 4890*x*y^3*z^2 - 9327*x*y^3*z + 6088*x*y^3 + 720*x*y^2*z^4 - 5400*x*y^2*z^3 + 15417*x*y^2*z^2 - 20049*x*y^2*z + 10029*x*y^2 - 315*x*y*z^5 + 2955*x*y*z^4 - 11229*x*y*z^3 + 21834*x*y*z^2 - 21771*x*y*z + 8901*x*y + 57*x*z^6 - 642*x*z^5 + 3045*x*z^4 - 7873*x*z^3 + 11742*x*z^2 - 9582*x*z + 3350*x - 3*y^7 + 24*y^6*z - 45*y^6 - 81*y^5*z^2 + 303*y^5*z - 291*y^5 + 150*y^4*z^3 - 840*y^4*z^2 + 1605*y^4*z - 1051*y^4 - 165*y^3*z^4 + 1230*y^3*z^3 - 3510*y^3*z^2 + 4571*y^3*z - 2290*y^3 + 108*y^2*z^5 - 1005*y^2*z^4 + 3810*y^2*z^3 - 7407*y^2*z^2 + 7386*y^2*z - 3021*y^2 - 39*y*z^6 + 435*y*z^5 - 2055*y*z^4 + 5305*y*z^3 - 7902*y*z^2 + 6441*y*z - 2248*y + 6*z^7 - 78*z^6 + 441*z^5 - 1418*z^4 + 2806*z^3 - 3420*z^2 + 2381*z - 1, -16*x^7 + 88*x^6*y - 104*x^6*z + 184*x^6 - 204*x^5*y^2 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1893*x*y*z^4 + 7020*x*y*z^3 - 13422*x*y*z^2 + 13224*x*y*z - 5362*x*y - 38*x*z^6 + 411*x*z^5 - 1902*x*z^4 + 4836*x*z^3 - 7128*x*z^2 + 5770*x*z - 2006*x + 2*y^7 - 16*y^6*z + 29*y^6 + 54*y^5*z^2 - 195*y^5*z + 183*y^5 - 100*y^4*z^3 + 540*y^4*z^2 - 1008*y^4*z + 649*y^4 + 110*y^3*z^4 - 790*y^3*z^3 + 2202*y^3*z^2 - 2820*y^3*z + 1396*y^3 - 72*y^2*z^5 + 645*y^2*z^4 - 2388*y^2*z^3 + 4566*y^2*z^2 - 4500*y^2*z + 1826*y^2 + 26*y*z^6 - 279*y*z^5 + 1287*y*z^4 - 3268*y*z^3 + 4812*y*z^2 - 3892*y*z + 1352*y - 4*z^7 + 50*z^6 - 276*z^5 + 873*z^4 - 1708*z^3 + 2066*z^2 - 1432*z + 1, 8*x^7 - 44*x^6*y + 52*x^6*z - 128*x^6 + 102*x^5*y^2 - 240*x^5*y*z + 588*x^5*y + 138*x^5*z^2 - 696*x^5*z + 738*x^5 - 129*x^4*y^3 + 453*x^4*y^2*z - 1104*x^4*y^2 - 519*x^4*y*z^2 + 2592*x^4*y*z - 2769*x^4*y + 195*x^4*z^3 - 1488*x^4*z^2 + 3219*x^4*z - 2216*x^4 + 96*x^3*y^4 - 447*x^3*y^3*z + 1083*x^3*y^3 + 765*x^3*y^2*z^2 - 3780*x^3*y^2*z + 4068*x^3*y^2 - 573*x^3*y*z^3 + 4311*x^3*y*z^2 - 9345*x^3*y*z + 6520*x^3*y + 159*x^3*z^4 - 1614*x^3*z^3 + 5277*x^3*z^2 - 7381*x^3*z + 3847*x^3 - 42*x^2*y^5 + 243*x^2*y^4*z - 585*x^2*y^4 - 552*x^2*y^3*z^2 + 2697*x^2*y^3*z - 2922*x^2*y^3 + 618*x^2*y^2*z^3 - 4581*x^2*y^2*z^2 + 9948*x^2*y^2*z - 7017*x^2*y^2 - 342*x^2*y*z^4 + 3411*x^2*y*z^3 - 11130*x^2*y*z^2 + 15675*x^2*y*z - 8262*x^2*y + 75*x^2*z^5 - 942*x^2*z^4 + 4104*x^2*z^3 - 8658*x^2*z^2 + 9111*x^2*z - 3856*x^2 + 10*x*y^6 - 69*x*y^5*z + 165*x*y^5 + 195*x*y^4*z^2 - 942*x*y^4*z + 1026*x*y^4 - 290*x*y^3*z^3 + 2118*x*y^3*z^2 - 4605*x*y^3*z + 3271*x*y^3 + 240*x*y^2*z^4 - 2352*x*y^2*z^3 + 7659*x*y^2*z^2 - 10830*x*y^2*z + 5739*x*y^2 - 105*x*y*z^5 + 1293*x*y*z^4 - 5607*x*y*z^3 + 11847*x*y*z^2 - 12501*x*y*z + 5318*x*y + 19*x*z^6 - 282*x*z^5 + 1527*x*z^4 - 4288*x*z^3 + 6762*x*z^2 - 5735*x*z + 2056*x - y^7 + 8*y^6*z - 19*y^6 - 27*y^5*z^2 + 129*y^5*z - 141*y^5 + 50*y^4*z^3 - 360*y^4*z^2 + 783*y^4*z - 558*y^4 - 55*y^3*z^4 + 530*y^3*z^3 - 1722*y^3*z^2 + 2438*y^3*z - 1293*y^3 + 36*y^2*z^5 - 435*y^2*z^4 + 1878*y^2*z^3 - 3966*y^2*z^2 + 4182*y^2*z - 1777*y^2 - 13*y*z^6 + 189*y*z^5 - 1017*y*z^4 + 2850*y*z^3 - 4485*y*z^2 + 3794*y*z - 1353*y + 2*z^7 - 34*z^6 + 219*z^5 - 764*z^4 + 1596*z^3 - 2017*z^2 + 1433*z - 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -2, 1, -1 ], [ -1, 1, -1 ], [ -1, 1, -2 ] ], "source_shift": [ 1, 2, 1 ], "target_matrix": [ [ -3, -2, -2 ], [ 2, 1, 1 ], [ -1, -2, -1 ] ], "target_shift": [ 728, -437, 439 ], "collision_preimages": [ [ "-1", "-15/4", "-3/4" ], [ "-7/2", "-16", "-9" ], [ "3/2", "-5", "-6" ] ], "collision_image": [ "2915/4", "-875/2", "1757/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0355496760021197 ], "surface_features": [ 3.5341199249764723, 4.457170099784619, 3.1885066338181143, 0.0, 3.6535552160184417, 4873 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C208", 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"certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ 1, -1, -2 ], [ 1, -1, -1 ], [ 1, -2, -1 ] ], "source_shift": [ 2, 2, 1 ], "target_matrix": [ [ 2, 1, 1 ], [ 1, 1, 1 ], [ 2, 2, 3 ] ], "target_shift": [ -1397, -952, -1877 ], "collision_preimages": [ [ "-11/4", "-3/4", "0" ], [ "-15", "-9", "-5/2" ], [ "-4", "-6", "5/2" ] ], "collision_image": [ "-2795/2", "-3809/4", "-3755/2" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0605884639990109 ], "surface_features": [ 3.6510461021703624, 4.550472957106564, 3.2558752733914664, 0.0, 3.7861396712770703, 4892 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C214", "pair": 214, "family": "counterexample", "mapping": "F(x,y,z)=(x^7 - 10*x^6*y + 8*x^6*z - 22*x^6 + 42*x^5*y^2 - 69*x^5*y*z + 189*x^5*y + 27*x^5*z^2 - 150*x^5*z + 195*x^5 - 96*x^4*y^3 + 243*x^4*y^2*z - 666*x^4*y^2 - 195*x^4*y*z^2 + 1077*x^4*y*z - 1413*x^4*y + 50*x^4*z^3 - 420*x^4*z^2 + 1098*x^4*z - 925*x^4 + 129*x^3*y^4 - 447*x^3*y^3*z + 1233*x^3*y^3 + 552*x^3*y^2*z^2 - 3042*x^3*y^2*z + 4041*x^3*y^2 - 290*x^3*y*z^3 + 2418*x^3*y*z^2 - 6366*x^3*y*z + 5431*x^3*y + 55*x^3*z^4 - 620*x^3*z^3 + 2442*x^3*z^2 - 4135*x^3*z + 2573*x^3 - 102*x^2*y^5 + 453*x^2*y^4*z - 1266*x^2*y^4 - 765*x^2*y^3*z^2 + 4227*x^2*y^3*z - 5703*x^2*y^3 + 618*x^2*y^2*z^3 - 5130*x^2*y^2*z^2 + 13638*x^2*y^2*z - 11811*x^2*y^2 - 240*x^2*y*z^4 + 2682*x^2*y*z^3 - 10620*x^2*y*z^2 + 18174*x^2*y*z - 11469*x^2*y + 36*x^2*z^5 - 510*x^2*z^4 + 2688*x^2*z^3 - 6855*x^2*z^2 + 8580*x^2*z - 4237*x^2 + 44*x*y^6 - 240*x*y^5*z + 684*x*y^5 + 519*x*y^4*z^2 - 2892*x*y^4*z + 3972*x*y^4 - 573*x*y^3*z^3 + 4755*x*y^3*z^2 - 12798*x*y^3*z + 11272*x*y^3 + 342*x*y^2*z^4 - 3798*x*y^2*z^3 + 15153*x*y^2*z^2 - 26259*x*y^2*z + 16839*x*y^2 - 105*x*y*z^5 + 1473*x*y*z^4 - 7794*x*y*z^3 + 20055*x*y*z^2 - 25407*x*y*z + 12739*x*y + 13*x*z^6 - 222*x*z^5 + 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10380*x*y^3*z - 8578*x*y^3 - 342*x*y^2*z^4 + 3315*x*y^2*z^3 - 12093*x*y^2*z^2 + 19677*x*y^2*z - 12045*x*y^2 + 105*x*y*z^5 - 1269*x*y*z^4 + 6126*x*y*z^3 - 14805*x*y*z^2 + 17919*x*y*z - 8677*x*y - 13*x*z^6 + 189*x*z^5 - 1137*x*z^4 + 3643*x*z^3 - 6567*x*z^2 + 6311*x*z - 2517*x + 8*y^7 - 52*y^6*z + 140*y^6 + 138*y^5*z^2 - 708*y^5*z + 912*y^5 - 195*y^4*z^3 + 1455*y^4*z^2 - 3648*y^4*z + 3067*y^4 + 159*y^3*z^4 - 1554*y^3*z^3 + 5736*y^3*z^2 - 9465*y^3*z + 5889*y^3 - 75*y^2*z^5 + 909*y^2*z^4 - 4422*y^2*z^3 + 10800*y^2*z^2 - 13245*y^2*z + 6523*y^2 + 19*y*z^6 - 276*y*z^5 + 1668*y*z^4 - 5386*y*z^3 + 9810*y*z^2 - 9550*y*z + 3872*y - 2*z^7 + 34*z^6 - 246*z^5 + 988*z^4 - 2384*z^3 + 3457*z^2 - 2784*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -1, 1, -1 ], [ -1, 2, -1 ], [ -1, 1, -2 ] ], "source_shift": [ 2, 2, 1 ], "target_matrix": [ [ -1, -2, -1 ], [ -1, -1, -2 ], [ 1, 1, 1 ] ], "target_shift": [ 1488, 924, -952 ], "collision_preimages": [ [ "11/4", "0", "-3/4" ], [ "5", "-5/2", "-13/2" ], [ "14", "5/2", "-17/2" ] ], "collision_image": [ "5953/4", "3697/4", "-3809/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0354217419990164 ], "surface_features": [ 3.538977415987282, 4.419278183187153, 3.0580462303952816, 0.0, 3.6718358213552857, 4835 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C161", "pair": 161, "family": "counterexample", "mapping": "F(x,y,z)=(-27*x^7 + 162*x^6*y + 270*x^6*z - 459*x^6 - 414*x^5*y^2 - 1377*x^5*y*z + 2241*x^5*y - 1134*x^5*z^2 + 3807*x^5*z - 2016*x^5 + 584*x^4*y^3 + 2907*x^4*y^2*z - 4536*x^4*y^2 + 4779*x^4*y*z^2 - 15336*x^4*y*z + 8076*x^4*y + 2592*x^4*z^3 - 12879*x^4*z^2 + 13752*x^4*z - 4011*x^4 - 491*x^3*y^4 - 3251*x^3*y^3*z + 4873*x^3*y^3 - 8001*x^3*y^2*z^2 + 24597*x^3*y^2*z - 12867*x^3*y^2 - 8667*x^3*y*z^3 + 41121*x^3*y*z^2 - 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8790*x*y^3 - 14274*x*y^2*z^4 + 73881*x*y^2*z^3 - 88590*x*y^2*z^2 + 43692*x*y^2*z - 7938*x*y^2 - 9072*x*y*z^5 + 60588*x*y*z^4 - 97374*x*y*z^3 + 72036*x*y*z^2 - 26100*x*y*z + 3840*x*y - 2376*x*z^6 + 19764*x*z^5 - 39942*x*z^4 + 39401*x*z^3 - 21348*x*z^2 + 6243*x*z - 778*x + 16*y^7 + 184*y^6*z - 212*y^6 + 900*y^5*z^2 - 2112*y^5*z + 828*y^5 + 2426*y^4*z^3 - 8721*y^4*z^2 + 6864*y^4*z - 1680*y^4 + 3890*y^3*z^4 - 19105*y^3*z^3 + 22653*y^3*z^2 - 11086*y^3*z + 2000*y^3 + 3708*y^2*z^5 - 23418*y^2*z^4 + 37206*y^2*z^3 - 27306*y^2*z^2 + 9822*y^2*z - 1433*y^2 + 1944*y*z^6 - 15228*y*z^5 + 30414*y*z^4 - 29759*y*z^3 + 16005*y*z^2 - 4641*y*z + 575*y + 432*z^7 - 4104*z^6 + 9900*z^5 - 12110*z^4 + 8656*z^3 - 3741*z^2 + 920*z - 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ 1, -1, -2 ], [ 3, -2, -3 ], [ 1, -1, -1 ] ], "source_shift": [ 1, 1, 2 ], "target_matrix": [ [ -1, -2, -1 ], [ 1, 1, 1 ], [ -2, -1, -1 ] ], "target_shift": [ 118, -72, 102 ], "collision_preimages": [ [ "9/4", "23/4", "-5/4" ], [ "-7", "-16", "9/2" ], [ "-2", "-13", "13/2" ] ], "collision_image": [ "473/4", "-289/4", "205/2" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0639730590010004 ], "surface_features": [ 4.169983671419947, 5.060739379578298, 3.8441042306975133, 0.0, 4.2875064806770045, 5116 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B167", "pair": 167, "family": "bijection", "mapping": "F(x,y,z)=(2*x^7 - 15*x^6*y - 14*x^6*z + 31*x^6 + 48*x^5*y^2 + 90*x^5*y*z - 199*x^5*y + 42*x^5*z^2 - 186*x^5*z + 205*x^5 - 85*x^4*y^3 - 240*x^4*y^2*z + 530*x^4*y^2 - 225*x^4*y*z^2 + 995*x^4*y*z - 1095*x^4*y - 70*x^4*z^3 + 465*x^4*z^2 - 1025*x^4*z + 768*x^4 + 90*x^3*y^4 + 340*x^3*y^3*z - 750*x^3*y^3 + 480*x^3*y^2*z^2 - 2120*x^3*y^2*z + 2330*x^3*y^2 + 300*x^3*y*z^3 - 1990*x^3*y*z^2 + 4380*x^3*y*z - 3305*x^3*y + 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- 20*y^6*z + 45*y^6 - 57*y^5*z^2 + 257*y^5*z - 288*y^5 - 90*y^4*z^3 + 610*y^4*z^2 - 1370*y^4*z + 1104*y^4 - 85*y^3*z^4 + 770*y^3*z^3 - 2600*y^3*z^2 + 4107*y^3*z - 2643*y^3 - 48*y^2*z^5 + 545*y^2*z^4 - 2460*y^2*z^3 + 5751*y^2*z^2 - 7144*y^2*z + 3783*y^2 - 15*y*z^6 + 205*y*z^5 - 1160*y*z^4 + 3585*y*z^3 - 6512*y*z^2 + 6633*y*z - 2932*y - 2*z^7 + 32*z^6 - 218*z^5 + 838*z^4 - 1994*z^3 + 2958*z^2 - 2525*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ 1, -1, -1 ], [ 2, -1, -1 ], [ 1, -2, -1 ] ], "source_shift": [ 2, 1, 2 ], "target_matrix": [ [ 1, 1, 1 ], [ 2, 1, 1 ], [ 1, 2, 1 ] ], "target_shift": [ -838, -1580, -943 ], "inverse_recipe": { "p": 6, "q": 1, "f": "x**6 + x**5 + x**3", "g": "x", "h": "y*z + z**4 + z**3", "polynomials": { "f": [ [ [ 6, 0, 0 ], 1 ], [ [ 3, 0, 0 ], 1 ], [ [ 5, 0, 0 ], 1 ] ], "g": [ [ [ 1, 0, 0 ], 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1233*x*y*z^5 + 15525*x*y*z^4 + 57846*x*y*z^3 + 94449*x*y*z^2 + 70797*x*y*z + 19883*x*y + 162*x*z^6 + 2646*x*z^5 + 12981*x*z^4 + 29629*x*z^3 + 34902*x*z^2 + 20514*x*z + 4745*x + 64*y^7 + 448*y^6*z + 800*y^6 + 1296*y^5*z^2 + 4944*y^5*z + 3888*y^5 + 2000*y^4*z^3 + 12336*y^4*z^2 + 20256*y^4*z + 9880*y^4 + 1772*y^3*z^4 + 15832*y^3*z^3 + 40872*y^3*z^2 + 41444*y^3*z + 14524*y^3 + 900*y^2*z^5 + 10962*y^2*z^4 + 39696*y^2*z^3 + 63018*y^2*z^2 + 45936*y^2*z + 12550*y^2 + 243*y*z^6 + 3861*y*z^5 + 18435*y*z^4 + 40891*y*z^3 + 46713*y*z^2 + 26613*y*z + 5973*y + 27*z^7 + 540*z^6 + 3258*z^5 + 9478*z^4 + 15152*z^3 + 13575*z^2 + 6370*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ 1, 2, 1 ], [ 2, 2, 3 ], [ 1, 1, 1 ] ], "source_shift": [ 2, 2, 2 ], "target_matrix": [ [ 2, 1, 1 ], [ 1, 2, 1 ], [ 1, 1, 1 ] ], "target_shift": [ -1789, -1905, -1219 ], "collision_preimages": [ [ "-5", "1/4", "5/2" ], [ "45/2", "-11/2", "-25/2" ], [ "43/2", "-15/2", "-19/2" ] ], "collision_image": [ "-3579/2", "-7621/4", "-4877/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0302698550003697 ], "surface_features": [ 4.36443294751419, 5.219877798594039, 4.001474096691733, 0.0, 4.480440801112559, 5159 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B239", "pair": 239, "family": "bijection", "mapping": "F(x,y,z)=(-3*x^7 + 21*x^6*y + 21*x^6*z - 54*x^6 - 63*x^5*y^2 - 126*x^5*y*z + 324*x^5*y - 63*x^5*z^2 + 324*x^5*z - 417*x^5 + 105*x^4*y^3 + 315*x^4*y^2*z - 810*x^4*y^2 + 315*x^4*y*z^2 - 1620*x^4*y*z + 2088*x^4*y + 105*x^4*z^3 - 810*x^4*z^2 + 2085*x^4*z - 1801*x^4 - 105*x^3*y^4 - 420*x^3*y^3*z + 1080*x^3*y^3 - 630*x^3*y^2*z^2 + 3240*x^3*y^2*z - 4182*x^3*y^2 - 420*x^3*y*z^3 + 3240*x^3*y*z^2 - 8352*x^3*y*z + 7240*x^3*y - 105*x^3*z^4 + 1080*x^3*z^3 - 4170*x^3*z^2 + 7201*x^3*z - 4702*x^3 + 63*x^2*y^5 + 315*x^2*y^4*z - 810*x^2*y^4 + 630*x^2*y^3*z^2 - 3240*x^2*y^3*z + 4188*x^2*y^3 + 630*x^2*y^2*z^3 - 4860*x^2*y^2*z^2 + 12546*x^2*y^2*z - 10914*x^2*y^2 + 315*x^2*y*z^4 - 3240*x^2*y*z^3 + 12528*x^2*y*z^2 - 21711*x^2*y*z + 14262*x^2*y + 63*x^2*z^5 - 810*x^2*z^4 + 4170*x^2*z^3 - 10797*x^2*z^2 + 14085*x^2*z - 7409*x^2 - 21*x*y^6 - 126*x*y^5*z + 324*x*y^5 - 315*x*y^4*z^2 + 1620*x*y^4*z - 2097*x*y^4 - 420*x*y^3*z^3 + 3240*x*y^3*z^2 - 8376*x*y^3*z + 7312*x*y^3 - 315*x*y^2*z^4 + 3240*x*y^2*z^3 - 12546*x*y^2*z^2 + 21819*x*y^2*z - 14418*x*y^2 - 126*x*y*z^5 + 1620*x*y*z^4 - 8352*x*y*z^3 + 21702*x*y*z^2 - 28482*x*y*z + 15121*x*y - 21*x*z^6 + 324*x*z^5 - 2085*x*z^4 + 7195*x*z^3 - 14064*x*z^2 + 14767*x*z - 6503*x + 3*y^7 + 21*y^6*z - 54*y^6 + 63*y^5*z^2 - 324*y^5*z + 420*y^5 + 105*y^4*z^3 - 810*y^4*z^2 + 2097*y^4*z - 1837*y^4 + 105*y^3*z^4 - 1080*y^3*z^3 + 4188*y^3*z^2 - 7309*y^3*z + 4858*y^3 + 63*y^2*z^5 - 810*y^2*z^4 + 4182*y^2*z^3 - 10905*y^2*z^2 + 14397*y^2*z - 7715*y^2 + 21*y*z^6 - 324*y*z^5 + 2088*y*z^4 - 7231*y*z^3 + 14220*y*z^2 - 15067*y*z + 6716*y + 3*z^7 - 54*z^6 + 417*z^5 - 1798*z^4 + 4681*z^3 - 7358*z^2 + 6460*z - 1, 3*x^7 - 21*x^6*y - 21*x^6*z + 54*x^6 + 63*x^5*y^2 + 126*x^5*y*z - 324*x^5*y + 63*x^5*z^2 - 324*x^5*z + 417*x^5 - 105*x^4*y^3 - 315*x^4*y^2*z + 810*x^4*y^2 - 315*x^4*y*z^2 + 1620*x^4*y*z - 2088*x^4*y - 105*x^4*z^3 + 810*x^4*z^2 - 2085*x^4*z + 1802*x^4 + 105*x^3*y^4 + 420*x^3*y^3*z - 1080*x^3*y^3 + 630*x^3*y^2*z^2 - 3240*x^3*y^2*z + 4182*x^3*y^2 + 420*x^3*y*z^3 - 3240*x^3*y*z^2 + 8352*x^3*y*z - 7244*x^3*y + 105*x^3*z^4 - 1080*x^3*z^3 + 4170*x^3*z^2 - 7205*x^3*z + 4711*x^3 - 63*x^2*y^5 - 315*x^2*y^4*z + 810*x^2*y^4 - 630*x^2*y^3*z^2 + 3240*x^2*y^3*z - 4188*x^2*y^3 - 630*x^2*y^2*z^3 + 4860*x^2*y^2*z^2 - 12546*x^2*y^2*z + 10920*x^2*y^2 - 315*x^2*y*z^4 + 3240*x^2*y*z^3 - 12528*x^2*y*z^2 + 21723*x^2*y*z - 14289*x^2*y - 63*x^2*z^5 + 810*x^2*z^4 - 4170*x^2*z^3 + 10803*x^2*z^2 - 14112*x^2*z + 7440*x^2 + 21*x*y^6 + 126*x*y^5*z - 324*x*y^5 + 315*x*y^4*z^2 - 1620*x*y^4*z + 2097*x*y^4 + 420*x*y^3*z^3 - 3240*x*y^3*z^2 + 8376*x*y^3*z - 7316*x*y^3 + 315*x*y^2*z^4 - 3240*x*y^2*z^3 + 12546*x*y^2*z^2 - 21831*x*y^2*z + 14445*x*y^2 + 126*x*y*z^5 - 1620*x*y*z^4 + 8352*x*y*z^3 - 21714*x*y*z^2 + 28536*x*y*z - 15183*x*y + 21*x*z^6 - 324*x*z^5 + 2085*x*z^4 - 7199*x*z^3 + 14091*x*z^2 - 14829*x*z + 6553*x - 3*y^7 - 21*y^6*z + 54*y^6 - 63*y^5*z^2 + 324*y^5*z - 420*y^5 - 105*y^4*z^3 + 810*y^4*z^2 - 2097*y^4*z + 1838*y^4 - 105*y^3*z^4 + 1080*y^3*z^3 - 4188*y^3*z^2 + 7313*y^3*z - 4867*y^3 - 63*y^2*z^5 + 810*y^2*z^4 - 4182*y^2*z^3 + 10911*y^2*z^2 - 14424*y^2*z + 7746*y^2 - 21*y*z^6 + 324*y*z^5 - 2088*y*z^4 + 7235*y*z^3 - 14247*y*z^2 + 15129*y*z - 6765*y - 3*z^7 + 54*z^6 - 417*z^5 + 1799*z^4 - 4690*z^3 + 7389*z^2 - 6509*z + 1, -6*x^7 + 42*x^6*y + 42*x^6*z - 108*x^6 - 126*x^5*y^2 - 252*x^5*y*z + 648*x^5*y - 126*x^5*z^2 + 648*x^5*z - 834*x^5 + 210*x^4*y^3 + 630*x^4*y^2*z - 1620*x^4*y^2 + 630*x^4*y*z^2 - 3240*x^4*y*z + 4176*x^4*y + 210*x^4*z^3 - 1620*x^4*z^2 + 4170*x^4*z - 3603*x^4 - 210*x^3*y^4 - 840*x^3*y^3*z + 2160*x^3*y^3 - 1260*x^3*y^2*z^2 + 6480*x^3*y^2*z - 8364*x^3*y^2 - 840*x^3*y*z^3 + 6480*x^3*y*z^2 - 16704*x^3*y*z + 14484*x^3*y - 210*x^3*z^4 + 2160*x^3*z^3 - 8340*x^3*z^2 + 14406*x^3*z - 9411*x^3 + 126*x^2*y^5 + 630*x^2*y^4*z - 1620*x^2*y^4 + 1260*x^2*y^3*z^2 - 6480*x^2*y^3*z + 8376*x^2*y^3 + 1260*x^2*y^2*z^3 - 9720*x^2*y^2*z^2 + 25092*x^2*y^2*z - 21834*x^2*y^2 + 630*x^2*y*z^4 - 6480*x^2*y*z^3 + 25056*x^2*y*z^2 - 43434*x^2*y*z + 28545*x^2*y + 126*x^2*z^5 - 1620*x^2*z^4 + 8340*x^2*z^3 - 21600*x^2*z^2 + 28191*x^2*z - 14835*x^2 - 42*x*y^6 - 252*x*y^5*z + 648*x*y^5 - 630*x*y^4*z^2 + 3240*x*y^4*z - 4194*x*y^4 - 840*x*y^3*z^3 + 6480*x*y^3*z^2 - 16752*x*y^3*z + 14628*x*y^3 - 630*x*y^2*z^4 + 6480*x*y^2*z^3 - 25092*x*y^2*z^2 + 43650*x*y^2*z - 28857*x*y^2 - 252*x*y*z^5 + 3240*x*y*z^4 - 16704*x*y*z^3 + 43416*x*y*z^2 - 57006*x*y*z + 30276*x*y - 42*x*z^6 + 648*x*z^5 - 4170*x*z^4 + 14394*x*z^3 - 28149*x*z^2 + 29568*x*z - 13022*x + 6*y^7 + 42*y^6*z - 108*y^6 + 126*y^5*z^2 - 648*y^5*z + 840*y^5 + 210*y^4*z^3 - 1620*y^4*z^2 + 4194*y^4*z - 3675*y^4 + 210*y^3*z^4 - 2160*y^3*z^3 + 8376*y^3*z^2 - 14622*y^3*z + 9723*y^3 + 126*y^2*z^5 - 1620*y^2*z^4 + 8364*y^2*z^3 - 21816*y^2*z^2 + 28815*y^2*z - 15447*y^2 + 42*y*z^6 - 648*y*z^5 + 4176*y*z^4 - 14466*y*z^3 + 28461*y*z^2 - 30168*y*z + 13445*y + 6*z^7 - 108*z^6 + 834*z^5 - 3597*z^4 + 9369*z^3 - 14733*z^2 + 12935*z - 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ 1, -1, -1 ], [ 2, -1, -1 ], [ 1, -2, -1 ] ], "source_shift": [ 2, 2, 2 ], "target_matrix": [ [ -1, -1, -1 ], [ 1, 2, 1 ], [ -2, -3, -3 ] ], "target_shift": [ 2439, -2469, 4881 ], "inverse_recipe": { "p": 4, "q": 3, "f": "x**4 + x**3 + x**2", "g": "x**3 + x**2", "h": "y*z + 2*z**2", "polynomials": { "f": [ [ [ 4, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ], [ [ 3, 0, 0 ], 1 ] ], "g": [ [ [ 3, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 0, 2 ], 2 ] ] }, "h_multiplier": 3 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 0.9967721180000808 ], "surface_features": [ 3.7532112005132605, 4.755920568475218, 3.2556342664585873, 0.0, 3.920791907911018, 4904 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B094", "pair": 94, "family": "bijection", "mapping": "F(x,y,z)=(-256*x^7 + 896*x^6*y - 896*x^6*z + 1920*x^6 - 1344*x^5*y^2 + 2688*x^5*y*z - 5760*x^5*y - 1344*x^5*z^2 + 5760*x^5*z - 6176*x^5 + 1120*x^4*y^3 - 3360*x^4*y^2*z + 7200*x^4*y^2 + 3360*x^4*y*z^2 - 14400*x^4*y*z + 15456*x^4*y - 1120*x^4*z^3 + 7200*x^4*z^2 - 15456*x^4*z + 11024*x^4 - 560*x^3*y^4 + 2240*x^3*y^3*z - 4800*x^3*y^3 - 3360*x^3*y^2*z^2 + 14400*x^3*y^2*z - 15472*x^3*y^2 + 2240*x^3*y*z^3 - 14400*x^3*y*z^2 + 30944*x^3*y*z - 22136*x^3*y - 560*x^3*z^4 + 4800*x^3*z^3 - 15472*x^3*z^2 + 22152*x^3*z - 11816*x^3 + 168*x^2*y^5 - 840*x^2*y^4*z + 1800*x^2*y^4 + 1680*x^2*y^3*z^2 - 7200*x^2*y^3*z + 7744*x^2*y^3 - 1680*x^2*y^2*z^3 + 10800*x^2*y^2*z^2 - 23232*x^2*y^2*z + 16668*x^2*y^2 + 840*x^2*y*z^4 - 7200*x^2*y*z^3 + 23232*x^2*y*z^2 - 33360*x^2*y*z + 17892*x^2*y - 168*x^2*z^5 + 1800*x^2*z^4 - 7744*x^2*z^3 + 16692*x^2*z^2 - 17940*x^2*z + 7644*x^2 - 28*x*y^6 + 168*x*y^5*z - 360*x*y^5 - 420*x*y^4*z^2 + 1800*x*y^4*z - 1938*x*y^4 + 560*x*y^3*z^3 - 3600*x*y^3*z^2 + 7752*x*y^3*z - 5578*x*y^3 - 420*x*y^2*z^4 + 3600*x*y^2*z^3 - 11628*x*y^2*z^2 + 16746*x*y^2*z - 9030*x*y^2 + 168*x*y*z^5 - 1800*x*y*z^4 + 7752*x*y*z^3 - 16758*x*y*z^2 + 18108*x*y*z - 7784*x*y - 28*x*z^6 + 360*x*z^5 - 1938*x*z^4 + 5590*x*z^3 - 9078*x*z^2 + 7834*x*z - 2789*x + 2*y^7 - 14*y^6*z + 30*y^6 + 42*y^5*z^2 - 180*y^5*z + 194*y^5 - 70*y^4*z^3 + 450*y^4*z^2 - 970*y^4*z + 700*y^4 + 70*y^3*z^4 - 600*y^3*z^3 + 1940*y^3*z^2 - 2802*y^3*z + 1519*y^3 - 42*y^2*z^5 + 450*y^2*z^4 - 1940*y^2*z^3 + 4206*y^2*z^2 - 4569*y^2*z + 1982*y^2 + 14*y*z^6 - 180*y*z^5 + 970*y*z^4 - 2806*y*z^3 + 4581*y*z^2 - 3990*y*z + 1439*y - 2*z^7 + 30*z^6 - 194*z^5 + 702*z^4 - 1531*z^3 + 2008*z^2 - 1453*z + 1, -256*x^7 + 896*x^6*y - 896*x^6*z + 1920*x^6 - 1344*x^5*y^2 + 2688*x^5*y*z - 5760*x^5*y - 1344*x^5*z^2 + 5760*x^5*z - 6176*x^5 + 1120*x^4*y^3 - 3360*x^4*y^2*z + 7200*x^4*y^2 + 3360*x^4*y*z^2 - 14400*x^4*y*z + 15456*x^4*y - 1120*x^4*z^3 + 7200*x^4*z^2 - 15456*x^4*z + 11056*x^4 - 560*x^3*y^4 + 2240*x^3*y^3*z - 4800*x^3*y^3 - 3360*x^3*y^2*z^2 + 14400*x^3*y^2*z - 15472*x^3*y^2 + 2240*x^3*y*z^3 - 14400*x^3*y*z^2 + 30944*x^3*y*z - 22200*x^3*y - 560*x^3*z^4 + 4800*x^3*z^3 - 15472*x^3*z^2 + 22216*x^3*z - 11928*x^3 + 168*x^2*y^5 - 840*x^2*y^4*z + 1800*x^2*y^4 + 1680*x^2*y^3*z^2 - 7200*x^2*y^3*z + 7744*x^2*y^3 - 1680*x^2*y^2*z^3 + 10800*x^2*y^2*z^2 - 23232*x^2*y^2*z + 16716*x^2*y^2 + 840*x^2*y*z^4 - 7200*x^2*y*z^3 + 23232*x^2*y*z^2 - 33456*x^2*y*z + 18060*x^2*y - 168*x^2*z^5 + 1800*x^2*z^4 - 7744*x^2*z^3 + 16740*x^2*z^2 - 18108*x^2*z + 7788*x^2 - 28*x*y^6 + 168*x*y^5*z - 360*x*y^5 - 420*x*y^4*z^2 + 1800*x*y^4*z - 1938*x*y^4 + 560*x*y^3*z^3 - 3600*x*y^3*z^2 + 7752*x*y^3*z - 5594*x*y^3 - 420*x*y^2*z^4 + 3600*x*y^2*z^3 - 11628*x*y^2*z^2 + 16794*x*y^2*z - 9114*x*y^2 + 168*x*y*z^5 - 1800*x*y*z^4 + 7752*x*y*z^3 - 16806*x*y*z^2 + 18276*x*y*z - 7928*x*y - 28*x*z^6 + 360*x*z^5 - 1938*x*z^4 + 5606*x*z^3 - 9162*x*z^2 + 7978*x*z - 2869*x + 2*y^7 - 14*y^6*z + 30*y^6 + 42*y^5*z^2 - 180*y^5*z + 194*y^5 - 70*y^4*z^3 + 450*y^4*z^2 - 970*y^4*z + 702*y^4 + 70*y^3*z^4 - 600*y^3*z^3 + 1940*y^3*z^2 - 2810*y^3*z + 1533*y^3 - 42*y^2*z^5 + 450*y^2*z^4 - 1940*y^2*z^3 + 4218*y^2*z^2 - 4611*y^2*z + 2018*y^2 + 14*y*z^6 - 180*y*z^5 + 970*y*z^4 - 2814*y*z^3 + 4623*y*z^2 - 4062*y*z + 1479*y - 2*z^7 + 30*z^6 - 194*z^5 + 704*z^4 - 1545*z^3 + 2044*z^2 - 1495*z + 1, -128*x^7 + 448*x^6*y - 448*x^6*z + 960*x^6 - 672*x^5*y^2 + 1344*x^5*y*z - 2880*x^5*y - 672*x^5*z^2 + 2880*x^5*z - 3088*x^5 + 560*x^4*y^3 - 1680*x^4*y^2*z + 3600*x^4*y^2 + 1680*x^4*y*z^2 - 7200*x^4*y*z + 7728*x^4*y - 560*x^4*z^3 + 3600*x^4*z^2 - 7728*x^4*z + 5512*x^4 - 280*x^3*y^4 + 1120*x^3*y^3*z - 2400*x^3*y^3 - 1680*x^3*y^2*z^2 + 7200*x^3*y^2*z - 7736*x^3*y^2 + 1120*x^3*y*z^3 - 7200*x^3*y*z^2 + 15472*x^3*y*z - 11068*x^3*y - 280*x^3*z^4 + 2400*x^3*z^3 - 7736*x^3*z^2 + 11076*x^3*z - 5912*x^3 + 84*x^2*y^5 - 420*x^2*y^4*z + 900*x^2*y^4 + 840*x^2*y^3*z^2 - 3600*x^2*y^3*z + 3872*x^2*y^3 - 840*x^2*y^2*z^3 + 5400*x^2*y^2*z^2 - 11616*x^2*y^2*z + 8334*x^2*y^2 + 420*x^2*y*z^4 - 3600*x^2*y*z^3 + 11616*x^2*y*z^2 - 16680*x^2*y*z + 8952*x^2*y - 84*x^2*z^5 + 900*x^2*z^4 - 3872*x^2*z^3 + 8346*x^2*z^2 - 8976*x^2*z + 3834*x^2 - 14*x*y^6 + 84*x*y^5*z - 180*x*y^5 - 210*x*y^4*z^2 + 900*x*y^4*z - 969*x*y^4 + 280*x*y^3*z^3 - 1800*x*y^3*z^2 + 3876*x*y^3*z - 2789*x*y^3 - 210*x*y^2*z^4 + 1800*x*y^2*z^3 - 5814*x*y^2*z^2 + 8373*x*y^2*z - 4518*x*y^2 + 84*x*y*z^5 - 900*x*y*z^4 + 3876*x*y*z^3 - 8379*x*y*z^2 + 9060*x*y*z - 3904*x*y - 14*x*z^6 + 180*x*z^5 - 969*x*z^4 + 2795*x*z^3 - 4542*x*z^2 + 3929*x*z - 1407*x + y^7 - 7*y^6*z + 15*y^6 + 21*y^5*z^2 - 90*y^5*z + 97*y^5 - 35*y^4*z^3 + 225*y^4*z^2 - 485*y^4*z + 350*y^4 + 35*y^3*z^4 - 300*y^3*z^3 + 970*y^3*z^2 - 1401*y^3*z + 760*y^3 - 21*y^2*z^5 + 225*y^2*z^4 - 970*y^2*z^3 + 2103*y^2*z^2 - 2286*y^2*z + 994*y^2 + 7*y*z^6 - 90*y*z^5 + 485*y*z^4 - 1403*y*z^3 + 2292*y*z^2 - 2001*y*z + 726*y - z^7 + 15*z^6 - 97*z^5 + 351*z^4 - 766*z^3 + 1007*z^2 - 733*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -2, 1, -1 ], [ -1, 1, -1 ], [ -1, 1, -2 ] ], "source_shift": [ 2, 1, 2 ], "target_matrix": [ [ 2, 3, 2 ], [ 2, 1, 1 ], [ 1, 2, 1 ] ], "target_shift": [ -444, -462, -226 ], "inverse_recipe": { "p": 3, "q": 4, "f": "x**3", "g": "x**4", "h": "y**2 + y*z", "polynomials": { "f": [ [ [ 3, 0, 0 ], 1 ] ], "g": [ [ [ 4, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 2, 0 ], 1 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0710510420030914 ], "surface_features": [ 3.664851343175664, 4.524474015473596, 3.269512944217916, 0.0, 3.7929700753523887, 4901 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C064", "pair": 64, "family": "counterexample", "mapping": "F(x,y,z)=(216*x^7 - 1188*x^6*y + 972*x^6*z - 1944*x^6 + 2754*x^5*y^2 - 4536*x^5*y*z + 9396*x^5*y + 1854*x^5*z^2 - 7272*x^5*z + 5130*x^5 - 3483*x^4*y^3 + 8667*x^4*y^2*z - 18630*x^4*y^2 - 7137*x^4*y*z^2 + 29016*x^4*y*z - 20763*x^4*y + 1945*x^4*z^3 - 11262*x^4*z^2 + 15849*x^4*z - 6594*x^4 + 2592*x^3*y^4 - 8667*x^3*y^3*z + 19359*x^3*y^3 + 10791*x^3*y^2*z^2 - 45558*x^3*y^2*z + 33075*x^3*y^2 - 5929*x^3*y*z^3 + 35607*x^3*y*z^2 - 50859*x^3*y*z + 21552*x^3*y + 1213*x^3*z^4 - 9244*x^3*z^3 + 19482*x^3*z^2 - 16293*x^3*z + 4830*x^3 - 1134*x^2*y^5 + 4779*x^2*y^4*z - 11097*x^2*y^4 - 8001*x^2*y^3*z^2 + 35118*x^2*y^3*z - 25866*x^2*y^3 + 6651*x^2*y^2*z^3 - 41499*x^2*y^2*z^2 + 60165*x^2*y^2*z - 25983*x^2*y^2 - 2745*x^2*y*z^4 + 21708*x^2*y*z^3 - 46455*x^2*y*z^2 + 39585*x^2*y*z - 11997*x^2*y + 450*x^2*z^5 - 4242*x^2*z^4 + 11910*x^2*z^3 - 15021*x^2*z^2 + 8988*x^2*z - 2091*x^2 + 270*x*y^6 - 1377*x*y^5*z + 3321*x*y^5 + 2907*x*y^4*z^2 - 13266*x*y^4*z + 9909*x*y^4 - 3251*x*y^3*z^3 + 21093*x*y^3*z^2 - 31032*x*y^3*z + 13656*x*y^3 + 2031*x*y^2*z^4 - 16692*x*y^2*z^3 + 36267*x*y^2*z^2 - 31494*x*y^2*z + 9765*x*y^2 - 672*x*y*z^5 + 6576*x*y*z^4 - 18753*x*y*z^3 + 24102*x*y*z^2 - 14748*x*y*z + 3528*x*y + 92*x*z^6 - 1032*x*z^5 + 3621*x*z^4 - 6123*x*z^3 + 5547*x*z^2 - 2616*x*z + 505*x - 27*y^7 + 162*y^6*z - 405*y^6 - 414*y^5*z^2 + 1962*y^5*z - 1485*y^5 + 584*y^4*z^3 - 3939*y^4*z^2 + 5877*y^4*z - 2631*y^4 - 491*y^3*z^4 + 4196*y^3*z^3 - 9252*y^3*z^2 + 8181*y^3*z - 2595*y^3 + 246*y^2*z^5 - 2502*y^2*z^4 + 7245*y^2*z^3 - 9486*y^2*z^2 + 5937*y^2*z - 1464*y^2 - 68*y*z^6 + 792*y*z^5 - 2823*y*z^4 + 4864*y*z^3 - 4506*y*z^2 + 2187*y*z - 433*y + 8*z^7 - 104*z^6 + 438*z^5 - 931*z^4 + 1135*z^3 - 814*z^2 + 319*z - 1, -216*x^7 + 1188*x^6*y - 972*x^6*z + 2160*x^6 - 2754*x^5*y^2 + 4536*x^5*y*z - 10368*x^5*y - 1854*x^5*z^2 + 8100*x^5*z - 6786*x^5 + 3483*x^4*y^3 - 8667*x^4*y^2*z + 20412*x^4*y^2 + 7137*x^4*y*z^2 - 32076*x^4*y*z + 27207*x^4*y - 1945*x^4*z^3 + 12564*x^4*z^2 - 20913*x^4*z + 9480*x^4 - 2592*x^3*y^4 + 8667*x^3*y^3*z - 21060*x^3*y^3 - 10791*x^3*y^2*z^2 + 49977*x^3*y^2*z - 42885*x^3*y^2 + 5929*x^3*y*z^3 - 39402*x^3*y*z^2 + 66435*x^3*y*z - 30693*x^3*y - 1213*x^3*z^4 + 10321*x^3*z^3 - 25632*x^3*z^2 + 23346*x^3*z - 7326*x^3 + 1134*x^2*y^5 - 4779*x^2*y^4*z + 11988*x^2*y^4 + 8001*x^2*y^3*z^2 - 38232*x^2*y^3*z + 33147*x^2*y^3 - 6651*x^2*y^2*z^3 + 45549*x^2*y^2*z^2 - 77715*x^2*y^2*z + 36612*x^2*y^2 + 2745*x^2*y*z^4 - 24030*x^2*y*z^3 + 60468*x^2*y*z^2 - 56151*x^2*y*z + 18012*x^2*y - 450*x^2*z^5 + 4737*x^2*z^4 - 15618*x^2*z^3 + 21444*x^2*z^2 - 13578*x^2*z + 3297*x^2 - 270*x*y^6 + 1377*x*y^5*z - 3564*x*y^5 - 2907*x*y^4*z^2 + 14337*x*y^4*z - 12537*x*y^4 + 3251*x*y^3*z^3 - 22968*x*y^3*z^2 + 39594*x*y^3*z - 19014*x*y^3 - 2031*x*y^2*z^4 + 18321*x*y^2*z^3 - 46653*x*y^2*z^2 + 44172*x*y^2*z - 14493*x*y^2 + 672*x*y*z^5 - 7278*x*y*z^4 + 24315*x*y*z^3 - 34041*x*y*z^2 + 22038*x*y*z - 5496*x*y - 92*x*z^6 + 1152*x*z^5 - 4731*x*z^4 + 8706*x*z^3 - 8343*x*z^2 + 4104*x*z - 817*x + 27*y^7 - 162*y^6*z + 432*y^6 + 414*y^5*z^2 - 2106*y^5*z + 1854*y^5 - 584*y^4*z^3 + 4257*y^4*z^2 - 7401*y^4*z + 3615*y^4 + 491*y^3*z^4 - 4568*y^3*z^3 + 11751*y^3*z^2 - 11331*y^3*z + 3801*y^3 - 246*y^2*z^5 + 2745*y^2*z^4 - 9279*y^2*z^3 + 13239*y^2*z^2 - 8763*y^2*z + 2250*y^2 + 68*y*z^6 - 876*y*z^5 + 3645*y*z^4 - 6838*y*z^3 + 6699*y*z^2 - 3387*y*z + 691*y - 8*z^7 + 116*z^6 - 570*z^5 + 1318*z^4 - 1699*z^3 + 1270*z^2 - 513*z + 1, 216*x^7 - 1188*x^6*y + 972*x^6*z - 1944*x^6 + 2754*x^5*y^2 - 4536*x^5*y*z + 9396*x^5*y + 1854*x^5*z^2 - 7272*x^5*z + 5130*x^5 - 3483*x^4*y^3 + 8667*x^4*y^2*z - 18630*x^4*y^2 - 7137*x^4*y*z^2 + 29016*x^4*y*z - 20763*x^4*y + 1945*x^4*z^3 - 11262*x^4*z^2 + 15849*x^4*z - 6586*x^4 + 2592*x^3*y^4 - 8667*x^3*y^3*z + 19359*x^3*y^3 + 10791*x^3*y^2*z^2 - 45558*x^3*y^2*z + 33075*x^3*y^2 - 5929*x^3*y*z^3 + 35607*x^3*y*z^2 - 50859*x^3*y*z + 21532*x^3*y + 1213*x^3*z^4 - 9244*x^3*z^3 + 19482*x^3*z^2 - 16273*x^3*z + 4774*x^3 - 1134*x^2*y^5 + 4779*x^2*y^4*z - 11097*x^2*y^4 - 8001*x^2*y^3*z^2 + 35118*x^2*y^3*z - 25866*x^2*y^3 + 6651*x^2*y^2*z^3 - 41499*x^2*y^2*z^2 + 60165*x^2*y^2*z - 25965*x^2*y^2 - 2745*x^2*y*z^4 + 21708*x^2*y*z^3 - 46455*x^2*y*z^2 + 39549*x^2*y*z - 11889*x^2*y + 450*x^2*z^5 - 4242*x^2*z^4 + 11910*x^2*z^3 - 15003*x^2*z^2 + 8892*x^2*z - 2025*x^2 + 270*x*y^6 - 1377*x*y^5*z + 3321*x*y^5 + 2907*x*y^4*z^2 - 13266*x*y^4*z + 9909*x*y^4 - 3251*x*y^3*z^3 + 21093*x*y^3*z^2 - 31032*x*y^3*z + 13649*x*y^3 + 2031*x*y^2*z^4 - 16692*x*y^2*z^3 + 36267*x*y^2*z^2 - 31473*x*y^2*z + 9699*x*y^2 - 672*x*y*z^5 + 6576*x*y*z^4 - 18753*x*y*z^3 + 24081*x*y*z^2 - 14628*x*y*z + 3441*x*y + 92*x*z^6 - 1032*x*z^5 + 3621*x*z^4 - 6116*x*z^3 + 5493*x*z^2 - 2541*x*z + 481*x - 27*y^7 + 162*y^6*z - 405*y^6 - 414*y^5*z^2 + 1962*y^5*z - 1485*y^5 + 584*y^4*z^3 - 3939*y^4*z^2 + 5877*y^4*z - 2630*y^4 - 491*y^3*z^4 + 4196*y^3*z^3 - 9252*y^3*z^2 + 8177*y^3*z - 2582*y^3 + 246*y^2*z^5 - 2502*y^2*z^4 + 7245*y^2*z^3 - 9480*y^2*z^2 + 5901*y^2*z - 1437*y^2 - 68*y*z^6 + 792*y*z^5 - 2823*y*z^4 + 4860*y*z^3 - 4473*y*z^2 + 2139*y*z - 416*y + 8*z^7 - 104*z^6 + 438*z^5 - 930*z^4 + 1125*z^3 - 793*z^2 + 305*z - 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -2, 1, -1 ], [ -3, 3, -2 ], [ -1, 1, -1 ] ], "source_shift": [ 1, 1, 1 ], "target_matrix": [ [ -1, -1, -1 ], [ 1, 2, 1 ], [ -1, -1, -2 ] ], "target_shift": [ 53, -87, 51 ], "collision_preimages": [ [ "-1/4", "5/4", "11/4" ], [ "11/2", "-8", "-19" ], [ "15/2", "-3", "-16" ] ], "collision_image": [ "213/4", "-349/4", "205/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.01964685799976 ], "surface_features": [ 4.034799854831283, 4.890504851342802, 3.7101173651118162, 0.0, 4.1359681211619, 5072 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C056", "pair": 56, "family": "counterexample", "mapping": "F(x,y,z)=(-48*x^7 + 264*x^6*y - 248*x^6*z + 364*x^6 - 612*x^5*y^2 + 1152*x^5*y*z - 1644*x^5*y - 540*x^5*z^2 + 1560*x^5*z - 1152*x^5 + 774*x^4*y^3 - 2190*x^4*y^2*z + 3051*x^4*y^2 + 2058*x^4*y*z^2 - 5790*x^4*y*z + 4224*x^4*y - 642*x^4*z^3 + 2739*x^4*z^2 - 4020*x^4*z + 2044*x^4 - 576*x^3*y^4 + 2178*x^3*y^3*z - 2976*x^3*y^3 - 3078*x^3*y^2*z^2 + 8475*x^3*y^2*z - 6114*x^3*y^2 + 1926*x^3*y*z^3 - 8022*x^3*y*z^2 + 11628*x^3*y*z - 5854*x^3*y - 450*x^3*z^4 + 2523*x^3*z^3 - 5514*x^3*z^2 + 5570*x^3*z - 2196*x^3 + 252*x^2*y^5 - 1194*x^2*y^4*z + 1608*x^2*y^4 + 2256*x^2*y^3*z^2 - 6111*x^2*y^3*z + 4365*x^2*y^3 - 2124*x^2*y^2*z^3 + 8685*x^2*y^2*z^2 - 12450*x^2*y^2*z + 6207*x^2*y^2 + 996*x^2*y*z^4 - 5469*x^2*y*z^3 + 11805*x^2*y*z^2 - 11796*x^2*y*z + 4608*x^2*y - 186*x^2*z^5 + 1287*x^2*z^4 - 3720*x^2*z^3 + 5589*x^2*z^2 - 4368*x^2*z + 1427*x^2 - 60*x*y^6 + 342*x*y^5*z - 456*x*y^5 - 810*x*y^4*z^2 + 2169*x*y^4*z - 1536*x*y^4 + 1020*x*y^3*z^3 - 4116*x*y^3*z^2 + 5844*x*y^3*z - 2888*x*y^3 - 720*x*y^2*z^4 + 3894*x*y^2*z^3 - 8316*x*y^2*z^2 + 8229*x*y^2*z - 3186*x*y^2 + 270*x*y*z^5 - 1836*x*y*z^4 + 5244*x*y*z^3 - 7794*x*y*z^2 + 6030*x*y*z - 1946*x*y - 42*x*z^6 + 345*x*z^5 - 1236*x*z^4 + 2453*x*z^3 - 2844*x*z^2 + 1829*x*z - 512*x + 6*y^7 - 40*y^6*z + 53*y^6 + 114*y^5*z^2 - 303*y^5*z + 213*y^5 - 180*y^4*z^3 + 720*y^4*z^2 - 1014*y^4*z 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6552*x*y*z^3 + 10224*x*y*z^2 - 8106*x*y*z + 2642*x*y + 42*x*z^6 - 396*x*z^5 + 1542*x*z^4 - 3215*x*z^3 + 3822*x*z^2 - 2483*x*z + 698*x - 6*y^7 + 40*y^6*z - 62*y^6 - 114*y^5*z^2 + 354*y^5*z - 270*y^5 + 180*y^4*z^3 - 840*y^4*z^2 + 1284*y^4*z - 659*y^4 - 170*y^3*z^4 + 1060*y^3*z^3 - 2436*y^3*z^2 + 2503*y^3*z - 981*y^3 + 96*y^2*z^5 - 750*y^2*z^4 + 2304*y^2*z^3 - 3555*y^2*z^2 + 2784*y^2*z - 899*y^2 - 30*y*z^6 + 282*y*z^5 - 1086*y*z^4 + 2237*y*z^3 - 2625*y*z^2 + 1687*y*z - 473*y + 4*z^7 - 44*z^6 + 204*z^5 - 526*z^4 + 822*z^3 - 788*z^2 + 436*z - 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -1, 1, -1 ], [ -2, 1, -1 ], [ -3, 3, -2 ] ], "source_shift": [ 1, 1, 1 ], "target_matrix": [ [ 2, 1, 1 ], [ -1, -1, -1 ], [ -2, -2, -3 ] ], "target_shift": [ -75, 53, 105 ], "collision_preimages": [ [ "0", "3/4", "7/4" ], [ "5/2", "8", "11/2" ], [ "-5/2", "7", "23/2" ] ], 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3888*y^4*z - 2783*y^4 + 192*y^3*z^4 - 1692*y^3*z^3 + 5520*y^3*z^2 - 7910*y^3*z + 4226*y^3 + 84*y^2*z^5 - 900*y^2*z^4 + 3867*y^2*z^3 - 8298*y^2*z^2 + 8913*y^2*z - 3851*y^2 + 20*y*z^6 - 252*y*z^5 + 1338*y*z^4 - 3814*y*z^3 + 6156*y*z^2 - 5354*y*z + 1970*y + 2*z^7 - 29*z^6 + 183*z^5 - 649*z^4 + 1396*z^3 - 1826*z^2 + 1352*z - 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ 2, -1, -1 ], [ 1, -2, -1 ], [ 1, -1, -1 ] ], "source_shift": [ 1, 2, 1 ], "target_matrix": [ [ -1, -2, -1 ], [ 1, 1, 1 ], [ -2, -1, -1 ] ], "target_shift": [ 439, -290, 437 ], "collision_preimages": [ [ "1/4", "3/4", "3/4" ], [ "-11/2", "9", "-20" ], [ "-15/2", "6", "-19" ] ], "collision_image": [ "1757/4", "-1161/4", "875/2" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.1018184429995017 ], "surface_features": [ 3.5662377480667797, 4.394574246649326, 3.2337573629655103, 0.0, 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8952*x*z^3 + 12156*x*z^2 + 8656*x*z + 2487*x + 12*y^7 + 84*y^6*z + 132*y^6 + 252*y^5*z^2 + 804*y^5*z + 630*y^5 + 420*y^4*z^3 + 2040*y^4*z^2 + 3294*y^4*z + 1752*y^4 + 420*y^3*z^4 + 2760*y^3*z^3 + 6876*y^3*z^2 + 7608*y^3*z + 3120*y^3 + 252*y^2*z^5 + 2100*y^2*z^4 + 7164*y^2*z^3 + 12384*y^2*z^2 + 10644*y^2*z + 3584*y^2 + 84*y*z^6 + 852*y*z^5 + 3726*y*z^4 + 8952*y*z^3 + 12144*y*z^2 + 8608*y*z + 2457*y + 12*z^7 + 144*z^6 + 774*z^5 + 2424*z^4 + 4644*z^3 + 5240*z^2 + 3155*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ 1, 1, 1 ], [ 1, 1, 2 ], [ 2, 1, 1 ] ], "source_shift": [ 1, 2, 1 ], "target_matrix": [ [ -1, -1, -1 ], [ -1, -1, -2 ], [ 3, 2, 3 ] ], "target_shift": [ 259, 251, -776 ], "inverse_recipe": { "p": 2, "q": 5, "f": "x**2", "g": "x**5 + x**4 + x**2", "h": "2*y**3 + y*z", "polynomials": { "f": [ [ [ 2, 0, 0 ], 1 ] ], "g": [ [ [ 5, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ], [ [ 4, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 3, 0 ], 2 ] ] }, "h_multiplier": 4 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0326982130027318 ], "surface_features": [ 3.4356187756900907, 4.393890939002799, 3.079543007402906, 0.0, 3.59467041567649, 4846 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C186", "pair": 186, "family": "counterexample", "mapping": "F(x,y,z)=(-16*x^7 + 88*x^6*y + 136*x^6*z - 284*x^6 - 204*x^5*y^2 - 624*x^5*y*z + 1260*x^5*y - 468*x^5*z^2 + 1920*x^5*z - 1800*x^5 + 258*x^4*y^3 + 1170*x^4*y^2*z - 2295*x^4*y^2 + 1734*x^4*y*z^2 - 6894*x^4*y*z + 6432*x^4*y + 838*x^4*z^3 - 5067*x^4*z^2 + 9648*x^4*z - 5871*x^4 - 192*x^3*y^4 - 1146*x^3*y^3*z + 2196*x^3*y^3 - 2514*x^3*y^2*z^2 + 9753*x^3*y^2*z - 9060*x^3*y^2 - 2398*x^3*y*z^3 + 14130*x^3*y*z^2 - 26748*x^3*y*z + 16350*x^3*y - 838*x^3*z^4 + 6665*x^3*z^3 - 19296*x^3*z^2 + 24103*x^3*z - 11010*x^3 + 84*x^2*y^5 + 618*x^2*y^4*z - 1164*x^2*y^4 + 1782*x^2*y^3*z^2 - 6795*x^2*y^3*z + 6291*x^2*y^3 + 2514*x^2*y^2*z^3 - 14562*x^2*y^2*z^2 + 27435*x^2*y^2*z - 16830*x^2*y^2 + 1734*x^2*y*z^4 - 13563*x^2*y*z^3 + 38997*x^2*y*z^2 - 48795*x^2*y*z + 22467*x^2*y + 468*x^2*z^5 - 4632*x^2*z^4 + 18045*x^2*z^3 - 34530*x^2*z^2 + 32481*x^2*z - 12041*x^2 - 20*x*y^6 - 174*x*y^5*z + 324*x*y^5 - 618*x*y^4*z^2 + 2331*x*y^4*z - 2154*x*y^4 - 1146*x*y^3*z^3 + 6570*x*y^3*z^2 - 12342*x*y^3*z + 7596*x*y^3 - 1170*x*y^2*z^4 + 9063*x*y^2*z^3 - 25950*x*y^2*z^2 + 32523*x*y^2*z - 15072*x*y^2 - 624*x*y*z^5 + 6120*x*y*z^4 - 23718*x*y*z^3 + 45366*x*y*z^2 - 42852*x*y*z + 16022*x*y - 136*x*z^6 + 1620*x*z^5 - 7956*x*z^4 + 20610*x*z^3 - 29718*x*z^2 + 22661*x*z - 7154*x + 2*y^7 + 20*y^6*z - 37*y^6 + 84*y^5*z^2 - 315*y^5*z + 291*y^5 + 192*y^4*z^3 - 1095*y^4*z^2 + 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30090*x*y^2*z^2 - 39495*x*y^2*z + 18888*x*y^2 + 624*x*y*z^5 - 6684*x*y*z^4 + 27672*x*y*z^3 - 55506*x*y*z^2 + 54144*x*y*z - 20660*x*y + 136*x*z^6 - 1776*x*z^5 + 9336*x*z^4 - 25394*x*z^3 + 37842*x*z^2 - 29462*x*z + 9412*x - 2*y^7 - 20*y^6*z + 40*y^6 - 84*y^5*z^2 + 342*y^5*z - 336*y^5 - 192*y^4*z^3 + 1194*y^4*z^2 - 2388*y^4*z + 1543*y^4 - 258*y^3*z^4 + 2178*y^3*z^3 - 6648*y^3*z^2 + 8745*y^3*z - 4205*y^3 - 204*y^2*z^5 + 2190*y^2*z^4 - 9060*y^2*z^3 + 18180*y^2*z^2 - 17793*y^2*z + 6836*y^2 - 88*y*z^6 + 1152*y*z^5 - 6048*y*z^4 + 16430*y*z^3 - 24510*y*z^2 + 19162*y*z - 6174*y - 16*z^7 + 248*z^6 - 1584*z^5 + 5452*z^4 - 10996*z^3 + 13088*z^2 - 8584*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ 1, -1, -2 ], [ 2, -1, -1 ], [ 1, -1, -1 ] ], "source_shift": [ 2, 2, 2 ], "target_matrix": [ [ -2, -1, -1 ], [ -1, -1, -1 ], [ 2, 2, 3 ] ], "target_shift": [ 1789, 1219, -2403 ], "collision_preimages": [ [ "1/4", "11/4", "-1/4" ], [ "-8", "-18", "11/2" ], [ "-5", "-17", "15/2" ] ], "collision_image": [ "3579/2", "4877/4", "-4807/2" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 0.9997674480000569 ], "surface_features": [ 3.887964384669162, 4.768734271659541, 3.432568465297358, 0.0, 4.027587543432798, 4978 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C113", "pair": 113, "family": "counterexample", "mapping": "F(x,y,z)=(3*x^7 + 20*x^6*y + 29*x^6*z + 31*x^6 + 57*x^5*y^2 + 165*x^5*y*z + 177*x^5*y + 117*x^5*z^2 + 255*x^5*z + 135*x^5 + 90*x^4*y^3 + 390*x^4*y^2*z + 420*x^4*y^2 + 552*x^4*y*z^2 + 1209*x^4*y*z + 642*x^4*y + 255*x^4*z^3 + 855*x^4*z^2 + 930*x^4*z + 328*x^4 + 85*x^3*y^4 + 490*x^3*y^3*z + 530*x^3*y^3 + 1038*x^3*y^2*z^2 + 2286*x^3*y^2*z + 1218*x^3*y^2 + 957*x^3*y*z^3 + 3231*x^3*y*z^2 + 3528*x^3*y*z + 1246*x^3*y + 324*x^3*z^4 + 1497*x^3*z^3 + 2514*x^3*z^2 + 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41877*x*z^5 + 70200*x*z^4 + 61560*x*z^3 + 29847*x*z^2 + 7648*x*z + 824*x + 128*y^7 + 1344*y^6*z + 960*y^6 + 6048*y^5*z^2 + 8576*y^5*z + 2976*y^5 + 15120*y^4*z^3 + 31920*y^4*z^2 + 22000*y^4*z + 4960*y^4 + 22680*y^3*z^4 + 63360*y^3*z^3 + 65040*y^3*z^2 + 29120*y^3*z + 4812*y^3 + 20412*y^2*z^5 + 70740*y^2*z^4 + 96120*y^2*z^3 + 64080*y^2*z^2 + 21016*y^2*z + 2738*y^2 + 10206*y*z^6 + 42120*y*z^5 + 71010*y*z^4 + 62640*y*z^3 + 30567*y*z^2 + 7893*y*z + 861*y + 2187*z^7 + 10449*z^6 + 20979*z^5 + 22950*z^4 + 14805*z^3 + 5678*z^2 + 1219*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ 2, 2, 3 ], [ 1, 2, 1 ], [ 1, 1, 2 ] ], "source_shift": [ 2, 1, 2 ], "target_matrix": [ [ 1, 2, 1 ], [ 1, 1, 2 ], [ 1, 1, 1 ] ], "target_shift": [ -119, -42, -114 ], "inverse_recipe": { "p": 2, "q": 5, "f": "x**2", "g": "x**5 + x", "h": "y*z", "polynomials": { "f": [ [ [ 2, 0, 0 ], 1 ] ], "g": [ [ [ 5, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.044252203002543 ], "surface_features": [ 4.61296318684176, 5.2854672904600575, 4.326028468337008, 0.0, 4.675696094124643, 5294 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B097", "pair": 97, "family": "bijection", "mapping": "F(x,y,z)=(-2187*x^7 + 10206*x^6*y + 10206*x^6*z - 5589*x^6 - 20412*x^5*y^2 - 40824*x^5*y*z + 22518*x^5*y - 20412*x^5*z^2 + 22518*x^5*z - 7236*x^5 + 22680*x^4*y^3 + 68040*x^4*y^2*z - 37800*x^4*y^2 + 68040*x^4*y*z^2 - 75600*x^4*y*z + 24435*x^4*y + 22680*x^4*z^3 - 37800*x^4*z^2 + 24516*x^4*z - 5796*x^4 - 15120*x^3*y^4 - 60480*x^3*y^3*z + 33840*x^3*y^3 - 90720*x^3*y^2*z^2 + 101520*x^3*y^2*z - 33003*x^3*y^2 - 60480*x^3*y*z^3 + 101520*x^3*y*z^2 - 66222*x^3*y*z + 15756*x^3*y - 15120*x^3*z^4 + 33840*x^3*z^3 - 33219*x^3*z^2 + 15882*x^3*z - 3040*x^3 + 6048*x^2*y^5 + 30240*x^2*y^4*z - 17040*x^2*y^4 + 60480*x^2*y^3*z^2 - 68160*x^2*y^3*z + 22286*x^2*y^3 + 60480*x^2*y^2*z^3 - 102240*x^2*y^2*z^2 + 67074*x^2*y^2*z - 16061*x^2*y^2 + 30240*x^2*y*z^4 - 68160*x^2*y*z^3 + 67290*x^2*y*z^2 - 32380*x^2*y*z + 6247*x^2*y + 6048*x^2*z^5 - 17040*x^2*z^4 + 22502*x^2*z^3 - 16319*x^2*z^2 + 6350*x^2*z - 1053*x^2 - 1344*x*y^6 - 8064*x*y^5*z + 4576*x*y^5 - 20160*x*y^4*z^2 + 22880*x*y^4*z - 7524*x*y^4 - 26880*x*y^3*z^3 + 45760*x*y^3*z^2 - 30192*x*y^3*z + 7276*x*y^3 - 20160*x*y^2*z^4 + 45760*x*y^2*z^3 - 45432*x*y^2*z^2 + 22004*x*y^2*z - 4279*x*y^2 - 8064*x*y*z^5 + 22880*x*y*z^4 - 30384*x*y*z^3 + 22180*x*y*z^2 - 8700*x*y*z + 1458*x*y - 1344*x*z^6 + 4576*x*z^5 - 7620*x*z^4 + 7452*x*z^3 - 4421*x*z^2 + 1500*x*z - 227*x + 128*y^7 + 896*y^6*z - 512*y^6 + 2688*y^5*z^2 - 3072*y^5*z + 1016*y^5 + 4480*y^4*z^3 - 7680*y^4*z^2 + 5096*y^4*z - 1236*y^4 + 4480*y^3*z^4 - 10240*y^3*z^3 + 10224*y^3*z^2 - 4984*y^3*z + 977*y^3 + 2688*y^2*z^5 - 7680*y^2*z^4 + 10256*y^2*z^3 - 7536*y^2*z^2 + 2980*y^2*z - 505*y^2 + 896*y*z^6 - 3072*y*z^5 + 5144*y*z^4 - 5064*y*z^3 + 3029*y*z^2 - 1040*y*z + 160*y + 128*z^7 - 512*z^6 + 1032*z^5 - 1276*z^4 + 1026*z^3 - 535*z^2 + 167*z - 1, -2187*x^7 + 10206*x^6*y + 10206*x^6*z - 5589*x^6 - 20412*x^5*y^2 - 40824*x^5*y*z + 22518*x^5*y - 20412*x^5*z^2 + 22518*x^5*z - 7236*x^5 + 22680*x^4*y^3 + 68040*x^4*y^2*z - 37800*x^4*y^2 + 68040*x^4*y*z^2 - 75600*x^4*y*z + 24435*x^4*y + 22680*x^4*z^3 - 37800*x^4*z^2 + 24516*x^4*z - 5796*x^4 - 15120*x^3*y^4 - 60480*x^3*y^3*z + 33840*x^3*y^3 - 90720*x^3*y^2*z^2 + 101520*x^3*y^2*z - 33003*x^3*y^2 - 60480*x^3*y*z^3 + 101520*x^3*y*z^2 - 66222*x^3*y*z + 15756*x^3*y - 15120*x^3*z^4 + 33840*x^3*z^3 - 33219*x^3*z^2 + 15882*x^3*z - 3040*x^3 + 6048*x^2*y^5 + 30240*x^2*y^4*z - 17040*x^2*y^4 + 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16319*x^2*z^2 - 6242*x^2*z + 999*x^2 + 1344*x*y^6 + 8064*x*y^5*z - 4576*x*y^5 + 20160*x*y^4*z^2 - 22880*x*y^4*z + 7524*x*y^4 + 26880*x*y^3*z^3 - 45760*x*y^3*z^2 + 30192*x*y^3*z - 7276*x*y^3 + 20160*x*y^2*z^4 - 45760*x*y^2*z^3 + 45432*x*y^2*z^2 - 22004*x*y^2*z + 4207*x*y^2 + 8064*x*y*z^5 - 22880*x*y*z^4 + 30384*x*y*z^3 - 22180*x*y*z^2 + 8556*x*y*z - 1386*x*y + 1344*x*z^6 - 4576*x*z^5 + 7620*x*z^4 - 7452*x*z^3 + 4349*x*z^2 - 1428*x*z + 201*x - 128*y^7 - 896*y^6*z + 512*y^6 - 2688*y^5*z^2 + 3072*y^5*z - 1016*y^5 - 4480*y^4*z^3 + 7680*y^4*z^2 - 5096*y^4*z + 1236*y^4 - 4480*y^3*z^4 + 10240*y^3*z^3 - 10224*y^3*z^2 + 4984*y^3*z - 961*y^3 - 2688*y^2*z^5 + 7680*y^2*z^4 - 10256*y^2*z^3 + 7536*y^2*z^2 - 2932*y^2*z + 481*y^2 - 896*y*z^6 + 3072*y*z^5 - 5144*y*z^4 + 5064*y*z^3 - 2981*y*z^2 + 992*y*z - 142*y - 128*z^7 + 512*z^6 - 1032*z^5 + 1276*z^4 - 1010*z^3 + 511*z^2 - 147*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ 3, -2, -2 ], [ 1, -1, -1 ], [ 1, -1, -2 ] ], "source_shift": [ 1, 2, 1 ], "target_matrix": [ [ -1, -1, -1 ], [ -1, -2, -1 ], [ 1, 1, 2 ] ], "target_shift": [ 24, 27, -18 ], "inverse_recipe": { "p": 2, "q": 3, "f": "x**2", "g": "x**3 + x", "h": "y**2*z", "polynomials": { "f": [ [ [ 2, 0, 0 ], 1 ] ], "g": [ [ [ 3, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "h": [ [ [ 0, 2, 1 ], 1 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0487123309976596 ], "surface_features": [ 4.31844630566003, 5.009620840814325, 4.008855563996213, 0.0, 4.388998697143906, 5193 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C096", "pair": 96, "family": "counterexample", "mapping": "F(x,y,z)=(x^7 - 8*x^6*y + 10*x^6*z - 16*x^6 + 27*x^5*y^2 - 69*x^5*y*z + 108*x^5*y + 42*x^5*z^2 - 135*x^5*z + 105*x^5 - 50*x^4*y^3 + 195*x^4*y^2*z - 300*x^4*y^2 - 243*x^4*y*z^2 + 765*x^4*y*z - 585*x^4*y + 96*x^4*z^3 - 465*x^4*z^2 + 726*x^4*z - 369*x^4 + 55*x^3*y^4 - 290*x^3*y^3*z + 440*x^3*y^3 + 552*x^3*y^2*z^2 - 1710*x^3*y^2*z + 1290*x^3*y^2 - 447*x^3*y*z^3 + 2124*x^3*y*z^2 - 3261*x^3*y*z + 1637*x^3*y + 129*x^3*z^4 - 837*x^3*z^3 + 1965*x^3*z^2 - 1999*x^3*z + 754*x^3 - 36*x^2*y^5 + 240*x^2*y^4*z - 360*x^2*y^4 - 618*x^2*y^3*z^2 + 1890*x^2*y^3*z - 1410*x^2*y^3 + 765*x^2*y^2*z^3 - 3582*x^2*y^2*z^2 + 5427*x^2*y^2*z - 2697*x^2*y^2 - 453*x^2*y*z^4 + 2889*x^2*y*z^3 - 6672*x^2*y*z^2 + 6705*x^2*y*z - 2508*x^2*y + 102*x^2*z^5 - 831*x^2*z^4 + 2604*x^2*z^3 - 3969*x^2*z^2 + 2985*x^2*z - 898*x^2 + 13*x*y^6 - 105*x*y^5*z + 156*x*y^5 + 342*x*y^4*z^2 - 1035*x*y^4*z + 765*x*y^4 - 573*x*y^3*z^3 + 2652*x*y^3*z^2 - 3975*x*y^3*z + 1959*x*y^3 + 519*x*y^2*z^4 - 3267*x*y^2*z^3 + 7449*x*y^2*z^2 - 7413*x*y^2*z + 2754*x*y^2 - 240*x*y*z^5 + 1926*x*y*z^4 - 5940*x*y*z^3 + 8946*x*y*z^2 - 6675*x*y*z + 2000*x*y + 44*x*z^6 - 432*x*z^5 + 1692*x*z^4 - 3429*x*z^3 + 3846*x*z^2 - 2294*x*z + 577*x - 2*y^7 + 19*y^6*z - 28*y^6 - 75*y^5*z^2 + 225*y^5*z - 165*y^5 + 159*y^4*z^3 - 729*y^4*z^2 + 1083*y^4*z - 530*y^4 - 195*y^3*z^4 + 1215*y^3*z^3 - 2742*y^3*z^2 + 2707*y^3*z - 1000*y^3 + 138*y^2*z^5 - 1095*y^2*z^4 + 3336*y^2*z^3 - 4977*y^2*z^2 + 3690*y^2*z - 1102*y^2 - 52*y*z^6 + 504*y*z^5 - 1944*y*z^4 + 3893*y*z^3 - 4332*y*z^2 + 2576*y*z - 650*y + 8*z^7 - 92*z^6 + 432*z^5 - 1090*z^4 + 1618*z^3 - 1432*z^2 + 712*z - 1, -x^7 + 8*x^6*y - 10*x^6*z + 16*x^6 - 27*x^5*y^2 + 69*x^5*y*z - 108*x^5*y - 42*x^5*z^2 + 135*x^5*z - 105*x^5 + 50*x^4*y^3 - 195*x^4*y^2*z + 300*x^4*y^2 + 243*x^4*y*z^2 - 765*x^4*y*z + 585*x^4*y - 96*x^4*z^3 + 465*x^4*z^2 - 726*x^4*z + 368*x^4 - 55*x^3*y^4 + 290*x^3*y^3*z - 440*x^3*y^3 - 552*x^3*y^2*z^2 + 1710*x^3*y^2*z - 1290*x^3*y^2 + 447*x^3*y*z^3 - 2124*x^3*y*z^2 + 3261*x^3*y*z - 1632*x^3*y - 129*x^3*z^4 + 837*x^3*z^3 - 1965*x^3*z^2 + 1992*x^3*z - 744*x^3 + 36*x^2*y^5 - 240*x^2*y^4*z + 360*x^2*y^4 + 618*x^2*y^3*z^2 - 1890*x^2*y^3*z + 1410*x^2*y^3 - 765*x^2*y^2*z^3 + 3582*x^2*y^2*z^2 - 5427*x^2*y^2*z + 2688*x^2*y^2 + 453*x^2*y*z^4 - 2889*x^2*y*z^3 + 6672*x^2*y*z^2 - 6678*x^2*y*z + 2472*x^2*y - 102*x^2*z^5 + 831*x^2*z^4 - 2604*x^2*z^3 + 3951*x^2*z^2 - 2934*x^2*z + 865*x^2 - 13*x*y^6 + 105*x*y^5*z - 156*x*y^5 - 342*x*y^4*z^2 + 1035*x*y^4*z - 765*x*y^4 + 573*x*y^3*z^3 - 2652*x*y^3*z^2 + 3975*x*y^3*z - 1952*x*y^3 - 519*x*y^2*z^4 + 3267*x*y^2*z^3 - 7449*x*y^2*z^2 + 7380*x*y^2*z - 2712*x*y^2 + 240*x*y*z^5 - 1926*x*y*z^4 + 5940*x*y*z^3 - 8898*x*y*z^2 + 6549*x*y*z - 1922*x*y - 44*x*z^6 + 432*x*z^5 - 1692*x*z^4 + 3409*x*z^3 - 3762*x*z^2 + 2186*x*z - 535*x + 2*y^7 - 19*y^6*z + 28*y^6 + 75*y^5*z^2 - 225*y^5*z + 165*y^5 - 159*y^4*z^3 + 729*y^4*z^2 - 1083*y^4*z + 528*y^4 + 195*y^3*z^4 - 1215*y^3*z^3 + 2742*y^3*z^2 - 2694*y^3*z + 984*y^3 - 138*y^2*z^5 + 1095*y^2*z^4 - 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1, -2 ], [ -1, 1, -1 ], [ -1, 2, -1 ] ], "source_shift": [ 2, 1, 1 ], "target_matrix": [ [ -1, -1, -1 ], [ 1, 1, 2 ], [ -2, -1, -1 ] ], "target_shift": [ 155, -139, 212 ], "collision_preimages": [ [ "-1/4", "-1/4", "1" ], [ "12", "8", "-3/2" ], [ "1", "5", "7/2" ] ], "collision_image": [ "621/4", "-557/4", "425/2" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.115334702997643 ], "surface_features": [ 3.2645686091534305, 4.130526745384164, 2.929929560084588, 0.0, 3.3646695079030065, 4790 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C013", "pair": 13, "family": "counterexample", "mapping": "F(x,y,z)=(-2*x^7 - 13*x^6*y - 19*x^6*z - 29*x^6 - 36*x^5*y^2 - 105*x^5*y*z - 162*x^5*y - 75*x^5*z^2 - 234*x^5*z - 156*x^5 - 55*x^4*y^3 - 240*x^4*y^2*z - 375*x^4*y^2 - 342*x^4*y*z^2 - 1083*x^4*y*z - 732*x^4*y - 159*x^4*z^3 - 768*x^4*z^2 - 1068*x^4*z - 444*x^4 - 50*x^3*y^4 - 290*x^3*y^3*z - 460*x^3*y^3 - 618*x^3*y^2*z^2 - 1992*x^3*y^2*z - 1368*x^3*y^2 - 573*x^3*y*z^3 - 2826*x^3*y*z^2 - 3996*x^3*y*z - 1673*x^3*y - 195*x^3*z^4 - 1314*x^3*z^3 - 2868*x^3*z^2 - 2462*x^3*z - 753*x^3 - 27*x^2*y^5 - 195*x^2*y^4*z - 315*x^2*y^4 - 552*x^2*y^3*z^2 - 1818*x^2*y^3*z - 1272*x^2*y^3 - 765*x^2*y^2*z^3 - 3870*x^2*y^2*z^2 - 5580*x^2*y^2*z - 2355*x^2*y^2 - 519*x^2*y*z^4 - 3603*x^2*y*z^3 - 8025*x^2*y*z^2 - 6945*x^2*y*z - 2130*x^2*y - 138*x^2*z^5 - 1239*x^2*z^4 - 3783*x^2*z^3 - 5031*x^2*z^2 - 3162*x^2*z - 778*x^2 - 8*x*y^6 - 69*x*y^5*z - 114*x*y^5 - 243*x*y^4*z^2 - 822*x*y^4*z - 588*x*y^4 - 447*x*y^3*z^3 - 2334*x*y^3*z^2 - 3444*x*y^3*z - 1467*x*y^3 - 453*x*y^2*z^4 - 3264*x*y^2*z^3 - 7446*x*y^2*z^2 - 6504*x*y^2*z - 2001*x*y^2 - 240*x*y*z^5 - 2250*x*y*z^4 - 7041*x*y*z^3 - 9453*x*y*z^2 - 5958*x*y*z - 1463*x*y - 52*x*z^6 - 612*x*z^5 - 2457*x*z^4 - 4503*x*z^3 - 4353*x*z^2 - 2188*x*z - 459*x - y^7 - 10*y^6*z - 17*y^6 - 42*y^5*z^2 - 147*y^5*z - 108*y^5 - 96*y^4*z^3 - 522*y^4*z^2 - 792*y^4*z - 341*y^4 - 129*y^3*z^4 - 975*y^3*z^3 - 2289*y^3*z^2 - 2021*y^3*z - 624*y^3 - 102*y^2*z^5 - 1011*y^2*z^4 - 3258*y^2*z^3 - 4422*y^2*z^2 - 2796*y^2*z - 685*y^2 - 44*y*z^6 - 552*y*z^5 - 2283*y*z^4 - 4231*y*z^3 - 4104*y*z^2 - 2059*y*z - 428*y - 8*z^7 - 124*z^6 - 630*z^5 - 1492*z^4 - 1971*z^3 - 1519*z^2 - 642*z - 1, -2*x^7 - 13*x^6*y - 19*x^6*z - 23*x^6 - 36*x^5*y^2 - 105*x^5*y*z - 129*x^5*y - 75*x^5*z^2 - 189*x^5*z - 111*x^5 - 55*x^4*y^3 - 240*x^4*y^2*z - 300*x^4*y^2 - 342*x^4*y*z^2 - 879*x^4*y*z - 522*x^4*y - 159*x^4*z^3 - 633*x^4*z^2 - 771*x^4*z - 297*x^4 - 50*x^3*y^4 - 290*x^3*y^3*z - 370*x^3*y^3 - 618*x^3*y^2*z^2 - 1626*x^3*y^2*z - 978*x^3*y^2 - 573*x^3*y*z^3 - 2343*x^3*y*z^2 - 2892*x^3*y*z - 1121*x^3*y - 195*x^3*z^4 - 1107*x^3*z^3 - 2103*x^3*z^2 - 1670*x^3*z - 483*x^3 - 27*x^2*y^5 - 195*x^2*y^4*z - 255*x^2*y^4 - 552*x^2*y^3*z^2 - 1494*x^2*y^3*z - 912*x^2*y^3 - 765*x^2*y^2*z^3 - 3231*x^2*y^2*z^2 - 4050*x^2*y^2*z - 1581*x^2*y^2 - 519*x^2*y*z^4 - 3057*x^2*y*z^3 - 5901*x^2*y*z^2 - 4719*x^2*y*z - 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+ 1008*x*z^5 + 3132*x*z^4 + 4934*x*z^3 + 4302*x*z^2 + 2009*x*z + 399*x + 2*y^7 + 20*y^6*z + 25*y^6 + 84*y^5*z^2 + 222*y^5*z + 126*y^5 + 192*y^4*z^3 + 810*y^4*z^2 + 945*y^4*z + 350*y^4 + 258*y^3*z^4 + 1554*y^3*z^3 + 2796*y^3*z^2 + 2120*y^3*z + 590*y^3 + 204*y^2*z^5 + 1653*y^2*z^4 + 4077*y^2*z^3 + 4746*y^2*z^2 + 2703*y^2*z + 614*y^2 + 88*y*z^6 + 924*y*z^5 + 2928*y*z^4 + 4650*y*z^3 + 4062*y*z^2 + 1892*y*z + 372*y + 16*z^7 + 212*z^6 + 828*z^5 + 1680*z^4 + 2000*z^3 + 1433*z^2 + 575*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ 1, 1, 1 ], [ 1, 1, 2 ], [ 2, 1, 1 ] ], "source_shift": [ 1, 1, 2 ], "target_matrix": [ [ -1, -2, -1 ], [ -1, -1, -1 ], [ 2, 1, 1 ] ], "target_shift": [ 118, 72, -102 ], "collision_preimages": [ [ "-5/4", "1/4", "0" ], [ "9/2", "-2", "-5/2" ], [ "13/2", "-11", "5/2" ] ], "collision_image": [ "473/4", "289/4", "-205/2" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.000660773999698 ], "surface_features": [ 3.1906055753393137, 4.004751155591001, 2.88024177589548, 0.0, 3.2838693365310303, 4773 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C177", "pair": 177, "family": "counterexample", "mapping": "F(x,y,z)=(-2*x^7 + 19*x^6*y + 13*x^6*z - 35*x^6 - 75*x^5*y^2 - 105*x^5*y*z + 285*x^5*y - 36*x^5*z^2 + 195*x^5*z - 261*x^5 + 159*x^4*y^3 + 342*x^4*y^2*z - 936*x^4*y^2 + 240*x^4*y*z^2 - 1311*x^4*y*z + 1764*x^4*y + 55*x^4*z^3 - 450*x^4*z^2 + 1212*x^4*z - 1079*x^4 - 195*x^3*y^4 - 573*x^3*y^3*z + 1581*x^3*y^3 - 618*x^3*y^2*z^2 + 3405*x^3*y^2*z - 4605*x^3*y^2 - 290*x^3*y*z^3 + 2394*x^3*y*z^2 - 6486*x^3*y*z + 5791*x^3*y - 50*x^3*z^4 + 550*x^3*z^3 - 2238*x^3*z^2 + 4008*x^3*z - 2676*x^3 + 138*x^2*y^5 + 519*x^2*y^4*z - 1443*x^2*y^4 + 765*x^2*y^3*z^2 - 4251*x^2*y^3*z + 5778*x^2*y^3 + 552*x^2*y^2*z^3 - 4599*x^2*y^2*z^2 + 12534*x^2*y^2*z - 11217*x^2*y^2 + 195*x^2*y*z^4 - 2166*x^2*y*z^3 + 8874*x^2*y*z^2 - 15951*x^2*y*z + 10653*x^2*y + 27*x^2*z^5 - 375*x^2*z^4 + 2052*x^2*z^3 - 5550*x^2*z^2 + 7449*x^2*z - 3988*x^2 - 52*x*y^6 - 240*x*y^5*z + 672*x*y^5 - 453*x*y^4*z^2 + 2538*x*y^4*z - 3468*x*y^4 - 447*x*y^3*z^3 + 3759*x*y^3*z^2 - 10308*x*y^3*z + 9245*x*y^3 - 243*x*y^2*z^4 + 2727*x*y^2*z^3 - 11253*x*y^2*z^2 + 20298*x*y^2*z - 13545*x*y^2 - 69*x*y*z^5 + 969*x*y*z^4 - 5346*x*y*z^3 + 14529*x*y*z^2 - 19521*x*y*z + 10421*x*y - 8*x*z^6 + 135*x*z^5 - 933*x*z^4 + 3392*x*z^3 - 6870*x*z^2 + 7385*x*z - 3313*x + 8*y^7 + 44*y^6*z - 124*y^6 + 102*y^5*z^2 - 576*y^5*z + 792*y^5 + 129*y^4*z^3 - 1095*y^4*z^2 + 3024*y^4*z - 2722*y^4 + 96*y^3*z^4 - 1089*y^3*z^3 + 4530*y^3*z^2 - 8205*y^3*z + 5472*y^3 + 42*y^2*z^5 - 597*y^2*z^4 + 3324*y^2*z^3 - 9081*y^2*z^2 + 12207*y^2*z - 6484*y^2 + 10*y*z^6 - 171*y*z^5 + 1194*y*z^4 - 4369*y*z^3 + 8868*y*z^2 - 9509*y*z + 4236*y + z^7 - 20*z^6 + 168*z^5 - 771*z^4 + 2097*z^3 - 3397*z^2 + 3054*z - 1, -2*x^7 + 19*x^6*y + 13*x^6*z - 35*x^6 - 75*x^5*y^2 - 105*x^5*y*z + 285*x^5*y - 36*x^5*z^2 + 195*x^5*z - 261*x^5 + 159*x^4*y^3 + 342*x^4*y^2*z - 936*x^4*y^2 + 240*x^4*y*z^2 - 1311*x^4*y*z + 1764*x^4*y + 55*x^4*z^3 - 450*x^4*z^2 + 1212*x^4*z - 1081*x^4 - 195*x^3*y^4 - 573*x^3*y^3*z + 1581*x^3*y^3 - 618*x^3*y^2*z^2 + 3405*x^3*y^2*z - 4605*x^3*y^2 - 290*x^3*y*z^3 + 2394*x^3*y*z^2 - 6486*x^3*y*z + 5804*x^3*y - 50*x^3*z^4 + 550*x^3*z^3 - 2238*x^3*z^2 + 4015*x^3*z - 2693*x^3 + 138*x^2*y^5 + 519*x^2*y^4*z - 1443*x^2*y^4 + 765*x^2*y^3*z^2 - 4251*x^2*y^3*z + 5778*x^2*y^3 + 552*x^2*y^2*z^3 - 4599*x^2*y^2*z^2 + 12534*x^2*y^2*z - 11247*x^2*y^2 + 195*x^2*y*z^4 - 2166*x^2*y*z^3 + 8874*x^2*y*z^2 - 15984*x^2*y*z + 10734*x^2*y + 27*x^2*z^5 - 375*x^2*z^4 + 2052*x^2*z^3 - 5559*x^2*z^2 + 7494*x^2*z - 4042*x^2 - 52*x*y^6 - 240*x*y^5*z + 672*x*y^5 - 453*x*y^4*z^2 + 2538*x*y^4*z - 3468*x*y^4 - 447*x*y^3*z^3 + 3759*x*y^3*z^2 - 10308*x*y^3*z + 9273*x*y^3 - 243*x*y^2*z^4 + 2727*x*y^2*z^3 - 11253*x*y^2*z^2 + 20346*x*y^2*z - 13665*x*y^2 - 69*x*y*z^5 + 969*x*y*z^4 - 5346*x*y*z^3 + 14556*x*y*z^2 - 19659*x*y*z + 10589*x*y - 8*x*z^6 + 135*x*z^5 - 933*x*z^4 + 3397*x*z^3 - 6909*x*z^2 + 7481*x*z - 3387*x + 8*y^7 + 44*y^6*z - 124*y^6 + 102*y^5*z^2 - 576*y^5*z + 792*y^5 + 129*y^4*z^3 - 1095*y^4*z^2 + 3024*y^4*z - 2730*y^4 + 96*y^3*z^4 - 1089*y^3*z^3 + 4530*y^3*z^2 - 8225*y^3*z + 5524*y^3 + 42*y^2*z^5 - 597*y^2*z^4 + 3324*y^2*z^3 - 9099*y^2*z^2 + 12303*y^2*z - 6604*y^2 + 10*y*z^6 - 171*y*z^5 + 1194*y*z^4 - 4376*y*z^3 + 8925*y*z^2 - 9653*y*z + 4348*y + z^7 - 20*z^6 + 168*z^5 - 772*z^4 + 2108*z^3 - 3439*z^2 + 3120*z - 1, 2*x^7 - 19*x^6*y - 13*x^6*z + 41*x^6 + 75*x^5*y^2 + 105*x^5*y*z - 336*x^5*y + 36*x^5*z^2 - 228*x^5*z + 336*x^5 - 159*x^4*y^3 - 342*x^4*y^2*z + 1110*x^4*y^2 - 240*x^4*y*z^2 + 1542*x^4*y*z - 2289*x^4*y - 55*x^4*z^3 + 525*x^4*z^2 - 1557*x^4*z + 1483*x^4 + 195*x^3*y^4 + 573*x^3*y^3*z - 1884*x^3*y^3 + 618*x^3*y^2*z^2 - 4026*x^3*y^2*z + 6018*x^3*y^2 + 290*x^3*y*z^3 - 2808*x^3*y*z^2 + 8394*x^3*y*z - 8030*x^3*y + 50*x^3*z^4 - 640*x^3*z^3 + 2868*x^3*z^2 - 5497*x^3*z + 3869*x^3 - 138*x^2*y^5 - 519*x^2*y^4*z + 1725*x^2*y^4 - 765*x^2*y^3*z^2 + 5046*x^2*y^3*z - 7593*x^2*y^3 - 552*x^2*y^2*z^3 + 5418*x^2*y^2*z^2 - 16323*x^2*y^2*z + 15675*x^2*y^2 - 195*x^2*y*z^4 + 2532*x^2*y*z^3 - 11448*x^2*y*z^2 + 22056*x^2*y*z - 15558*x^2*y - 27*x^2*z^5 + 435*x^2*z^4 - 2622*x^2*z^3 + 7593*x^2*z^2 - 10746*x^2*z + 6016*x^2 + 52*x*y^6 + 240*x*y^5*z - 804*x*y^5 + 453*x*y^4*z^2 - 3018*x*y^4*z + 4572*x*y^4 + 447*x*y^3*z^3 - 4440*x*y^3*z^2 + 13482*x*y^3*z - 12993*x*y^3 + 243*x*y^2*z^4 - 3198*x*y^2*z^3 + 14592*x*y^2*z^2 - 28254*x*y^2*z + 19953*x*y^2 + 69*x*y*z^5 - 1128*x*y*z^4 + 6870*x*y*z^3 - 20022*x*y*z^2 + 28419*x*y*z - 15908*x*y + 8*x*z^6 - 156*x*z^5 + 1188*x*z^4 - 4627*x*z^3 + 9885*x*z^2 - 11114*x*z + 5182*x - 8*y^7 - 44*y^6*z + 148*y^6 - 102*y^5*z^2 + 684*y^5*z - 1044*y^5 - 129*y^4*z^3 + 1293*y^4*z^2 - 3960*y^4*z + 3834*y^4 - 96*y^3*z^4 + 1278*y^3*z^3 - 5889*y^3*z^2 + 11465*y^3*z - 8104*y^3 - 42*y^2*z^5 + 696*y^2*z^4 - 4287*y^2*z^3 + 12579*y^2*z^2 - 17895*y^2*z + 9994*y^2 - 10*y*z^6 + 198*y*z^5 - 1527*y*z^4 + 5996*y*z^3 - 12861*y*z^2 + 14462*y*z - 6719*y - z^7 + 23*z^6 - 213*z^5 + 1048*z^4 - 3008*z^3 + 5098*z^2 - 4765*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ 1, -2, -1 ], [ 1, -1, -1 ], [ 2, -1, -1 ] ], "source_shift": [ 2, 2, 2 ], "target_matrix": [ [ -1, -1, -2 ], [ -1, -1, -1 ], [ 1, 2, 1 ] ], "target_shift": [ 1183, 1219, -1905 ], "collision_preimages": [ [ "-1/4", "0", "7/4" ], [ "8", "-5/2", "14" ], [ "5", "5/2", "3" ] ], "collision_image": [ "4733/4", "4877/4", "-7621/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.1357407760006026 ], "surface_features": [ 3.566750676258555, 4.453608792024852, 3.0835026198302673, 0.0, 3.7032675842044473, 4845 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B084", "pair": 84, "family": "bijection", "mapping": "F(x,y,z)=(-2*x^7 - 28*x^6*y - 14*x^6*z - 16*x^6 - 168*x^5*y^2 - 168*x^5*y*z - 192*x^5*y - 42*x^5*z^2 - 96*x^5*z - 60*x^5 - 560*x^4*y^3 - 840*x^4*y^2*z - 960*x^4*y^2 - 420*x^4*y*z^2 - 960*x^4*y*z - 598*x^4*y - 70*x^4*z^3 - 240*x^4*z^2 - 300*x^4*z - 144*x^4 - 1120*x^3*y^4 - 2240*x^3*y^3*z - 2560*x^3*y^3 - 1680*x^3*y^2*z^2 - 3840*x^3*y^2*z - 2384*x^3*y^2 - 560*x^3*y*z^3 - 1920*x^3*y*z^2 - 2392*x^3*y*z - 1134*x^3*y - 70*x^3*z^4 - 320*x^3*z^3 - 600*x^3*z^2 - 574*x^3*z - 231*x^3 - 1344*x^2*y^5 - 3360*x^2*y^4*z - 3840*x^2*y^4 - 3360*x^2*y^3*z^2 - 7680*x^2*y^3*z - 4752*x^2*y^3 - 1680*x^2*y^2*z^3 - 5760*x^2*y^2*z^2 - 7152*x^2*y^2*z - 3348*x^2*y^2 - 420*x^2*y*z^4 - 1920*x^2*y*z^3 - 3588*x^2*y*z^2 - 3390*x^2*y*z - 1342*x^2*y - 42*x^2*z^5 - 240*x^2*z^4 - 600*x^2*z^3 - 858*x^2*z^2 - 687*x^2*z - 251*x^2 - 896*x*y^6 - 2688*x*y^5*z - 3072*x*y^5 - 3360*x*y^4*z^2 - 7680*x*y^4*z - 4736*x*y^4 - 2240*x*y^3*z^3 - 7680*x*y^3*z^2 - 9504*x*y^3*z - 4392*x*y^3 - 840*x*y^2*z^4 - 3840*x*y^2*z^3 - 7152*x*y^2*z^2 - 6672*x*y^2*z - 2596*x*y^2 - 168*x*y*z^5 - 960*x*y*z^4 - 2392*x*y*z^3 - 3378*x*y*z^2 - 2660*x*y*z - 936*x*y - 14*x*z^6 - 96*x*z^5 - 300*x*z^4 - 570*x*z^3 - 681*x*z^2 - 492*x*z - 171*x - 256*y^7 - 896*y^6*z - 1024*y^6 - 1344*y^5*z^2 - 3072*y^5*z - 1888*y^5 - 1120*y^4*z^3 - 3840*y^4*z^2 - 4736*y^4*z - 2160*y^4 - 560*y^3*z^4 - 2560*y^3*z^3 - 4752*y^3*z^2 - 4376*y^3*z - 1672*y^3 - 168*y^2*z^5 - 960*y^2*z^4 - 2384*y^2*z^3 - 3324*y^2*z^2 - 2572*y^2*z - 876*y^2 - 28*y*z^6 - 192*y*z^5 - 598*y*z^4 - 1122*y*z^3 - 1318*y*z^2 - 918*y*z - 301*y - 2*z^7 - 16*z^6 - 60*z^5 - 142*z^4 - 225*z^3 - 241*z^2 - 165*z - 1, -x^7 - 14*x^6*y - 7*x^6*z - 8*x^6 - 84*x^5*y^2 - 84*x^5*y*z - 96*x^5*y - 21*x^5*z^2 - 48*x^5*z - 30*x^5 - 280*x^4*y^3 - 420*x^4*y^2*z - 480*x^4*y^2 - 210*x^4*y*z^2 - 480*x^4*y*z - 299*x^4*y - 35*x^4*z^3 - 120*x^4*z^2 - 150*x^4*z - 71*x^4 - 560*x^3*y^4 - 1120*x^3*y^3*z - 1280*x^3*y^3 - 840*x^3*y^2*z^2 - 1920*x^3*y^2*z - 1192*x^3*y^2 - 280*x^3*y*z^3 - 960*x^3*y*z^2 - 1196*x^3*y*z - 559*x^3*y - 35*x^3*z^4 - 160*x^3*z^3 - 300*x^3*z^2 - 283*x^3*z - 113*x^3 - 672*x^2*y^5 - 1680*x^2*y^4*z - 1920*x^2*y^4 - 1680*x^2*y^3*z^2 - 3840*x^2*y^3*z - 2376*x^2*y^3 - 840*x^2*y^2*z^3 - 2880*x^2*y^2*z^2 - 3576*x^2*y^2*z - 1650*x^2*y^2 - 210*x^2*y*z^4 - 960*x^2*y*z^3 - 1794*x^2*y*z^2 - 1671*x^2*y*z - 656*x^2*y - 21*x^2*z^5 - 120*x^2*z^4 - 300*x^2*z^3 - 423*x^2*z^2 - 336*x^2*z - 123*x^2 - 448*x*y^6 - 1344*x*y^5*z - 1536*x*y^5 - 1680*x*y^4*z^2 - 3840*x*y^4*z - 2368*x*y^4 - 1120*x*y^3*z^3 - 3840*x*y^3*z^2 - 4752*x*y^3*z - 2164*x*y^3 - 420*x*y^2*z^4 - 1920*x*y^2*z^3 - 3576*x*y^2*z^2 - 3288*x*y^2*z - 1268*x*y^2 - 84*x*y*z^5 - 480*x*y*z^4 - 1196*x*y*z^3 - 1665*x*y*z^2 - 1300*x*y*z - 458*x*y - 7*x*z^6 - 48*x*z^5 - 150*x*z^4 - 281*x*z^3 - 333*x*z^2 - 241*x*z - 84*x - 128*y^7 - 448*y^6*z - 512*y^6 - 672*y^5*z^2 - 1536*y^5*z - 944*y^5 - 560*y^4*z^3 - 1920*y^4*z^2 - 2368*y^4*z - 1064*y^4 - 280*y^3*z^4 - 1280*y^3*z^3 - 2376*y^3*z^2 - 2156*y^3*z - 816*y^3 - 84*y^2*z^5 - 480*y^2*z^4 - 1192*y^2*z^3 - 1638*y^2*z^2 - 1256*y^2*z - 428*y^2 - 14*y*z^6 - 96*y*z^5 - 299*y*z^4 - 553*y*z^3 - 644*y*z^2 - 449*y*z - 149*y - z^7 - 8*z^6 - 30*z^5 - 70*z^4 - 110*z^3 - 118*z^2 - 82*z - 1, x^7 + 14*x^6*y + 7*x^6*z + 8*x^6 + 84*x^5*y^2 + 84*x^5*y*z + 96*x^5*y + 21*x^5*z^2 + 48*x^5*z + 30*x^5 + 280*x^4*y^3 + 420*x^4*y^2*z + 480*x^4*y^2 + 210*x^4*y*z^2 + 480*x^4*y*z + 299*x^4*y + 35*x^4*z^3 + 120*x^4*z^2 + 150*x^4*z + 71*x^4 + 560*x^3*y^4 + 1120*x^3*y^3*z + 1280*x^3*y^3 + 840*x^3*y^2*z^2 + 1920*x^3*y^2*z + 1192*x^3*y^2 + 280*x^3*y*z^3 + 960*x^3*y*z^2 + 1196*x^3*y*z + 559*x^3*y + 35*x^3*z^4 + 160*x^3*z^3 + 300*x^3*z^2 + 283*x^3*z + 112*x^3 + 672*x^2*y^5 + 1680*x^2*y^4*z + 1920*x^2*y^4 + 1680*x^2*y^3*z^2 + 3840*x^2*y^3*z + 2376*x^2*y^3 + 840*x^2*y^2*z^3 + 2880*x^2*y^2*z^2 + 3576*x^2*y^2*z + 1650*x^2*y^2 + 210*x^2*y*z^4 + 960*x^2*y*z^3 + 1794*x^2*y*z^2 + 1671*x^2*y*z + 650*x^2*y + 21*x^2*z^5 + 120*x^2*z^4 + 300*x^2*z^3 + 423*x^2*z^2 + 333*x^2*z + 120*x^2 + 448*x*y^6 + 1344*x*y^5*z + 1536*x*y^5 + 1680*x*y^4*z^2 + 3840*x*y^4*z + 2368*x*y^4 + 1120*x*y^3*z^3 + 3840*x*y^3*z^2 + 4752*x*y^3*z + 2164*x*y^3 + 420*x*y^2*z^4 + 1920*x*y^2*z^3 + 3576*x*y^2*z^2 + 3288*x*y^2*z + 1256*x*y^2 + 84*x*y*z^5 + 480*x*y*z^4 + 1196*x*y*z^3 + 1665*x*y*z^2 + 1288*x*y*z + 446*x*y + 7*x*z^6 + 48*x*z^5 + 150*x*z^4 + 281*x*z^3 + 330*x*z^2 + 235*x*z + 79*x + 128*y^7 + 448*y^6*z + 512*y^6 + 672*y^5*z^2 + 1536*y^5*z + 944*y^5 + 560*y^4*z^3 + 1920*y^4*z^2 + 2368*y^4*z + 1064*y^4 + 280*y^3*z^4 + 1280*y^3*z^3 + 2376*y^3*z^2 + 2156*y^3*z + 808*y^3 + 84*y^2*z^5 + 480*y^2*z^4 + 1192*y^2*z^3 + 1638*y^2*z^2 + 1244*y^2*z + 416*y^2 + 14*y*z^6 + 96*y*z^5 + 299*y*z^4 + 553*y*z^3 + 638*y*z^2 + 437*y*z + 140*y + z^7 + 8*z^6 + 30*z^5 + 70*z^4 + 109*z^3 + 115*z^2 + 77*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ 1, 2, 1 ], [ 1, 1, 1 ], [ 2, 1, 1 ] ], "source_shift": [ 1, 1, 1 ], "target_matrix": [ [ -2, -1, -1 ], [ -1, -2, -1 ], [ 1, 1, 1 ] ], "target_shift": [ 52, 26, -23 ], "inverse_recipe": { "p": 3, "q": 4, "f": "x**3 + x", "g": "x**4 + x**2 + x", "h": "y**2 + y*z + y + z", "polynomials": { "f": [ [ [ 3, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "g": [ [ [ 4, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 2, 0 ], 1 ], [ [ 0, 0, 1 ], 1 ], [ [ 0, 1, 0 ], 1 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 0.9845086530003755 ], "surface_features": [ 3.0723469278304147, 3.9779064276371185, 2.7481880270062002, 0.0, 3.174449378928746, 4730 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B226", "pair": 226, "family": "bijection", "mapping": "F(x,y,z)=(4*x^7 + 28*x^6*y - 28*x^6*z - 60*x^6 + 84*x^5*y^2 - 168*x^5*y*z - 360*x^5*y + 84*x^5*z^2 + 360*x^5*z + 392*x^5 + 140*x^4*y^3 - 420*x^4*y^2*z - 900*x^4*y^2 + 420*x^4*y*z^2 + 1800*x^4*y*z + 1960*x^4*y - 140*x^4*z^3 - 900*x^4*z^2 - 1964*x^4*z - 1458*x^4 + 140*x^3*y^4 - 560*x^3*y^3*z - 1200*x^3*y^3 + 840*x^3*y^2*z^2 + 3600*x^3*y^2*z + 3920*x^3*y^2 - 560*x^3*y*z^3 - 3600*x^3*y*z^2 - 7856*x^3*y*z - 5828*x^3*y + 140*x^3*z^4 + 1200*x^3*z^3 + 3936*x^3*z^2 + 5864*x^3*z + 3347*x^3 + 84*x^2*y^5 - 420*x^2*y^4*z - 900*x^2*y^4 + 840*x^2*y^3*z^2 + 3600*x^2*y^3*z + 3920*x^2*y^3 - 840*x^2*y^2*z^3 - 5400*x^2*y^2*z^2 - 11784*x^2*y^2*z - 8736*x^2*y^2 + 420*x^2*y*z^4 + 3600*x^2*y*z^3 + 11808*x^2*y*z^2 + 17580*x^2*y*z + 10017*x^2*y - 84*x^2*z^5 - 900*x^2*z^4 - 3944*x^2*z^3 - 8844*x^2*z^2 - 10141*x^2*z - 4752*x^2 + 28*x*y^6 - 168*x*y^5*z - 360*x*y^5 + 420*x*y^4*z^2 + 1800*x*y^4*z + 1960*x*y^4 - 560*x*y^3*z^3 - 3600*x*y^3*z^2 - 7856*x*y^3*z - 5820*x*y^3 + 420*x*y^2*z^4 + 3600*x*y^2*z^3 + 11808*x*y^2*z^2 + 17568*x*y^2*z + 9993*x*y^2 - 168*x*y*z^5 - 1800*x*y*z^4 - 7888*x*y*z^3 - 17676*x*y*z^2 - 20234*x*y*z - 9452*x*y + 28*x*z^6 + 360*x*z^5 + 1976*x*z^4 + 5928*x*z^3 + 10241*x*z^2 + 9652*x*z + 3874*x + 4*y^7 - 28*y^6*z - 60*y^6 + 84*y^5*z^2 + 360*y^5*z + 392*y^5 - 140*y^4*z^3 - 900*y^4*z^2 - 1964*y^4*z - 1454*y^4 + 140*y^3*z^4 + 1200*y^3*z^3 + 3936*y^3*z^2 + 5852*y^3*z + 3323*y^3 - 84*y^2*z^5 - 900*y^2*z^4 - 3944*y^2*z^3 - 8832*y^2*z^2 - 10093*y^2*z - 4700*y^2 + 28*y*z^6 + 360*y*z^5 + 1976*y*z^4 + 5924*y*z^3 + 10217*y*z^2 + 9596*y*z + 3828*y - 4*z^7 - 60*z^6 - 396*z^5 - 1490*z^4 - 3447*z^3 - 4896*z^2 - 3949*z - 1, 12*x^7 + 84*x^6*y - 84*x^6*z - 180*x^6 + 252*x^5*y^2 - 504*x^5*y*z - 1080*x^5*y + 252*x^5*z^2 + 1080*x^5*z + 1176*x^5 + 420*x^4*y^3 - 1260*x^4*y^2*z - 2700*x^4*y^2 + 1260*x^4*y*z^2 + 5400*x^4*y*z + 5880*x^4*y - 420*x^4*z^3 - 2700*x^4*z^2 - 5892*x^4*z - 4376*x^4 + 420*x^3*y^4 - 1680*x^3*y^3*z - 3600*x^3*y^3 + 2520*x^3*y^2*z^2 + 10800*x^3*y^2*z + 11760*x^3*y^2 - 1680*x^3*y*z^3 - 10800*x^3*y*z^2 - 23568*x^3*y*z - 17492*x^3*y + 420*x^3*z^4 + 3600*x^3*z^3 + 11808*x^3*z^2 + 17600*x^3*z + 10059*x^3 + 252*x^2*y^5 - 1260*x^2*y^4*z - 2700*x^2*y^4 + 2520*x^2*y^3*z^2 + 10800*x^2*y^3*z + 11760*x^2*y^3 - 2520*x^2*y^2*z^3 - 16200*x^2*y^2*z^2 - 35352*x^2*y^2*z - 26220*x^2*y^2 + 1260*x^2*y*z^4 + 10800*x^2*y*z^3 + 35424*x^2*y*z^2 + 52764*x^2*y*z + 30105*x^2*y - 252*x^2*z^5 - 2700*x^2*z^4 - 11832*x^2*z^3 - 26544*x^2*z^2 - 30477*x^2*z - 14318*x^2 + 84*x*y^6 - 504*x*y^5*z - 1080*x*y^5 + 1260*x*y^4*z^2 + 5400*x*y^4*z + 5880*x*y^4 - 1680*x*y^3*z^3 - 10800*x*y^3*z^2 - 23568*x*y^3*z - 17468*x*y^3 + 1260*x*y^2*z^4 + 10800*x*y^2*z^3 + 35424*x*y^2*z^2 + 52728*x*y^2*z + 30033*x*y^2 - 504*x*y*z^5 - 5400*x*y*z^4 - 23664*x*y*z^3 - 53052*x*y*z^2 - 60810*x*y*z - 28480*x*y + 84*x*z^6 + 1080*x*z^5 + 5928*x*z^4 + 17792*x*z^3 + 30777*x*z^2 + 29080*x*z + 11722*x + 12*y^7 - 84*y^6*z - 180*y^6 + 252*y^5*z^2 + 1080*y^5*z + 1176*y^5 - 420*y^4*z^3 - 2700*y^4*z^2 - 5892*y^4*z - 4364*y^4 + 420*y^3*z^4 + 3600*y^3*z^3 + 11808*y^3*z^2 + 17564*y^3*z + 9987*y^3 - 252*y^2*z^5 - 2700*y^2*z^4 - 11832*y^2*z^3 - 26508*y^2*z^2 - 30333*y^2*z - 14162*y^2 + 84*y*z^6 + 1080*y*z^5 + 5928*y*z^4 + 17780*y*z^3 + 30705*y*z^2 + 28912*y*z + 11582*y - 12*z^7 - 180*z^6 - 1188*z^5 - 4472*z^4 - 10359*z^3 - 14750*z^2 - 11945*z - 1, -4*x^7 - 28*x^6*y + 28*x^6*z + 60*x^6 - 84*x^5*y^2 + 168*x^5*y*z + 360*x^5*y - 84*x^5*z^2 - 360*x^5*z - 392*x^5 - 140*x^4*y^3 + 420*x^4*y^2*z + 900*x^4*y^2 - 420*x^4*y*z^2 - 1800*x^4*y*z - 1960*x^4*y + 140*x^4*z^3 + 900*x^4*z^2 + 1964*x^4*z + 1458*x^4 - 140*x^3*y^4 + 560*x^3*y^3*z + 1200*x^3*y^3 - 840*x^3*y^2*z^2 - 3600*x^3*y^2*z - 3920*x^3*y^2 + 560*x^3*y*z^3 + 3600*x^3*y*z^2 + 7856*x^3*y*z + 5828*x^3*y - 140*x^3*z^4 - 1200*x^3*z^3 - 3936*x^3*z^2 - 5864*x^3*z - 3348*x^3 - 84*x^2*y^5 + 420*x^2*y^4*z + 900*x^2*y^4 - 840*x^2*y^3*z^2 - 3600*x^2*y^3*z - 3920*x^2*y^3 + 840*x^2*y^2*z^3 + 5400*x^2*y^2*z^2 + 11784*x^2*y^2*z + 8736*x^2*y^2 - 420*x^2*y*z^4 - 3600*x^2*y*z^3 - 11808*x^2*y*z^2 - 17580*x^2*y*z - 10020*x^2*y + 84*x^2*z^5 + 900*x^2*z^4 + 3944*x^2*z^3 + 8844*x^2*z^2 + 10144*x^2*z + 4758*x^2 - 28*x*y^6 + 168*x*y^5*z + 360*x*y^5 - 420*x*y^4*z^2 - 1800*x*y^4*z - 1960*x*y^4 + 560*x*y^3*z^3 + 3600*x*y^3*z^2 + 7856*x*y^3*z + 5820*x*y^3 - 420*x*y^2*z^4 - 3600*x*y^2*z^3 - 11808*x*y^2*z^2 - 17568*x*y^2*z - 9996*x*y^2 + 168*x*y*z^5 + 1800*x*y*z^4 + 7888*x*y*z^3 + 17676*x*y*z^2 + 20240*x*y*z + 9464*x*y - 28*x*z^6 - 360*x*z^5 - 1976*x*z^4 - 5928*x*z^3 - 10244*x*z^2 - 9664*x*z - 3887*x - 4*y^7 + 28*y^6*z + 60*y^6 - 84*y^5*z^2 - 360*y^5*z - 392*y^5 + 140*y^4*z^3 + 900*y^4*z^2 + 1964*y^4*z + 1454*y^4 - 140*y^3*z^4 - 1200*y^3*z^3 - 3936*y^3*z^2 - 5852*y^3*z - 3324*y^3 + 84*y^2*z^5 + 900*y^2*z^4 + 3944*y^2*z^3 + 8832*y^2*z^2 + 10096*y^2*z + 4706*y^2 - 28*y*z^6 - 360*y*z^5 - 1976*y*z^4 - 5924*y*z^3 - 10220*y*z^2 - 9608*y*z - 3841*y + 4*z^7 + 60*z^6 + 396*z^5 + 1490*z^4 + 3448*z^3 + 4902*z^2 + 3963*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -1, -1, 1 ], [ -1, -1, 2 ], [ -2, -1, 1 ] ], "source_shift": [ 2, 2, 2 ], "target_matrix": [ [ -1, -1, -1 ], [ -3, -3, -2 ], [ 1, 2, 1 ] ], "target_shift": [ 1391, 4235, -1401 ], "inverse_recipe": { "p": 3, "q": 4, "f": "x**3", "g": "x**4 + x**3 + x**2", "h": "y*z + 2*z", "polynomials": { "f": [ [ [ 3, 0, 0 ], 1 ] ], "g": [ [ [ 4, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ], [ [ 3, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 0, 1 ], 2 ] ] }, "h_multiplier": 4 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0665227549980045 ], "surface_features": [ 3.758616713975927, 4.783975003412671, 3.2931414834509307, 0.0, 3.9503634126395157, 4925 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B206", "pair": 206, "family": "bijection", "mapping": "F(x,y,z)=(-512*x^7 + 1792*x^6*y - 1792*x^6*z + 3584*x^6 - 2688*x^5*y^2 + 5376*x^5*y*z - 10752*x^5*y - 2688*x^5*z^2 + 10752*x^5*z - 11072*x^5 + 2240*x^4*y^3 - 6720*x^4*y^2*z + 13440*x^4*y^2 + 6720*x^4*y*z^2 - 26880*x^4*y*z + 27712*x^4*y - 2240*x^4*z^3 + 13440*x^4*z^2 - 27712*x^4*z + 19712*x^4 - 1120*x^3*y^4 + 4480*x^3*y^3*z - 8960*x^3*y^3 - 6720*x^3*y^2*z^2 + 26880*x^3*y^2*z - 27744*x^3*y^2 + 4480*x^3*y*z^3 - 26880*x^3*y*z^2 + 55488*x^3*y*z - 39600*x^3*y - 1120*x^3*z^4 + 8960*x^3*z^3 - 27744*x^3*z^2 + 39568*x^3*z - 21920*x^3 + 336*x^2*y^5 - 1680*x^2*y^4*z + 3360*x^2*y^4 + 3360*x^2*y^3*z^2 - 13440*x^2*y^3*z + 13888*x^2*y^3 - 3360*x^2*y^2*z^3 + 20160*x^2*y^2*z^2 - 41664*x^2*y^2*z + 29832*x^2*y^2 + 1680*x^2*y*z^4 - 13440*x^2*y*z^3 + 41664*x^2*y*z^2 - 59616*x^2*y*z + 33224*x^2*y - 336*x^2*z^5 + 3360*x^2*z^4 - 13888*x^2*z^3 + 29784*x^2*z^2 - 33128*x^2*z + 15260*x^2 - 56*x*y^6 + 336*x*y^5*z - 672*x*y^5 - 840*x*y^4*z^2 + 3360*x*y^4*z - 3476*x*y^4 + 1120*x*y^3*z^3 - 6720*x*y^3*z^2 + 13904*x*y^3*z - 9988*x*y^3 - 840*x*y^2*z^4 + 6720*x*y^2*z^3 - 20856*x*y^2*z^2 + 29940*x*y^2*z - 16784*x*y^2 + 336*x*y*z^5 - 3360*x*y*z^4 + 13904*x*y*z^3 - 29916*x*y*z^2 + 33472*x*y*z - 15572*x*y - 56*x*z^6 + 672*x*z^5 - 3476*x*z^4 + 9964*x*z^3 - 16688*x*z^2 + 15464*x*z - 6184*x + 4*y^7 - 28*y^6*z + 56*y^6 + 84*y^5*z^2 - 336*y^5*z + 348*y^5 - 140*y^4*z^3 + 840*y^4*z^2 - 1740*y^4*z + 1254*y^4 + 140*y^3*z^4 - 1120*y^3*z^3 + 3480*y^3*z^2 - 5012*y^3*z + 2826*y^3 - 84*y^2*z^5 + 840*y^2*z^4 - 3480*y^2*z^3 + 7512*y^2*z^2 - 8454*y^2*z + 3974*y^2 + 28*y*z^6 - 336*y*z^5 + 1740*y*z^4 - 5004*y*z^3 + 8430*y*z^2 - 7892*y*z + 3212*y - 4*z^7 + 56*z^6 - 348*z^5 + 1250*z^4 - 2802*z^3 + 3918*z^2 - 3162*z + 1, -256*x^7 + 896*x^6*y - 896*x^6*z + 1792*x^6 - 1344*x^5*y^2 + 2688*x^5*y*z - 5376*x^5*y - 1344*x^5*z^2 + 5376*x^5*z - 5536*x^5 + 1120*x^4*y^3 - 3360*x^4*y^2*z + 6720*x^4*y^2 + 3360*x^4*y*z^2 - 13440*x^4*y*z + 13856*x^4*y - 1120*x^4*z^3 + 6720*x^4*z^2 - 13856*x^4*z + 9856*x^4 - 560*x^3*y^4 + 2240*x^3*y^3*z - 4480*x^3*y^3 - 3360*x^3*y^2*z^2 + 13440*x^3*y^2*z - 13872*x^3*y^2 + 2240*x^3*y*z^3 - 13440*x^3*y*z^2 + 27744*x^3*y*z - 19800*x^3*y - 560*x^3*z^4 + 4480*x^3*z^3 - 13872*x^3*z^2 + 19784*x^3*z - 10952*x^3 + 168*x^2*y^5 - 840*x^2*y^4*z + 1680*x^2*y^4 + 1680*x^2*y^3*z^2 - 6720*x^2*y^3*z + 6944*x^2*y^3 - 1680*x^2*y^2*z^3 + 10080*x^2*y^2*z^2 - 20832*x^2*y^2*z + 14916*x^2*y^2 + 840*x^2*y*z^4 - 6720*x^2*y*z^3 + 20832*x^2*y*z^2 - 29808*x^2*y*z + 16600*x^2*y - 168*x^2*z^5 + 1680*x^2*z^4 - 6944*x^2*z^3 + 14892*x^2*z^2 - 16552*x^2*z + 7606*x^2 - 28*x*y^6 + 168*x*y^5*z - 336*x*y^5 - 420*x*y^4*z^2 + 1680*x*y^4*z - 1738*x*y^4 + 560*x*y^3*z^3 - 3360*x*y^3*z^2 + 6952*x*y^3*z - 4994*x*y^3 - 420*x*y^2*z^4 + 3360*x*y^2*z^3 - 10428*x*y^2*z^2 + 14970*x*y^2*z - 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26880*x^3*y*z^2 + 55488*x^3*y*z - 39568*x^3*y - 1120*x^3*z^4 + 8960*x^3*z^3 - 27744*x^3*z^2 + 39536*x^3*z - 21888*x^3 + 336*x^2*y^5 - 1680*x^2*y^4*z + 3360*x^2*y^4 + 3360*x^2*y^3*z^2 - 13440*x^2*y^3*z + 13888*x^2*y^3 - 3360*x^2*y^2*z^3 + 20160*x^2*y^2*z^2 - 41664*x^2*y^2*z + 29808*x^2*y^2 + 1680*x^2*y*z^4 - 13440*x^2*y*z^3 + 41664*x^2*y*z^2 - 59568*x^2*y*z + 33176*x^2*y - 336*x^2*z^5 + 3360*x^2*z^4 - 13888*x^2*z^3 + 29760*x^2*z^2 - 33080*x^2*z + 15256*x^2 - 56*x*y^6 + 336*x*y^5*z - 672*x*y^5 - 840*x*y^4*z^2 + 3360*x*y^4*z - 3476*x*y^4 + 1120*x*y^3*z^3 - 6720*x*y^3*z^2 + 13904*x*y^3*z - 9980*x*y^3 - 840*x*y^2*z^4 + 6720*x*y^2*z^3 - 20856*x*y^2*z^2 + 29916*x*y^2*z - 16760*x*y^2 + 336*x*y*z^5 - 3360*x*y*z^4 + 13904*x*y*z^3 - 29892*x*y*z^2 + 33424*x*y*z - 15568*x*y - 56*x*z^6 + 672*x*z^5 - 3476*x*z^4 + 9956*x*z^3 - 16664*x*z^2 + 15460*x*z - 6217*x + 4*y^7 - 28*y^6*z + 56*y^6 + 84*y^5*z^2 - 336*y^5*z + 348*y^5 - 140*y^4*z^3 + 840*y^4*z^2 - 1740*y^4*z + 1253*y^4 + 140*y^3*z^4 - 1120*y^3*z^3 + 3480*y^3*z^2 - 5008*y^3*z + 2822*y^3 - 84*y^2*z^5 + 840*y^2*z^4 - 3480*y^2*z^3 + 7506*y^2*z^2 - 8442*y^2*z + 3973*y^2 + 28*y*z^6 - 336*y*z^5 + 1740*y*z^4 - 5000*y*z^3 + 8418*y*z^2 - 7890*y*z + 3229*y - 4*z^7 + 56*z^6 - 348*z^5 + 1249*z^4 - 2798*z^3 + 3917*z^2 - 3180*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -2, 1, -1 ], [ -1, 2, -1 ], [ -1, 1, -1 ] ], "source_shift": [ 2, 2, 2 ], "target_matrix": [ [ 2, 2, 3 ], [ 1, 1, 2 ], [ 2, 1, 1 ] ], "target_shift": [ -1131, -553, -1155 ], "inverse_recipe": { "p": 4, "q": 3, "f": "x**4 + x**2 + x", "g": "x**3 + x", "h": "y*z", "polynomials": { "f": [ [ [ 4, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "g": [ [ [ 3, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ] ] }, "h_multiplier": 2 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0159261850021721 ], "surface_features": [ 3.928100275530377, 4.775362833216149, 3.533772058384718, 0.0, 4.048022231526453, 5012 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B112", "pair": 112, "family": "bijection", "mapping": "F(x,y,z)=(x^7 + 7*x^6*y - 14*x^6*z - 16*x^6 + 21*x^5*y^2 - 84*x^5*y*z - 95*x^5*y + 84*x^5*z^2 + 189*x^5*z + 104*x^5 + 35*x^4*y^3 - 210*x^4*y^2*z - 235*x^4*y^2 + 420*x^4*y*z^2 + 935*x^4*y*z + 510*x^4*y - 280*x^4*z^3 - 930*x^4*z^2 - 1010*x^4*z - 361*x^4 + 35*x^3*y^4 - 280*x^3*y^3*z - 310*x^3*y^3 + 840*x^3*y^2*z^2 + 1850*x^3*y^2*z + 1000*x^3*y^2 - 1120*x^3*y*z^3 - 3680*x^3*y*z^2 - 3960*x^3*y*z - 1404*x^3*y + 560*x^3*z^4 + 2440*x^3*z^3 + 3920*x^3*z^2 + 2768*x^3*z + 731*x^3 + 21*x^2*y^5 - 210*x^2*y^4*z - 230*x^2*y^4 + 840*x^2*y^3*z^2 + 1830*x^2*y^3*z + 980*x^2*y^3 - 1680*x^2*y^2*z^3 - 5460*x^2*y^2*z^2 - 5820*x^2*y^2*z - 2046*x^2*y^2 + 1680*x^2*y*z^4 + 7240*x^2*y*z^3 + 11520*x^2*y*z^2 + 8064*x^2*y*z + 2112*x^2*y - 672*x^2*z^5 - 3600*x^2*z^4 - 7600*x^2*z^3 - 7944*x^2*z^2 - 4142*x^2*z - 873*x^2 + 7*x*y^6 - 84*x*y^5*z - 91*x*y^5 + 420*x*y^4*z^2 + 905*x*y^4*z + 480*x*y^4 - 1120*x*y^3*z^3 - 3600*x*y^3*z^2 - 3800*x*y^3*z - 1324*x*y^3 + 1680*x*y^2*z^4 + 7160*x*y^2*z^3 + 11280*x*y^2*z^2 + 7824*x*y^2*z + 2031*x*y^2 - 1344*x*y*z^5 - 7120*x*y*z^4 - 14880*x*y*z^3 - 15408*x*y*z^2 - 7960*x*y*z - 1661*x*y + 448*x*z^6 + 2832*x*z^5 + 7360*x*z^4 + 10112*x*z^3 + 7796*x*z^2 + 3231*x*z + 575*x + y^7 - 14*y^6*z - 15*y^6 + 84*y^5*z^2 + 179*y^5*z + 94*y^5 - 280*y^4*z^3 - 890*y^4*z^2 - 930*y^4*z - 321*y^4 + 560*y^3*z^4 + 2360*y^3*z^3 + 3680*y^3*z^2 + 2528*y^3*z + 650*y^3 - 672*y^2*z^5 - 3520*y^2*z^4 - 7280*y^2*z^3 - 7464*y^2*z^2 - 3818*y^2*z - 788*y^2 + 448*y*z^6 + 2800*y*z^5 + 7200*y*z^4 + 9792*y*z^3 + 7472*y*z^2 + 3062*y*z + 536*y - 128*z^7 - 928*z^6 - 2848*z^5 - 4816*z^4 - 4872*z^3 - 2973*z^2 - 1029*z - 1, -3*x^7 - 21*x^6*y + 42*x^6*z + 48*x^6 - 63*x^5*y^2 + 252*x^5*y*z + 285*x^5*y - 252*x^5*z^2 - 567*x^5*z - 312*x^5 - 105*x^4*y^3 + 630*x^4*y^2*z + 705*x^4*y^2 - 1260*x^4*y*z^2 - 2805*x^4*y*z - 1530*x^4*y + 840*x^4*z^3 + 2790*x^4*z^2 + 3030*x^4*z + 1083*x^4 - 105*x^3*y^4 + 840*x^3*y^3*z + 930*x^3*y^3 - 2520*x^3*y^2*z^2 - 5550*x^3*y^2*z - 3000*x^3*y^2 + 3360*x^3*y*z^3 + 11040*x^3*y*z^2 + 11880*x^3*y*z + 4212*x^3*y - 1680*x^3*z^4 - 7320*x^3*z^3 - 11760*x^3*z^2 - 8304*x^3*z - 2193*x^3 - 63*x^2*y^5 + 630*x^2*y^4*z + 690*x^2*y^4 - 2520*x^2*y^3*z^2 - 5490*x^2*y^3*z - 2940*x^2*y^3 + 5040*x^2*y^2*z^3 + 16380*x^2*y^2*z^2 + 17460*x^2*y^2*z + 6138*x^2*y^2 - 5040*x^2*y*z^4 - 21720*x^2*y*z^3 - 34560*x^2*y*z^2 - 24192*x^2*y*z - 6336*x^2*y + 2016*x^2*z^5 + 10800*x^2*z^4 + 22800*x^2*z^3 + 23832*x^2*z^2 + 12426*x^2*z + 2618*x^2 - 21*x*y^6 + 252*x*y^5*z + 273*x*y^5 - 1260*x*y^4*z^2 - 2715*x*y^4*z - 1440*x*y^4 + 3360*x*y^3*z^3 + 10800*x*y^3*z^2 + 11400*x*y^3*z + 3972*x*y^3 - 5040*x*y^2*z^4 - 21480*x*y^2*z^3 - 33840*x*y^2*z^2 - 23472*x*y^2*z - 6093*x*y^2 + 4032*x*y*z^5 + 21360*x*y*z^4 + 44640*x*y*z^3 + 46224*x*y*z^2 + 23880*x*y*z + 4981*x*y - 1344*x*z^6 - 8496*x*z^5 - 22080*x*z^4 - 30336*x*z^3 - 23388*x*z^2 - 9689*x*z - 1719*x - 3*y^7 + 42*y^6*z + 45*y^6 - 252*y^5*z^2 - 537*y^5*z - 282*y^5 + 840*y^4*z^3 + 2670*y^4*z^2 + 2790*y^4*z + 963*y^4 - 1680*y^3*z^4 - 7080*y^3*z^3 - 11040*y^3*z^2 - 7584*y^3*z - 1950*y^3 + 2016*y^2*z^5 + 10560*y^2*z^4 + 21840*y^2*z^3 + 22392*y^2*z^2 + 11454*y^2*z + 2363*y^2 - 1344*y*z^6 - 8400*y*z^5 - 21600*y*z^4 - 29376*y*z^3 - 22416*y*z^2 - 9182*y*z - 1603*y + 384*z^7 + 2784*z^6 + 8544*z^5 + 14448*z^4 + 14616*z^3 + 8915*z^2 + 3078*z + 1, 2*x^7 + 14*x^6*y - 28*x^6*z - 32*x^6 + 42*x^5*y^2 - 168*x^5*y*z - 190*x^5*y + 168*x^5*z^2 + 378*x^5*z + 210*x^5 + 70*x^4*y^3 - 420*x^4*y^2*z - 470*x^4*y^2 + 840*x^4*y*z^2 + 1870*x^4*y*z + 1030*x^4*y - 560*x^4*z^3 - 1860*x^4*z^2 - 2040*x^4*z - 742*x^4 + 70*x^3*y^4 - 560*x^3*y^3*z - 620*x^3*y^3 + 1680*x^3*y^2*z^2 + 3700*x^3*y^2*z + 2020*x^3*y^2 - 2240*x^3*y*z^3 - 7360*x^3*y*z^2 - 8000*x^3*y*z - 2888*x^3*y + 1120*x^3*z^4 + 4880*x^3*z^3 + 7920*x^3*z^2 + 5696*x^3*z + 1542*x^3 + 42*x^2*y^5 - 420*x^2*y^4*z - 460*x^2*y^4 + 1680*x^2*y^3*z^2 + 3660*x^2*y^3*z + 1980*x^2*y^3 - 3360*x^2*y^2*z^3 - 10920*x^2*y^2*z^2 - 11760*x^2*y^2*z - 4212*x^2*y^2 + 3360*x^2*y*z^4 + 14480*x^2*y*z^3 + 23280*x^2*y*z^2 + 16608*x^2*y*z + 4464*x^2*y - 1344*x^2*z^5 - 7200*x^2*z^4 - 15360*x^2*z^3 - 16368*x^2*z^2 - 8764*x^2*z - 1907*x^2 + 14*x*y^6 - 168*x*y^5*z - 182*x*y^5 + 840*x*y^4*z^2 + 1810*x*y^4*z + 970*x*y^4 - 2240*x*y^3*z^3 - 7200*x*y^3*z^2 - 7680*x*y^3*z - 2728*x*y^3 + 3360*x*y^2*z^4 + 14320*x*y^2*z^3 + 22800*x*y^2*z^2 + 16128*x*y^2*z + 4302*x*y^2 - 2688*x*y*z^5 - 14240*x*y*z^4 - 30080*x*y*z^3 - 31776*x*y*z^2 - 16880*x*y*z - 3644*x*y + 896*x*z^6 + 5664*x*z^5 + 14880*x*z^4 + 20864*x*z^3 + 16552*x*z^2 + 7106*x*z + 1314*x + 2*y^7 - 28*y^6*z - 30*y^6 + 168*y^5*z^2 + 358*y^5*z + 190*y^5 - 560*y^4*z^3 - 1780*y^4*z^2 - 1880*y^4*z - 662*y^4 + 1120*y^3*z^4 + 4720*y^3*z^3 + 7440*y^3*z^2 + 5216*y^3*z + 1380*y^3 - 1344*y^2*z^5 - 7040*y^2*z^4 - 14720*y^2*z^3 - 15408*y^2*z^2 - 8116*y^2*z - 1737*y^2 + 896*y*z^6 + 5600*y*z^5 + 14560*y*z^4 + 20224*y*z^3 + 15904*y*z^2 + 6768*y*z + 1237*y - 256*z^7 - 1856*z^6 - 5760*z^5 - 9952*z^4 - 10384*z^3 - 6590*z^2 - 2387*z - 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -1, -1, 2 ], [ -2, -1, 1 ], [ -1, -1, 1 ] ], "source_shift": [ 2, 1, 2 ], "target_matrix": [ [ -1, -1, -1 ], [ 3, 2, 3 ], [ -2, -1, -1 ] ], "target_shift": [ 158, -471, 388 ], "inverse_recipe": { "p": 2, "q": 5, "f": "x**2", "g": "x**5 + x**2", "h": "y*z + z", "polynomials": { "f": [ [ [ 2, 0, 0 ], 1 ] ], "g": [ [ [ 5, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 0, 1 ], 1 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0688935540019884 ], "surface_features": [ 3.720365372049232, 4.664867524500122, 3.306532247519607, 0.0, 3.8711591349106604, 4919 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C028", "pair": 28, "family": "counterexample", "mapping": "F(x,y,z)=(2*x^7 + 19*x^6*y - 13*x^6*z - 34*x^6 + 75*x^5*y^2 - 105*x^5*y*z - 276*x^5*y + 36*x^5*z^2 + 189*x^5*z + 246*x^5 + 159*x^4*y^3 - 342*x^4*y^2*z - 903*x^4*y^2 + 240*x^4*y*z^2 + 1266*x^4*y*z + 1650*x^4*y - 55*x^4*z^3 - 435*x^4*z^2 - 1137*x^4*z - 986*x^4 + 195*x^3*y^4 - 573*x^3*y^3*z - 1518*x^3*y^3 + 618*x^3*y^2*z^2 + 3273*x^3*y^2*z + 4266*x^3*y^2 - 290*x^3*y*z^3 - 2304*x^3*y*z^2 - 6030*x^3*y*z - 5221*x^3*y + 50*x^3*z^4 + 530*x^3*z^3 + 2088*x^3*z^2 + 3636*x^3*z + 2368*x^3 + 138*x^2*y^5 - 519*x^2*y^4*z - 1377*x^2*y^4 + 765*x^2*y^3*z^2 + 4062*x^2*y^3*z + 5286*x^2*y^3 - 552*x^2*y^2*z^3 - 4401*x^2*y^2*z^2 - 11517*x^2*y^2*z - 9936*x^2*y^2 + 195*x^2*y*z^4 + 2076*x^2*y*z^3 + 8190*x^2*y*z^2 + 14241*x^2*y*z + 9231*x^2*y - 27*x^2*z^5 - 360*x^2*z^4 - 1902*x^2*z^3 - 4992*x^2*z^2 - 6525*x^2*z - 3409*x^2 + 52*x*y^6 - 240*x*y^5*z - 636*x*y^5 + 453*x*y^4*z^2 + 2406*x*y^4*z + 3120*x*y^4 - 447*x*y^3*z^3 - 3570*x*y^3*z^2 - 9324*x*y^3*z - 7997*x*y^3 + 243*x*y^2*z^4 + 2595*x*y^2*z^3 + 10236*x*y^2*z^2 + 17736*x*y^2*z + 11409*x*y^2 - 69*x*y*z^5 - 924*x*y*z^4 - 4890*x*y*z^3 - 12819*x*y*z^2 - 16677*x*y*z - 8636*x*y + 8*x*z^6 + 129*x*z^5 + 858*x*z^4 + 3020*x*z^3 + 5946*x*z^2 + 6227*x*z + 2722*x + 8*y^7 - 44*y^6*z - 116*y^6 + 102*y^5*z^2 + 540*y^5*z + 696*y^5 - 129*y^4*z^3 - 1029*y^4*z^2 - 2676*y^4*z - 2278*y^4 + 96*y^3*z^4 + 1026*y^3*z^3 + 4038*y^3*z^2 + 6957*y^3*z + 4432*y^3 - 42*y^2*z^5 - 564*y^2*z^4 - 2985*y^2*z^3 - 7800*y^2*z^2 - 10071*y^2*z - 5146*y^2 + 10*y*z^6 + 162*y*z^5 + 1080*y*z^4 + 3799*y*z^3 + 7446*y*z^2 + 7724*y*z + 3324*y - z^7 - 19*z^6 - 153*z^5 - 678*z^4 - 1789*z^3 - 2818*z^2 - 2463*z - 1, 4*x^7 + 38*x^6*y - 26*x^6*z - 62*x^6 + 150*x^5*y^2 - 210*x^5*y*z - 501*x^5*y + 72*x^5*z^2 + 345*x^5*z + 420*x^5 + 318*x^4*y^3 - 684*x^4*y^2*z - 1632*x^4*y^2 + 480*x^4*y*z^2 + 2301*x^4*y*z + 2799*x^4*y - 110*x^4*z^3 - 795*x^4*z^2 - 1944*x^4*z - 1606*x^4 + 390*x^3*y^4 - 1146*x^3*y^3*z - 2733*x^3*y^3 + 1236*x^3*y^2*z^2 + 5925*x^3*y^2*z + 7194*x^3*y^2 - 580*x^3*y*z^3 - 4194*x^3*y*z^2 - 10248*x^3*y*z - 8447*x^3*y + 100*x^3*z^4 + 970*x^3*z^3 + 3576*x^3*z^2 + 5931*x^3*z + 3734*x^3 + 276*x^2*y^5 - 1038*x^2*y^4*z - 2472*x^2*y^4 + 1530*x^2*y^3*z^2 + 7329*x^2*y^3*z + 8871*x^2*y^3 - 1104*x^2*y^2*z^3 - 7983*x^2*y^2*z^2 - 19470*x^2*y^2*z - 15984*x^2*y^2 + 390*x^2*y*z^4 + 3786*x^2*y*z^3 + 13950*x^2*y*z^2 + 23085*x^2*y*z + 14469*x^2*y - 54*x^2*z^5 - 660*x^2*z^4 - 3264*x^2*z^3 - 8157*x^2*z^2 - 10302*x^2*z - 5264*x^2 + 104*x*y^6 - 480*x*y^5*z - 1140*x*y^5 + 906*x*y^4*z^2 + 4332*x*y^4*z + 5220*x*y^4 - 894*x*y^3*z^3 - 6459*x*y^3*z^2 - 15702*x*y^3*z - 12814*x*y^3 + 486*x*y^2*z^4 + 4719*x*y^2*z^3 + 17358*x*y^2*z^2 + 28608*x*y^2*z + 17802*x*y^2 - 138*x*y*z^5 - 1689*x*y*z^4 - 8352*x*y*z^3 - 20829*x*y*z^2 - 26184*x*y*z - 13273*x*y + 16*x*z^6 + 237*x*z^5 + 1476*x*z^4 + 4945*x*z^3 + 9402*x*z^2 + 9625*x*z + 4150*x + 16*y^7 - 88*y^6*z - 208*y^6 + 204*y^5*z^2 + 972*y^5*z + 1164*y^5 - 258*y^4*z^3 - 1860*y^4*z^2 - 4500*y^4*z - 3644*y^4 + 192*y^3*z^4 + 1863*y^3*z^3 + 6831*y^3*z^2 + 11190*y^3*z + 6896*y^3 - 84*y^2*z^5 - 1029*y^2*z^4 - 5082*y^2*z^3 - 12624*y^2*z^2 - 15750*y^2*z - 7886*y^2 + 20*y*z^6 + 297*y*z^5 + 1851*y*z^4 + 6191*y*z^3 + 11715*y*z^2 + 11887*y*z + 5051*y - 2*z^7 - 35*z^6 - 264*z^5 - 1113*z^4 - 2834*z^3 - 4361*z^2 - 3758*z - 1, -4*x^7 - 38*x^6*y + 26*x^6*z + 68*x^6 - 150*x^5*y^2 + 210*x^5*y*z + 552*x^5*y - 72*x^5*z^2 - 378*x^5*z - 492*x^5 - 318*x^4*y^3 + 684*x^4*y^2*z + 1806*x^4*y^2 - 480*x^4*y*z^2 - 2532*x^4*y*z - 3300*x^4*y + 110*x^4*z^3 + 870*x^4*z^2 + 2274*x^4*z + 1974*x^4 - 390*x^3*y^4 + 1146*x^3*y^3*z + 3036*x^3*y^3 - 1236*x^3*y^2*z^2 - 6546*x^3*y^2*z - 8532*x^3*y^2 + 580*x^3*y*z^3 + 4608*x^3*y*z^2 + 12060*x^3*y*z + 10455*x^3*y - 100*x^3*z^4 - 1060*x^3*z^3 - 4176*x^3*z^2 - 7279*x^3*z - 4752*x^3 - 276*x^2*y^5 + 1038*x^2*y^4*z + 2754*x^2*y^4 - 1530*x^2*y^3*z^2 - 8124*x^2*y^3*z - 10572*x^2*y^3 + 1104*x^2*y^2*z^3 + 8802*x^2*y^2*z^2 + 23034*x^2*y^2*z + 19902*x^2*y^2 - 390*x^2*y*z^4 - 4152*x^2*y*z^3 - 16380*x^2*y*z^2 - 28515*x^2*y*z - 18537*x^2*y + 54*x^2*z^5 + 720*x^2*z^4 + 3804*x^2*z^3 + 9993*x^2*z^2 + 13092*x^2*z + 6866*x^2 - 104*x*y^6 + 480*x*y^5*z + 1272*x*y^5 - 906*x*y^4*z^2 - 4812*x*y^4*z - 6240*x*y^4 + 894*x*y^3*z^3 + 7140*x*y^3*z^2 + 18648*x*y^3*z + 16022*x*y^3 - 486*x*y^2*z^4 - 5190*x*y^2*z^3 - 20472*x*y^2*z^2 - 35520*x*y^2*z - 22926*x*y^2 + 138*x*y*z^5 + 1848*x*y*z^4 + 9780*x*y*z^3 + 25665*x*y*z^2 + 33480*x*y*z + 17416*x*y - 16*x*z^6 - 258*x*z^5 - 1716*x*z^4 - 6045*x*z^3 - 11928*x*z^2 - 12538*x*z - 5506*x - 16*y^7 + 88*y^6*z + 232*y^6 - 204*y^5*z^2 - 1080*y^5*z - 1392*y^5 + 258*y^4*z^3 + 2058*y^4*z^2 + 5352*y^4*z + 4564*y^4 - 192*y^3*z^4 - 2052*y^3*z^3 - 8076*y^3*z^2 - 13934*y^3*z - 8908*y^3 + 84*y^2*z^5 + 1128*y^2*z^4 + 5970*y^2*z^3 + 15618*y^2*z^2 + 20226*y^2*z + 10388*y^2 - 20*y*z^6 - 324*y*z^5 - 2160*y*z^4 - 7605*y*z^3 - 14943*y*z^2 - 15568*y*z - 6736*y + 2*z^7 + 38*z^6 + 306*z^5 + 1357*z^4 + 3588*z^3 + 5672*z^2 + 4980*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -1, -2, 1 ], [ -1, -1, 1 ], [ -2, -1, 1 ] ], "source_shift": [ 2, 2, 1 ], "target_matrix": [ [ -1, -1, -2 ], [ -2, -1, -1 ], [ 2, 2, 3 ] ], "target_shift": [ 924, 1397, -1877 ], "collision_preimages": [ [ "-3/4", "0", "-11/4" ], [ "-9", "-5/2", "-15" ], [ "-6", "5/2", "-4" ] ], "collision_image": [ "3697/4", "2795/2", "-3755/2" ] }, "probe_target": "No", "jev_answers": [ "Maybe" ], "seconds": [ 1.1115117369990912 ], "surface_features": [ 3.6506202040464775, 4.550472957106564, 3.2546688990549204, 0.0, 3.7859555102062106, 4893 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B071", "pair": 71, "family": "bijection", "mapping": "F(x,y,z)=(-6*x^7 - 42*x^6*y + 42*x^6*z + 96*x^6 - 126*x^5*y^2 + 252*x^5*y*z + 582*x^5*y - 126*x^5*z^2 - 576*x^5*z - 661*x^5 - 210*x^4*y^3 + 630*x^4*y^2*z + 1470*x^4*y^2 - 630*x^4*y*z^2 - 2910*x^4*y*z - 3365*x^4*y + 210*x^4*z^3 + 1440*x^4*z^2 + 3305*x^4*z + 2548*x^4 - 210*x^3*y^4 + 840*x^3*y^3*z + 1980*x^3*y^3 - 1260*x^3*y^2*z^2 - 5880*x^3*y^2*z - 6850*x^3*y^2 + 840*x^3*y*z^3 + 5820*x^3*y*z^2 + 13460*x^3*y*z + 10438*x^3*y - 210*x^3*z^4 - 1920*x^3*z^3 - 6610*x^3*z^2 - 10192*x^3*z - 5987*x^3 - 126*x^2*y^5 + 630*x^2*y^4*z + 1500*x^2*y^4 - 1260*x^2*y^3*z^2 - 5940*x^2*y^3*z - 6970*x^2*y^3 + 1260*x^2*y^2*z^3 + 8820*x^2*y^2*z^2 + 20550*x^2*y^2*z + 16026*x^2*y^2 - 630*x^2*y*z^4 - 5820*x^2*y*z^3 - 20190*x^2*y*z^2 - 31314*x^2*y*z - 18483*x^2*y + 126*x^2*z^5 + 1440*x^2*z^4 + 6610*x^2*z^3 + 15288*x^2*z^2 + 17955*x^2*z + 8682*x^2 - 42*x*y^6 + 252*x*y^5*z + 606*x*y^5 - 630*x*y^4*z^2 - 3000*x*y^4*z - 3545*x*y^4 + 840*x*y^3*z^3 + 5940*x*y^3*z^2 + 13940*x*y^3*z + 10930*x*y^3 - 630*x*y^2*z^4 - 5880*x*y^2*z^3 - 20550*x*y^2*z^2 - 32052*x*y^2*z - 19005*x*y^2 + 252*x*y*z^5 + 2910*x*y*z^4 + 13460*x*y*z^3 + 31314*x*y*z^2 + 36954*x*y*z + 17964*x*y - 42*x*z^6 - 576*x*z^5 - 3305*x*z^4 - 10192*x*z^3 - 17949*x*z^2 - 17328*x*z - 7284*x - 6*y^7 + 42*y^6*z + 102*y^6 - 126*y^5*z^2 - 606*y^5*z - 721*y^5 + 210*y^4*z^3 + 1500*y^4*z^2 + 3545*y^4*z + 2794*y^4 - 210*y^3*z^4 - 1980*y^3*z^3 - 6970*y^3*z^2 - 10930*y^3*z - 6509*y^3 + 126*y^2*z^5 + 1470*y^2*z^4 + 6850*y^2*z^3 + 16026*y^2*z^2 + 18999*y^2*z + 9282*y^2 - 42*y*z^6 - 582*y*z^5 - 3365*y*z^4 - 10438*y*z^3 - 18471*y*z^2 - 17922*y*z - 7581*y + 6*z^7 + 96*z^6 + 661*z^5 + 2548*z^4 + 5981*z^3 + 8646*z^2 + 7229*z + 1, -3*x^7 - 21*x^6*y + 21*x^6*z + 48*x^6 - 63*x^5*y^2 + 126*x^5*y*z + 291*x^5*y - 63*x^5*z^2 - 288*x^5*z - 331*x^5 - 105*x^4*y^3 + 315*x^4*y^2*z + 735*x^4*y^2 - 315*x^4*y*z^2 - 1455*x^4*y*z - 1685*x^4*y + 105*x^4*z^3 + 720*x^4*z^2 + 1655*x^4*z + 1279*x^4 - 105*x^3*y^4 + 420*x^3*y^3*z + 990*x^3*y^3 - 630*x^3*y^2*z^2 - 2940*x^3*y^2*z - 3430*x^3*y^2 + 420*x^3*y*z^3 + 2910*x^3*y*z^2 + 6740*x^3*y*z + 5239*x^3*y - 105*x^3*z^4 - 960*x^3*z^3 - 3310*x^3*z^2 - 5116*x^3*z - 3014*x^3 - 63*x^2*y^5 + 315*x^2*y^4*z + 750*x^2*y^4 - 630*x^2*y^3*z^2 - 2970*x^2*y^3*z - 3490*x^2*y^3 + 630*x^2*y^2*z^3 + 4410*x^2*y^2*z^2 + 10290*x^2*y^2*z + 8043*x^2*y^2 - 315*x^2*y*z^4 - 2910*x^2*y*z^3 - 10110*x^2*y*z^2 - 15717*x^2*y*z - 9303*x^2*y + 63*x^2*z^5 + 720*x^2*z^4 + 3310*x^2*z^3 + 7674*x^2*z^2 + 9039*x^2*z + 4383*x^2 - 21*x*y^6 + 126*x*y^5*z + 303*x*y^5 - 315*x*y^4*z^2 - 1500*x*y^4*z - 1775*x*y^4 + 420*x*y^3*z^3 + 2970*x*y^3*z^2 + 6980*x*y^3*z + 5485*x*y^3 - 315*x*y^2*z^4 - 2940*x*y^2*z^3 - 10290*x*y^2*z^2 - 16086*x*y^2*z - 9564*x*y^2 + 126*x*y*z^5 + 1455*x*y*z^4 + 6740*x*y*z^3 + 15717*x*y*z^2 + 18600*x*y*z + 9066*x*y - 21*x*z^6 - 288*x*z^5 - 1655*x*z^4 - 5116*x*z^3 - 9036*x*z^2 - 8748*x*z - 3683*x - 3*y^7 + 21*y^6*z + 51*y^6 - 63*y^5*z^2 - 303*y^5*z - 361*y^5 + 105*y^4*z^3 + 750*y^4*z^2 + 1775*y^4*z + 1402*y^4 - 105*y^3*z^4 - 990*y^3*z^3 - 3490*y^3*z^2 - 5485*y^3*z - 3275*y^3 + 63*y^2*z^5 + 735*y^2*z^4 + 3430*y^2*z^3 + 8043*y^2*z^2 + 9561*y^2*z + 4683*y^2 - 21*y*z^6 - 291*y*z^5 - 1685*y*z^4 - 5239*y*z^3 - 9297*y*z^2 - 9045*y*z - 3830*y + 3*z^7 + 48*z^6 + 331*z^5 + 1279*z^4 + 3011*z^3 + 4365*z^2 + 3655*z + 1, -3*x^7 - 21*x^6*y + 21*x^6*z + 48*x^6 - 63*x^5*y^2 + 126*x^5*y*z + 291*x^5*y - 63*x^5*z^2 - 288*x^5*z - 331*x^5 - 105*x^4*y^3 + 315*x^4*y^2*z + 735*x^4*y^2 - 315*x^4*y*z^2 - 1455*x^4*y*z - 1685*x^4*y + 105*x^4*z^3 + 720*x^4*z^2 + 1655*x^4*z + 1279*x^4 - 105*x^3*y^4 + 420*x^3*y^3*z + 990*x^3*y^3 - 630*x^3*y^2*z^2 - 2940*x^3*y^2*z - 3430*x^3*y^2 + 420*x^3*y*z^3 + 2910*x^3*y*z^2 + 6740*x^3*y*z + 5239*x^3*y - 105*x^3*z^4 - 960*x^3*z^3 - 3310*x^3*z^2 - 5116*x^3*z - 3014*x^3 - 63*x^2*y^5 + 315*x^2*y^4*z + 750*x^2*y^4 - 630*x^2*y^3*z^2 - 2970*x^2*y^3*z - 3490*x^2*y^3 + 630*x^2*y^2*z^3 + 4410*x^2*y^2*z^2 + 10290*x^2*y^2*z + 8043*x^2*y^2 - 315*x^2*y*z^4 - 2910*x^2*y*z^3 - 10110*x^2*y*z^2 - 15717*x^2*y*z - 9303*x^2*y + 63*x^2*z^5 + 720*x^2*z^4 + 3310*x^2*z^3 + 7674*x^2*z^2 + 9039*x^2*z + 4381*x^2 - 21*x*y^6 + 126*x*y^5*z + 303*x*y^5 - 315*x*y^4*z^2 - 1500*x*y^4*z - 1775*x*y^4 + 420*x*y^3*z^3 + 2970*x*y^3*z^2 + 6980*x*y^3*z + 5485*x*y^3 - 315*x*y^2*z^4 - 2940*x*y^2*z^3 - 10290*x*y^2*z^2 - 16086*x*y^2*z - 9564*x*y^2 + 126*x*y*z^5 + 1455*x*y*z^4 + 6740*x*y*z^3 + 15717*x*y*z^2 + 18600*x*y*z + 9062*x*y - 21*x*z^6 - 288*x*z^5 - 1655*x*z^4 - 5116*x*z^3 - 9036*x*z^2 - 8744*x*z - 3671*x - 3*y^7 + 21*y^6*z + 51*y^6 - 63*y^5*z^2 - 303*y^5*z - 361*y^5 + 105*y^4*z^3 + 750*y^4*z^2 + 1775*y^4*z + 1402*y^4 - 105*y^3*z^4 - 990*y^3*z^3 - 3490*y^3*z^2 - 5485*y^3*z - 3275*y^3 + 63*y^2*z^5 + 735*y^2*z^4 + 3430*y^2*z^3 + 8043*y^2*z^2 + 9561*y^2*z + 4681*y^2 - 21*y*z^6 - 291*y*z^5 - 1685*y*z^4 - 5239*y*z^3 - 9297*y*z^2 - 9041*y*z - 3816*y + 3*z^7 + 48*z^6 + 331*z^5 + 1279*z^4 + 3011*z^3 + 4363*z^2 + 3643*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -1, -1, 1 ], [ -2, -1, 1 ], [ -1, -2, 1 ] ], "source_shift": [ 2, 2, 2 ], "target_matrix": [ [ 2, 1, 1 ], [ 1, 1, 1 ], [ 1, 1, 2 ] ], "target_shift": [ -2729, -1377, -1361 ], "inverse_recipe": { "p": 5, "q": 2, "f": "x**5 + x**3", "g": "x**2 + x", "h": "y*z + y + z**2 + z", "polynomials": { "f": [ [ [ 5, 0, 0 ], 1 ], [ [ 3, 0, 0 ], 1 ] ], "g": [ [ [ 2, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 0, 2 ], 1 ], [ [ 0, 1, 0 ], 1 ], [ [ 0, 0, 1 ], 1 ] ] }, "h_multiplier": 3 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 0.9769142859986459 ], "surface_features": [ 3.6097369572492926, 4.567661454473898, 3.1760912590556813, 0.0, 3.7576442839401, 4867 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B076", "pair": 76, "family": "bijection", "mapping": "F(x,y,z)=(1536*x^7 - 5376*x^6*y - 5376*x^6*z + 9216*x^6 + 8064*x^5*y^2 + 16128*x^5*y*z - 29184*x^5*y + 8064*x^5*z^2 - 28416*x^5*z + 23616*x^5 - 6720*x^4*y^3 - 20160*x^4*y^2*z + 38400*x^4*y^2 - 20160*x^4*y*z^2 + 74880*x^4*y*z - 65280*x^4*y - 6720*x^4*z^3 + 36480*x^4*z^2 - 62208*x^4*z + 33792*x^4 + 3360*x^3*y^4 + 13440*x^3*y^3*z - 26880*x^3*y^3 + 20160*x^3*y^2*z^2 - 78720*x^3*y^2*z + 71904*x^3*y^2 + 13440*x^3*y*z^3 - 76800*x^3*y*z^2 + 137280*x^3*y*z - 77760*x^3*y + 3360*x^3*z^4 - 24960*x^3*z^3 + 65472*x^3*z^2 - 72864*x^3*z + 29616*x^3 - 1008*x^2*y^5 - 5040*x^2*y^4*z + 10560*x^2*y^4 - 10080*x^2*y^3*z^2 + 41280*x^2*y^3*z - 39456*x^2*y^3 - 10080*x^2*y^2*z^3 + 60480*x^2*y^2*z^2 - 113184*x^2*y^2*z + 66816*x^2*y^2 - 5040*x^2*y*z^4 + 39360*x^2*y*z^3 - 108144*x^2*y*z^2 + 125472*x^2*y*z - 53064*x^2*y - 1008*x^2*z^5 + 9600*x^2*z^4 - 34416*x^2*z^3 + 58848*x^2*z^2 - 49032*x^2*z + 16236*x^2 + 168*x*y^6 + 1008*x*y^5*z - 2208*x*y^5 + 2520*x*y^4*z^2 - 10800*x*y^4*z + 10788*x*y^4 + 3360*x*y^3*z^3 - 21120*x*y^3*z^2 + 41328*x*y^3*z - 25392*x*y^3 + 2520*x*y^2*z^4 - 20640*x*y^2*z^3 + 59328*x*y^2*z^2 - 71688*x*y^2*z + 31572*x*y^2 + 1008*x*y*z^5 - 10080*x*y*z^4 + 37824*x*y*z^3 - 67392*x*y*z^2 + 58488*x*y*z - 20268*x*y + 168*x*z^6 - 1968*x*z^5 + 9036*x*z^4 - 21096*x*z^3 + 27048*x*z^2 - 18444*x*z + 5278*x - 12*y^7 - 84*y^6*z + 192*y^6 - 252*y^5*z^2 + 1128*y^5*z - 1176*y^5 - 420*y^4*z^3 + 2760*y^4*z^2 - 5640*y^4*z + 3600*y^4 - 420*y^3*z^4 + 3600*y^3*z^3 - 10812*y^3*z^2 + 13584*y^3*z - 6246*y^3 - 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6720*x^4*y^2 + 30240*x^4*y*z^2 + 20160*x^4*y*z + 3504*x^4*y + 15120*x^4*z^3 + 15120*x^4*z^2 + 5248*x^4*z + 648*x^4 + 4480*x^3*y^4 + 26880*x^3*y^3*z + 8960*x^3*y^3 + 60480*x^3*y^2*z^2 + 40320*x^3*y^2*z + 6976*x^3*y^2 + 60480*x^3*y*z^3 + 60480*x^3*y*z^2 + 20896*x^3*y*z + 2560*x^3*y + 22680*x^3*z^4 + 30240*x^3*z^3 + 15648*x^3*z^2 + 3820*x^3*z + 404*x^3 + 2688*x^2*y^5 + 20160*x^2*y^4*z + 6720*x^2*y^4 + 60480*x^2*y^3*z^2 + 40320*x^2*y^3*z + 6944*x^2*y^3 + 90720*x^2*y^2*z^3 + 90720*x^2*y^2*z^2 + 31200*x^2*y^2*z + 3792*x^2*y^2 + 68040*x^2*y*z^4 + 90720*x^2*y*z^3 + 46728*x^2*y*z^2 + 11316*x^2*y*z + 1188*x^2*y + 20412*x^2*z^5 + 34020*x^2*z^4 + 23328*x^2*z^3 + 8442*x^2*z^2 + 1764*x^2*z + 184*x^2 + 896*x*y^6 + 8064*x*y^5*z + 2688*x*y^5 + 30240*x*y^4*z^2 + 20160*x*y^4*z + 3456*x*y^4 + 60480*x*y^3*z^3 + 60480*x*y^3*z^2 + 20704*x*y^3*z + 2496*x*y^3 + 68040*x*y^2*z^4 + 90720*x*y^2*z^3 + 46512*x*y^2*z^2 + 11172*x*y^2*z + 1164*x*y^2 + 40824*x*y*z^5 + 68040*x*y*z^4 + 46440*x*y*z^3 + 16668*x*y*z^2 + 3456*x*y*z + 357*x*y + 10206*x*z^6 + 20412*x*z^5 + 17388*x*z^4 + 8289*x*z^3 + 2565*x*z^2 + 526*x*z + 61*x + 128*y^7 + 1344*y^6*z + 448*y^6 + 6048*y^5*z^2 + 4032*y^5*z + 688*y^5 + 15120*y^4*z^3 + 15120*y^4*z^2 + 5152*y^4*z + 616*y^4 + 22680*y^3*z^4 + 30240*y^3*z^3 + 15432*y^3*z^2 + 3676*y^3*z + 380*y^3 + 20412*y^2*z^5 + 34020*y^2*z^4 + 23112*y^2*z^3 + 8226*y^2*z^2 + 1692*y^2*z + 173*y^2 + 10206*y*z^6 + 20412*y*z^5 + 17307*y*z^4 + 8181*y*z^3 + 2511*y*z^2 + 510*y*z + 55*y + 2187*z^7 + 5103*z^6 + 5184*z^5 + 3051*z^4 + 1242*z^3 + 376*z^2 + 79*z + 1, -128*x^7 - 896*x^6*y - 1344*x^6*z - 448*x^6 - 2688*x^5*y^2 - 8064*x^5*y*z - 2688*x^5*y - 6048*x^5*z^2 - 4032*x^5*z - 704*x^5 - 4480*x^4*y^3 - 20160*x^4*y^2*z - 6720*x^4*y^2 - 30240*x^4*y*z^2 - 20160*x^4*y*z - 3504*x^4*y - 15120*x^4*z^3 - 15120*x^4*z^2 - 5248*x^4*z - 616*x^4 - 4480*x^3*y^4 - 26880*x^3*y^3*z - 8960*x^3*y^3 - 60480*x^3*y^2*z^2 - 40320*x^3*y^2*z - 6976*x^3*y^2 - 60480*x^3*y*z^3 - 60480*x^3*y*z^2 - 20896*x^3*y*z - 2432*x^3*y - 22680*x^3*z^4 - 30240*x^3*z^3 - 15648*x^3*z^2 - 3628*x^3*z - 340*x^3 - 2688*x^2*y^5 - 20160*x^2*y^4*z - 6720*x^2*y^4 - 60480*x^2*y^3*z^2 - 40320*x^2*y^3*z - 6944*x^2*y^3 - 90720*x^2*y^2*z^3 - 90720*x^2*y^2*z^2 - 31200*x^2*y^2*z - 3600*x^2*y^2 - 68040*x^2*y*z^4 - 90720*x^2*y*z^3 - 46728*x^2*y*z^2 - 10740*x^2*y*z - 996*x^2*y - 20412*x^2*z^5 - 34020*x^2*z^4 - 23328*x^2*z^3 - 8010*x^2*z^2 - 1476*x^2*z - 136*x^2 - 896*x*y^6 - 8064*x*y^5*z - 2688*x*y^5 - 30240*x*y^4*z^2 - 20160*x*y^4*z - 3456*x*y^4 - 60480*x*y^3*z^3 - 60480*x*y^3*z^2 - 20704*x*y^3*z - 2368*x*y^3 - 68040*x*y^2*z^4 - 90720*x*y^2*z^3 - 46512*x*y^2*z^2 - 10596*x*y^2*z - 972*x*y^2 - 40824*x*y*z^5 - 68040*x*y*z^4 - 46440*x*y*z^3 - 15804*x*y*z^2 - 2880*x*y*z - 261*x*y - 10206*x*z^6 - 20412*x*z^5 - 17388*x*z^4 - 7857*x*z^3 - 2133*x*z^2 - 382*x*z - 39*x - 128*y^7 - 1344*y^6*z - 448*y^6 - 6048*y^5*z^2 - 4032*y^5*z - 688*y^5 - 15120*y^4*z^3 - 15120*y^4*z^2 - 5152*y^4*z - 584*y^4 - 22680*y^3*z^4 - 30240*y^3*z^3 - 15432*y^3*z^2 - 3484*y^3*z - 316*y^3 - 20412*y^2*z^5 - 34020*y^2*z^4 - 23112*y^2*z^3 - 7794*y^2*z^2 - 1404*y^2*z - 125*y^2 - 10206*y*z^6 - 20412*y*z^5 - 17307*y*z^4 - 7749*y*z^3 - 2079*y*z^2 - 366*y*z - 33*y - 2187*z^7 - 5103*z^6 - 5184*z^5 - 2889*z^4 - 1026*z^3 - 268*z^2 - 47*z - 1, -256*x^7 - 1792*x^6*y - 2688*x^6*z - 896*x^6 - 5376*x^5*y^2 - 16128*x^5*y*z - 5376*x^5*y - 12096*x^5*z^2 - 8064*x^5*z - 1408*x^5 - 8960*x^4*y^3 - 40320*x^4*y^2*z - 13440*x^4*y^2 - 60480*x^4*y*z^2 - 40320*x^4*y*z - 7008*x^4*y - 30240*x^4*z^3 - 30240*x^4*z^2 - 10496*x^4*z - 1328*x^4 - 8960*x^3*y^4 - 53760*x^3*y^3*z - 17920*x^3*y^3 - 120960*x^3*y^2*z^2 - 80640*x^3*y^2*z - 13952*x^3*y^2 - 120960*x^3*y*z^3 - 120960*x^3*y*z^2 - 41792*x^3*y*z - 5248*x^3*y - 45360*x^3*z^4 - 60480*x^3*z^3 - 31296*x^3*z^2 - 7832*x^3*z - 864*x^3 - 5376*x^2*y^5 - 40320*x^2*y^4*z - 13440*x^2*y^4 - 120960*x^2*y^3*z^2 - 80640*x^2*y^3*z - 13888*x^2*y^3 - 181440*x^2*y^2*z^3 - 181440*x^2*y^2*z^2 - 62400*x^2*y^2*z - 7776*x^2*y^2 - 136080*x^2*y*z^4 - 181440*x^2*y*z^3 - 93456*x^2*y*z^2 - 23208*x^2*y*z - 2544*x^2*y - 40824*x^2*z^5 - 68040*x^2*z^4 - 46656*x^2*z^3 - 17316*x^2*z^2 - 3780*x^2*z - 404*x^2 - 1792*x*y^6 - 16128*x*y^5*z - 5376*x*y^5 - 60480*x*y^4*z^2 - 40320*x*y^4*z - 6912*x*y^4 - 120960*x*y^3*z^3 - 120960*x*y^3*z^2 - 41408*x*y^3*z - 5120*x*y^3 - 136080*x*y^2*z^4 - 181440*x*y^2*z^3 - 93024*x*y^2*z^2 - 22920*x*y^2*z - 2496*x*y^2 - 81648*x*y*z^5 - 136080*x*y*z^4 - 92880*x*y*z^3 - 34200*x*y*z^2 - 7416*x*y*z - 786*x*y - 20412*x*z^6 - 40824*x*z^5 - 34776*x*z^4 - 17010*x*z^3 - 5508*x*z^2 - 1160*x*z - 136*x - 256*y^7 - 2688*y^6*z - 896*y^6 - 12096*y^5*z^2 - 8064*y^5*z - 1376*y^5 - 30240*y^4*z^3 - 30240*y^4*z^2 - 10304*y^4*z - 1264*y^4 - 45360*y^3*z^4 - 60480*y^3*z^3 - 30864*y^3*z^2 - 7544*y^3*z - 816*y^3 - 40824*y^2*z^5 - 68040*y^2*z^4 - 46224*y^2*z^3 - 16884*y^2*z^2 - 3636*y^2*z - 382*y^2 - 20412*y*z^6 - 40824*y*z^5 - 34614*y*z^4 - 16794*y*z^3 - 5400*y*z^2 - 1128*y*z - 125*y - 4374*z^7 - 10206*z^6 - 10368*z^5 - 6264*z^4 - 2673*z^3 - 833*z^2 - 180*z - 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ 2, 2, 3 ], [ 2, 1, 1 ], [ 1, 1, 1 ] ], "source_shift": [ 1, 1, 1 ], "target_matrix": [ [ 1, 1, 1 ], [ -1, -1, -2 ], [ -2, -1, -1 ] ], "target_shift": [ -9, 3, 23 ], "inverse_recipe": { "p": 3, "q": 4, "f": "x**3", "g": "x**4 + x", "h": "y*z + 2*y + z", "polynomials": { "f": [ [ [ 3, 0, 0 ], 1 ] ], "g": [ [ [ 4, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 0, 1 ], 1 ], [ [ 0, 1, 0 ], 2 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 0.9839652080008818 ], "surface_features": [ 4.37232866714182, 5.258733037212813, 3.9523080096621253, 0.0, 4.516153781138056, 5168 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C118", "pair": 118, "family": "counterexample", "mapping": "F(x,y,z)=(-8*x^7 - 44*x^6*y + 68*x^6*z + 116*x^6 - 102*x^5*y^2 + 312*x^5*y*z + 516*x^5*y - 234*x^5*z^2 - 792*x^5*z - 612*x^5 - 129*x^4*y^3 + 585*x^4*y^2*z + 945*x^4*y^2 - 867*x^4*y*z^2 - 2862*x^4*y*z - 2220*x^4*y + 419*x^4*z^3 + 2121*x^4*z^2 + 3408*x^4*z + 1667*x^4 - 96*x^3*y^4 + 573*x^3*y^3*z + 912*x^3*y^3 - 1257*x^3*y^2*z^2 - 4089*x^3*y^2*z - 3177*x^3*y^2 + 1199*x^3*y*z^3 + 5982*x^3*y*z^2 + 9594*x^3*y*z + 4778*x^3*y - 419*x^3*z^4 - 2849*x^3*z^3 - 7077*x^3*z^2 - 7327*x^3*z - 2636*x^3 - 42*x^2*y^5 + 309*x^2*y^4*z + 489*x^2*y^4 - 891*x^2*y^3*z^2 - 2886*x^2*y^3*z - 2244*x^2*y^3 + 1257*x^2*y^2*z^3 + 6255*x^2*y^2*z^2 + 10005*x^2*y^2*z + 5055*x^2*y^2 - 867*x^2*y*z^4 - 5892*x^2*y*z^3 - 14526*x^2*y*z^2 - 15213*x^2*y*z - 5610*x^2*y + 234*x^2*z^5 + 2034*x^2*z^4 + 6849*x^2*z^3 + 11163*x^2*z^2 + 8571*x^2*z + 2473*x^2 - 10*x*y^6 + 87*x*y^5*z + 138*x*y^5 - 309*x*y^4*z^2 - 1005*x*y^4*z - 783*x*y^4 + 573*x*y^3*z^3 + 2868*x*y^3*z^2 + 4587*x*y^3*z + 2343*x*y^3 - 585*x*y^2*z^4 - 4005*x*y^2*z^3 - 9852*x*y^2*z^2 - 10389*x*y^2*z - 3909*x*y^2 + 312*x*y*z^5 + 2736*x*y*z^4 + 9183*x*y*z^3 + 14973*x*y*z^2 + 11691*x*y*z + 3472*x*y - 68*x*z^6 - 732*x*z^5 - 3135*x*z^4 - 6994*x*z^3 - 8502*x*z^2 - 5266*x*z - 1291*x - y^7 + 10*y^6*z + 16*y^6 - 42*y^5*z^2 - 138*y^5*z - 108*y^5 + 96*y^4*z^3 + 486*y^4*z^2 + 780*y^4*z + 402*y^4 - 129*y^3*z^4 - 894*y^3*z^3 - 2205*y^3*z^2 - 2338*y^3*z - 894*y^3 + 102*y^2*z^5 + 906*y^2*z^4 + 3048*y^2*z^3 + 4977*y^2*z^2 + 3933*y^2*z + 1195*y^2 - 44*y*z^6 - 480*y*z^5 - 2061*y*z^4 - 4591*y*z^3 - 5613*y*z^2 - 3539*y*z - 896*y + 8*z^7 + 104*z^6 + 546*z^5 + 1550*z^4 + 2593*z^3 + 2539*z^2 + 1344*z + 1, 16*x^7 + 88*x^6*y - 136*x^6*z - 220*x^6 + 204*x^5*y^2 - 624*x^5*y*z - 972*x^5*y + 468*x^5*z^2 + 1488*x^5*z + 1080*x^5 + 258*x^4*y^3 - 1170*x^4*y^2*z - 1767*x^4*y^2 + 1734*x^4*y*z^2 + 5334*x^4*y*z + 3888*x^4*y - 838*x^4*z^3 - 3939*x^4*z^2 - 5940*x^4*z - 2783*x^4 + 192*x^3*y^4 - 1146*x^3*y^3*z - 1692*x^3*y^3 + 2514*x^3*y^2*z^2 + 7557*x^3*y^2*z + 5520*x^3*y^2 - 2398*x^3*y*z^3 - 11010*x^3*y*z^2 - 16584*x^3*y*z - 7910*x^3*y + 838*x^3*z^4 + 5221*x^3*z^3 + 12168*x^3*z^2 + 12051*x^3*z + 4226*x^3 + 84*x^2*y^5 - 618*x^2*y^4*z - 900*x^2*y^4 + 1782*x^2*y^3*z^2 + 5289*x^2*y^3*z + 3867*x^2*y^3 - 2514*x^2*y^2*z^3 - 11412*x^2*y^2*z^2 - 17148*x^2*y^2*z - 8298*x^2*y^2 + 1734*x^2*y*z^4 + 10701*x^2*y*z^3 + 24759*x^2*y*z^2 + 24807*x^2*y*z + 8913*x^2*y - 468*x^2*z^5 - 3678*x^2*z^4 - 11610*x^2*z^3 - 18078*x^2*z^2 - 13512*x^2*z - 3851*x^2 + 20*x*y^6 - 174*x*y^5*z - 252*x*y^5 + 618*x*y^4*z^2 + 1827*x*y^4*z + 1338*x*y^4 - 1146*x*y^3*z^3 - 5190*x*y^3*z^2 - 7794*x*y^3*z - 3814*x*y^3 + 1170*x*y^2*z^4 + 7215*x*y^2*z^3 + 16644*x*y^2*z^2 + 16800*x*y^2*z + 6156*x*y^2 - 624*x*y*z^5 - 4908*x*y*z^4 - 15426*x*y*z^3 - 24054*x*y*z^2 - 18270*x*y*z - 5354*x*y + 136*x*z^6 + 1308*x*z^5 + 5238*x*z^4 + 11167*x*z^3 + 13182*x*z^2 + 8048*x*z + 1970*x + 2*y^7 - 20*y^6*z - 29*y^6 + 84*y^5*z^2 + 249*y^5*z + 183*y^5 - 192*y^4*z^3 - 873*y^4*z^2 - 1314*y^4*z - 649*y^4 + 258*y^3*z^4 + 1599*y^3*z^3 + 3693*y^3*z^2 + 3750*y^3*z + 1396*y^3 - 204*y^2*z^5 - 1614*y^2*z^4 - 5076*y^2*z^3 - 7932*y^2*z^2 - 6096*y^2*z - 1826*y^2 + 88*y*z^6 + 852*y*z^5 + 3414*y*z^4 + 7273*y*z^3 + 8637*y*z^2 + 5362*y*z + 1352*y - 16*z^7 - 184*z^6 - 900*z^5 - 2442*z^4 - 3964*z^3 - 3815*z^2 - 2006*z - 1, 8*x^7 + 44*x^6*y - 68*x^6*z - 128*x^6 + 102*x^5*y^2 - 312*x^5*y*z - 576*x^5*y + 234*x^5*z^2 + 888*x^5*z + 756*x^5 + 129*x^4*y^3 - 585*x^4*y^2*z - 1068*x^4*y^2 + 867*x^4*y*z^2 + 3252*x^4*y*z + 2772*x^4*y - 419*x^4*z^3 - 2424*x^4*z^2 - 4284*x^4*z - 2219*x^4 + 96*x^3*y^4 - 573*x^3*y^3*z - 1044*x^3*y^3 + 1257*x^3*y^2*z^2 + 4710*x^3*y^2*z + 4011*x^3*y^2 - 1199*x^3*y*z^3 - 6936*x^3*y*z^2 - 12198*x^3*y*z - 6428*x^3*y + 419*x^3*z^4 + 3326*x^3*z^3 + 9063*x^3*z^2 + 9937*x^3*z + 3692*x^3 + 42*x^2*y^5 - 309*x^2*y^4*z - 567*x^2*y^4 + 891*x^2*y^3*z^2 + 3369*x^2*y^3*z + 2865*x^2*y^3 - 1257*x^2*y^2*z^3 - 7353*x^2*y^2*z^2 - 12867*x^2*y^2*z - 6873*x^2*y^2 + 867*x^2*y*z^4 + 6975*x^2*y*z^3 + 18819*x^2*y*z^2 + 20853*x^2*y*z + 7944*x^2*y - 234*x^2*z^5 - 2424*x^2*z^4 - 8937*x^2*z^3 - 15429*x^2*z^2 - 12249*x^2*z - 3592*x^2 + 10*x*y^6 - 87*x*y^5*z - 162*x*y^5 + 309*x*y^4*z^2 + 1188*x*y^4*z + 1011*x*y^4 - 573*x*y^3*z^3 - 3414*x*y^3*z^2 - 5967*x*y^3*z - 3219*x*y^3 + 585*x*y^2*z^4 + 4800*x*y^2*z^3 + 12912*x*y^2*z^2 + 14388*x*y^2*z + 5595*x*y^2 - 312*x*y*z^5 - 3300*x*y*z^4 - 12123*x*y*z^3 - 20901*x*y*z^2 - 16887*x*y*z - 5104*x*y + 68*x*z^6 + 888*x*z^5 + 4167*x*z^4 + 9835*x*z^3 + 12396*x*z^2 + 7825*x*z + 1923*x + y^7 - 10*y^6*z - 19*y^6 + 42*y^5*z^2 + 165*y^5*z + 141*y^5 - 96*y^4*z^3 - 585*y^4*z^2 - 1026*y^4*z - 558*y^4 + 129*y^3*z^4 + 1083*y^3*z^3 + 2922*y^3*z^2 + 3271*y^3*z + 1293*y^3 - 102*y^2*z^5 - 1104*y^2*z^4 - 4068*y^2*z^3 - 7017*y^2*z^2 - 5739*y^2*z - 1777*y^2 + 44*y*z^6 + 588*y*z^5 + 2769*y*z^4 + 6520*y*z^3 + 8262*y*z^2 + 5318*y*z + 1353*y - 8*z^7 - 128*z^6 - 738*z^5 - 2216*z^4 - 3847*z^3 - 3856*z^2 - 2056*z - 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -1, -1, 2 ], [ -2, -1, 1 ], [ -1, -1, 1 ] ], "source_shift": [ 1, 2, 1 ], "target_matrix": [ [ 1, 1, 1 ], [ -2, -1, -1 ], [ -1, -2, -1 ] ], "target_shift": [ -290, 437, 439 ], "collision_preimages": [ [ "3/4", "3/4", "1/4" ], [ "9", "-20", "-11/2" ], [ "6", "-19", "-15/2" ] ], "collision_image": [ "-1161/4", "875/2", "1757/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0679339739981515 ], "surface_features": [ 3.5662377480667797, 4.394574246649326, 3.2337573629655103, 0.0, 3.6764469920731777, 4883 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B106", "pair": 106, "family": "bijection", "mapping": "F(x,y,z)=(2187*x^7 + 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86400*x*y^3*z^2 - 93568*x*y^3*z - 34320*x*y^3 + 20160*x*y^2*z^4 + 86400*x*y^2*z^3 + 140352*x*y^2*z^2 + 102924*x*y^2*z + 28784*x*y^2 - 8064*x*y*z^5 - 43200*x*y*z^4 - 93568*x*y*z^3 - 102888*x*y*z^2 - 57484*x*y*z - 13019*x*y + 1344*x*z^6 + 8640*x*z^5 + 23392*x*z^4 + 34284*x*z^3 + 28700*x*z^2 + 12970*x*z + 2460*x + 128*y^7 - 896*y^6*z - 960*y^6 + 2688*y^5*z^2 + 5760*y^5*z + 3120*y^5 - 4480*y^4*z^3 - 14400*y^4*z^2 - 15600*y^4*z - 5728*y^4 + 4480*y^3*z^4 + 19200*y^3*z^3 + 31200*y^3*z^2 + 22904*y^3*z + 6420*y^3 - 2688*y^2*z^5 - 14400*y^2*z^4 - 31200*y^2*z^3 - 34344*y^2*z^2 - 19232*y^2*z - 4374*y^2 + 896*y*z^6 + 5760*y*z^5 + 15600*y*z^4 + 22888*y*z^3 + 19204*y*z^2 + 8715*y*z + 1665*y - 128*z^7 - 960*z^6 - 3120*z^5 - 5720*z^4 - 6392*z^3 - 4341*z^2 - 1651*z - 1, -4374*x^7 - 20412*x^6*y + 20412*x^6*z + 21870*x^6 - 40824*x^5*y^2 + 81648*x^5*y*z + 87480*x^5*y - 40824*x^5*z^2 - 87480*x^5*z - 47304*x^5 - 45360*x^4*y^3 + 136080*x^4*y^2*z + 145800*x^4*y^2 - 136080*x^4*y*z^2 - 291600*x^4*y*z - 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4480*y^4*z^3 - 14400*y^4*z^2 - 15600*y^4*z - 5744*y^4 + 4480*y^3*z^4 + 19200*y^3*z^3 + 31200*y^3*z^2 + 22968*y^3*z + 6500*y^3 - 2688*y^2*z^5 - 14400*y^2*z^4 - 31200*y^2*z^3 - 34440*y^2*z^2 - 19472*y^2*z - 4530*y^2 + 896*y*z^6 + 5760*y*z^5 + 15600*y*z^4 + 22952*y*z^3 + 19444*y*z^2 + 9027*y*z + 1805*y - 128*z^7 - 960*z^6 - 3120*z^5 - 5736*z^4 - 6472*z^3 - 4497*z^2 - 1790*z - 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -3, -2, 2 ], [ -1, -2, 1 ], [ -1, -1, 1 ] ], "source_shift": [ 2, 2, 1 ], "target_matrix": [ [ -1, -1, -2 ], [ 2, 2, 3 ], [ -1, -2, -1 ] ], "target_shift": [ 270, -567, 318 ], "inverse_recipe": { "p": 4, "q": 3, "f": "x**4 + x**2", "g": "x**3 + x**2", "h": "y*z + z", "polynomials": { "f": [ [ [ 4, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "g": [ [ [ 3, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 0, 1 ], 1 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.072731592998025 ], "surface_features": [ 4.779586411308603, 5.624191307452686, 4.494154594018442, 0.0, 4.875928665281686, 5363 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B110", "pair": 110, "family": "bijection", "mapping": "F(x,y,z)=(-2*x^7 + 27*x^6*y - 15*x^6*z + 25*x^6 - 156*x^5*y^2 + 174*x^5*y*z - 288*x^5*y - 48*x^5*z^2 + 162*x^5*z - 134*x^5 + 500*x^4*y^3 - 840*x^4*y^2*z + 1380*x^4*y^2 + 465*x^4*y*z^2 - 1560*x^4*y*z + 1279*x^4*y - 85*x^4*z^3 + 435*x^4*z^2 - 731*x^4*z + 397*x^4 - 960*x^3*y^4 + 2160*x^3*y^3*z - 3520*x^3*y^3 - 1800*x^3*y^2*z^2 + 6000*x^3*y^2*z - 4872*x^3*y^2 + 660*x^3*y*z^3 - 3360*x^3*y*z^2 + 5604*x^3*y*z - 3008*x^3*y - 90*x^3*z^4 + 620*x^3*z^3 - 1584*x^3*z^2 + 1756*x^3*z - 698*x^3 + 1104*x^2*y^5 - 3120*x^2*y^4*z + 5040*x^2*y^4 + 3480*x^2*y^3*z^2 - 11520*x^2*y^3*z + 9256*x^2*y^3 - 1920*x^2*y^2*z^3 + 9720*x^2*y^2*z^2 - 16080*x^2*y^2*z + 8520*x^2*y^2 + 525*x^2*y*z^4 - 3600*x^2*y*z^3 + 9138*x^2*y*z^2 - 10032*x^2*y*z + 3923*x^2*y - 57*x^2*z^5 + 495*x^2*z^4 - 1706*x^2*z^3 + 2886*x^2*z^2 - 2359*x^2*z + 731*x^2 - 704*x*y^6 + 2400*x*y^5*z - 3840*x*y^5 - 3360*x*y^4*z^2 + 11040*x*y^4*z - 8768*x*y^4 + 2480*x*y^3*z^3 - 12480*x*y^3*z^2 + 20464*x*y^3*z - 10688*x*y^3 - 1020*x*y^2*z^4 + 6960*x*y^2*z^3 - 17544*x*y^2*z^2 + 19056*x*y^2*z - 7316*x*y^2 + 222*x*y*z^5 - 1920*x*y*z^4 + 6580*x*y*z^3 - 11040*x*y*z^2 + 8906*x*y*z - 2687*x*y - 20*x*z^6 + 210*x*z^5 - 914*x*z^4 + 2092*x*z^3 - 2624*x*z^2 + 1697*x*z - 420*x + 192*y^7 - 768*y^6*z + 1216*y^6 + 1296*y^5*z^2 - 4224*y^5*z + 3312*y^5 - 1200*y^4*z^3 + 6000*y^4*z^2 - 9744*y^4*z + 5008*y^4 + 660*y^3*z^4 - 4480*y^3*z^3 + 11208*y^3*z^2 - 12032*y^3*z + 4524*y^3 - 216*y^2*z^5 + 1860*y^2*z^4 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6504*x^2*z^2 + 5726*x^2*z - 1992*x^2 + 1408*x*y^6 - 4800*x*y^5*z + 8064*x*y^5 + 6720*x*y^4*z^2 - 23040*x*y^4*z + 19456*x*y^4 - 4960*x*y^3*z^3 + 25920*x*y^3*z^2 - 44768*x*y^3*z + 25280*x*y^3 + 2040*x*y^2*z^4 - 14400*x*y^2*z^3 + 37968*x*y^2*z^2 - 43968*x*y^2*z + 18664*x*y^2 - 444*x*y*z^5 + 3960*x*y*z^4 - 14120*x*y*z^3 + 25008*x*y*z^2 - 21844*x*y*z + 7494*x*y + 40*x*z^6 - 432*x*z^5 + 1948*x*z^4 - 4672*x*z^3 + 6256*x*z^2 - 4454*x*z + 1300*x - 384*y^7 + 1536*y^6*z - 2560*y^6 - 2592*y^5*z^2 + 8832*y^5*z - 7392*y^5 + 2400*y^4*z^3 - 12480*y^4*z^2 + 21408*y^4*z - 11968*y^4 - 1320*y^3*z^4 + 9280*y^3*z^3 - 24336*y^3*z^2 + 27968*y^3*z - 11736*y^3 + 432*y^2*z^5 - 3840*y^2*z^4 + 13632*y^2*z^3 - 24000*y^2*z^2 + 20784*y^2*z - 7024*y^2 - 78*y*z^6 + 840*y*z^5 - 3774*y*z^4 + 9008*y*z^3 - 11982*y*z^2 + 8428*y*z - 2409*y + 6*z^7 - 76*z^6 + 414*z^5 - 1252*z^4 + 2262*z^3 - 2464*z^2 + 1487*z - 1, 6*x^7 - 81*x^6*y + 45*x^6*z - 77*x^6 + 468*x^5*y^2 - 522*x^5*y*z + 888*x^5*y + 144*x^5*z^2 - 498*x^5*z + 426*x^5 - 1500*x^4*y^3 + 2520*x^4*y^2*z - 4260*x^4*y^2 - 1395*x^4*y*z^2 + 4800*x^4*y*z - 4077*x^4*y + 255*x^4*z^3 - 1335*x^4*z^2 + 2313*x^4*z - 1313*x^4 + 2880*x^3*y^4 - 6480*x^3*y^3*z + 10880*x^3*y^3 + 5400*x^3*y^2*z^2 - 18480*x^3*y^2*z + 15576*x^3*y^2 - 1980*x^3*y*z^3 + 10320*x^3*y*z^2 - 17772*x^3*y*z + 10000*x^3*y + 270*x^3*z^4 - 1900*x^3*z^3 + 4992*x^3*z^2 - 5756*x^3*z + 2430*x^3 - 3312*x^2*y^5 + 9360*x^2*y^4*z - 15600*x^2*y^4 - 10440*x^2*y^3*z^2 + 35520*x^2*y^3*z - 29688*x^2*y^3 + 5760*x^2*y^2*z^3 - 29880*x^2*y^2*z^2 + 51120*x^2*y^2*z - 28488*x^2*y^2 - 1575*x^2*y*z^4 + 11040*x^2*y*z^3 - 28854*x^2*y*z^2 + 33024*x^2*y*z - 13785*x^2*y + 171*x^2*z^5 - 1515*x^2*z^4 + 5358*x^2*z^3 - 9390*x^2*z^2 + 8085*x^2*z - 2723*x^2 + 2112*x*y^6 - 7200*x*y^5*z + 11904*x*y^5 + 10080*x*y^4*z^2 - 34080*x*y^4*z + 28224*x*y^4 - 7440*x*y^3*z^3 + 38400*x*y^3*z^2 - 65232*x*y^3*z + 35968*x*y^3 + 3060*x*y^2*z^4 - 21360*x*y^2*z^3 + 55512*x*y^2*z^2 - 63024*x*y^2*z + 25980*x*y^2 - 666*x*y*z^5 + 5880*x*y*z^4 - 20700*x*y*z^3 + 36048*x*y*z^2 - 30750*x*y*z + 10181*x*y + 60*x*z^6 - 642*x*z^5 + 2862*x*z^4 - 6764*x*z^3 + 8880*x*z^2 - 6151*x*z + 1722*x - 576*y^7 + 2304*y^6*z - 3776*y^6 - 3888*y^5*z^2 + 13056*y^5*z - 10704*y^5 + 3600*y^4*z^3 - 18480*y^4*z^2 + 31152*y^4*z - 16976*y^4 - 1980*y^3*z^4 + 13760*y^3*z^3 - 35544*y^3*z^2 + 40000*y^3*z - 16260*y^3 + 648*y^2*z^5 - 5700*y^2*z^4 + 19968*y^2*z^3 - 34536*y^2*z^2 + 29160*y^2*z - 9476*y^2 - 117*y*z^6 + 1248*y*z^5 - 5541*y*z^4 + 13024*y*z^3 - 16965*y*z^2 + 11582*y*z - 3157*y + 9*z^7 - 113*z^6 + 609*z^5 - 1817*z^4 + 3225*z^3 - 3431*z^2 + 2004*z - 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -1, 2, -1 ], [ -1, 1, -2 ], [ -2, 3, -2 ] ], "source_shift": [ 2, 1, 1 ], "target_matrix": [ [ 1, 1, 1 ], [ -2, -1, -1 ], [ -3, -3, -2 ] ], "target_shift": [ -100, 372, 476 ], "inverse_recipe": { "p": 1, "q": 6, "f": "x", "g": "x**6 + x**4 + x**2 + x", "h": "y**2 + y*z", "polynomials": { "f": [ [ [ 1, 0, 0 ], 1 ] ], "g": [ [ [ 6, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ], [ [ 4, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 2, 0 ], 1 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0214364099992963 ], "surface_features": [ 3.8883643902162164, 4.814460694108082, 3.5232260419657013, 0.0, 4.028804532203014, 5004 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C219", "pair": 219, "family": "counterexample", "mapping": "F(x,y,z)=(x^7 - 8*x^6*y + 10*x^6*z - 19*x^6 + 27*x^5*y^2 - 69*x^5*y*z + 129*x^5*y + 42*x^5*z^2 - 162*x^5*z + 153*x^5 - 50*x^4*y^3 + 195*x^4*y^2*z - 360*x^4*y^2 - 243*x^4*y*z^2 + 924*x^4*y*z - 858*x^4*y + 96*x^4*z^3 - 564*x^4*z^2 + 1080*x^4*z - 678*x^4 + 55*x^3*y^4 - 290*x^3*y^3*z + 530*x^3*y^3 + 552*x^3*y^2*z^2 - 2076*x^3*y^2*z + 1902*x^3*y^2 - 447*x^3*y*z^3 + 2595*x^3*y*z^2 - 4890*x^3*y*z + 3020*x^3*y + 129*x^3*z^4 - 1026*x^3*z^3 + 2985*x^3*z^2 - 3799*x^3*z + 1789*x^3 - 36*x^2*y^5 + 240*x^2*y^4*z - 435*x^2*y^4 - 618*x^2*y^3*z^2 + 2304*x^2*y^3*z - 2088*x^2*y^3 + 765*x^2*y^2*z^3 - 4401*x^2*y^2*z^2 + 8190*x^2*y^2*z - 4992*x^2*y^2 - 453*x^2*y*z^4 + 3570*x^2*y*z^3 - 10236*x^2*y*z^2 + 12819*x^2*y*z - 5946*x^2*y + 102*x^2*z^5 - 1029*x^2*z^4 + 4038*x^2*z^3 - 7800*x^2*z^2 + 7446*x^2*z - 2818*x^2 + 13*x*y^6 - 105*x*y^5*z + 189*x*y^5 + 342*x*y^4*z^2 - 1266*x*y^4*z + 1137*x*y^4 - 573*x*y^3*z^3 + 3273*x*y^3*z^2 - 6030*x*y^3*z + 3636*x*y^3 + 519*x*y^2*z^4 - 4062*x*y^2*z^3 + 11517*x*y^2*z^2 - 14241*x*y^2*z + 6525*x*y^2 - 240*x*y*z^5 + 2406*x*y*z^4 - 9324*x*y*z^3 + 17736*x*y*z^2 - 16677*x*y*z + 6227*x*y + 44*x*z^6 - 540*x*z^5 + 2676*x*z^4 - 6957*x*z^3 + 10071*x*z^2 - 7724*x*z + 2463*x - 2*y^7 + 19*y^6*z - 34*y^6 - 75*y^5*z^2 + 276*y^5*z - 246*y^5 + 159*y^4*z^3 - 903*y^4*z^2 + 1650*y^4*z - 986*y^4 - 195*y^3*z^4 + 1518*y^3*z^3 - 4266*y^3*z^2 + 5221*y^3*z - 2368*y^3 + 138*y^2*z^5 - 1377*y^2*z^4 + 5286*y^2*z^3 - 9936*y^2*z^2 + 9231*y^2*z - 3409*y^2 - 52*y*z^6 + 636*y*z^5 - 3120*y*z^4 + 7997*y*z^3 - 11409*y*z^2 + 8636*y*z - 2722*y + 8*z^7 - 116*z^6 + 696*z^5 - 2278*z^4 + 4432*z^3 - 5146*z^2 + 3324*z - 1, -3*x^7 + 24*x^6*y - 30*x^6*z + 54*x^6 - 81*x^5*y^2 + 207*x^5*y*z - 366*x^5*y - 126*x^5*z^2 + 459*x^5*z - 417*x^5 + 150*x^4*y^3 - 585*x^4*y^2*z + 1020*x^4*y^2 + 729*x^4*y*z^2 - 2613*x^4*y*z + 2334*x^4*y - 288*x^4*z^3 + 1593*x^4*z^2 - 2931*x^4*z + 1791*x^4 - 165*x^3*y^4 + 870*x^3*y^3*z - 1500*x^3*y^3 - 1656*x^3*y^2*z^2 + 5862*x^3*y^2*z - 5166*x^3*y^2 + 1341*x^3*y*z^3 - 7314*x^3*y*z^2 + 13242*x^3*y*z - 7965*x^3*y - 387*x^3*z^4 + 2889*x^3*z^3 - 8067*x^3*z^2 + 9990*x^3*z - 4623*x^3 + 108*x^2*y^5 - 720*x^2*y^4*z + 1230*x^2*y^4 + 1854*x^2*y^3*z^2 - 6498*x^2*y^3*z + 5664*x^2*y^3 - 2295*x^2*y^2*z^3 + 12384*x^2*y^2*z^2 - 22140*x^2*y^2*z + 13149*x^2*y^2 + 1359*x^2*y*z^4 - 10029*x^2*y*z^3 + 27594*x^2*y*z^2 - 33648*x^2*y*z + 15348*x^2*y - 306*x^2*z^5 + 2889*x^2*z^4 - 10869*x^2*z^3 + 20424*x^2*z^2 - 19161*x^2*z + 7179*x^2 - 39*x*y^6 + 315*x*y^5*z - 534*x*y^5 - 1026*x*y^4*z^2 + 3567*x*y^4*z - 3081*x*y^4 + 1719*x*y^3*z^3 - 9198*x*y^3*z^2 + 16278*x*y^3*z - 9567*x*y^3 - 1557*x*y^2*z^4 + 11391*x*y^2*z^3 - 30987*x*y^2*z^2 + 37326*x*y^2*z - 16827*x*y^2 + 720*x*y*z^5 - 6738*x*y*z^4 + 25026*x*y*z^3 - 46344*x*y*z^2 + 42861*x*y*z - 15852*x*y - 132*x*z^6 + 1512*x*z^5 - 7176*x*z^4 + 18147*x*z^3 - 25821*x*z^2 + 19611*x*z - 6221*x + 6*y^7 - 57*y^6*z + 96*y^6 + 225*y^5*z^2 - 777*y^5*z + 666*y^5 - 477*y^4*z^3 + 2535*y^4*z^2 - 4449*y^4*z + 2592*y^4 + 585*y^3*z^4 - 4251*y^3*z^3 + 11460*y^3*z^2 - 13668*y^3*z + 6102*y^3 - 414*y^2*z^5 + 3849*y^2*z^4 - 14157*y^2*z^3 + 25920*y^2*z^2 - 23700*y^2*z + 8673*y^2 + 156*y*z^6 - 1776*y*z^5 + 8340*y*z^4 - 20811*y*z^3 + 29211*y*z^2 - 21909*y*z + 6872*y - 24*z^7 + 324*z^6 - 1860*z^5 + 5922*z^4 - 11328*z^3 + 13032*z^2 - 8375*z + 1, x^7 - 8*x^6*y + 10*x^6*z - 19*x^6 + 27*x^5*y^2 - 69*x^5*y*z + 129*x^5*y + 42*x^5*z^2 - 162*x^5*z + 153*x^5 - 50*x^4*y^3 + 195*x^4*y^2*z - 360*x^4*y^2 - 243*x^4*y*z^2 + 924*x^4*y*z - 858*x^4*y + 96*x^4*z^3 - 564*x^4*z^2 + 1080*x^4*z - 679*x^4 + 55*x^3*y^4 - 290*x^3*y^3*z + 530*x^3*y^3 + 552*x^3*y^2*z^2 - 2076*x^3*y^2*z + 1902*x^3*y^2 - 447*x^3*y*z^3 + 2595*x^3*y*z^2 - 4890*x^3*y*z + 3025*x^3*y + 129*x^3*z^4 - 1026*x^3*z^3 + 2985*x^3*z^2 - 3806*x^3*z + 1799*x^3 - 36*x^2*y^5 + 240*x^2*y^4*z - 435*x^2*y^4 - 618*x^2*y^3*z^2 + 2304*x^2*y^3*z - 2088*x^2*y^3 + 765*x^2*y^2*z^3 - 4401*x^2*y^2*z^2 + 8190*x^2*y^2*z - 5001*x^2*y^2 - 453*x^2*y*z^4 + 3570*x^2*y*z^3 - 10236*x^2*y*z^2 + 12846*x^2*y*z - 5982*x^2*y + 102*x^2*z^5 - 1029*x^2*z^4 + 4038*x^2*z^3 - 7818*x^2*z^2 + 7497*x^2*z - 2854*x^2 + 13*x*y^6 - 105*x*y^5*z + 189*x*y^5 + 342*x*y^4*z^2 - 1266*x*y^4*z + 1137*x*y^4 - 573*x*y^3*z^3 + 3273*x*y^3*z^2 - 6030*x*y^3*z + 3643*x*y^3 + 519*x*y^2*z^4 - 4062*x*y^2*z^3 + 11517*x*y^2*z^2 - 14274*x*y^2*z + 6567*x*y^2 - 240*x*y*z^5 + 2406*x*y*z^4 - 9324*x*y*z^3 + 17784*x*y*z^2 - 16803*x*y*z + 6311*x*y + 44*x*z^6 - 540*x*z^5 + 2676*x*z^4 - 6977*x*z^3 + 10155*x*z^2 - 7844*x*z + 2517*x - 2*y^7 + 19*y^6*z - 34*y^6 - 75*y^5*z^2 + 276*y^5*z - 246*y^5 + 159*y^4*z^3 - 903*y^4*z^2 + 1650*y^4*z - 988*y^4 - 195*y^3*z^4 + 1518*y^3*z^3 - 4266*y^3*z^2 + 5234*y^3*z - 2384*y^3 + 138*y^2*z^5 - 1377*y^2*z^4 + 5286*y^2*z^3 - 9966*y^2*z^2 + 9306*y^2*z - 3457*y^2 - 52*y*z^6 + 636*y*z^5 - 3120*y*z^4 + 8025*y*z^3 - 11517*y*z^2 + 8780*y*z - 2784*y + 8*z^7 - 116*z^6 + 696*z^5 - 2286*z^4 + 4476*z^3 - 5242*z^2 + 3412*z - 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -1, 1, -2 ], [ -1, 1, -1 ], [ -1, 2, -1 ] ], "source_shift": [ 2, 2, 1 ], "target_matrix": [ [ -1, -1, -2 ], [ 3, 2, 3 ], [ -1, -1, -1 ] ], "target_shift": [ 924, -2322, 952 ], "collision_preimages": [ [ "11/4", "3/4", "0" ], [ "15", "9", "-5/2" ], [ "4", "6", "5/2" ] ], "collision_image": [ "3697/4", "-9291/4", "3809/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0168292910020682 ], "surface_features": [ 3.6506202040464775, 4.665993515491, 3.2272437815030623, 0.0, 3.8338561811999594, 4874 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C029", "pair": 29, "family": "counterexample", "mapping": "F(x,y,z)=(-2*x^7 - 21*x^6*y + 20*x^6*z + 23*x^6 - 93*x^5*y^2 + 183*x^5*y*z + 207*x^5*y - 84*x^5*z^2 - 198*x^5*z - 111*x^5 - 225*x^4*y^3 + 687*x^4*y^2*z + 765*x^4*y^2 - 651*x^4*y*z^2 - 1506*x^4*y*z - 828*x^4*y + 192*x^4*z^3 + 699*x^4*z^2 + 804*x^4*z + 295*x^4 - 321*x^3*y^4 + 1353*x^3*y^3*z + 1485*x^3*y^3 - 1989*x^3*y^2*z^2 - 4518*x^3*y^2*z - 2433*x^3*y^2 + 1209*x^3*y*z^3 + 4311*x^3*y*z^2 + 4866*x^3*y*z + 1745*x^3*y - 258*x^3*z^4 - 1296*x^3*z^3 - 2295*x^3*z^2 - 1729*x^3*z - 472*x^3 - 270*x^2*y^5 + 1473*x^2*y^4*z + 1596*x^2*y^4 - 2991*x^2*y^3*z^2 - 6678*x^2*y^3*z - 3519*x^2*y^3 + 2817*x^2*y^2*z^3 + 9837*x^2*y^2*z^2 + 10878*x^2*y^2*z + 3804*x^2*y^2 - 1236*x^2*y*z^4 - 6072*x^2*y*z^3 - 10563*x^2*y*z^2 - 7791*x^2*y*z - 2067*x^2*y + 204*x^2*z^5 + 1332*x^2*z^4 + 3228*x^2*z^3 + 3741*x^2*z^2 + 2103*x^2*z + 463*x^2 - 124*x*y^6 + 840*x*y^5*z + 900*x*y^5 - 2211*x*y^4*z^2 - 4860*x*y^4*z - 2505*x*y^4 + 2875*x*y^3*z^3 + 9837*x*y^3*z^2 + 10641*x*y^3*z + 3620*x*y^3 - 1950*x*y^2*z^4 - 9360*x*y^2*z^3 - 15966*x*y^2*z^2 - 11505*x*y^2*z - 2955*x*y^2 + 660*x*y*z^5 + 4212*x*y*z^4 + 10041*x*y*z^3 + 11409*x*y*z^2 + 6252*x*y*z + 1331*x*y - 88*x*z^6 - 720*x*z^5 - 2238*x*z^4 - 3544*x*z^3 - 3069*x*z^2 - 1393*x*z - 266*x - 24*y^7 + 196*y^6*z + 208*y^6 - 642*y^5*z^2 - 1392*y^5*z - 702*y^5 + 1083*y^4*z^3 + 3636*y^4*z^2 + 3843*y^4*z + 1268*y^4 - 1012*y^3*z^4 - 4744*y^3*z^3 - 7920*y^3*z^2 - 5567*y^3*z - 1379*y^3 + 528*y^2*z^5 + 3288*y^2*z^4 + 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88*x*z^6 - 720*x*z^5 - 2238*x*z^4 - 3552*x*z^3 - 3108*x*z^2 - 1441*x*z - 281*x - 24*y^7 + 196*y^6*z + 208*y^6 - 642*y^5*z^2 - 1392*y^5*z - 702*y^5 + 1083*y^4*z^3 + 3636*y^4*z^2 + 3843*y^4*z + 1292*y^4 - 1012*y^3*z^4 - 4744*y^3*z^3 - 7920*y^3*z^2 - 5619*y^3*z - 1443*y^3 + 528*y^2*z^5 + 3288*y^2*z^4 + 7695*y^2*z^3 + 8592*y^2*z^2 + 4671*y^2*z + 1000*y^2 - 144*y*z^6 - 1152*y*z^5 - 3528*y*z^4 - 5503*y*z^3 - 4716*y*z^2 - 2128*y*z - 405*y + 16*z^7 + 160*z^6 + 612*z^5 + 1242*z^4 + 1482*z^3 + 1057*z^2 + 419*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -1, -2, 1 ], [ -1, -1, 2 ], [ -2, -3, 2 ] ], "source_shift": [ 1, 1, 2 ], "target_matrix": [ [ 1, 1, 2 ], [ 2, 1, 1 ], [ 1, 1, 1 ] ], "target_shift": [ -69, -102, -72 ], "collision_preimages": [ [ "7/4", "-1/4", "1/4" ], [ "-16", "9/2", "-7" ], [ "-21", "17/2", "-6" ] ], "collision_image": [ "-277/4", "-205/2", "-289/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0619932139998127 ], "surface_features": [ 3.5209211002864995, 4.407101845235177, 3.238798562713917, 0.0, 3.605962357406199, 4924 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C081", "pair": 81, "family": "counterexample", "mapping": "F(x,y,z)=(-8*x^7 + 44*x^6*y - 68*x^6*z + 148*x^6 - 102*x^5*y^2 + 312*x^5*y*z - 660*x^5*y - 234*x^5*z^2 + 1008*x^5*z - 996*x^5 + 129*x^4*y^3 - 585*x^4*y^2*z + 1209*x^4*y^2 + 867*x^4*y*z^2 - 3642*x^4*y*z + 3588*x^4*y - 419*x^4*z^3 + 2685*x^4*z^2 - 5412*x^4*z + 3438*x^4 - 96*x^3*y^4 + 573*x^3*y^3*z - 1164*x^3*y^3 - 1257*x^3*y^2*z^2 + 5187*x^3*y^2*z - 5097*x^3*y^2 + 1199*x^3*y*z^3 - 7542*x^3*y*z^2 + 15138*x^3*y*z - 9666*x^3*y - 419*x^3*z^4 + 3571*x^3*z^3 - 10989*x^3*z^2 + 14360*x^3*z - 6762*x^3 + 42*x^2*y^5 - 309*x^2*y^4*z + 621*x^2*y^4 + 891*x^2*y^3*z^2 - 3639*x^2*y^3*z + 3570*x^2*y^3 - 1257*x^2*y^2*z^3 + 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1048*y^4 - 129*y^3*z^4 + 1278*y^3*z^3 - 4287*y^3*z^2 + 5996*y^3*z - 3008*y^3 + 102*y^2*z^5 - 1293*y^2*z^4 + 5889*y^2*z^3 - 12579*y^2*z^2 + 12861*y^2*z - 5098*y^2 - 44*y*z^6 + 684*y*z^5 - 3960*y*z^4 + 11465*y*z^3 - 17895*y*z^2 + 14462*y*z - 4765*y + 8*z^7 - 148*z^6 + 1044*z^5 - 3834*z^4 + 8104*z^3 - 9994*z^2 + 6719*z - 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -1, 1, -2 ], [ -2, 1, -1 ], [ -1, 1, -1 ] ], "source_shift": [ 2, 2, 2 ], "target_matrix": [ [ 1, 1, 2 ], [ -1, -1, -1 ], [ -1, -2, -1 ] ], "target_shift": [ -1183, 1219, 1905 ], "collision_preimages": [ [ "-1/4", "-11/4", "-1/4" ], [ "8", "18", "11/2" ], [ "5", "17", "15/2" ] ], "collision_image": [ "-4733/4", "4877/4", "7621/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0911689700005809 ], "surface_features": [ 3.740218778188363, 4.59469081818349, 3.3998467127129226, 0.0, 3.8724040679259666, 4911 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C036", "pair": 36, "family": "counterexample", "mapping": "F(x,y,z)=(-2*x^7 - 19*x^6*y - 13*x^6*z - 28*x^6 - 75*x^5*y^2 - 105*x^5*y*z - 231*x^5*y - 36*x^5*z^2 - 156*x^5*z - 165*x^5 - 159*x^4*y^3 - 342*x^4*y^2*z - 768*x^4*y^2 - 240*x^4*y*z^2 - 1062*x^4*y*z - 1143*x^4*y - 55*x^4*z^3 - 360*x^4*z^2 - 768*x^4*z - 545*x^4 - 195*x^3*y^4 - 573*x^3*y^3*z - 1311*x^3*y^3 - 618*x^3*y^2*z^2 - 2790*x^3*y^2*z - 3054*x^3*y^2 - 290*x^3*y*z^3 - 1938*x^3*y*z^2 - 4209*x^3*y*z - 3037*x^3*y - 50*x^3*z^4 - 440*x^3*z^3 - 1422*x^3*z^2 - 2037*x^3*z - 1098*x^3 - 138*x^2*y^5 - 519*x^2*y^4*z - 1206*x^2*y^4 - 765*x^2*y^3*z^2 - 3516*x^2*y^3*z - 3909*x^2*y^3 - 552*x^2*y^2*z^3 - 3762*x^2*y^2*z^2 - 8313*x^2*y^2*z - 6093*x^2*y^2 - 195*x^2*y*z^4 - 1752*x^2*y*z^3 - 5769*x^2*y*z^2 - 8412*x^2*y*z - 4614*x^2*y - 27*x^2*z^5 - 300*x^2*z^4 - 1308*x^2*z^3 - 2841*x^2*z^2 - 3090*x^2*z - 1354*x^2 - 52*x*y^6 - 240*x*y^5*z - 564*x*y^5 - 453*x*y^4*z^2 - 2112*x*y^4*z - 2379*x*y^4 - 447*x*y^3*z^3 - 3099*x*y^3*z^2 - 6957*x*y^3*z - 5170*x*y^3 - 243*x*y^2*z^4 - 2226*x*y^2*z^3 - 7464*x*y^2*z^2 - 11070*x*y^2*z - 6171*x*y^2 - 69*x*y*z^5 - 783*x*y*z^4 - 3483*x*y*z^3 - 7713*x*y*z^2 - 8547*x*y*z - 3812*x*y - 8*x*z^6 - 108*x*z^5 - 597*x*z^4 - 1751*x*z^3 - 2886*x*z^2 - 2549*x*z - 949*x - 8*y^7 - 44*y^6*z - 104*y^6 - 102*y^5*z^2 - 480*y^5*z - 546*y^5 - 129*y^4*z^3 - 906*y^4*z^2 - 2061*y^4*z - 1550*y^4 - 96*y^3*z^4 - 894*y^3*z^3 - 3048*y^3*z^2 - 4591*y^3*z - 2593*y^3 - 42*y^2*z^5 - 486*y^2*z^4 - 2205*y^2*z^3 - 4977*y^2*z^2 - 5613*y^2*z - 2539*y^2 - 10*y*z^6 - 138*y*z^5 - 780*y*z^4 - 2338*y*z^3 - 3933*y*z^2 - 3539*y*z - 1344*y - z^7 - 16*z^6 - 108*z^5 - 402*z^4 - 894*z^3 - 1195*z^2 - 896*z - 1, 2*x^7 + 19*x^6*y + 13*x^6*z + 28*x^6 + 75*x^5*y^2 + 105*x^5*y*z + 231*x^5*y + 36*x^5*z^2 + 156*x^5*z + 165*x^5 + 159*x^4*y^3 + 342*x^4*y^2*z + 768*x^4*y^2 + 240*x^4*y*z^2 + 1062*x^4*y*z + 1143*x^4*y + 55*x^4*z^3 + 360*x^4*z^2 + 768*x^4*z + 543*x^4 + 195*x^3*y^4 + 573*x^3*y^3*z + 1311*x^3*y^3 + 618*x^3*y^2*z^2 + 2790*x^3*y^2*z + 3054*x^3*y^2 + 290*x^3*y*z^3 + 1938*x^3*y*z^2 + 4209*x^3*y*z + 3024*x^3*y + 50*x^3*z^4 + 440*x^3*z^3 + 1422*x^3*z^2 + 2030*x^3*z + 1088*x^3 + 138*x^2*y^5 + 519*x^2*y^4*z + 1206*x^2*y^4 + 765*x^2*y^3*z^2 + 3516*x^2*y^3*z + 3909*x^2*y^3 + 552*x^2*y^2*z^3 + 3762*x^2*y^2*z^2 + 8313*x^2*y^2*z + 6063*x^2*y^2 + 195*x^2*y*z^4 + 1752*x^2*y*z^3 + 5769*x^2*y*z^2 + 8379*x^2*y*z + 4566*x^2*y + 27*x^2*z^5 + 300*x^2*z^4 + 1308*x^2*z^3 + 2832*x^2*z^2 + 3063*x^2*z + 1333*x^2 + 52*x*y^6 + 240*x*y^5*z + 564*x*y^5 + 453*x*y^4*z^2 + 2112*x*y^4*z + 2379*x*y^4 + 447*x*y^3*z^3 + 3099*x*y^3*z^2 + 6957*x*y^3*z + 5142*x*y^3 + 243*x*y^2*z^4 + 2226*x*y^2*z^3 + 7464*x*y^2*z^2 + 11022*x*y^2*z + 6099*x*y^2 + 69*x*y*z^5 + 783*x*y*z^4 + 3483*x*y*z^3 + 7686*x*y*z^2 + 8463*x*y*z + 3743*x*y + 8*x*z^6 + 108*x*z^5 + 597*x*z^4 + 1746*x*z^3 + 2862*x*z^2 + 2510*x*z + 931*x + 8*y^7 + 44*y^6*z + 104*y^6 + 102*y^5*z^2 + 480*y^5*z + 546*y^5 + 129*y^4*z^3 + 906*y^4*z^2 + 2061*y^4*z + 1542*y^4 + 96*y^3*z^4 + 894*y^3*z^3 + 3048*y^3*z^2 + 4571*y^3*z + 2561*y^3 + 42*y^2*z^5 + 486*y^2*z^4 + 2205*y^2*z^3 + 4959*y^2*z^2 + 5553*y^2*z + 2485*y^2 + 10*y*z^6 + 138*y*z^5 + 780*y*z^4 + 2331*y*z^3 + 3897*y*z^2 + 3476*y*z + 1314*y + z^7 + 16*z^6 + 108*z^5 + 401*z^4 + 887*z^3 + 1177*z^2 + 879*z + 1, -6*x^7 - 57*x^6*y - 39*x^6*z - 78*x^6 - 225*x^5*y^2 - 315*x^5*y*z - 642*x^5*y - 108*x^5*z^2 - 435*x^5*z - 441*x^5 - 477*x^4*y^3 - 1026*x^4*y^2*z - 2130*x^4*y^2 - 720*x^4*y*z^2 - 2955*x^4*y*z - 3045*x^4*y - 165*x^4*z^3 - 1005*x^4*z^2 - 2055*x^4*z - 1418*x^4 - 585*x^3*y^4 - 1719*x^3*y^3*z - 3630*x^3*y^3 - 1854*x^3*y^2*z^2 - 7749*x^3*y^2*z - 8112*x^3*y^2 - 870*x^3*y*z^3 - 5400*x^3*y*z^2 - 11229*x^3*y*z - 7873*x^3*y - 150*x^3*z^4 - 1230*x^3*z^3 - 3810*x^3*z^2 - 5305*x^3*z - 2806*x^3 - 414*x^2*y^5 - 1557*x^2*y^4*z - 3336*x^2*y^4 - 2295*x^2*y^3*z^2 - 9753*x^2*y^3*z - 10359*x^2*y^3 - 1656*x^2*y^2*z^3 - 10467*x^2*y^2*z^2 - 22122*x^2*y^2*z - 15744*x^2*y^2 - 585*x^2*y*z^4 - 4890*x^2*y*z^3 - 15417*x^2*y*z^2 - 21834*x^2*y*z - 11742*x^2*y - 81*x^2*z^5 - 840*x^2*z^4 - 3510*x^2*z^3 - 7407*x^2*z^2 - 7902*x^2*z - 3420*x^2 - 156*x*y^6 - 720*x*y^5*z - 1560*x*y^5 - 1359*x*y^4*z^2 - 5856*x*y^4*z - 6297*x*y^4 - 1341*x*y^3*z^3 - 8616*x*y^3*z^2 - 18483*x*y^3*z - 13327*x*y^3 - 729*x*y^2*z^4 - 6207*x*y^2*z^3 - 19908*x*y^2*z^2 - 28653*x*y^2*z - 15645*x*y^2 - 207*x*y*z^5 - 2190*x*y*z^4 - 9327*x*y*z^3 - 20049*x*y*z^2 - 21771*x*y*z - 9582*x*y - 24*x*z^6 - 303*x*z^5 - 1605*x*z^4 - 4571*x*z^3 - 7386*x*z^2 - 6441*x*z - 2381*x - 24*y^7 - 132*y^6*z - 288*y^6 - 306*y^5*z^2 - 1332*y^5*z - 1446*y^5 - 387*y^4*z^3 - 2520*y^4*z^2 - 5475*y^4*z - 3992*y^4 - 288*y^3*z^4 - 2493*y^3*z^3 - 8124*y^3*z^2 - 11864*y^3*z - 6557*y^3 - 126*y^2*z^5 - 1359*y^2*z^4 - 5898*y^2*z^3 - 12909*y^2*z^2 - 14250*y^2*z - 6354*y^2 - 30*y*z^6 - 387*y*z^5 - 2094*y*z^4 - 6088*y*z^3 - 10029*y*z^2 - 8901*y*z - 3350*y - 3*z^7 - 45*z^6 - 291*z^5 - 1051*z^4 - 2290*z^3 - 3021*z^2 - 2248*z - 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ 1, 2, 1 ], [ 1, 1, 1 ], [ 2, 1, 1 ] ], "source_shift": [ 1, 2, 1 ], "target_matrix": [ [ -1, -1, -1 ], [ 1, 1, 2 ], [ -3, -2, -2 ] ], "target_shift": [ 290, -285, 728 ], "collision_preimages": [ [ "3/4", "1", "-15/4" ], [ "9", "7/2", "-16" ], [ "6", "-3/2", "-5" ] ], "collision_image": [ "1161/4", "-1141/4", "2915/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0407180879992666 ], "surface_features": [ 3.4485418613420395, 4.457170099784619, 3.079362164393046, 0.0, 3.6140428249094865, 4832 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C146", "pair": 146, "family": "counterexample", "mapping": "F(x,y,z)=(-x^7 + 10*x^6*y - 8*x^6*z + 17*x^6 - 42*x^5*y^2 + 69*x^5*y*z - 150*x^5*y - 27*x^5*z^2 + 114*x^5*z - 117*x^5 + 96*x^4*y^3 - 243*x^4*y^2*z + 543*x^4*y^2 + 195*x^4*y*z^2 - 840*x^4*y*z + 879*x^4*y - 50*x^4*z^3 + 315*x^4*z^2 - 645*x^4*z + 428*x^4 - 129*x^3*y^4 + 447*x^3*y^3*z - 1032*x^3*y^3 - 552*x^3*y^2*z^2 + 2436*x^3*y^2*z - 2604*x^3*y^2 + 290*x^3*y*z^3 - 1860*x^3*y*z^2 + 3879*x^3*y*z - 2627*x^3*y - 55*x^3*z^4 + 460*x^3*z^3 - 1410*x^3*z^2 + 1872*x^3*z - 904*x^3 + 102*x^2*y^5 - 453*x^2*y^4*z + 1086*x^2*y^4 + 765*x^2*y^3*z^2 - 3474*x^2*y^3*z + 3801*x^2*y^3 - 618*x^2*y^2*z^3 + 4050*x^2*y^2*z^2 - 8616*x^2*y^2*z + 5961*x^2*y^2 + 240*x^2*y*z^4 - 2040*x^2*y*z^3 + 6363*x^2*y*z^2 - 8619*x^2*y*z + 4260*x^2*y - 36*x^2*z^5 + 375*x^2*z^4 - 1530*x^2*z^3 + 3048*x^2*z^2 - 2952*x^2*z + 1105*x^2 - 44*x*y^6 + 240*x*y^5*z - 600*x*y^5 - 519*x*y^4*z^2 + 2436*x*y^4*z - 2733*x*y^4 + 573*x*y^3*z^3 - 3852*x*y^3*z^2 + 8373*x*y^3*z - 5924*x*y^3 - 342*x*y^2*z^4 + 2964*x*y^2*z^3 - 9420*x*y^2*z^2 + 13026*x*y^2*z - 6591*x*y^2 + 105*x*y*z^5 - 1110*x*y*z^4 + 4605*x*y*z^3 - 9357*x*y*z^2 + 9279*x*y*z - 3571*x*y - 13*x*z^6 + 162*x*z^5 - 825*x*z^4 + 2192*x*z^3 - 3192*x*z^2 + 2402*x*z - 727*x + 8*y^7 - 52*y^6*z + 136*y^6 + 138*y^5*z^2 - 672*y^5*z + 774*y^5 - 195*y^4*z^3 + 1350*y^4*z^2 - 3003*y^4*z + 2174*y^4 + 159*y^3*z^4 - 1410*y^3*z^3 + 4572*y^3*z^2 - 6458*y^3*z + 3349*y^3 - 75*y^2*z^5 + 807*y^2*z^4 - 3408*y^2*z^3 + 7065*y^2*z^2 - 7170*y^2*z + 2836*y^2 + 19*y*z^6 - 240*y*z^5 + 1242*y*z^4 - 3365*y*z^3 + 5019*y*z^2 - 3889*y*z + 1210*y - 2*z^7 + 29*z^6 - 177*z^5 + 588*z^4 - 1144*z^3 + 1297*z^2 - 792*z + 1, x^7 - 10*x^6*y + 8*x^6*z - 17*x^6 + 42*x^5*y^2 - 69*x^5*y*z + 150*x^5*y + 27*x^5*z^2 - 114*x^5*z + 117*x^5 - 96*x^4*y^3 + 243*x^4*y^2*z - 543*x^4*y^2 - 195*x^4*y*z^2 + 840*x^4*y*z - 879*x^4*y + 50*x^4*z^3 - 315*x^4*z^2 + 645*x^4*z - 429*x^4 + 129*x^3*y^4 - 447*x^3*y^3*z + 1032*x^3*y^3 + 552*x^3*y^2*z^2 - 2436*x^3*y^2*z + 2604*x^3*y^2 - 290*x^3*y*z^3 + 1860*x^3*y*z^2 - 3879*x^3*y*z + 2631*x^3*y + 55*x^3*z^4 - 460*x^3*z^3 + 1410*x^3*z^2 - 1877*x^3*z + 915*x^3 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1242*y*z^4 + 3372*y*z^3 - 5067*y*z^2 + 3991*y*z - 1276*y + 2*z^7 - 29*z^6 + 177*z^5 - 590*z^4 + 1161*z^3 - 1348*z^2 + 854*z - 1, x^7 - 10*x^6*y + 8*x^6*z - 20*x^6 + 42*x^5*y^2 - 69*x^5*y*z + 174*x^5*y + 27*x^5*z^2 - 135*x^5*z + 156*x^5 - 96*x^4*y^3 + 243*x^4*y^2*z - 621*x^4*y^2 - 195*x^4*y*z^2 + 981*x^4*y*z - 1155*x^4*y + 50*x^4*z^3 - 375*x^4*z^2 + 864*x^4*z - 621*x^4 + 129*x^3*y^4 - 447*x^3*y^3*z + 1164*x^3*y^3 + 552*x^3*y^2*z^2 - 2805*x^3*y^2*z + 3369*x^3*y^2 - 290*x^3*y*z^3 + 2184*x^3*y*z^2 - 5121*x^3*y*z + 3759*x^3*y + 55*x^3*z^4 - 550*x^3*z^3 + 1896*x^3*z^2 - 2726*x^3*z + 1404*x^3 - 102*x^2*y^5 + 453*x^2*y^4*z - 1209*x^2*y^4 - 765*x^2*y^3*z^2 + 3945*x^2*y^3*z - 4839*x^2*y^3 + 618*x^2*y^2*z^3 - 4689*x^2*y^2*z^2 + 11199*x^2*y^2*z - 8406*x^2*y^2 - 240*x^2*y*z^4 + 2406*x^2*y*z^3 - 8433*x^2*y*z^2 + 12375*x^2*y*z - 6510*x^2*y + 36*x^2*z^5 - 450*x^2*z^4 + 2064*x^2*z^3 - 4452*x^2*z^2 + 4602*x^2*z - 1840*x^2 + 44*x*y^6 - 240*x*y^5*z + 660*x*y^5 + 519*x*y^4*z^2 - 2730*x*y^4*z + 3423*x*y^4 - 573*x*y^3*z^3 + 4398*x*y^3*z^2 - 10707*x*y^3*z + 8226*x*y^3 + 342*x*y^2*z^4 - 3447*x*y^2*z^3 + 12291*x*y^2*z^2 - 18426*x*y^2*z + 9915*x*y^2 - 105*x*y*z^5 + 1314*x*y*z^4 - 6123*x*y*z^3 + 13473*x*y*z^2 - 14223*x*y*z + 5809*x*y + 13*x*z^6 - 195*x*z^5 + 1116*x*z^4 - 3210*x*z^3 + 4992*x*z^2 - 4019*x*z + 1310*x - 8*y^7 + 52*y^6*z - 148*y^6 - 138*y^5*z^2 + 744*y^5*z - 954*y^5 + 195*y^4*z^3 - 1521*y^4*z^2 + 3777*y^4*z - 2970*y^4 - 159*y^3*z^4 + 1617*y^3*z^3 - 5868*y^3*z^2 + 8992*y^3*z - 4959*y^3 + 75*y^2*z^5 - 942*y^2*z^4 + 4461*y^2*z^3 - 10020*y^2*z^2 + 10812*y^2*z - 4519*y^2 - 19*y*z^6 + 285*y*z^5 - 1656*y*z^4 + 4857*y*z^3 - 7713*y*z^2 + 6346*y*z - 2106*y + 2*z^7 - 35*z^6 + 240*z^5 - 863*z^4 + 1794*z^3 - 2179*z^2 + 1437*z - 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -1, 1, -1 ], [ -1, 2, -1 ], [ -1, 1, -2 ] ], "source_shift": [ 2, 1, 2 ], "target_matrix": [ [ 1, 1, 2 ], [ -1, -1, -1 ], [ -1, -2, -1 ] ], "target_shift": [ -204, 228, 397 ], "collision_preimages": [ [ "11/4", "1", "1/4" ], [ "5", "-3/2", "-11/2" ], [ "14", "7/2", "-15/2" ] ], "collision_image": [ "-817/4", "913/4", "1589/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0380577779978921 ], "surface_features": [ 3.361207735059829, 4.265431066826713, 2.9588027033995026, 0.0, 3.48127147381885, 4793 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B043", "pair": 43, "family": "bijection", "mapping": "F(x,y,z)=(384*x^7 + 1344*x^6*y + 1344*x^6*z + 1728*x^6 + 2016*x^5*y^2 + 4032*x^5*y*z + 5376*x^5*y + 2016*x^5*z^2 + 5280*x^5*z + 3296*x^5 + 1680*x^4*y^3 + 5040*x^4*y^2*z + 6960*x^4*y^2 + 5040*x^4*y*z^2 + 13680*x^4*y*z + 8720*x^4*y + 1680*x^4*z^3 + 6720*x^4*z^2 + 8480*x^4*z + 3488*x^4 + 840*x^3*y^4 + 3360*x^3*y^3*z + 4800*x^3*y^3 + 5040*x^3*y^2*z^2 + 14160*x^3*y^2*z + 9200*x^3*y^2 + 3360*x^3*y*z^3 + 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12024*x*y*z^2 + 7788*x*y*z + 2216*x*y + 84*x*z^6 + 708*x*z^5 + 2310*x*z^4 + 3888*x*z^3 + 3750*x*z^2 + 2078*x*z + 521*x + 6*y^7 + 42*y^6*z + 66*y^6 + 126*y^5*z^2 + 390*y^5*z + 267*y^5 + 210*y^4*z^3 + 960*y^4*z^2 + 1305*y^4*z + 561*y^4 + 210*y^3*z^4 + 1260*y^3*z^3 + 2550*y^3*z^2 + 2184*y^3*z + 720*y^3 + 126*y^2*z^5 + 930*y^2*z^4 + 2490*y^2*z^3 + 3186*y^2*z^2 + 2088*y^2*z + 635*y^2 + 42*y*z^6 + 366*y*z^5 + 1215*y*z^4 + 2064*y*z^3 + 2016*y*z^2 + 1180*y*z + 329*y + 6*z^7 + 60*z^6 + 237*z^5 + 501*z^4 + 648*z^3 + 551*z^2 + 288*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ 2, 1, 1 ], [ 1, 2, 1 ], [ 2, 3, 2 ] ], "source_shift": [ 1, 1, 2 ], "target_matrix": [ [ 1, 1, 2 ], [ 1, 1, 1 ], [ 2, 3, 2 ] ], "target_shift": [ -23, -31, -66 ], "inverse_recipe": { "p": 5, "q": 2, "f": "x**5 + x**2", "g": "x**2 + x", "h": "y*z", "polynomials": { "f": [ [ [ 5, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "g": [ [ [ 2, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ] ] }, "h_multiplier": 3 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0868160079990048 ], "surface_features": [ 3.6174153800155584, 4.556302500767287, 3.280350693046006, 0.0, 3.741374331115158, 4923 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B166", "pair": 166, "family": "bijection", "mapping": "F(x,y,z)=(-2187*x^7 - 15309*x^6*y + 10206*x^6*z + 11178*x^6 - 45927*x^5*y^2 + 61236*x^5*y*z + 67068*x^5*y - 20412*x^5*z^2 - 44793*x^5*z - 24381*x^5 - 76545*x^4*y^3 + 153090*x^4*y^2*z + 167670*x^4*y^2 - 102060*x^4*y*z^2 - 223965*x^4*y*z - 121905*x^4*y + 22680*x^4*z^3 + 74790*x^4*z^2 + 81621*x^4*z + 29700*x^4 - 76545*x^3*y^4 + 204120*x^3*y^3*z + 223560*x^3*y^3 - 204120*x^3*y^2*z^2 - 447930*x^3*y^2*z - 243810*x^3*y^2 + 90720*x^3*y*z^3 + 299160*x^3*y*z^2 + 326484*x^3*y*z + 118800*x^3*y - 15120*x^3*z^4 - 66600*x^3*z^3 - 109296*x^3*z^2 - 79794*x^3*z - 22024*x^3 - 45927*x^2*y^5 + 153090*x^2*y^4*z + 167670*x^2*y^4 - 204120*x^2*y^3*z^2 - 447930*x^2*y^3*z - 243810*x^2*y^3 + 136080*x^2*y^2*z^3 + 448740*x^2*y^2*z^2 + 489726*x^2*y^2*z + 178200*x^2*y^2 - 45360*x^2*y*z^4 - 199800*x^2*y*z^3 - 327888*x^2*y*z^2 - 239382*x^2*y*z - 66063*x^2*y + 6048*x^2*z^5 + 33360*x^2*z^4 + 73176*x^2*z^3 + 80391*x^2*z^2 + 44553*x^2*z + 10029*x^2 - 15309*x*y^6 + 61236*x*y^5*z + 67068*x*y^5 - 102060*x*y^4*z^2 - 223965*x*y^4*z - 121905*x*y^4 + 90720*x*y^3*z^3 + 299160*x*y^3*z^2 + 326484*x*y^3*z + 118800*x*y^3 - 45360*x*y^2*z^4 - 199800*x*y^2*z^3 - 327888*x*y^2*z^2 - 239382*x*y^2*z - 66054*x*y^2 + 12096*x*y*z^5 + 66720*x*y*z^4 + 146352*x*y*z^3 + 160782*x*y*z^2 + 89094*x*y*z + 20045*x*y - 1344*x*z^6 - 8912*x*z^5 - 24496*x*z^4 - 35996*x*z^3 - 30043*x*z^2 - 13591*x*z - 2618*x - 2187*y^7 + 10206*y^6*z + 11178*y^6 - 20412*y^5*z^2 - 44793*y^5*z - 24381*y^5 + 22680*y^4*z^3 + 74790*y^4*z^2 + 81621*y^4*z + 29700*y^4 - 15120*y^3*z^4 - 66600*y^3*z^3 - 109296*y^3*z^2 - 79794*y^3*z - 22015*y^3 + 6048*y^2*z^5 + 33360*y^2*z^4 + 73176*y^2*z^3 + 80391*y^2*z^2 + 44541*y^2*z + 10016*y^2 - 1344*y*z^6 - 8912*y*z^5 - 24496*y*z^4 - 35996*y*z^3 - 30039*y*z^2 - 13582*y*z - 2614*y + 128*z^7 + 992*z^6 + 3280*z^5 + 6044*z^4 + 6753*z^3 + 4605*z^2 + 1785*z + 1, 4374*x^7 + 30618*x^6*y - 20412*x^6*z - 22356*x^6 + 91854*x^5*y^2 - 122472*x^5*y*z - 134136*x^5*y + 40824*x^5*z^2 + 89586*x^5*z + 49248*x^5 + 153090*x^4*y^3 - 306180*x^4*y^2*z - 335340*x^4*y^2 + 204120*x^4*y*z^2 + 447930*x^4*y*z + 246240*x^4*y - 45360*x^4*z^3 - 149580*x^4*z^2 - 164862*x^4*z - 61020*x^4 + 153090*x^3*y^4 - 408240*x^3*y^3*z - 447120*x^3*y^3 + 408240*x^3*y^2*z^2 + 895860*x^3*y^2*z + 492480*x^3*y^2 - 181440*x^3*y*z^3 - 598320*x^3*y*z^2 - 659448*x^3*y*z - 244080*x^3*y + 30240*x^3*z^4 + 133200*x^3*z^3 + 220752*x^3*z^2 + 163908*x^3*z + 46208*x^3 + 91854*x^2*y^5 - 306180*x^2*y^4*z - 335340*x^2*y^4 + 408240*x^2*y^3*z^2 + 895860*x^2*y^3*z + 492480*x^2*y^3 - 272160*x^2*y^2*z^3 - 897480*x^2*y^2*z^2 - 989172*x^2*y^2*z - 366120*x^2*y^2 + 90720*x^2*y*z^4 + 399600*x^2*y*z^3 + 662256*x^2*y*z^2 + 491724*x^2*y*z + 138606*x^2*y - 12096*x^2*z^5 - 66720*x^2*z^4 - 147792*x^2*z^3 - 165102*x^2*z^2 - 93426*x^2*z - 21507*x^2 + 30618*x*y^6 - 122472*x*y^5*z - 134136*x*y^5 + 204120*x*y^4*z^2 + 447930*x*y^4*z + 246240*x*y^4 - 181440*x*y^3*z^3 - 598320*x*y^3*z^2 - 659448*x*y^3*z - 244080*x*y^3 + 90720*x*y^2*z^4 + 399600*x*y^2*z^3 + 662256*x*y^2*z^2 + 491724*x*y^2*z + 138588*x*y^2 - 24192*x*y*z^5 - 133440*x*y*z^4 - 295584*x*y*z^3 - 330204*x*y*z^2 - 186828*x*y*z - 42988*x*y + 2688*x*z^6 + 17824*x*z^5 + 49472*x*z^4 + 73912*x*z^3 + 62966*x*z^2 + 29114*x*z + 5731*x + 4374*y^7 - 20412*y^6*z - 22356*y^6 + 40824*y^5*z^2 + 89586*y^5*z + 49248*y^5 - 45360*y^4*z^3 - 149580*y^4*z^2 - 164862*y^4*z - 61020*y^4 + 30240*y^3*z^4 + 133200*y^3*z^3 + 220752*y^3*z^2 + 163908*y^3*z + 46190*y^3 - 12096*y^2*z^5 - 66720*y^2*z^4 - 147792*y^2*z^3 - 165102*y^2*z^2 - 93402*y^2*z - 21481*y^2 + 2688*y*z^6 + 17824*y*z^5 + 49472*y*z^4 + 73912*y*z^3 + 62958*y*z^2 + 29096*y*z + 5721*y - 256*z^7 - 1984*z^6 - 6624*z^5 - 12408*z^4 - 14146*z^3 - 9854*z^2 - 3899*z - 1, 6561*x^7 + 45927*x^6*y - 30618*x^6*z - 33534*x^6 + 137781*x^5*y^2 - 183708*x^5*y*z - 201204*x^5*y + 61236*x^5*z^2 + 134379*x^5*z + 73629*x^5 + 229635*x^4*y^3 - 459270*x^4*y^2*z - 503010*x^4*y^2 + 306180*x^4*y*z^2 + 671895*x^4*y*z + 368145*x^4*y - 68040*x^4*z^3 - 224370*x^4*z^2 - 246483*x^4*z - 90720*x^4 + 229635*x^3*y^4 - 612360*x^3*y^3*z - 670680*x^3*y^3 + 612360*x^3*y^2*z^2 + 1343790*x^3*y^2*z + 736290*x^3*y^2 - 272160*x^3*y*z^3 - 897480*x^3*y*z^2 - 985932*x^3*y*z - 362880*x^3*y + 45360*x^3*z^4 + 199800*x^3*z^3 + 330048*x^3*z^2 + 243702*x^3*z + 68232*x^3 + 137781*x^2*y^5 - 459270*x^2*y^4*z - 503010*x^2*y^4 + 612360*x^2*y^3*z^2 + 1343790*x^2*y^3*z + 736290*x^2*y^3 - 408240*x^2*y^2*z^3 - 1346220*x^2*y^2*z^2 - 1478898*x^2*y^2*z - 544320*x^2*y^2 + 136080*x^2*y*z^4 + 599400*x^2*y*z^3 + 990144*x^2*y*z^2 + 731106*x^2*y*z + 204669*x^2*y - 18144*x^2*z^5 - 100080*x^2*z^4 - 220968*x^2*z^3 - 245493*x^2*z^2 - 137979*x^2*z - 31545*x^2 + 45927*x*y^6 - 183708*x*y^5*z - 201204*x*y^5 + 306180*x*y^4*z^2 + 671895*x*y^4*z + 368145*x*y^4 - 272160*x*y^3*z^3 - 897480*x*y^3*z^2 - 985932*x*y^3*z - 362880*x*y^3 + 136080*x*y^2*z^4 + 599400*x*y^2*z^3 + 990144*x*y^2*z^2 + 731106*x*y^2*z + 204642*x*y^2 - 36288*x*y*z^5 - 200160*x*y*z^4 - 441936*x*y*z^3 - 490986*x*y*z^2 - 275922*x*y*z - 63051*x*y + 4032*x*z^6 + 26736*x*z^5 + 73968*x*z^4 + 109908*x*z^3 + 93009*x*z^2 + 42717*x*z + 8362*x + 6561*y^7 - 30618*y^6*z - 33534*y^6 + 61236*y^5*z^2 + 134379*y^5*z + 73629*y^5 - 68040*y^4*z^3 - 224370*y^4*z^2 - 246483*y^4*z - 90720*y^4 + 45360*y^3*z^4 + 199800*y^3*z^3 + 330048*y^3*z^2 + 243702*y^3*z + 68205*y^3 - 18144*y^2*z^5 - 100080*y^2*z^4 - 220968*y^2*z^3 - 245493*y^2*z^2 - 137943*y^2*z - 31506*y^2 + 4032*y*z^6 + 26736*y*z^5 + 73968*y*z^4 + 109908*y*z^3 + 92997*y*z^2 + 42690*y*z + 8348*y - 384*z^7 - 2976*z^6 - 9904*z^5 - 18452*z^4 - 20899*z^3 - 14463*z^2 - 5693*z - 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -3, -3, 2 ], [ -1, -1, 1 ], [ -2, -1, 1 ] ], "source_shift": [ 2, 1, 1 ], "target_matrix": [ [ 1, 1, 1 ], [ -2, -1, -1 ], [ -3, -3, -2 ] ], "target_shift": [ -304, 678, 988 ], "inverse_recipe": { "p": 2, "q": 5, "f": "x**2", "g": "x**5 + x**2", "h": "y**3 + y**2 + y*z + z", "polynomials": { "f": [ [ [ 2, 0, 0 ], 1 ] ], "g": [ [ [ 5, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 0, 1 ], 1 ], [ [ 0, 3, 0 ], 1 ], [ [ 0, 2, 0 ], 1 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.078117408000253 ], "surface_features": [ 5.283037254278428, 6.169938221620461, 4.960408934924756, 0.0, 5.395149397519738, 5531 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C138", "pair": 138, "family": "counterexample", "mapping": "F(x,y,z)=(-16*x^7 - 136*x^6*y + 88*x^6*z + 176*x^6 - 468*x^5*y^2 + 624*x^5*y*z + 1308*x^5*y - 204*x^5*z^2 - 828*x^5*z - 804*x^5 - 838*x^4*y^3 + 1734*x^4*y^2*z + 3852*x^4*y^2 - 1170*x^4*y*z^2 - 4992*x^4*y*z - 5028*x^4*y + 258*x^4*z^3 + 1596*x^4*z^2 + 3144*x^4*z + 2008*x^4 - 838*x^3*y^4 + 2398*x^3*y^3*z + 5711*x^3*y^3 - 2514*x^3*y^2*z^2 - 11397*x^3*y^2*z - 11961*x^3*y^2 + 1146*x^3*y*z^3 + 7473*x^3*y*z^2 + 15294*x^3*y*z + 10130*x^3*y - 192*x^3*z^4 - 1611*x^3*z^3 - 4833*x^3*z^2 - 6278*x^3*z - 2992*x^3 - 468*x^2*y^5 + 1734*x^2*y^4*z + 4461*x^2*y^4 - 2514*x^2*y^3*z^2 - 12222*x^2*y^3*z - 13416*x^2*y^3 + 1782*x^2*y^2*z^3 + 12366*x^2*y^2*z^2 + 26397*x^2*y^2*z + 18177*x^2*y^2 - 618*x^2*y*z^4 - 5478*x^2*y*z^3 - 17094*x^2*y*z^2 - 23052*x^2*y*z - 11436*x^2*y + 84*x^2*z^5 + 897*x^2*z^4 + 3645*x^2*z^3 + 7227*x^2*z^2 + 7032*x^2*z + 2690*x^2 - 136*x*y^6 + 624*x*y^5*z + 1728*x*y^5 - 1170*x*y^4*z^2 - 6114*x*y^4*z - 7032*x*y^4 + 1146*x*y^3*z^3 + 8511*x*y^3*z^2 + 18996*x*y^3*z + 13622*x*y^3 - 618*x*y^2*z^4 - 5829*x*y^2*z^3 - 18975*x*y^2*z^2 - 26610*x*y^2*z - 13746*x*y^2 + 174*x*y*z^5 + 1965*x*y*z^4 + 8310*x*y*z^3 + 17112*x*y*z^2 + 17334*x*y*z + 6941*x*y - 20*x*z^6 - 261*x*z^5 - 1347*x*z^4 - 3624*x*z^3 - 5400*x*z^2 - 4237*x*z - 1373*x - 16*y^7 + 88*y^6*z + 260*y^6 - 204*y^5*z^2 - 1140*y^5*z - 1368*y^5 + 258*y^4*z^3 + 2049*y^4*z^2 + 4776*y^4*z + 3556*y^4 - 192*y^3*z^4 - 1932*y^3*z^3 - 6564*y^3*z^2 - 9562*y^3*z - 5144*y^3 + 84*y^2*z^5 + 1008*y^2*z^4 + 4443*y^2*z^3 + 9501*y^2*z^2 + 10014*y^2*z 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312*x*y^5*z + 1008*x*y^5 - 585*x*y^4*z^2 - 3642*x*y^4*z - 5145*x*y^4 + 573*x*y^3*z^3 + 5187*x*y^3*z^2 + 14313*x*y^3*z + 12032*x*y^3 - 309*x*y^2*z^4 - 3639*x*y^2*z^3 - 14712*x*y^2*z^2 - 24144*x*y^2*z - 13971*x*y^2 + 87*x*y*z^5 + 1257*x*y*z^4 + 6621*x*y*z^3 + 15921*x*y*z^2 + 18027*x*y*z + 7852*x*y - 10*x*z^6 - 171*x*z^5 - 1101*x*z^4 - 3451*x*z^3 - 5736*x*z^2 - 4892*x*z - 1697*x - 8*y^7 + 44*y^6*z + 148*y^6 - 102*y^5*z^2 - 660*y^5*z - 954*y^5 + 129*y^4*z^3 + 1209*y^4*z^2 + 3423*y^4*z + 2970*y^4 - 96*y^3*z^4 - 1164*y^3*z^3 - 4839*y^3*z^2 - 8226*y^3*z - 4959*y^3 + 42*y^2*z^5 + 621*y^2*z^4 + 3369*y^2*z^3 + 8406*y^2*z^2 + 9915*y^2*z + 4519*y^2 - 10*y*z^6 - 174*y*z^5 - 1155*y*z^4 - 3759*y*z^3 - 6510*y*z^2 - 5809*y*z - 2106*y + z^7 + 20*z^6 + 156*z^5 + 621*z^4 + 1404*z^3 + 1840*z^2 + 1310*z + 1, -8*x^7 - 68*x^6*y + 44*x^6*z + 100*x^6 - 234*x^5*y^2 + 312*x^5*y*z + 732*x^5*y - 102*x^5*z^2 - 468*x^5*z - 504*x^5 - 419*x^4*y^3 + 867*x^4*y^2*z + 2121*x^4*y^2 - 585*x^4*y*z^2 - 2778*x^4*y*z - 3096*x^4*y + 129*x^4*z^3 + 897*x^4*z^2 + 1956*x^4*z + 1342*x^4 - 419*x^3*y^4 + 1199*x^3*y^3*z + 3094*x^3*y^3 - 1257*x^3*y^2*z^2 - 6240*x^3*y^2*z - 7218*x^3*y^2 + 573*x^3*y*z^3 + 4134*x^3*y*z^2 + 9336*x^3*y*z + 6651*x^3*y - 96*x^3*z^4 - 900*x^3*z^3 - 2982*x^3*z^2 - 4161*x^3*z - 2082*x^3 - 234*x^2*y^5 + 867*x^2*y^4*z + 2382*x^2*y^4 - 1257*x^2*y^3*z^2 - 6588*x^2*y^3*z - 7920*x^2*y^3 + 891*x^2*y^2*z^3 + 6732*x^2*y^2*z^2 + 15777*x^2*y^2*z + 11685*x^2*y^2 - 309*x^2*y*z^4 - 3012*x^2*y*z^3 - 10338*x^2*y*z^2 - 14985*x^2*y*z - 7830*x^2*y + 42*x^2*z^5 + 498*x^2*z^4 + 2229*x^2*z^3 + 4746*x^2*z^2 + 4854*x^2*z + 1918*x^2 - 68*x*y^6 + 312*x*y^5*z + 912*x*y^5 - 585*x*y^4*z^2 - 3252*x*y^4*z - 4059*x*y^4 + 573*x*y^3*z^3 + 4566*x*y^3*z^2 + 11103*x*y^3*z + 8549*x*y^3 - 309*x*y^2*z^4 - 3156*x*y^2*z^3 - 11229*x*y^2*z^2 - 16911*x*y^2*z - 9216*x*y^2 + 87*x*y*z^5 + 1074*x*y*z^4 + 4977*x*y*z^3 + 11004*x*y*z^2 + 11745*x*y*z + 4879*x*y - 10*x*z^6 - 144*x*z^5 - 816*x*z^4 - 2356*x*z^3 - 3693*x*z^2 - 2999*x*z - 994*x - 8*y^7 + 44*y^6*z + 136*y^6 - 102*y^5*z^2 - 600*y^5*z - 774*y^5 + 129*y^4*z^3 + 1086*y^4*z^2 + 2733*y^4*z + 2175*y^4 - 96*y^3*z^4 - 1032*y^3*z^3 - 3801*y^3*z^2 - 5928*y^3*z - 3363*y^3 + 42*y^2*z^5 + 543*y^2*z^4 + 2604*y^2*z^3 + 5967*y^2*z^2 + 6630*y^2*z + 2887*y^2 - 10*y*z^6 - 150*y*z^5 - 879*y*z^4 - 2631*y*z^3 - 4296*y*z^2 - 3661*y*z - 1276*y + z^7 + 17*z^6 + 117*z^5 + 429*z^4 + 915*z^3 + 1144*z^2 + 781*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -2, -1, 1 ], [ -1, -2, 1 ], [ -1, -1, 1 ] ], "source_shift": [ 2, 1, 2 ], "target_matrix": [ [ 2, 1, 1 ], [ 1, 2, 1 ], [ 1, 1, 1 ] ], "target_shift": [ -312, -397, -228 ], "collision_preimages": [ [ "-1/4", "-5/4", "-15/4" ], [ "11/2", "7", "17" ], [ "15/2", "4", "16" ] ], "collision_image": [ "-625/2", "-1589/4", "-913/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.022490542000014 ], "surface_features": [ 3.5887816644246855, 4.425044874551388, 3.2336305801644634, 0.0, 3.703030236595741, 4894 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C128", "pair": 128, "family": "counterexample", "mapping": "F(x,y,z)=(-8*x^7 - 44*x^6*y + 68*x^6*z + 128*x^6 - 102*x^5*y^2 + 312*x^5*y*z + 564*x^5*y - 234*x^5*z^2 - 852*x^5*z - 702*x^5 - 129*x^4*y^3 + 585*x^4*y^2*z + 1020*x^4*y^2 - 867*x^4*y*z^2 - 3036*x^4*y*z - 2469*x^4*y + 419*x^4*z^3 + 2208*x^4*z^2 + 3627*x^4*z + 1904*x^4 - 96*x^3*y^4 + 573*x^3*y^3*z + 969*x^3*y^3 - 1257*x^3*y^2*z^2 - 4263*x^3*y^2*z - 3423*x^3*y^2 + 1199*x^3*y*z^3 + 6111*x^3*y*z^2 + 9891*x^3*y*z + 5153*x^3*y - 419*x^3*z^4 - 2849*x^3*z^3 - 6966*x^3*z^2 - 7309*x^3*z - 2815*x^3 - 42*x^2*y^5 + 309*x^2*y^4*z + 510*x^2*y^4 - 891*x^2*y^3*z^2 - 2949*x^2*y^3*z - 2340*x^2*y^3 + 1257*x^2*y^2*z^3 + 6255*x^2*y^2*z^2 + 9987*x^2*y^2*z + 5157*x^2*y^2 - 867*x^2*y*z^4 - 5763*x^2*y*z^3 - 13869*x^2*y*z^2 - 14379*x^2*y*z - 5487*x^2*y + 234*x^2*z^5 + 1947*x^2*z^4 + 6264*x^2*z^3 + 9750*x^2*z^2 + 7437*x^2*z + 2257*x^2 - 10*x*y^6 + 87*x*y^5*z + 141*x*y^5 - 309*x*y^4*z^2 - 1005*x*y^4*z - 789*x*y^4 + 573*x*y^3*z^3 + 2805*x*y^3*z^2 + 4425*x*y^3*z + 2264*x*y^3 - 585*x*y^2*z^4 - 3831*x*y^2*z^3 - 9096*x*y^2*z^2 - 9324*x*y^2*z - 3522*x*y^2 + 312*x*y*z^5 + 2562*x*y*z^4 + 8124*x*y*z^3 + 12483*x*y*z^2 + 9399*x*y*z + 2812*x*y - 68*x*z^6 - 672*x*z^5 - 2664*x*z^4 - 5443*x*z^3 - 6102*x*z^2 - 3604*x*z - 892*x - y^7 + 10*y^6*z + 16*y^6 - 42*y^5*z^2 - 135*y^5*z - 105*y^5 + 96*y^4*z^3 + 465*y^4*z^2 + 726*y^4*z + 368*y^4 - 129*y^3*z^4 - 837*y^3*z^3 - 1965*y^3*z^2 - 1992*y^3*z - 744*y^3 + 102*y^2*z^5 + 831*y^2*z^4 + 2604*y^2*z^3 + 3951*y^2*z^2 + 2934*y^2*z + 865*y^2 - 44*y*z^6 - 432*y*z^5 - 1692*y*z^4 - 3409*y*z^3 - 3762*y*z^2 - 2186*y*z - 535*y + 8*z^7 + 92*z^6 + 432*z^5 + 1082*z^4 + 1574*z^3 + 1348*z^2 + 648*z + 1, -16*x^7 - 88*x^6*y + 136*x^6*z + 244*x^6 - 204*x^5*y^2 + 624*x^5*y*z + 1068*x^5*y - 468*x^5*z^2 - 1608*x^5*z - 1236*x^5 - 258*x^4*y^3 + 1170*x^4*y^2*z + 1917*x^4*y^2 - 1734*x^4*y*z^2 - 5682*x^4*y*z - 4296*x^4*y + 838*x^4*z^3 + 4113*x^4*z^2 + 6252*x^4*z + 3100*x^4 - 192*x^3*y^4 + 1146*x^3*y^3*z + 1806*x^3*y^3 - 2514*x^3*y^2*z^2 - 7905*x^3*y^2*z - 5883*x^3*y^2 + 2398*x^3*y*z^3 + 11268*x^3*y*z^2 + 16830*x^3*y*z + 8278*x^3*y - 838*x^3*z^4 - 5221*x^3*z^3 - 11727*x^3*z^2 - 11594*x^3*z - 4304*x^3 - 84*x^2*y^5 + 618*x^2*y^4*z + 942*x^2*y^4 - 1782*x^2*y^3*z^2 - 5415*x^2*y^3*z - 3972*x^2*y^3 + 2514*x^2*y^2*z^3 + 11412*x^2*y^2*z^2 + 16779*x^2*y^2*z + 8181*x^2*y^2 - 1734*x^2*y*z^4 - 10443*x^2*y*z^3 - 23058*x^2*y*z^2 - 22542*x^2*y*z - 8304*x^2*y + 468*x^2*z^5 + 3504*x^2*z^4 + 10311*x^2*z^3 + 15117*x^2*z^2 + 11136*x^2*z + 3329*x^2 - 20*x*y^6 + 174*x*y^5*z + 258*x*y^5 - 618*x*y^4*z^2 - 1827*x*y^4*z - 1323*x*y^4 + 1146*x*y^3*z^3 + 5064*x*y^3*z^2 + 7344*x*y^3*z + 3550*x*y^3 - 1170*x*y^2*z^4 - 6867*x*y^2*z^3 - 14943*x*y^2*z^2 - 14460*x*y^2*z - 5286*x*y^2 + 624*x*y*z^5 + 4560*x*y*z^4 + 13218*x*y*z^3 + 19164*x*y*z^2 + 13986*x*y*z + 4142*x*y - 136*x*z^6 - 1188*x*z^5 - 4296*x*z^4 - 8282*x*z^3 - 9024*x*z^2 - 5303*x*z - 1323*x - 2*y^7 + 20*y^6*z + 29*y^6 - 84*y^5*z^2 - 243*y^5*z - 174*y^5 + 192*y^4*z^3 + 831*y^4*z^2 + 1191*y^4*z + 571*y^4 - 258*y^3*z^4 - 1485*y^3*z^3 - 3192*y^3*z^2 - 3060*y^3*z - 1110*y^3 + 204*y^2*z^5 + 1464*y^2*z^4 + 4191*y^2*z^3 + 6015*y^2*z^2 + 4350*y^2*z + 1277*y^2 - 88*y*z^6 - 756*y*z^5 - 2700*y*z^4 - 5150*y*z^3 - 5556*y*z^2 - 3235*y*z - 802*y + 16*z^7 + 160*z^6 + 684*z^5 + 1624*z^4 + 2320*z^3 + 2006*z^2 + 983*z + 1, -8*x^7 - 44*x^6*y + 68*x^6*z + 128*x^6 - 102*x^5*y^2 + 312*x^5*y*z + 564*x^5*y - 234*x^5*z^2 - 852*x^5*z - 702*x^5 - 129*x^4*y^3 + 585*x^4*y^2*z + 1020*x^4*y^2 - 867*x^4*y*z^2 - 3036*x^4*y*z - 2469*x^4*y + 419*x^4*z^3 + 2208*x^4*z^2 + 3627*x^4*z + 1905*x^4 - 96*x^3*y^4 + 573*x^3*y^3*z + 969*x^3*y^3 - 1257*x^3*y^2*z^2 - 4263*x^3*y^2*z - 3423*x^3*y^2 + 1199*x^3*y*z^3 + 6111*x^3*y*z^2 + 9891*x^3*y*z + 5157*x^3*y - 419*x^3*z^4 - 2849*x^3*z^3 - 6966*x^3*z^2 - 7316*x^3*z - 2828*x^3 - 42*x^2*y^5 + 309*x^2*y^4*z + 510*x^2*y^4 - 891*x^2*y^3*z^2 - 2949*x^2*y^3*z - 2340*x^2*y^3 + 1257*x^2*y^2*z^3 + 6255*x^2*y^2*z^2 + 9987*x^2*y^2*z + 5163*x^2*y^2 - 867*x^2*y*z^4 - 5763*x^2*y*z^3 - 13869*x^2*y*z^2 - 14400*x^2*y*z - 5523*x^2*y + 234*x^2*z^5 + 1947*x^2*z^4 + 6264*x^2*z^3 + 9768*x^2*z^2 + 7500*x^2*z + 2302*x^2 - 10*x*y^6 + 87*x*y^5*z + 141*x*y^5 - 309*x*y^4*z^2 - 1005*x*y^4*z - 789*x*y^4 + 573*x*y^3*z^3 + 2805*x*y^3*z^2 + 4425*x*y^3*z + 2268*x*y^3 - 585*x*y^2*z^4 - 3831*x*y^2*z^3 - 9096*x*y^2*z^2 - 9345*x*y^2*z - 3555*x*y^2 + 312*x*y*z^5 + 2562*x*y*z^4 + 8124*x*y*z^3 + 12519*x*y*z^2 + 9513*x*y*z + 2890*x*y - 68*x*z^6 - 672*x*z^5 - 2664*x*z^4 - 5463*x*z^3 - 6198*x*z^2 - 3736*x*z - 946*x - y^7 + 10*y^6*z + 16*y^6 - 42*y^5*z^2 - 135*y^5*z - 105*y^5 + 96*y^4*z^3 + 465*y^4*z^2 + 726*y^4*z + 369*y^4 - 129*y^3*z^4 - 837*y^3*z^3 - 1965*y^3*z^2 - 1999*y^3*z - 754*y^3 + 102*y^2*z^5 + 831*y^2*z^4 + 2604*y^2*z^3 + 3969*y^2*z^2 + 2985*y^2*z + 898*y^2 - 44*y*z^6 - 432*y*z^5 - 1692*y*z^4 - 3429*y*z^3 - 3846*y*z^2 - 2294*y*z - 577*y + 8*z^7 + 92*z^6 + 432*z^5 + 1090*z^4 + 1618*z^3 + 1432*z^2 + 712*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -1, -1, 2 ], [ -2, -1, 1 ], [ -1, -1, 1 ] ], "source_shift": [ 2, 1, 1 ], "target_matrix": [ [ 1, 1, 2 ], [ 2, 1, 1 ], [ 1, 1, 1 ] ], "target_shift": [ -139, -212, -155 ], "collision_preimages": [ [ "-1/4", "3/4", "-3/4" ], [ "8", "-20", "-13/2" ], [ "5", "-19", "-17/2" ] ], "collision_image": [ "-557/4", "-425/2", "-621/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.073133436999342 ], "surface_features": [ 3.4836954411328183, 4.362821634847992, 3.113943352306837, 0.0, 3.595287700346951, 4855 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C027", "pair": 27, "family": "counterexample", "mapping": "F(x,y,z)=(-16*x^7 + 144*x^6*y + 88*x^6*z - 244*x^6 - 528*x^5*y^2 - 660*x^5*y*z + 1776*x^5*y - 204*x^5*z^2 + 1116*x^5*z - 1296*x^5 + 1012*x^4*y^3 + 1950*x^4*y^2*z - 5073*x^4*y^2 + 1236*x^4*y*z^2 - 6546*x^4*y*z + 7506*x^4*y + 258*x^4*z^3 - 2085*x^4*z^2 + 4812*x^4*z - 3527*x^4 - 1083*x^3*y^4 - 2875*x^3*y^3*z + 7207*x^3*y^3 - 2817*x^3*y^2*z^2 + 14388*x^3*y^2*z - 16299*x^3*y^2 - 1209*x^3*y*z^3 + 9435*x^3*y*z^2 - 21483*x^3*y*z + 15724*x^3*y - 192*x^3*z^4 + 2034*x^3*z^3 - 6990*x^3*z^2 + 10231*x^3*z - 5557*x^3 + 642*x^2*y^5 + 2211*x^2*y^4*z - 5334*x^2*y^4 + 2991*x^2*y^3*z^2 - 14697*x^2*y^3*z + 16443*x^2*y^3 + 1989*x^2*y^2*z^3 - 14940*x^2*y^2*z^2 + 33561*x^2*y^2*z - 24456*x^2*y^2 + 651*x^2*y*z^4 - 6645*x^2*y*z^3 + 22506*x^2*y*z^2 - 32808*x^2*y*z + 17805*x^2*y + 84*x^2*z^5 - 1092*x^2*z^4 + 4962*x^2*z^3 - 10857*x^2*z^2 + 11775*x^2*z - 5122*x^2 - 196*x*y^6 - 840*x*y^5*z + 1956*x*y^5 - 1473*x*y^4*z^2 + 6972*x*y^4*z - 7668*x*y^4 - 1353*x*y^3*z^3 + 9777*x*y^3*z^2 - 21630*x*y^3*z + 15520*x*y^3 - 687*x*y^2*z^4 + 6744*x*y^2*z^3 - 22497*x*y^2*z^2 + 32454*x*y^2*z - 17388*x*y^2 - 183*x*y*z^5 + 2289*x*y*z^4 - 10239*x*y*z^3 + 22224*x*y*z^2 - 23919*x*y*z + 10333*x*y - 20*x*z^6 + 306*x*z^5 - 1722*x*z^4 + 4995*x*z^3 - 8082*x*z^2 + 6998*x*z - 2553*x + 24*y^7 + 124*y^6*z - 280*y^6 + 270*y^5*z^2 - 1236*y^5*z + 1332*y^5 + 321*y^4*z^3 - 2238*y^4*z^2 + 4860*y^4*z - 3413*y^4 + 225*y^3*z^4 - 2127*y^3*z^3 + 6975*y^3*z^2 - 9881*y^3*z + 5177*y^3 + 93*y^2*z^5 - 1119*y^2*z^4 + 4923*y^2*z^3 - 10533*y^2*z^2 + 11142*y^2*z - 4705*y^2 + 21*y*z^6 - 309*y*z^5 + 1710*y*z^4 - 4903*y*z^3 + 7833*y*z^2 - 6682*y*z + 2386*y + 2*z^7 - 35*z^6 + 234*z^5 - 842*z^4 + 1802*z^3 - 2320*z^2 + 1673*z - 1, -32*x^7 + 288*x^6*y + 176*x^6*z - 392*x^6 - 1056*x^5*y^2 - 1320*x^5*y*z + 2880*x^5*y - 408*x^5*z^2 + 1800*x^5*z - 1896*x^5 + 2024*x^4*y^3 + 3900*x^4*y^2*z - 8322*x^4*y^2 + 2472*x^4*y*z^2 - 10668*x^4*y*z + 11124*x^4*y + 516*x^4*z^3 - 3378*x^4*z^2 + 7080*x^4*z 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48*y^7 + 248*y^6*z - 488*y^6 + 540*y^5*z^2 - 2136*y^5*z + 2112*y^5 + 642*y^4*z^3 - 3834*y^4*z^2 + 7632*y^4*z - 5071*y^4 + 450*y^3*z^4 - 3612*y^3*z^3 + 10848*y^3*z^2 - 14531*y^3*z + 7337*y^3 + 186*y^2*z^5 - 1884*y^2*z^4 + 7584*y^2*z^3 - 15333*y^2*z^2 + 15615*y^2*z - 6431*y^2 + 42*y*z^6 - 516*y*z^5 + 2610*y*z^4 - 7067*y*z^3 + 10857*y*z^2 - 9017*y*z + 3177*y + 4*z^7 - 58*z^6 + 354*z^5 - 1202*z^4 + 2471*z^3 - 3092*z^2 + 2195*z - 1, 16*x^7 - 144*x^6*y - 88*x^6*z + 196*x^6 + 528*x^5*y^2 + 660*x^5*y*z - 1440*x^5*y + 204*x^5*z^2 - 900*x^5*z + 948*x^5 - 1012*x^4*y^3 - 1950*x^4*y^2*z + 4161*x^4*y^2 - 1236*x^4*y*z^2 + 5334*x^4*y*z - 5562*x^4*y - 258*x^4*z^3 + 1689*x^4*z^2 - 3540*x^4*z + 2447*x^4 + 1083*x^3*y^4 + 2875*x^3*y^3*z - 5995*x^3*y^3 + 2817*x^3*y^2*z^2 - 11874*x^3*y^2*z + 12264*x^3*y^2 + 1209*x^3*y*z^3 - 7731*x^3*y*z^2 + 16029*x^3*y*z - 11074*x^3*y + 192*x^3*z^4 - 1656*x^3*z^3 + 5175*x^3*z^2 - 7147*x^3*z + 3730*x^3 - 642*x^2*y^5 - 2211*x^2*y^4*z + 4509*x^2*y^4 - 2991*x^2*y^3*z^2 + 12312*x^2*y^3*z - 12588*x^2*y^3 - 1989*x^2*y^2*z^3 + 12411*x^2*y^2*z^2 - 25452*x^2*y^2*z + 17517*x^2*y^2 - 651*x^2*y*z^4 + 5478*x^2*y*z^3 - 16917*x^2*y*z^2 + 23289*x^2*y*z - 12156*x^2*y - 84*x^2*z^5 + 894*x^2*z^4 - 3699*x^2*z^3 + 7641*x^2*z^2 - 7968*x^2*z + 3370*x^2 + 196*x*y^6 + 840*x*y^5*z - 1680*x*y^5 + 1473*x*y^4*z^2 - 5934*x*y^4*z + 5976*x*y^4 + 1353*x*y^3*z^3 - 8247*x*y^3*z^2 + 16692*x*y^3*z - 11326*x*y^3 + 687*x*y^2*z^4 - 5640*x*y^2*z^3 + 17196*x*y^2*z^2 - 23448*x*y^2*z + 12108*x*y^2 + 183*x*y*z^5 - 1899*x*y*z^4 + 7755*x*y*z^3 - 15906*x*y*z^2 + 16485*x*y*z - 6943*x*y + 20*x*z^6 - 252*x*z^5 + 1293*x*z^4 - 3543*x*z^3 + 5517*x*z^2 - 4652*x*z + 1670*x - 24*y^7 - 124*y^6*z + 244*y^6 - 270*y^5*z^2 + 1068*y^5*z - 1056*y^5 - 321*y^4*z^3 + 1917*y^4*z^2 - 3816*y^4*z + 2537*y^4 - 225*y^3*z^4 + 1806*y^3*z^3 - 5424*y^3*z^2 + 7271*y^3*z - 3677*y^3 - 93*y^2*z^5 + 942*y^2*z^4 - 3792*y^2*z^3 + 7674*y^2*z^2 - 7830*y^2*z + 3232*y^2 - 21*y*z^6 + 258*y*z^5 - 1305*y*z^4 + 3538*y*z^3 - 5448*y*z^2 + 4537*y*z - 1601*y - 2*z^7 + 29*z^6 - 177*z^5 + 602*z^4 - 1241*z^3 + 1558*z^2 - 1108*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ 2, -1, -1 ], [ 1, -2, -1 ], [ 2, -3, -2 ] ], "source_shift": [ 1, 2, 2 ], "target_matrix": [ [ -1, -2, -1 ], [ -2, -2, -3 ], [ 1, 1, 1 ] ], "target_shift": [ 519, 681, -343 ], "collision_preimages": [ [ "-3/4", "7/4", "-9/4" ], [ "-8", "23/2", "-55/2" ], [ "-7", "11/2", "-35/2" ] ], "collision_image": [ "2077/4", "1363/2", "-1373/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0557911169998988 ], "surface_features": [ 3.821227580161073, 4.706751910228168, 3.4961682741749778, 0.0, 3.934673119293732, 5007 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C137", "pair": 137, "family": "counterexample", "mapping": "F(x,y,z)=(8*x^7 + 68*x^6*y - 64*x^6*z - 124*x^6 + 246*x^5*y^2 - 468*x^5*y*z - 936*x^5*y + 216*x^5*z^2 + 804*x^5*z + 630*x^5 + 491*x^4*y^3 - 1416*x^4*y^2*z - 2931*x^4*y^2 + 1320*x^4*y*z^2 + 5052*x^4*y*z + 4017*x^4*y - 400*x^4*z^3 - 2160*x^4*z^2 - 3324*x^4*z - 1492*x^4 + 584*x^3*y^4 - 2269*x^3*y^3*z - 4873*x^3*y^3 + 3204*x^3*y^2*z^2 + 12639*x^3*y^2*z + 10194*x^3*y^2 - 1960*x^3*y*z^3 - 10848*x^3*y*z^2 - 16935*x^3*y*z - 7705*x^3*y + 440*x^3*z^4 + 3080*x^3*z^3 + 6996*x^3*z^2 + 6240*x^3*z + 1971*x^3 + 414*x^2*y^5 - 2031*x^2*y^4*z - 4536*x^2*y^4 + 3861*x^2*y^3*z^2 + 15735*x^2*y^3*z + 12867*x^2*y^3 - 3576*x^2*y^2*z^3 - 20331*x^2*y^2*z^2 - 32187*x^2*y^2*z - 14844*x^2*y^2 + 1620*x^2*y*z^4 + 11592*x^2*y*z^3 + 26703*x^2*y*z^2 + 24138*x^2*y*z + 7722*x^2*y - 288*x^2*z^5 - 2460*x^2*z^4 - 7344*x^2*z^3 - 9768*x^2*z^2 - 6162*x^2*z - 1519*x^2 + 162*x*y^6 - 963*x*y^5*z - 2241*x*y^5 + 2310*x*y^4*z^2 + 9747*x*y^4*z + 8076*x*y^4 - 2879*x*y^3*z^3 - 16851*x*y^3*z^2 - 27042*x*y^3*z - 12639*x*y^3 + 1974*x*y^2*z^4 + 14469*x*y^2*z^3 + 33792*x*y^2*z^2 + 30957*x*y^2*z + 10032*x*y^2 - 708*x*y*z^5 - 6168*x*y*z^4 - 18669*x*y*z^3 - 25161*x*y*z^2 - 16074*x*y*z - 4017*x*y + 104*x*z^6 + 1044*x*z^5 + 3846*x*z^4 + 6784*x*z^3 + 6411*x*z^2 + 3167*x*z + 642*x + 27*y^7 - 189*y^6*z - 459*y^6 + 549*y^5*z^2 + 2403*y^5*z + 2016*y^5 - 863*y^4*z^3 - 5211*y^4*z^2 - 8472*y^4*z - 4011*y^4 + 796*y^3*z^4 + 5989*y^3*z^3 + 14175*y^3*z^2 + 13158*y^3*z + 4320*y^3 - 432*y^2*z^5 - 3846*y^2*z^4 - 11799*y^2*z^3 - 16113*y^2*z^2 - 10425*y^2*z - 2643*y^2 + 128*y*z^6 + 1308*y*z^5 + 4884*y*z^4 + 8728*y*z^3 + 8352*y*z^2 + 4182*y*z + 856*y - 16*z^7 - 184*z^6 - 804*z^5 - 1764*z^4 - 2220*z^3 - 1648*z^2 - 673*z - 1, 24*x^7 + 204*x^6*y - 192*x^6*z - 324*x^6 + 738*x^5*y^2 - 1404*x^5*y*z - 2472*x^5*y + 648*x^5*z^2 + 2076*x^5*z + 1314*x^5 + 1473*x^4*y^3 - 4248*x^4*y^2*z - 7821*x^4*y^2 + 3960*x^4*y*z^2 + 13188*x^4*y*z + 8475*x^4*y - 1200*x^4*z^3 - 5520*x^4*z^2 - 6900*x^4*z - 2737*x^4 + 1752*x^3*y^4 - 6807*x^3*y^3*z - 13131*x^3*y^3 + 9612*x^3*y^2*z^2 + 33345*x^3*y^2*z + 21762*x^3*y^2 - 5880*x^3*y*z^3 - 28032*x^3*y*z^2 - 35565*x^3*y*z - 14287*x^3*y + 1320*x^3*z^4 + 7800*x^3*z^3 + 14460*x^3*z^2 + 11417*x^3*z + 3320*x^3 + 1242*x^2*y^5 - 6093*x^2*y^4*z - 12336*x^2*y^4 + 11583*x^2*y^3*z^2 + 41937*x^2*y^3*z + 27801*x^2*y^3 - 10728*x^2*y^2*z^3 - 53109*x^2*y^2*z^2 - 68409*x^2*y^2*z - 27834*x^2*y^2 + 4860*x^2*y*z^4 + 29688*x^2*y*z^3 + 55845*x^2*y*z^2 + 44649*x^2*y*z + 13146*x^2*y - 864*x^2*z^5 - 6180*x^2*z^4 - 15120*x^2*z^3 - 17829*x^2*z^2 - 10362*x^2*z - 2409*x^2 + 486*x*y^6 - 2889*x*y^5*z - 6147*x*y^5 + 6930*x*y^4*z^2 + 26229*x*y^4*z + 17664*x*y^4 - 8637*x*y^3*z^3 - 44481*x*y^3*z^2 - 58182*x*y^3*z - 23977*x*y^3 + 5922*x*y^2*z^4 + 37467*x*y^2*z^3 + 71532*x*y^2*z^2 + 57918*x*y^2*z + 17271*x*y^2 - 2124*x*y*z^5 - 15672*x*y*z^4 - 38895*x*y*z^3 - 46437*x*y*z^2 - 27324*x*y*z - 6441*x*y + 312*x*z^6 + 2604*x*z^5 + 7890*x*z^4 + 12355*x*z^3 + 10764*x*z^2 + 5016*x*z + 975*x + 81*y^7 - 567*y^6*z - 1269*y^6 + 1647*y^5*z^2 + 6525*y^5*z + 4464*y^5 - 2589*y^4*z^3 - 13893*y^4*z^2 - 18456*y^4*z - 7702*y^4 + 2388*y^3*z^4 + 15675*y^3*z^3 + 30381*y^3*z^2 + 24911*y^3*z + 7526*y^3 - 1296*y^2*z^5 - 9882*y^2*z^4 - 24885*y^2*z^3 - 30084*y^2*z^2 - 17925*y^2*z - 4287*y^2 + 384*y*z^6 + 3300*y*z^5 + 10140*y*z^4 + 16075*y*z^3 + 14178*y*z^2 + 6699*y*z + 1316*y - 48*z^7 - 456*z^6 - 1644*z^5 - 3206*z^4 - 3722*z^3 - 2607*z^2 - 1021*z - 1, -16*x^7 - 136*x^6*y + 128*x^6*z + 212*x^6 - 492*x^5*y^2 + 936*x^5*y*z + 1620*x^5*y - 432*x^5*z^2 - 1356*x^5*z - 828*x^5 - 982*x^4*y^3 + 2832*x^4*y^2*z + 5133*x^4*y^2 - 2640*x^4*y*z^2 - 8628*x^4*y*z - 5352*x^4*y + 800*x^4*z^3 + 3600*x^4*z^2 + 4344*x^4*z + 1680*x^4 - 1168*x^3*y^4 + 4538*x^3*y^3*z + 8630*x^3*y^3 - 6408*x^3*y^2*z^2 - 21849*x^3*y^2*z - 13773*x^3*y^2 + 3920*x^3*y*z^3 + 18312*x^3*y*z^2 + 22440*x^3*y*z + 8790*x^3*y - 880*x^3*z^4 - 5080*x^3*z^3 - 9096*x^3*z^2 - 7004*x^3*z - 2000*x^3 - 828*x^2*y^5 + 4062*x^2*y^4*z + 8118*x^2*y^4 - 7722*x^2*y^3*z^2 - 27519*x^2*y^3*z - 17634*x^2*y^3 + 7152*x^2*y^2*z^3 + 34749*x^2*y^2*z^2 + 43260*x^2*y^2*z + 17166*x^2*y^2 - 3240*x^2*y*z^4 - 19368*x^2*y*z^3 - 35208*x^2*y*z^2 - 27456*x^2*y*z - 7938*x^2*y + 576*x^2*z^5 + 4020*x^2*z^4 + 9504*x^2*z^3 + 10932*x^2*z^2 + 6240*x^2*z + 1433*x^2 - 324*x*y^6 + 1926*x*y^5*z + 4050*x*y^5 - 4620*x*y^4*z^2 - 17235*x*y^4*z - 11229*x*y^4 + 5758*x*y^3*z^3 + 29148*x*y^3*z^2 + 36876*x*y^3*z + 14824*x*y^3 - 3948*x*y^2*z^4 - 24483*x*y^2*z^3 - 45201*x*y^2*z^2 - 35703*x*y^2*z - 10455*x*y^2 + 1416*x*y*z^5 + 10212*x*y*z^4 + 24504*x*y*z^3 + 28542*x*y*z^2 + 16494*x*y*z + 3840*x*y - 208*x*z^6 - 1692*x*z^5 - 4956*x*z^4 - 7572*x*z^3 - 6480*x*z^2 - 2983*x*z - 575*x - 54*y^7 + 378*y^6*z + 837*y^6 - 1098*y^5*z^2 - 4293*y^5*z - 2844*y^5 + 1726*y^4*z^3 + 9117*y^4*z^2 + 11724*y^4*z + 4774*y^4 - 1592*y^3*z^4 - 10259*y^3*z^3 - 19242*y^3*z^2 - 15395*y^3*z - 4568*y^3 + 864*y^2*z^5 + 6450*y^2*z^4 + 15714*y^2*z^3 + 18537*y^2*z^2 + 10848*y^2*z + 2562*y^2 - 256*y*z^6 - 2148*y*z^5 - 6384*y*z^4 - 9876*y*z^3 - 8556*y*z^2 - 3993*y*z - 778*y + 32*z^7 + 296*z^6 + 1032*z^5 + 1964*z^4 + 2240*z^3 + 1550*z^2 + 602*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -1, -1, 1 ], [ -2, -3, 2 ], [ -1, -1, 2 ] ], "source_shift": [ 1, 1, 2 ], "target_matrix": [ [ -1, -2, -1 ], [ -3, -2, -2 ], [ 2, 1, 1 ] ], "target_shift": [ 118, 175, -102 ], "collision_preimages": [ [ "3/4", "-1", "-5/4" ], [ "2", "5/2", "9/2" ], [ "13", "-9/2", "13/2" ] ], "collision_image": [ "473/4", "703/4", "-205/2" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0053169449965935 ], "surface_features": [ 3.9666079902793836, 4.854500367877103, 3.6582976503081897, 0.0, 4.077489911845597, 5062 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B116", "pair": 116, "family": "bijection", "mapping": "F(x,y,z)=(-192*x^7 + 704*x^6*y + 704*x^6*z - 1568*x^6 - 1104*x^5*y^2 - 2208*x^5*y*z + 4912*x^5*y - 1104*x^5*z^2 + 4912*x^5*z - 5440*x^5 + 960*x^4*y^3 + 2880*x^4*y^2*z - 6400*x^4*y^2 + 2880*x^4*y*z^2 - 12800*x^4*y*z + 14160*x^4*y + 960*x^4*z^3 - 6400*x^4*z^2 + 14160*x^4*z - 10424*x^4 - 500*x^3*y^4 - 2000*x^3*y^3*z + 4440*x^3*y^3 - 3000*x^3*y^2*z^2 + 13320*x^3*y^2*z - 14720*x^3*y^2 - 2000*x^3*y*z^3 + 13320*x^3*y*z^2 - 29440*x^3*y*z + 21652*x^3*y - 500*x^3*z^4 + 4440*x^3*z^3 - 14720*x^3*z^2 + 21652*x^3*z - 11964*x^3 + 156*x^2*y^5 + 780*x^2*y^4*z - 1730*x^2*y^4 + 1560*x^2*y^3*z^2 - 6920*x^2*y^3*z + 7640*x^2*y^3 + 1560*x^2*y^2*z^3 - 10380*x^2*y^2*z^2 + 22920*x^2*y^2*z - 16842*x^2*y^2 + 780*x^2*y*z^4 - 6920*x^2*y*z^3 + 22920*x^2*y*z^2 - 33684*x^2*y*z + 18600*x^2*y + 156*x^2*z^5 - 1730*x^2*z^4 + 7640*x^2*z^3 - 16842*x^2*z^2 + 18600*x^2*z - 8286*x^2 - 27*x*y^6 - 162*x*y^5*z + 359*x*y^5 - 405*x*y^4*z^2 + 1795*x*y^4*z - 1980*x*y^4 - 540*x*y^3*z^3 + 3590*x*y^3*z^2 - 7920*x*y^3*z + 5815*x*y^3 - 405*x*y^2*z^4 + 3590*x*y^2*z^3 - 11880*x*y^2*z^2 + 17445*x*y^2*z - 9627*x*y^2 - 162*x*y*z^5 + 1795*x*y*z^4 - 7920*x*y*z^3 + 17445*x*y*z^2 - 19254*x*y*z + 8583*x*y - 27*x*z^6 + 359*x*z^5 - 1980*x*z^4 + 5815*x*z^3 - 9627*x*z^2 + 8580*x*z - 3245*x + 2*y^7 + 14*y^6*z - 31*y^6 + 42*y^5*z^2 - 186*y^5*z + 205*y^5 + 70*y^4*z^3 - 465*y^4*z^2 + 1025*y^4*z - 752*y^4 + 70*y^3*z^4 - 620*y^3*z^3 + 2050*y^3*z^2 - 3008*y^3*z + 1659*y^3 + 42*y^2*z^5 - 465*y^2*z^4 + 2050*y^2*z^3 - 4512*y^2*z^2 + 4977*y^2*z - 2221*y^2 + 14*y*z^6 - 186*y*z^5 + 1025*y*z^4 - 3008*y*z^3 + 4977*y*z^2 - 4440*y*z + 1688*y + 2*z^7 - 31*z^6 + 205*z^5 - 752*z^4 + 1659*z^3 - 2219*z^2 + 1683*z - 1, -192*x^7 + 704*x^6*y + 704*x^6*z - 1504*x^6 - 1104*x^5*y^2 - 2208*x^5*y*z + 4720*x^5*y - 1104*x^5*z^2 + 4720*x^5*z - 5024*x^5 + 960*x^4*y^3 + 2880*x^4*y^2*z - 6160*x^4*y^2 + 2880*x^4*y*z^2 - 12320*x^4*y*z + 13120*x^4*y + 960*x^4*z^3 - 6160*x^4*z^2 + 13120*x^4*z - 9304*x^4 - 500*x^3*y^4 - 2000*x^3*y^3*z + 4280*x^3*y^3 - 3000*x^3*y^2*z^2 + 12840*x^3*y^2*z - 13680*x^3*y^2 - 2000*x^3*y*z^3 + 12840*x^3*y*z^2 - 27360*x^3*y*z + 19412*x^3*y - 500*x^3*z^4 + 4280*x^3*z^3 - 13680*x^3*z^2 + 19412*x^3*z - 10356*x^3 + 156*x^2*y^5 + 780*x^2*y^4*z - 1670*x^2*y^4 + 1560*x^2*y^3*z^2 - 6680*x^2*y^3*z + 7120*x^2*y^3 + 1560*x^2*y^2*z^3 - 10020*x^2*y^2*z^2 + 21360*x^2*y^2*z - 15162*x^2*y^2 + 780*x^2*y*z^4 - 6680*x^2*y*z^3 + 21360*x^2*y*z^2 - 30324*x^2*y*z + 16188*x^2*y + 156*x^2*z^5 - 1670*x^2*z^4 + 7120*x^2*z^3 - 15162*x^2*z^2 + 16188*x^2*z - 6978*x^2 - 27*x*y^6 - 162*x*y^5*z + 347*x*y^5 - 405*x*y^4*z^2 + 1735*x*y^4*z - 1850*x*y^4 - 540*x*y^3*z^3 + 3470*x*y^3*z^2 - 7400*x*y^3*z + 5255*x*y^3 - 405*x*y^2*z^4 + 3470*x*y^2*z^3 - 11100*x*y^2*z^2 + 15765*x*y^2*z - 8421*x*y^2 - 162*x*y*z^5 + 1735*x*y*z^4 - 7400*x*y*z^3 + 15765*x*y*z^2 - 16842*x*y*z + 7275*x*y - 27*x*z^6 + 347*x*z^5 - 1850*x*z^4 + 5255*x*z^3 - 8421*x*z^2 + 7272*x*z - 2660*x + 2*y^7 + 14*y^6*z - 30*y^6 + 42*y^5*z^2 - 180*y^5*z + 192*y^5 + 70*y^4*z^3 - 450*y^4*z^2 + 960*y^4*z - 682*y^4 + 70*y^3*z^4 - 600*y^3*z^3 + 1920*y^3*z^2 - 2728*y^3*z + 1458*y^3 + 42*y^2*z^5 - 450*y^2*z^4 + 1920*y^2*z^3 - 4092*y^2*z^2 + 4374*y^2*z - 1894*y^2 + 14*y*z^6 - 180*y*z^5 + 960*y*z^4 - 2728*y*z^3 + 4374*y*z^2 - 3786*y*z + 1393*y + 2*z^7 - 30*z^6 + 192*z^5 - 682*z^4 + 1458*z^3 - 1892*z^2 + 1389*z - 1, 192*x^7 - 704*x^6*y - 704*x^6*z + 1504*x^6 + 1104*x^5*y^2 + 2208*x^5*y*z - 4720*x^5*y + 1104*x^5*z^2 - 4720*x^5*z + 5024*x^5 - 960*x^4*y^3 - 2880*x^4*y^2*z + 6160*x^4*y^2 - 2880*x^4*y*z^2 + 12320*x^4*y*z - 13120*x^4*y - 960*x^4*z^3 + 6160*x^4*z^2 - 13120*x^4*z + 9304*x^4 + 500*x^3*y^4 + 2000*x^3*y^3*z - 4280*x^3*y^3 + 3000*x^3*y^2*z^2 - 12840*x^3*y^2*z + 13680*x^3*y^2 + 2000*x^3*y*z^3 - 12840*x^3*y*z^2 + 27360*x^3*y*z - 19412*x^3*y + 500*x^3*z^4 - 4280*x^3*z^3 + 13680*x^3*z^2 - 19412*x^3*z + 10356*x^3 - 156*x^2*y^5 - 780*x^2*y^4*z + 1670*x^2*y^4 - 1560*x^2*y^3*z^2 + 6680*x^2*y^3*z - 7120*x^2*y^3 - 1560*x^2*y^2*z^3 + 10020*x^2*y^2*z^2 - 21360*x^2*y^2*z + 15162*x^2*y^2 - 780*x^2*y*z^4 + 6680*x^2*y*z^3 - 21360*x^2*y*z^2 + 30324*x^2*y*z - 16188*x^2*y - 156*x^2*z^5 + 1670*x^2*z^4 - 7120*x^2*z^3 + 15162*x^2*z^2 - 16188*x^2*z + 6978*x^2 + 27*x*y^6 + 162*x*y^5*z - 347*x*y^5 + 405*x*y^4*z^2 - 1735*x*y^4*z + 1850*x*y^4 + 540*x*y^3*z^3 - 3470*x*y^3*z^2 + 7400*x*y^3*z - 5255*x*y^3 + 405*x*y^2*z^4 - 3470*x*y^2*z^3 + 11100*x*y^2*z^2 - 15765*x*y^2*z + 8421*x*y^2 + 162*x*y*z^5 - 1735*x*y*z^4 + 7400*x*y*z^3 - 15765*x*y*z^2 + 16842*x*y*z - 7275*x*y + 27*x*z^6 - 347*x*z^5 + 1850*x*z^4 - 5255*x*z^3 + 8421*x*z^2 - 7272*x*z + 2666*x - 2*y^7 - 14*y^6*z + 30*y^6 - 42*y^5*z^2 + 180*y^5*z - 192*y^5 - 70*y^4*z^3 + 450*y^4*z^2 - 960*y^4*z + 682*y^4 - 70*y^3*z^4 + 600*y^3*z^3 - 1920*y^3*z^2 + 2728*y^3*z - 1458*y^3 - 42*y^2*z^5 + 450*y^2*z^4 - 1920*y^2*z^3 + 4092*y^2*z^2 - 4374*y^2*z + 1894*y^2 - 14*y*z^6 + 180*y*z^5 - 960*y*z^4 + 2728*y*z^3 - 4374*y*z^2 + 3786*y*z - 1397*y - 2*z^7 + 30*z^6 - 192*z^5 + 682*z^4 - 1458*z^3 + 1892*z^2 - 1393*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ 2, -1, -1 ], [ 1, -2, -1 ], [ 1, -1, -1 ] ], "source_shift": [ 2, 2, 1 ], "target_matrix": [ [ -1, -2, -1 ], [ -1, -1, -2 ], [ 1, 1, 1 ] ], "target_shift": [ 564, 446, -452 ], "inverse_recipe": { "p": 6, "q": 1, "f": "x**6 + x**5 + x**3 + x**2 + x", "g": "x", "h": "y*z + z**2", "polynomials": { "f": [ [ [ 6, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ], [ [ 3, 0, 0 ], 1 ], [ [ 5, 0, 0 ], 1 ] ], "g": [ [ [ 1, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 0, 2 ], 1 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0288046019995818 ], "surface_features": [ 3.7128460436316097, 4.527423658682064, 3.3010299956639813, 0.0, 3.8138950744385616, 4946 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B169", "pair": 169, "family": "bijection", "mapping": "F(x,y,z)=(24*x^7 - 336*x^6*y - 168*x^6*z + 192*x^6 + 2016*x^5*y^2 + 2016*x^5*y*z - 2304*x^5*y + 504*x^5*z^2 - 1152*x^5*z + 768*x^5 - 6720*x^4*y^3 - 10080*x^4*y^2*z + 11520*x^4*y^2 - 5040*x^4*y*z^2 + 11520*x^4*y*z - 7584*x^4*y - 840*x^4*z^3 + 2880*x^4*z^2 - 3816*x^4*z + 1942*x^4 + 13440*x^3*y^4 + 26880*x^3*y^3*z - 30720*x^3*y^3 + 20160*x^3*y^2*z^2 - 46080*x^3*y^2*z + 29952*x^3*y^2 + 6720*x^3*y*z^3 - 23040*x^3*y*z^2 + 30144*x^3*y*z - 14984*x^3*y + 840*x^3*z^4 - 3840*x^3*z^3 + 7584*x^3*z^2 - 7624*x^3*z + 3135*x^3 - 16128*x^2*y^5 - 40320*x^2*y^4*z + 46080*x^2*y^4 - 40320*x^2*y^3*z^2 + 92160*x^2*y^3*z - 59136*x^2*y^3 - 20160*x^2*y^2*z^3 + 69120*x^2*y^2*z^2 - 89280*x^2*y^2*z + 43296*x^2*y^2 - 5040*x^2*y*z^4 + 23040*x^2*y*z^3 - 44928*x^2*y*z^2 + 44088*x^2*y*z - 17634*x^2*y - 504*x^2*z^5 + 2880*x^2*z^4 - 7536*x^2*z^3 + 11220*x^2*z^2 - 9093*x^2*z + 3319*x^2 + 10752*x*y^6 + 32256*x*y^5*z - 36864*x*y^5 + 40320*x*y^4*z^2 - 92160*x*y^4*z + 58368*x*y^4 + 26880*x*y^3*z^3 - 92160*x*y^3*z^2 + 117504*x*y^3*z - 55520*x*y^3 + 10080*x*y^2*z^4 - 46080*x*y^2*z^3 + 88704*x*y^2*z^2 - 84864*x*y^2*z + 32916*x*y^2 + 2016*x*y*z^5 - 11520*x*y*z^4 + 29760*x*y*z^3 - 43224*x*y*z^2 + 34020*x*y*z - 11620*x*y + 168*x*z^6 - 1152*x*z^5 + 3744*x*z^4 - 7336*x*z^3 + 8781*x*z^2 - 6182*x*z + 1977*x - 3072*y^7 - 10752*y^6*z + 12288*y^6 - 16128*y^5*z^2 + 36864*y^5*z - 23040*y^5 - 13440*y^4*z^3 + 46080*y^4*z^2 - 57984*y^4*z + 26656*y^4 - 6720*y^3*z^4 + 30720*y^3*z^3 - 58368*y^3*z^2 + 54368*y^3*z - 20376*y^3 - 2016*y^2*z^5 + 11520*y^2*z^4 - 29376*y^2*z^3 + 41568*y^2*z^2 - 31668*y^2*z + 10252*y^2 - 336*y*z^6 + 2304*y*z^5 - 7392*y*z^4 + 14120*y*z^3 - 16386*y*z^2 + 10876*y*z - 3188*y - 24*z^7 + 192*z^6 - 744*z^5 + 1798*z^4 - 2823*z^3 + 2887*z^2 - 1762*z + 1, 12*x^7 - 168*x^6*y - 84*x^6*z + 96*x^6 + 1008*x^5*y^2 + 1008*x^5*y*z - 1152*x^5*y + 252*x^5*z^2 - 576*x^5*z + 384*x^5 - 3360*x^4*y^3 - 5040*x^4*y^2*z + 5760*x^4*y^2 - 2520*x^4*y*z^2 + 5760*x^4*y*z - 3792*x^4*y - 420*x^4*z^3 + 1440*x^4*z^2 - 1908*x^4*z + 970*x^4 + 6720*x^3*y^4 + 13440*x^3*y^3*z - 15360*x^3*y^3 + 10080*x^3*y^2*z^2 - 23040*x^3*y^2*z + 14976*x^3*y^2 + 3360*x^3*y*z^3 - 11520*x^3*y*z^2 + 15072*x^3*y*z - 7484*x^3*y + 420*x^3*z^4 - 1920*x^3*z^3 + 3792*x^3*z^2 - 3808*x^3*z + 1564*x^3 - 8064*x^2*y^5 - 20160*x^2*y^4*z + 23040*x^2*y^4 - 20160*x^2*y^3*z^2 + 46080*x^2*y^3*z - 29568*x^2*y^3 - 10080*x^2*y^2*z^3 + 34560*x^2*y^2*z^2 - 44640*x^2*y^2*z + 21624*x^2*y^2 - 2520*x^2*y*z^4 + 11520*x^2*y*z^3 - 22464*x^2*y*z^2 + 22020*x^2*y*z - 8796*x^2*y - 252*x^2*z^5 + 1440*x^2*z^4 - 3768*x^2*z^3 + 5604*x^2*z^2 - 4536*x^2*z + 1654*x^2 + 5376*x*y^6 + 16128*x*y^5*z - 18432*x*y^5 + 20160*x*y^4*z^2 - 46080*x*y^4*z + 29184*x*y^4 + 13440*x*y^3*z^3 - 46080*x*y^3*z^2 + 58752*x*y^3*z - 27728*x*y^3 + 5040*x*y^2*z^4 - 23040*x*y^2*z^3 + 44352*x*y^2*z^2 - 42384*x*y^2*z + 16416*x*y^2 + 1008*x*y*z^5 - 5760*x*y*z^4 + 14880*x*y*z^3 - 21588*x*y*z^2 + 16968*x*y*z - 5788*x*y + 84*x*z^6 - 576*x*z^5 + 1872*x*z^4 - 3664*x*z^3 + 4380*x*z^2 - 3080*x*z + 987*x - 1536*y^7 - 5376*y^6*z + 6144*y^6 - 8064*y^5*z^2 + 18432*y^5*z - 11520*y^5 - 6720*y^4*z^3 + 23040*y^4*z^2 - 28992*y^4*z + 13312*y^4 - 3360*y^3*z^4 + 15360*y^3*z^3 - 29184*y^3*z^2 + 27152*y^3*z - 10160*y^3 - 1008*y^2*z^5 + 5760*y^2*z^4 - 14688*y^2*z^3 + 20760*y^2*z^2 - 15792*y^2*z + 5104*y^2 - 168*y*z^6 + 1152*y*z^5 - 3696*y*z^4 + 7052*y*z^3 - 8172*y*z^2 + 5416*y*z - 1588*y - 12*z^7 + 96*z^6 - 372*z^5 + 898*z^4 - 1408*z^3 + 1438*z^2 - 879*z + 1, 12*x^7 - 168*x^6*y - 84*x^6*z + 96*x^6 + 1008*x^5*y^2 + 1008*x^5*y*z - 1152*x^5*y + 252*x^5*z^2 - 576*x^5*z + 384*x^5 - 3360*x^4*y^3 - 5040*x^4*y^2*z + 5760*x^4*y^2 - 2520*x^4*y*z^2 + 5760*x^4*y*z - 3792*x^4*y - 420*x^4*z^3 + 1440*x^4*z^2 - 1908*x^4*z + 970*x^4 + 6720*x^3*y^4 + 13440*x^3*y^3*z - 15360*x^3*y^3 + 10080*x^3*y^2*z^2 - 23040*x^3*y^2*z + 14976*x^3*y^2 + 3360*x^3*y*z^3 - 11520*x^3*y*z^2 + 15072*x^3*y*z - 7484*x^3*y + 420*x^3*z^4 - 1920*x^3*z^3 + 3792*x^3*z^2 - 3808*x^3*z + 1563*x^3 - 8064*x^2*y^5 - 20160*x^2*y^4*z + 23040*x^2*y^4 - 20160*x^2*y^3*z^2 + 46080*x^2*y^3*z - 29568*x^2*y^3 - 10080*x^2*y^2*z^3 + 34560*x^2*y^2*z^2 - 44640*x^2*y^2*z + 21624*x^2*y^2 - 2520*x^2*y*z^4 + 11520*x^2*y*z^3 - 22464*x^2*y*z^2 + 22020*x^2*y*z - 8790*x^2*y - 252*x^2*z^5 + 1440*x^2*z^4 - 3768*x^2*z^3 + 5604*x^2*z^2 - 4533*x^2*z + 1651*x^2 + 5376*x*y^6 + 16128*x*y^5*z - 18432*x*y^5 + 20160*x*y^4*z^2 - 46080*x*y^4*z + 29184*x*y^4 + 13440*x*y^3*z^3 - 46080*x*y^3*z^2 + 58752*x*y^3*z - 27728*x*y^3 + 5040*x*y^2*z^4 - 23040*x*y^2*z^3 + 44352*x*y^2*z^2 - 42384*x*y^2*z + 16404*x*y^2 + 1008*x*y*z^5 - 5760*x*y*z^4 + 14880*x*y*z^3 - 21588*x*y*z^2 + 16956*x*y*z - 5776*x*y + 84*x*z^6 - 576*x*z^5 + 1872*x*z^4 - 3664*x*z^3 + 4377*x*z^2 - 3074*x*z + 980*x - 1536*y^7 - 5376*y^6*z + 6144*y^6 - 8064*y^5*z^2 + 18432*y^5*z - 11520*y^5 - 6720*y^4*z^3 + 23040*y^4*z^2 - 28992*y^4*z + 13312*y^4 - 3360*y^3*z^4 + 15360*y^3*z^3 - 29184*y^3*z^2 + 27152*y^3*z - 10152*y^3 - 1008*y^2*z^5 + 5760*y^2*z^4 - 14688*y^2*z^3 + 20760*y^2*z^2 - 15780*y^2*z + 5092*y^2 - 168*y*z^6 + 1152*y*z^5 - 3696*y*z^4 + 7052*y*z^3 - 8166*y*z^2 + 5404*y*z - 1578*y - 12*z^7 + 96*z^6 - 372*z^5 + 898*z^4 - 1407*z^3 + 1435*z^2 - 873*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ 1, -2, -1 ], [ 3, -2, -2 ], [ 2, -1, -1 ] ], "source_shift": [ 1, 2, 1 ], "target_matrix": [ [ 2, 1, 1 ], [ 1, 2, 1 ], [ 1, 1, 1 ] ], "target_shift": [ -475, -238, -234 ], "inverse_recipe": { "p": 3, "q": 4, "f": "x**3 + x", "g": "x**4 + x**3 + x**2 + x", "h": "y*z", "polynomials": { "f": [ [ [ 3, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "g": [ [ [ 4, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ], [ [ 3, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ] ] }, "h_multiplier": 12 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0346894390022499 ], "surface_features": [ 4.16240167324911, 5.070052650849111, 3.856970433482492, 0.0, 4.263315748929992, 5132 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C024", "pair": 24, "family": "counterexample", "mapping": "F(x,y,z)=(-2*x^7 - 13*x^6*y - 19*x^6*z - 34*x^6 - 36*x^5*y^2 - 105*x^5*y*z - 189*x^5*y - 75*x^5*z^2 - 276*x^5*z - 225*x^5 - 55*x^4*y^3 - 240*x^4*y^2*z - 435*x^4*y^2 - 342*x^4*y*z^2 - 1269*x^4*y*z - 1041*x^4*y - 159*x^4*z^3 - 909*x^4*z^2 - 1545*x^4*z - 779*x^4 - 50*x^3*y^4 - 290*x^3*y^3*z - 530*x^3*y^3 - 618*x^3*y^2*z^2 - 2316*x^3*y^2*z - 1914*x^3*y^2 - 573*x^3*y*z^3 - 3315*x^3*y*z^2 - 5679*x^3*y*z - 2874*x^3*y - 195*x^3*z^4 - 1554*x^3*z^3 - 4140*x^3*z^2 - 4356*x^3*z - 1552*x^3 - 27*x^2*y^5 - 195*x^2*y^4*z - 360*x^2*y^4 - 552*x^2*y^3*z^2 - 2094*x^2*y^3*z - 1746*x^2*y^3 - 765*x^2*y^2*z^3 - 4491*x^2*y^2*z^2 - 7767*x^2*y^2*z - 3948*x^2*y^2 - 519*x^2*y*z^4 - 4209*x^2*y*z^3 - 11328*x^2*y*z^2 - 11970*x^2*y*z - 4266*x^2*y - 138*x^2*z^5 - 1455*x^2*z^4 - 5415*x^2*z^3 - 8928*x^2*z^2 - 6615*x^2*z - 1789*x^2 - 8*x*y^6 - 69*x*y^5*z - 129*x*y^5 - 243*x*y^4*z^2 - 936*x*y^4*z - 789*x*y^4 - 447*x*y^3*z^3 - 2673*x*y^3*z^2 - 4677*x*y^3*z - 2390*x*y^3 - 453*x*y^2*z^4 - 3756*x*y^2*z^3 - 10236*x*y^2*z^2 - 10872*x*y^2*z - 3876*x*y^2 - 240*x*y*z^5 - 2598*x*y*z^4 - 9801*x*y*z^3 - 16242*x*y*z^2 - 12027*x*y*z - 3239*x*y - 52*x*z^6 - 708*x*z^5 - 3465*x*z^4 - 7963*x*z^3 - 9198*x*z^2 - 5143*x*z - 1098*x - y^7 - 10*y^6*z - 19*y^6 - 42*y^5*z^2 - 165*y^5*z - 141*y^5 - 96*y^4*z^3 - 588*y^4*z^2 - 1044*y^4*z - 537*y^4 - 129*y^3*z^4 - 1101*y^3*z^3 - 3048*y^3*z^2 - 3258*y^3*z - 1162*y^3 - 102*y^2*z^5 - 1143*y^2*z^4 - 4386*y^2*z^3 - 7314*y^2*z^2 - 5412*y^2*z - 1450*y^2 - 44*y*z^6 - 624*y*z^5 - 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440*x^3*y^3 - 618*x^3*y^2*z^2 - 1950*x^3*y^2*z - 1410*x^3*y^2 - 573*x^3*y*z^3 - 2832*x^3*y*z^2 - 4245*x^3*y*z - 1959*x^3*y - 195*x^3*z^4 - 1347*x^3*z^3 - 3144*x^3*z^2 - 3009*x^3*z - 1000*x^3 - 27*x^2*y^5 - 195*x^2*y^4*z - 300*x^2*y^4 - 552*x^2*y^3*z^2 - 1770*x^2*y^3*z - 1290*x^2*y^3 - 765*x^2*y^2*z^3 - 3852*x^2*y^2*z^2 - 5823*x^2*y^2*z - 2697*x^2*y^2 - 519*x^2*y*z^4 - 3663*x^2*y*z^3 - 8628*x^2*y*z^2 - 8286*x^2*y*z - 2754*x^2*y - 138*x^2*z^5 - 1284*x^2*z^4 - 4194*x^2*z^3 - 6270*x^2*z^2 - 4326*x^2*z - 1102*x^2 - 8*x*y^6 - 69*x*y^5*z - 108*x*y^5 - 243*x*y^4*z^2 - 795*x*y^4*z - 585*x*y^4 - 447*x*y^3*z^3 - 2304*x*y^3*z^2 - 3519*x*y^3*z - 1637*x*y^3 - 453*x*y^2*z^4 - 3285*x*y^2*z^3 - 7824*x*y^2*z^2 - 7545*x*y^2*z - 2508*x*y^2 - 240*x*y*z^5 - 2304*x*y*z^4 - 7617*x*y*z^3 - 11433*x*y*z^2 - 7881*x*y*z - 2000*x*y - 52*x*z^6 - 636*x*z^5 - 2739*x*z^4 - 5692*x*z^3 - 6108*x*z^2 - 3220*x*z - 650*x - y^7 - 10*y^6*z - 16*y^6 - 42*y^5*z^2 - 141*y^5*z - 105*y^5 - 96*y^4*z^3 - 510*y^4*z^2 - 789*y^4*z - 369*y^4 - 129*y^3*z^4 - 969*y^3*z^3 - 2340*y^3*z^2 - 2268*y^3*z - 754*y^3 - 102*y^2*z^5 - 1020*y^2*z^4 - 3423*y^2*z^3 - 5163*y^2*z^2 - 3555*y^2*z - 898*y^2 - 44*y*z^6 - 564*y*z^5 - 2469*y*z^4 - 5157*y*z^3 - 5523*y*z^2 - 2890*y*z - 577*y - 8*z^7 - 128*z^6 - 702*z^5 - 1905*z^4 - 2828*z^3 - 2302*z^2 - 946*z - 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ 1, 1, 1 ], [ 1, 1, 2 ], [ 2, 1, 1 ] ], "source_shift": [ 2, 1, 1 ], "target_matrix": [ [ -1, -2, -1 ], [ 1, 1, 2 ], [ -1, -1, -1 ] ], "target_shift": [ 270, -139, 155 ], "collision_preimages": [ [ "3/4", "-15/4", "1" ], [ "13/2", "-6", "-3/2" ], [ "17/2", "-15", "7/2" ] ], "collision_image": [ "1081/4", "-557/4", "621/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.031759494002472 ], "surface_features": [ 3.309681021373021, 4.21063950615415, 2.9003671286564705, 0.0, 3.4258772805585957, 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618*x^2*y^4*z - 1008*x^2*y^4 + 1782*x^2*y^3*z^2 + 5829*x^2*y^3*z + 4443*x^2*y^3 - 2514*x^2*y^2*z^3 - 12366*x^2*y^2*z^2 - 18975*x^2*y^2*z - 9501*x^2*y^2 + 1734*x^2*y*z^4 + 11397*x^2*y*z^3 + 26397*x^2*y*z^2 + 26610*x^2*y*z + 10014*x^2*y - 468*x^2*z^5 - 3852*x^2*z^4 - 11961*x^2*z^3 - 18177*x^2*z^2 - 13746*x^2*z - 4193*x^2 + 20*x*y^6 - 174*x*y^5*z - 276*x*y^5 + 618*x*y^4*z^2 + 1965*x*y^4*z + 1482*x*y^4 - 1146*x*y^3*z^3 - 5478*x*y^3*z^2 - 8310*x*y^3*z - 4138*x*y^3 + 1170*x*y^2*z^4 + 7473*x*y^2*z^3 + 17094*x*y^2*z^2 + 17112*x*y^2*z + 6414*x*y^2 - 624*x*y*z^5 - 4992*x*y*z^4 - 15294*x*y*z^3 - 23052*x*y*z^2 - 17334*x*y*z - 5264*x*y + 136*x*z^6 + 1308*x*z^5 + 5028*x*z^4 + 10130*x*z^3 + 11436*x*z^2 + 6941*x*z + 1788*x + 2*y^7 - 20*y^6*z - 31*y^6 + 84*y^5*z^2 + 261*y^5*z + 195*y^5 - 192*y^4*z^3 - 897*y^4*z^2 - 1347*y^4*z - 667*y^4 + 258*y^3*z^4 + 1611*y^3*z^3 + 3645*y^3*z^2 + 3624*y^3*z + 1352*y^3 - 204*y^2*z^5 - 1596*y^2*z^4 - 4833*y^2*z^3 - 7227*y^2*z^2 - 5400*y^2*z - 1631*y^2 + 88*y*z^6 + 828*y*z^5 + 3144*y*z^4 + 6278*y*z^3 + 7032*y*z^2 + 4237*y*z + 1087*y - 16*z^7 - 176*z^6 - 804*z^5 - 2008*z^4 - 2992*z^3 - 2690*z^2 - 1373*z - 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -1, -1, 2 ], [ -2, -1, 1 ], [ -1, -1, 1 ] ], "source_shift": [ 2, 1, 2 ], "target_matrix": [ [ -1, -1, -1 ], [ 3, 2, 3 ], [ -2, -1, -1 ] ], "target_shift": [ 228, -517, 312 ], "collision_preimages": [ [ "-5/4", "15/4", "1/4" ], [ "7", "-17", "-11/2" ], [ "4", "-16", "-15/2" ] ], "collision_image": [ "913/4", "-2071/4", "625/2" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0477421049981785 ], "surface_features": [ 3.7077860869404446, 4.638489256954637, 3.3507324517161297, 0.0, 3.849271752214216, 4940 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B165", "pair": 165, "family": "bijection", "mapping": "F(x,y,z)=(3*x^7 + 42*x^6*y - 21*x^6*z - 30*x^6 + 252*x^5*y^2 - 252*x^5*y*z - 357*x^5*y + 63*x^5*z^2 + 180*x^5*z + 119*x^5 + 840*x^4*y^3 - 1260*x^4*y^2*z - 1770*x^4*y^2 + 630*x^4*y*z^2 + 1785*x^4*y*z + 1172*x^4*y - 105*x^4*z^3 - 450*x^4*z^2 - 595*x^4*z - 252*x^4 + 1680*x^3*y^4 - 3360*x^3*y^3*z - 4680*x^3*y^3 + 2520*x^3*y^2*z^2 + 7080*x^3*y^2*z + 4616*x^3*y^2 - 840*x^3*y*z^3 - 3570*x^3*y*z^2 - 4688*x^3*y*z - 1974*x^3*y + 105*x^3*z^4 + 600*x^3*z^3 + 1190*x^3*z^2 + 1008*x^3*z + 343*x^3 + 2016*x^2*y^5 - 5040*x^2*y^4*z - 6960*x^2*y^4 + 5040*x^2*y^3*z^2 + 14040*x^2*y^3*z + 9088*x^2*y^3 - 2520*x^2*y^2*z^3 - 10620*x^2*y^2*z^2 - 13848*x^2*y^2*z - 5796*x^2*y^2 + 630*x^2*y*z^4 + 3570*x^2*y*z^3 + 7032*x^2*y*z^2 + 5922*x^2*y*z + 1989*x^2*y - 63*x^2*z^5 - 450*x^2*z^4 - 1190*x^2*z^3 - 1512*x^2*z^2 - 1026*x^2*z - 359*x^2 + 1344*x*y^6 - 4032*x*y^5*z - 5520*x*y^5 + 5040*x*y^4*z^2 + 13920*x*y^4*z + 8944*x*y^4 - 3360*x*y^3*z^3 - 14040*x*y^3*z^2 - 18176*x*y^3*z - 7560*x*y^3 + 1260*x*y^2*z^4 + 7080*x*y^2*z^3 + 13848*x*y^2*z^2 + 11592*x*y^2*z + 3840*x*y^2 - 252*x*y*z^5 - 1785*x*y*z^4 - 4688*x*y*z^3 - 5922*x*y*z^2 - 3966*x*y*z - 1319*x*y + 21*x*z^6 + 180*x*z^5 + 595*x*z^4 + 1008*x*z^3 + 1023*x*z^2 + 706*x*z + 262*x + 384*y^7 - 1344*y^6*z - 1824*y^6 + 2016*y^5*z^2 + 5520*y^5*z + 3520*y^5 - 1680*y^4*z^3 - 6960*y^4*z^2 - 8944*y^4*z - 3696*y^4 + 840*y^3*z^4 + 4680*y^3*z^3 + 9088*y^3*z^2 + 7560*y^3*z + 2468*y^3 - 252*y^2*z^5 - 1770*y^2*z^4 - 4616*y^2*z^3 - 5796*y^2*z^2 - 3828*y^2*z - 1217*y^2 + 42*y*z^6 + 357*y*z^5 + 1172*y*z^4 + 1974*y*z^3 + 1977*y*z^2 + 1298*y*z + 458*y - 3*z^7 - 30*z^6 - 119*z^5 - 252*z^4 - 340*z^3 - 347*z^2 - 257*z - 1, x^7 + 14*x^6*y - 7*x^6*z - 10*x^6 + 84*x^5*y^2 - 84*x^5*y*z - 119*x^5*y + 21*x^5*z^2 + 60*x^5*z + 39*x^5 + 280*x^4*y^3 - 420*x^4*y^2*z - 590*x^4*y^2 + 210*x^4*y*z^2 + 595*x^4*y*z + 384*x^4*y - 35*x^4*z^3 - 150*x^4*z^2 - 195*x^4*z - 80*x^4 + 560*x^3*y^4 - 1120*x^3*y^3*z - 1560*x^3*y^3 + 840*x^3*y^2*z^2 + 2360*x^3*y^2*z + 1512*x^3*y^2 - 280*x^3*y*z^3 - 1190*x^3*y*z^2 - 1536*x^3*y*z - 626*x^3*y + 35*x^3*z^4 + 200*x^3*z^3 + 390*x^3*z^2 + 320*x^3*z + 105*x^3 + 672*x^2*y^5 - 1680*x^2*y^4*z - 2320*x^2*y^4 + 1680*x^2*y^3*z^2 + 4680*x^2*y^3*z + 2976*x^2*y^3 - 840*x^2*y^2*z^3 - 3540*x^2*y^2*z^2 - 4536*x^2*y^2*z - 1836*x^2*y^2 + 210*x^2*y*z^4 + 1190*x^2*y*z^3 + 2304*x^2*y*z^2 + 1878*x^2*y*z + 607*x^2*y - 21*x^2*z^5 - 150*x^2*z^4 - 390*x^2*z^3 - 480*x^2*z^2 - 314*x^2*z - 109*x^2 + 448*x*y^6 - 1344*x*y^5*z - 1840*x*y^5 + 1680*x*y^4*z^2 + 4640*x*y^4*z + 2928*x*y^4 - 1120*x*y^3*z^3 - 4680*x*y^3*z^2 - 5952*x*y^3*z - 2392*x*y^3 + 420*x*y^2*z^4 + 2360*x*y^2*z^3 + 4536*x*y^2*z^2 + 3672*x*y^2*z + 1168*x*y^2 - 84*x*y*z^5 - 595*x*y*z^4 - 1536*x*y*z^3 - 1878*x*y*z^2 - 1210*x*y*z - 397*x*y + 7*x*z^6 + 60*x*z^5 + 195*x*z^4 + 320*x*z^3 + 313*x*z^2 + 214*x*z + 80*x + 128*y^7 - 448*y^6*z - 608*y^6 + 672*y^5*z^2 + 1840*y^5*z + 1152*y^5 - 560*y^4*z^3 - 2320*y^4*z^2 - 2928*y^4*z - 1168*y^4 + 280*y^3*z^4 + 1560*y^3*z^3 + 2976*y^3*z^2 + 2392*y^3*z + 748*y^3 - 84*y^2*z^5 - 590*y^2*z^4 - 1512*y^2*z^3 - 1836*y^2*z^2 - 1164*y^2*z - 363*y^2 + 14*y*z^6 + 119*y*z^5 + 384*y*z^4 + 626*y*z^3 + 603*y*z^2 + 390*y*z + 140*y - z^7 - 10*z^6 - 39*z^5 - 80*z^4 - 104*z^3 - 105*z^2 - 79*z - 1, -2*x^7 - 28*x^6*y + 14*x^6*z + 20*x^6 - 168*x^5*y^2 + 168*x^5*y*z + 238*x^5*y - 42*x^5*z^2 - 120*x^5*z - 80*x^5 - 560*x^4*y^3 + 840*x^4*y^2*z + 1180*x^4*y^2 - 420*x^4*y*z^2 - 1190*x^4*y*z - 788*x^4*y + 70*x^4*z^3 + 300*x^4*z^2 + 400*x^4*z + 172*x^4 - 1120*x^3*y^4 + 2240*x^3*y^3*z + 3120*x^3*y^3 - 1680*x^3*y^2*z^2 - 4720*x^3*y^2*z - 3104*x^3*y^2 + 560*x^3*y*z^3 + 2380*x^3*y*z^2 + 3152*x^3*y*z + 1348*x^3*y - 70*x^3*z^4 - 400*x^3*z^3 - 800*x^3*z^2 - 688*x^3*z - 238*x^3 - 1344*x^2*y^5 + 3360*x^2*y^4*z + 4640*x^2*y^4 - 3360*x^2*y^3*z^2 - 9360*x^2*y^3*z - 6112*x^2*y^3 + 1680*x^2*y^2*z^3 + 7080*x^2*y^2*z^2 + 9312*x^2*y^2*z + 3960*x^2*y^2 - 420*x^2*y*z^4 - 2380*x^2*y*z^3 - 4728*x^2*y*z^2 - 4044*x^2*y*z - 1382*x^2*y + 42*x^2*z^5 + 300*x^2*z^4 + 800*x^2*z^3 + 1032*x^2*z^2 + 712*x^2*z + 249*x^2 - 896*x*y^6 + 2688*x*y^5*z + 3680*x*y^5 - 3360*x*y^4*z^2 - 9280*x*y^4*z - 6016*x*y^4 + 2240*x*y^3*z^3 + 9360*x*y^3*z^2 + 12224*x*y^3*z + 5168*x*y^3 - 840*x*y^2*z^4 - 4720*x*y^2*z^3 - 9312*x*y^2*z^2 - 7920*x*y^2*z - 2672*x*y^2 + 168*x*y*z^5 + 1190*x*y*z^4 + 3152*x*y*z^3 + 4044*x*y*z^2 + 2756*x*y*z + 918*x*y - 14*x*z^6 - 120*x*z^5 - 400*x*z^4 - 688*x*z^3 - 710*x*z^2 - 490*x*z - 178*x - 256*y^7 + 896*y^6*z + 1216*y^6 - 1344*y^5*z^2 - 3680*y^5*z - 2368*y^5 + 1120*y^4*z^3 + 4640*y^4*z^2 + 6016*y^4*z + 2528*y^4 - 560*y^3*z^4 - 3120*y^3*z^3 - 6112*y^3*z^2 - 5168*y^3*z - 1720*y^3 + 168*y^2*z^5 + 1180*y^2*z^4 + 3104*y^2*z^3 + 3960*y^2*z^2 + 2664*y^2*z + 850*y^2 - 28*y*z^6 - 238*y*z^5 - 788*y*z^4 - 1348*y*z^3 - 1374*y*z^2 - 904*y*z - 311*y + 2*z^7 + 20*z^6 + 80*z^5 + 172*z^4 + 236*z^3 + 241*z^2 + 174*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -1, -2, 1 ], [ -1, -1, 1 ], [ -2, -1, 1 ] ], "source_shift": [ 1, 1, 2 ], "target_matrix": [ [ -3, -3, -2 ], [ -1, -1, -1 ], [ 2, 1, 1 ] ], "target_shift": [ 85, 25, -56 ], "inverse_recipe": { "p": 2, "q": 5, "f": "x**2 + x", "g": "x**5 + x**4", "h": "2*y**2 + y*z", "polynomials": { "f": [ [ [ 2, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "g": [ [ [ 5, 0, 0 ], 1 ], [ [ 4, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 2, 0 ], 2 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "Maybe" ], "seconds": [ 1.040090477999911 ], "surface_features": [ 3.3188062428922267, 4.259498314030925, 3.0034605321095067, 0.0, 3.4541341634298983, 4811 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C248", "pair": 248, "family": "counterexample", "mapping": "F(x,y,z)=(27*x^7 + 162*x^6*y - 189*x^6*z - 378*x^6 + 414*x^5*y^2 - 963*x^5*y*z - 1917*x^5*y + 549*x^5*z^2 + 2214*x^5*z + 2124*x^5 + 584*x^4*y^3 - 2031*x^4*y^2*z - 4023*x^4*y^2 + 2310*x^4*y*z^2 + 9252*x^4*y*z + 8850*x^4*y - 863*x^4*z^3 - 5229*x^4*z^2 - 10047*x^4*z - 6296*x^4 + 491*x^3*y^4 - 2269*x^3*y^3*z - 4471*x^3*y^3 + 3861*x^3*y^2*z^2 + 15354*x^3*y^2*z + 14640*x^3*y^2 - 2879*x^3*y*z^3 - 17295*x^3*y*z^2 - 33066*x^3*y*z - 20680*x^3*y + 796*x^3*z^4 + 6412*x^3*z^3 + 18426*x^3*z^2 + 23019*x^3*z + 10722*x^3 + 246*x^2*y^5 - 1416*x^2*y^4*z - 2775*x^2*y^4 + 3204*x^2*y^3*z^2 + 12648*x^2*y^3*z + 12018*x^2*y^3 - 3576*x^2*y^2*z^3 - 21294*x^2*y^2*z^2 - 40503*x^2*y^2*z - 25269*x^2*y^2 + 1974*x^2*y*z^4 + 15744*x^2*y*z^3 + 44952*x^2*y*z^2 + 55950*x^2*y*z + 26001*x^2*y - 432*x^2*z^5 - 4323*x^2*z^4 - 16467*x^2*z^3 - 30681*x^2*z^2 - 28416*x^2*z - 10549*x^2 + 68*x*y^6 - 468*x*y^5*z - 912*x*y^5 + 1320*x*y^4*z^2 + 5172*x*y^4*z + 4896*x*y^4 - 1960*x*y^3*z^3 - 11568*x*y^3*z^2 - 21888*x*y^3*z - 13615*x*y^3 + 1620*x*y^2*z^4 + 12792*x*y^2*z^3 + 36288*x*y^2*z^2 + 44985*x*y^2*z + 20841*x*y^2 - 708*x*y*z^5 - 7008*x*y*z^4 - 26496*x*y*z^3 - 49125*x*y*z^2 - 45321*x*y*z - 16763*x*y + 128*x*z^6 + 1524*x*z^5 + 7200*x*z^4 + 17755*x*z^3 + 24480*x*z^2 + 18032*x*z + 5576*x + 8*y^7 - 64*y^6*z - 124*y^6 + 216*y^5*z^2 + 840*y^5*z + 792*y^5 - 400*y^4*z^3 - 2340*y^4*z^2 - 4404*y^4*z - 2730*y^4 + 440*y^3*z^4 + 3440*y^3*z^3 + 9696*y^3*z^2 + 11968*y^3*z + 5524*y^3 - 288*y^2*z^5 - 2820*y^2*z^4 - 10584*y^2*z^3 - 19524*y^2*z^2 - 17934*y^2*z - 6604*y^2 + 104*y*z^6 + 1224*y*z^5 + 5736*y*z^4 + 14064*y*z^3 + 19296*y*z^2 + 14144*y*z + 4348*y - 16*z^7 - 220*z^6 - 1236*z^5 - 3778*z^4 - 6886*z^3 - 7540*z^2 - 4615*z - 1, -81*x^7 - 486*x^6*y + 567*x^6*z + 1053*x^6 - 1242*x^5*y^2 + 2889*x^5*y*z + 5346*x^5*y - 1647*x^5*z^2 - 6156*x^5*z - 5643*x^5 - 1752*x^4*y^3 + 6093*x^4*y^2*z + 11232*x^4*y^2 - 6930*x^4*y*z^2 - 25758*x^4*y*z - 23553*x^4*y + 2589*x^4*z^3 + 14526*x^4*z^2 + 26658*x^4*z + 16215*x^4 - 1473*x^3*y^4 + 6807*x^3*y^3*z + 12498*x^3*y^3 - 11583*x^3*y^2*z^2 - 42804*x^3*y^2*z - 39033*x^3*y^2 + 8637*x^3*y*z^3 + 48114*x^3*y*z^2 + 87912*x^3*y*z + 53373*x^3*y - 2388*x^3*z^4 - 17808*x^3*z^3 - 48879*x^3*z^2 - 59283*x^3*z - 27057*x^3 - 738*x^2*y^5 + 4248*x^2*y^4*z + 7767*x^2*y^4 - 9612*x^2*y^3*z^2 - 35310*x^2*y^3*z - 32103*x^2*y^3 + 10728*x^2*y^2*z^3 + 59328*x^2*y^2*z^2 + 107910*x^2*y^2*z + 65361*x^2*y^2 - 5922*x^2*y*z^4 - 43794*x^2*y*z^3 - 119511*x^2*y*z^2 - 144438*x^2*y*z - 65772*x^2*y + 1296*x^2*z^5 + 12009*x^2*z^4 + 43704*x^2*z^3 + 79077*x^2*z^2 + 71787*x^2*z + 26280*x^2 - 204*x*y^6 + 1404*x*y^5*z + 2556*x*y^5 - 3960*x*y^4*z^2 - 14460*x*y^4*z - 13104*x*y^4 + 5880*x*y^3*z^3 + 32280*x*y^3*z^2 + 58440*x*y^3*z + 35297*x*y^3 - 4860*x*y^2*z^4 - 35640*x*y^2*z^3 - 96696*x*y^2*z^2 - 116415*x*y^2*z - 52851*x*y^2 + 2124*x*y*z^5 + 19500*x*y*z^4 + 70488*x*y*z^3 + 126939*x*y*z^2 + 114795*x*y*z + 41874*x*y - 384*x*z^6 - 4236*x*z^5 - 19128*x*z^4 - 45821*x*z^3 - 61944*x*z^2 - 45012*x*z - 13789*x - 24*y^7 + 192*y^6*z + 348*y^6 - 648*y^5*z^2 - 2352*y^5*z - 2124*y^5 + 1200*y^4*z^3 + 6540*y^4*z^2 + 11784*y^4*z + 7094*y^4 - 1320*y^3*z^4 - 9600*y^3*z^3 - 25896*y^3*z^2 - 31048*y^3*z - 14044*y^3 + 864*y^2*z^5 + 7860*y^2*z^4 + 28224*y^2*z^3 + 50580*y^2*z^2 + 45546*y^2*z + 16542*y^2 - 312*y*z^6 - 3408*y*z^5 - 15276*y*z^4 - 36392*y*z^3 - 48960*y*z^2 - 35406*y*z - 10785*y + 48*z^7 + 612*z^6 + 3288*z^5 + 9766*z^4 + 17458*z^3 + 18864*z^2 + 11444*z + 1, 27*x^7 + 162*x^6*y - 189*x^6*z - 459*x^6 + 414*x^5*y^2 - 963*x^5*y*z - 2322*x^5*y + 549*x^5*z^2 + 2700*x^5*z + 2853*x^5 + 584*x^4*y^3 - 2031*x^4*y^2*z - 4860*x^4*y^2 + 2310*x^4*y*z^2 + 11250*x^4*y*z + 11847*x^4*y - 863*x^4*z^3 - 6390*x^4*z^2 - 13530*x^4*z - 8996*x^4 + 491*x^3*y^4 - 2269*x^3*y^3*z - 5386*x^3*y^3 + 3861*x^3*y^2*z^2 + 18612*x^3*y^2*z + 19527*x^3*y^2 - 2879*x^3*y*z^3 - 21066*x^3*y*z^2 - 44352*x^3*y*z - 29428*x^3*y + 796*x^3*z^4 + 7840*x^3*z^3 + 24825*x^3*z^2 + 32901*x^3*z + 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27588*y^2*z^2 - 26370*y^2*z - 9994*y^2 + 104*y*z^6 + 1488*y*z^5 + 7668*y*z^4 + 19920*y*z^3 + 28428*y*z^2 + 21434*y*z + 6719*y - 16*z^7 - 268*z^6 - 1656*z^5 - 5362*z^4 - 10162*z^3 - 11440*z^2 - 7136*z - 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -3, -2, 2 ], [ -1, -1, 1 ], [ -1, -1, 2 ] ], "source_shift": [ 2, 2, 2 ], "target_matrix": [ [ -1, -1, -1 ], [ 3, 2, 2 ], [ -1, -2, -1 ] ], "target_shift": [ 1219, -3009, 1905 ], "collision_preimages": [ [ "-2", "15/4", "-1/4" ], [ "-6", "35/2", "8" ], [ "2", "7/2", "5" ] ], "collision_image": [ "4877/4", "-12039/4", "7621/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0172859790000075 ], "surface_features": [ 4.212633141922308, 5.159681466219914, 3.843170464519898, 0.0, 4.355791354628929, 5116 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C078", "pair": 78, "family": "counterexample", "mapping": "F(x,y,z)=(x^7 - 10*x^6*y - 8*x^6*z + 14*x^6 + 42*x^5*y^2 + 69*x^5*y*z - 123*x^5*y + 27*x^5*z^2 - 93*x^5*z + 78*x^5 - 96*x^4*y^3 - 243*x^4*y^2*z + 444*x^4*y^2 - 195*x^4*y*z^2 + 681*x^4*y*z - 579*x^4*y - 50*x^4*z^3 + 255*x^4*z^2 - 423*x^4*z + 230*x^4 + 129*x^3*y^4 + 447*x^3*y^3*z - 843*x^3*y^3 + 552*x^3*y^2*z^2 - 1965*x^3*y^2*z + 1695*x^3*y^2 + 290*x^3*y*z^3 - 1494*x^3*y*z^2 + 2508*x^3*y*z - 1379*x^3*y + 55*x^3*z^4 - 370*x^3*z^3 + 912*x^3*z^2 - 987*x^3*z + 402*x^3 - 102*x^2*y^5 - 453*x^2*y^4*z + 888*x^2*y^4 - 765*x^2*y^3*z^2 + 2793*x^2*y^3*z - 2445*x^2*y^3 - 618*x^2*y^2*z^3 + 3231*x^2*y^2*z^2 - 5493*x^2*y^2*z + 3054*x^2*y^2 - 240*x^2*y*z^4 + 1626*x^2*y*z^3 - 4050*x^2*y*z^2 + 4428*x^2*y*z - 1824*x^2*y - 36*x^2*z^5 + 300*x^2*z^4 - 978*x^2*z^3 + 1581*x^2*z^2 - 1287*x^2*z + 427*x^2 + 44*x*y^6 + 240*x*y^5*z - 492*x*y^5 + 519*x*y^4*z^2 - 1956*x*y^4*z + 1737*x*y^4 + 573*x*y^3*z^3 - 3057*x*y^3*z^2 + 5265*x*y^3*z - 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"certificate_data": { "source_matrix": [ [ 1, -1, -1 ], [ 1, -2, -1 ], [ 1, -1, -2 ] ], "source_shift": [ 1, 1, 2 ], "target_matrix": [ [ 1, 1, 1 ], [ 2, 1, 1 ], [ 1, 2, 1 ] ], "target_shift": [ -72, -102, -118 ], "collision_preimages": [ [ "1/4", "0", "5/4" ], [ "-2", "5/2", "-9/2" ], [ "-11", "-5/2", "-13/2" ] ], "collision_image": [ "-289/4", "-205/2", "-473/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.020317212001828 ], "surface_features": [ 3.1906055753393137, 4.004751155591001, 2.88024177589548, 0.0, 3.2838693365310303, 4771 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B006", "pair": 6, "family": "bijection", "mapping": "F(x,y,z)=(-6*x^7 + 42*x^6*y + 84*x^6*z - 54*x^6 - 126*x^5*y^2 - 504*x^5*y*z + 324*x^5*y - 504*x^5*z^2 + 648*x^5*z - 216*x^5 + 210*x^4*y^3 + 1260*x^4*y^2*z - 810*x^4*y^2 + 2520*x^4*y*z^2 - 3240*x^4*y*z + 1074*x^4*y + 1680*x^4*z^3 - 3240*x^4*z^2 + 2142*x^4*z - 540*x^4 - 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3360*x*y^4*z^2 - 14080*x*y^4*z - 14880*x*y^4 + 2240*x*y^3*z^3 + 14160*x*y^3*z^2 + 30080*x*y^3*z + 21440*x*y^3 - 840*x*y^2*z^4 - 7120*x*y^2*z^3 - 22800*x*y^2*z^2 - 32640*x*y^2*z - 17616*x*y^2 + 168*x*y*z^5 + 1790*x*y*z^4 + 7680*x*y*z^3 + 16560*x*y*z^2 + 17948*x*y*z + 7796*x*y - 14*x*z^6 - 180*x*z^5 - 970*x*z^4 - 2800*x*z^3 - 4570*x*z^2 - 3994*x*z - 1446*x - 256*y^7 + 896*y^6*z + 1856*y^6 - 1344*y^5*z^2 - 5600*y^5*z - 5888*y^5 + 1120*y^4*z^3 + 7040*y^4*z^2 + 14880*y^4*z + 10560*y^4 - 560*y^3*z^4 - 4720*y^3*z^3 - 15040*y^3*z^2 - 21440*y^3*z - 11528*y^3 + 168*y^2*z^5 + 1780*y^2*z^4 + 7600*y^2*z^3 + 16320*y^2*z^2 + 17624*y^2*z + 7626*y^2 - 28*y*z^6 - 358*y*z^5 - 1920*y*z^4 - 5520*y*z^3 - 8978*y*z^2 - 7814*y*z - 2816*y + 2*z^7 + 30*z^6 + 194*z^5 + 700*z^4 + 1524*z^3 + 2002*z^2 + 1460*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -1, -2, 1 ], [ -1, -1, 2 ], [ -1, -1, 1 ] ], "source_shift": [ 2, 1, 1 ], "target_matrix": [ [ 2, 1, 1 ], [ 1, 1, 1 ], [ 2, 2, 3 ] ], "target_shift": [ -432, -227, -445 ], "inverse_recipe": { "p": 5, "q": 2, "f": "x**5", "g": "x**2", "h": "y*z + y + z", "polynomials": { "f": [ [ [ 5, 0, 0 ], 1 ] ], "g": [ [ [ 2, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 0, 1 ], 1 ], [ [ 0, 1, 0 ], 1 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0447859740015701 ], "surface_features": [ 3.6545388377646586, 4.513750150081823, 3.2607866686549762, 0.0, 3.779929651771887, 4901 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C221", "pair": 221, "family": "counterexample", "mapping": "F(x,y,z)=(-16*x^7 - 136*x^6*y - 88*x^6*z - 248*x^6 - 468*x^5*y^2 - 624*x^5*y*z - 1740*x^5*y - 204*x^5*z^2 - 1140*x^5*z - 1476*x^5 - 838*x^4*y^3 - 1734*x^4*y^2*z - 4806*x^4*y^2 - 1170*x^4*y*z^2 - 6468*x^4*y*z - 8394*x^4*y - 258*x^4*z^3 - 2142*x^4*z^2 - 5514*x^4*z - 4534*x^4 - 838*x^3*y^4 - 2398*x^3*y^3*z - 6665*x^3*y^3 - 2514*x^3*y^2*z^2 - 13821*x^3*y^2*z - 18030*x^3*y^2 - 1146*x^3*y*z^3 - 9411*x^3*y*z^2 - 24258*x^3*y*z - 19967*x^3*y - 192*x^3*z^4 - 2103*x^3*z^3 - 8064*x^3*z^2 - 13181*x^3*z - 7880*x^3 - 468*x^2*y^5 - 1734*x^2*y^4*z - 4893*x^2*y^4 - 2514*x^2*y^3*z^2 - 13872*x^2*y^3*z - 18189*x^2*y^3 - 1782*x^2*y^2*z^3 - 14562*x^2*y^2*z^2 - 37638*x^2*y^2*z - 30849*x^2*y^2 - 618*x^2*y*z^4 - 6696*x^2*y*z^3 - 25671*x^2*y*z^2 - 41850*x^2*y*z - 24849*x^2*y - 84*x^2*z^5 - 1137*x^2*z^4 - 5766*x^2*z^3 - 14025*x^2*z^2 - 16632*x^2*z - 7802*x^2 - 136*x*y^6 - 624*x*y^5*z - 1800*x*y^5 - 1170*x*y^4*z^2 - 6546*x*y^4*z - 8544*x*y^4 - 1146*x*y^3*z^3 - 9405*x*y^3*z^2 - 24294*x*y^3*z - 19592*x*y^3 - 618*x*y^2*z^4 - 6669*x*y^2*z^3 - 25554*x*y^2*z^2 - 41232*x*y^2*z - 24042*x*y^2 - 174*x*y*z^5 - 2331*x*y*z^4 - 11796*x*y*z^3 - 28491*x*y*z^2 - 33360*x*y*z - 15389*x*y - 20*x*z^6 - 321*x*z^5 - 2016*x*z^4 - 6473*x*z^3 - 11388*x*z^2 - 10561*x*z - 4090*x - 16*y^7 - 88*y^6*z - 260*y^6 - 204*y^5*z^2 - 1164*y^5*z - 1500*y^5 - 258*y^4*z^3 - 2145*y^4*z^2 - 5496*y^4*z - 4330*y^4 - 192*y^3*z^4 - 2082*y^3*z^3 - 7941*y^3*z^2 - 12574*y^3*z - 7138*y^3 - 84*y^2*z^5 - 1122*y^2*z^4 - 5658*y^2*z^3 - 13473*y^2*z^2 - 15438*y^2*z - 6923*y^2 - 20*y*z^6 - 318*y*z^5 - 1989*y*z^4 - 6318*y*z^3 - 10929*y*z^2 - 9914*y*z - 3721*y - 2*z^7 - 37*z^6 - 276*z^5 - 1095*z^4 - 2536*z^3 - 3479*z^2 - 2648*z - 1, -8*x^7 - 68*x^6*y - 44*x^6*z - 112*x^6 - 234*x^5*y^2 - 312*x^5*y*z - 792*x^5*y - 102*x^5*z^2 - 516*x^5*z - 630*x^5 - 419*x^4*y^3 - 867*x^4*y^2*z - 2208*x^4*y^2 - 585*x^4*y*z^2 - 2952*x^4*y*z - 3615*x^4*y - 129*x^4*z^3 - 972*x^4*z^2 - 2361*x^4*z - 1868*x^4 - 419*x^3*y^4 - 1199*x^3*y^3*z - 3094*x^3*y^3 - 1257*x^3*y^2*z^2 - 6369*x^3*y^2*z - 7842*x^3*y^2 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3516*y^3*z^2 - 5366*y^3*z - 2975*y^3 - 42*y^2*z^5 - 522*y^2*z^4 - 2490*y^2*z^3 - 5715*y^2*z^2 - 6393*y^2*z - 2824*y^2 - 10*y*z^6 - 147*y*z^5 - 870*y*z^4 - 2664*y*z^3 - 4497*y*z^2 - 4015*y*z - 1495*y - z^7 - 17*z^6 - 120*z^5 - 459*z^4 - 1037*z^3 - 1399*z^2 - 1055*z - 1, 16*x^7 + 136*x^6*y + 88*x^6*z + 200*x^6 + 468*x^5*y^2 + 624*x^5*y*z + 1428*x^5*y + 204*x^5*z^2 + 924*x^5*z + 1044*x^5 + 838*x^4*y^3 + 1734*x^4*y^2*z + 4026*x^4*y^2 + 1170*x^4*y*z^2 + 5340*x^4*y*z + 6066*x^4*y + 258*x^4*z^3 + 1746*x^4*z^2 + 3930*x^4*z + 2946*x^4 + 838*x^3*y^4 + 2398*x^3*y^3*z + 5711*x^3*y^3 + 2514*x^3*y^2*z^2 + 11655*x^3*y^2*z + 13338*x^3*y^2 + 1146*x^3*y*z^3 + 7821*x^3*y*z^2 + 17682*x^3*y*z + 13273*x^3*y + 192*x^3*z^4 + 1725*x^3*z^3 + 5796*x^3*z^2 + 8647*x^3*z + 4848*x^3 + 468*x^2*y^5 + 1734*x^2*y^4*z + 4287*x^2*y^4 + 2514*x^2*y^3*z^2 + 11964*x^2*y^3*z + 13779*x^2*y^3 + 1782*x^2*y^2*z^3 + 12366*x^2*y^2*z^2 + 28098*x^2*y^2*z + 21021*x^2*y^2 + 618*x^2*y*z^4 + 5604*x^2*y*z^3 + 18885*x^2*y*z^2 + 28122*x^2*y*z + 15687*x^2*y + 84*x^2*z^5 + 939*x^2*z^4 + 4182*x^2*z^3 + 9297*x^2*z^2 + 10356*x^2*z + 4652*x^2 + 136*x*y^6 + 624*x*y^5*z + 1608*x*y^5 + 1170*x*y^4*z^2 + 5766*x*y^4*z + 6624*x*y^4 + 1146*x*y^3*z^3 + 8163*x*y^3*z^2 + 18570*x*y^3*z + 13707*x*y^3 + 618*x*y^2*z^4 + 5703*x*y^2*z^3 + 19254*x*y^2*z^2 + 28425*x*y^2*z + 15624*x*y^2 + 174*x*y*z^5 + 1965*x*y*z^4 + 8760*x*y*z^3 + 19362*x*y*z^2 + 21348*x*y*z + 9470*x*y + 20*x*z^6 + 267*x*z^5 + 1476*x*z^4 + 4338*x*z^3 + 7182*x*z^2 + 6394*x*z + 2414*x + 16*y^7 + 88*y^6*z + 236*y^6 + 204*y^5*z^2 + 1044*y^5*z + 1188*y^5 + 258*y^4*z^3 + 1899*y^4*z^2 + 4296*y^4*z + 3107*y^4 + 192*y^3*z^4 + 1818*y^3*z^3 + 6123*y^3*z^2 + 8894*y^3*z + 4773*y^3 + 84*y^2*z^5 + 966*y^2*z^4 + 4302*y^2*z^3 + 9393*y^2*z^2 + 10164*y^2*z + 4400*y^2 + 20*y*z^6 + 270*y*z^5 + 1491*y*z^4 + 4342*y*z^3 + 7086*y*z^2 + 6194*y*z + 2282*y + 2*z^7 + 31*z^6 + 204*z^5 + 742*z^4 + 1620*z^3 + 2138*z^2 + 1592*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ 2, 1, 1 ], [ 1, 2, 1 ], [ 1, 1, 1 ] ], "source_shift": [ 1, 2, 2 ], "target_matrix": [ [ -2, -3, -2 ], [ -1, -1, -1 ], [ 2, 1, 1 ] ], "target_shift": [ 863, 343, -517 ], "collision_preimages": [ [ "5/4", "1/4", "-15/4" ], [ "-9/2", "-8", "17" ], [ "-13/2", "-5", "16" ] ], "collision_image": [ "1727/2", "1373/4", "-1035/2" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0675667809991864 ], "surface_features": [ 3.7172813772512714, 4.621695462329279, 3.3377586714934173, 0.0, 3.8551409471416966, 4938 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C023", "pair": 23, "family": "counterexample", "mapping": "F(x,y,z)=(54*x^7 - 297*x^6*y - 459*x^6*z + 459*x^6 + 693*x^5*y^2 + 2115*x^5*y*z - 2133*x^5*y + 1584*x^5*z^2 - 3213*x^5*z + 1467*x^5 - 889*x^4*y^3 - 4017*x^4*y^2*z + 4095*x^4*y^2 - 5934*x^4*y*z^2 + 12195*x^4*y*z - 5658*x^4*y - 2860*x^4*z^3 + 8910*x^4*z^2 - 8403*x^4*z + 2393*x^4 + 677*x^3*y^4 + 4025*x^3*y^3*z - 4159*x^3*y^3 + 8796*x^3*y^2*z^2 - 18375*x^3*y^2*z + 8673*x^3*y^2 + 8360*x^3*y*z^3 - 26568*x^3*y*z^2 + 25512*x^3*y*z - 7407*x^3*y + 2912*x^3*z^4 - 12568*x^3*z^3 + 18432*x^3*z^2 - 10962*x^3*z + 2262*x^3 - 306*x^2*y^5 - 2244*x^2*y^4*z + 2358*x^2*y^4 - 6450*x^2*y^3*z^2 + 13749*x^2*y^3*z - 6609*x^2*y^3 - 9072*x^2*y^2*z^3 + 29538*x^2*y^2*z^2 - 28902*x^2*y^2*z + 8538*x^2*y^2 - 6240*x^2*y*z^4 + 27720*x^2*y*z^3 - 41424*x^2*y*z^2 + 24984*x^2*y*z - 5247*x^2*y - 1680*x^2*z^5 + 9600*x^2*z^4 - 19464*x^2*z^3 + 17892*x^2*z^2 - 7722*x^2*z + 1309*x^2 + 76*x*y^6 + 660*x*y^5*z - 708*x*y^5 + 2340*x*y^4*z^2 - 5112*x*y^4*z + 2505*x*y^4 + 4332*x*y^3*z^3 - 14520*x*y^3*z^2 + 14487*x*y^3*z - 4348*x*y^3 + 4416*x*y^2*z^4 - 20292*x*y^2*z^3 + 30912*x*y^2*z^2 - 18891*x*y^2*z + 4029*x*y^2 + 2352*x*y*z^5 - 13968*x*y*z^4 + 28848*x*y*z^3 - 26820*x*y*z^2 + 11712*x*y*z - 2009*x*y + 512*x*z^6 - 3792*x*z^5 + 9936*x*z^4 - 12456*x*z^3 + 8304*x*z^2 - 2914*x*z + 456*x - 8*y^7 - 80*y^6*z + 88*y^6 - 336*y^5*z^2 + 756*y^5*z - 378*y^5 - 768*y^4*z^3 + 2664*y^4*z^2 - 2712*y^4*z + 826*y^4 - 1032*y^3*z^4 + 4932*y^3*z^3 - 7662*y^3*z^2 + 4741*y^3*z - 1025*y^3 - 816*y^2*z^5 + 5064*y^2*z^4 - 10656*y^2*z^3 + 10014*y^2*z^2 - 4419*y^2*z + 766*y^2 - 352*y*z^6 + 2736*y*z^5 - 7296*y*z^4 + 9232*y*z^3 - 6204*y*z^2 + 2194*y*z - 343*y - 64*z^7 + 608*z^6 - 1968*z^5 + 3136*z^4 - 2840*z^3 + 1528*z^2 - 478*z + 1, 108*x^7 - 594*x^6*y - 918*x^6*z + 918*x^6 + 1386*x^5*y^2 + 4230*x^5*y*z - 4266*x^5*y + 3168*x^5*z^2 - 6426*x^5*z + 2934*x^5 - 1778*x^4*y^3 - 8034*x^4*y^2*z + 8190*x^4*y^2 - 11868*x^4*y*z^2 + 24390*x^4*y*z - 11316*x^4*y - 5720*x^4*z^3 + 17820*x^4*z^2 - 16806*x^4*z + 4840*x^4 + 1354*x^3*y^4 + 8050*x^3*y^3*z - 8318*x^3*y^3 + 17592*x^3*y^2*z^2 - 36750*x^3*y^2*z + 17346*x^3*y^2 + 16720*x^3*y*z^3 - 53136*x^3*y*z^2 + 51024*x^3*y*z - 14949*x^3*y + 5824*x^3*z^4 - 25136*x^3*z^3 + 36864*x^3*z^2 - 22059*x^3*z + 4659*x^3 - 612*x^2*y^5 - 4488*x^2*y^4*z + 4716*x^2*y^4 - 12900*x^2*y^3*z^2 + 27498*x^2*y^3*z - 13218*x^2*y^3 - 18144*x^2*y^2*z^3 + 59076*x^2*y^2*z^2 - 57804*x^2*y^2*z + 17202*x^2*y^2 - 12480*x^2*y*z^4 + 55440*x^2*y*z^3 - 82848*x^2*y*z^2 + 50220*x^2*y*z - 10764*x^2*y - 3360*x^2*z^5 + 19200*x^2*z^4 - 38928*x^2*z^3 + 35910*x^2*z^2 - 15741*x^2*z + 2735*x^2 + 152*x*y^6 + 1320*x*y^5*z - 1416*x*y^5 + 4680*x*y^4*z^2 - 10224*x*y^4*z + 5010*x*y^4 + 8664*x*y^3*z^3 - 29040*x*y^3*z^2 + 28974*x*y^3*z - 8748*x*y^3 + 8832*x*y^2*z^4 - 40584*x*y^2*z^3 + 61824*x*y^2*z^2 - 37938*x*y^2*z + 8238*x*y^2 + 4704*x*y*z^5 - 27936*x*y*z^4 + 57696*x*y*z^3 - 53796*x*y*z^2 + 23820*x*y*z - 4177*x*y + 1024*x*z^6 - 7584*x*z^5 + 19872*x*z^4 - 24964*x*z^3 + 16824*x*z^2 - 6005*x*z + 947*x - 16*y^7 - 160*y^6*z + 176*y^6 - 672*y^5*z^2 + 1512*y^5*z - 756*y^5 - 1536*y^4*z^3 + 5328*y^4*z^2 - 5424*y^4*z + 1660*y^4 - 2064*y^3*z^4 + 9864*y^3*z^3 - 15324*y^3*z^2 + 9514*y^3*z - 2090*y^3 - 1632*y^2*z^5 + 10128*y^2*z^4 - 21312*y^2*z^3 + 20076*y^2*z^2 - 8970*y^2*z + 1586*y^2 - 704*y*z^6 + 5472*y*z^5 - 14592*y*z^4 + 18496*y*z^3 - 12552*y*z^2 + 4508*y*z - 710*y - 128*z^7 + 1216*z^6 - 3936*z^5 + 6280*z^4 - 5732*z^3 + 3122*z^2 - 983*z + 1, 54*x^7 - 297*x^6*y - 459*x^6*z + 621*x^6 + 693*x^5*y^2 + 2115*x^5*y*z - 2862*x^5*y + 1584*x^5*z^2 - 4266*x^5*z + 2196*x^5 - 889*x^4*y^3 - 4017*x^4*y^2*z + 5445*x^4*y^2 - 5934*x^4*y*z^2 + 16029*x^4*y*z - 8412*x^4*y - 2860*x^4*z^3 + 11556*x^4*z^2 - 12372*x^4*z + 3878*x^4 + 677*x^3*y^4 + 4025*x^3*y^3*z - 5476*x^3*y^3 + 8796*x^3*y^2*z^2 - 23892*x^3*y^2*z + 12804*x^3*y^2 + 8360*x^3*y*z^3 - 34056*x^3*y*z^2 + 37284*x^3*y*z - 11862*x^3*y + 2912*x^3*z^4 - 15856*x^3*z^3 + 26640*x^3*z^2 - 17253*x^3*z + 3951*x^3 - 306*x^2*y^5 - 2244*x^2*y^4*z + 3072*x^2*y^4 - 6450*x^2*y^3*z^2 + 17673*x^2*y^3*z - 9687*x^2*y^3 - 9072*x^2*y^2*z^3 + 37404*x^2*y^2*z^2 - 41925*x^2*y^2*z + 13527*x^2*y^2 - 6240*x^2*y*z^4 + 34536*x^2*y*z^3 - 59424*x^2*y*z^2 + 38958*x^2*y*z - 9015*x^2*y - 1680*x^2*z^5 + 11760*x^2*z^4 - 27600*x^2*z^3 + 27498*x^2*z^2 - 12915*x^2*z + 2404*x^2 + 76*x*y^6 + 660*x*y^5*z - 912*x*y^5 + 2340*x*y^4*z^2 - 6492*x*y^4*z + 3645*x*y^4 + 4332*x*y^3*z^3 - 18156*x*y^3*z^2 + 20859*x*y^3*z - 6821*x*y^3 + 4416*x*y^2*z^4 - 24960*x*y^2*z^3 + 44016*x*y^2*z^2 - 29202*x*y^2*z + 6822*x*y^2 + 2352*x*y*z^5 - 16896*x*y*z^4 + 40608*x*y*z^3 - 40896*x*y*z^2 + 19356*x*y*z - 3620*x*y + 512*x*z^6 - 4512*x*z^5 + 13824*x*z^4 - 18748*x*z^3 + 13440*x*z^2 - 5071*x*z + 825*x - 8*y^7 - 80*y^6*z + 112*y^6 - 336*y^5*z^2 + 948*y^5*z - 546*y^5 - 768*y^4*z^3 + 3288*y^4*z^2 - 3876*y^4*z + 1284*y^4 - 1032*y^3*z^4 + 5988*y^3*z^3 - 10830*y^3*z^2 + 7269*y^3*z - 1713*y^3 - 816*y^2*z^5 + 6048*y^2*z^4 - 14892*y^2*z^3 + 15156*y^2*z^2 - 7224*y^2*z + 1357*y^2 - 352*y*z^6 + 3216*y*z^5 - 10080*y*z^4 + 13800*y*z^3 - 9948*y*z^2 + 3766*y*z - 611*y - 64*z^7 + 704*z^6 - 2688*z^5 + 4632*z^4 - 4476*z^3 + 2554*z^2 - 825*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ 3, -2, -2 ], [ 1, -1, -2 ], [ 2, -1, -1 ] ], "source_shift": [ 1, 1, 2 ], "target_matrix": [ [ 1, 1, 2 ], [ 2, 2, 3 ], [ 1, 2, 1 ] ], "target_shift": [ -69, -142, -118 ], "collision_preimages": [ [ "-7/2", "-7", "9/4" ], [ "9", "31/2", "-2" ], [ "11", "49/2", "-7" ] ], "collision_image": [ "-277/4", "-285/2", "-473/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0365385330005665 ], "surface_features": [ 4.014289936525114, 4.918282028754778, 3.7068457481036043, 0.0, 4.116586023296126, 5070 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B018", "pair": 18, "family": "bijection", "mapping": "F(x,y,z)=(128*x^7 + 448*x^6*y - 448*x^6*z - 1152*x^6 + 672*x^5*y^2 - 1344*x^5*y*z - 3552*x^5*y + 672*x^5*z^2 + 3488*x^5*z + 4504*x^5 + 560*x^4*y^3 - 1680*x^4*y^2*z - 4560*x^4*y^2 + 1680*x^4*y*z^2 + 8960*x^4*y*z + 11868*x^4*y - 560*x^4*z^3 - 4400*x^4*z^2 - 11468*x^4*z - 9924*x^4 + 280*x^3*y^4 - 1120*x^3*y^3*z - 3120*x^3*y^3 + 1680*x^3*y^2*z^2 + 9200*x^3*y^2*z + 12494*x^3*y^2 - 1120*x^3*y*z^3 - 9040*x^3*y*z^2 - 24164*x^3*y*z - 21428*x^3*y + 280*x^3*z^4 + 2960*x^3*z^3 + 11678*x^3*z^2 + 20404*x^3*z + 13291*x^3 + 84*x^2*y^5 - 420*x^2*y^4*z - 1200*x^2*y^4 + 840*x^2*y^3*z^2 + 4720*x^2*y^3*z + 6569*x^2*y^3 - 840*x^2*y^2*z^3 - 6960*x^2*y^2*z^2 - 19071*x^2*y^2*z - 17325*x^2*y^2 + 420*x^2*y*z^4 + 4560*x^2*y*z^3 + 18447*x^2*y*z^2 + 33026*x^2*y*z + 22048*x^2*y - 84*x^2*z^5 - 1120*x^2*z^4 - 5945*x^2*z^3 - 15729*x^2*z^2 - 20702*x^2*z - 10804*x^2 + 14*x*y^6 - 84*x*y^5*z - 246*x*y^5 + 210*x*y^4*z^2 + 1210*x*y^4*z + 1725*x*y^4 - 280*x*y^3*z^3 - 2380*x*y^3*z^2 - 6682*x*y^3*z - 6216*x*y^3 + 210*x*y^2*z^4 + 2340*x*y^2*z^3 + 9702*x*y^2*z^2 + 17792*x*y^2*z + 12175*x*y^2 - 84*x*y*z^5 - 1150*x*y*z^4 - 6258*x*y*z^3 - 16964*x*y*z^2 - 22888*x*y*z - 12260*x*y + 14*x*z^6 + 226*x*z^5 + 1513*x*z^4 + 5388*x*z^3 + 10748*x*z^2 + 11348*x*z + 4929*x + y^7 - 7*y^6*z - 21*y^6 + 21*y^5*z^2 + 124*y^5*z + 181*y^5 - 35*y^4*z^3 - 305*y^4*z^2 - 877*y^4*z - 835*y^4 + 35*y^3*z^4 + 400*y^3*z^3 + 1699*y^3*z^2 + 3190*y^3*z + 2238*y^3 - 21*y^2*z^5 - 295*y^2*z^4 - 1645*y^2*z^3 - 4567*y^2*z^2 - 6318*y^2*z - 3477*y^2 + 7*y*z^6 + 116*y*z^5 + 796*y*z^4 + 2904*y*z^3 + 5940*y*z^2 + 6442*y*z + 2880*y - z^7 - 19*z^6 - 154*z^5 - 692*z^4 - 1860*z^3 - 2981*z^2 - 2623*z - 1, 128*x^7 + 448*x^6*y - 448*x^6*z - 1152*x^6 + 672*x^5*y^2 - 1344*x^5*y*z - 3552*x^5*y + 672*x^5*z^2 + 3488*x^5*z + 4504*x^5 + 560*x^4*y^3 - 1680*x^4*y^2*z - 4560*x^4*y^2 + 1680*x^4*y*z^2 + 8960*x^4*y*z + 11868*x^4*y - 560*x^4*z^3 - 4400*x^4*z^2 - 11468*x^4*z - 9924*x^4 + 280*x^3*y^4 - 1120*x^3*y^3*z - 3120*x^3*y^3 + 1680*x^3*y^2*z^2 + 9200*x^3*y^2*z + 12494*x^3*y^2 - 1120*x^3*y*z^3 - 9040*x^3*y*z^2 - 24164*x^3*y*z - 21428*x^3*y + 280*x^3*z^4 + 2960*x^3*z^3 + 11678*x^3*z^2 + 20404*x^3*z + 13307*x^3 + 84*x^2*y^5 - 420*x^2*y^4*z - 1200*x^2*y^4 + 840*x^2*y^3*z^2 + 4720*x^2*y^3*z + 6569*x^2*y^3 - 840*x^2*y^2*z^3 - 6960*x^2*y^2*z^2 - 19071*x^2*y^2*z - 17325*x^2*y^2 + 420*x^2*y*z^4 + 4560*x^2*y*z^3 + 18447*x^2*y*z^2 + 33026*x^2*y*z + 22072*x^2*y - 84*x^2*z^5 - 1120*x^2*z^4 - 5945*x^2*z^3 - 15729*x^2*z^2 - 20726*x^2*z - 10860*x^2 + 14*x*y^6 - 84*x*y^5*z - 246*x*y^5 + 210*x*y^4*z^2 + 1210*x*y^4*z + 1725*x*y^4 - 280*x*y^3*z^3 - 2380*x*y^3*z^2 - 6682*x*y^3*z - 6216*x*y^3 + 210*x*y^2*z^4 + 2340*x*y^2*z^3 + 9702*x*y^2*z^2 + 17792*x*y^2*z + 12187*x*y^2 - 84*x*y*z^5 - 1150*x*y*z^4 - 6258*x*y*z^3 - 16964*x*y*z^2 - 22912*x*y*z - 12316*x*y + 14*x*z^6 + 226*x*z^5 + 1513*x*z^4 + 5388*x*z^3 + 10760*x*z^2 + 11404*x*z + 4999*x + y^7 - 7*y^6*z - 21*y^6 + 21*y^5*z^2 + 124*y^5*z + 181*y^5 - 35*y^4*z^3 - 305*y^4*z^2 - 877*y^4*z - 835*y^4 + 35*y^3*z^4 + 400*y^3*z^3 + 1699*y^3*z^2 + 3190*y^3*z + 2240*y^3 - 21*y^2*z^5 - 295*y^2*z^4 - 1645*y^2*z^3 - 4567*y^2*z^2 - 6324*y^2*z - 3491*y^2 + 7*y*z^6 + 116*y*z^5 + 796*y*z^4 + 2904*y*z^3 + 5946*y*z^2 + 6470*y*z + 2916*y - z^7 - 19*z^6 - 154*z^5 - 692*z^4 - 1862*z^3 - 2995*z^2 - 2659*z - 1, -256*x^7 - 896*x^6*y + 896*x^6*z + 2304*x^6 - 1344*x^5*y^2 + 2688*x^5*y*z + 7104*x^5*y - 1344*x^5*z^2 - 6976*x^5*z - 9008*x^5 - 1120*x^4*y^3 + 3360*x^4*y^2*z + 9120*x^4*y^2 - 3360*x^4*y*z^2 - 17920*x^4*y*z - 23736*x^4*y + 1120*x^4*z^3 + 8800*x^4*z^2 + 22936*x^4*z + 19848*x^4 - 560*x^3*y^4 + 2240*x^3*y^3*z + 6240*x^3*y^3 - 3360*x^3*y^2*z^2 - 18400*x^3*y^2*z - 24988*x^3*y^2 + 2240*x^3*y*z^3 + 18080*x^3*y*z^2 + 48328*x^3*y*z + 42856*x^3*y - 560*x^3*z^4 - 5920*x^3*z^3 - 23356*x^3*z^2 - 40808*x^3*z - 26614*x^3 - 168*x^2*y^5 + 840*x^2*y^4*z + 2400*x^2*y^4 - 1680*x^2*y^3*z^2 - 9440*x^2*y^3*z - 13138*x^2*y^3 + 1680*x^2*y^2*z^3 + 13920*x^2*y^2*z^2 + 38142*x^2*y^2*z + 34650*x^2*y^2 - 840*x^2*y*z^4 - 9120*x^2*y*z^3 - 36894*x^2*y*z^2 - 66052*x^2*y*z - 44144*x^2*y + 168*x^2*z^5 + 2240*x^2*z^4 + 11890*x^2*z^3 + 31458*x^2*z^2 + 41452*x^2*z + 21724*x^2 - 28*x*y^6 + 168*x*y^5*z + 492*x*y^5 - 420*x*y^4*z^2 - 2420*x*y^4*z - 3450*x*y^4 + 560*x*y^3*z^3 + 4760*x*y^3*z^2 + 13364*x*y^3*z + 12432*x*y^3 - 420*x*y^2*z^4 - 4680*x*y^2*z^3 - 19404*x*y^2*z^2 - 35584*x*y^2*z - 24374*x*y^2 + 168*x*y*z^5 + 2300*x*y*z^4 + 12516*x*y*z^3 + 33928*x*y*z^2 + 45824*x*y*z + 24636*x*y - 28*x*z^6 - 452*x*z^5 - 3026*x*z^4 - 10776*x*z^3 - 21520*x*z^2 - 22812*x*z - 10009*x - 2*y^7 + 14*y^6*z + 42*y^6 - 42*y^5*z^2 - 248*y^5*z - 362*y^5 + 70*y^4*z^3 + 610*y^4*z^2 + 1754*y^4*z + 1670*y^4 - 70*y^3*z^4 - 800*y^3*z^3 - 3398*y^3*z^2 - 6380*y^3*z - 4480*y^3 + 42*y^2*z^5 + 590*y^2*z^4 + 3290*y^2*z^3 + 9134*y^2*z^2 + 12648*y^2*z + 6983*y^2 - 14*y*z^6 - 232*y*z^5 - 1592*y*z^4 - 5808*y*z^3 - 11892*y*z^2 - 12942*y*z - 5839*y + 2*z^7 + 38*z^6 + 308*z^5 + 1384*z^4 + 3724*z^3 + 5991*z^2 + 5324*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -2, -1, 1 ], [ -1, -2, 1 ], [ -1, -1, 1 ] ], "source_shift": [ 2, 2, 1 ], "target_matrix": [ [ -1, -1, -2 ], [ -1, -1, -1 ], [ 2, 3, 2 ] ], "target_shift": [ 973, 1003, -2015 ], "inverse_recipe": { "p": 2, "q": 3, "f": "x**2 + x", "g": "x**3 + x**2 + x", "h": "y**2*z + y**2", "polynomials": { "f": [ [ [ 2, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "g": [ [ [ 3, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "h": [ [ [ 0, 2, 1 ], 1 ], [ [ 0, 2, 0 ], 1 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0642983539983106 ], "surface_features": [ 3.827056975732771, 4.819885972227815, 3.376576957056512, 0.0, 3.988395950685406, 4936 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C158", "pair": 158, "family": "counterexample", "mapping": "F(x,y,z)=(-8*x^7 - 68*x^6*y + 44*x^6*z + 128*x^6 - 234*x^5*y^2 + 312*x^5*y*z + 888*x^5*y - 102*x^5*z^2 - 588*x^5*z - 738*x^5 - 419*x^4*y^3 + 867*x^4*y^2*z + 2424*x^4*y^2 - 585*x^4*y*z^2 - 3300*x^4*y*z - 4167*x^4*y + 129*x^4*z^3 + 1104*x^4*z^2 + 2769*x^4*z + 2216*x^4 - 419*x^3*y^4 + 1199*x^3*y^3*z + 3326*x^3*y^3 - 1257*x^3*y^2*z^2 - 6975*x^3*y^2*z - 8937*x^3*y^2 + 573*x^3*y*z^3 + 4800*x^3*y*z^2 + 12123*x^3*y*z + 9835*x^3*y - 96*x^3*z^4 - 1083*x^3*z^3 - 4068*x^3*z^2 - 6520*x^3*z - 3847*x^3 - 234*x^2*y^5 + 867*x^2*y^4*z + 2424*x^2*y^4 - 1257*x^2*y^3*z^2 - 6936*x^2*y^3*z - 9063*x^2*y^3 + 891*x^2*y^2*z^3 + 7353*x^2*y^2*z^2 + 18819*x^2*y^2*z + 15429*x^2*y^2 - 309*x^2*y*z^4 - 3414*x^2*y*z^3 - 12912*x^2*y*z^2 - 20901*x^2*y*z - 12396*x^2*y + 42*x^2*z^5 + 585*x^2*z^4 + 2922*x^2*z^3 + 7017*x^2*z^2 + 8262*x^2*z + 3856*x^2 - 68*x*y^6 + 312*x*y^5*z + 888*x*y^5 - 585*x*y^4*z^2 - 3252*x*y^4*z - 4284*x*y^4 + 573*x*y^3*z^3 + 4710*x*y^3*z^2 + 12198*x*y^3*z + 9937*x*y^3 - 309*x*y^2*z^4 - 3369*x*y^2*z^3 - 12867*x*y^2*z^2 - 20853*x*y^2*z - 12249*x*y^2 + 87*x*y*z^5 + 1188*x*y*z^4 + 5967*x*y*z^3 + 14388*x*y*z^2 + 16887*x*y*z + 7825*x*y - 10*x*z^6 - 165*x*z^5 - 1026*x*z^4 - 3271*x*z^3 - 5739*x*z^2 - 5318*x*z - 2056*x - 8*y^7 + 44*y^6*z + 128*y^6 - 102*y^5*z^2 - 576*y^5*z - 756*y^5 + 129*y^4*z^3 + 1068*y^4*z^2 + 2772*y^4*z + 2219*y^4 - 96*y^3*z^4 - 1044*y^3*z^3 - 4011*y^3*z^2 - 6428*y^3*z - 3692*y^3 + 42*y^2*z^5 + 567*y^2*z^4 + 2865*y^2*z^3 + 6873*y^2*z^2 + 7944*y^2*z + 3592*y^2 - 10*y*z^6 - 162*y*z^5 - 1011*y*z^4 - 3219*y*z^3 - 5595*y*z^2 - 5104*y*z - 1923*y + z^7 + 19*z^6 + 141*z^5 + 558*z^4 + 1293*z^3 + 1777*z^2 + 1353*z + 1, -16*x^7 - 136*x^6*y + 88*x^6*z + 232*x^6 - 468*x^5*y^2 + 624*x^5*y*z + 1620*x^5*y - 204*x^5*z^2 - 1068*x^5*z - 1284*x^5 - 838*x^4*y^3 + 1734*x^4*y^2*z + 4458*x^4*y^2 - 1170*x^4*y*z^2 - 6036*x^4*y*z - 7302*x^4*y + 258*x^4*z^3 + 2010*x^4*z^2 + 4830*x^4*z + 3766*x^4 - 838*x^3*y^4 + 2398*x^3*y^3*z + 6175*x^3*y^3 - 2514*x^3*y^2*z^2 - 12867*x^3*y^2*z - 15786*x^3*y^2 + 1146*x^3*y*z^3 + 8805*x^3*y*z^2 + 21306*x^3*y*z + 16829*x^3*y - 192*x^3*z^4 - 1977*x^3*z^3 - 7116*x^3*z^2 - 11111*x^3*z - 6440*x^3 - 468*x^2*y^5 + 1734*x^2*y^4*z + 4545*x^2*y^4 - 2514*x^2*y^3*z^2 - 12918*x^2*y^3*z - 16140*x^2*y^3 + 1782*x^2*y^2*z^3 + 13608*x^2*y^2*z^2 + 33345*x^2*y^2*z + 26592*x^2*y^2 - 618*x^2*y*z^4 - 6282*x^2*y*z^3 - 22764*x^2*y*z^2 - 35874*x^2*y*z - 20898*x^2*y + 84*x^2*z^5 + 1071*x^2*z^4 + 5127*x^2*z^3 + 11994*x^2*z^2 + 13875*x^2*z + 6395*x^2 - 136*x*y^6 + 624*x*y^5*z + 1680*x*y^5 - 1170*x*y^4*z^2 - 6114*x*y^4*z - 7692*x*y^4 + 1146*x*y^3*z^3 + 8799*x*y^3*z^2 + 21792*x*y^3*z + 17264*x*y^3 - 618*x*y^2*z^4 - 6255*x*y^2*z^3 - 22872*x*y^2*z^2 - 36066*x*y^2*z - 20820*x*y^2 + 174*x*y*z^5 + 2193*x*y*z^4 + 10554*x*y*z^3 + 24777*x*y*z^2 + 28578*x*y*z + 13091*x*y - 20*x*z^6 - 303*x*z^5 - 1806*x*z^4 - 5609*x*z^3 - 9672*x*z^2 - 8857*x*z - 3400*x - 16*y^7 + 88*y^6*z + 244*y^6 - 204*y^5*z^2 - 1092*y^5*z - 1368*y^5 + 258*y^4*z^3 + 2013*y^4*z^2 + 4992*y^4*z + 3886*y^4 - 192*y^3*z^4 - 1956*y^3*z^3 - 7188*y^3*z^2 - 11206*y^3*z - 6328*y^3 + 84*y^2*z^5 + 1056*y^2*z^4 + 5109*y^2*z^3 + 11928*y^2*z^2 + 13554*y^2*z + 6065*y^2 - 20*y*z^6 - 300*y*z^5 - 1794*y*z^4 - 5562*y*z^3 - 9504*y*z^2 - 8576*y*z - 3214*y + 2*z^7 + 35*z^6 + 249*z^5 + 960*z^4 + 2187*z^3 + 2972*z^2 + 2249*z + 1, -16*x^7 - 136*x^6*y + 88*x^6*z + 184*x^6 - 468*x^5*y^2 + 624*x^5*y*z + 1308*x^5*y - 204*x^5*z^2 - 852*x^5*z - 900*x^5 - 838*x^4*y^3 + 1734*x^4*y^2*z + 3678*x^4*y^2 - 1170*x^4*y*z^2 - 4908*x^4*y*z - 5238*x^4*y + 258*x^4*z^3 + 1614*x^4*z^2 + 3414*x^4*z + 2442*x^4 - 838*x^3*y^4 + 2398*x^3*y^3*z + 5221*x^3*y^3 - 2514*x^3*y^2*z^2 - 10701*x^3*y^2*z - 11610*x^3*y^2 + 1146*x^3*y*z^3 + 7215*x^3*y*z^2 + 15426*x^3*y*z + 11167*x^3*y - 192*x^3*z^4 - 1599*x^3*z^3 - 5076*x^3*z^2 - 7273*x^3*z - 3964*x^3 - 468*x^2*y^5 + 1734*x^2*y^4*z + 3939*x^2*y^4 - 2514*x^2*y^3*z^2 - 11010*x^2*y^3*z - 12168*x^2*y^3 + 1782*x^2*y^2*z^3 + 11412*x^2*y^2*z^2 + 24759*x^2*y^2*z + 18078*x^2*y^2 - 618*x^2*y*z^4 - 5190*x^2*y*z^3 - 16644*x^2*y*z^2 - 24054*x^2*y*z - 13182*x^2*y + 84*x^2*z^5 + 873*x^2*z^4 + 3693*x^2*z^3 + 7932*x^2*z^2 + 8637*x^2*z + 3815*x^2 - 136*x*y^6 + 624*x*y^5*z + 1488*x*y^5 - 1170*x*y^4*z^2 - 5334*x*y^4*z - 5940*x*y^4 + 1146*x*y^3*z^3 + 7557*x*y^3*z^2 + 16584*x*y^3*z + 12051*x*y^3 - 618*x*y^2*z^4 - 5289*x*y^2*z^3 - 17148*x*y^2*z^2 - 24807*x*y^2*z - 13512*x*y^2 + 174*x*y*z^5 + 1827*x*y*z^4 + 7794*x*y*z^3 + 16800*x*y*z^2 + 18270*x*y*z + 8048*x*y - 20*x*z^6 - 249*x*z^5 - 1314*x*z^4 - 3750*x*z^3 - 6096*x*z^2 - 5362*x*z - 2006*x - 16*y^7 + 88*y^6*z + 220*y^6 - 204*y^5*z^2 - 972*y^5*z - 1080*y^5 + 258*y^4*z^3 + 1767*y^4*z^2 + 3888*y^4*z + 2783*y^4 - 192*y^3*z^4 - 1692*y^3*z^3 - 5520*y^3*z^2 - 7910*y^3*z - 4226*y^3 + 84*y^2*z^5 + 900*y^2*z^4 + 3867*y^2*z^3 + 8298*y^2*z^2 + 8913*y^2*z + 3851*y^2 - 20*y*z^6 - 252*y*z^5 - 1338*y*z^4 - 3814*y*z^3 - 6156*y*z^2 - 5354*y*z - 1970*y + 2*z^7 + 29*z^6 + 183*z^5 + 649*z^4 + 1396*z^3 + 1826*z^2 + 1352*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -2, -1, 1 ], [ -1, -2, 1 ], [ -1, -1, 1 ] ], "source_shift": [ 1, 2, 1 ], "target_matrix": [ [ 1, 2, 1 ], [ 2, 3, 2 ], [ 2, 1, 1 ] ], "target_shift": [ -439, -730, -437 ], "collision_preimages": [ [ "-1/4", "3/4", "-3/4" ], [ "11/2", "9", "20" ], [ "15/2", "6", "19" ] ], "collision_image": [ "-1757/4", "-1461/2", "-875/2" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.044965183999011 ], "surface_features": [ 3.6905726007544644, 4.55709797835226, 3.326643036102635, 0.0, 3.808351187321654, 4937 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B007", "pair": 7, "family": "bijection", "mapping": "F(x,y,z)=(2*x^7 - 27*x^6*y - 15*x^6*z + 27*x^6 + 156*x^5*y^2 + 174*x^5*y*z - 312*x^5*y + 48*x^5*z^2 - 174*x^5*z + 156*x^5 - 500*x^4*y^3 - 840*x^4*y^2*z + 1500*x^4*y^2 - 465*x^4*y*z^2 + 1680*x^4*y*z - 1500*x^4*y - 85*x^4*z^3 + 465*x^4*z^2 - 840*x^4*z + 502*x^4 + 960*x^3*y^4 + 2160*x^3*y^3*z - 3840*x^3*y^3 + 1800*x^3*y^2*z^2 - 6480*x^3*y^2*z + 5760*x^3*y^2 + 660*x^3*y*z^3 - 3600*x^3*y*z^2 + 6480*x^3*y*z - 3855*x^3*y + 90*x^3*z^4 - 660*x^3*z^3 + 1800*x^3*z^2 - 2169*x^3*z + 975*x^3 - 1104*x^2*y^5 - 3120*x^2*y^4*z + 5520*x^2*y^4 - 3480*x^2*y^3*z^2 + 12480*x^2*y^3*z - 11040*x^2*y^3 - 1920*x^2*y^2*z^3 + 10440*x^2*y^2*z^2 - 18720*x^2*y^2*z + 11082*x^2*y^2 - 525*x^2*y*z^4 + 3840*x^2*y*z^3 - 10440*x^2*y*z^2 + 12531*x^2*y*z - 5604*x^2*y - 57*x^2*z^5 + 525*x^2*z^4 - 1920*x^2*z^3 + 3495*x^2*z^2 - 3171*x^2*z + 1152*x^2 + 704*x*y^6 + 2400*x*y^5*z - 4224*x*y^5 + 3360*x*y^4*z^2 - 12000*x*y^4*z + 10560*x*y^4 + 2480*x*y^3*z^3 - 13440*x*y^3*z^2 + 24000*x*y^3*z - 14132*x*y^3 + 1020*x*y^2*z^4 - 7440*x*y^2*z^3 + 20160*x*y^2*z^2 - 24096*x*y^2*z + 10716*x*y^2 + 222*x*y*z^5 - 2040*x*y*z^4 + 7440*x*y*z^3 - 13497*x*y*z^2 + 12192*x*y*z - 4399*x*y + 20*x*z^6 - 222*x*z^5 + 1020*x*z^4 - 2491*x*z^3 + 3417*x*z^2 - 2511*x*z + 778*x - 192*y^7 - 768*y^6*z + 1344*y^6 - 1296*y^5*z^2 + 4608*y^5*z - 4032*y^5 - 1200*y^4*z^3 + 6480*y^4*z^2 - 11520*y^4*z + 6744*y^4 - 660*y^3*z^4 + 4800*y^3*z^3 - 12960*y^3*z^2 + 15420*y^3*z - 6816*y^3 - 216*y^2*z^5 + 1980*y^2*z^4 - 7200*y^2*z^3 + 13014*y^2*z^2 - 11700*y^2*z + 4191*y^2 - 39*y*z^6 + 432*y*z^5 - 1980*y*z^4 + 4821*y*z^3 - 6588*y*z^2 + 4812*y*z - 1475*y - 3*z^7 + 39*z^6 - 216*z^5 + 663*z^4 - 1221*z^3 + 1359*z^2 - 856*z + 1, 2*x^7 - 27*x^6*y - 15*x^6*z + 25*x^6 + 156*x^5*y^2 + 174*x^5*y*z - 288*x^5*y + 48*x^5*z^2 - 162*x^5*z + 132*x^5 - 500*x^4*y^3 - 840*x^4*y^2*z + 1380*x^4*y^2 - 465*x^4*y*z^2 + 1560*x^4*y*z - 1260*x^4*y - 85*x^4*z^3 + 435*x^4*z^2 - 720*x^4*z + 382*x^4 + 960*x^3*y^4 + 2160*x^3*y^3*z - 3520*x^3*y^3 + 1800*x^3*y^2*z^2 - 6000*x^3*y^2*z + 4800*x^3*y^2 + 660*x^3*y*z^3 - 3360*x^3*y*z^2 + 5520*x^3*y*z - 2895*x^3*y + 90*x^3*z^4 - 620*x^3*z^3 + 1560*x^3*z^2 - 1689*x^3*z + 653*x^3 - 1104*x^2*y^5 - 3120*x^2*y^4*z + 5040*x^2*y^4 - 3480*x^2*y^3*z^2 + 11520*x^2*y^3*z - 9120*x^2*y^3 - 1920*x^2*y^2*z^3 + 9720*x^2*y^2*z^2 - 15840*x^2*y^2*z + 8202*x^2*y^2 - 525*x^2*y*z^4 + 3600*x^2*y*z^3 - 9000*x^2*y*z^2 + 9651*x^2*y*z - 3672*x^2*y - 57*x^2*z^5 + 495*x^2*z^4 - 1680*x^2*z^3 + 2775*x^2*z^2 - 2205*x^2*z + 660*x^2 + 704*x*y^6 + 2400*x*y^5*z - 3840*x*y^5 + 3360*x*y^4*z^2 - 11040*x*y^4*z + 8640*x*y^4 + 2480*x*y^3*z^3 - 12480*x*y^3*z^2 + 20160*x*y^3*z - 10292*x*y^3 + 1020*x*y^2*z^4 - 6960*x*y^2*z^3 + 17280*x*y^2*z^2 - 18336*x*y^2*z + 6852*x*y^2 + 222*x*y*z^5 - 1920*x*y*z^4 + 6480*x*y*z^3 - 10617*x*y*z^2 + 8328*x*y*z - 2431*x*y + 20*x*z^6 - 210*x*z^5 + 900*x*z^4 - 2011*x*z^3 + 2451*x*z^2 - 1527*x*z + 364*x - 192*y^7 - 768*y^6*z + 1216*y^6 - 1296*y^5*z^2 + 4224*y^5*z - 3264*y^5 - 1200*y^4*z^3 + 6000*y^4*z^2 - 9600*y^4*z + 4824*y^4 - 660*y^3*z^4 + 4480*y^3*z^3 - 11040*y^3*z^2 + 11580*y^3*z - 4240*y^3 - 216*y^2*z^5 + 1860*y^2*z^4 - 6240*y^2*z^3 + 10134*y^2*z^2 - 7836*y^2*z + 2223*y^2 - 39*y*z^6 + 408*y*z^5 - 1740*y*z^4 + 3861*y*z^3 - 4656*y*z^2 + 2844*y*z - 649*y - 3*z^7 + 37*z^6 - 192*z^5 + 543*z^4 - 899*z^3 + 867*z^2 - 442*z + 1, 4*x^7 - 54*x^6*y - 30*x^6*z + 56*x^6 + 312*x^5*y^2 + 348*x^5*y*z - 648*x^5*y + 96*x^5*z^2 - 360*x^5*z + 336*x^5 - 1000*x^4*y^3 - 1680*x^4*y^2*z + 3120*x^4*y^2 - 930*x^4*y*z^2 + 3480*x^4*y*z - 3240*x^4*y - 170*x^4*z^3 + 960*x^4*z^2 - 1800*x^4*z + 1124*x^4 + 1920*x^3*y^4 + 4320*x^3*y^3*z - 8000*x^3*y^3 + 3600*x^3*y^2*z^2 - 13440*x^3*y^2*z + 12480*x^3*y^2 + 1320*x^3*y*z^3 - 7440*x^3*y*z^2 + 13920*x^3*y*z - 8670*x^3*y + 180*x^3*z^4 - 1360*x^3*z^3 + 3840*x^3*z^2 - 4818*x^3*z + 2272*x^3 - 2208*x^2*y^5 - 6240*x^2*y^4*z + 11520*x^2*y^4 - 6960*x^2*y^3*z^2 + 25920*x^2*y^3*z - 24000*x^2*y^3 - 3840*x^2*y^2*z^3 + 21600*x^2*y^2*z^2 - 40320*x^2*y^2*z + 25044*x^2*y^2 - 1050*x^2*y*z^4 + 7920*x^2*y*z^3 - 22320*x^2*y*z^2 + 27942*x^2*y*z - 13140*x^2*y - 114*x^2*z^5 + 1080*x^2*z^4 - 4080*x^2*z^3 + 7710*x^2*z^2 - 7308*x^2*z + 2796*x^2 + 1408*x*y^6 + 4800*x*y^5*z - 8832*x*y^5 + 6720*x*y^4*z^2 - 24960*x*y^4*z + 23040*x*y^4 + 4960*x*y^3*z^3 - 27840*x*y^3*z^2 + 51840*x*y^3*z - 32104*x*y^3 + 2040*x*y^2*z^4 - 15360*x*y^2*z^3 + 43200*x*y^2*z^2 - 53952*x*y^2*z + 25296*x*y^2 + 444*x*y*z^5 - 4200*x*y*z^4 + 15840*x*y*z^3 - 29874*x*y*z^2 + 28248*x*y*z - 10766*x*y + 40*x*z^6 - 456*x*z^5 + 2160*x*z^4 - 5462*x*z^3 + 7800*x*z^2 - 6006*x*z + 1968*x - 384*y^7 - 1536*y^6*z + 2816*y^6 - 2592*y^5*z^2 + 9600*y^5*z - 8832*y^5 - 2400*y^4*z^3 + 13440*y^4*z^2 - 24960*y^4*z + 15408*y^4 - 1320*y^3*z^4 + 9920*y^3*z^3 - 27840*y^3*z^2 + 34680*y^3*z - 16208*y^3 - 432*y^2*z^5 + 4080*y^2*z^4 - 15360*y^2*z^3 + 28908*y^2*z^2 - 27264*y^2*z + 10350*y^2 - 78*y*z^6 + 888*y*z^5 - 4200*y*z^4 + 10602*y*z^3 - 15108*y*z^2 + 11592*y*z - 3773*y - 6*z^7 + 80*z^6 - 456*z^5 + 1446*z^4 - 2764*z^3 + 3210*z^2 - 2123*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ 1, -2, -1 ], [ 1, -1, -2 ], [ 2, -3, -2 ] ], "source_shift": [ 2, 1, 2 ], "target_matrix": [ [ 1, 1, 1 ], [ 1, 1, 2 ], [ 2, 1, 1 ] ], "target_shift": [ -235, -83, -620 ], "inverse_recipe": { "p": 1, "q": 6, "f": "x", "g": "x**6 + x**3 + x", "h": "y*z + y + 2*z", "polynomials": { "f": [ [ [ 1, 0, 0 ], 1 ] ], "g": [ [ [ 6, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ], [ [ 3, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 0, 1 ], 2 ], [ [ 0, 1, 0 ], 1 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0207242629985558 ], "surface_features": [ 3.7580263863705605, 4.73200754860863, 3.3830070418225384, 0.0, 3.9102721016589075, 4958 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B017", "pair": 17, "family": "bijection", "mapping": "F(x,y,z)=(-5*x^7 + 35*x^6*y + 35*x^6*z - 45*x^6 - 105*x^5*y^2 - 210*x^5*y*z + 270*x^5*y - 105*x^5*z^2 + 275*x^5*z - 168*x^5 + 175*x^4*y^3 + 525*x^4*y^2*z - 675*x^4*y^2 + 525*x^4*y*z^2 - 1375*x^4*y*z + 840*x^4*y + 175*x^4*z^3 - 700*x^4*z^2 + 865*x^4*z - 350*x^4 - 175*x^3*y^4 - 700*x^3*y^3*z + 900*x^3*y^3 - 1050*x^3*y^2*z^2 + 2750*x^3*y^2*z - 1680*x^3*y^2 - 700*x^3*y*z^3 + 2800*x^3*y*z^2 - 3460*x^3*y*z + 1400*x^3*y - 175*x^3*z^4 + 950*x^3*z^3 - 1780*x^3*z^2 + 1455*x^3*z - 453*x^3 + 105*x^2*y^5 + 525*x^2*y^4*z - 675*x^2*y^4 + 1050*x^2*y^3*z^2 - 2750*x^2*y^3*z + 1680*x^2*y^3 + 1050*x^2*y^2*z^3 - 4200*x^2*y^2*z^2 + 5190*x^2*y^2*z - 2100*x^2*y^2 + 525*x^2*y*z^4 - 2850*x^2*y*z^3 + 5340*x^2*y*z^2 - 4365*x^2*y*z + 1354*x^2*y + 105*x^2*z^5 - 725*x^2*z^4 + 1830*x^2*z^3 - 2265*x^2*z^2 + 1419*x^2*z - 390*x^2 - 35*x*y^6 - 210*x*y^5*z + 270*x*y^5 - 525*x*y^4*z^2 + 1375*x*y^4*z - 840*x*y^4 - 700*x*y^3*z^3 + 2800*x*y^3*z^2 - 3460*x*y^3*z + 1400*x*y^3 - 525*x*y^2*z^4 + 2850*x*y^2*z^3 - 5340*x*y^2*z^2 + 4365*x*y^2*z - 1349*x*y^2 - 210*x*y*z^5 + 1450*x*y*z^4 - 3660*x*y*z^3 + 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105*x^2*y^5 - 525*x^2*y^4*z + 675*x^2*y^4 - 1050*x^2*y^3*z^2 + 2750*x^2*y^3*z - 1660*x^2*y^3 - 1050*x^2*y^2*z^3 + 4200*x^2*y^2*z^2 - 5130*x^2*y^2*z + 2040*x^2*y^2 - 525*x^2*y*z^4 + 2850*x^2*y*z^3 - 5280*x^2*y*z^2 + 4245*x^2*y*z - 1288*x^2*y - 105*x^2*z^5 + 725*x^2*z^4 - 1810*x^2*z^3 + 2205*x^2*z^2 - 1353*x^2*z + 364*x^2 + 35*x*y^6 + 210*x*y^5*z - 270*x*y^5 + 525*x*y^4*z^2 - 1375*x*y^4*z + 830*x*y^4 + 700*x*y^3*z^3 - 2800*x*y^3*z^2 + 3420*x*y^3*z - 1360*x*y^3 + 525*x*y^2*z^4 - 2850*x*y^2*z^3 + 5280*x*y^2*z^2 - 4245*x*y^2*z + 1283*x*y^2 + 210*x*y*z^5 - 1450*x*y*z^4 + 3620*x*y*z^3 - 4410*x*y*z^2 + 2696*x*y*z - 708*x*y + 35*x*z^6 - 295*x*z^5 + 930*x*z^4 - 1525*x*z^3 + 1413*x*z^2 - 758*x*z + 195*x - 5*y^7 - 35*y^6*z + 45*y^6 - 105*y^5*z^2 + 275*y^5*z - 166*y^5 - 175*y^4*z^3 + 700*y^4*z^2 - 855*y^4*z + 340*y^4 - 175*y^3*z^4 + 950*y^3*z^3 - 1760*y^3*z^2 + 1415*y^3*z - 426*y^3 - 105*y^2*z^5 + 725*y^2*z^4 - 1810*y^2*z^3 + 2205*y^2*z^2 - 1343*y^2*z + 344*y^2 - 35*y*z^6 + 295*y*z^5 - 930*y*z^4 + 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-2 ], [ 2, -1, -1 ] ], "source_shift": [ 1, 1, 2 ], "target_matrix": [ [ -1, -1, -1 ], [ 1, 1, 2 ], [ -3, -2, -3 ] ], "target_shift": [ 55, -47, 164 ], "inverse_recipe": { "p": 2, "q": 5, "f": "x**2 + x", "g": "x**5 + x**3", "h": "y*z", "polynomials": { "f": [ [ [ 2, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "g": [ [ [ 5, 0, 0 ], 1 ], [ [ 3, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ] ] }, "h_multiplier": 5 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "Maybe" ], "seconds": [ 1.0513124429999152 ], "surface_features": [ 3.270080735925755, 4.204662511748219, 2.942008053022313, 0.0, 3.417472690545536, 4827 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B117", "pair": 117, "family": "bijection", "mapping": "F(x,y,z)=(-2*x^7 - 28*x^6*y - 14*x^6*z - 22*x^6 - 168*x^5*y^2 - 168*x^5*y*z - 260*x^5*y - 42*x^5*z^2 - 136*x^5*z - 110*x^5 - 560*x^4*y^3 - 840*x^4*y^2*z - 1280*x^4*y^2 - 420*x^4*y*z^2 - 1340*x^4*y*z - 1064*x^4*y - 70*x^4*z^3 - 350*x^4*z^2 - 584*x^4*z - 320*x^4 - 1120*x^3*y^4 - 2240*x^3*y^3*z - 3360*x^3*y^3 - 1680*x^3*y^2*z^2 - 5280*x^3*y^2*z - 4114*x^3*y^2 - 560*x^3*y*z^3 - 2760*x^3*y*z^2 - 4524*x^3*y*z - 2424*x^3*y - 70*x^3*z^4 - 480*x^3*z^3 - 1238*x^3*z^2 - 1404*x^3*z - 578*x^3 - 1344*x^2*y^5 - 3360*x^2*y^4*z - 4960*x^2*y^4 - 3360*x^2*y^3*z^2 - 10400*x^2*y^3*z - 7948*x^2*y^3 - 1680*x^2*y^2*z^3 - 8160*x^2*y^2*z^2 - 13134*x^2*y^2*z - 6882*x^2*y^2 - 420*x^2*y*z^4 - 2840*x^2*y*z^3 - 7200*x^2*y*z^2 - 7984*x^2*y*z - 3200*x^2*y - 42*x^2*z^5 - 370*x^2*z^4 - 1310*x^2*z^3 - 2306*x^2*z^2 - 1972*x^2*z - 645*x^2 - 896*x*y^6 - 2688*x*y^5*z - 3904*x*y^5 - 3360*x*y^4*z^2 - 10240*x*y^4*z - 7672*x*y^4 - 2240*x*y^3*z^3 - 10720*x*y^3*z^2 - 16936*x*y^3*z - 8680*x*y^3 - 840*x*y^2*z^4 - 5600*x*y^2*z^3 - 13950*x*y^2*z^2 - 15124*x*y^2*z - 5908*x*y^2 - 168*x*y*z^5 - 1460*x*y*z^4 - 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1238*x^3*z^2 + 1404*x^3*z + 580*x^3 + 1344*x^2*y^5 + 3360*x^2*y^4*z + 4960*x^2*y^4 + 3360*x^2*y^3*z^2 + 10400*x^2*y^3*z + 7948*x^2*y^3 + 1680*x^2*y^2*z^3 + 8160*x^2*y^2*z^2 + 13134*x^2*y^2*z + 6882*x^2*y^2 + 420*x^2*y*z^4 + 2840*x^2*y*z^3 + 7200*x^2*y*z^2 + 7984*x^2*y*z + 3212*x^2*y + 42*x^2*z^5 + 370*x^2*z^4 + 1310*x^2*z^3 + 2306*x^2*z^2 + 1978*x^2*z + 653*x^2 + 896*x*y^6 + 2688*x*y^5*z + 3904*x*y^5 + 3360*x*y^4*z^2 + 10240*x*y^4*z + 7672*x*y^4 + 2240*x*y^3*z^3 + 10720*x*y^3*z^2 + 16936*x*y^3*z + 8680*x*y^3 + 840*x*y^2*z^4 + 5600*x*y^2*z^3 + 13950*x*y^2*z^2 + 15124*x*y^2*z + 5932*x*y^2 + 168*x*y*z^5 + 1460*x*y*z^4 + 5084*x*y*z^3 + 8752*x*y*z^2 + 7300*x*y*z + 2336*x*y + 14*x*z^6 + 152*x*z^5 + 692*x*z^4 + 1680*x*z^3 + 2248*x*z^2 + 1546*x*z + 422*x + 256*y^7 + 896*y^6*z + 1280*y^6 + 1344*y^5*z^2 + 4032*y^5*z + 2960*y^5 + 1120*y^4*z^3 + 5280*y^4*z^2 + 8184*y^4*z + 4104*y^4 + 560*y^3*z^4 + 3680*y^3*z^3 + 9004*y^3*z^2 + 9544*y^3*z + 3652*y^3 + 168*y^2*z^5 + 1440*y^2*z^4 + 4930*y^2*z^3 + 8298*y^2*z^2 + 6742*y^2*z + 2098*y^2 + 28*y*z^6 + 300*y*z^5 + 1344*y*z^4 + 3192*y*z^3 + 4144*y*z^2 + 2752*y*z + 725*y + 2*z^7 + 26*z^6 + 146*z^5 + 458*z^4 + 852*z^3 + 919*z^2 + 525*z + 1, -2*x^7 - 28*x^6*y - 14*x^6*z - 22*x^6 - 168*x^5*y^2 - 168*x^5*y*z - 260*x^5*y - 42*x^5*z^2 - 136*x^5*z - 110*x^5 - 560*x^4*y^3 - 840*x^4*y^2*z - 1280*x^4*y^2 - 420*x^4*y*z^2 - 1340*x^4*y*z - 1064*x^4*y - 70*x^4*z^3 - 350*x^4*z^2 - 584*x^4*z - 320*x^4 - 1120*x^3*y^4 - 2240*x^3*y^3*z - 3360*x^3*y^3 - 1680*x^3*y^2*z^2 - 5280*x^3*y^2*z - 4114*x^3*y^2 - 560*x^3*y*z^3 - 2760*x^3*y*z^2 - 4524*x^3*y*z - 2424*x^3*y - 70*x^3*z^4 - 480*x^3*z^3 - 1238*x^3*z^2 - 1404*x^3*z - 580*x^3 - 1344*x^2*y^5 - 3360*x^2*y^4*z - 4960*x^2*y^4 - 3360*x^2*y^3*z^2 - 10400*x^2*y^3*z - 7948*x^2*y^3 - 1680*x^2*y^2*z^3 - 8160*x^2*y^2*z^2 - 13134*x^2*y^2*z - 6882*x^2*y^2 - 420*x^2*y*z^4 - 2840*x^2*y*z^3 - 7200*x^2*y*z^2 - 7984*x^2*y*z - 3212*x^2*y - 42*x^2*z^5 - 370*x^2*z^4 - 1310*x^2*z^3 - 2306*x^2*z^2 - 1978*x^2*z - 654*x^2 - 896*x*y^6 - 2688*x*y^5*z - 3904*x*y^5 - 3360*x*y^4*z^2 - 10240*x*y^4*z - 7672*x*y^4 - 2240*x*y^3*z^3 - 10720*x*y^3*z^2 - 16936*x*y^3*z - 8680*x*y^3 - 840*x*y^2*z^4 - 5600*x*y^2*z^3 - 13950*x*y^2*z^2 - 15124*x*y^2*z - 5932*x*y^2 - 168*x*y*z^5 - 1460*x*y*z^4 - 5084*x*y*z^3 - 8752*x*y*z^2 - 7300*x*y*z - 2340*x*y - 14*x*z^6 - 152*x*z^5 - 692*x*z^4 - 1680*x*z^3 - 2248*x*z^2 - 1548*x*z - 425*x - 256*y^7 - 896*y^6*z - 1280*y^6 - 1344*y^5*z^2 - 4032*y^5*z - 2960*y^5 - 1120*y^4*z^3 - 5280*y^4*z^2 - 8184*y^4*z - 4104*y^4 - 560*y^3*z^4 - 3680*y^3*z^3 - 9004*y^3*z^2 - 9544*y^3*z - 3652*y^3 - 168*y^2*z^5 - 1440*y^2*z^4 - 4930*y^2*z^3 - 8298*y^2*z^2 - 6742*y^2*z - 2102*y^2 - 28*y*z^6 - 300*y*z^5 - 1344*y*z^4 - 3192*y*z^3 - 4144*y*z^2 - 2756*y*z - 730*y - 2*z^7 - 26*z^6 - 146*z^5 - 458*z^4 - 852*z^3 - 920*z^2 - 529*z - 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ 1, 2, 1 ], [ 1, 1, 2 ], [ 1, 1, 1 ] ], "source_shift": [ 1, 2, 1 ], "target_matrix": [ [ -1, -1, -2 ], [ 1, 1, 1 ], [ -1, -2, -1 ] ], "target_shift": [ 113, -121, 124 ], "inverse_recipe": { "p": 2, "q": 3, "f": "x**2", "g": "x**3 + x**2 + x", "h": "y**3 + y**2*z", "polynomials": { "f": [ [ [ 2, 0, 0 ], 1 ] ], "g": [ [ [ 3, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "h": [ [ [ 0, 2, 1 ], 1 ], [ [ 0, 3, 0 ], 1 ] ] }, "h_multiplier": 2 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.001550049000798 ], "surface_features": [ 3.4827342727411437, 4.228810845011355, 3.161368002234975, 0.0, 3.552571404936247, 4880 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C087", "pair": 87, "family": "counterexample", "mapping": "F(x,y,z)=(-64*x^7 + 352*x^6*y + 512*x^6*z - 608*x^6 - 816*x^5*y^2 - 2352*x^5*y*z + 2880*x^5*y - 1680*x^5*z^2 + 4080*x^5*z - 1968*x^5 + 1032*x^4*y^3 + 4416*x^4*y^2*z - 5592*x^4*y^2 + 6240*x^4*y*z^2 - 15696*x^4*y*z + 7728*x^4*y + 2912*x^4*z^3 - 10944*x^4*z^2 + 10800*x^4*z - 3144*x^4 - 768*x^3*y^4 - 4332*x^3*y^3*z + 5688*x^3*y^3 - 9072*x^3*y^2*z^2 + 23700*x^3*y^2*z - 11928*x^3*y^2 - 8360*x^3*y*z^3 + 32688*x^3*y*z^2 - 33024*x^3*y*z + 9868*x^3*y - 2860*x^3*z^4 + 14920*x^3*z^3 - 22728*x^3*z^2 + 13716*x^3*z - 2892*x^3 + 336*x^2*y^5 + 2340*x^2*y^4*z - 3192*x^2*y^4 + 6450*x^2*y^3*z^2 - 17532*x^2*y^3*z + 9030*x^2*y^3 + 8796*x^2*y^2*z^3 - 35838*x^2*y^2*z^2 + 37104*x^2*y^2*z - 11400*x^2*y^2 + 5934*x^2*y*z^4 - 32304*x^2*y*z^3 + 50496*x^2*y*z^2 - 31380*x^2*y*z + 6828*x^2*y + 1584*x^2*z^5 - 10830*x^2*z^4 + 22752*x^2*z^3 - 21474*x^2*z^2 + 9480*x^2*z - 1594*x^2 - 80*x*y^6 - 660*x*y^5*z + 936*x*y^5 - 2244*x*y^4*z^2 + 6348*x*y^4*z - 3348*x*y^4 - 4025*x*y^3*z^3 + 17082*x*y^3*z^2 - 18129*x*y^3*z + 5733*x*y^3 - 4017*x*y^2*z^4 + 22791*x*y^2*z^3 - 36552*x*y^2*z^2 + 23406*x*y^2*z - 5262*x*y^2 - 2115*x*y*z^5 + 15072*x*y*z^4 - 32508*x*y*z^3 + 31647*x*y*z^2 - 14454*x*y*z + 2525*x*y - 459*x*z^6 + 3951*x*z^5 - 10755*x*z^4 + 14163*x*z^3 - 9867*x*z^2 + 3513*x*z - 505*x + 8*y^7 + 76*y^6*z - 112*y^6 + 306*y^5*z^2 - 900*y^5*z + 486*y^5 + 677*y^4*z^3 - 2988*y^4*z^2 + 3249*y^4*z - 1057*y^4 + 889*y^3*z^4 - 5245*y^3*z^3 + 8622*y^3*z^2 - 5682*y^3*z + 1320*y^3 + 693*y^2*z^5 - 5133*y^2*z^4 + 11349*y^2*z^3 - 11370*y^2*z^2 + 5370*y^2*z - 976*y^2 + 297*y*z^6 - 2655*y*z^5 + 7407*y*z^4 - 10032*y*z^3 + 7230*y*z^2 - 2682*y*z + 400*y + 54*z^7 - 567*z^6 + 1917*z^5 - 3291*z^4 + 3219*z^3 - 1830*z^2 + 557*z - 1, 64*x^7 - 352*x^6*y - 512*x^6*z + 608*x^6 + 816*x^5*y^2 + 2352*x^5*y*z - 2880*x^5*y + 1680*x^5*z^2 - 4080*x^5*z + 1968*x^5 - 1032*x^4*y^3 - 4416*x^4*y^2*z + 5592*x^4*y^2 - 6240*x^4*y*z^2 + 15696*x^4*y*z - 7728*x^4*y - 2912*x^4*z^3 + 10944*x^4*z^2 - 10800*x^4*z + 3136*x^4 + 768*x^3*y^4 + 4332*x^3*y^3*z - 5688*x^3*y^3 + 9072*x^3*y^2*z^2 - 23700*x^3*y^2*z + 11928*x^3*y^2 + 8360*x^3*y*z^3 - 32688*x^3*y*z^2 + 33024*x^3*y*z - 9848*x^3*y + 2860*x^3*z^4 - 14920*x^3*z^3 + 22728*x^3*z^2 - 13688*x^3*z + 2840*x^3 - 336*x^2*y^5 - 2340*x^2*y^4*z + 3192*x^2*y^4 - 6450*x^2*y^3*z^2 + 17532*x^2*y^3*z - 9030*x^2*y^3 - 8796*x^2*y^2*z^3 + 35838*x^2*y^2*z^2 - 37104*x^2*y^2*z + 11382*x^2*y^2 - 5934*x^2*y*z^4 + 32304*x^2*y*z^3 - 50496*x^2*y*z^2 + 31332*x^2*y*z - 6732*x^2*y - 1584*x^2*z^5 + 10830*x^2*z^4 - 22752*x^2*z^3 + 21444*x^2*z^2 - 9360*x^2*z + 1528*x^2 + 80*x*y^6 + 660*x*y^5*z - 936*x*y^5 + 2244*x*y^4*z^2 - 6348*x*y^4*z + 3348*x*y^4 + 4025*x*y^3*z^3 - 17082*x*y^3*z^2 + 18129*x*y^3*z - 5726*x*y^3 + 4017*x*y^2*z^4 - 22791*x*y^2*z^3 + 36552*x*y^2*z^2 - 23379*x*y^2*z + 5205*x*y^2 + 2115*x*y*z^5 - 15072*x*y*z^4 + 32508*x*y*z^3 - 31614*x*y*z^2 + 14316*x*y*z - 2444*x*y + 459*x*z^6 - 3951*x*z^5 + 10755*x*z^4 - 14150*x*z^3 + 9786*x*z^2 - 3414*x*z + 478*x - 8*y^7 - 76*y^6*z + 112*y^6 - 306*y^5*z^2 + 900*y^5*z - 486*y^5 - 677*y^4*z^3 + 2988*y^4*z^2 - 3249*y^4*z + 1056*y^4 - 889*y^3*z^4 + 5245*y^3*z^3 - 8622*y^3*z^2 + 5677*y^3*z - 1309*y^3 - 693*y^2*z^5 + 5133*y^2*z^4 - 11349*y^2*z^3 + 11361*y^2*z^2 - 5331*y^2*z + 952*y^2 - 297*y*z^6 + 2655*y*z^5 - 7407*y*z^4 + 10025*y*z^3 - 7185*y*z^2 + 2625*y*z - 383*y - 54*z^7 + 567*z^6 - 1917*z^5 + 3289*z^4 - 3202*z^3 + 1797*z^2 - 536*z + 1, -192*x^7 + 1056*x^6*y + 1536*x^6*z - 1728*x^6 - 2448*x^5*y^2 - 7056*x^5*y*z + 8208*x^5*y - 5040*x^5*z^2 + 11616*x^5*z - 5184*x^5 + 3096*x^4*y^3 + 13248*x^4*y^2*z - 15984*x^4*y^2 + 18720*x^4*y*z^2 - 44832*x^4*y*z + 20448*x^4*y + 8736*x^4*z^3 - 31248*x^4*z^2 + 28608*x^4*z - 7952*x^4 - 2304*x^3*y^4 - 12996*x^3*y^3*z + 16308*x^3*y^3 - 27216*x^3*y^2*z^2 + 67920*x^3*y^2*z - 31716*x^3*y^2 - 25080*x^3*y*z^3 + 93672*x^3*y*z^2 - 87936*x^3*y*z + 25088*x^3*y - 8580*x^3*z^4 + 42768*x^3*z^3 - 60624*x^3*z^2 + 34936*x^3*z - 7144*x^3 + 1008*x^2*y^5 + 7020*x^2*y^4*z - 9180*x^2*y^4 + 19350*x^2*y^3*z^2 - 50412*x^2*y^3*z + 24138*x^2*y^3 + 26388*x^2*y^2*z^3 - 103068*x^2*y^2*z^2 + 99360*x^2*y^2*z - 29148*x^2*y^2 + 17802*x^2*y*z^4 - 92952*x^2*y*z^3 + 135504*x^2*y*z^2 - 80412*x^2*y*z + 16956*x^2*y + 4752*x^2*z^5 - 31188*x^2*z^4 + 61200*x^2*z^3 - 55164*x^2*z^2 + 23592*x^2*z - 3888*x^2 - 240*x*y^6 - 1980*x*y^5*z + 2700*x*y^5 - 6732*x*y^4*z^2 + 18312*x*y^4*z - 9000*x*y^4 - 12075*x*y^3*z^3 + 49293*x*y^3*z^2 - 48837*x*y^3*z + 14749*x*y^3 - 12051*x*y^2*z^4 + 65808*x*y^2*z^3 - 98703*x*y^2*z^2 + 60372*x*y^2*z - 13146*x*y^2 - 6345*x*y*z^5 + 43557*x*y*z^4 - 88020*x*y*z^3 + 81864*x*y*z^2 - 36204*x*y*z + 6192*x*y - 1377*x*z^6 + 11430*x*z^5 - 29208*x*z^4 + 36754*x*z^3 - 24786*x*z^2 + 8634*x*z - 1222*x + 24*y^7 + 228*y^6*z - 324*y^6 + 918*y^5*z^2 - 2604*y^5*z + 1314*y^5 + 2031*y^4*z^3 - 8649*y^4*z^2 + 8805*y^4*z - 2737*y^4 + 2667*y^3*z^4 - 15192*y^3*z^3 + 23427*y^3*z^2 - 14756*y^3*z + 3320*y^3 + 2079*y^2*z^5 - 14880*y^2*z^4 + 30924*y^2*z^3 - 29622*y^2*z^2 + 13548*y^2*z - 2409*y^2 + 891*y*z^6 - 7704*y*z^5 + 20244*y*z^4 - 26227*y*z^3 + 18303*y*z^2 - 6639*y*z + 975*y + 162*z^7 - 1647*z^6 + 5256*z^5 - 8636*z^4 + 8180*z^3 - 4545*z^2 + 1362*z - 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ 2, -1, -1 ], [ 2, -2, -3 ], [ 1, -1, -2 ] ], "source_shift": [ 1, 1, 2 ], "target_matrix": [ [ -1, -1, -1 ], [ 1, 1, 2 ], [ -3, -2, -2 ] ], "target_shift": [ 72, -69, 175 ], "collision_preimages": [ [ "-9/4", "-7", "7/2" ], [ "7", "51/2", "-23/2" ], [ "2", "29/2", "-17/2" ] ], "collision_image": [ "289/4", "-277/4", "703/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0822937580014695 ], "surface_features": [ 4.122110797252636, 5.131952115522506, 3.793511005792858, 0.0, 4.283850423118304, 5097 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C062", "pair": 62, "family": "counterexample", "mapping": "F(x,y,z)=(-54*x^7 + 432*x^6*y - 324*x^6*z + 783*x^6 - 1458*x^5*y^2 + 2214*x^5*y*z - 4941*x^5*y - 828*x^5*z^2 + 3780*x^5*z - 2601*x^5 + 2700*x^4*y^3 - 6210*x^4*y^2*z + 12960*x^4*y^2 + 4698*x^4*y*z^2 - 19899*x^4*y*z + 13500*x^4*y - 1168*x^4*z^3 + 7560*x^4*z^2 - 10230*x^4*z + 4279*x^4 - 2970*x^3*y^4 + 9180*x^3*y^3*z - 18090*x^3*y^3 - 10512*x^3*y^2*z^2 + 41796*x^3*y^2*z - 27990*x^3*y^2 + 5282*x^3*y*z^3 - 31869*x^3*y*z^2 + 42528*x^3*y*z - 17687*x^3*y - 982*x^3*z^4 + 8020*x^3*z^3 - 16005*x^3*z^2 + 13216*x^3*z - 3997*x^3 + 1944*x^2*y^5 - 7560*x^2*y^4*z + 14175*x^2*y^4 + 11628*x^2*y^3*z^2 - 43794*x^2*y^3*z + 28980*x^2*y^3 - 8838*x^2*y^2*z^3 + 50247*x^2*y^2*z^2 - 66204*x^2*y^2*z + 27387*x^2*y^2 + 3318*x^2*y*z^4 - 25377*x^2*y*z^3 + 49956*x^2*y*z^2 - 41019*x^2*y*z + 12384*x^2*y - 492*x^2*z^5 + 4761*x^2*z^4 - 12456*x^2*z^3 + 15225*x^2*z^2 - 9084*x^2*z + 2169*x^2 - 702*x*y^6 + 3294*x*y^5*z - 5913*x*y^5 - 6372*x*y^4*z^2 + 22896*x*y^4*z - 14985*x*y^4 + 6502*x*y^3*z^3 - 35127*x*y^3*z^2 + 45744*x*y^3*z - 18829*x*y^3 - 3690*x*y^2*z^4 + 26694*x*y^2*z^3 - 51897*x*y^2*z^2 + 42390*x*y^2*z - 12777*x*y^2 + 1104*x*y*z^5 - 10050*x*y*z^4 + 25944*x*y*z^3 - 31536*x*y*z^2 + 18786*x*y*z - 4491*x*y - 136*x*z^6 + 1500*x*z^5 - 4824*x*z^4 + 7758*x*z^3 - 6852*x*z^2 + 3222*x*z - 625*x + 108*y^7 - 594*y^6*z + 1026*y^6 + 1386*y^5*z^2 - 4779*y^5*z + 3096*y^5 - 1778*y^4*z^3 + 9189*y^4*z^2 - 11838*y^4*z + 4850*y^4 + 1354*y^3*z^4 - 9337*y^3*z^3 + 17946*y^3*z^2 - 14587*y^3*z + 4390*y^3 - 612*y^2*z^5 + 5289*y^2*z^4 - 13488*y^2*z^3 + 16311*y^2*z^2 - 9702*y^2*z + 2322*y^2 + 152*y*z^6 - 1584*y*z^5 + 5028*y*z^4 - 8042*y*z^3 + 7092*y*z^2 - 3339*y*z + 652*y - 16*z^7 + 196*z^6 - 744*z^5 + 1476*z^4 - 1716*z^3 + 1193*z^2 - 458*z + 1, 27*x^7 - 216*x^6*y + 162*x^6*z - 405*x^6 + 729*x^5*y^2 - 1107*x^5*y*z + 2565*x^5*y + 414*x^5*z^2 - 1962*x^5*z + 1485*x^5 - 1350*x^4*y^3 + 3105*x^4*y^2*z - 6750*x^4*y^2 - 2349*x^4*y*z^2 + 10368*x^4*y*z - 7722*x^4*y + 584*x^4*z^3 - 3939*x^4*z^2 + 5877*x^4*z - 2631*x^4 + 1485*x^3*y^4 - 4590*x^3*y^3*z + 9450*x^3*y^3 + 5256*x^3*y^2*z^2 - 21852*x^3*y^2*z + 16038*x^3*y^2 - 2641*x^3*y*z^3 + 16671*x^3*y*z^2 - 24480*x^3*y*z + 10887*x^3*y + 491*x^3*z^4 - 4196*x^3*z^3 + 9252*x^3*z^2 - 8181*x^3*z + 2595*x^3 - 972*x^2*y^5 + 3780*x^2*y^4*z - 7425*x^2*y^4 - 5814*x^2*y^3*z^2 + 22968*x^2*y^3*z - 16632*x^2*y^3 + 4419*x^2*y^2*z^3 - 26379*x^2*y^2*z^2 + 38178*x^2*y^2*z - 16875*x^2*y^2 - 1659*x^2*y*z^4 + 13332*x^2*y*z^3 - 28938*x^2*y*z^2 + 25422*x^2*y*z - 8046*x^2*y + 246*x^2*z^5 - 2502*x^2*z^4 + 7245*x^2*z^3 - 9486*x^2*z^2 + 5937*x^2*z - 1464*x^2 + 351*x*y^6 - 1647*x*y^5*z + 3105*x*y^5 + 3186*x*y^4*z^2 - 12042*x*y^4*z + 8613*x*y^4 - 3251*x*y^3*z^3 + 18501*x*y^3*z^2 - 26424*x*y^3*z + 11613*x*y^3 + 1845*x*y^2*z^4 - 14076*x*y^2*z^3 + 30120*x*y^2*z^2 - 26301*x*y^2*z + 8307*x*y^2 - 552*x*y*z^5 + 5304*x*y*z^4 - 15123*x*y*z^3 + 19674*x*y*z^2 - 12288*x*y*z + 3033*x*y + 68*x*z^6 - 792*x*z^5 + 2823*x*z^4 - 4864*x*z^3 + 4506*x*z^2 - 2187*x*z + 433*x - 54*y^7 + 297*y^6*z - 540*y^6 - 693*y^5*z^2 + 2520*y^5*z - 1782*y^5 + 889*y^4*z^3 - 4854*y^4*z^2 + 6849*y^4*z - 2994*y^4 - 677*y^3*z^4 + 4940*y^3*z^3 - 10434*y^3*z^2 + 9060*y^3*z - 2856*y^3 + 306*y^2*z^5 - 2802*y^2*z^4 + 7878*y^2*z^3 - 10188*y^2*z^2 + 6351*y^2*z - 1569*y^2 - 76*y*z^6 + 840*y*z^5 - 2949*y*z^4 + 5049*y*z^3 - 4668*y*z^2 + 2268*y*z - 452*y + 8*z^7 - 104*z^6 + 438*z^5 - 931*z^4 + 1135*z^3 - 814*z^2 + 319*z - 1, 27*x^7 - 216*x^6*y + 162*x^6*z - 405*x^6 + 729*x^5*y^2 - 1107*x^5*y*z + 2565*x^5*y + 414*x^5*z^2 - 1962*x^5*z + 1485*x^5 - 1350*x^4*y^3 + 3105*x^4*y^2*z - 6750*x^4*y^2 - 2349*x^4*y*z^2 + 10368*x^4*y*z - 7722*x^4*y + 584*x^4*z^3 - 3939*x^4*z^2 + 5877*x^4*z - 2630*x^4 + 1485*x^3*y^4 - 4590*x^3*y^3*z + 9450*x^3*y^3 + 5256*x^3*y^2*z^2 - 21852*x^3*y^2*z + 16038*x^3*y^2 - 2641*x^3*y*z^3 + 16671*x^3*y*z^2 - 24480*x^3*y*z + 10882*x^3*y + 491*x^3*z^4 - 4196*x^3*z^3 + 9252*x^3*z^2 - 8177*x^3*z + 2582*x^3 - 972*x^2*y^5 + 3780*x^2*y^4*z - 7425*x^2*y^4 - 5814*x^2*y^3*z^2 + 22968*x^2*y^3*z - 16632*x^2*y^3 + 4419*x^2*y^2*z^3 - 26379*x^2*y^2*z^2 + 38178*x^2*y^2*z - 16866*x^2*y^2 - 1659*x^2*y*z^4 + 13332*x^2*y*z^3 - 28938*x^2*y*z^2 + 25407*x^2*y*z - 8004*x^2*y + 246*x^2*z^5 - 2502*x^2*z^4 + 7245*x^2*z^3 - 9480*x^2*z^2 + 5901*x^2*z - 1437*x^2 + 351*x*y^6 - 1647*x*y^5*z + 3105*x*y^5 + 3186*x*y^4*z^2 - 12042*x*y^4*z + 8613*x*y^4 - 3251*x*y^3*z^3 + 18501*x*y^3*z^2 - 26424*x*y^3*z + 11606*x*y^3 + 1845*x*y^2*z^4 - 14076*x*y^2*z^3 + 30120*x*y^2*z^2 - 26283*x*y^2*z + 8262*x*y^2 - 552*x*y*z^5 + 5304*x*y*z^4 - 15123*x*y*z^3 + 19659*x*y*z^2 - 12210*x*y*z + 2976*x*y + 68*x*z^6 - 792*x*z^5 + 2823*x*z^4 - 4860*x*z^3 + 4473*x*z^2 - 2139*x*z + 416*x - 54*y^7 + 297*y^6*z - 540*y^6 - 693*y^5*z^2 + 2520*y^5*z - 1782*y^5 + 889*y^4*z^3 - 4854*y^4*z^2 + 6849*y^4*z - 2992*y^4 - 677*y^3*z^4 + 4940*y^3*z^3 - 10434*y^3*z^2 + 9053*y^3*z - 2840*y^3 + 306*y^2*z^5 - 2802*y^2*z^4 + 7878*y^2*z^3 - 10179*y^2*z^2 + 6309*y^2*z - 1539*y^2 - 76*y*z^6 + 840*y*z^5 - 2949*y*z^4 + 5044*y*z^3 - 4632*y*z^2 + 2217*y*z - 434*y + 8*z^7 - 104*z^6 + 438*z^5 - 930*z^4 + 1125*z^3 - 793*z^2 + 305*z - 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -1, 1, -1 ], [ -3, 3, -2 ], [ -1, 2, -1 ] ], "source_shift": [ 1, 1, 1 ], "target_matrix": [ [ 2, 1, 1 ], [ -1, -1, -1 ], [ -1, -1, -2 ] ], "target_shift": [ -75, 53, 51 ], "collision_preimages": [ [ "-5/4", "-1/4", "2" ], [ "8", "11/2", "-5/2" ], [ "3", "15/2", "13/2" ] ], "collision_image": [ "-151/2", "213/4", "205/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0719519669983129 ], "surface_features": [ 3.909912446562701, 4.8208842300053645, 3.6369390072874714, 0.0, 4.017736044528219, 5039 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C021", "pair": 21, "family": "counterexample", "mapping": "F(x,y,z)=(24*x^7 - 204*x^6*y - 240*x^6*z + 276*x^6 + 738*x^5*y^2 + 1764*x^5*y*z - 2016*x^5*y + 1008*x^5*z^2 - 2400*x^5*z + 1308*x^5 - 1473*x^4*y^3 - 5364*x^4*y^2*z + 6093*x^4*y^2 - 6228*x^4*y*z^2 + 14736*x^4*y*z - 7980*x^4*y - 2304*x^4*z^3 + 8556*x^4*z^2 - 9564*x^4*z + 3350*x^4 + 1752*x^3*y^4 + 8637*x^3*y^3*z - 9753*x^3*y^3 + 15282*x^3*y^2*z^2 - 35934*x^3*y^2*z + 19335*x^3*y^2 + 11484*x^3*y*z^3 - 42396*x^3*y*z^2 + 47112*x^3*y*z - 16361*x^3*y + 3096*x^3*z^4 - 16008*x^3*z^3 + 27540*x^3*z^2 - 19742*x^3*z + 5074*x^3 - 1242*x^2*y^5 - 7767*x^2*y^4*z + 8721*x^2*y^4 - 18612*x^2*y^3*z^2 + 43497*x^2*y^3*z - 23256*x^2*y^3 - 21312*x^2*y^2*z^3 + 78210*x^2*y^2*z^2 - 86391*x^2*y^2*z + 29739*x^2*y^2 - 11664*x^2*y*z^4 + 60012*x^2*y*z^3 - 102708*x^2*y*z^2 + 73044*x^2*y*z - 18558*x^2*y - 2448*x^2*z^5 + 16584*x^2*z^4 - 39036*x^2*z^3 + 42924*x^2*z^2 - 22602*x^2*z + 4608*x^2 + 486*x*y^6 + 3699*x*y^5*z - 4131*x*y^5 + 11250*x*y^4*z^2 - 26136*x*y^4*z + 13887*x*y^4 + 17448*x*y^3*z^3 - 63648*x*y^3*z^2 + 69885*x*y^3*z - 23844*x*y^3 + 14544*x*y^2*z^4 - 74448*x*y^2*z^3 + 126738*x*y^2*z^2 - 89394*x*y^2*z + 22437*x*y^2 + 6192*x*y*z^5 - 41808*x*y*z^4 + 97980*x*y*z^3 - 106968*x*y*z^2 + 55743*x*y*z - 11196*x*y + 1056*x*z^6 - 9024*x*z^5 + 27240*x*z^4 - 40808*x*z^3 + 32958*x*z^2 - 13800*x*z + 2367*x - 81*y^7 - 729*y^6*z + 810*y^6 - 2700*y^5*z^2 + 6237*y^5*z - 3294*y^5 - 5316*y^4*z^3 + 19278*y^4*z^2 - 21042*y^4*z + 7116*y^4 - 6000*y^3*z^4 + 30552*y^3*z^3 - 51732*y^3*z^2 + 36183*y^3*z - 8967*y^3 - 3888*y^2*z^5 + 26160*y^2*z^4 - 61032*y^2*z^3 + 66126*y^2*z^2 - 34080*y^2*z + 6735*y^2 - 1344*y*z^6 + 11472*y*z^5 - 34512*y*z^4 + 51380*y*z^3 - 41112*y*z^2 + 16989*y*z - 2866*y - 192*z^7 + 2016*z^6 - 7488*z^5 + 14312*z^4 - 15736*z^3 + 10122*z^2 - 3563*z + 1, 16*x^7 - 136*x^6*y - 160*x^6*z + 176*x^6 + 492*x^5*y^2 + 1176*x^5*y*z - 1284*x^5*y + 672*x^5*z^2 - 1536*x^5*z + 804*x^5 - 982*x^4*y^3 - 3576*x^4*y^2*z + 3876*x^4*y^2 - 4152*x^4*y*z^2 + 9420*x^4*y*z - 4896*x^4*y - 1536*x^4*z^3 + 5496*x^4*z^2 - 5904*x^4*z + 2008*x^4 + 1168*x^3*y^4 + 5758*x^3*y^3*z - 6197*x^3*y^3 + 10188*x^3*y^2*z^2 - 22944*x^3*y^2*z + 11841*x^3*y^2 + 7656*x^3*y*z^3 - 27204*x^3*y*z^2 + 29028*x^3*y*z - 9786*x^3*y + 2064*x^3*z^4 - 10320*x^3*z^3 + 17076*x^3*z^2 - 11884*x^3*z + 2992*x^3 - 828*x^2*y^5 - 5178*x^2*y^4*z + 5535*x^2*y^4 - 12408*x^2*y^3*z^2 + 27741*x^2*y^3*z - 14217*x^2*y^3 - 14208*x^2*y^2*z^3 + 50130*x^2*y^2*z^2 - 53130*x^2*y^2*z + 17751*x^2*y^2 - 7776*x^2*y*z^4 + 38652*x^2*y*z^3 - 63564*x^2*y*z^2 + 43872*x^2*y*z - 10920*x^2*y - 1632*x^2*z^5 + 10728*x^2*z^4 - 24312*x^2*z^3 + 25956*x^2*z^2 - 13380*x^2*z + 2690*x^2 + 324*x*y^6 + 2466*x*y^5*z - 2619*x*y^5 + 7500*x*y^4*z^2 - 16650*x*y^4*z + 8475*x*y^4 + 11632*x*y^3*z^3 - 40752*x*y^3*z^2 + 42900*x*y^3*z - 14204*x*y^3 + 9696*x*y^2*z^4 - 47904*x*y^2*z^3 + 78288*x*y^2*z^2 - 53574*x*y^2*z + 13176*x*y^2 + 4128*x*y*z^5 - 27024*x*y*z^4 + 60912*x*y*z^3 - 64536*x*y*z^2 + 32922*x*y*z - 6523*x*y + 704*x*z^6 - 5856*x*z^5 + 17040*x*z^4 - 24792*x*z^3 + 19596*x*z^2 - 8084*x*z + 1373*x - 54*y^7 - 486*y^6*z + 513*y^6 - 1800*y^5*z^2 + 3969*y^5*z - 2007*y^5 - 3544*y^4*z^3 + 12330*y^4*z^2 - 12894*y^4*z + 4231*y^4 - 4000*y^3*z^4 + 19640*y^3*z^3 - 31896*y^3*z^2 + 21638*y^3*z - 5256*y^3 - 2592*y^2*z^5 + 16896*y^2*z^4 - 37872*y^2*z^3 + 39804*y^2*z^2 - 20082*y^2*z + 3917*y^2 - 896*y*z^6 + 7440*y*z^5 - 21552*y*z^4 + 31144*y*z^3 - 24384*y*z^2 + 9929*y*z - 1659*y - 128*z^7 + 1312*z^6 - 4704*z^5 + 8736*z^4 - 9400*z^3 + 5954*z^2 - 2074*z + 1, 8*x^7 - 68*x^6*y - 80*x^6*z + 124*x^6 + 246*x^5*y^2 + 588*x^5*y*z - 912*x^5*y + 336*x^5*z^2 - 1056*x^5*z + 708*x^5 - 491*x^4*y^3 - 1788*x^4*y^2*z + 2775*x^4*y^2 - 2076*x^4*y*z^2 + 6528*x^4*y*z - 4356*x^4*y - 768*x^4*z^3 + 3684*x^4*z^2 - 5076*x^4*z + 2026*x^4 + 584*x^3*y^4 + 2879*x^3*y^3*z - 4471*x^3*y^3 + 5094*x^3*y^2*z^2 - 16026*x^3*y^2*z + 10641*x^3*y^2 + 3828*x^3*y*z^3 - 18372*x^3*y*z^2 + 25224*x^3*y*z - 9983*x^3*y + 1032*x^3*z^4 - 6744*x^3*z^3 + 14316*x^3*z^2 - 11722*x^3*z + 3306*x^3 - 414*x^2*y^5 - 2589*x^2*y^4*z + 4023*x^2*y^4 - 6204*x^2*y^3*z^2 + 19527*x^2*y^3*z - 12900*x^2*y^3 - 7104*x^2*y^2*z^3 + 34110*x^2*y^2*z^2 - 46653*x^2*y^2*z + 18303*x^2*y^2 - 3888*x^2*y*z^4 + 25428*x^2*y*z^3 - 53868*x^2*y*z^2 + 43776*x^2*y*z - 12210*x^2*y - 816*x^2*z^5 + 6840*x^2*z^4 - 19860*x^2*z^3 + 24996*x^2*z^2 - 14454*x^2*z + 3172*x^2 + 162*x*y^6 + 1233*x*y^5*z - 1917*x*y^5 + 3750*x*y^4*z^2 - 11808*x*y^4*z + 7761*x*y^4 + 5816*x*y^3*z^3 - 27936*x*y^3*z^2 + 38055*x*y^3*z - 14797*x*y^3 + 4848*x*y^2*z^4 - 31728*x*y^2*z^3 + 67062*x*y^2*z^2 - 54066*x*y^2*z + 14904*x*y^2 + 2064*x*y*z^5 - 17328*x*y*z^4 + 50292*x*y*z^3 - 62892*x*y*z^2 + 36009*x*y*z - 7796*x*y + 352*x*z^6 - 3648*x*z^5 + 13560*x*z^4 - 23288*x*z^3 + 20670*x*z^2 - 9304*x*z + 1697*x - 27*y^7 - 243*y^6*z + 378*y^6 - 900*y^5*z^2 + 2835*y^5*z - 1854*y^5 - 1772*y^4*z^3 + 8514*y^4*z^2 - 11550*y^4*z + 4451*y^4 - 2000*y^3*z^4 + 13096*y^3*z^3 - 27612*y^3*z^2 + 22078*y^3*z - 6012*y^3 - 1296*y^2*z^5 + 10896*y^2*z^4 - 31608*y^2*z^3 + 39252*y^2*z^2 - 22236*y^2*z + 4744*y^2 - 448*y*z^6 + 4656*y*z^5 - 17328*y*z^4 + 29608*y*z^3 - 26052*y*z^2 + 11587*y*z - 2084*y - 64*z^7 + 800*z^6 - 3648*z^5 + 8000*z^4 - 9672*z^3 + 6682*z^2 - 2493*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ 2, -3, -2 ], [ 1, -1, -2 ], [ 1, -1, -1 ] ], "source_shift": [ 2, 1, 2 ], "target_matrix": [ [ 3, 2, 2 ], [ 2, 1, 1 ], [ 1, 2, 1 ] ], "target_shift": [ -541, -312, -397 ], "collision_preimages": [ [ "-6", "-5/2", "-5/4" ], [ "43/2", "10", "7" ], [ "41/2", "12", "4" ] ], "collision_image": [ "-2167/4", "-625/2", "-1589/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0317018770001596 ], "surface_features": [ 4.203819189905263, 5.1029068494215615, 3.9053640687668922, 0.0, 4.307297067433867, 5142 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C224", "pair": 224, "family": "counterexample", "mapping": "F(x,y,z)=(2*x^7 + 13*x^6*y - 19*x^6*z - 29*x^6 + 36*x^5*y^2 - 105*x^5*y*z - 162*x^5*y + 75*x^5*z^2 + 234*x^5*z + 177*x^5 + 55*x^4*y^3 - 240*x^4*y^2*z - 375*x^4*y^2 + 342*x^4*y*z^2 + 1083*x^4*y*z + 828*x^4*y - 159*x^4*z^3 - 768*x^4*z^2 - 1191*x^4*z - 602*x^4 + 50*x^3*y^4 - 290*x^3*y^3*z - 460*x^3*y^3 + 618*x^3*y^2*z^2 + 1992*x^3*y^2*z + 1542*x^3*y^2 - 573*x^3*y*z^3 - 2826*x^3*y*z^2 - 4443*x^3*y*z - 2265*x^3*y + 195*x^3*z^4 + 1314*x^3*z^3 + 3150*x^3*z^2 + 3252*x^3*z + 1241*x^3 + 27*x^2*y^5 - 195*x^2*y^4*z - 315*x^2*y^4 + 552*x^2*y^3*z^2 + 1818*x^2*y^3*z + 1428*x^2*y^3 - 765*x^2*y^2*z^3 - 3870*x^2*y^2*z^2 - 6183*x^2*y^2*z - 3183*x^2*y^2 + 519*x^2*y*z^4 + 3603*x^2*y*z^3 + 8790*x^2*y*z^2 + 9168*x^2*y*z + 3519*x^2*y - 138*x^2*z^5 - 1239*x^2*z^4 - 4104*x^2*z^3 - 6498*x^2*z^2 - 5040*x^2*z - 1558*x^2 + 8*x*y^6 - 69*x*y^5*z - 114*x*y^5 + 243*x*y^4*z^2 + 822*x*y^4*z + 657*x*y^4 - 447*x*y^3*z^3 - 2334*x*y^3*z^2 - 3801*x*y^3*z - 1979*x*y^3 + 453*x*y^2*z^4 + 3264*x*y^2*z^3 + 8130*x*y^2*z^2 + 8580*x*y^2*z + 3315*x*y^2 - 240*x*y*z^5 - 2250*x*y*z^4 - 7620*x*y*z^3 - 12213*x*y*z^2 - 9537*x*y*z - 2957*x*y + 52*x*z^6 + 612*x*z^5 + 2640*x*z^4 + 5705*x*z^3 + 6744*x*z^2 + 4219*x*z + 1108*x + y^7 - 10*y^6*z - 17*y^6 + 42*y^5*z^2 + 147*y^5*z + 120*y^5 - 96*y^4*z^3 - 522*y^4*z^2 - 870*y^4*z - 459*y^4 + 129*y^3*z^4 + 975*y^3*z^3 + 2490*y^3*z^2 + 2664*y^3*z + 1037*y^3 - 102*y^2*z^5 - 1011*y^2*z^4 - 3516*y^2*z^3 - 5715*y^2*z^2 - 4497*y^2*z - 1399*y^2 + 44*y*z^6 + 552*y*z^5 + 2448*y*z^4 + 5366*y*z^3 + 6393*y*z^2 + 4015*y*z + 1055*y - 8*z^7 - 124*z^6 - 672*z^5 - 1859*z^4 - 2975*z^3 - 2824*z^2 - 1495*z - 1, 2*x^7 + 13*x^6*y - 19*x^6*z - 35*x^6 + 36*x^5*y^2 - 105*x^5*y*z - 195*x^5*y + 75*x^5*z^2 + 279*x^5*z + 234*x^5 + 55*x^4*y^3 - 240*x^4*y^2*z - 450*x^4*y^2 + 342*x^4*y*z^2 + 1287*x^4*y*z + 1092*x^4*y - 159*x^4*z^3 - 903*x^4*z^2 - 1554*x^4*z - 842*x^4 + 50*x^3*y^4 - 290*x^3*y^3*z - 550*x^3*y^3 + 618*x^3*y^2*z^2 + 2358*x^3*y^2*z + 2028*x^3*y^2 - 573*x^3*y*z^3 - 3309*x^3*y*z^2 - 5781*x^3*y*z - 3162*x^3*y + 195*x^3*z^4 + 1521*x^3*z^3 + 4053*x^3*z^2 + 4491*x^3*z + 1802*x^3 + 27*x^2*y^5 - 195*x^2*y^4*z - 375*x^2*y^4 + 552*x^2*y^3*z^2 + 2142*x^2*y^3*z + 1872*x^2*y^3 - 765*x^2*y^2*z^3 - 4509*x^2*y^2*z^2 - 8019*x^2*y^2*z - 4434*x^2*y^2 + 519*x^2*y*z^4 + 4149*x^2*y*z^3 + 11274*x^2*y*z^2 + 12636*x^2*y*z + 5103*x^2*y - 138*x^2*z^5 - 1410*x^2*z^4 - 5205*x^2*z^3 - 8859*x^2*z^2 - 7227*x^2*z - 2320*x^2 + 8*x*y^6 - 69*x*y^5*z - 135*x*y^5 + 243*x*y^4*z^2 + 963*x*y^4*z + 858*x*y^4 - 447*x*y^3*z^3 - 2703*x*y^3*z^2 - 4911*x*y^3*z - 2750*x*y^3 + 453*x*y^2*z^4 + 3735*x*y^2*z^3 + 10389*x*y^2*z^2 + 11799*x*y^2*z + 4800*x*y^2 - 240*x*y*z^5 - 2544*x*y*z^4 - 9630*x*y*z^3 - 16617*x*y*z^2 - 13659*x*y*z - 4400*x*y + 52*x*z^6 + 684*x*z^5 + 3300*x*z^4 + 7679*x*z^3 + 9552*x*z^2 + 6202*x*z + 1673*x + y^7 - 10*y^6*z - 20*y^6 + 42*y^5*z^2 + 171*y^5*z + 156*y^5 - 96*y^4*z^3 - 600*y^4*z^2 - 1119*y^4*z - 636*y^4 + 129*y^3*z^4 + 1107*y^3*z^3 + 3168*y^3*z^2 + 3654*y^3*z + 1499*y^3 - 102*y^2*z^5 - 1134*y^2*z^4 - 4425*y^2*z^3 - 7758*y^2*z^2 - 6432*y^2*z - 2080*y^2 + 44*y*z^6 + 612*y*z^5 + 3048*y*z^4 + 7208*y*z^3 + 9045*y*z^2 + 5899*y*z + 1593*y - 8*z^7 - 136*z^6 - 828*z^5 - 2471*z^4 - 4163*z^3 - 4099*z^2 - 2226*z - 1, -4*x^7 - 26*x^6*y + 38*x^6*z + 64*x^6 - 72*x^5*y^2 + 210*x^5*y*z + 357*x^5*y - 150*x^5*z^2 - 513*x^5*z - 411*x^5 - 110*x^4*y^3 + 480*x^4*y^2*z + 825*x^4*y^2 - 684*x^4*y*z^2 - 2370*x^4*y*z - 1920*x^4*y + 318*x^4*z^3 + 1671*x^4*z^2 + 2745*x^4*z + 1442*x^4 - 100*x^3*y^4 + 580*x^3*y^3*z + 1010*x^3*y^3 - 1236*x^3*y^2*z^2 - 4350*x^3*y^2*z - 3570*x^3*y^2 + 1146*x^3*y*z^3 + 6135*x^3*y*z^2 + 10224*x^3*y*z + 5420*x^3*y - 390*x^3*z^4 - 2835*x^3*z^3 - 7203*x^3*z^2 - 7736*x^3*z - 3032*x^3 - 54*x^2*y^5 + 390*x^2*y^4*z + 690*x^2*y^4 - 1104*x^2*y^3*z^2 - 3960*x^2*y^3*z - 3300*x^2*y^3 + 1530*x^2*y^2*z^3 + 8379*x^2*y^2*z^2 + 14202*x^2*y^2*z + 7608*x^2*y^2 - 1038*x^2*y*z^4 - 7752*x^2*y*z^3 - 20064*x^2*y*z^2 - 21786*x^2*y*z - 8592*x^2*y + 276*x^2*z^5 + 2649*x^2*z^4 + 9309*x^2*z^3 + 15348*x^2*z^2 + 12234*x^2*z + 3854*x^2 - 16*x*y^6 + 138*x*y^5*z + 249*x*y^5 - 486*x*y^4*z^2 - 1785*x*y^4*z - 1515*x*y^4 + 894*x*y^3*z^3 + 5037*x*y^3*z^2 + 8712*x*y^3*z + 4724*x*y^3 - 906*x*y^2*z^4 - 6999*x*y^2*z^3 - 18519*x*y^2*z^2 - 20364*x*y^2*z - 8088*x*y^2 + 480*x*y*z^5 + 4794*x*y*z^4 + 17250*x*y*z^3 + 28815*x*y*z^2 + 23136*x*y*z + 7312*x*y - 104*x*z^6 - 1296*x*z^5 - 5940*x*z^4 - 13379*x*z^3 - 16263*x*z^2 - 10370*x*z - 2760*x - 2*y^7 + 20*y^6*z + 37*y^6 - 84*y^5*z^2 - 318*y^5*z - 276*y^5 + 192*y^4*z^3 + 1122*y^4*z^2 + 1989*y^4*z + 1094*y^4 - 258*y^3*z^4 - 2082*y^3*z^3 - 5658*y^3*z^2 - 6314*y^3*z - 2528*y^3 + 204*y^2*z^5 + 2145*y^2*z^4 + 7941*y^2*z^3 + 13467*y^2*z^2 + 10902*y^2*z + 3458*y^2 - 88*y*z^6 - 1164*y*z^5 - 5496*y*z^4 - 12570*y*z^3 - 15408*y*z^2 - 9866*y*z - 2628*y + 16*z^7 + 260*z^6 + 1500*z^5 + 4329*z^4 + 7127*z^3 + 6896*z^2 + 3698*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -1, -1, 1 ], [ -1, -1, 2 ], [ -2, -1, 1 ] ], "source_shift": [ 1, 2, 2 ], "target_matrix": [ [ -1, -1, -1 ], [ -1, -2, -1 ], [ 2, 3, 3 ] ], "target_shift": [ 343, 519, -857 ], "collision_preimages": [ [ "5/4", "-5/4", "-1" ], [ "-9/2", "1", "-7/2" ], [ "-13/2", "10", "3/2" ] ], "collision_image": [ "1373/4", "2077/4", "-1715/2" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0900535830005538 ], "surface_features": [ 3.4892508668539, 4.4596186239173745, 3.0925452076056064, 0.0, 3.6351918408281025, 4839 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B037", "pair": 37, "family": "bijection", "mapping": "F(x,y,z)=(2187*x^7 - 10206*x^6*y + 10206*x^6*z - 5832*x^6 + 20412*x^5*y^2 - 40824*x^5*y*z + 23328*x^5*y + 20412*x^5*z^2 - 23328*x^5*z + 6642*x^5 - 22680*x^4*y^3 + 68040*x^4*y^2*z - 38880*x^4*y^2 - 68040*x^4*y*z^2 + 77760*x^4*y*z - 22248*x^4*y + 22680*x^4*z^3 - 38880*x^4*z^2 + 22167*x^4*z - 4455*x^4 + 15120*x^3*y^4 - 60480*x^3*y^3*z + 34560*x^3*y^3 + 90720*x^3*y^2*z^2 - 103680*x^3*y^2*z + 29808*x^3*y^2 - 60480*x^3*y*z^3 + 103680*x^3*y*z^2 - 59400*x^3*y*z + 12015*x^3*y + 15120*x^3*z^4 - 34560*x^3*z^3 + 29592*x^3*z^2 - 11907*x^3*z + 1908*x^3 - 6048*x^2*y^5 + 30240*x^2*y^4*z - 17280*x^2*y^4 - 60480*x^2*y^3*z^2 + 69120*x^2*y^3*z - 19968*x^2*y^3 + 60480*x^2*y^2*z^3 - 103680*x^2*y^2*z^2 + 59688*x^2*y^2*z - 12150*x^2*y^2 - 30240*x^2*y*z^4 + 69120*x^2*y*z^3 - 59472*x^2*y*z^2 + 24084*x^2*y*z - 3876*x^2*y + 6048*x^2*z^5 - 17280*x^2*z^4 + 19752*x^2*z^3 - 11934*x^2*z^2 + 3822*x^2*z - 488*x^2 + 1344*x*y^6 - 8064*x*y^5*z + 4608*x*y^5 + 20160*x*y^4*z^2 - 23040*x*y^4*z + 6688*x*y^4 - 26880*x*y^3*z^3 + 46080*x*y^3*z^2 - 26656*x*y^3*z + 5460*x*y^3 + 20160*x*y^2*z^4 - 46080*x*y^2*z^3 + 39840*x*y^2*z^2 - 16236*x*y^2*z + 2624*x*y^2 - 8064*x*y*z^5 + 23040*x*y*z^4 - 26464*x*y*z^3 + 16092*x*y*z^2 - 5176*x*y*z + 668*x*y + 1344*x*z^6 - 4608*x*z^5 + 6592*x*z^4 - 5316*x*z^3 + 2552*x*z^2 - 651*x*z + 71*x - 128*y^7 + 896*y^6*z - 512*y^6 - 2688*y^5*z^2 + 3072*y^5*z - 896*y^5 + 4480*y^4*z^3 - 7680*y^4*z^2 + 4464*y^4*z - 920*y^4 - 4480*y^3*z^4 + 10240*y^3*z^3 - 8896*y^3*z^2 + 3648*y^3*z - 592*y^3 + 2688*y^2*z^5 - 7680*y^2*z^4 + 8864*y^2*z^3 - 5424*y^2*z^2 + 1752*y^2*z - 228*y^2 - 896*y*z^6 + 3072*y*z^5 - 4416*y*z^4 + 3584*y*z^3 - 1728*y*z^2 + 445*y*z - 46*y + 128*z^7 - 512*z^6 + 880*z^5 - 888*z^4 + 568*z^3 - 217*z^2 + 46*z - 1, 4374*x^7 - 20412*x^6*y + 20412*x^6*z - 11664*x^6 + 40824*x^5*y^2 - 81648*x^5*y*z + 46656*x^5*y + 40824*x^5*z^2 - 46656*x^5*z + 13284*x^5 - 45360*x^4*y^3 + 136080*x^4*y^2*z - 77760*x^4*y^2 - 136080*x^4*y*z^2 + 155520*x^4*y*z - 44496*x^4*y + 45360*x^4*z^3 - 77760*x^4*z^2 + 44334*x^4*z - 8829*x^4 + 30240*x^3*y^4 - 120960*x^3*y^3*z + 69120*x^3*y^3 + 181440*x^3*y^2*z^2 - 207360*x^3*y^2*z + 59616*x^3*y^2 - 120960*x^3*y*z^3 + 207360*x^3*y*z^2 - 118800*x^3*y*z + 23814*x^3*y + 30240*x^3*z^4 - 69120*x^3*z^3 + 59184*x^3*z^2 - 23598*x^3*z + 3870*x^3 - 12096*x^2*y^5 + 60480*x^2*y^4*z - 34560*x^2*y^4 - 120960*x^2*y^3*z^2 + 138240*x^2*y^3*z - 39936*x^2*y^3 + 120960*x^2*y^2*z^3 - 207360*x^2*y^2*z^2 + 119376*x^2*y^2*z - 24084*x^2*y^2 - 60480*x^2*y*z^4 + 138240*x^2*y*z^3 - 118944*x^2*y*z^2 + 47736*x^2*y*z - 7860*x^2*y + 12096*x^2*z^5 - 34560*x^2*z^4 + 39504*x^2*z^3 - 23652*x^2*z^2 + 7752*x^2*z - 1138*x^2 + 2688*x*y^6 - 16128*x*y^5*z + 9216*x*y^5 + 40320*x*y^4*z^2 - 46080*x*y^4*z + 13376*x*y^4 - 53760*x*y^3*z^3 + 92160*x*y^3*z^2 - 53312*x*y^3*z + 10824*x*y^3 + 40320*x*y^2*z^4 - 92160*x*y^2*z^3 + 79680*x*y^2*z^2 - 32184*x*y^2*z + 5320*x*y^2 - 16128*x*y*z^5 + 46080*x*y*z^4 - 52928*x*y*z^3 + 31896*x*y*z^2 - 10496*x*y*z + 1552*x*y + 2688*x*z^6 - 9216*x*z^5 + 13184*x*z^4 - 10536*x*z^3 + 5176*x*z^2 - 1518*x*z + 221*x - 256*y^7 + 1792*y^6*z - 1024*y^6 - 5376*y^5*z^2 + 6144*y^5*z - 1792*y^5 + 8960*y^4*z^3 - 15360*y^4*z^2 + 8928*y^4*z - 1824*y^4 - 8960*y^3*z^4 + 20480*y^3*z^3 - 17792*y^3*z^2 + 7232*y^3*z - 1200*y^3 + 5376*y^2*z^5 - 15360*y^2*z^4 + 17728*y^2*z^3 - 10752*y^2*z^2 + 3552*y^2*z - 528*y^2 - 1792*y*z^6 + 6144*y*z^5 - 8832*y*z^4 + 7104*y*z^3 - 3504*y*z^2 + 1034*y*z - 153*y + 256*z^7 - 1024*z^6 + 1760*z^5 - 1760*z^4 + 1152*z^3 - 506*z^2 + 147*z - 1, -6561*x^7 + 30618*x^6*y - 30618*x^6*z + 17496*x^6 - 61236*x^5*y^2 + 122472*x^5*y*z - 69984*x^5*y - 61236*x^5*z^2 + 69984*x^5*z - 19926*x^5 + 68040*x^4*y^3 - 204120*x^4*y^2*z + 116640*x^4*y^2 + 204120*x^4*y*z^2 - 233280*x^4*y*z + 66744*x^4*y - 68040*x^4*z^3 + 116640*x^4*z^2 - 66501*x^4*z + 13284*x^4 - 45360*x^3*y^4 + 181440*x^3*y^3*z - 103680*x^3*y^3 - 272160*x^3*y^2*z^2 + 311040*x^3*y^2*z - 89424*x^3*y^2 + 181440*x^3*y*z^3 - 311040*x^3*y*z^2 + 178200*x^3*y*z - 35829*x^3*y - 45360*x^3*z^4 + 103680*x^3*z^3 - 88776*x^3*z^2 + 35505*x^3*z - 5832*x^3 + 18144*x^2*y^5 - 90720*x^2*y^4*z + 51840*x^2*y^4 + 181440*x^2*y^3*z^2 - 207360*x^2*y^3*z + 59904*x^2*y^3 - 181440*x^2*y^2*z^3 + 311040*x^2*y^2*z^2 - 179064*x^2*y^2*z + 36234*x^2*y^2 + 90720*x^2*y*z^4 - 207360*x^2*y*z^3 + 178416*x^2*y*z^2 - 71820*x^2*y*z + 11844*x^2*y - 18144*x^2*z^5 + 51840*x^2*z^4 - 59256*x^2*z^3 + 35586*x^2*z^2 - 11682*x^2*z + 1698*x^2 - 4032*x*y^6 + 24192*x*y^5*z - 13824*x*y^5 - 60480*x*y^4*z^2 + 69120*x*y^4*z - 20064*x*y^4 + 80640*x*y^3*z^3 - 138240*x*y^3*z^2 + 79968*x*y^3*z - 16284*x*y^3 - 60480*x*y^2*z^4 + 138240*x*y^2*z^3 - 119520*x*y^2*z^2 + 48420*x*y^2*z - 8016*x*y^2 + 24192*x*y*z^5 - 69120*x*y*z^4 + 79392*x*y*z^3 - 47988*x*y*z^2 + 15816*x*y*z - 2316*x*y - 4032*x*z^6 + 13824*x*z^5 - 19776*x*z^4 + 15852*x*z^3 - 7800*x*z^2 + 2265*x*z - 324*x + 384*y^7 - 2688*y^6*z + 1536*y^6 + 8064*y^5*z^2 - 9216*y^5*z + 2688*y^5 - 13440*y^4*z^3 + 23040*y^4*z^2 - 13392*y^4*z + 2744*y^4 + 13440*y^3*z^4 - 30720*y^3*z^3 + 26688*y^3*z^2 - 10880*y^3*z + 1808*y^3 - 8064*y^2*z^5 + 23040*y^2*z^4 - 26592*y^2*z^3 + 16176*y^2*z^2 - 5352*y^2*z + 788*y^2 + 2688*y*z^6 - 9216*y*z^5 + 13248*y*z^4 - 10688*y*z^3 + 5280*y*z^2 - 1543*y*z + 223*y - 384*z^7 + 1536*z^6 - 2640*z^5 + 2648*z^4 - 1736*z^3 + 755*z^2 - 215*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -3, 2, -2 ], [ -2, 1, -1 ], [ -1, 2, -1 ] ], "source_shift": [ 1, 1, 1 ], "target_matrix": [ [ -1, -1, -2 ], [ -2, -1, -1 ], [ 3, 2, 2 ] ], "target_shift": [ 3, 22, -32 ], "inverse_recipe": { "p": 4, "q": 3, "f": "x**4 + x", "g": "x**3 + x**2", "h": "y*z + z", "polynomials": { "f": [ [ [ 4, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "g": [ [ [ 3, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 0, 1 ], 1 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0717972029997327 ], "surface_features": [ 4.600188407587466, 5.492816243246181, 4.224843716755171, 0.0, 4.74401983125335, 5263 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C097", "pair": 97, "family": "counterexample", "mapping": "F(x,y,z)=(-27*x^7 + 162*x^6*y + 189*x^6*z - 324*x^6 - 414*x^5*y^2 - 963*x^5*y*z + 1638*x^5*y - 549*x^5*z^2 + 1908*x^5*z - 1512*x^5 + 584*x^4*y^3 + 2031*x^4*y^2*z - 3426*x^4*y^2 + 2310*x^4*y*z^2 - 7941*x^4*y*z + 6264*x^4*y + 863*x^4*z^3 - 4515*x^4*z^2 + 7191*x^4*z - 3706*x^4 - 491*x^3*y^4 - 2269*x^3*y^3*z + 3794*x^3*y^3 - 3861*x^3*y^2*z^2 + 13122*x^3*y^2*z - 10299*x^3*y^2 - 2879*x^3*y*z^3 + 14862*x^3*y*z^2 - 23475*x^3*y*z + 12078*x^3*y - 796*x^3*z^4 + 5534*x^3*z^3 - 13176*x^3*z^2 + 13501*x^3*z - 5264*x^3 + 246*x^2*y^5 + 1416*x^2*y^4*z - 2346*x^2*y^4 + 3204*x^2*y^3*z^2 - 10761*x^2*y^3*z + 8400*x^2*y^3 + 3576*x^2*y^2*z^3 - 18207*x^2*y^2*z^2 + 28515*x^2*y^2*z - 14631*x^2*y^2 + 1974*x^2*y*z^4 - 13515*x^2*y*z^3 + 31830*x^2*y*z^2 - 32445*x^2*y*z + 12597*x^2*y + 432*x^2*z^5 - 3723*x^2*z^4 + 11715*x^2*z^3 - 17814*x^2*z^2 + 13671*x^2*z - 4273*x^2 - 68*x*y^6 - 468*x*y^5*z + 768*x*y^5 - 1320*x*y^4*z^2 + 4380*x*y^4*z - 3399*x*y^4 - 1960*x*y^3*z^3 + 9840*x*y^3*z^2 - 15279*x*y^3*z + 7809*x*y^3 - 1620*x*y^2*z^4 + 10920*x*y^2*z^3 - 25443*x*y^2*z^2 + 25785*x*y^2*z - 9945*x*y^2 - 708*x*y*z^5 + 6000*x*y*z^4 - 18645*x*y*z^3 + 28143*x*y*z^2 - 21441*x*y*z + 6617*x*y - 128*x*z^6 + 1308*x*z^5 - 5082*x*z^4 + 10167*x*z^3 - 11496*x*z^2 + 7004*x*z - 1792*x + 8*y^7 + 64*y^6*z - 104*y^6 + 216*y^5*z^2 - 708*y^5*z + 546*y^5 + 400*y^4*z^3 - 1980*y^4*z^2 + 3048*y^4*z - 1550*y^4 + 440*y^3*z^4 - 2920*y^3*z^3 + 6732*y^3*z^2 - 6779*y^3*z + 2593*y^3 + 288*y^2*z^5 - 2400*y^2*z^4 + 7368*y^2*z^3 - 11037*y^2*z^2 + 8337*y^2*z - 2539*y^2 + 104*y*z^6 - 1044*y*z^5 + 4002*y*z^4 - 7937*y*z^3 + 8895*y*z^2 - 5351*y*z + 1344*y + 16*z^7 - 188*z^6 + 864*z^5 - 2129*z^4 + 3151*z^3 - 2812*z^2 + 1397*z - 1, -27*x^7 + 162*x^6*y + 189*x^6*z - 405*x^6 - 414*x^5*y^2 - 963*x^5*y*z + 2043*x^5*y - 549*x^5*z^2 + 2394*x^5*z - 2079*x^5 + 584*x^4*y^3 + 2031*x^4*y^2*z - 4263*x^4*y^2 + 2310*x^4*y*z^2 - 9939*x^4*y*z + 8586*x^4*y + 863*x^4*z^3 - 5676*x^4*z^2 + 9918*x^4*z - 5380*x^4 - 491*x^3*y^4 - 2269*x^3*y^3*z + 4709*x^3*y^3 - 3861*x^3*y^2*z^2 + 16380*x^3*y^2*z - 14070*x^3*y^2 - 2879*x^3*y*z^3 + 18633*x^3*y*z^2 - 32259*x^3*y*z + 17469*x^3*y - 796*x^3*z^4 + 6962*x^3*z^3 - 18189*x^3*z^2 + 19621*x^3*z - 7913*x^3 + 246*x^2*y^5 + 1416*x^2*y^4*z - 2904*x^2*y^4 + 3204*x^2*y^3*z^2 - 13395*x^2*y^3*z + 11436*x^2*y^3 + 3576*x^2*y^2*z^3 - 22761*x^2*y^2*z^2 + 39036*x^2*y^2*z - 21081*x^2*y^2 + 1974*x^2*y*z^4 - 16953*x^2*y*z^3 + 43764*x^2*y*z^2 - 46947*x^2*y*z + 18855*x^2*y + 432*x^2*z^5 - 4683*x^2*z^4 + 16164*x^2*z^3 - 25866*x^2*z^2 + 20508*x^2*z - 6565*x^2 - 68*x*y^6 - 468*x*y^5*z + 948*x*y^5 - 1320*x*y^4*z^2 + 5436*x*y^4*z - 4611*x*y^4 - 1960*x*y^3*z^3 + 12264*x*y^3*z^2 - 20835*x*y^3*z + 11208*x*y^3 - 1620*x*y^2*z^4 + 13656*x*y^2*z^3 - 34839*x*y^2*z^2 + 37149*x*y^2*z - 14820*x*y^2 - 708*x*y*z^5 + 7524*x*y*z^4 - 25617*x*y*z^3 + 40674*x*y*z^2 - 32013*x*y*z + 10106*x*y - 128*x*z^6 + 1644*x*z^5 - 7002*x*z^4 + 14733*x*z^3 - 17193*x*z^2 + 10700*x*z - 2759*x + 8*y^7 + 64*y^6*z - 128*y^6 + 216*y^5*z^2 - 876*y^5*z + 738*y^5 + 400*y^4*z^3 - 2460*y^4*z^2 + 4140*y^4*z - 2216*y^4 + 440*y^3*z^4 - 3640*y^3*z^3 + 9180*y^3*z^2 - 9725*y^3*z + 3847*y^3 + 288*y^2*z^5 - 3000*y^2*z^4 + 10080*y^2*z^3 - 15879*y^2*z^2 + 12390*y^2*z - 3856*y^2 + 104*y*z^6 - 1308*y*z^5 + 5490*y*z^4 - 11447*y*z^3 + 13239*y*z^2 - 8129*y*z + 2056*y + 16*z^7 - 236*z^6 + 1188*z^5 - 3077*z^4 + 4696*z^3 - 4273*z^2 + 2136*z - 1, 27*x^7 - 162*x^6*y - 189*x^6*z + 324*x^6 + 414*x^5*y^2 + 963*x^5*y*z - 1638*x^5*y + 549*x^5*z^2 - 1908*x^5*z + 1512*x^5 - 584*x^4*y^3 - 2031*x^4*y^2*z + 3426*x^4*y^2 - 2310*x^4*y*z^2 + 7941*x^4*y*z - 6264*x^4*y - 863*x^4*z^3 + 4515*x^4*z^2 - 7191*x^4*z + 3679*x^4 + 491*x^3*y^4 + 2269*x^3*y^3*z - 3794*x^3*y^3 + 3861*x^3*y^2*z^2 - 13122*x^3*y^2*z + 10299*x^3*y^2 + 2879*x^3*y*z^3 - 14862*x^3*y*z^2 + 23475*x^3*y*z - 11997*x^3*y + 796*x^3*z^4 - 5534*x^3*z^3 + 13176*x^3*z^2 - 13393*x^3*z + 5183*x^3 - 246*x^2*y^5 - 1416*x^2*y^4*z + 2346*x^2*y^4 - 3204*x^2*y^3*z^2 + 10761*x^2*y^3*z - 8400*x^2*y^3 - 3576*x^2*y^2*z^3 + 18207*x^2*y^2*z^2 - 28515*x^2*y^2*z + 14541*x^2*y^2 - 1974*x^2*y*z^4 + 13515*x^2*y*z^3 - 31830*x^2*y*z^2 + 32211*x^2*y*z - 12417*x^2*y - 432*x^2*z^5 + 3723*x^2*z^4 - 11715*x^2*z^3 + 17670*x^2*z^2 - 13464*x^2*z + 4165*x^2 + 68*x*y^6 + 468*x*y^5*z - 768*x*y^5 + 1320*x*y^4*z^2 - 4380*x*y^4*z + 3399*x*y^4 + 1960*x*y^3*z^3 - 9840*x*y^3*z^2 + 15279*x*y^3*z - 7765*x*y^3 + 1620*x*y^2*z^4 - 10920*x*y^2*z^3 + 25443*x*y^2*z^2 - 25617*x*y^2*z + 9813*x*y^2 + 708*x*y*z^5 - 6000*x*y*z^4 + 18645*x*y*z^3 - 27939*x*y*z^2 + 21141*x*y*z - 6464*x*y + 128*x*z^6 - 1308*x*z^5 + 5082*x*z^4 - 10087*x*z^3 + 11328*x*z^2 - 6842*x*z + 1749*x - 8*y^7 - 64*y^6*z + 104*y^6 - 216*y^5*z^2 + 708*y^5*z - 546*y^5 - 400*y^4*z^3 + 1980*y^4*z^2 - 3048*y^4*z + 1542*y^4 - 440*y^3*z^4 + 2920*y^3*z^3 - 6732*y^3*z^2 + 6739*y^3*z - 2561*y^3 - 288*y^2*z^5 + 2400*y^2*z^4 - 7368*y^2*z^3 + 10965*y^2*z^2 - 8229*y^2*z + 2485*y^2 - 104*y*z^6 + 1044*y*z^5 - 4002*y*z^4 + 7881*y*z^3 - 8775*y*z^2 + 5237*y*z - 1314*y - 16*z^7 + 188*z^6 - 864*z^5 + 2113*z^4 - 3107*z^3 + 2752*z^2 - 1366*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ 3, -2, -2 ], [ 1, -1, -1 ], [ 1, -1, -2 ] ], "source_shift": [ 1, 2, 1 ], "target_matrix": [ [ -1, -1, -1 ], [ -1, -2, -1 ], [ 1, 1, 2 ] ], "target_shift": [ 290, 439, -285 ], "collision_preimages": [ [ "3", "23/4", "-3/4" ], [ "7", "39/2", "-9" ], [ "-1", "11/2", "-6" ] ], "collision_image": [ "1161/4", "1757/4", "-1141/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0650087050016737 ], "surface_features": [ 3.8424332744894487, 4.671607845269368, 3.5347873586294916, 0.0, 3.9315868619991616, 5023 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B103", "pair": 103, "family": "bijection", "mapping": "F(x,y,z)=(1792*x^7 - 6272*x^6*y + 6272*x^6*z - 13440*x^6 + 9408*x^5*y^2 - 18816*x^5*y*z + 40320*x^5*y + 9408*x^5*z^2 - 40320*x^5*z + 44352*x^5 - 7840*x^4*y^3 + 23520*x^4*y^2*z - 50400*x^4*y^2 - 23520*x^4*y*z^2 + 100800*x^4*y*z - 111104*x^4*y + 7840*x^4*z^3 - 50400*x^4*z^2 + 111328*x^4*z - 83408*x^4 + 3920*x^3*y^4 - 15680*x^3*y^3*z + 33600*x^3*y^3 + 23520*x^3*y^2*z^2 - 100800*x^3*y^2*z + 111328*x^3*y^2 - 15680*x^3*y*z^3 + 100800*x^3*y*z^2 - 223104*x^3*y*z + 167768*x^3*y + 3920*x^3*z^4 - 33600*x^3*z^3 + 111776*x^3*z^2 - 168776*x^3*z + 96928*x^3 - 1176*x^2*y^5 + 5880*x^2*y^4*z - 12600*x^2*y^4 - 11760*x^2*y^3*z^2 + 50400*x^2*y^3*z - 55776*x^2*y^3 + 11760*x^2*y^2*z^3 - 75600*x^2*y^2*z^2 + 167664*x^2*y^2*z - 126540*x^2*y^2 - 5880*x^2*y*z^4 + 50400*x^2*y*z^3 - 168000*x^2*y*z^2 + 254592*x^2*y*z - 146988*x^2*y + 1176*x^2*z^5 - 12600*x^2*z^4 + 56112*x^2*z^3 - 128052*x^2*z^2 + 148780*x^2*z - 70220*x^2 + 196*x*y^6 - 1176*x*y^5*z + 2520*x*y^5 + 2940*x*y^4*z^2 - 12600*x*y^4*z + 13972*x*y^4 - 3920*x*y^3*z^3 + 25200*x*y^3*z^2 - 56000*x*y^3*z + 42418*x*y^3 + 2940*x*y^2*z^4 - 25200*x*y^2*z^3 + 84168*x*y^2*z^2 - 128010*x*y^2*z + 74292*x*y^2 - 1176*x*y*z^5 + 12600*x*y*z^4 - 56224*x*y*z^3 + 128766*x*y*z^2 - 150376*x*y*z + 71522*x*y + 196*x*z^6 - 2520*x*z^5 + 14084*x*z^4 - 43174*x*z^3 + 76084*x*z^2 - 73076*x*z + 29794*x - 14*y^7 + 98*y^6*z - 210*y^6 - 294*y^5*z^2 + 1260*y^5*z - 1400*y^5 + 490*y^4*z^3 - 3150*y^4*z^2 + 7014*y^4*z - 5332*y^4 - 490*y^3*z^4 + 4200*y^3*z^3 - 14056*y^3*z^2 + 21454*y^3*z - 12515*y^3 + 294*y^2*z^5 - 3150*y^2*z^4 + 14084*y^2*z^3 - 32370*y^2*z^2 + 37993*y^2*z - 18213*y^2 - 98*y*z^6 + 1260*y*z^5 - 7056*y*z^4 + 21706*y*z^3 - 38441*y*z^2 + 37224*y*z - 15373*y + 14*z^7 - 210*z^6 + 1414*z^5 - 5458*z^4 + 12963*z^3 - 19025*z^2 + 15960*z - 1, -896*x^7 + 3136*x^6*y - 3136*x^6*z + 6720*x^6 - 4704*x^5*y^2 + 9408*x^5*y*z - 20160*x^5*y - 4704*x^5*z^2 + 20160*x^5*z - 22176*x^5 + 3920*x^4*y^3 - 11760*x^4*y^2*z + 25200*x^4*y^2 + 11760*x^4*y*z^2 - 50400*x^4*y*z + 55552*x^4*y - 3920*x^4*z^3 + 25200*x^4*z^2 - 55664*x^4*z + 41688*x^4 - 1960*x^3*y^4 + 7840*x^3*y^3*z - 16800*x^3*y^3 - 11760*x^3*y^2*z^2 + 50400*x^3*y^2*z - 55664*x^3*y^2 + 7840*x^3*y*z^3 - 50400*x^3*y*z^2 + 111552*x^3*y*z - 83852*x^3*y - 1960*x^3*z^4 + 16800*x^3*z^3 - 55888*x^3*z^2 + 84356*x^3*z - 48412*x^3 + 588*x^2*y^5 - 2940*x^2*y^4*z + 6300*x^2*y^4 + 5880*x^2*y^3*z^2 - 25200*x^2*y^3*z + 27888*x^2*y^3 - 5880*x^2*y^2*z^3 + 37800*x^2*y^2*z^2 - 83832*x^2*y^2*z + 63246*x^2*y^2 + 2940*x^2*y*z^4 - 25200*x^2*y*z^3 + 84000*x^2*y*z^2 - 127248*x^2*y*z + 73416*x^2*y - 588*x^2*z^5 + 6300*x^2*z^4 - 28056*x^2*z^3 + 64002*x^2*z^2 - 74312*x^2*z + 35052*x^2 - 98*x*y^6 + 588*x*y^5*z - 1260*x*y^5 - 1470*x*y^4*z^2 + 6300*x*y^4*z - 6986*x*y^4 + 1960*x*y^3*z^3 - 12600*x*y^3*z^2 + 28000*x*y^3*z - 21201*x*y^3 - 1470*x*y^2*z^4 + 12600*x*y^2*z^3 - 42084*x*y^2*z^2 + 63981*x*y^2*z - 37107*x*y^2 + 588*x*y*z^5 - 6300*x*y*z^4 + 28112*x*y*z^3 - 64359*x*y*z^2 + 75110*x*y*z - 35703*x*y - 98*x*z^6 + 1260*x*z^5 - 7042*x*z^4 + 21579*x*z^3 - 38003*x*z^2 + 36480*x*z - 14876*x + 7*y^7 - 49*y^6*z + 105*y^6 + 147*y^5*z^2 - 630*y^5*z + 700*y^5 - 245*y^4*z^3 + 1575*y^4*z^2 - 3507*y^4*z + 2665*y^4 + 245*y^3*z^4 - 2100*y^3*z^3 + 7028*y^3*z^2 - 10723*y^3*z + 6251*y^3 - 147*y^2*z^5 + 1575*y^2*z^4 - 7042*y^2*z^3 + 16179*y^2*z^2 - 18977*y^2*z + 9092*y^2 + 49*y*z^6 - 630*y*z^5 + 3528*y*z^4 - 10849*y*z^3 + 19201*y*z^2 - 18583*y*z + 7677*y - 7*z^7 + 105*z^6 - 707*z^5 + 2728*z^4 - 6475*z^3 + 9498*z^2 - 7971*z + 1, 1792*x^7 - 6272*x^6*y + 6272*x^6*z - 13440*x^6 + 9408*x^5*y^2 - 18816*x^5*y*z + 40320*x^5*y + 9408*x^5*z^2 - 40320*x^5*z + 44352*x^5 - 7840*x^4*y^3 + 23520*x^4*y^2*z - 50400*x^4*y^2 - 23520*x^4*y*z^2 + 100800*x^4*y*z - 111104*x^4*y + 7840*x^4*z^3 - 50400*x^4*z^2 + 111328*x^4*z - 83376*x^4 + 3920*x^3*y^4 - 15680*x^3*y^3*z + 33600*x^3*y^3 + 23520*x^3*y^2*z^2 - 100800*x^3*y^2*z + 111328*x^3*y^2 - 15680*x^3*y*z^3 + 100800*x^3*y*z^2 - 223104*x^3*y*z + 167704*x^3*y + 3920*x^3*z^4 - 33600*x^3*z^3 + 111776*x^3*z^2 - 168712*x^3*z + 96816*x^3 - 1176*x^2*y^5 + 5880*x^2*y^4*z - 12600*x^2*y^4 - 11760*x^2*y^3*z^2 + 50400*x^2*y^3*z - 55776*x^2*y^3 + 11760*x^2*y^2*z^3 - 75600*x^2*y^2*z^2 + 167664*x^2*y^2*z - 126492*x^2*y^2 - 5880*x^2*y*z^4 + 50400*x^2*y*z^3 - 168000*x^2*y*z^2 + 254496*x^2*y*z - 146820*x^2*y + 1176*x^2*z^5 - 12600*x^2*z^4 + 56112*x^2*z^3 - 128004*x^2*z^2 + 148612*x^2*z - 70076*x^2 + 196*x*y^6 - 1176*x*y^5*z + 2520*x*y^5 + 2940*x*y^4*z^2 - 12600*x*y^4*z + 13972*x*y^4 - 3920*x*y^3*z^3 + 25200*x*y^3*z^2 - 56000*x*y^3*z + 42402*x*y^3 + 2940*x*y^2*z^4 - 25200*x*y^2*z^3 + 84168*x*y^2*z^2 - 127962*x*y^2*z + 74208*x*y^2 - 1176*x*y*z^5 + 12600*x*y*z^4 - 56224*x*y*z^3 + 128718*x*y*z^2 - 150208*x*y*z + 71378*x*y + 196*x*z^6 - 2520*x*z^5 + 14084*x*z^4 - 43158*x*z^3 + 76000*x*z^2 - 72932*x*z + 29716*x - 14*y^7 + 98*y^6*z - 210*y^6 - 294*y^5*z^2 + 1260*y^5*z - 1400*y^5 + 490*y^4*z^3 - 3150*y^4*z^2 + 7014*y^4*z - 5330*y^4 - 490*y^3*z^4 + 4200*y^3*z^3 - 14056*y^3*z^2 + 21446*y^3*z - 12501*y^3 + 294*y^2*z^5 - 3150*y^2*z^4 + 14084*y^2*z^3 - 32358*y^2*z^2 + 37951*y^2*z - 18177*y^2 - 98*y*z^6 + 1260*y*z^5 - 7056*y*z^4 + 21698*y*z^3 - 38399*y*z^2 + 37152*y*z - 15335*y + 14*z^7 - 210*z^6 + 1414*z^5 - 5456*z^4 + 12949*z^3 - 18989*z^2 + 15922*z - 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -2, 1, -1 ], [ -2, 2, -3 ], [ -1, 1, -2 ] ], "source_shift": [ 2, 2, 2 ], "target_matrix": [ [ -2, -1, -1 ], [ 1, 2, 1 ], [ -2, -3, -2 ] ], "target_shift": [ 5795, -2897, 5779 ], "inverse_recipe": { "p": 3, "q": 4, "f": "x**3 + x**2 + x", "g": "x**4 + x**2 + x", "h": "y*z + 2*y", "polynomials": { "f": [ [ [ 3, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "g": [ [ [ 4, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 1, 0 ], 2 ] ] }, "h_multiplier": 7 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0698903380034608 ], "surface_features": [ 4.542506571524565, 5.405844752772304, 4.147861748487238, 0.0, 4.669576225943303, 5227 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C045", "pair": 45, "family": "counterexample", "mapping": "F(x,y,z)=(16*x^7 - 136*x^6*y + 152*x^6*z - 256*x^6 + 492*x^5*y^2 - 1104*x^5*y*z + 1944*x^5*y + 612*x^5*z^2 - 2088*x^5*z + 1404*x^5 - 982*x^4*y^3 + 3318*x^4*y^2*z - 6120*x^4*y^2 - 3690*x^4*y*z^2 + 13152*x^4*y*z - 9102*x^4*y + 1354*x^4*z^3 - 7032*x^4*z^2 + 9642*x^4*z - 3809*x^4 + 1168*x^3*y^4 - 5282*x^3*y^3*z + 10222*x^3*y^3 + 8838*x^3*y^2*z^2 - 32958*x^3*y^2*z + 23502*x^3*y^2 - 6502*x^3*y*z^3 + 35250*x^3*y*z^2 - 49782*x^3*y*z + 20162*x^3*y + 1778*x^3*z^4 - 12514*x^3*z^3 + 26280*x^3*z^2 - 21231*x^3*z + 5643*x^3 - 828*x^2*y^5 + 4698*x^2*y^4*z - 9552*x^2*y^4 - 10512*x^2*y^3*z^2 + 41064*x^2*y^3*z - 30204*x^2*y^3 + 11628*x^2*y^2*z^3 - 65880*x^2*y^2*z^2 + 95928*x^2*y^2*z - 39876*x^2*y^2 - 6372*x^2*y*z^4 + 46776*x^2*y*z^3 - 101244*x^2*y*z^2 + 83955*x^2*y*z - 22848*x^2*y + 1386*x^2*z^5 - 12408*x^2*z^4 + 35520*x^2*z^3 - 44079*x^2*z^2 + 24012*x^2*z - 4559*x^2 + 324*x*y^6 - 2214*x*y^5*z + 4734*x*y^5 + 6210*x*y^4*z^2 - 25434*x*y^4*z + 19314*x*y^4 - 9180*x*y^3*z^3 + 54396*x*y^3*z^2 - 81738*x*y^3*z + 34916*x*y^3 + 7560*x*y^2*z^4 - 57924*x*y^2*z^3 + 129330*x*y^2*z^2 - 110211*x*y^2*z + 30753*x*y^2 - 3294*x*y*z^5 + 30726*x*y*z^4 - 90702*x*y*z^3 + 115674*x*y*z^2 - 64632*x*y*z + 12534*x*y + 594*x*z^6 - 6498*x*z^5 + 23796*x*z^4 - 40379*x*z^3 + 33879*x*z^2 - 13182*x*z + 1838*x - 54*y^7 + 432*y^6*z - 972*y^6 - 1458*y^5*z^2 + 6264*y^5*z - 4914*y^5 + 2700*y^4*z^3 - 16740*y^4*z^2 + 25974*y^4*z - 11417*y^4 - 2970*y^3*z^4 + 23760*y^3*z^3 - 54756*y^3*z^2 + 48013*y^3*z - 13760*y^3 + 1944*y^2*z^5 - 18900*y^2*z^4 + 57564*y^2*z^3 - 75537*y^2*z^2 + 43362*y^2*z - 8595*y^2 - 702*y*z^6 + 7992*y*z^5 - 30186*y*z^4 + 52703*y*z^3 - 45444*y*z^2 + 18090*y*z - 2564*y + 108*z^7 - 1404*z^6 + 6318*z^5 - 13762*z^4 + 15842*z^3 - 9495*z^2 + 2702*z - 1, -8*x^7 + 68*x^6*y - 76*x^6*z + 140*x^6 - 246*x^5*y^2 + 552*x^5*y*z - 1056*x^5*y - 306*x^5*z^2 + 1140*x^5*z - 870*x^5 + 491*x^4*y^3 - 1659*x^4*y^2*z + 3303*x^4*y^2 + 1845*x^4*y*z^2 - 7134*x^4*y*z + 5589*x^4*y - 677*x^4*z^3 + 3831*x^4*z^2 - 5943*x^4*z + 2616*x^4 - 584*x^3*y^4 + 2641*x^3*y^3*z - 5483*x^3*y^3 - 4419*x^3*y^2*z^2 + 17766*x^3*y^2*z - 14298*x^3*y^2 + 3251*x^3*y*z^3 - 19083*x^3*y*z^2 + 30399*x^3*y*z - 13731*x^3*y - 889*x^3*z^4 + 6800*x^3*z^3 - 16101*x^3*z^2 + 14497*x^3*z - 4193*x^3 + 414*x^2*y^5 - 2349*x^2*y^4*z + 5094*x^2*y^4 + 5256*x^2*y^3*z^2 - 22005*x^2*y^3*z + 18204*x^2*y^3 - 5814*x^2*y^2*z^3 + 35451*x^2*y^2*z^2 - 58026*x^2*y^2*z + 26916*x^2*y^2 + 3186*x^2*y*z^4 - 25263*x^2*y*z^3 + 61440*x^2*y*z^2 - 56814*x^2*y*z + 16857*x^2*y - 693*x^2*z^5 + 6723*x^2*z^4 - 21618*x^2*z^3 + 29898*x^2*z^2 - 17754*x^2*z + 3622*x^2 - 162*x*y^6 + 1107*x*y^5*z - 2511*x*y^5 - 3105*x*y^4*z^2 + 13554*x*y^4*z - 11532*x*y^4 + 4590*x*y^3*z^3 - 29106*x*y^3*z^2 + 48975*x*y^3*z - 23346*x*y^3 - 3780*x*y^2*z^4 + 31104*x*y^2*z^3 - 77733*x*y^2*z^2 + 73872*x*y^2*z - 22524*x*y^2 + 1647*x*y*z^5 - 16551*x*y*z^4 + 54669*x*y*z^3 - 77706*x*y*z^2 + 47433*x*y*z - 9882*x*y - 297*x*z^6 + 3510*x*z^5 - 14379*x*z^4 + 27180*x*z^3 - 24909*x*z^2 + 10419*x*z - 1569*x + 27*y^7 - 216*y^6*z + 513*y^6 + 729*y^5*z^2 - 3321*y^5*z + 2907*y^5 - 1350*y^4*z^3 + 8910*y^4*z^2 - 15417*y^4*z + 7557*y^4 + 1485*y^3*z^4 - 12690*y^3*z^3 + 32598*y^3*z^2 - 31854*y^3*z + 10002*y^3 - 972*y^2*z^5 + 10125*y^2*z^4 - 34362*y^2*z^3 + 50220*y^2*z^2 - 31578*y^2*z + 6726*y^2 + 351*y*z^6 - 4293*y*z^5 + 18063*y*z^4 - 35106*y*z^3 + 33150*y*z^2 - 14187*y*z + 2167*y - 54*z^7 + 756*z^6 - 3789*z^5 + 9183*z^4 - 11574*z^3 + 7461*z^2 - 2294*z + 1, 8*x^7 - 68*x^6*y + 76*x^6*z - 128*x^6 + 246*x^5*y^2 - 552*x^5*y*z + 972*x^5*y + 306*x^5*z^2 - 1044*x^5*z + 702*x^5 - 491*x^4*y^3 + 1659*x^4*y^2*z - 3060*x^4*y^2 - 1845*x^4*y*z^2 + 6576*x^4*y*z - 4551*x^4*y + 677*x^4*z^3 - 3516*x^4*z^2 + 4821*x^4*z - 1904*x^4 + 584*x^3*y^4 - 2641*x^3*y^3*z + 5111*x^3*y^3 + 4419*x^3*y^2*z^2 - 16479*x^3*y^2*z + 11751*x^3*y^2 - 3251*x^3*y*z^3 + 17625*x^3*y*z^2 - 24891*x^3*y*z + 10079*x^3*y + 889*x^3*z^4 - 6257*x^3*z^3 + 13140*x^3*z^2 - 10613*x^3*z + 2815*x^3 - 414*x^2*y^5 + 2349*x^2*y^4*z - 4776*x^2*y^4 - 5256*x^2*y^3*z^2 + 20532*x^2*y^3*z - 15102*x^2*y^3 + 5814*x^2*y^2*z^3 - 32940*x^2*y^2*z^2 + 47964*x^2*y^2*z - 19935*x^2*y^2 - 3186*x^2*y*z^4 + 23388*x^2*y*z^3 - 50622*x^2*y*z^2 + 41970*x^2*y*z - 11403*x^2*y + 693*x^2*z^5 - 6204*x^2*z^4 + 17760*x^2*z^3 - 22035*x^2*z^2 + 11982*x^2*z - 2257*x^2 + 162*x*y^6 - 1107*x*y^5*z + 2367*x*y^5 + 3105*x*y^4*z^2 - 12717*x*y^4*z + 9657*x*y^4 - 4590*x*y^3*z^3 + 27198*x*y^3*z^2 - 40869*x*y^3*z + 17456*x*y^3 + 3780*x*y^2*z^4 - 28962*x*y^2*z^3 + 64665*x*y^2*z^2 - 55098*x*y^2*z + 15354*x*y^2 - 1647*x*y*z^5 + 15363*x*y*z^4 - 45351*x*y*z^3 + 57828*x*y*z^2 - 32265*x*y*z + 6216*x*y + 297*x*z^6 - 3249*x*z^5 + 11898*x*z^4 - 20186*x*z^3 + 16911*x*z^2 - 6534*x*z + 892*x - 27*y^7 + 216*y^6*z - 486*y^6 - 729*y^5*z^2 + 3132*y^5*z - 2457*y^5 + 1350*y^4*z^3 - 8370*y^4*z^2 + 12987*y^4*z - 5708*y^4 - 1485*y^3*z^4 + 11880*y^3*z^3 - 27378*y^3*z^2 + 24004*y^3*z - 6872*y^3 + 972*y^2*z^5 - 9450*y^2*z^4 + 28782*y^2*z^3 - 37764*y^2*z^2 + 21654*y^2*z - 4269*y^2 - 351*y*z^6 + 3996*y*z^5 - 15093*y*z^4 + 26348*y*z^3 - 22692*y*z^2 + 8982*y*z - 1249*y + 54*z^7 - 702*z^6 + 3159*z^5 - 6880*z^4 + 7910*z^3 - 4713*z^2 + 1314*z - 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -1, 1, -1 ], [ -2, 3, -3 ], [ -1, 1, -2 ] ], "source_shift": [ 2, 1, 1 ], "target_matrix": [ [ -2, -2, -3 ], [ 1, 2, 1 ], [ -1, -1, -2 ] ], "target_shift": [ 295, -270, 139 ], "collision_preimages": [ [ "5", "9/4", "-3/4" ], [ "1/2", "-7", "-13/2" ], [ "19/2", "-2", "-17/2" ] ], "collision_image": [ "591/2", "-1081/4", "557/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0575598940013151 ], "surface_features": [ 4.195584651648189, 5.111699277573551, 3.822788990355016, 0.0, 4.323916842330134, 5117 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B023", "pair": 23, "family": "bijection", "mapping": "F(x,y,z)=(2187*x^7 - 10206*x^6*y - 10206*x^6*z + 5832*x^6 + 20412*x^5*y^2 + 40824*x^5*y*z - 23328*x^5*y + 20412*x^5*z^2 - 23328*x^5*z + 6966*x^5 - 22680*x^4*y^3 - 68040*x^4*y^2*z + 38880*x^4*y^2 - 68040*x^4*y*z^2 + 77760*x^4*y*z - 23193*x^4*y - 22680*x^4*z^3 + 38880*x^4*z^2 - 23193*x^4*z + 5130*x^4 + 15120*x^3*y^4 + 60480*x^3*y^3*z - 34560*x^3*y^3 + 90720*x^3*y^2*z^2 - 103680*x^3*y^2*z + 30888*x^3*y^2 + 60480*x^3*y*z^3 - 103680*x^3*y*z^2 + 61776*x^3*y*z - 13635*x^3*y + 15120*x^3*z^4 - 34560*x^3*z^3 + 30888*x^3*z^2 - 13662*x^3*z + 2529*x^3 - 6048*x^2*y^5 - 30240*x^2*y^4*z + 17280*x^2*y^4 - 60480*x^2*y^3*z^2 + 69120*x^2*y^3*z - 20568*x^2*y^3 - 60480*x^2*y^2*z^3 + 103680*x^2*y^2*z^2 - 61704*x^2*y^2*z + 13590*x^2*y^2 - 30240*x^2*y*z^4 + 69120*x^2*y*z^3 - 61704*x^2*y*z^2 + 27234*x^2*y*z - 5028*x^2*y - 6048*x^2*z^5 + 17280*x^2*z^4 - 20568*x^2*z^3 + 13644*x^2*z^2 - 5055*x^2*z + 831*x^2 + 1344*x*y^6 + 8064*x*y^5*z - 4608*x*y^5 + 20160*x*y^4*z^2 - 23040*x*y^4*z + 6848*x*y^4 + 26880*x*y^3*z^3 - 46080*x*y^3*z^2 + 27392*x*y^3*z - 6020*x*y^3 + 20160*x*y^2*z^4 - 46080*x*y^2*z^3 + 41088*x*y^2*z^2 - 18096*x*y^2*z + 3332*x*y^2 + 8064*x*y*z^5 - 23040*x*y*z^4 + 27392*x*y*z^3 - 18132*x*y*z^2 + 6700*x*y*z - 1097*x*y + 1344*x*z^6 - 4608*x*z^5 + 6848*x*z^4 - 6056*x*z^3 + 3368*x*z^2 - 1108*x*z + 179*x - 128*y^7 - 896*y^6*z + 512*y^6 - 2688*y^5*z^2 + 3072*y^5*z - 912*y^5 - 4480*y^4*z^3 + 7680*y^4*z^2 - 4560*y^4*z + 1000*y^4 - 4480*y^3*z^4 + 10240*y^3*z^3 - 9120*y^3*z^2 + 4008*y^3*z - 736*y^3 - 2688*y^2*z^5 + 7680*y^2*z^4 - 9120*y^2*z^3 + 6024*y^2*z^2 - 2220*y^2*z + 362*y^2 - 896*y*z^6 + 3072*y*z^5 - 4560*y*z^4 + 4024*y*z^3 - 2232*y*z^2 + 731*y*z - 118*y - 128*z^7 + 512*z^6 - 912*z^5 + 1008*z^4 - 748*z^3 + 369*z^2 - 122*z + 1, 4374*x^7 - 20412*x^6*y - 20412*x^6*z + 11664*x^6 + 40824*x^5*y^2 + 81648*x^5*y*z - 46656*x^5*y + 40824*x^5*z^2 - 46656*x^5*z + 13932*x^5 - 45360*x^4*y^3 - 136080*x^4*y^2*z + 77760*x^4*y^2 - 136080*x^4*y*z^2 + 155520*x^4*y*z - 46386*x^4*y - 45360*x^4*z^3 + 77760*x^4*z^2 - 46386*x^4*z + 10260*x^4 + 30240*x^3*y^4 + 120960*x^3*y^3*z - 69120*x^3*y^3 + 181440*x^3*y^2*z^2 - 207360*x^3*y^2*z + 61776*x^3*y^2 + 120960*x^3*y*z^3 - 207360*x^3*y*z^2 + 123552*x^3*y*z - 27270*x^3*y + 30240*x^3*z^4 - 69120*x^3*z^3 + 61776*x^3*z^2 - 27324*x^3*z + 5112*x^3 - 12096*x^2*y^5 - 60480*x^2*y^4*z + 34560*x^2*y^4 - 120960*x^2*y^3*z^2 + 138240*x^2*y^3*z - 41136*x^2*y^3 - 120960*x^2*y^2*z^3 + 207360*x^2*y^2*z^2 - 123408*x^2*y^2*z + 27180*x^2*y^2 - 60480*x^2*y*z^4 + 138240*x^2*y*z^3 - 123408*x^2*y*z^2 + 54468*x^2*y*z - 10164*x^2*y - 12096*x^2*z^5 + 34560*x^2*z^4 - 41136*x^2*z^3 + 27288*x^2*z^2 - 10218*x^2*z + 1716*x^2 + 2688*x*y^6 + 16128*x*y^5*z - 9216*x*y^5 + 40320*x*y^4*z^2 - 46080*x*y^4*z + 13696*x*y^4 + 53760*x*y^3*z^3 - 92160*x*y^3*z^2 + 54784*x*y^3*z - 12040*x*y^3 + 40320*x*y^2*z^4 - 92160*x*y^2*z^3 + 82176*x*y^2*z^2 - 36192*x*y^2*z + 6736*x*y^2 + 16128*x*y*z^5 - 46080*x*y*z^4 + 54784*x*y*z^3 - 36264*x*y*z^2 + 13544*x*y*z - 2266*x*y + 2688*x*z^6 - 9216*x*z^5 + 13696*x*z^4 - 12112*x*z^3 + 6808*x*z^2 - 2288*x*z + 380*x - 256*y^7 - 1792*y^6*z + 1024*y^6 - 5376*y^5*z^2 + 6144*y^5*z - 1824*y^5 - 8960*y^4*z^3 + 15360*y^4*z^2 - 9120*y^4*z + 2000*y^4 - 8960*y^3*z^4 + 20480*y^3*z^3 - 18240*y^3*z^2 + 8016*y^3*z - 1488*y^3 - 5376*y^2*z^5 + 15360*y^2*z^4 - 18240*y^2*z^3 + 12048*y^2*z^2 - 4488*y^2*z + 748*y^2 - 1792*y*z^6 + 6144*y*z^5 - 9120*y*z^4 + 8048*y*z^3 - 4512*y*z^2 + 1510*y*z - 250*y - 256*z^7 + 1024*z^6 - 1824*z^5 + 2016*z^4 - 1512*z^3 + 762*z^2 - 258*z + 1, 2187*x^7 - 10206*x^6*y - 10206*x^6*z + 5832*x^6 + 20412*x^5*y^2 + 40824*x^5*y*z - 23328*x^5*y + 20412*x^5*z^2 - 23328*x^5*z + 6966*x^5 - 22680*x^4*y^3 - 68040*x^4*y^2*z + 38880*x^4*y^2 - 68040*x^4*y*z^2 + 77760*x^4*y*z - 23193*x^4*y - 22680*x^4*z^3 + 38880*x^4*z^2 - 23193*x^4*z + 5211*x^4 + 15120*x^3*y^4 + 60480*x^3*y^3*z - 34560*x^3*y^3 + 90720*x^3*y^2*z^2 - 103680*x^3*y^2*z + 30888*x^3*y^2 + 60480*x^3*y*z^3 - 103680*x^3*y*z^2 + 61776*x^3*y*z - 13851*x^3*y + 15120*x^3*z^4 - 34560*x^3*z^3 + 30888*x^3*z^2 - 13878*x^3*z + 2691*x^3 - 6048*x^2*y^5 - 30240*x^2*y^4*z + 17280*x^2*y^4 - 60480*x^2*y^3*z^2 + 69120*x^2*y^3*z - 20568*x^2*y^3 - 60480*x^2*y^2*z^3 + 103680*x^2*y^2*z^2 - 61704*x^2*y^2*z + 13806*x^2*y^2 - 30240*x^2*y*z^4 + 69120*x^2*y*z^3 - 61704*x^2*y*z^2 + 27666*x^2*y*z - 5352*x^2*y - 6048*x^2*z^5 + 17280*x^2*z^4 - 20568*x^2*z^3 + 13860*x^2*z^2 - 5379*x^2*z + 948*x^2 + 1344*x*y^6 + 8064*x*y^5*z - 4608*x*y^5 + 20160*x*y^4*z^2 - 23040*x*y^4*z + 6848*x*y^4 + 26880*x*y^3*z^3 - 46080*x*y^3*z^2 + 27392*x*y^3*z - 6116*x*y^3 + 20160*x*y^2*z^4 - 46080*x*y^2*z^3 + 41088*x*y^2*z^2 - 18384*x*y^2*z + 3548*x*y^2 + 8064*x*y*z^5 - 23040*x*y*z^4 + 27392*x*y*z^3 - 18420*x*y*z^2 + 7132*x*y*z - 1253*x*y + 1344*x*z^6 - 4608*x*z^5 + 6848*x*z^4 - 6152*x*z^3 + 3584*x*z^2 - 1264*x*z + 223*x - 128*y^7 - 896*y^6*z + 512*y^6 - 2688*y^5*z^2 + 3072*y^5*z - 912*y^5 - 4480*y^4*z^3 + 7680*y^4*z^2 - 4560*y^4*z + 1016*y^4 - 4480*y^3*z^4 + 10240*y^3*z^3 - 9120*y^3*z^2 + 4072*y^3*z - 784*y^3 - 2688*y^2*z^5 + 7680*y^2*z^4 - 9120*y^2*z^3 + 6120*y^2*z^2 - 2364*y^2*z + 414*y^2 - 896*y*z^6 + 3072*y*z^5 - 4560*y*z^4 + 4088*y*z^3 - 2376*y*z^2 + 835*y*z - 147*y - 128*z^7 + 512*z^6 - 912*z^5 + 1024*z^4 - 796*z^3 + 421*z^2 - 152*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ 3, -2, -2 ], [ 1, -1, -2 ], [ 2, -1, -1 ] ], "source_shift": [ 1, 1, 2 ], "target_matrix": [ [ 1, 1, 2 ], [ 2, 2, 3 ], [ 1, 2, 1 ] ], "target_shift": [ -19, -45, -29 ], "inverse_recipe": { "p": 4, "q": 3, "f": "x**4 + x**2 + x", "g": "x**3", "h": "y*z + z**2 + 2*z", "polynomials": { "f": [ [ [ 4, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "g": [ [ [ 3, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 0, 1 ], 2 ], [ [ 0, 0, 2 ], 1 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0154908700023952 ], "surface_features": [ 4.4321888542963315, 5.3167249841904995, 4.132483825000562, 0.0, 4.556664971859582, 5214 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C229", "pair": 229, "family": "counterexample", "mapping": "F(x,y,z)=(-x^7 + 8*x^6*y - 10*x^6*z + 17*x^6 - 27*x^5*y^2 + 69*x^5*y*z - 114*x^5*y - 42*x^5*z^2 + 147*x^5*z - 120*x^5 + 50*x^4*y^3 - 195*x^4*y^2*z + 315*x^4*y^2 + 243*x^4*y*z^2 - 828*x^4*y*z + 657*x^4*y - 96*x^4*z^3 + 519*x^4*z^2 - 873*x^4*z + 459*x^4 - 55*x^3*y^4 + 290*x^3*y^3*z - 460*x^3*y^3 - 552*x^3*y^2*z^2 + 1842*x^3*y^2*z - 1428*x^3*y^2 + 447*x^3*y*z^3 - 2358*x^3*y*z^2 + 3855*x^3*y*z - 1979*x^3*y - 129*x^3*z^4 + 957*x^3*z^3 - 2487*x^3*z^2 + 2701*x^3*z - 1037*x^3 + 36*x^2*y^5 - 240*x^2*y^4*z + 375*x^2*y^4 + 618*x^2*y^3*z^2 - 2028*x^2*y^3*z + 1542*x^2*y^3 - 765*x^2*y^2*z^3 + 3960*x^2*y^2*z^2 - 6327*x^2*y^2*z + 3183*x^2*y^2 + 453*x^2*y*z^4 - 3288*x^2*y*z^3 + 8310*x^2*y*z^2 - 8802*x^2*y*z + 3315*x^2*y - 102*x^2*z^5 + 972*x^2*z^4 - 3465*x^2*z^3 + 5826*x^2*z^2 - 4632*x^2*z + 1399*x^2 - 13*x*y^6 + 105*x*y^5*z - 162*x*y^5 - 342*x*y^4*z^2 + 1107*x*y^4*z - 828*x*y^4 + 573*x*y^3*z^3 - 2922*x*y^3*z^2 + 4581*x*y^3*z - 2265*x*y^3 - 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preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -1, 1, -2 ], [ -1, 1, -1 ], [ -1, 2, -1 ] ], "source_shift": [ 1, 2, 2 ], "target_matrix": [ [ 1, 1, 1 ], [ 2, 1, 1 ], [ 1, 2, 1 ] ], "target_shift": [ -343, -517, -519 ], "collision_preimages": [ [ "11/4", "-1/4", "-1" ], [ "15", "8", "-7/2" ], [ "4", "5", "3/2" ] ], "collision_image": [ "-1373/4", "-1035/2", "-2077/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0299436679997598 ], "surface_features": [ 3.4368602806370303, 4.307774068517087, 3.059184617631371, 0.0, 3.55155826639374, 4830 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C240", "pair": 240, "family": "counterexample", "mapping": "F(x,y,z)=(-8*x^7 + 44*x^6*y + 52*x^6*z - 88*x^6 - 102*x^5*y^2 - 240*x^5*y*z + 408*x^5*y - 138*x^5*z^2 + 468*x^5*z - 378*x^5 + 129*x^4*y^3 + 453*x^4*y^2*z - 774*x^4*y^2 + 519*x^4*y*z^2 - 1764*x^4*y*z + 1437*x^4*y + 195*x^4*z^3 - 990*x^4*z^2 + 1599*x^4*z - 826*x^4 - 96*x^3*y^4 - 447*x^3*y^3*z + 768*x^3*y^3 - 765*x^3*y^2*z^2 + 2607*x^3*y^2*z - 2142*x^3*y^2 - 573*x^3*y*z^3 + 2910*x^3*y*z^2 - 4737*x^3*y*z + 2481*x^3*y - 159*x^3*z^4 + 1071*x^3*z^3 - 2595*x^3*z^2 + 2702*x^3*z - 1025*x^3 + 42*x^2*y^5 + 243*x^2*y^4*z - 420*x^2*y^4 + 552*x^2*y^3*z^2 - 1887*x^2*y^3*z + 1563*x^2*y^3 + 618*x^2*y^2*z^3 - 3141*x^2*y^2*z^2 + 5151*x^2*y^2*z - 2733*x^2*y^2 + 342*x^2*y*z^4 - 2301*x^2*y*z^3 + 5613*x^2*y*z^2 - 5925*x^2*y*z + 2298*x^2*y + 75*x^2*z^5 - 627*x^2*z^4 + 2025*x^2*z^3 - 3192*x^2*z^2 + 2475*x^2*z - 766*x^2 - 10*x*y^6 - 69*x*y^5*z + 120*x*y^5 - 195*x*y^4*z^2 + 669*x*y^4*z - 558*x*y^4 - 290*x*y^3*z^3 + 1476*x*y^3*z^2 - 2436*x*y^3*z + 1307*x*y^3 - 240*x*y^2*z^4 + 1614*x*y^2*z^3 - 3960*x*y^2*z^2 + 4227*x*y^2*z - 1671*x*y^2 - 105*x*y*z^5 + 876*x*y*z^4 - 2844*x*y*z^3 + 4533*x*y*z^2 - 3582*x*y*z + 1139*x*y - 19*x*z^6 + 189*x*z^5 - 762*x*z^4 + 1613*x*z^3 - 1911*x*z^2 + 1223*x*z - 343*x + y^7 + 8*y^6*z - 14*y^6 + 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2751*x^2*y^2 + 342*x^2*y*z^4 - 2301*x^2*y*z^3 + 5613*x^2*y*z^2 - 5973*x^2*y*z + 2370*x^2*y + 75*x^2*z^5 - 627*x^2*z^4 + 2025*x^2*z^3 - 3222*x^2*z^2 + 2559*x^2*z - 820*x^2 - 10*x*y^6 - 69*x*y^5*z + 120*x*y^5 - 195*x*y^4*z^2 + 669*x*y^4*z - 558*x*y^4 - 290*x*y^3*z^3 + 1476*x*y^3*z^2 - 2436*x*y^3*z + 1314*x*y^3 - 240*x*y^2*z^4 + 1614*x*y^2*z^3 - 3960*x*y^2*z^2 + 4254*x*y^2*z - 1713*x*y^2 - 105*x*y*z^5 + 876*x*y*z^4 - 2844*x*y*z^3 + 4566*x*y*z^2 - 3678*x*y*z + 1202*x*y - 19*x*z^6 + 189*x*z^5 - 762*x*z^4 + 1626*x*z^3 - 1965*x*z^2 + 1292*x*z - 367*x + y^7 + 8*y^6*z - 14*y^6 + 27*y^5*z^2 - 93*y^5*z + 78*y^5 + 50*y^4*z^3 - 255*y^4*z^2 + 423*y^4*z - 230*y^4 + 55*y^3*z^4 - 370*y^3*z^3 + 912*y^3*z^2 - 987*y^3*z + 402*y^3 + 36*y^2*z^5 - 300*y^2*z^4 + 978*y^2*z^3 - 1581*y^2*z^2 + 1287*y^2*z - 427*y^2 + 13*y*z^6 - 129*y*z^5 + 522*y*z^4 - 1121*y*z^3 + 1368*y*z^2 - 911*y*z + 262*y + 2*z^7 - 23*z^6 + 111*z^5 - 297*z^4 + 483*z^3 - 484*z^2 + 281*z - 1, 16*x^7 - 88*x^6*y - 104*x^6*z + 200*x^6 + 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210*x*y*z^5 - 1983*x*y*z^4 + 6864*x*y*z^3 - 11409*x*y*z^2 + 9372*x*y*z - 3097*x*y + 38*x*z^6 - 429*x*z^5 + 1842*x*z^4 - 4067*x*z^3 + 5010*x*z^2 - 3331*x*z + 954*x - 2*y^7 - 16*y^6*z + 31*y^6 - 54*y^5*z^2 + 207*y^5*z - 186*y^5 - 100*y^4*z^3 + 570*y^4*z^2 - 1011*y^4*z + 570*y^4 - 110*y^3*z^4 + 830*y^3*z^3 - 2184*y^3*z^2 + 2449*y^3*z - 1018*y^3 - 72*y^2*z^5 + 675*y^2*z^4 - 2346*y^2*z^3 + 3927*y^2*z^2 - 3261*y^2*z + 1094*y^2 - 26*y*z^6 + 291*y*z^5 - 1254*y*z^4 + 2787*y*z^3 - 3468*y*z^2 + 2335*y*z - 676*y - 4*z^7 + 52*z^6 - 267*z^5 + 739*z^4 - 1225*z^3 + 1241*z^2 - 725*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ 2, -1, -1 ], [ 1, -1, -1 ], [ 1, -1, -2 ] ], "source_shift": [ 1, 1, 2 ], "target_matrix": [ [ -1, -1, -2 ], [ -1, -1, -1 ], [ 2, 3, 3 ] ], "target_shift": [ 69, 72, -188 ], "collision_preimages": [ [ "0", "-1/4", "5/4" ], [ "5/2", "12", "-7" ], [ "-5/2", "1", "-4" ] ], "collision_image": [ "277/4", "289/4", "-377/2" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0750881660023879 ], "surface_features": [ 3.2048590112948268, 4.1758306041622495, 2.8836614351536176, 0.0, 3.3525961852722217, 4769 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B108", "pair": 108, "family": "bijection", "mapping": "F(x,y,z)=(3*x^7 - 21*x^6*y + 21*x^6*z - 42*x^6 + 63*x^5*y^2 - 126*x^5*y*z + 246*x^5*y + 63*x^5*z^2 - 246*x^5*z + 249*x^5 - 105*x^4*y^3 + 315*x^4*y^2*z - 600*x^4*y^2 - 315*x^4*y*z^2 + 1200*x^4*y*z - 1191*x^4*y + 105*x^4*z^3 - 600*x^4*z^2 + 1188*x^4*z - 837*x^4 + 105*x^3*y^4 - 420*x^3*y^3*z + 780*x^3*y^3 + 630*x^3*y^2*z^2 - 2340*x^3*y^2*z + 2277*x^3*y^2 - 420*x^3*y*z^3 + 2340*x^3*y*z^2 - 4542*x^3*y*z + 3144*x^3*y + 105*x^3*z^4 - 780*x^3*z^3 + 2265*x^3*z^2 - 3114*x^3*z + 1726*x^3 - 63*x^2*y^5 + 315*x^2*y^4*z - 570*x^2*y^4 - 630*x^2*y^3*z^2 + 2280*x^2*y^3*z - 2175*x^2*y^3 + 630*x^2*y^2*z^3 - 3420*x^2*y^2*z^2 + 6507*x^2*y^2*z - 4422*x^2*y^2 - 315*x^2*y*z^4 + 2280*x^2*y*z^3 - 6489*x^2*y*z^2 + 8760*x^2*y*z - 4740*x^2*y + 63*x^2*z^5 - 570*x^2*z^4 + 2157*x^2*z^3 - 4338*x^2*z^2 + 4641*x^2*z - 2080*x^2 + 21*x*y^6 - 126*x*y^5*z + 222*x*y^5 + 315*x*y^4*z^2 - 1110*x*y^4*z + 1038*x*y^4 - 420*x*y^3*z^3 + 2220*x*y^3*z^2 - 4140*x*y^3*z + 2760*x*y^3 + 315*x*y^2*z^4 - 2220*x*y^2*z^3 + 6192*x*y^2*z^2 - 8202*x*y^2*z + 4332*x*y^2 - 126*x*y*z^5 + 1110*x*y*z^4 - 4116*x*y*z^3 + 8124*x*y*z^2 - 8496*x*y*z + 3710*x*y + 21*x*z^6 - 222*x*z^5 + 1026*x*z^4 - 2682*x*z^3 + 4164*x*z^2 - 3590*x*z + 1316*x - 3*y^7 + 21*y^6*z - 36*y^6 - 63*y^5*z^2 + 216*y^5*z - 198*y^5 + 105*y^4*z^3 - 540*y^4*z^2 + 987*y^4*z - 645*y^4 - 105*y^3*z^4 + 720*y^3*z^3 - 1968*y^3*z^2 + 2556*y^3*z - 1318*y^3 + 63*y^2*z^5 - 540*y^2*z^4 + 1962*y^2*z^3 - 3798*y^2*z^2 + 3882*y^2*z - 1651*y^2 - 21*y*z^6 + 216*y*z^5 - 978*y*z^4 + 2508*y*z^3 - 3810*y*z^2 + 3206*y*z - 1145*y + 3*z^7 - 36*z^6 + 195*z^5 - 621*z^4 + 1246*z^3 - 1555*z^2 + 1099*z - 1, 9*x^7 - 63*x^6*y + 63*x^6*z - 126*x^6 + 189*x^5*y^2 - 378*x^5*y*z + 738*x^5*y + 189*x^5*z^2 - 738*x^5*z + 747*x^5 - 315*x^4*y^3 + 945*x^4*y^2*z - 1800*x^4*y^2 - 945*x^4*y*z^2 + 3600*x^4*y*z - 3573*x^4*y + 315*x^4*z^3 - 1800*x^4*z^2 + 3564*x^4*z - 2511*x^4 + 315*x^3*y^4 - 1260*x^3*y^3*z + 2340*x^3*y^3 + 1890*x^3*y^2*z^2 - 7020*x^3*y^2*z + 6831*x^3*y^2 - 1260*x^3*y*z^3 + 7020*x^3*y*z^2 - 13626*x^3*y*z + 9432*x^3*y + 315*x^3*z^4 - 2340*x^3*z^3 + 6795*x^3*z^2 - 9342*x^3*z + 5178*x^3 - 189*x^2*y^5 + 945*x^2*y^4*z - 1710*x^2*y^4 - 1890*x^2*y^3*z^2 + 6840*x^2*y^3*z - 6525*x^2*y^3 + 1890*x^2*y^2*z^3 - 10260*x^2*y^2*z^2 + 19521*x^2*y^2*z - 13266*x^2*y^2 - 945*x^2*y*z^4 + 6840*x^2*y*z^3 - 19467*x^2*y*z^2 + 26280*x^2*y*z - 14220*x^2*y + 189*x^2*z^5 - 1710*x^2*z^4 + 6471*x^2*z^3 - 13014*x^2*z^2 + 13923*x^2*z - 6239*x^2 + 63*x*y^6 - 378*x*y^5*z + 666*x*y^5 + 945*x*y^4*z^2 - 3330*x*y^4*z + 3114*x*y^4 - 1260*x*y^3*z^3 + 6660*x*y^3*z^2 - 12420*x*y^3*z + 8280*x*y^3 + 945*x*y^2*z^4 - 6660*x*y^2*z^3 + 18576*x*y^2*z^2 - 24606*x*y^2*z + 12996*x*y^2 - 378*x*y*z^5 + 3330*x*y*z^4 - 12348*x*y*z^3 + 24372*x*y*z^2 - 25488*x*y*z + 11128*x*y + 63*x*z^6 - 666*x*z^5 + 3078*x*z^4 - 8046*x*z^3 + 12492*x*z^2 - 10768*x*z + 3943*x - 9*y^7 + 63*y^6*z - 108*y^6 - 189*y^5*z^2 + 648*y^5*z - 594*y^5 + 315*y^4*z^3 - 1620*y^4*z^2 + 2961*y^4*z - 1935*y^4 - 315*y^3*z^4 + 2160*y^3*z^3 - 5904*y^3*z^2 + 7668*y^3*z - 3954*y^3 + 189*y^2*z^5 - 1620*y^2*z^4 + 5886*y^2*z^3 - 11394*y^2*z^2 + 11646*y^2*z - 4952*y^2 - 63*y*z^6 + 648*y*z^5 - 2934*y*z^4 + 7524*y*z^3 - 11430*y*z^2 + 9616*y*z - 3431*y + 9*z^7 - 108*z^6 + 585*z^5 - 1863*z^4 + 3738*z^3 - 4664*z^2 + 3293*z - 1, -6*x^7 + 42*x^6*y - 42*x^6*z + 84*x^6 - 126*x^5*y^2 + 252*x^5*y*z - 492*x^5*y - 126*x^5*z^2 + 492*x^5*z - 498*x^5 + 210*x^4*y^3 - 630*x^4*y^2*z + 1200*x^4*y^2 + 630*x^4*y*z^2 - 2400*x^4*y*z + 2382*x^4*y - 210*x^4*z^3 + 1200*x^4*z^2 - 2376*x^4*z + 1674*x^4 - 210*x^3*y^4 + 840*x^3*y^3*z - 1560*x^3*y^3 - 1260*x^3*y^2*z^2 + 4680*x^3*y^2*z - 4554*x^3*y^2 + 840*x^3*y*z^3 - 4680*x^3*y*z^2 + 9084*x^3*y*z - 6288*x^3*y - 210*x^3*z^4 + 1560*x^3*z^3 - 4530*x^3*z^2 + 6228*x^3*z - 3454*x^3 + 126*x^2*y^5 - 630*x^2*y^4*z + 1140*x^2*y^4 + 1260*x^2*y^3*z^2 - 4560*x^2*y^3*z + 4350*x^2*y^3 - 1260*x^2*y^2*z^3 + 6840*x^2*y^2*z^2 - 13014*x^2*y^2*z + 8844*x^2*y^2 + 630*x^2*y*z^4 - 4560*x^2*y*z^3 + 12978*x^2*y*z^2 - 17520*x^2*y*z + 9486*x^2*y - 126*x^2*z^5 + 1140*x^2*z^4 - 4314*x^2*z^3 + 8676*x^2*z^2 - 9288*x^2*z + 4165*x^2 - 42*x*y^6 + 252*x*y^5*z - 444*x*y^5 - 630*x*y^4*z^2 + 2220*x*y^4*z - 2076*x*y^4 + 840*x*y^3*z^3 - 4440*x*y^3*z^2 + 8280*x*y^3*z - 5520*x*y^3 - 630*x*y^2*z^4 + 4440*x*y^2*z^3 - 12384*x*y^2*z^2 + 16404*x*y^2*z - 8670*x*y^2 + 252*x*y*z^5 - 2220*x*y*z^4 + 8232*x*y*z^3 - 16248*x*y*z^2 + 17004*x*y*z - 7430*x*y - 42*x*z^6 + 444*x*z^5 - 2052*x*z^4 + 5364*x*z^3 - 8334*x*z^2 + 7190*x*z - 2637*x + 6*y^7 - 42*y^6*z + 72*y^6 + 126*y^5*z^2 - 432*y^5*z + 396*y^5 - 210*y^4*z^3 + 1080*y^4*z^2 - 1974*y^4*z + 1290*y^4 + 210*y^3*z^4 - 1440*y^3*z^3 + 3936*y^3*z^2 - 5112*y^3*z + 2638*y^3 - 126*y^2*z^5 + 1080*y^2*z^4 - 3924*y^2*z^3 + 7596*y^2*z^2 - 7770*y^2*z + 3307*y^2 + 42*y*z^6 - 432*y*z^5 + 1956*y*z^4 - 5016*y*z^3 + 7626*y*z^2 - 6422*y*z + 2298*y - 6*z^7 + 72*z^6 - 390*z^5 + 1242*z^4 - 2494*z^3 + 3115*z^2 - 2204*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -1, 1, -1 ], [ -2, 1, -1 ], [ -1, 2, -1 ] ], "source_shift": [ 1, 2, 1 ], "target_matrix": [ [ -1, -1, -1 ], [ -3, -2, -3 ], [ 2, 1, 1 ] ], "target_shift": [ 334, 1000, -671 ], "inverse_recipe": { "p": 2, "q": 3, "f": "x**2 + x", "g": "x**3 + x", "h": "y**3 + y**2*z", "polynomials": { "f": [ [ [ 2, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "g": [ [ [ 3, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "h": [ [ [ 0, 2, 1 ], 1 ], [ [ 0, 3, 0 ], 1 ] ] }, "h_multiplier": 3 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "Maybe" ], "seconds": [ 1.0628821890022664 ], "surface_features": [ 3.523610999552628, 4.419625360887743, 3.192428055331207, 0.0, 3.656721394013795, 4870 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B075", "pair": 75, "family": "bijection", "mapping": "F(x,y,z)=(-24*x^7 + 312*x^6*y + 160*x^6*z - 161*x^6 - 1728*x^5*y^2 - 1776*x^5*y*z + 1788*x^5*y - 456*x^5*z^2 + 918*x^5*z - 462*x^5 + 5280*x^4*y^3 + 8160*x^4*y^2*z - 8220*x^4*y^2 + 4200*x^4*y*z^2 - 8460*x^4*y*z + 4260*x^4*y + 720*x^4*z^3 - 2175*x^4*z^2 + 2190*x^4*z - 735*x^4 - 9600*x^3*y^4 - 19840*x^3*y^3*z + 20000*x^3*y^3 - 15360*x^3*y^2*z^2 + 30960*x^3*y^2*z - 15600*x^3*y^2 - 5280*x^3*y*z^3 + 15960*x^3*y*z^2 - 16080*x^3*y*z + 5400*x^3*y - 680*x^3*z^4 + 2740*x^3*z^3 - 4140*x^3*z^2 + 2780*x^3*z - 724*x^3 + 10368*x^2*y^5 + 26880*x^2*y^4*z - 27120*x^2*y^4 + 27840*x^2*y^3*z^2 - 56160*x^2*y^3*z + 28320*x^2*y^3 + 14400*x^2*y^2*z^3 - 43560*x^2*y^2*z^2 + 43920*x^2*y^2*z - 14760*x^2*y^2 + 3720*x^2*y*z^4 - 15000*x^2*y*z^3 + 22680*x^2*y*z^2 - 15240*x^2*y*z + 3960*x^2*y + 384*x^2*z^5 - 1935*x^2*z^4 + 3900*x^2*z^3 - 3930*x^2*z^2 + 2044*x^2*z - 632*x^2 - 6144*x*y^6 - 19200*x*y^5*z + 19392*x*y^5 - 24960*x*y^4*z^2 + 50400*x*y^4*z - 25440*x*y^4 - 17280*x*y^3*z^3 + 52320*x*y^3*z^2 - 52800*x*y^3*z + 17760*x*y^3 - 6720*x*y^2*z^4 + 27120*x*y^2*z^3 - 41040*x*y^2*z^2 + 27600*x*y^2*z - 7152*x*y^2 - 1392*x*y*z^5 + 7020*x*y*z^4 - 14160*x*y*z^3 + 14280*x*y*z^2 - 7408*x*y*z + 2000*x*y - 120*x*z^6 + 726*x*z^5 - 1830*x*z^4 + 2460*x*z^3 - 1916*x*z^2 + 1096*x*z - 409*x + 1536*y^7 + 5632*y^6*z - 5696*y^6 + 8832*y^5*z^2 - 17856*y^5*z + 9024*y^5 + 7680*y^4*z^3 - 23280*y^4*z^2 + 23520*y^4*z - 7920*y^4 + 4000*y^3*z^4 - 16160*y^3*z^3 + 24480*y^3*z^2 - 16480*y^3*z + 4256*y^3 + 1248*y^2*z^5 - 6300*y^2*z^4 + 12720*y^2*z^3 - 12840*y^2*z^2 + 6640*y^2*z - 1640*y^2 + 216*y*z^6 - 1308*y*z^5 + 3300*y*z^4 - 4440*y*z^3 + 3448*y*z^2 - 1768*y*z + 546*y + 16*z^7 - 113*z^6 + 342*z^5 - 575*z^4 + 596*z^3 - 480*z^2 + 324*z - 1, 48*x^7 - 624*x^6*y - 320*x^6*z + 321*x^6 + 3456*x^5*y^2 + 3552*x^5*y*z - 3564*x^5*y + 912*x^5*z^2 - 1830*x^5*z + 918*x^5 - 10560*x^4*y^3 - 16320*x^4*y^2*z + 16380*x^4*y^2 - 8400*x^4*y*z^2 + 16860*x^4*y*z - 8460*x^4*y - 1440*x^4*z^3 + 4335*x^4*z^2 - 4350*x^4*z + 1455*x^4 + 19200*x^3*y^4 + 39680*x^3*y^3*z - 39840*x^3*y^3 + 30720*x^3*y^2*z^2 - 61680*x^3*y^2*z + 30960*x^3*y^2 + 10560*x^3*y*z^3 - 31800*x^3*y*z^2 + 31920*x^3*y*z - 10680*x^3*y + 1360*x^3*z^4 - 5460*x^3*z^3 + 8220*x^3*z^2 - 5500*x^3*z + 1428*x^3 - 20736*x^2*y^5 - 53760*x^2*y^4*z + 54000*x^2*y^4 - 55680*x^2*y^3*z^2 + 111840*x^2*y^3*z - 56160*x^2*y^3 - 28800*x^2*y^2*z^3 + 86760*x^2*y^2*z^2 - 87120*x^2*y^2*z + 29160*x^2*y^2 - 7440*x^2*y*z^4 + 29880*x^2*y*z^3 - 45000*x^2*y*z^2 + 30120*x^2*y*z - 7800*x^2*y - 768*x^2*z^5 + 3855*x^2*z^4 - 7740*x^2*z^3 + 7770*x^2*z^2 - 4028*x^2*z + 1248*x^2 + 12288*x*y^6 + 38400*x*y^5*z - 38592*x*y^5 + 49920*x*y^4*z^2 - 100320*x*y^4*z + 50400*x*y^4 + 34560*x*y^3*z^3 - 104160*x*y^3*z^2 + 104640*x*y^3*z - 35040*x*y^3 + 13440*x*y^2*z^4 - 54000*x*y^2*z^3 + 81360*x*y^2*z^2 - 54480*x*y^2*z + 14064*x*y^2 + 2784*x*y*z^5 - 13980*x*y*z^4 + 28080*x*y*z^3 - 28200*x*y*z^2 + 14576*x*y*z - 3936*x*y + 240*x*z^6 - 1446*x*z^5 + 3630*x*z^4 - 4860*x*z^3 + 3772*x*z^2 - 2160*x*z + 824*x - 3072*y^7 - 11264*y^6*z + 11328*y^6 - 17664*y^5*z^2 + 35520*y^5*z - 17856*y^5 - 15360*y^4*z^3 + 46320*y^4*z^2 - 46560*y^4*z + 15600*y^4 - 8000*y^3*z^4 + 32160*y^3*z^3 - 48480*y^3*z^2 + 32480*y^3*z - 8352*y^3 - 2496*y^2*z^5 + 12540*y^2*z^4 - 25200*y^2*z^3 + 25320*y^2*z^2 - 13040*y^2*z + 3216*y^2 - 432*y*z^6 + 2604*y*z^5 - 6540*y*z^4 + 8760*y*z^3 - 6776*y*z^2 + 3472*y*z - 1090*y - 32*z^7 + 225*z^6 - 678*z^5 + 1135*z^4 - 1172*z^3 + 944*z^2 - 649*z + 1, -24*x^7 + 312*x^6*y + 160*x^6*z - 161*x^6 - 1728*x^5*y^2 - 1776*x^5*y*z + 1788*x^5*y - 456*x^5*z^2 + 918*x^5*z - 462*x^5 + 5280*x^4*y^3 + 8160*x^4*y^2*z - 8220*x^4*y^2 + 4200*x^4*y*z^2 - 8460*x^4*y*z + 4260*x^4*y + 720*x^4*z^3 - 2175*x^4*z^2 + 2190*x^4*z - 735*x^4 - 9600*x^3*y^4 - 19840*x^3*y^3*z + 20000*x^3*y^3 - 15360*x^3*y^2*z^2 + 30960*x^3*y^2*z - 15600*x^3*y^2 - 5280*x^3*y*z^3 + 15960*x^3*y*z^2 - 16080*x^3*y*z + 5400*x^3*y - 680*x^3*z^4 + 2740*x^3*z^3 - 4140*x^3*z^2 + 2780*x^3*z - 724*x^3 + 10368*x^2*y^5 + 26880*x^2*y^4*z - 27120*x^2*y^4 + 27840*x^2*y^3*z^2 - 56160*x^2*y^3*z + 28320*x^2*y^3 + 14400*x^2*y^2*z^3 - 43560*x^2*y^2*z^2 + 43920*x^2*y^2*z - 14760*x^2*y^2 + 3720*x^2*y*z^4 - 15000*x^2*y*z^3 + 22680*x^2*y*z^2 - 15240*x^2*y*z + 3960*x^2*y + 384*x^2*z^5 - 1935*x^2*z^4 + 3900*x^2*z^3 - 3930*x^2*z^2 + 2044*x^2*z - 632*x^2 - 6144*x*y^6 - 19200*x*y^5*z + 19392*x*y^5 - 24960*x*y^4*z^2 + 50400*x*y^4*z - 25440*x*y^4 - 17280*x*y^3*z^3 + 52320*x*y^3*z^2 - 52800*x*y^3*z + 17760*x*y^3 - 6720*x*y^2*z^4 + 27120*x*y^2*z^3 - 41040*x*y^2*z^2 + 27600*x*y^2*z - 7152*x*y^2 - 1392*x*y*z^5 + 7020*x*y*z^4 - 14160*x*y*z^3 + 14280*x*y*z^2 - 7408*x*y*z + 2000*x*y - 120*x*z^6 + 726*x*z^5 - 1830*x*z^4 + 2460*x*z^3 - 1916*x*z^2 + 1096*x*z - 415*x + 1536*y^7 + 5632*y^6*z - 5696*y^6 + 8832*y^5*z^2 - 17856*y^5*z + 9024*y^5 + 7680*y^4*z^3 - 23280*y^4*z^2 + 23520*y^4*z - 7920*y^4 + 4000*y^3*z^4 - 16160*y^3*z^3 + 24480*y^3*z^2 - 16480*y^3*z + 4256*y^3 + 1248*y^2*z^5 - 6300*y^2*z^4 + 12720*y^2*z^3 - 12840*y^2*z^2 + 6640*y^2*z - 1640*y^2 + 216*y*z^6 - 1308*y*z^5 + 3300*y*z^4 - 4440*y*z^3 + 3448*y*z^2 - 1768*y*z + 552*y + 16*z^7 - 113*z^6 + 342*z^5 - 575*z^4 + 596*z^3 - 480*z^2 + 328*z - 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ 1, -2, -1 ], [ 3, -2, -2 ], [ 2, -1, -1 ] ], "source_shift": [ 1, 2, 1 ], "target_matrix": [ [ -1, -1, -2 ], [ 2, 1, 1 ], [ -1, -1, -1 ] ], "target_shift": [ 109, -226, 113 ], "inverse_recipe": { "p": 6, "q": 1, "f": "x**6 + x**2 + x", "g": "x", "h": "y*z + z**2", "polynomials": { "f": [ [ [ 6, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "g": [ [ [ 1, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 0, 2 ], 1 ] ] }, "h_multiplier": 8 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0951884870009962 ], "surface_features": [ 4.148033710213243, 5.0485971584016065, 3.8273692730538253, 0.0, 4.268030644147073, 5118 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B219", "pair": 219, "family": "bijection", "mapping": "F(x,y,z)=(x^7 - 7*x^6*y + 14*x^6*z - 15*x^6 + 21*x^5*y^2 - 84*x^5*y*z + 91*x^5*y + 84*x^5*z^2 - 179*x^5*z + 96*x^5 - 35*x^4*y^3 + 210*x^4*y^2*z - 230*x^4*y^2 - 420*x^4*y*z^2 + 905*x^4*y*z - 490*x^4*y + 280*x^4*z^3 - 890*x^4*z^2 + 950*x^4*z - 340*x^4 + 35*x^3*y^4 - 280*x^3*y^3*z + 310*x^3*y^3 + 840*x^3*y^2*z^2 - 1830*x^3*y^2*z + 1000*x^3*y^2 - 1120*x^3*y*z^3 + 3600*x^3*y*z^2 - 3880*x^3*y*z + 1400*x^3*y + 560*x^3*z^4 - 2360*x^3*z^3 + 3760*x^3*z^2 - 2680*x^3*z + 722*x^3 - 21*x^2*y^5 + 210*x^2*y^4*z - 235*x^2*y^4 - 840*x^2*y^3*z^2 + 1850*x^2*y^3*z - 1020*x^2*y^3 + 1680*x^2*y^2*z^3 - 5460*x^2*y^2*z^2 + 5940*x^2*y^2*z - 2160*x^2*y^2 - 1680*x^2*y*z^4 + 7160*x^2*y*z^3 - 11520*x^2*y*z^2 + 8280*x^2*y*z - 2246*x^2*y + 672*x^2*z^5 - 3520*x^2*z^4 + 7440*x^2*z^3 - 7920*x^2*z^2 + 4251*x^2*z - 922*x^2 + 7*x*y^6 - 84*x*y^5*z + 95*x*y^5 + 420*x*y^4*z^2 - 935*x*y^4*z + 520*x*y^4 - 1120*x*y^3*z^3 + 3680*x*y^3*z^2 - 4040*x*y^3*z + 1480*x*y^3 + 1680*x*y^2*z^4 - 7240*x*y^2*z^3 + 11760*x*y^2*z^2 - 8520*x*y^2*z + 2326*x*y^2 - 1344*x*y*z^5 + 7120*x*y*z^4 - 15200*x*y*z^3 + 16320*x*y*z^2 - 8822*x*y*z + 1926*x*y + 448*x*z^6 - 2800*x*z^5 + 7360*x*z^4 - 10400*x*z^3 + 8340*x*z^2 - 3601*x*z + 653*x - y^7 + 14*y^6*z - 16*y^6 - 84*y^5*z^2 + 189*y^5*z - 106*y^5 + 280*y^4*z^3 - 930*y^4*z^2 + 1030*y^4*z - 380*y^4 - 560*y^3*z^4 + 2440*y^3*z^3 - 4000*y^3*z^2 + 2920*y^3*z - 802*y^3 + 672*y^2*z^5 - 3600*y^2*z^4 + 7760*y^2*z^3 - 8400*y^2*z^2 + 4571*y^2*z - 1004*y^2 - 448*y*z^6 + 2832*y*z^5 - 7520*y*z^4 + 10720*y*z^3 - 8660*y*z^2 + 3764*y*z - 685*y + 128*z^7 - 928*z^6 + 2912*z^5 - 5120*z^4 + 5452*z^3 - 3515*z^2 + 1268*z - 1, -3*x^7 + 21*x^6*y - 42*x^6*z + 45*x^6 - 63*x^5*y^2 + 252*x^5*y*z - 273*x^5*y - 252*x^5*z^2 + 537*x^5*z - 287*x^5 + 105*x^4*y^3 - 630*x^4*y^2*z + 690*x^4*y^2 + 1260*x^4*y*z^2 - 2715*x^4*y*z + 1465*x^4*y - 840*x^4*z^3 + 2670*x^4*z^2 - 2840*x^4*z + 1010*x^4 - 105*x^3*y^4 + 840*x^3*y^3*z - 930*x^3*y^3 - 2520*x^3*y^2*z^2 + 5490*x^3*y^2*z - 2990*x^3*y^2 + 3360*x^3*y*z^3 - 10800*x^3*y*z^2 + 11600*x^3*y*z - 4160*x^3*y - 1680*x^3*z^4 + 7080*x^3*z^3 - 11240*x^3*z^2 + 7960*x^3*z - 2126*x^3 + 63*x^2*y^5 - 630*x^2*y^4*z + 705*x^2*y^4 + 2520*x^2*y^3*z^2 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31840*y*z^3 + 25500*y*z^2 - 10996*y*z + 2009*y - 384*z^7 + 2784*z^6 - 8704*z^5 + 15200*z^4 - 16036*z^3 + 10249*z^2 - 3695*z + 1, x^7 - 7*x^6*y + 14*x^6*z - 15*x^6 + 21*x^5*y^2 - 84*x^5*y*z + 91*x^5*y + 84*x^5*z^2 - 179*x^5*z + 96*x^5 - 35*x^4*y^3 + 210*x^4*y^2*z - 230*x^4*y^2 - 420*x^4*y*z^2 + 905*x^4*y*z - 490*x^4*y + 280*x^4*z^3 - 890*x^4*z^2 + 950*x^4*z - 340*x^4 + 35*x^3*y^4 - 280*x^3*y^3*z + 310*x^3*y^3 + 840*x^3*y^2*z^2 - 1830*x^3*y^2*z + 1000*x^3*y^2 - 1120*x^3*y*z^3 + 3600*x^3*y*z^2 - 3880*x^3*y*z + 1400*x^3*y + 560*x^3*z^4 - 2360*x^3*z^3 + 3760*x^3*z^2 - 2680*x^3*z + 722*x^3 - 21*x^2*y^5 + 210*x^2*y^4*z - 235*x^2*y^4 - 840*x^2*y^3*z^2 + 1850*x^2*y^3*z - 1020*x^2*y^3 + 1680*x^2*y^2*z^3 - 5460*x^2*y^2*z^2 + 5940*x^2*y^2*z - 2160*x^2*y^2 - 1680*x^2*y*z^4 + 7160*x^2*y*z^3 - 11520*x^2*y*z^2 + 8280*x^2*y*z - 2246*x^2*y + 672*x^2*z^5 - 3520*x^2*z^4 + 7440*x^2*z^3 - 7920*x^2*z^2 + 4251*x^2*z - 924*x^2 + 7*x*y^6 - 84*x*y^5*z + 95*x*y^5 + 420*x*y^4*z^2 - 935*x*y^4*z + 520*x*y^4 - 1120*x*y^3*z^3 + 3680*x*y^3*z^2 - 4040*x*y^3*z + 1480*x*y^3 + 1680*x*y^2*z^4 - 7240*x*y^2*z^3 + 11760*x*y^2*z^2 - 8520*x*y^2*z + 2326*x*y^2 - 1344*x*y*z^5 + 7120*x*y*z^4 - 15200*x*y*z^3 + 16320*x*y*z^2 - 8822*x*y*z + 1930*x*y + 448*x*z^6 - 2800*x*z^5 + 7360*x*z^4 - 10400*x*z^3 + 8340*x*z^2 - 3609*x*z + 663*x - y^7 + 14*y^6*z - 16*y^6 - 84*y^5*z^2 + 189*y^5*z - 106*y^5 + 280*y^4*z^3 - 930*y^4*z^2 + 1030*y^4*z - 380*y^4 - 560*y^3*z^4 + 2440*y^3*z^3 - 4000*y^3*z^2 + 2920*y^3*z - 802*y^3 + 672*y^2*z^5 - 3600*y^2*z^4 + 7760*y^2*z^3 - 8400*y^2*z^2 + 4571*y^2*z - 1006*y^2 - 448*y*z^6 + 2832*y*z^5 - 7520*y*z^4 + 10720*y*z^3 - 8660*y*z^2 + 3772*y*z - 697*y + 128*z^7 - 928*z^6 + 2912*z^5 - 5120*z^4 + 5452*z^3 - 3523*z^2 + 1286*z - 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -1, 1, -2 ], [ -1, 1, -1 ], [ -1, 2, -1 ] ], "source_shift": [ 2, 2, 1 ], "target_matrix": [ [ -1, -1, -2 ], [ 3, 2, 3 ], [ -1, -1, -1 ] ], "target_shift": [ 197, -587, 207 ], "inverse_recipe": { "p": 5, "q": 2, "f": "x**5 + x", "g": "x**2", "h": "y*z", "polynomials": { "f": [ [ [ 5, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "g": [ [ [ 2, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0376668860008067 ], "surface_features": [ 3.6548260532301633, 4.68556261115823, 3.225309281725863, 0.0, 3.844415349328263, 4878 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C030", "pair": 30, "family": "counterexample", "mapping": "F(x,y,z)=(216*x^7 - 1188*x^6*y + 972*x^6*z - 2160*x^6 + 2754*x^5*y^2 - 4536*x^5*y*z + 10368*x^5*y + 1854*x^5*z^2 - 8100*x^5*z + 7326*x^5 - 3483*x^4*y^3 + 8667*x^4*y^2*z - 20412*x^4*y^2 - 7137*x^4*y*z^2 + 32076*x^4*y*z - 29259*x^4*y + 1945*x^4*z^3 - 12564*x^4*z^2 + 22569*x^4*z - 12384*x^4 + 2592*x^3*y^4 - 8667*x^3*y^3*z + 21060*x^3*y^3 + 10791*x^3*y^2*z^2 - 49977*x^3*y^2*z + 45936*x^3*y^2 - 5929*x^3*y*z^3 + 39402*x^3*y*z^2 - 71385*x^3*y*z + 39411*x^3*y + 1213*x^3*z^4 - 10321*x^3*z^3 + 27639*x^3*z^2 - 30264*x^3*z + 11895*x^3 - 1134*x^2*y^5 + 4779*x^2*y^4*z - 11988*x^2*y^4 - 8001*x^2*y^3*z^2 + 38232*x^2*y^3*z - 35361*x^2*y^3 + 6651*x^2*y^2*z^3 - 45549*x^2*y^2*z^2 + 83142*x^2*y^2*z - 46143*x^2*y^2 - 2745*x^2*y*z^4 + 24030*x^2*y*z^3 - 64896*x^2*y*z^2 + 71487*x^2*y*z - 28317*x^2*y + 450*x^2*z^5 - 4737*x^2*z^4 + 16821*x^2*z^3 - 27573*x^2*z^2 + 21696*x^2*z - 6699*x^2 + 270*x*y^6 - 1377*x*y^5*z + 3564*x*y^5 + 2907*x*y^4*z^2 - 14337*x*y^4*z + 13320*x*y^4 - 3251*x*y^3*z^3 + 22968*x*y^3*z^2 - 42177*x*y^3*z + 23499*x*y^3 + 2031*x*y^2*z^4 - 18321*x*y^2*z^3 + 49839*x*y^2*z^2 - 55170*x*y^2*z + 21999*x*y^2 - 672*x*y*z^5 + 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"source_shift": [ 2, 1, 2 ], "target_matrix": [ [ 1, 2, 1 ], [ 1, 1, 1 ], [ 3, 3, 2 ] ], "target_shift": [ -397, -228, -710 ], "collision_preimages": [ [ "1", "-1/4", "-11/4" ], [ "-3/2", "11/2", "-5" ], [ "7/2", "15/2", "-14" ] ], "collision_image": [ "-1589/4", "-913/4", "-2843/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0649517469973944 ], "surface_features": [ 3.564618398375829, 4.592598628906131, 3.13656203658998, 0.0, 3.7415991068357113, 4856 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B203", "pair": 203, "family": "bijection", "mapping": "F(x,y,z)=(x^7 - 14*x^6*y - 7*x^6*z + 9*x^6 + 84*x^5*y^2 + 84*x^5*y*z - 108*x^5*y + 21*x^5*z^2 - 54*x^5*z + 37*x^5 - 280*x^4*y^3 - 420*x^4*y^2*z + 540*x^4*y^2 - 210*x^4*y*z^2 + 540*x^4*y*z - 369*x^4*y - 35*x^4*z^3 + 135*x^4*z^2 - 185*x^4*z + 89*x^4 + 560*x^3*y^4 + 1120*x^3*y^3*z - 1440*x^3*y^3 + 840*x^3*y^2*z^2 - 2160*x^3*y^2*z + 1472*x^3*y^2 + 280*x^3*y*z^3 - 1080*x^3*y*z^2 + 1476*x^3*y*z - 704*x^3*y + 35*x^3*z^4 - 180*x^3*z^3 + 370*x^3*z^2 - 355*x^3*z + 136*x^3 - 672*x^2*y^5 - 1680*x^2*y^4*z + 2160*x^2*y^4 - 1680*x^2*y^3*z^2 + 4320*x^2*y^3*z - 2936*x^2*y^3 - 840*x^2*y^2*z^3 + 3240*x^2*y^2*z^2 - 4416*x^2*y^2*z + 2088*x^2*y^2 - 210*x^2*y*z^4 + 1080*x^2*y*z^3 - 2214*x^2*y*z^2 + 2106*x^2*y*z - 794*x^2*y - 21*x^2*z^5 + 135*x^2*z^4 - 370*x^2*z^3 + 531*x^2*z^2 - 404*x^2*z + 135*x^2 + 448*x*y^6 + 1344*x*y^5*z - 1728*x*y^5 + 1680*x*y^4*z^2 - 4320*x*y^4*z + 2928*x*y^4 + 1120*x*y^3*z^3 - 4320*x*y^3*z^2 + 5872*x*y^3*z - 2752*x*y^3 + 420*x*y^2*z^4 - 2160*x*y^2*z^3 + 4416*x*y^2*z^2 - 4164*x*y^2*z + 1544*x*y^2 + 84*x*y*z^5 - 540*x*y*z^4 + 1476*x*y*z^3 - 2100*x*y*z^2 + 1572*x*y*z - 508*x*y + 7*x*z^6 - 54*x*z^5 + 185*x*z^4 - 353*x*z^3 + 400*x*z^2 - 263*x*z + 83*x - 128*y^7 - 448*y^6*z + 576*y^6 - 672*y^5*z^2 + 1728*y^5*z - 1168*y^5 - 560*y^4*z^3 + 2160*y^4*z^2 - 2928*y^4*z + 1360*y^4 - 280*y^3*z^4 + 1440*y^3*z^3 - 2936*y^3*z^2 + 2744*y^3*z - 1000*y^3 - 84*y^2*z^5 + 540*y^2*z^4 - 1472*y^2*z^3 + 2076*y^2*z^2 - 1528*y^2*z + 479*y^2 - 14*y*z^6 + 108*y*z^5 - 369*y*z^4 + 698*y*z^3 - 778*y*z^2 + 495*y*z - 150*y - z^7 + 9*z^6 - 37*z^5 + 88*z^4 - 132*z^3 + 128*z^2 - 80*z + 1, x^7 - 14*x^6*y - 7*x^6*z + 9*x^6 + 84*x^5*y^2 + 84*x^5*y*z - 108*x^5*y + 21*x^5*z^2 - 54*x^5*z + 37*x^5 - 280*x^4*y^3 - 420*x^4*y^2*z + 540*x^4*y^2 - 210*x^4*y*z^2 + 540*x^4*y*z - 369*x^4*y - 35*x^4*z^3 + 135*x^4*z^2 - 185*x^4*z + 89*x^4 + 560*x^3*y^4 + 1120*x^3*y^3*z - 1440*x^3*y^3 + 840*x^3*y^2*z^2 - 2160*x^3*y^2*z + 1472*x^3*y^2 + 280*x^3*y*z^3 - 1080*x^3*y*z^2 + 1476*x^3*y*z - 704*x^3*y + 35*x^3*z^4 - 180*x^3*z^3 + 370*x^3*z^2 - 355*x^3*z + 135*x^3 - 672*x^2*y^5 - 1680*x^2*y^4*z + 2160*x^2*y^4 - 1680*x^2*y^3*z^2 + 4320*x^2*y^3*z - 2936*x^2*y^3 - 840*x^2*y^2*z^3 + 3240*x^2*y^2*z^2 - 4416*x^2*y^2*z + 2088*x^2*y^2 - 210*x^2*y*z^4 + 1080*x^2*y*z^3 - 2214*x^2*y*z^2 + 2106*x^2*y*z - 788*x^2*y - 21*x^2*z^5 + 135*x^2*z^4 - 370*x^2*z^3 + 531*x^2*z^2 - 401*x^2*z + 131*x^2 + 448*x*y^6 + 1344*x*y^5*z - 1728*x*y^5 + 1680*x*y^4*z^2 - 4320*x*y^4*z + 2928*x*y^4 + 1120*x*y^3*z^3 - 4320*x*y^3*z^2 + 5872*x*y^3*z - 2752*x*y^3 + 420*x*y^2*z^4 - 2160*x*y^2*z^3 + 4416*x*y^2*z^2 - 4164*x*y^2*z + 1532*x*y^2 + 84*x*y*z^5 - 540*x*y*z^4 + 1476*x*y*z^3 - 2100*x*y*z^2 + 1560*x*y*z - 492*x*y + 7*x*z^6 - 54*x*z^5 + 185*x*z^4 - 353*x*z^3 + 397*x*z^2 - 255*x*z + 76*x - 128*y^7 - 448*y^6*z + 576*y^6 - 672*y^5*z^2 + 1728*y^5*z - 1168*y^5 - 560*y^4*z^3 + 2160*y^4*z^2 - 2928*y^4*z + 1360*y^4 - 280*y^3*z^4 + 1440*y^3*z^3 - 2936*y^3*z^2 + 2744*y^3*z - 992*y^3 - 84*y^2*z^5 + 540*y^2*z^4 - 1472*y^2*z^3 + 2076*y^2*z^2 - 1516*y^2*z + 463*y^2 - 14*y*z^6 + 108*y*z^5 - 369*y*z^4 + 698*y*z^3 - 772*y*z^2 + 479*y*z - 137*y - z^7 + 9*z^6 - 37*z^5 + 88*z^4 - 131*z^3 + 124*z^2 - 73*z + 1, 2*x^7 - 28*x^6*y - 14*x^6*z + 18*x^6 + 168*x^5*y^2 + 168*x^5*y*z - 216*x^5*y + 42*x^5*z^2 - 108*x^5*z + 74*x^5 - 560*x^4*y^3 - 840*x^4*y^2*z + 1080*x^4*y^2 - 420*x^4*y*z^2 + 1080*x^4*y*z - 738*x^4*y - 70*x^4*z^3 + 270*x^4*z^2 - 370*x^4*z + 180*x^4 + 1120*x^3*y^4 + 2240*x^3*y^3*z - 2880*x^3*y^3 + 1680*x^3*y^2*z^2 - 4320*x^3*y^2*z + 2944*x^3*y^2 + 560*x^3*y*z^3 - 2160*x^3*y*z^2 + 2952*x^3*y*z - 1424*x^3*y + 70*x^3*z^4 - 360*x^3*z^3 + 740*x^3*z^2 - 718*x^3*z + 279*x^3 - 1344*x^2*y^5 - 3360*x^2*y^4*z + 4320*x^2*y^4 - 3360*x^2*y^3*z^2 + 8640*x^2*y^3*z - 5872*x^2*y^3 - 1680*x^2*y^2*z^3 + 6480*x^2*y^2*z^2 - 8832*x^2*y^2*z + 4224*x^2*y^2 - 420*x^2*y*z^4 + 2160*x^2*y*z^3 - 4428*x^2*y*z^2 + 4260*x^2*y*z - 1630*x^2*y - 42*x^2*z^5 + 270*x^2*z^4 - 740*x^2*z^3 + 1074*x^2*z^2 - 829*x^2*z + 278*x^2 + 896*x*y^6 + 2688*x*y^5*z - 3456*x*y^5 + 3360*x*y^4*z^2 - 8640*x*y^4*z + 5856*x*y^4 + 2240*x*y^3*z^3 - 8640*x*y^3*z^2 + 11744*x*y^3*z - 5568*x*y^3 + 840*x*y^2*z^4 - 4320*x*y^2*z^3 + 8832*x*y^2*z^2 - 8424*x*y^2*z + 3172*x*y^2 + 168*x*y*z^5 - 1080*x*y*z^4 + 2952*x*y*z^3 - 4248*x*y*z^2 + 3228*x*y*z - 1048*x*y + 14*x*z^6 - 108*x*z^5 + 370*x*z^4 - 714*x*z^3 + 821*x*z^2 - 542*x*z + 167*x - 256*y^7 - 896*y^6*z + 1152*y^6 - 1344*y^5*z^2 + 3456*y^5*z - 2336*y^5 - 1120*y^4*z^3 + 4320*y^4*z^2 - 5856*y^4*z + 2752*y^4 - 560*y^3*z^4 + 2880*y^3*z^3 - 5872*y^3*z^2 + 5552*y^3*z - 2056*y^3 - 168*y^2*z^5 + 1080*y^2*z^4 - 2944*y^2*z^3 + 4200*y^2*z^2 - 3140*y^2*z + 990*y^2 - 28*y*z^6 + 216*y*z^5 - 738*y*z^4 + 1412*y*z^3 - 1598*y*z^2 + 1022*y*z - 299*y - 2*z^7 + 18*z^6 - 74*z^5 + 178*z^4 - 271*z^3 + 264*z^2 - 159*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ 1, -2, -1 ], [ 1, -1, -1 ], [ 2, -1, -1 ] ], "source_shift": [ 1, 1, 1 ], "target_matrix": [ [ 1, 2, 1 ], [ 1, 1, 1 ], [ 2, 1, 1 ] ], "target_shift": [ -24, -20, -45 ], "inverse_recipe": { "p": 3, "q": 4, "f": "x**3 + x**2 + x", "g": "x**4 + x**3 + x**2", "h": "y*z + y + z", "polynomials": { "f": [ [ [ 3, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "g": [ [ [ 4, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ], [ [ 3, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 0, 1 ], 1 ], [ [ 0, 1, 0 ], 1 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "Maybe" ], "seconds": [ 1.0668636209993565 ], "surface_features": [ 3.1389365685138095, 4.069816042571995, 2.8273692730538253, 0.0, 3.251570205141714, 4749 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C228", "pair": 228, "family": "counterexample", "mapping": "F(x,y,z)=(-8*x^7 - 68*x^6*y + 44*x^6*z + 100*x^6 - 234*x^5*y^2 + 312*x^5*y*z + 732*x^5*y - 102*x^5*z^2 - 468*x^5*z - 504*x^5 - 419*x^4*y^3 + 867*x^4*y^2*z + 2121*x^4*y^2 - 585*x^4*y*z^2 - 2778*x^4*y*z - 3096*x^4*y + 129*x^4*z^3 + 897*x^4*z^2 + 1956*x^4*z + 1342*x^4 - 419*x^3*y^4 + 1199*x^3*y^3*z + 3094*x^3*y^3 - 1257*x^3*y^2*z^2 - 6240*x^3*y^2*z - 7218*x^3*y^2 + 573*x^3*y*z^3 + 4134*x^3*y*z^2 + 9336*x^3*y*z + 6651*x^3*y - 96*x^3*z^4 - 900*x^3*z^3 - 2982*x^3*z^2 - 4161*x^3*z - 2082*x^3 - 234*x^2*y^5 + 867*x^2*y^4*z + 2382*x^2*y^4 - 1257*x^2*y^3*z^2 - 6588*x^2*y^3*z - 7920*x^2*y^3 + 891*x^2*y^2*z^3 + 6732*x^2*y^2*z^2 + 15777*x^2*y^2*z + 11685*x^2*y^2 - 309*x^2*y*z^4 - 3012*x^2*y*z^3 - 10338*x^2*y*z^2 - 14985*x^2*y*z - 7830*x^2*y + 42*x^2*z^5 + 498*x^2*z^4 + 2229*x^2*z^3 + 4746*x^2*z^2 + 4854*x^2*z + 1918*x^2 - 68*x*y^6 + 312*x*y^5*z + 912*x*y^5 - 585*x*y^4*z^2 - 3252*x*y^4*z - 4059*x*y^4 + 573*x*y^3*z^3 + 4566*x*y^3*z^2 + 11103*x*y^3*z + 8549*x*y^3 - 309*x*y^2*z^4 - 3156*x*y^2*z^3 - 11229*x*y^2*z^2 - 16911*x*y^2*z - 9216*x*y^2 + 87*x*y*z^5 + 1074*x*y*z^4 + 4977*x*y*z^3 + 11004*x*y*z^2 + 11745*x*y*z + 4879*x*y - 10*x*z^6 - 144*x*z^5 - 816*x*z^4 - 2356*x*z^3 - 3693*x*z^2 - 2999*x*z - 994*x - 8*y^7 + 44*y^6*z + 136*y^6 - 102*y^5*z^2 - 600*y^5*z - 774*y^5 + 129*y^4*z^3 + 1086*y^4*z^2 + 2733*y^4*z + 2175*y^4 - 96*y^3*z^4 - 1032*y^3*z^3 - 3801*y^3*z^2 - 5928*y^3*z - 3363*y^3 + 42*y^2*z^5 + 543*y^2*z^4 + 2604*y^2*z^3 + 5967*y^2*z^2 + 6630*y^2*z + 2887*y^2 - 10*y*z^6 - 150*y*z^5 - 879*y*z^4 - 2631*y*z^3 - 4296*y*z^2 - 3661*y*z - 1276*y + z^7 + 17*z^6 + 117*z^5 + 429*z^4 + 915*z^3 + 1144*z^2 + 781*z + 1, -8*x^7 - 68*x^6*y + 44*x^6*z + 124*x^6 - 234*x^5*y^2 + 312*x^5*y*z + 888*x^5*y - 102*x^5*z^2 - 576*x^5*z - 708*x^5 - 419*x^4*y^3 + 867*x^4*y^2*z + 2511*x^4*y^2 - 585*x^4*y*z^2 - 3342*x^4*y*z - 4260*x^4*y + 129*x^4*z^3 + 1095*x^4*z^2 + 2724*x^4*z + 2026*x^4 - 419*x^3*y^4 + 1199*x^3*y^3*z + 3571*x^3*y^3 - 1257*x^3*y^2*z^2 - 7323*x^3*y^2*z - 9693*x^3*y^2 + 573*x^3*y*z^3 + 4929*x^3*y*z^2 + 12714*x^3*y*z + 9843*x^3*y - 96*x^3*z^4 - 1089*x^3*z^3 - 4113*x^3*z^2 - 6225*x^3*z - 3306*x^3 - 234*x^2*y^5 + 867*x^2*y^4*z + 2685*x^2*y^4 - 1257*x^2*y^3*z^2 - 7542*x^2*y^3*z - 10344*x^2*y^3 + 891*x^2*y^2*z^3 + 7830*x^2*y^2*z^2 + 20934*x^2*y^2*z + 16896*x^2*y^2 - 309*x^2*y*z^4 - 3558*x^2*y*z^3 - 13920*x^2*y*z^2 - 21939*x^2*y*z - 12162*x^2*y + 42*x^2*z^5 + 597*x^2*z^4 + 3042*x^2*z^3 + 7029*x^2*z^2 + 7626*x^2*z + 3172*x^2 - 68*x*y^6 + 312*x*y^5*z + 1008*x*y^5 - 585*x*y^4*z^2 - 3642*x*y^4*z - 5145*x*y^4 + 573*x*y^3*z^3 + 5187*x*y^3*z^2 + 14313*x*y^3*z + 12032*x*y^3 - 309*x*y^2*z^4 - 3639*x*y^2*z^3 - 14712*x*y^2*z^2 - 24144*x*y^2*z - 13971*x*y^2 + 87*x*y*z^5 + 1257*x*y*z^4 + 6621*x*y*z^3 + 15921*x*y*z^2 + 18027*x*y*z + 7852*x*y - 10*x*z^6 - 171*x*z^5 - 1101*x*z^4 - 3451*x*z^3 - 5736*x*z^2 - 4892*x*z - 1697*x - 8*y^7 + 44*y^6*z + 148*y^6 - 102*y^5*z^2 - 660*y^5*z - 954*y^5 + 129*y^4*z^3 + 1209*y^4*z^2 + 3423*y^4*z + 2970*y^4 - 96*y^3*z^4 - 1164*y^3*z^3 - 4839*y^3*z^2 - 8226*y^3*z - 4959*y^3 + 42*y^2*z^5 + 621*y^2*z^4 + 3369*y^2*z^3 + 8406*y^2*z^2 + 9915*y^2*z + 4519*y^2 - 10*y*z^6 - 174*y*z^5 - 1155*y*z^4 - 3759*y*z^3 - 6510*y*z^2 - 5809*y*z - 2106*y + z^7 + 20*z^6 + 156*z^5 + 621*z^4 + 1404*z^3 + 1840*z^2 + 1310*z + 1, -8*x^7 - 68*x^6*y + 44*x^6*z + 100*x^6 - 234*x^5*y^2 + 312*x^5*y*z + 732*x^5*y - 102*x^5*z^2 - 468*x^5*z - 504*x^5 - 419*x^4*y^3 + 867*x^4*y^2*z + 2121*x^4*y^2 - 585*x^4*y*z^2 - 2778*x^4*y*z - 3096*x^4*y + 129*x^4*z^3 + 897*x^4*z^2 + 1956*x^4*z + 1334*x^4 - 419*x^3*y^4 + 1199*x^3*y^3*z + 3094*x^3*y^3 - 1257*x^3*y^2*z^2 - 6240*x^3*y^2*z - 7218*x^3*y^2 + 573*x^3*y*z^3 + 4134*x^3*y*z^2 + 9336*x^3*y*z + 6631*x^3*y - 96*x^3*z^4 - 900*x^3*z^3 - 2982*x^3*z^2 - 4141*x^3*z - 2030*x^3 - 234*x^2*y^5 + 867*x^2*y^4*z + 2382*x^2*y^4 - 1257*x^2*y^3*z^2 - 6588*x^2*y^3*z - 7920*x^2*y^3 + 891*x^2*y^2*z^3 + 6732*x^2*y^2*z^2 + 15777*x^2*y^2*z + 11667*x^2*y^2 - 309*x^2*y*z^4 - 3012*x^2*y*z^3 - 10338*x^2*y*z^2 - 14949*x^2*y*z - 7722*x^2*y + 42*x^2*z^5 + 498*x^2*z^4 + 2229*x^2*z^3 + 4728*x^2*z^2 + 4758*x^2*z + 1810*x^2 - 68*x*y^6 + 312*x*y^5*z + 912*x*y^5 - 585*x*y^4*z^2 - 3252*x*y^4*z - 4059*x*y^4 + 573*x*y^3*z^3 + 4566*x*y^3*z^2 + 11103*x*y^3*z + 8542*x*y^3 - 309*x*y^2*z^4 - 3156*x*y^2*z^3 - 11229*x*y^2*z^2 - 16890*x*y^2*z - 9147*x*y^2 + 87*x*y*z^5 + 1074*x*y*z^4 + 4977*x*y*z^3 + 10983*x*y*z^2 + 11619*x*y*z + 4723*x*y - 10*x*z^6 - 144*x*z^5 - 816*x*z^4 - 2349*x*z^3 - 3636*x*z^2 - 2867*x*z - 906*x - 8*y^7 + 44*y^6*z + 136*y^6 - 102*y^5*z^2 - 600*y^5*z - 774*y^5 + 129*y^4*z^3 + 1086*y^4*z^2 + 2733*y^4*z + 2174*y^4 - 96*y^3*z^4 - 1032*y^3*z^3 - 3801*y^3*z^2 - 5924*y^3*z - 3349*y^3 + 42*y^2*z^5 + 543*y^2*z^4 + 2604*y^2*z^3 + 5961*y^2*z^2 + 6591*y^2*z + 2836*y^2 - 10*y*z^6 - 150*y*z^5 - 879*y*z^4 - 2627*y*z^3 - 4260*y*z^2 - 3571*y*z - 1210*y + z^7 + 17*z^6 + 117*z^5 + 428*z^4 + 904*z^3 + 1105*z^2 + 727*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -2, -1, 1 ], [ -1, -2, 1 ], [ -1, -1, 1 ] ], "source_shift": [ 2, 1, 2 ], "target_matrix": [ [ 1, 1, 1 ], [ 1, 2, 1 ], [ 1, 1, 2 ] ], "target_shift": [ -228, -397, -204 ], "collision_preimages": [ [ "-1/4", "-5/4", "-15/4" ], [ "11/2", "7", "17" ], [ "15/2", "4", "16" ] ], "collision_image": [ "-913/4", "-1589/4", "-817/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0596365759993205 ], "surface_features": [ 3.5131191547493397, 4.382809222431515, 3.0993352776859577, 0.0, 3.6292772790630763, 4855 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B140", "pair": 140, "family": "bijection", "mapping": "F(x,y,z)=(3*x^7 + 21*x^6*y + 21*x^6*z + 36*x^6 + 63*x^5*y^2 + 126*x^5*y*z + 222*x^5*y + 63*x^5*z^2 + 216*x^5*z + 192*x^5 + 105*x^4*y^3 + 315*x^4*y^2*z + 570*x^4*y^2 + 315*x^4*y*z^2 + 1110*x^4*y*z + 1005*x^4*y + 105*x^4*z^3 + 540*x^4*z^2 + 957*x^4*z + 591*x^4 + 105*x^3*y^4 + 420*x^3*y^3*z + 780*x^3*y^3 + 630*x^3*y^2*z^2 + 2280*x^3*y^2*z + 2103*x^3*y^2 + 420*x^3*y*z^3 + 2220*x^3*y*z^2 + 4008*x^3*y*z + 2508*x^3*y + 105*x^3*z^4 + 720*x^3*z^3 + 1908*x^3*z^2 + 2340*x^3*z + 1129*x^3 + 63*x^2*y^5 + 315*x^2*y^4*z + 600*x^2*y^4 + 630*x^2*y^3*z^2 + 2340*x^2*y^3*z + 2199*x^2*y^3 + 630*x^2*y^2*z^3 + 3420*x^2*y^2*z^2 + 6291*x^2*y^2*z + 3984*x^2*y^2 + 315*x^2*y*z^4 + 2220*x^2*y*z^3 + 5994*x^2*y*z^2 + 7446*x^2*y*z + 3633*x^2*y + 63*x^2*z^5 + 540*x^2*z^4 + 1902*x^2*z^3 + 3474*x^2*z^2 + 3315*x^2*z + 1320*x^2 + 21*x*y^6 + 126*x*y^5*z + 246*x*y^5 + 315*x*y^4*z^2 + 1200*x*y^4*z + 1149*x*y^4 + 420*x*y^3*z^3 + 2340*x*y^3*z^2 + 4386*x*y^3*z + 2808*x*y^3 + 315*x*y^2*z^4 + 2280*x*y^2*z^3 + 6273*x*y^2*z^2 + 7884*x*y^2*z + 3879*x*y^2 + 126*x*y*z^5 + 1110*x*y*z^4 + 3984*x*y*z^3 + 7368*x*y*z^2 + 7098*x*y*z + 2856*x*y + 21*x*z^6 + 216*x*z^5 + 948*x*z^4 + 2292*x*z^3 + 3243*x*z^2 + 2544*x*z + 857*x + 3*y^7 + 21*y^6*z + 42*y^6 + 63*y^5*z^2 + 246*y^5*z + 240*y^5 + 105*y^4*z^3 + 600*y^4*z^2 + 1146*y^4*z + 741*y^4 + 105*y^3*z^4 + 780*y^3*z^3 + 2187*y^3*z^2 + 2778*y^3*z + 1372*y^3 + 63*y^2*z^5 + 570*y^2*z^4 + 2085*y^2*z^3 + 3900*y^2*z^2 + 3780*y^2*z + 1527*y^2 + 21*y*z^6 + 222*y*z^5 + 993*y*z^4 + 2430*y*z^3 + 3465*y*z^2 + 2736*y*z + 924*y + 3*z^7 + 36*z^6 + 189*z^5 + 567*z^4 + 1057*z^3 + 1224*z^2 + 808*z + 1, -3*x^7 - 21*x^6*y - 21*x^6*z - 36*x^6 - 63*x^5*y^2 - 126*x^5*y*z - 222*x^5*y - 63*x^5*z^2 - 216*x^5*z - 192*x^5 - 105*x^4*y^3 - 315*x^4*y^2*z - 570*x^4*y^2 - 315*x^4*y*z^2 - 1110*x^4*y*z - 1005*x^4*y - 105*x^4*z^3 - 540*x^4*z^2 - 957*x^4*z - 591*x^4 - 105*x^3*y^4 - 420*x^3*y^3*z - 780*x^3*y^3 - 630*x^3*y^2*z^2 - 2280*x^3*y^2*z - 2103*x^3*y^2 - 420*x^3*y*z^3 - 2220*x^3*y*z^2 - 4008*x^3*y*z - 2508*x^3*y - 105*x^3*z^4 - 720*x^3*z^3 - 1908*x^3*z^2 - 2340*x^3*z - 1129*x^3 - 63*x^2*y^5 - 315*x^2*y^4*z - 600*x^2*y^4 - 630*x^2*y^3*z^2 - 2340*x^2*y^3*z - 2199*x^2*y^3 - 630*x^2*y^2*z^3 - 3420*x^2*y^2*z^2 - 6291*x^2*y^2*z - 3984*x^2*y^2 - 315*x^2*y*z^4 - 2220*x^2*y*z^3 - 5994*x^2*y*z^2 - 7446*x^2*y*z - 3633*x^2*y - 63*x^2*z^5 - 540*x^2*z^4 - 1902*x^2*z^3 - 3474*x^2*z^2 - 3315*x^2*z - 1322*x^2 - 21*x*y^6 - 126*x*y^5*z - 246*x*y^5 - 315*x*y^4*z^2 - 1200*x*y^4*z - 1149*x*y^4 - 420*x*y^3*z^3 - 2340*x*y^3*z^2 - 4386*x*y^3*z - 2808*x*y^3 - 315*x*y^2*z^4 - 2280*x*y^2*z^3 - 6273*x*y^2*z^2 - 7884*x*y^2*z - 3879*x*y^2 - 126*x*y*z^5 - 1110*x*y*z^4 - 3984*x*y*z^3 - 7368*x*y*z^2 - 7098*x*y*z - 2860*x*y - 21*x*z^6 - 216*x*z^5 - 948*x*z^4 - 2292*x*z^3 - 3243*x*z^2 - 2548*x*z - 865*x - 3*y^7 - 21*y^6*z - 42*y^6 - 63*y^5*z^2 - 246*y^5*z - 240*y^5 - 105*y^4*z^3 - 600*y^4*z^2 - 1146*y^4*z - 741*y^4 - 105*y^3*z^4 - 780*y^3*z^3 - 2187*y^3*z^2 - 2778*y^3*z - 1372*y^3 - 63*y^2*z^5 - 570*y^2*z^4 - 2085*y^2*z^3 - 3900*y^2*z^2 - 3780*y^2*z - 1529*y^2 - 21*y*z^6 - 222*y*z^5 - 993*y*z^4 - 2430*y*z^3 - 3465*y*z^2 - 2740*y*z - 934*y - 3*z^7 - 36*z^6 - 189*z^5 - 567*z^4 - 1057*z^3 - 1226*z^2 - 816*z - 1, -3*x^7 - 21*x^6*y - 21*x^6*z - 36*x^6 - 63*x^5*y^2 - 126*x^5*y*z - 222*x^5*y - 63*x^5*z^2 - 216*x^5*z - 192*x^5 - 105*x^4*y^3 - 315*x^4*y^2*z - 570*x^4*y^2 - 315*x^4*y*z^2 - 1110*x^4*y*z - 1005*x^4*y - 105*x^4*z^3 - 540*x^4*z^2 - 957*x^4*z - 591*x^4 - 105*x^3*y^4 - 420*x^3*y^3*z - 780*x^3*y^3 - 630*x^3*y^2*z^2 - 2280*x^3*y^2*z - 2103*x^3*y^2 - 420*x^3*y*z^3 - 2220*x^3*y*z^2 - 4008*x^3*y*z - 2508*x^3*y - 105*x^3*z^4 - 720*x^3*z^3 - 1908*x^3*z^2 - 2340*x^3*z - 1130*x^3 - 63*x^2*y^5 - 315*x^2*y^4*z - 600*x^2*y^4 - 630*x^2*y^3*z^2 - 2340*x^2*y^3*z - 2199*x^2*y^3 - 630*x^2*y^2*z^3 - 3420*x^2*y^2*z^2 - 6291*x^2*y^2*z - 3984*x^2*y^2 - 315*x^2*y*z^4 - 2220*x^2*y*z^3 - 5994*x^2*y*z^2 - 7446*x^2*y*z - 3636*x^2*y - 63*x^2*z^5 - 540*x^2*z^4 - 1902*x^2*z^3 - 3474*x^2*z^2 - 3318*x^2*z - 1326*x^2 - 21*x*y^6 - 126*x*y^5*z - 246*x*y^5 - 315*x*y^4*z^2 - 1200*x*y^4*z - 1149*x*y^4 - 420*x*y^3*z^3 - 2340*x*y^3*z^2 - 4386*x*y^3*z - 2808*x*y^3 - 315*x*y^2*z^4 - 2280*x*y^2*z^3 - 6273*x*y^2*z^2 - 7884*x*y^2*z - 3882*x*y^2 - 126*x*y*z^5 - 1110*x*y*z^4 - 3984*x*y*z^3 - 7368*x*y*z^2 - 7104*x*y*z - 2868*x*y - 21*x*z^6 - 216*x*z^5 - 948*x*z^4 - 2292*x*z^3 - 3246*x*z^2 - 2556*x*z - 873*x - 3*y^7 - 21*y^6*z - 42*y^6 - 63*y^5*z^2 - 246*y^5*z - 240*y^5 - 105*y^4*z^3 - 600*y^4*z^2 - 1146*y^4*z - 741*y^4 - 105*y^3*z^4 - 780*y^3*z^3 - 2187*y^3*z^2 - 2778*y^3*z - 1373*y^3 - 63*y^2*z^5 - 570*y^2*z^4 - 2085*y^2*z^3 - 3900*y^2*z^2 - 3783*y^2*z - 1533*y^2 - 21*y*z^6 - 222*y*z^5 - 993*y*z^4 - 2430*y*z^3 - 3468*y*z^2 - 2748*y*z - 941*y - 3*z^7 - 36*z^6 - 189*z^5 - 567*z^4 - 1058*z^3 - 1230*z^2 - 823*z - 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ 1, 1, 1 ], [ 2, 1, 1 ], [ 1, 2, 1 ] ], "source_shift": [ 1, 1, 2 ], "target_matrix": [ [ 1, 1, 2 ], [ -1, -1, -1 ], [ -1, -2, -1 ] ], "target_shift": [ -228, 236, 240 ], "inverse_recipe": { "p": 3, "q": 2, "f": "x**3 + x**2 + x", "g": "x**2 + x", "h": "y*z**2 + z**2", "polynomials": { "f": [ [ [ 3, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "g": [ [ [ 2, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 2 ], 1 ], [ [ 0, 0, 2 ], 1 ] ] }, "h_multiplier": 3 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.107472322000831 ], "surface_features": [ 3.1816975286924434, 3.8967466156074058, 2.8920946026904804, 0.0, 3.2562346972657044, 4778 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B083", "pair": 83, "family": "bijection", "mapping": "F(x,y,z)=(-192*x^7 - 768*x^6*y + 704*x^6*z + 1376*x^6 - 1296*x^5*y^2 + 2400*x^5*y*z + 4752*x^5*y - 1104*x^5*z^2 - 4336*x^5*z - 4240*x^5 - 1200*x^4*y^3 + 3360*x^4*y^2*z + 6720*x^4*y^2 - 3120*x^4*y*z^2 - 12400*x^4*y*z - 12304*x^4*y + 960*x^4*z^3 + 5680*x^4*z^2 + 11168*x^4*z + 7345*x^4 - 660*x^3*y^4 + 2480*x^3*y^3*z + 5000*x^3*y^3 - 3480*x^3*y^2*z^2 - 13960*x^3*y^2*z - 14008*x^3*y^2 + 2160*x^3*y*z^3 + 12920*x^3*y*z^2 + 25744*x^3*y*z + 17348*x^3*y - 500*x^3*z^4 - 3960*x^3*z^3 - 11736*x^3*z^2 - 15576*x^3*z - 7888*x^3 - 216*x^2*y^5 + 1020*x^2*y^4*z + 2070*x^2*y^4 - 1920*x^2*y^3*z^2 - 7760*x^2*y^3*z - 7856*x^2*y^3 + 1800*x^2*y^2*z^3 + 10860*x^2*y^2*z^2 + 21864*x^2*y^2*z + 15126*x^2*y^2 - 840*x^2*y*z^4 - 6720*x^2*y*z^3 - 20160*x^2*y*z^2 - 27432*x^2*y*z - 14544*x^2*y + 156*x^2*z^5 + 1550*x^2*z^4 + 6152*x^2*z^3 + 12360*x^2*z^2 + 12736*x^2*z + 5449*x^2 - 39*x*y^6 + 222*x*y^5*z + 453*x*y^5 - 525*x*y^4*z^2 - 2135*x*y^4*z - 2177*x*y^4 + 660*x*y^3*z^3 + 4010*x*y^3*z^2 + 8140*x*y^3*z + 5828*x*y^3 - 465*x*y^2*z^4 - 3750*x*y^2*z^3 - 11358*x*y^2*z^2 - 15912*x*y^2*z - 8952*x*y^2 + 174*x*y*z^5 + 1745*x*y*z^4 + 7004*x*y*z^3 + 14448*x*y*z^2 + 15664*x*y*z + 7231*x*y - 27*x*z^6 - 323*x*z^5 - 1609*x*z^4 - 4352*x*z^3 - 6856*x*z^2 - 6044*x*z - 2346*x - 3*y^7 + 20*y^6*z + 41*y^6 - 57*y^5*z^2 - 233*y^5*z - 239*y^5 + 90*y^4*z^3 + 550*y^4*z^2 + 1124*y^4*z + 847*y^4 - 85*y^3*z^4 - 690*y^3*z^3 - 2106*y^3*z^2 - 3072*y^3*z - 1880*y^3 + 48*y^2*z^5 + 485*y^2*z^4 + 1964*y^2*z^3 + 4188*y^2*z^2 + 4864*y^2*z + 2470*y^2 - 15*y*z^6 - 181*y*z^5 - 911*y*z^4 - 2536*y*z^3 - 4232*y*z^2 - 4058*y*z - 1735*y + 2*z^7 + 28*z^6 + 168*z^5 + 574*z^4 + 1232*z^3 + 1684*z^2 + 1361*z + 1, -576*x^7 - 2304*x^6*y + 2112*x^6*z + 4256*x^6 - 3888*x^5*y^2 + 7200*x^5*y*z + 14640*x^5*y - 3312*x^5*z^2 - 13392*x^5*z - 13552*x^5 - 3600*x^4*y^3 + 10080*x^4*y^2*z + 20640*x^4*y^2 - 9360*x^4*y*z^2 - 38160*x^4*y*z - 38992*x^4*y + 2880*x^4*z^3 + 17520*x^4*z^2 + 35584*x^4*z + 24307*x^4 - 1980*x^3*y^4 + 7440*x^3*y^3*z + 15320*x^3*y^3 - 10440*x^3*y^2*z^2 - 42840*x^3*y^2*z - 44104*x^3*y^2 + 6480*x^3*y*z^3 + 39720*x^3*y*z^2 + 81392*x^3*y*z + 56588*x^3*y - 1500*x^3*z^4 - 12200*x^3*z^3 - 37288*x^3*z^2 - 51272*x^3*z - 26992*x^3 - 648*x^2*y^5 + 3060*x^2*y^4*z + 6330*x^2*y^4 - 5760*x^2*y^3*z^2 - 23760*x^2*y^3*z - 24608*x^2*y^3 + 5400*x^2*y^2*z^3 + 33300*x^2*y^2*z^2 + 68712*x^2*y^2*z + 48786*x^2*y^2 - 2520*x^2*y*z^4 - 20640*x^2*y*z^3 - 63600*x^2*y*z^2 - 89112*x^2*y*z - 48624*x^2*y + 468*x^2*z^5 + 4770*x^2*z^4 + 19496*x^2*z^3 + 40488*x^2*z^2 + 43200*x^2*z + 19099*x^2 - 117*x*y^6 + 666*x*y^5*z + 1383*x*y^5 - 1575*x*y^4*z^2 - 6525*x*y^4*z - 6791*x*y^4 + 1980*x*y^3*z^3 + 12270*x*y^3*z^2 + 25460*x*y^3*z + 18620*x*y^3 - 1395*x*y^2*z^4 - 11490*x*y^2*z^3 - 35634*x*y^2*z^2 - 51144*x*y^2*z - 29352*x*y^2 + 522*x*y*z^5 + 5355*x*y*z^4 + 22052*x*y*z^3 + 46752*x*y*z^2 + 51984*x*y*z + 24445*x*y - 81*x*z^6 - 993*x*z^5 - 5087*x*z^4 - 14192*x*z^3 - 23064*x*z^2 - 20884*x*z - 8248*x - 9*y^7 + 60*y^6*z + 125*y^6 - 171*y^5*z^2 - 711*y^5*z - 743*y^5 + 270*y^4*z^3 + 1680*y^4*z^2 + 3502*y^4*z + 2683*y^4 - 255*y^3*z^4 - 2110*y^3*z^3 - 6578*y^3*z^2 - 9784*y^3*z - 6056*y^3 + 144*y^2*z^5 + 1485*y^2*z^4 + 6152*y^2*z^3 + 13416*y^2*z^2 + 15840*y^2*z + 8098*y^2 - 45*y*z^6 - 555*y*z^5 - 2863*y*z^4 - 8176*y*z^3 - 13944*y*z^2 - 13550*y*z - 5805*y + 6*z^7 + 86*z^6 + 530*z^5 + 1864*z^4 + 4112*z^3 + 5740*z^2 + 4687*z + 1, -384*x^7 - 1536*x^6*y + 1408*x^6*z + 2880*x^6 - 2592*x^5*y^2 + 4800*x^5*y*z + 9888*x^5*y - 2208*x^5*z^2 - 9056*x^5*z - 9312*x^5 - 2400*x^4*y^3 + 6720*x^4*y^2*z + 13920*x^4*y^2 - 6240*x^4*y*z^2 - 25760*x^4*y*z - 26688*x^4*y + 1920*x^4*z^3 + 11840*x^4*z^2 + 24416*x^4*z + 16962*x^4 - 1320*x^3*y^4 + 4960*x^3*y^3*z + 10320*x^3*y^3 - 6960*x^3*y^2*z^2 - 28880*x^3*y^2*z - 30096*x^3*y^2 + 4320*x^3*y*z^3 + 26800*x^3*y*z^2 + 55648*x^3*y*z + 39240*x^3*y - 1000*x^3*z^4 - 8240*x^3*z^3 - 25552*x^3*z^2 - 35696*x^3*z - 19104*x^3 - 432*x^2*y^5 + 2040*x^2*y^4*z + 4260*x^2*y^4 - 3840*x^2*y^3*z^2 - 16000*x^2*y^3*z - 16752*x^2*y^3 + 3600*x^2*y^2*z^3 + 22440*x^2*y^2*z^2 + 46848*x^2*y^2*z + 33660*x^2*y^2 - 1680*x^2*y*z^4 - 13920*x^2*y*z^3 - 43440*x^2*y*z^2 - 61680*x^2*y*z - 34080*x^2*y + 312*x^2*z^5 + 3220*x^2*z^4 + 13344*x^2*z^3 + 28128*x^2*z^2 + 30464*x^2*z + 13650*x^2 - 78*x*y^6 + 444*x*y^5*z + 930*x*y^5 - 1050*x*y^4*z^2 - 4390*x*y^4*z - 4614*x*y^4 + 1320*x*y^3*z^3 + 8260*x*y^3*z^2 + 17320*x*y^3*z + 12792*x*y^3 - 930*x*y^2*z^4 - 7740*x*y^2*z^3 - 24276*x*y^2*z^2 - 35232*x*y^2*z - 20400*x*y^2 + 348*x*y*z^5 + 3610*x*y*z^4 + 15048*x*y*z^3 + 32304*x*y*z^2 + 36320*x*y*z + 17214*x*y - 54*x*z^6 - 670*x*z^5 - 3478*x*z^4 - 9840*x*z^3 - 16208*x*z^2 - 14840*x*z - 5905*x - 6*y^7 + 40*y^6*z + 84*y^6 - 114*y^5*z^2 - 478*y^5*z - 504*y^5 + 180*y^4*z^3 + 1130*y^4*z^2 + 2378*y^4*z + 1836*y^4 - 170*y^3*z^4 - 1420*y^3*z^3 - 4472*y^3*z^2 - 6712*y^3*z - 4176*y^3 + 96*y^2*z^5 + 1000*y^2*z^4 + 4188*y^2*z^3 + 9228*y^2*z^2 + 10976*y^2*z + 5628*y^2 - 30*y*z^6 - 374*y*z^5 - 1952*y*z^4 - 5640*y*z^3 - 9712*y*z^2 - 9492*y*z - 4073*y + 4*z^7 + 58*z^6 + 362*z^5 + 1290*z^4 + 2880*z^3 + 4056*z^2 + 3328*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -2, -1, 1 ], [ -1, -2, 1 ], [ -1, -1, 1 ] ], "source_shift": [ 2, 2, 2 ], "target_matrix": [ [ 1, 2, 1 ], [ 3, 2, 2 ], [ 2, 1, 1 ] ], "target_shift": [ -497, -1709, -1215 ], "inverse_recipe": { "p": 1, "q": 6, "f": "x", "g": "x**6 + x**5 + x**4 + x", "h": "y**4 + y*z", "polynomials": { "f": [ [ [ 1, 0, 0 ], 1 ] ], "g": [ [ [ 6, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ], [ [ 4, 0, 0 ], 1 ], [ [ 5, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 4, 0 ], 1 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "Maybe" ], "seconds": [ 1.1058266069994716 ], "surface_features": [ 4.010533981881876, 4.949936190937765, 3.6324572921847245, 0.0, 4.152149627578431, 5037 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B010", "pair": 10, "family": "bijection", "mapping": "F(x,y,z)=(-4*x^7 + 28*x^6*y - 28*x^6*z + 36*x^6 - 84*x^5*y^2 + 168*x^5*y*z - 216*x^5*y - 84*x^5*z^2 + 216*x^5*z - 144*x^5 + 140*x^4*y^3 - 420*x^4*y^2*z + 540*x^4*y^2 + 420*x^4*y*z^2 - 1080*x^4*y*z + 720*x^4*y - 140*x^4*z^3 + 540*x^4*z^2 - 724*x^4*z + 357*x^4 - 140*x^3*y^4 + 560*x^3*y^3*z - 720*x^3*y^3 - 840*x^3*y^2*z^2 + 2160*x^3*y^2*z - 1440*x^3*y^2 + 560*x^3*y*z^3 - 2160*x^3*y*z^2 + 2896*x^3*y*z - 1432*x^3*y - 140*x^3*z^4 + 720*x^3*z^3 - 1456*x^3*z^2 + 1456*x^3*z - 577*x^3 + 84*x^2*y^5 - 420*x^2*y^4*z + 540*x^2*y^4 + 840*x^2*y^3*z^2 - 2160*x^2*y^3*z + 1440*x^2*y^3 - 840*x^2*y^2*z^3 + 3240*x^2*y^2*z^2 - 4344*x^2*y^2*z + 2154*x^2*y^2 + 420*x^2*y*z^4 - 2160*x^2*y*z^3 + 4368*x^2*y*z^2 - 4380*x^2*y*z + 1743*x^2*y - 84*x^2*z^5 + 540*x^2*z^4 - 1464*x^2*z^3 + 2226*x^2*z^2 - 1795*x^2*z + 610*x^2 - 28*x*y^6 + 168*x*y^5*z - 216*x*y^5 - 420*x*y^4*z^2 + 1080*x*y^4*z - 720*x*y^4 + 560*x*y^3*z^3 - 2160*x*y^3*z^2 + 2896*x*y^3*z - 1440*x*y^3 - 420*x*y^2*z^4 + 2160*x*y^2*z^3 - 4368*x*y^2*z^2 + 4392*x*y^2*z - 1755*x*y^2 + 168*x*y*z^5 - 1080*x*y*z^4 + 2928*x*y*z^3 - 4464*x*y*z^2 + 3614*x*y*z - 1240*x*y - 28*x*z^6 + 216*x*z^5 - 736*x*z^4 + 1512*x*z^3 - 1859*x*z^2 + 1304*x*z - 388*x + 4*y^7 - 28*y^6*z + 36*y^6 + 84*y^5*z^2 - 216*y^5*z + 144*y^5 - 140*y^4*z^3 + 540*y^4*z^2 - 724*y^4*z + 361*y^4 + 140*y^3*z^4 - 720*y^3*z^3 + 1456*y^3*z^2 - 1468*y^3*z + 589*y^3 - 84*y^2*z^5 + 540*y^2*z^4 - 1464*y^2*z^3 + 2238*y^2*z^2 - 1819*y^2*z + 630*y^2 + 28*y*z^6 - 216*y*z^5 + 736*y*z^4 - 1516*y*z^3 + 1871*y*z^2 - 1328*y*z + 405*y - 4*z^7 + 36*z^6 - 148*z^5 + 385*z^4 - 641*z^3 + 698*z^2 - 428*z + 1, 12*x^7 - 84*x^6*y + 84*x^6*z - 108*x^6 + 252*x^5*y^2 - 504*x^5*y*z + 648*x^5*y + 252*x^5*z^2 - 648*x^5*z + 432*x^5 - 420*x^4*y^3 + 1260*x^4*y^2*z - 1620*x^4*y^2 - 1260*x^4*y*z^2 + 3240*x^4*y*z - 2160*x^4*y + 420*x^4*z^3 - 1620*x^4*z^2 + 2172*x^4*z - 1070*x^4 + 420*x^3*y^4 - 1680*x^3*y^3*z + 2160*x^3*y^3 + 2520*x^3*y^2*z^2 - 6480*x^3*y^2*z + 4320*x^3*y^2 - 1680*x^3*y*z^3 + 6480*x^3*y*z^2 - 8688*x^3*y*z + 4292*x^3*y + 420*x^3*z^4 - 2160*x^3*z^3 + 4368*x^3*z^2 - 4364*x^3*z + 1734*x^3 - 252*x^2*y^5 + 1260*x^2*y^4*z - 1620*x^2*y^4 - 2520*x^2*y^3*z^2 + 6480*x^2*y^3*z - 4320*x^2*y^3 + 2520*x^2*y^2*z^3 - 9720*x^2*y^2*z^2 + 13032*x^2*y^2*z - 6456*x^2*y^2 - 1260*x^2*y*z^4 + 6480*x^2*y*z^3 - 13104*x^2*y*z^2 + 13128*x^2*y*z - 5238*x^2*y + 252*x^2*z^5 - 1620*x^2*z^4 + 4392*x^2*z^3 - 6672*x^2*z^2 + 5394*x^2*z - 1844*x^2 + 84*x*y^6 - 504*x*y^5*z + 648*x*y^5 + 1260*x*y^4*z^2 - 3240*x*y^4*z + 2160*x*y^4 - 1680*x*y^3*z^3 + 6480*x*y^3*z^2 - 8688*x*y^3*z + 4316*x*y^3 + 1260*x*y^2*z^4 - 6480*x*y^2*z^3 + 13104*x*y^2*z^2 - 13164*x*y^2*z + 5274*x*y^2 - 504*x*y*z^5 + 3240*x*y*z^4 - 8784*x*y*z^3 + 13380*x*y*z^2 - 10860*x*y*z + 3748*x*y + 84*x*z^6 - 648*x*z^5 + 2208*x*z^4 - 4532*x*z^3 + 5586*x*z^2 - 3940*x*z + 1193*x - 12*y^7 + 84*y^6*z - 108*y^6 - 252*y^5*z^2 + 648*y^5*z - 432*y^5 + 420*y^4*z^3 - 1620*y^4*z^2 + 2172*y^4*z - 1082*y^4 - 420*y^3*z^4 + 2160*y^3*z^3 - 4368*y^3*z^2 + 4400*y^3*z - 1770*y^3 + 252*y^2*z^5 - 1620*y^2*z^4 + 4392*y^2*z^3 - 6708*y^2*z^2 + 5466*y^2*z - 1904*y^2 - 84*y*z^6 + 648*y*z^5 - 2208*y*z^4 + 4544*y*z^3 - 5622*y*z^2 + 4012*y*z - 1243*y + 12*z^7 - 108*z^6 + 444*z^5 - 1154*z^4 + 1926*z^3 - 2108*z^2 + 1321*z - 1, 4*x^7 - 28*x^6*y + 28*x^6*z - 36*x^6 + 84*x^5*y^2 - 168*x^5*y*z + 216*x^5*y + 84*x^5*z^2 - 216*x^5*z + 144*x^5 - 140*x^4*y^3 + 420*x^4*y^2*z - 540*x^4*y^2 - 420*x^4*y*z^2 + 1080*x^4*y*z - 720*x^4*y + 140*x^4*z^3 - 540*x^4*z^2 + 724*x^4*z - 357*x^4 + 140*x^3*y^4 - 560*x^3*y^3*z + 720*x^3*y^3 + 840*x^3*y^2*z^2 - 2160*x^3*y^2*z + 1440*x^3*y^2 - 560*x^3*y*z^3 + 2160*x^3*y*z^2 - 2896*x^3*y*z + 1432*x^3*y + 140*x^3*z^4 - 720*x^3*z^3 + 1456*x^3*z^2 - 1456*x^3*z + 579*x^3 - 84*x^2*y^5 + 420*x^2*y^4*z - 540*x^2*y^4 - 840*x^2*y^3*z^2 + 2160*x^2*y^3*z - 1440*x^2*y^3 + 840*x^2*y^2*z^3 - 3240*x^2*y^2*z^2 + 4344*x^2*y^2*z - 2154*x^2*y^2 - 420*x^2*y*z^4 + 2160*x^2*y*z^3 - 4368*x^2*y*z^2 + 4380*x^2*y*z - 1749*x^2*y + 84*x^2*z^5 - 540*x^2*z^4 + 1464*x^2*z^3 - 2226*x^2*z^2 + 1801*x^2*z - 616*x^2 + 28*x*y^6 - 168*x*y^5*z + 216*x*y^5 + 420*x*y^4*z^2 - 1080*x*y^4*z + 720*x*y^4 - 560*x*y^3*z^3 + 2160*x*y^3*z^2 - 2896*x*y^3*z + 1440*x*y^3 + 420*x*y^2*z^4 - 2160*x*y^2*z^3 + 4368*x*y^2*z^2 - 4392*x*y^2*z + 1761*x*y^2 - 168*x*y*z^5 + 1080*x*y*z^4 - 2928*x*y*z^3 + 4464*x*y*z^2 - 3626*x*y*z + 1252*x*y + 28*x*z^6 - 216*x*z^5 + 736*x*z^4 - 1512*x*z^3 + 1865*x*z^2 - 1316*x*z + 398*x - 4*y^7 + 28*y^6*z - 36*y^6 - 84*y^5*z^2 + 216*y^5*z - 144*y^5 + 140*y^4*z^3 - 540*y^4*z^2 + 724*y^4*z - 361*y^4 - 140*y^3*z^4 + 720*y^3*z^3 - 1456*y^3*z^2 + 1468*y^3*z - 591*y^3 + 84*y^2*z^5 - 540*y^2*z^4 + 1464*y^2*z^3 - 2238*y^2*z^2 + 1825*y^2*z - 636*y^2 - 28*y*z^6 + 216*y*z^5 - 736*y*z^4 + 1516*y*z^3 - 1877*y*z^2 + 1340*y*z - 415*y + 4*z^7 - 36*z^6 + 148*z^5 - 385*z^4 + 643*z^3 - 704*z^2 + 440*z - 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -1, 1, -1 ], [ -1, 2, -1 ], [ -1, 1, -2 ] ], "source_shift": [ 1, 1, 1 ], "target_matrix": [ [ 1, 1, 2 ], [ -3, -2, -2 ], [ -1, -1, -1 ] ], "target_shift": [ -109, 348, 115 ], "inverse_recipe": { "p": 4, "q": 3, "f": "x**4 + x**3 + x**2 + x", "g": "x**3 + x", "h": "y*z + y + z**2", "polynomials": { "f": [ [ [ 4, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ], [ [ 3, 0, 0 ], 1 ] ], "g": [ [ [ 3, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 0, 2 ], 1 ], [ [ 0, 1, 0 ], 1 ] ] }, "h_multiplier": 4 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0389818380026554 ], "surface_features": [ 3.2341782980710967, 4.1264561134318045, 2.866877814337499, 0.0, 3.369163672798312, 4808 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C014", "pair": 14, "family": "counterexample", "mapping": "F(x,y,z)=(216*x^7 - 1188*x^6*y - 972*x^6*z + 1728*x^6 + 2610*x^5*y^2 + 4464*x^5*y*z - 7524*x^5*y + 1854*x^5*z^2 - 6444*x^5*z + 5346*x^5 - 3043*x^4*y^3 - 8031*x^4*y^2*z + 13128*x^4*y^2 - 6933*x^4*y*z^2 + 23088*x^4*y*z - 18459*x^4*y - 1945*x^4*z^3 + 9960*x^4*z^2 - 16281*x^4*z + 8488*x^4 + 2060*x^3*y^4 + 7393*x^3*y^3*z - 11869*x^3*y^3 + 9819*x^3*y^2*z^2 - 31905*x^3*y^2*z + 24879*x^3*y^2 + 5699*x^3*y*z^3 - 28203*x^3*y*z^2 + 44643*x^3*y*z - 22701*x^3*y + 1213*x^3*z^4 - 8167*x^3*z^3 + 19764*x^3*z^2 - 20387*x^3*z + 7631*x^3 - 816*x^2*y^5 - 3714*x^2*y^4*z + 5901*x^2*y^4 - 6696*x^2*y^3*z^2 + 21450*x^2*y^3*z - 16437*x^2*y^3 - 5964*x^2*y^2*z^3 + 28944*x^2*y^2*z^2 - 44826*x^2*y^2*z + 22368*x^2*y^2 - 2616*x^2*y*z^4 + 17142*x^2*y*z^3 - 40341*x^2*y*z^2 + 40674*x^2*y*z - 14958*x^2*y - 450*x^2*z^5 + 3747*x^2*z^4 - 11952*x^2*z^3 + 18306*x^2*z^2 - 13575*x^2*z + 3949*x^2 + 176*x*y^6 + 972*x*y^5*z - 1536*x*y^5 + 2220*x*y^4*z^2 - 7056*x*y^4*z + 5340*x*y^4 + 2680*x*y^3*z^3 - 12864*x*y^3*z^2 + 19620*x*y^3*z - 9645*x*y^3 + 1800*x*y^2*z^4 - 11616*x*y^2*z^3 + 26820*x*y^2*z^2 - 26571*x*y^2*z + 9627*x*y^2 + 636*x*y*z^5 - 5184*x*y*z^4 + 16140*x*y*z^3 - 24207*x*y*z^2 + 17649*x*y*z - 5074*x*y + 92*x*z^6 - 912*x*z^5 + 3600*x*z^4 - 7281*x*z^3 + 8022*x*z^2 - 4624*x*z + 1118*x - 16*y^7 - 104*y^6*z + 164*y^6 - 288*y^5*z^2 + 912*y^5*z - 684*y^5 - 440*y^4*z^3 + 2100*y^4*z^2 - 3168*y^4*z + 1538*y^4 - 400*y^3*z^4 + 2560*y^3*z^3 - 5832*y^3*z^2 + 5696*y^3*z - 2036*y^3 - 216*y^2*z^5 + 1740*y^2*z^4 - 5328*y^2*z^3 + 7860*y^2*z^2 - 5646*y^2*z + 1606*y^2 - 64*y*z^6 + 624*y*z^5 - 2412*y*z^4 + 4784*y*z^3 - 5184*y*z^2 + 2954*y*z - 713*y - 8*z^7 + 92*z^6 - 432*z^5 + 1082*z^4 - 1574*z^3 + 1348*z^2 - 648*z + 1, 216*x^7 - 1188*x^6*y - 972*x^6*z + 1728*x^6 + 2610*x^5*y^2 + 4464*x^5*y*z - 7524*x^5*y + 1854*x^5*z^2 - 6444*x^5*z + 5346*x^5 - 3043*x^4*y^3 - 8031*x^4*y^2*z + 13128*x^4*y^2 - 6933*x^4*y*z^2 + 23088*x^4*y*z - 18459*x^4*y - 1945*x^4*z^3 + 9960*x^4*z^2 - 16281*x^4*z + 8515*x^4 + 2060*x^3*y^4 + 7393*x^3*y^3*z - 11869*x^3*y^3 + 9819*x^3*y^2*z^2 - 31905*x^3*y^2*z + 24879*x^3*y^2 + 5699*x^3*y*z^3 - 28203*x^3*y*z^2 + 44643*x^3*y*z - 22809*x^3*y + 1213*x^3*z^4 - 8167*x^3*z^3 + 19764*x^3*z^2 - 20468*x^3*z + 7766*x^3 - 816*x^2*y^5 - 3714*x^2*y^4*z + 5901*x^2*y^4 - 6696*x^2*y^3*z^2 + 21450*x^2*y^3*z - 16437*x^2*y^3 - 5964*x^2*y^2*z^3 + 28944*x^2*y^2*z^2 - 44826*x^2*y^2*z + 22512*x^2*y^2 - 2616*x^2*y*z^4 + 17142*x^2*y*z^3 - 40341*x^2*y*z^2 + 40908*x^2*y*z - 15291*x^2*y - 450*x^2*z^5 + 3747*x^2*z^4 - 11952*x^2*z^3 + 18396*x^2*z^2 - 13854*x^2*z + 4138*x^2 + 176*x*y^6 + 972*x*y^5*z - 1536*x*y^5 + 2220*x*y^4*z^2 - 7056*x*y^4*z + 5340*x*y^4 + 2680*x*y^3*z^3 - 12864*x*y^3*z^2 + 19620*x*y^3*z - 9725*x*y^3 + 1800*x*y^2*z^4 - 11616*x*y^2*z^3 + 26820*x*y^2*z^2 - 26775*x*y^2*z + 9891*x*y^2 + 636*x*y*z^5 - 5184*x*y*z^4 + 16140*x*y*z^3 - 24375*x*y*z^2 + 18105*x*y*z - 5362*x*y + 92*x*z^6 - 912*x*z^5 + 3600*x*z^4 - 7325*x*z^3 + 8214*x*z^2 - 4876*x*z + 1216*x - 16*y^7 - 104*y^6*z + 164*y^6 - 288*y^5*z^2 + 912*y^5*z - 684*y^5 - 440*y^4*z^3 + 2100*y^4*z^2 - 3168*y^4*z + 1554*y^4 - 400*y^3*z^4 + 2560*y^3*z^3 - 5832*y^3*z^2 + 5752*y^3*z - 2104*y^3 - 216*y^2*z^5 + 1740*y^2*z^4 - 5328*y^2*z^3 + 7932*y^2*z^2 - 5826*y^2*z + 1714*y^2 - 64*y*z^6 + 624*y*z^5 - 2412*y*z^4 + 4824*y*z^3 - 5340*y*z^2 + 3146*y*z - 785*y - 8*z^7 + 92*z^6 - 432*z^5 + 1090*z^4 - 1618*z^3 + 1432*z^2 - 712*z + 1, 432*x^7 - 2376*x^6*y - 1944*x^6*z + 3780*x^6 + 5220*x^5*y^2 + 8928*x^5*y*z - 16668*x^5*y + 3708*x^5*z^2 - 14184*x^5*z + 12636*x^5 - 6086*x^4*y^3 - 16062*x^4*y^2*z + 29361*x^4*y^2 - 13866*x^4*y*z^2 + 51414*x^4*y*z - 43992*x^4*y - 3890*x^4*z^3 + 22053*x^4*z^2 - 38664*x^4*z + 21107*x^4 + 4120*x^3*y^4 + 14786*x^3*y^3*z - 26750*x^3*y^3 + 19638*x^3*y^2*z^2 - 71685*x^3*y^2*z + 59703*x^3*y^2 + 11398*x^3*y*z^3 - 63120*x^3*y*z^2 + 106854*x^3*y*z - 56886*x^3*y + 2426*x^3*z^4 - 18185*x^3*z^3 + 47151*x^3*z^2 - 50935*x^3*z + 19903*x^3 - 1632*x^2*y^5 - 7428*x^2*y^4*z + 13386*x^2*y^4 - 13392*x^2*y^3*z^2 + 48546*x^2*y^3*z - 39681*x^2*y^3 - 11928*x^2*y^2*z^3 + 65322*x^2*y^2*z^2 - 108003*x^2*y^2*z + 56457*x^2*y^2 - 5232*x^2*y*z^4 + 38550*x^2*y*z^3 - 96963*x^2*y*z^2 + 102414*x^2*y*z - 39405*x^2*y - 900*x^2*z^5 + 8388*x^2*z^4 - 28641*x^2*z^3 + 45957*x^2*z^2 - 35643*x^2*z + 10898*x^2 + 352*x*y^6 + 1944*x*y^5*z - 3504*x*y^5 + 4440*x*y^4*z^2 - 16068*x*y^4*z + 12960*x*y^4 + 5360*x*y^3*z^3 - 29232*x*y^3*z^2 + 47544*x*y^3*z - 24514*x*y^3 + 3600*x*y^2*z^4 - 26328*x*y^2*z^3 + 64872*x*y^2*z^2 - 67398*x*y^2*z + 25626*x*y^2 + 1272*x*y*z^5 - 11712*x*y*z^4 + 38952*x*y*z^3 - 61254*x*y*z^2 + 46842*x*y*z - 14204*x*y + 184*x*z^6 - 2052*x*z^5 + 8664*x*z^4 - 18370*x*z^3 + 21216*x*z^2 - 12899*x*z + 3277*x - 32*y^7 - 208*y^6*z + 376*y^6 - 576*y^5*z^2 + 2088*y^5*z - 1668*y^5 - 880*y^4*z^3 + 4800*y^4*z^2 - 7716*y^4*z + 3936*y^4 - 800*y^3*z^4 + 5840*y^3*z^3 - 14184*y^3*z^2 + 14552*y^3*z - 5480*y^3 - 432*y^2*z^5 + 3960*y^2*z^4 - 12936*y^2*z^3 + 20040*y^2*z^2 - 15156*y^2*z + 4568*y^2 - 128*y*z^6 + 1416*y*z^5 - 5844*y*z^4 + 12168*y*z^3 - 13872*y*z^2 + 8374*y*z - 2129*y - 16*z^7 + 208*z^6 - 1044*z^5 + 2744*z^4 - 4196*z^3 + 3806*z^2 - 1929*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ 3, -2, -2 ], [ 2, -1, -1 ], [ 1, -2, -1 ] ], "source_shift": [ 2, 1, 1 ], "target_matrix": [ [ 1, 1, 2 ], [ 1, 1, 1 ], [ 2, 3, 2 ] ], "target_shift": [ -139, -155, -426 ], "collision_preimages": [ [ "0", "1/4", "3/4" ], [ "-4", "-4", "-3/2" ], [ "4", "-9", "33/2" ] ], "collision_image": [ "-557/4", "-621/4", "-853/2" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0326188920007553 ], "surface_features": [ 4.092899907145188, 5.0334358190550095, 3.7557986569738304, 0.0, 4.2343489284159075, 5090 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B077", "pair": 77, "family": "bijection", "mapping": "F(x,y,z)=(-9*x^7 - 117*x^6*y + 60*x^6*z + 127*x^6 - 648*x^5*y^2 + 666*x^5*y*z + 1407*x^5*y - 171*x^5*z^2 - 723*x^5*z - 1489*x^5 - 1980*x^4*y^3 + 3060*x^4*y^2*z + 6450*x^4*y^2 - 1575*x^4*y*z^2 - 6645*x^4*y*z - 10615*x^4*y + 270*x^4*z^3 + 1710*x^4*z^2 + 6020*x^4*z + 6154*x^4 - 3600*x^3*y^4 + 7440*x^3*y^3*z + 15640*x^3*y^3 - 5760*x^3*y^2*z^2 - 24240*x^3*y^2*z - 32650*x^3*y^2 + 1980*x^3*y*z^3 + 12510*x^3*y*z^2 + 35920*x^3*y*z + 32843*x^3*y - 255*x^3*z^4 - 2150*x^3*z^3 - 10000*x^3*z^2 - 19153*x^3*z - 12324*x^3 - 3888*x^2*y^5 + 10080*x^2*y^4*z + 21120*x^2*y^4 - 10440*x^2*y^3*z^2 - 43800*x^2*y^3*z - 52970*x^2*y^3 + 5400*x^2*y^2*z^3 + 34020*x^2*y^2*z^2 + 85620*x^2*y^2*z + 71232*x^2*y^2 - 1395*x^2*y*z^4 - 11730*x^2*y*z^3 - 46500*x^2*y*z^2 - 80331*x^2*y*z - 48888*x^2*y + 144*x^2*z^5 + 1515*x^2*z^4 + 8500*x^2*z^3 + 22965*x^2*z^2 + 28620*x^2*z + 13431*x^2 - 2304*x*y^6 + 7200*x*y^5*z + 15024*x*y^5 - 9360*x*y^4*z^2 - 39120*x*y^4*z - 44285*x*y^4 + 6480*x*y^3*z^3 + 40680*x*y^3*z^2 + 94360*x*y^3*z + 73160*x*y^3 - 2520*x*y^2*z^4 - 21120*x*y^2*z^3 - 75720*x*y^2*z^2 - 120648*x*y^2*z - 69777*x*y^2 + 522*x*y*z^5 + 5475*x*y*z^4 + 27160*x*y*z^3 + 66993*x*y*z^2 + 79110*x*y*z + 35994*x*y - 45*x*z^6 - 567*x*z^5 - 3680*x*z^4 - 12547*x*z^3 - 22725*x*z^2 - 20955*x*z - 7811*x - 576*y^7 + 2112*y^6*z + 4384*y^6 - 3312*y^5*z^2 - 13776*y^5*z - 14969*y^5 + 2880*y^4*z^3 + 18000*y^4*z^2 + 39710*y^4*z + 29317*y^4 - 1500*y^3*z^4 - 12520*y^3*z^3 - 42200*y^3*z^2 - 63476*y^3*z - 35223*y^3 + 468*y^2*z^5 + 4890*y^2*z^4 + 22480*y^2*z^3 + 51846*y^2*z^2 + 58482*y^2*z + 25833*y^2 - 81*y*z^6 - 1017*y*z^5 - 6010*y*z^4 - 18965*y*z^3 - 32679*y*z^2 - 29223*y*z - 10692*y + 6*z^7 + 88*z^6 + 646*z^5 + 2626*z^4 + 6156*z^3 + 8364*z^2 + 6171*z + 1, -9*x^7 - 117*x^6*y + 60*x^6*z + 128*x^6 - 648*x^5*y^2 + 666*x^5*y*z + 1419*x^5*y - 171*x^5*z^2 - 729*x^5*z - 1502*x^5 - 1980*x^4*y^3 + 3060*x^4*y^2*z + 6510*x^4*y^2 - 1575*x^4*y*z^2 - 6705*x^4*y*z - 10745*x^4*y + 270*x^4*z^3 + 1725*x^4*z^2 + 6085*x^4*z + 6224*x^4 - 3600*x^3*y^4 + 7440*x^3*y^3*z + 15800*x^3*y^3 - 5760*x^3*y^2*z^2 - 24480*x^3*y^2*z - 33170*x^3*y^2 + 1980*x^3*y*z^3 + 12630*x^3*y*z^2 + 36440*x^3*y*z + 33403*x^3*y - 255*x^3*z^4 - 2170*x^3*z^3 - 10130*x^3*z^2 - 19433*x^3*z - 12525*x^3 - 3888*x^2*y^5 + 10080*x^2*y^4*z + 21360*x^2*y^4 - 10440*x^2*y^3*z^2 - 44280*x^2*y^3*z - 54010*x^2*y^3 + 5400*x^2*y^2*z^3 + 34380*x^2*y^2*z^2 + 87180*x^2*y^2*z + 72912*x^2*y^2 - 1395*x^2*y*z^4 - 11850*x^2*y*z^3 - 47280*x^2*y*z^2 - 82011*x^2*y*z - 50094*x^2*y + 144*x^2*z^5 + 1530*x^2*z^4 + 8630*x^2*z^3 + 23385*x^2*z^2 + 29223*x^2*z + 13758*x^2 - 2304*x*y^6 + 7200*x*y^5*z + 15216*x*y^5 - 9360*x*y^4*z^2 - 39600*x*y^4*z - 45325*x*y^4 + 6480*x*y^3*z^3 + 41160*x*y^3*z^2 + 96440*x*y^3*z + 75400*x*y^3 - 2520*x*y^2*z^4 - 21360*x*y^2*z^3 - 77280*x*y^2*z^2 - 124008*x*y^2*z - 72189*x*y^2 + 522*x*y*z^5 + 5535*x*y*z^4 + 27680*x*y*z^3 + 68673*x*y*z^2 + 81522*x*y*z + 37302*x*y - 45*x*z^6 - 573*x*z^5 - 3745*x*z^4 - 12827*x*z^3 - 23328*x*z^2 - 21609*x*z - 8103*x - 576*y^7 + 2112*y^6*z + 4448*y^6 - 3312*y^5*z^2 - 13968*y^5*z - 15385*y^5 + 2880*y^4*z^3 + 18240*y^4*z^2 + 40750*y^4*z + 30437*y^4 - 1500*y^3*z^4 - 12680*y^3*z^3 - 43240*y^3*z^2 - 65716*y^3*z - 36831*y^3 + 468*y^2*z^5 + 4950*y^2*z^4 + 23000*y^2*z^3 + 53526*y^2*z^2 + 60894*y^2*z + 27141*y^2 - 81*y*z^6 - 1029*y*z^5 - 6140*y*z^4 - 19525*y*z^3 - 33885*y*z^2 - 30531*y*z - 11272*y + 6*z^7 + 89*z^6 + 659*z^5 + 2696*z^4 + 6357*z^3 + 8691*z^2 + 6462*z + 1, -9*x^7 - 117*x^6*y + 60*x^6*z + 127*x^6 - 648*x^5*y^2 + 666*x^5*y*z + 1407*x^5*y - 171*x^5*z^2 - 723*x^5*z - 1489*x^5 - 1980*x^4*y^3 + 3060*x^4*y^2*z + 6450*x^4*y^2 - 1575*x^4*y*z^2 - 6645*x^4*y*z - 10615*x^4*y + 270*x^4*z^3 + 1710*x^4*z^2 + 6020*x^4*z + 6154*x^4 - 3600*x^3*y^4 + 7440*x^3*y^3*z + 15640*x^3*y^3 - 5760*x^3*y^2*z^2 - 24240*x^3*y^2*z - 32650*x^3*y^2 + 1980*x^3*y*z^3 + 12510*x^3*y*z^2 + 35920*x^3*y*z + 32843*x^3*y - 255*x^3*z^4 - 2150*x^3*z^3 - 10000*x^3*z^2 - 19153*x^3*z - 12324*x^3 - 3888*x^2*y^5 + 10080*x^2*y^4*z + 21120*x^2*y^4 - 10440*x^2*y^3*z^2 - 43800*x^2*y^3*z - 52970*x^2*y^3 + 5400*x^2*y^2*z^3 + 34020*x^2*y^2*z^2 + 85620*x^2*y^2*z + 71232*x^2*y^2 - 1395*x^2*y*z^4 - 11730*x^2*y*z^3 - 46500*x^2*y*z^2 - 80331*x^2*y*z - 48888*x^2*y + 144*x^2*z^5 + 1515*x^2*z^4 + 8500*x^2*z^3 + 22965*x^2*z^2 + 28620*x^2*z + 13431*x^2 - 2304*x*y^6 + 7200*x*y^5*z + 15024*x*y^5 - 9360*x*y^4*z^2 - 39120*x*y^4*z - 44285*x*y^4 + 6480*x*y^3*z^3 + 40680*x*y^3*z^2 + 94360*x*y^3*z + 73160*x*y^3 - 2520*x*y^2*z^4 - 21120*x*y^2*z^3 - 75720*x*y^2*z^2 - 120648*x*y^2*z - 69777*x*y^2 + 522*x*y*z^5 + 5475*x*y*z^4 + 27160*x*y*z^3 + 66993*x*y*z^2 + 79110*x*y*z + 35994*x*y - 45*x*z^6 - 567*x*z^5 - 3680*x*z^4 - 12547*x*z^3 - 22725*x*z^2 - 20955*x*z - 7805*x - 576*y^7 + 2112*y^6*z + 4384*y^6 - 3312*y^5*z^2 - 13776*y^5*z - 14969*y^5 + 2880*y^4*z^3 + 18000*y^4*z^2 + 39710*y^4*z + 29317*y^4 - 1500*y^3*z^4 - 12520*y^3*z^3 - 42200*y^3*z^2 - 63476*y^3*z - 35223*y^3 + 468*y^2*z^5 + 4890*y^2*z^4 + 22480*y^2*z^3 + 51846*y^2*z^2 + 58482*y^2*z + 25833*y^2 - 81*y*z^6 - 1017*y*z^5 - 6010*y*z^4 - 18965*y*z^3 - 32679*y*z^2 - 29223*y*z - 10686*y + 6*z^7 + 88*z^6 + 646*z^5 + 2626*z^4 + 6156*z^3 + 8364*z^2 + 6167*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -1, -2, 1 ], [ -3, -2, 2 ], [ -2, -1, 1 ] ], "source_shift": [ 2, 2, 1 ], "target_matrix": [ [ 1, 1, 1 ], [ 1, 2, 1 ], [ 1, 1, 2 ] ], "target_shift": [ -1925, -2037, -1919 ], "inverse_recipe": { "p": 6, "q": 1, "f": "x**6 + x**5 + x**3 + x**2 + x", "g": "x", "h": "y*z + z**5 + z**3", "polynomials": { "f": [ [ [ 6, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ], [ [ 3, 0, 0 ], 1 ], [ [ 5, 0, 0 ], 1 ] ], "g": [ [ [ 1, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 0, 5 ], 1 ], [ [ 0, 0, 3 ], 1 ] ] }, "h_multiplier": 3 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0575891549997323 ], "surface_features": [ 4.294312125937958, 5.093449703257268, 3.955471270547337, 0.0, 4.390213330698722, 5136 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C136", "pair": 136, "family": "counterexample", "mapping": "F(x,y,z)=(54*x^7 - 324*x^6*y - 486*x^6*z + 945*x^6 + 828*x^5*y^2 + 2466*x^5*y*z - 4626*x^5*y + 1800*x^5*z^2 - 6813*x^5*z + 4662*x^5 - 1168*x^4*y^3 - 5178*x^4*y^2*z + 9390*x^4*y^2 - 7500*x^4*y*z^2 + 27393*x^4*y*z - 18777*x^4*y - 3544*x^4*z^3 + 19542*x^4*z^2 - 27525*x^4*z + 11039*x^4 + 982*x^3*y^4 + 5758*x^3*y^3*z - 10118*x^3*y^3 + 12408*x^3*y^2*z^2 - 43863*x^3*y^2*z + 30084*x^3*y^2 + 11632*x^3*y*z^3 - 62016*x^3*y*z^2 + 87240*x^3*y*z - 35246*x^3*y + 4000*x^3*z^4 - 28568*x^3*z^3 + 61800*x^3*z^2 - 51227*x^3*z + 14674*x^3 - 492*x^2*y^5 - 3576*x^2*y^4*z + 6105*x^2*y^4 - 10188*x^2*y^3*z^2 + 34971*x^2*y^3*z - 23973*x^2*y^3 - 14208*x^2*y^2*z^3 + 73530*x^2*y^2*z^2 - 103179*x^2*y^2*z + 41949*x^2*y^2 - 9696*x^2*y*z^4 + 67224*x^2*y*z^3 - 144672*x^2*y*z^2 + 120507*x^2*y*z - 34833*x^2*y - 2592*x^2*z^5 + 22560*x^2*z^4 - 66024*x^2*z^3 + 84450*x^2*z^2 - 50025*x^2*z + 11343*x^2 + 136*x*y^6 + 1176*x*y^5*z - 1956*x*y^5 + 4152*x*y^4*z^2 - 13884*x*y^4*z + 9504*x*y^4 + 7656*x*y^3*z^3 - 38604*x*y^3*z^2 + 53988*x*y^3*z - 22066*x*y^3 + 7776*x*y^2*z^4 - 52548*x*y^2*z^3 + 112452*x*y^2*z^2 - 94020*x*y^2*z + 27390*x*y^2 + 4128*x*y*z^5 - 35040*x*y*z^4 + 101760*x*y*z^3 - 130380*x*y*z^2 + 77694*x*y*z - 17778*x*y + 896*x*z^6 - 9168*x*z^5 + 33792*x*z^4 - 58836*x*z^3 + 53676*x*z^2 - 25125*x*z + 4794*x - 16*y^7 - 160*y^6*z + 260*y^6 - 672*y^5*z^2 + 2196*y^5*z - 1500*y^5 - 1536*y^4*z^3 + 7572*y^4*z^2 - 10548*y^4*z + 4330*y^4 - 2064*y^3*z^4 + 13644*y^3*z^3 - 29028*y^3*z^2 + 24338*y^3*z - 7138*y^3 - 1632*y^2*z^5 + 13560*y^2*z^4 - 39084*y^2*z^3 + 50124*y^2*z^2 - 30018*y^2*z + 6923*y^2 - 704*y*z^6 + 7056*y*z^5 - 25776*y*z^4 + 44844*y*z^3 - 41040*y*z^2 + 19321*y*z - 3721*y - 128*z^7 + 1504*z^6 - 6672*z^5 + 14728*z^4 - 18260*z^3 + 13118*z^2 - 5163*z + 1, -27*x^7 + 162*x^6*y + 243*x^6*z - 459*x^6 - 414*x^5*y^2 - 1233*x^5*y*z + 2241*x^5*y - 900*x^5*z^2 + 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29712*x^3*y*z^2 - 39297*x^3*y*z + 15292*x^3*y - 2000*x^3*z^4 + 13648*x^3*z^3 - 27708*x^3*z^2 + 22099*x^3*z - 6185*x^3 + 246*x^2*y^5 + 1788*x^2*y^4*z - 2931*x^2*y^4 + 5094*x^2*y^3*z^2 - 16743*x^2*y^3*z + 10812*x^2*y^3 + 7104*x^2*y^2*z^3 - 35100*x^2*y^2*z^2 + 46320*x^2*y^2*z - 18135*x^2*y^2 + 4848*x^2*y*z^4 - 31992*x^2*y*z^3 + 64650*x^2*y*z^2 - 51804*x^2*y*z + 14625*x^2*y + 1296*x^2*z^5 - 10704*x^2*z^4 + 29376*x^2*z^3 - 36102*x^2*z^2 + 20868*x^2*z - 4665*x^2 - 68*x*y^6 - 588*x*y^5*z + 936*x*y^5 - 2076*x*y^4*z^2 + 6624*x*y^4*z - 4272*x*y^4 - 3828*x*y^3*z^3 + 18360*x*y^3*z^2 - 24156*x*y^3*z + 9506*x*y^3 - 3888*x*y^2*z^4 + 24912*x*y^2*z^3 - 50088*x*y^2*z^2 + 40281*x*y^2*z - 11457*x*y^2 - 2064*x*y*z^5 + 16560*x*y*z^4 - 45132*x*y*z^3 + 55560*x*y*z^2 - 32295*x*y*z + 7281*x*y - 448*x*z^6 + 4320*x*z^5 - 14928*x*z^4 + 24948*x*z^3 - 22176*x*z^2 + 10215*x*z - 1935*x + 8*y^7 + 80*y^6*z - 124*y^6 + 336*y^5*z^2 - 1044*y^5*z + 672*y^5 + 768*y^4*z^3 - 3588*y^4*z^2 + 4704*y^4*z - 1859*y^4 + 1032*y^3*z^4 - 6444*y^3*z^3 + 12888*y^3*z^2 - 10393*y^3*z + 2975*y^3 + 816*y^2*z^5 - 6384*y^2*z^4 + 17280*y^2*z^3 - 21294*y^2*z^2 + 12435*y^2*z - 2824*y^2 + 352*y*z^6 - 3312*y*z^5 + 11352*y*z^4 - 18960*y*z^3 + 16902*y*z^2 - 7826*y*z + 1495*y + 64*z^7 - 704*z^6 + 2928*z^5 - 6200*z^4 + 7480*z^3 - 5278*z^2 + 2057*z - 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ 1, -1, -2 ], [ 3, -2, -2 ], [ 1, -1, -1 ] ], "source_shift": [ 1, 2, 2 ], "target_matrix": [ [ 2, 3, 2 ], [ -1, -1, -2 ], [ -1, -1, -1 ] ], "target_shift": [ -863, 337, 343 ], "collision_preimages": [ [ "5/2", "6", "-5/4" ], [ "-25/2", "-43/2", "9/2" ], [ "-19/2", "-41/2", "13/2" ] ], "collision_image": [ "-1727/2", "1349/4", "1373/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0433811980001337 ], "surface_features": [ 4.195619329833383, 5.1603844853542, 3.836830286488879, 0.0, 4.340888346075545, 5125 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B066", "pair": 66, "family": "bijection", "mapping": "F(x,y,z)=(2*x^7 + 14*x^6*y - 14*x^6*z - 40*x^6 + 42*x^5*y^2 - 84*x^5*y*z - 240*x^5*y + 42*x^5*z^2 + 244*x^5*z + 338*x^5 + 70*x^4*y^3 - 210*x^4*y^2*z - 600*x^4*y^2 + 210*x^4*y*z^2 + 1220*x^4*y*z + 1688*x^4*y - 70*x^4*z^3 - 620*x^4*z^2 - 1740*x^4*z - 1576*x^4 + 70*x^3*y^4 - 280*x^3*y^3*z - 800*x^3*y^3 + 420*x^3*y^2*z^2 + 2440*x^3*y^2*z + 3372*x^3*y^2 - 280*x^3*y*z^3 - 2480*x^3*y*z^2 - 6952*x^3*y*z - 6276*x^3*y + 70*x^3*z^4 + 840*x^3*z^3 + 3582*x^3*z^2 + 6550*x^3*z + 4374*x^3 + 42*x^2*y^5 - 210*x^2*y^4*z - 600*x^2*y^4 + 420*x^2*y^3*z^2 + 2440*x^2*y^3*z + 3368*x^2*y^3 - 420*x^2*y^2*z^3 - 3720*x^2*y^2*z^2 - 10416*x^2*y^2*z - 9372*x^2*y^2 + 210*x^2*y*z^4 + 2520*x^2*y*z^3 + 10734*x^2*y*z^2 + 19562*x^2*y*z + 12988*x^2*y - 42*x^2*z^5 - 640*x^2*z^4 - 3686*x^2*z^3 - 10204*x^2*z^2 - 13714*x^2*z - 7191*x^2 + 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input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0249466210007085 ], "surface_features": [ 3.5284556780730023, 4.44015312977356, 3.1981069988734014, 0.0, 3.6383217827328944, 4893 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C051", "pair": 51, "family": "counterexample", "mapping": "F(x,y,z)=(-6*x^7 - 45*x^6*y - 60*x^6*z - 60*x^6 - 144*x^5*y^2 - 387*x^5*y*z - 384*x^5*y - 252*x^5*z^2 - 513*x^5*z - 264*x^5 - 255*x^4*y^3 - 1035*x^4*y^2*z - 1020*x^4*y^2 - 1359*x^4*y*z^2 - 2748*x^4*y*z - 1404*x^4*y - 576*x^4*z^3 - 1791*x^4*z^2 - 1869*x^4*z - 662*x^4 - 270*x^3*y^4 - 1470*x^3*y^3*z - 1440*x^3*y^3 - 2916*x^3*y^2*z^2 - 5862*x^3*y^2*z - 2976*x^3*y^2 - 2493*x^3*y*z^3 - 7710*x^3*y*z^2 - 7992*x^3*y*z - 2814*x^3*y - 774*x^3*z^4 - 3267*x^3*z^3 - 5178*x^3*z^2 - 3713*x^3*z - 1024*x^3 - 171*x^2*y^5 - 1170*x^2*y^4*z - 1140*x^2*y^4 - 3114*x^2*y^3*z^2 - 6228*x^2*y^3*z - 3144*x^2*y^3 - 4023*x^2*y^2*z^3 - 12384*x^2*y^2*z^2 - 12762*x^2*y^2*z - 4470*x^2*y^2 - 2520*x^2*y*z^4 - 10596*x^2*y*z^3 - 16695*x^2*y*z^2 - 11907*x^2*y*z - 3270*x^2*y - 612*x^2*z^5 - 3285*x^2*z^4 - 7017*x^2*z^3 - 7620*x^2*z^2 - 4245*x^2*z - 981*x^2 - 60*x*y^6 - 495*x*y^5*z - 480*x*y^5 - 1656*x*y^4*z^2 - 3297*x*y^4*z - 1656*x*y^4 - 2871*x*y^3*z^3 - 8802*x*y^3*z^2 - 9024*x*y^3*z - 3146*x*y^3 - 2718*x*y^2*z^4 - 11391*x*y^2*z^3 - 17856*x*y^2*z^2 - 12675*x*y^2*z - 3468*x*y^2 - 1332*x*y*z^5 - 7134*x*y*z^4 - 15165*x*y*z^3 - 16392*x*y*z^2 - 9102*x*y*z - 2100*x*y - 264*x*z^6 - 1728*x*z^5 - 4656*x*z^4 - 6784*x*z^3 - 5700*x*z^2 - 2655*x*z - 542*x - 9*y^7 - 87*y^6*z - 84*y^6 - 351*y^5*z^2 - 696*y^5*z - 348*y^5 - 765*y^4*z^3 - 2337*y^4*z^2 - 2385*y^4*z - 828*y^4 - 972*y^3*z^4 - 4062*y^3*z^3 - 6339*y^3*z^2 - 4481*y^3*z - 1222*y^3 - 720*y^2*z^5 - 3849*y^2*z^4 - 8148*y^2*z^3 - 8772*y^2*z^2 - 4857*y^2*z - 1119*y^2 - 288*y*z^6 - 1884*y*z^5 - 5058*y*z^4 - 7341*y*z^3 - 6156*y*z^2 - 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420*x^4*y*z^2 - 955*x^4*y*z - 525*x^4*y - 280*x^4*z^3 - 950*x^4*z^2 - 1040*x^4*z - 371*x^4 - 35*x^3*y^4 - 280*x^3*y^3*z - 320*x^3*y^3 - 840*x^3*y^2*z^2 - 1910*x^3*y^2*z - 1050*x^3*y^2 - 1120*x^3*y*z^3 - 3800*x^3*y*z^2 - 4160*x^3*y*z - 1484*x^3*y - 560*x^3*z^4 - 2520*x^3*z^3 - 4120*x^3*z^2 - 2928*x^3*z - 772*x^3 - 21*x^2*y^5 - 210*x^2*y^4*z - 240*x^2*y^4 - 840*x^2*y^3*z^2 - 1910*x^2*y^3*z - 1050*x^2*y^3 - 1680*x^2*y^2*z^3 - 5700*x^2*y^2*z^2 - 6240*x^2*y^2*z - 2226*x^2*y^2 - 1680*x^2*y*z^4 - 7560*x^2*y*z^3 - 12360*x^2*y*z^2 - 8784*x^2*y*z - 2317*x^2*y - 672*x^2*z^5 - 3760*x^2*z^4 - 8160*x^2*z^3 - 8664*x^2*z^2 - 4550*x^2*z - 963*x^2 - 7*x*y^6 - 84*x*y^5*z - 96*x*y^5 - 420*x*y^4*z^2 - 955*x*y^4*z - 525*x*y^4 - 1120*x*y^3*z^3 - 3800*x*y^3*z^2 - 4160*x*y^3*z - 1484*x*y^3 - 1680*x*y^2*z^4 - 7560*x*y^2*z^3 - 12360*x*y^2*z^2 - 8784*x*y^2*z - 2318*x*y^2 - 1344*x*y*z^5 - 7520*x*y*z^4 - 16320*x*y*z^3 - 17328*x*y*z^2 - 9104*x*y*z - 1932*x*y - 448*x*z^6 - 2992*x*z^5 - 8080*x*z^4 - 11392*x*z^3 - 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3200*z^5 + 5616*z^4 + 5848*z^3 + 3665*z^2 + 1308*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ 1, 1, 2 ], [ 1, 1, 1 ], [ 1, 2, 1 ] ], "source_shift": [ 2, 1, 1 ], "target_matrix": [ [ -3, -3, -2 ], [ -1, -2, -1 ], [ 1, 1, 1 ] ], "target_shift": [ 710, 217, -210 ], "inverse_recipe": { "p": 2, "q": 5, "f": "x**2 + x", "g": "x**5 + x**2 + x", "h": "y*z + y", "polynomials": { "f": [ [ [ 2, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "g": [ [ [ 5, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 1, 0 ], 1 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0275874959988869 ], "surface_features": [ 3.6832734870902337, 4.723816749815778, 3.225309281725863, 0.0, 3.8800428035215893, 4881 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C241", "pair": 241, "family": "counterexample", "mapping": "F(x,y,z)=(32*x^7 + 352*x^6*y - 208*x^6*z - 472*x^6 + 1632*x^5*y^2 - 1944*x^5*y*z - 4452*x^5*y + 576*x^5*z^2 + 2652*x^5*z + 2844*x^5 + 4120*x^4*y^3 - 7428*x^4*y^2*z - 17226*x^4*y^2 + 4440*x^4*y*z^2 + 20688*x^4*y*z + 22272*x^4*y - 880*x^4*z^3 - 6180*x^4*z^2 - 13380*x^4*z - 9132*x^4 + 6086*x^3*y^4 - 14786*x^3*y^3*z - 34895*x^3*y^3 + 13392*x^3*y^2*z^2 + 63435*x^3*y^2*z + 68715*x^3*y^2 - 5360*x^3*y*z^3 - 38232*x^3*y*z^2 - 83244*x^3*y*z - 56766*x^3*y + 800*x^3*z^4 + 7640*x^3*z^3 + 25080*x^3*z^2 + 34442*x^3*z + 16938*x^3 + 5220*x^2*y^5 - 16062*x^2*y^4*z - 38868*x^2*y^4 + 19638*x^2*y^3*z^2 + 95211*x^2*y^3*z + 104127*x^2*y^3 - 11928*x^2*y^2*z^3 - 86949*x^2*y^2*z^2 - 190965*x^2*y^2*z - 130308*x^2*y^2 + 3600*x^2*y*z^4 + 35088*x^2*y*z^3 + 116100*x^2*y*z^2 + 159480*x^2*y*z + 78036*x^2*y - 432*x^2*z^5 - 5280*x^2*z^4 - 23400*x^2*z^3 - 48534*x^2*z^2 - 47898*x^2*z - 18170*x^2 + 2376*x*y^6 - 8928*x*y^5*z - 22428*x*y^5 + 13866*x*y^4*z^2 + 69564*x*y^4*z + 77208*x*y^4 - 11398*x*y^3*z^3 - 85737*x*y^3*z^2 - 190806*x*y^3*z - 130586*x*y^3 + 5232*x*y^2*z^4 + 52497*x*y^2*z^3 + 175785*x*y^2*z^2 + 242004*x*y^2*z + 117942*x*y^2 - 1272*x*y*z^5 - 15972*x*y*z^4 - 71556*x*y*z^3 - 148662*x*y*z^2 - 146070*x*y*z - 54915*x*y + 128*x*z^6 + 1932*x*z^5 + 10860*x*z^4 + 30270*x*z^3 + 44982*x*z^2 + 34159*x*z + 10407*x + 432*y^7 - 1944*y^6*z - 5184*y^6 + 3708*y^5*z^2 + 19620*y^5*z + 22284*y^5 - 3890*y^4*z^3 - 30696*y^4*z^2 - 69720*y^4*z - 48028*y^4 + 2426*y^3*z^4 + 25421*y^3*z^3 + 86679*y^3*z^2 + 119962*y^3*z + 58324*y^3 - 900*y^2*z^5 - 11757*y^2*z^4 - 53535*y^2*z^3 - 111696*y^2*z^2 - 109410*y^2*z - 40794*y^2 + 184*y*z^6 + 2880*y*z^5 + 16428*y*z^4 + 45948*y*z^3 + 68034*y*z^2 + 51213*y*z + 15403*y - 16*z^7 - 292*z^6 - 2004*z^5 - 7046*z^4 - 14022*z^3 - 15989*z^2 - 9723*z - 1, -16*x^7 - 176*x^6*y + 104*x^6*z + 224*x^6 - 816*x^5*y^2 + 972*x^5*y*z + 2112*x^5*y - 288*x^5*z^2 - 1260*x^5*z - 1284*x^5 - 2060*x^4*y^3 + 3714*x^4*y^2*z + 8172*x^4*y^2 - 2220*x^4*y*z^2 - 9828*x^4*y*z - 10044*x^4*y + 440*x^4*z^3 + 2940*x^4*z^2 + 6048*x^4*z + 3952*x^4 - 3043*x^3*y^4 + 7393*x^3*y^3*z + 16564*x^3*y^3 - 6696*x^3*y^2*z^2 - 30147*x^3*y^2*z - 30969*x^3*y^2 + 2680*x^3*y*z^3 + 18192*x^3*y*z^2 + 37596*x^3*y*z + 24530*x^3*y - 400*x^3*z^4 - 3640*x^3*z^3 - 11352*x^3*z^2 - 14922*x^3*z - 7070*x^3 - 2610*x^2*y^5 + 8031*x^2*y^4*z + 18477*x^2*y^4 - 9819*x^2*y^3*z^2 - 45300*x^2*y^3*z - 46932*x^2*y^3 + 5964*x^2*y^2*z^3 + 41409*x^2*y^2*z^2 + 86226*x^2*y^2*z + 56259*x^2*y^2 - 1800*x^2*y*z^4 - 16728*x^2*y*z^3 - 52524*x^2*y*z^2 - 69012*x^2*y*z - 32520*x^2*y + 216*x^2*z^5 + 2520*x^2*z^4 + 10608*x^2*z^3 + 21054*x^2*z^2 + 20016*x^2*z + 7336*x^2 - 1188*x*y^6 + 4464*x*y^5*z + 10692*x*y^5 - 6933*x*y^4*z^2 - 33174*x*y^4*z - 34836*x*y^4 + 5699*x*y^3*z^3 + 40908*x*y^3*z^2 + 86208*x*y^3*z + 56373*x*y^3 - 2616*x*y^2*z^4 - 25065*x*y^2*z^3 - 79545*x*y^2*z^2 - 104670*x*y^2*z - 49107*x*y^2 + 636*x*y*z^5 + 7632*x*y*z^4 + 32436*x*y*z^3 + 64434*x*y*z^2 + 60963*x*y*z + 22134*x*y - 64*x*z^6 - 924*x*z^5 - 4932*x*z^4 - 13150*x*z^3 - 18822*x*z^2 - 13811*x*z - 4065*x - 216*y^7 + 972*y^6*z + 2484*y^6 - 1854*y^5*z^2 - 9396*y^5*z - 10080*y^5 + 1945*y^4*z^3 + 14697*y^4*z^2 + 31560*y^4*z + 20754*y^4 - 1213*y^3*z^4 - 12172*y^3*z^3 - 39276*y^3*z^2 - 51909*y^3*z - 24288*y^3 + 450*y^2*z^5 + 5631*y^2*z^4 + 24288*y^2*z^3 + 48411*y^2*z^2 + 45645*y^2*z + 16434*y^2 - 92*y*z^6 - 1380*y*z^5 - 7464*y*z^4 - 19952*y*z^3 - 28443*y*z^2 - 20682*y*z - 6010*y + 8*z^7 + 140*z^6 + 912*z^5 + 3066*z^4 + 5876*z^3 + 6475*z^2 + 3806*z + 1, 16*x^7 + 176*x^6*y - 104*x^6*z - 224*x^6 + 816*x^5*y^2 - 972*x^5*y*z - 2112*x^5*y + 288*x^5*z^2 + 1260*x^5*z + 1284*x^5 + 2060*x^4*y^3 - 3714*x^4*y^2*z - 8172*x^4*y^2 + 2220*x^4*y*z^2 + 9828*x^4*y*z + 10044*x^4*y - 440*x^4*z^3 - 2940*x^4*z^2 - 6048*x^4*z - 3954*x^4 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972*y^6*z - 2484*y^6 + 1854*y^5*z^2 + 9396*y^5*z + 10080*y^5 - 1945*y^4*z^3 - 14697*y^4*z^2 - 31560*y^4*z - 20762*y^4 + 1213*y^3*z^4 + 12172*y^3*z^3 + 39276*y^3*z^2 + 51929*y^3*z + 24356*y^3 - 450*y^2*z^5 - 5631*y^2*z^4 - 24288*y^2*z^3 - 48429*y^2*z^2 - 45765*y^2*z - 16578*y^2 + 92*y*z^6 + 1380*y*z^5 + 7464*y*z^4 + 19959*y*z^3 + 28512*y*z^2 + 20850*y*z + 6122*y - 8*z^7 - 140*z^6 - 912*z^5 - 3067*z^4 - 5889*z^3 - 6523*z^2 - 3872*z - 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -1, -2, 1 ], [ -2, -3, 2 ], [ -2, -1, 1 ] ], "source_shift": [ 2, 2, 1 ], "target_matrix": [ [ -2, -3, -2 ], [ 1, 1, 2 ], [ -1, -1, -1 ] ], "target_shift": [ 2441, -924, 952 ], "collision_preimages": [ [ "5/4", "2", "13/4" ], [ "-8", "-3/2", "-12" ], [ "-3", "11/2", "5" ] ], "collision_image": [ "4883/2", "-3697/4", "3809/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.026976963999914 ], "surface_features": [ 4.39027443406441, 5.383822544342294, 4.003460532109506, 0.0, 4.541954170892387, 5172 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B064", "pair": 64, "family": "bijection", "mapping": "F(x,y,z)=(768*x^7 - 2688*x^6*y + 2688*x^6*z - 4992*x^6 + 4032*x^5*y^2 - 8064*x^5*y*z + 15552*x^5*y + 4032*x^5*z^2 - 15168*x^5*z + 13008*x^5 - 3360*x^4*y^3 + 10080*x^4*y^2*z - 20160*x^4*y^2 - 10080*x^4*y*z^2 + 39360*x^4*y*z - 35016*x^4*y + 3360*x^4*z^3 - 19200*x^4*z^2 + 33384*x^4*z - 17712*x^4 + 1680*x^3*y^4 - 6720*x^3*y^3*z + 13920*x^3*y^3 + 10080*x^3*y^2*z^2 - 40800*x^3*y^2*z + 37620*x^3*y^2 - 6720*x^3*y*z^3 + 39840*x^3*y*z^2 - 71832*x^3*y*z + 39456*x^3*y + 1680*x^3*z^4 - 12960*x^3*z^3 + 34260*x^3*z^2 - 36912*x^3*z + 13826*x^3 - 504*x^2*y^5 + 2520*x^2*y^4*z - 5400*x^2*y^4 - 5040*x^2*y^3*z^2 + 21120*x^2*y^3*z - 20166*x^2*y^3 + 5040*x^2*y^2*z^3 - 30960*x^2*y^2*z^2 + 57834*x^2*y^2*z 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858*z - 1, -768*x^7 + 2688*x^6*y - 2688*x^6*z + 4992*x^6 - 4032*x^5*y^2 + 8064*x^5*y*z - 15552*x^5*y - 4032*x^5*z^2 + 15168*x^5*z - 13008*x^5 + 3360*x^4*y^3 - 10080*x^4*y^2*z + 20160*x^4*y^2 + 10080*x^4*y*z^2 - 39360*x^4*y*z + 35016*x^4*y - 3360*x^4*z^3 + 19200*x^4*z^2 - 33384*x^4*z + 17712*x^4 - 1680*x^3*y^4 + 6720*x^3*y^3*z - 13920*x^3*y^3 - 10080*x^3*y^2*z^2 + 40800*x^3*y^2*z - 37620*x^3*y^2 + 6720*x^3*y*z^3 - 39840*x^3*y*z^2 + 71832*x^3*y*z - 39456*x^3*y - 1680*x^3*z^4 + 12960*x^3*z^3 - 34260*x^3*z^2 + 36912*x^3*z - 13826*x^3 + 504*x^2*y^5 - 2520*x^2*y^4*z + 5400*x^2*y^4 + 5040*x^2*y^3*z^2 - 21120*x^2*y^3*z + 20166*x^2*y^3 - 5040*x^2*y^2*z^3 + 30960*x^2*y^2*z^2 - 57834*x^2*y^2*z + 32868*x^2*y^2 + 2520*x^2*y*z^4 - 20160*x^2*y*z^3 + 55242*x^2*y*z^2 - 61608*x^2*y*z + 23916*x^2*y - 504*x^2*z^5 + 4920*x^2*z^4 - 17574*x^2*z^3 + 28836*x^2*z^2 - 22008*x^2*z + 6294*x^2 - 84*x*y^6 + 504*x*y^5*z - 1116*x*y^5 - 1260*x*y^4*z^2 + 5460*x*y^4*z - 5394*x*y^4 + 1680*x*y^3*z^3 - 10680*x*y^3*z^2 + 20652*x*y^3*z - 12132*x*y^3 - 1260*x*y^2*z^4 + 10440*x*y^2*z^3 - 29628*x*y^2*z^2 + 34176*x*y^2*z - 13776*x*y^2 + 504*x*y*z^5 - 5100*x*y*z^4 + 18876*x*y*z^3 - 32052*x*y*z^2 + 25392*x*y*z - 7536*x*y - 84*x*z^6 + 996*x*z^5 - 4506*x*z^4 + 10008*x*z^3 - 11682*x*z^2 + 6816*x*z - 1564*x + 6*y^7 - 42*y^6*z + 96*y^6 + 126*y^5*z^2 - 564*y^5*z + 576*y^5 - 210*y^4*z^3 + 1380*y^4*z^2 - 2760*y^4*z + 1674*y^4 + 210*y^3*z^4 - 1800*y^3*z^3 + 5286*y^3*z^2 - 6300*y^3*z + 2644*y^3 - 126*y^2*z^5 + 1320*y^2*z^4 - 5058*y^2*z^3 + 8880*y^2*z^2 - 7320*y^2*z + 2262*y^2 + 42*y*z^6 - 516*y*z^5 + 2418*y*z^4 - 5556*y*z^3 + 6744*y*z^2 - 4092*y*z + 973*y - 6*z^7 + 84*z^6 - 462*z^5 + 1302*z^4 - 2068*z^3 + 1848*z^2 - 863*z + 1, 768*x^7 - 2688*x^6*y + 2688*x^6*z - 4992*x^6 + 4032*x^5*y^2 - 8064*x^5*y*z + 15552*x^5*y + 4032*x^5*z^2 - 15168*x^5*z + 13008*x^5 - 3360*x^4*y^3 + 10080*x^4*y^2*z - 20160*x^4*y^2 - 10080*x^4*y*z^2 + 39360*x^4*y*z - 35016*x^4*y + 3360*x^4*z^3 - 19200*x^4*z^2 + 33384*x^4*z - 17712*x^4 + 1680*x^3*y^4 - 6720*x^3*y^3*z + 13920*x^3*y^3 + 10080*x^3*y^2*z^2 - 40800*x^3*y^2*z + 37620*x^3*y^2 - 6720*x^3*y*z^3 + 39840*x^3*y*z^2 - 71832*x^3*y*z + 39456*x^3*y + 1680*x^3*z^4 - 12960*x^3*z^3 + 34260*x^3*z^2 - 36912*x^3*z + 13810*x^3 - 504*x^2*y^5 + 2520*x^2*y^4*z - 5400*x^2*y^4 - 5040*x^2*y^3*z^2 + 21120*x^2*y^3*z - 20166*x^2*y^3 + 5040*x^2*y^2*z^3 - 30960*x^2*y^2*z^2 + 57834*x^2*y^2*z - 32868*x^2*y^2 - 2520*x^2*y*z^4 + 20160*x^2*y*z^3 - 55242*x^2*y*z^2 + 61608*x^2*y*z - 23892*x^2*y + 504*x^2*z^5 - 4920*x^2*z^4 + 17574*x^2*z^3 - 28836*x^2*z^2 + 21984*x^2*z - 6258*x^2 + 84*x*y^6 - 504*x*y^5*z + 1116*x*y^5 + 1260*x*y^4*z^2 - 5460*x*y^4*z + 5394*x*y^4 - 1680*x*y^3*z^3 + 10680*x*y^3*z^2 - 20652*x*y^3*z + 12132*x*y^3 + 1260*x*y^2*z^4 - 10440*x*y^2*z^3 + 29628*x*y^2*z^2 - 34176*x*y^2*z + 13764*x*y^2 - 504*x*y*z^5 + 5100*x*y*z^4 - 18876*x*y*z^3 + 32052*x*y*z^2 - 25368*x*y*z + 7500*x*y + 84*x*z^6 - 996*x*z^5 + 4506*x*z^4 - 10008*x*z^3 + 11670*x*z^2 - 6780*x*z + 1533*x - 6*y^7 + 42*y^6*z - 96*y^6 - 126*y^5*z^2 + 564*y^5*z - 576*y^5 + 210*y^4*z^3 - 1380*y^4*z^2 + 2760*y^4*z - 1674*y^4 - 210*y^3*z^4 + 1800*y^3*z^3 - 5286*y^3*z^2 + 6300*y^3*z - 2642*y^3 + 126*y^2*z^5 - 1320*y^2*z^4 + 5058*y^2*z^3 - 8880*y^2*z^2 + 7314*y^2*z - 2253*y^2 - 42*y*z^6 + 516*y*z^5 - 2418*y*z^4 + 5556*y*z^3 - 6738*y*z^2 + 4074*y*z - 955*y + 6*z^7 - 84*z^6 + 462*z^5 - 1302*z^4 + 2066*z^3 - 1839*z^2 + 846*z - 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -2, 1, -1 ], [ -3, 3, -2 ], [ -1, 1, -1 ] ], "source_shift": [ 1, 1, 1 ], "target_matrix": [ [ -1, -1, -1 ], [ 1, 2, 1 ], [ -1, -1, -2 ] ], "target_shift": [ 159, -162, 153 ], "inverse_recipe": { "p": 2, "q": 3, "f": "x**2 + x", "g": "x**3 + x**2", "h": "y**2*z", "polynomials": { "f": [ [ [ 2, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "g": [ [ [ 3, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "h": [ [ [ 0, 2, 1 ], 1 ] ] }, "h_multiplier": 6 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0366720969977905 ], "surface_features": [ 4.06846179646329, 4.856317958551798, 3.727541257028556, 0.0, 4.167344989475084, 5074 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C011", "pair": 11, "family": "counterexample", "mapping": "F(x,y,z)=(-4*x^7 - 38*x^6*y + 26*x^6*z + 64*x^6 - 150*x^5*y^2 + 210*x^5*y*z + 531*x^5*y - 72*x^5*z^2 - 357*x^5*z - 411*x^5 - 318*x^4*y^3 + 684*x^4*y^2*z + 1776*x^4*y^2 - 480*x^4*y*z^2 - 2445*x^4*y*z - 2880*x^4*y + 110*x^4*z^3 + 825*x^4*z^2 + 1920*x^4*z + 1444*x^4 - 390*x^3*y^4 + 1146*x^3*y^3*z + 3051*x^3*y^3 - 1236*x^3*y^2*z^2 - 6465*x^3*y^2*z - 7797*x^3*y^2 + 580*x^3*y*z^3 + 4470*x^3*y*z^2 + 10656*x^3*y*z + 8189*x^3*y - 100*x^3*z^4 - 1010*x^3*z^3 - 3570*x^3*z^2 - 5427*x^3*z - 3043*x^3 - 276*x^2*y^5 + 1038*x^2*y^4*z + 2826*x^2*y^4 - 1530*x^2*y^3*z^2 - 8205*x^2*y^3*z - 10134*x^2*y^3 + 1104*x^2*y^2*z^3 + 8739*x^2*y^2*z^2 + 21360*x^2*y^2*z + 16782*x^2*y^2 - 390*x^2*y*z^4 - 4050*x^2*y*z^3 - 14688*x^2*y*z^2 - 22851*x^2*y*z - 13092*x^2*y + 54*x^2*z^5 + 690*x^2*z^4 + 3300*x^2*z^3 + 7617*x^2*z^2 + 8622*x^2*z + 3878*x^2 - 104*x*y^6 + 480*x*y^5*z + 1332*x*y^5 - 906*x*y^4*z^2 - 4968*x*y^4*z - 6282*x*y^4 + 894*x*y^3*z^3 + 7257*x*y^3*z^2 + 18198*x*y^3*z + 14621*x*y^3 - 486*x*y^2*z^4 - 5187*x*y^2*z^3 - 19329*x*y^2*z^2 - 30807*x*y^2*z - 18039*x*y^2 + 138*x*y*z^5 + 1815*x*y*z^4 + 8928*x*y*z^3 + 21135*x*y*z^2 + 24480*x*y*z + 11251*x*y - 16*x*z^6 - 249*x*z^5 - 1515*x*z^4 - 4729*x*z^3 - 8115*x*z^2 - 7357*x*z - 2781*x - 16*y^7 + 88*y^6*z + 248*y^6 - 204*y^5*z^2 - 1140*y^5*z - 1476*y^5 + 258*y^4*z^3 + 2142*y^4*z^2 + 5514*y^4*z + 4534*y^4 - 192*y^3*z^4 - 2103*y^3*z^3 - 8064*y^3*z^2 - 13181*y^3*z - 7880*y^3 + 84*y^2*z^5 + 1137*y^2*z^4 + 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2320*x^2 - 52*x*y^6 + 240*x*y^5*z + 732*x*y^5 - 453*x*y^4*z^2 - 2724*x*y^4*z - 3603*x*y^4 + 447*x*y^3*z^3 + 3969*x*y^3*z^2 + 10407*x*y^3*z + 8590*x*y^3 - 243*x*y^2*z^4 - 2829*x*y^2*z^3 - 11019*x*y^2*z^2 - 18039*x*y^2*z - 10752*x*y^2 + 69*x*y*z^5 + 987*x*y*z^4 + 5073*x*y*z^3 + 12333*x*y*z^2 + 14535*x*y*z + 6758*x*y - 8*x*z^6 - 135*x*z^5 - 858*x*z^4 - 2750*x*z^3 - 4800*x*z^2 - 4400*x*z - 1673*x - 8*y^7 + 44*y^6*z + 136*y^6 - 102*y^5*z^2 - 624*y^5*z - 846*y^5 + 129*y^4*z^3 + 1170*y^4*z^2 + 3153*y^4*z + 2666*y^4 - 96*y^3*z^4 - 1146*y^3*z^3 - 4599*y^3*z^2 - 7729*y^3*z - 4708*y^3 + 42*y^2*z^5 + 618*y^2*z^4 + 3279*y^2*z^3 + 8199*y^2*z^2 + 9903*y^2*z + 4705*y^2 - 10*y*z^6 - 174*y*z^5 - 1143*y*z^4 - 3772*y*z^3 - 6756*y*z^2 - 6341*y*z - 2473*y + z^7 + 20*z^6 + 156*z^5 + 636*z^4 + 1499*z^3 + 2080*z^2 + 1593*z + 1, -2*x^7 - 19*x^6*y + 13*x^6*z + 29*x^6 - 75*x^5*y^2 + 105*x^5*y*z + 240*x^5*y - 36*x^5*z^2 - 162*x^5*z - 177*x^5 - 159*x^4*y^3 + 342*x^4*y^2*z + 801*x^4*y^2 - 240*x^4*y*z^2 - 1107*x^4*y*z - 1236*x^4*y + 55*x^4*z^3 + 375*x^4*z^2 + 828*x^4*z + 600*x^4 - 195*x^3*y^4 + 573*x^3*y^3*z + 1374*x^3*y^3 - 618*x^3*y^2*z^2 - 2922*x^3*y^2*z - 3336*x^3*y^2 + 290*x^3*y*z^3 + 2028*x^3*y*z^2 + 4581*x^3*y*z + 3387*x^3*y - 50*x^3*z^4 - 460*x^3*z^3 - 1542*x^3*z^2 - 2258*x^3*z - 1230*x^3 - 138*x^2*y^5 + 519*x^2*y^4*z + 1272*x^2*y^4 - 765*x^2*y^3*z^2 - 3705*x^2*y^3*z - 4326*x^2*y^3 + 552*x^2*y^2*z^3 + 3960*x^2*y^2*z^2 + 9159*x^2*y^2*z + 6912*x^2*y^2 - 195*x^2*y*z^4 - 1842*x^2*y*z^3 - 6327*x^2*y*z^2 - 9468*x^2*y*z - 5259*x^2*y + 27*x^2*z^5 + 315*x^2*z^4 + 1428*x^2*z^3 + 3174*x^2*z^2 + 3489*x^2*z + 1534*x^2 - 52*x*y^6 + 240*x*y^5*z + 600*x*y^5 - 453*x*y^4*z^2 - 2244*x*y^4*z - 2679*x*y^4 + 447*x*y^3*z^3 + 3288*x*y^3*z^2 + 7791*x*y^3*z + 6003*x*y^3 - 243*x*y^2*z^4 - 2358*x*y^2*z^3 - 8310*x*y^2*z^2 - 12720*x*y^2*z - 7203*x*y^2 + 69*x*y*z^5 + 828*x*y*z^4 + 3855*x*y*z^3 + 8775*x*y*z^2 + 9849*x*y*z + 4412*x*y - 8*x*z^6 - 114*x*z^5 - 657*x*z^4 - 1974*x*z^3 - 3288*x*z^2 - 2912*x*z - 1087*x - 8*y^7 + 44*y^6*z + 112*y^6 - 102*y^5*z^2 - 516*y^5*z - 630*y^5 + 129*y^4*z^3 + 972*y^4*z^2 + 2361*y^4*z + 1860*y^4 - 96*y^3*z^4 - 957*y^3*z^3 - 3465*y^3*z^2 - 5432*y^3*z - 3132*y^3 + 42*y^2*z^5 + 519*y^2*z^4 + 2487*y^2*z^3 + 5808*y^2*z^2 + 6657*y^2*z + 3031*y^2 - 10*y*z^6 - 147*y*z^5 - 873*y*z^4 - 2694*y*z^3 - 4590*y*z^2 - 4145*y*z - 1581*y + z^7 + 17*z^6 + 120*z^5 + 458*z^4 + 1029*z^3 + 1378*z^2 + 1035*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -1, -2, 1 ], [ -1, -1, 1 ], [ -2, -1, 1 ] ], "source_shift": [ 1, 2, 2 ], "target_matrix": [ [ 2, 3, 2 ], [ 1, 2, 1 ], [ 1, 1, 2 ] ], "target_shift": [ -863, -519, -337 ], "collision_preimages": [ [ "1/4", "-1", "-11/4" ], [ "-8", "-7/2", "-15" ], [ "-5", "3/2", "-4" ] ], "collision_image": [ "-1727/2", "-2077/4", "-1349/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0216078250014107 ], "surface_features": [ 3.5091838474162667, 4.488649408577202, 3.092018470752797, 0.0, 3.661449938810335, 4844 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B056", "pair": 56, "family": "bijection", "mapping": "F(x,y,z)=(-6*x^7 + 42*x^6*y - 42*x^6*z + 84*x^6 - 126*x^5*y^2 + 252*x^5*y*z - 504*x^5*y - 126*x^5*z^2 + 492*x^5*z - 462*x^5 + 210*x^4*y^3 - 630*x^4*y^2*z + 1260*x^4*y^2 + 630*x^4*y*z^2 - 2460*x^4*y*z + 2304*x^4*y - 210*x^4*z^3 + 1200*x^4*z^2 - 2196*x^4*z + 1326*x^4 - 210*x^3*y^4 + 840*x^3*y^3*z - 1680*x^3*y^3 - 1260*x^3*y^2*z^2 + 4920*x^3*y^2*z - 4596*x^3*y^2 + 840*x^3*y*z^3 - 4800*x^3*y*z^2 + 8760*x^3*y*z - 5244*x^3*y - 210*x^3*z^4 + 1560*x^3*z^3 - 4170*x^3*z^2 + 4884*x^3*z - 2197*x^3 + 126*x^2*y^5 - 630*x^2*y^4*z + 1260*x^2*y^4 + 1260*x^2*y^3*z^2 - 4920*x^2*y^3*z + 4584*x^2*y^3 - 1260*x^2*y^2*z^3 + 7200*x^2*y^2*z^2 - 13104*x^2*y^2*z + 7776*x^2*y^2 + 630*x^2*y*z^4 - 4680*x^2*y*z^3 + 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42*y*z^6 + 444*y*z^5 - 1866*y*z^4 + 4068*y*z^3 - 4962*y*z^2 + 3160*y*z - 784*y + 6*z^7 - 72*z^6 + 354*z^5 - 942*z^4 + 1490*z^3 - 1376*z^2 + 664*z - 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -1, 1, -1 ], [ -2, 1, -1 ], [ -3, 3, -2 ] ], "source_shift": [ 1, 1, 1 ], "target_matrix": [ [ 2, 1, 1 ], [ -1, -1, -1 ], [ -2, -2, -3 ] ], "target_shift": [ -133, 66, 129 ], "inverse_recipe": { "p": 3, "q": 2, "f": "x**3 + x**2 + x", "g": "x**2", "h": "y*z**2 + z**2 + z", "polynomials": { "f": [ [ [ 3, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "g": [ [ [ 2, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 2 ], 1 ], [ [ 0, 0, 1 ], 1 ], [ [ 0, 0, 2 ], 1 ] ] }, "h_multiplier": 3 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0817968619994645 ], "surface_features": [ 3.3700003223979023, 4.160888516144974, 3.059752694209299, 0.0, 3.4778969079083386, 4834 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C204", "pair": 204, "family": "counterexample", "mapping": "F(x,y,z)=(x^7 + 10*x^6*y - 10*x^6*z - 17*x^6 + 42*x^5*y^2 - 87*x^5*y*z - 144*x^5*y + 42*x^5*z^2 + 150*x^5*z + 117*x^5 + 96*x^4*y^3 - 309*x^4*y^2*z - 498*x^4*y^2 + 309*x^4*y*z^2 + 1074*x^4*y*z + 816*x^4*y - 96*x^4*z^3 - 543*x^4*z^2 - 879*x^4*z - 429*x^4 + 129*x^3*y^4 - 573*x^3*y^3*z - 900*x^3*y^3 + 891*x^3*y^2*z^2 + 3012*x^3*y^2*z + 2229*x^3*y^2 - 573*x^3*y*z^3 - 3156*x^3*y*z^2 - 4977*x^3*y*z - 2356*x^3*y + 129*x^3*z^4 + 1032*x^3*z^3 + 2604*x^3*z^2 + 2631*x^3*z + 915*x^3 + 102*x^2*y^5 - 585*x^2*y^4*z - 897*x^2*y^4 + 1257*x^2*y^3*z^2 + 4134*x^2*y^3*z + 2982*x^2*y^3 - 1257*x^2*y^2*z^3 - 6732*x^2*y^2*z^2 - 10338*x^2*y^2*z - 4746*x^2*y^2 + 585*x^2*y*z^4 + 4566*x^2*y*z^3 + 11229*x^2*y*z^2 + 11004*x^2*y*z + 3693*x^2*y - 102*x^2*z^5 - 1086*x^2*z^4 - 3801*x^2*z^3 - 5967*x^2*z^2 - 4296*x^2*z - 1144*x^2 + 44*x*y^6 - 312*x*y^5*z - 468*x*y^5 + 867*x*y^4*z^2 + 2778*x*y^4*z + 1956*x*y^4 - 1199*x*y^3*z^3 - 6240*x*y^3*z^2 - 9336*x*y^3*z - 4161*x*y^3 + 867*x*y^2*z^4 + 6588*x*y^2*z^3 + 15777*x*y^2*z^2 + 14985*x*y^2*z + 4854*x*y^2 - 312*x*y*z^5 - 3252*x*y*z^4 - 11103*x*y*z^3 - 16911*x*y*z^2 - 11745*x*y*z - 2999*x*y + 44*x*z^6 + 600*x*z^5 + 2733*x*z^4 + 5928*x*z^3 + 6630*x*z^2 + 3661*x*z + 781*x + 8*y^7 - 68*y^6*z - 100*y^6 + 234*y^5*z^2 + 732*y^5*z + 504*y^5 - 419*y^4*z^3 - 2121*y^4*z^2 - 3096*y^4*z - 1342*y^4 + 419*y^3*z^4 + 3094*y^3*z^3 + 7218*y^3*z^2 + 6651*y^3*z + 2082*y^3 - 234*y^2*z^5 - 2382*y^2*z^4 - 7920*y^2*z^3 - 11685*y^2*z^2 - 7830*y^2*z - 1918*y^2 + 68*y*z^6 + 912*y*z^5 + 4059*y*z^4 + 8549*y*z^3 + 9216*y*z^2 + 4879*y*z + 994*y - 8*z^7 - 136*z^6 - 774*z^5 - 2175*z^4 - 3363*z^3 - 2887*z^2 - 1276*z - 1, 2*x^7 + 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9693*y^3*z^2 - 9843*y^3*z - 3306*y^3 + 234*y^2*z^5 + 2685*y^2*z^4 + 10344*y^2*z^3 + 16896*y^2*z^2 + 12162*y^2*z + 3172*y^2 - 68*y*z^6 - 1008*y*z^5 - 5145*y*z^4 - 12032*y*z^3 - 13971*y*z^2 - 7852*y*z - 1697*y + 8*z^7 + 148*z^6 + 954*z^5 + 2970*z^4 + 4959*z^3 + 4519*z^2 + 2106*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -1, -2, 1 ], [ -1, -1, 2 ], [ -1, -1, 1 ] ], "source_shift": [ 2, 1, 2 ], "target_matrix": [ [ -1, -1, -1 ], [ -2, -1, -1 ], [ 1, 2, 1 ] ], "target_shift": [ 228, 312, -397 ], "collision_preimages": [ [ "15/4", "-1/4", "5/4" ], [ "-17", "11/2", "-7" ], [ "-16", "15/2", "-4" ] ], "collision_image": [ "913/4", "625/2", "-1589/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0829467749972537 ], "surface_features": [ 3.5887816644246855, 4.425044874551388, 3.2336305801644634, 0.0, 3.703030236595741, 4892 ], 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600*x*y^5 + 519*x*y^4*z^2 + 2436*x*y^4*z + 2733*x*y^4 + 573*x*y^3*z^3 + 3852*x*y^3*z^2 + 8373*x*y^3*z + 5924*x*y^3 + 342*x*y^2*z^4 + 2964*x*y^2*z^3 + 9420*x*y^2*z^2 + 13026*x*y^2*z + 6591*x*y^2 + 105*x*y*z^5 + 1110*x*y*z^4 + 4605*x*y*z^3 + 9357*x*y*z^2 + 9279*x*y*z + 3571*x*y + 13*x*z^6 + 162*x*z^5 + 825*x*z^4 + 2192*x*z^3 + 3192*x*z^2 + 2402*x*z + 727*x + 8*y^7 + 52*y^6*z + 136*y^6 + 138*y^5*z^2 + 672*y^5*z + 774*y^5 + 195*y^4*z^3 + 1350*y^4*z^2 + 3003*y^4*z + 2174*y^4 + 159*y^3*z^4 + 1410*y^3*z^3 + 4572*y^3*z^2 + 6458*y^3*z + 3349*y^3 + 75*y^2*z^5 + 807*y^2*z^4 + 3408*y^2*z^3 + 7065*y^2*z^2 + 7170*y^2*z + 2836*y^2 + 19*y*z^6 + 240*y*z^5 + 1242*y*z^4 + 3365*y*z^3 + 5019*y*z^2 + 3889*y*z + 1210*y + 2*z^7 + 29*z^6 + 177*z^5 + 588*z^4 + 1144*z^3 + 1297*z^2 + 792*z + 1, x^7 + 10*x^6*y + 8*x^6*z + 17*x^6 + 42*x^5*y^2 + 69*x^5*y*z + 150*x^5*y + 27*x^5*z^2 + 114*x^5*z + 117*x^5 + 96*x^4*y^3 + 243*x^4*y^2*z + 543*x^4*y^2 + 195*x^4*y*z^2 + 840*x^4*y*z + 879*x^4*y + 50*x^4*z^3 + 315*x^4*z^2 + 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672*y^5*z + 774*y^5 + 195*y^4*z^3 + 1350*y^4*z^2 + 3003*y^4*z + 2175*y^4 + 159*y^3*z^4 + 1410*y^3*z^3 + 4572*y^3*z^2 + 6463*y^3*z + 3363*y^3 + 75*y^2*z^5 + 807*y^2*z^4 + 3408*y^2*z^3 + 7074*y^2*z^2 + 7215*y^2*z + 2887*y^2 + 19*y*z^6 + 240*y*z^5 + 1242*y*z^4 + 3372*y*z^3 + 5067*y*z^2 + 3991*y*z + 1276*y + 2*z^7 + 29*z^6 + 177*z^5 + 590*z^4 + 1161*z^3 + 1348*z^2 + 854*z + 1, x^7 + 10*x^6*y + 8*x^6*z + 20*x^6 + 42*x^5*y^2 + 69*x^5*y*z + 174*x^5*y + 27*x^5*z^2 + 135*x^5*z + 156*x^5 + 96*x^4*y^3 + 243*x^4*y^2*z + 621*x^4*y^2 + 195*x^4*y*z^2 + 981*x^4*y*z + 1155*x^4*y + 50*x^4*z^3 + 375*x^4*z^2 + 864*x^4*z + 621*x^4 + 129*x^3*y^4 + 447*x^3*y^3*z + 1164*x^3*y^3 + 552*x^3*y^2*z^2 + 2805*x^3*y^2*z + 3369*x^3*y^2 + 290*x^3*y*z^3 + 2184*x^3*y*z^2 + 5121*x^3*y*z + 3759*x^3*y + 55*x^3*z^4 + 550*x^3*z^3 + 1896*x^3*z^2 + 2726*x^3*z + 1404*x^3 + 102*x^2*y^5 + 453*x^2*y^4*z + 1209*x^2*y^4 + 765*x^2*y^3*z^2 + 3945*x^2*y^3*z + 4839*x^2*y^3 + 618*x^2*y^2*z^3 + 4689*x^2*y^2*z^2 + 11199*x^2*y^2*z + 8406*x^2*y^2 + 240*x^2*y*z^4 + 2406*x^2*y*z^3 + 8433*x^2*y*z^2 + 12375*x^2*y*z + 6510*x^2*y + 36*x^2*z^5 + 450*x^2*z^4 + 2064*x^2*z^3 + 4452*x^2*z^2 + 4602*x^2*z + 1840*x^2 + 44*x*y^6 + 240*x*y^5*z + 660*x*y^5 + 519*x*y^4*z^2 + 2730*x*y^4*z + 3423*x*y^4 + 573*x*y^3*z^3 + 4398*x*y^3*z^2 + 10707*x*y^3*z + 8226*x*y^3 + 342*x*y^2*z^4 + 3447*x*y^2*z^3 + 12291*x*y^2*z^2 + 18426*x*y^2*z + 9915*x*y^2 + 105*x*y*z^5 + 1314*x*y*z^4 + 6123*x*y*z^3 + 13473*x*y*z^2 + 14223*x*y*z + 5809*x*y + 13*x*z^6 + 195*x*z^5 + 1116*x*z^4 + 3210*x*z^3 + 4992*x*z^2 + 4019*x*z + 1310*x + 8*y^7 + 52*y^6*z + 148*y^6 + 138*y^5*z^2 + 744*y^5*z + 954*y^5 + 195*y^4*z^3 + 1521*y^4*z^2 + 3777*y^4*z + 2970*y^4 + 159*y^3*z^4 + 1617*y^3*z^3 + 5868*y^3*z^2 + 8992*y^3*z + 4959*y^3 + 75*y^2*z^5 + 942*y^2*z^4 + 4461*y^2*z^3 + 10020*y^2*z^2 + 10812*y^2*z + 4519*y^2 + 19*y*z^6 + 285*y*z^5 + 1656*y*z^4 + 4857*y*z^3 + 7713*y*z^2 + 6346*y*z + 2106*y + 2*z^7 + 35*z^6 + 240*z^5 + 863*z^4 + 1794*z^3 + 2179*z^2 + 1437*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ 1, 1, 1 ], [ 1, 2, 1 ], [ 1, 1, 2 ] ], "source_shift": [ 2, 1, 2 ], "target_matrix": [ [ 1, 1, 2 ], [ 1, 1, 1 ], [ 1, 2, 1 ] ], "target_shift": [ -204, -228, -397 ], "collision_preimages": [ [ "-11/4", "1", "-1/4" ], [ "-5", "-3/2", "11/2" ], [ "-14", "7/2", "15/2" ] ], "collision_image": [ "-817/4", "-913/4", "-1589/4" ] }, "probe_target": "No", "jev_answers": [ "Maybe" ], "seconds": [ 1.0233211269987805 ], "surface_features": [ 3.361207735059829, 4.265431066826713, 2.9588027033995026, 0.0, 3.48127147381885, 4792 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C004", "pair": 4, "family": "counterexample", "mapping": "F(x,y,z)=(-16*x^7 + 88*x^6*y + 144*x^6*z - 176*x^6 - 204*x^5*y^2 - 660*x^5*y*z + 804*x^5*y - 528*x^5*z^2 + 1296*x^5*z - 804*x^5 + 258*x^4*y^3 + 1236*x^4*y^2*z - 1512*x^4*y^2 + 1950*x^4*y*z^2 - 4824*x^4*y*z + 3000*x^4*y + 1012*x^4*z^3 - 3804*x^4*z^2 + 4788*x^4*z - 2004*x^4 - 192*x^3*y^4 - 1209*x^3*y^3*z + 1497*x^3*y^3 - 2817*x^3*y^2*z^2 + 7086*x^3*y^2*z - 4425*x^3*y^2 - 2875*x^3*y*z^3 + 11049*x^3*y*z^2 - 13962*x^3*y*z + 5892*x^3*y - 1083*x^3*z^4 + 5672*x^3*z^3 - 10869*x^3*z^2 + 9322*x^3*z - 2974*x^3 + 84*x^2*y^5 + 651*x^2*y^4*z - 822*x^2*y^4 + 1989*x^2*y^3*z^2 - 5127*x^2*y^3*z + 3225*x^2*y^3 + 2991*x^2*y^2*z^3 - 11844*x^2*y^2*z^2 + 15090*x^2*y^2*z - 6414*x^2*y^2 + 2211*x^2*y*z^4 - 12003*x^2*y*z^3 + 23229*x^2*y*z^2 - 20028*x^2*y*z + 6468*x^2*y + 642*x^2*z^5 - 4500*x^2*z^4 + 11760*x^2*z^3 - 15396*x^2*z^2 + 10134*x^2*z - 2650*x^2 - 20*x*y^6 - 183*x*y^5*z + 237*x*y^5 - 687*x*y^4*z^2 + 1824*x*y^4*z - 1161*x*y^4 - 1353*x*y^3*z^3 + 5541*x*y^3*z^2 - 7161*x*y^3*z + 3066*x*y^3 - 1473*x*y^2*z^4 + 8298*x*y^2*z^3 - 16350*x*y^2*z^2 + 14181*x*y^2*z - 4623*x*y^2 - 840*x*y*z^5 + 6120*x*y*z^4 - 16377*x*y*z^3 + 21540*x*y*z^2 - 14265*x*y*z + 3787*x*y - 196*x*z^6 + 1776*x*z^5 - 6075*x*z^4 + 10743*x*z^3 - 10806*x*z^2 + 5860*x*z - 1323*x + 2*y^7 + 21*y^6*z - 28*y^6 + 93*y^5*z^2 - 255*y^5*z + 165*y^5 + 225*y^4*z^3 - 954*y^4*z^2 + 1257*y^4*z - 543*y^4 + 321*y^3*z^4 - 1875*y^3*z^3 + 3780*y^3*z^2 - 3308*y^3*z + 1088*y^3 + 270*y^2*z^5 - 2040*y^2*z^4 + 5607*y^2*z^3 - 7449*y^2*z^2 + 4968*y^2*z - 1333*y^2 + 124*y*z^6 - 1164*y*z^5 + 4101*y*z^4 - 7350*y*z^3 + 7437*y*z^2 - 4055*y*z + 931*y + 24*z^7 - 272*z^6 + 1182*z^5 - 2684*z^4 + 3653*z^3 - 3019*z^2 + 1418*z - 1, -32*x^7 + 176*x^6*y + 288*x^6*z - 328*x^6 - 408*x^5*y^2 - 1320*x^5*y*z + 1488*x^5*y - 1056*x^5*z^2 + 2388*x^5*z - 1404*x^5 + 516*x^4*y^3 + 2472*x^4*y^2*z - 2778*x^4*y^2 + 3900*x^4*y*z^2 - 8820*x^4*y*z + 5196*x^4*y + 2024*x^4*z^3 - 6918*x^4*z^2 + 8244*x^4*z - 3330*x^4 - 384*x^3*y^4 - 2418*x^3*y^3*z + 2730*x^3*y^3 - 5634*x^3*y^2*z^2 + 12855*x^3*y^2*z - 7599*x^3*y^2 - 5750*x^3*y*z^3 + 19932*x^3*y*z^2 - 23826*x^3*y*z + 9708*x^3*y - 2166*x^3*z^4 + 10171*x^3*z^3 - 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11760*x^2*z^3 + 15438*x^2*z^2 - 10200*x^2*z + 2677*x^2 + 20*x*y^6 + 183*x*y^5*z - 237*x*y^5 + 687*x*y^4*z^2 - 1824*x*y^4*z + 1161*x*y^4 + 1353*x*y^3*z^3 - 5541*x*y^3*z^2 + 7161*x*y^3*z - 3074*x*y^3 + 1473*x*y^2*z^4 - 8298*x*y^2*z^3 + 16350*x*y^2*z^2 - 14226*x*y^2*z + 4656*x*y^2 + 840*x*y*z^5 - 6120*x*y*z^4 + 16377*x*y*z^3 - 21624*x*y*z^2 + 14385*x*y*z - 3835*x*y + 196*x*z^6 - 1776*x*z^5 + 6075*x*z^4 - 10795*x*z^3 + 10914*x*z^2 - 5947*x*z + 1344*x - 2*y^7 - 21*y^6*z + 28*y^6 - 93*y^5*z^2 + 255*y^5*z - 165*y^5 - 225*y^4*z^3 + 954*y^4*z^2 - 1257*y^4*z + 545*y^4 - 321*y^3*z^4 + 1875*y^3*z^3 - 3780*y^3*z^2 + 3323*y^3*z - 1098*y^3 - 270*y^2*z^5 + 2040*y^2*z^4 - 5607*y^2*z^3 + 7491*y^2*z^2 - 5022*y^2*z + 1354*y^2 - 124*y*z^6 + 1164*y*z^5 - 4101*y*z^4 + 7402*y*z^3 - 7533*y*z^2 + 4130*y*z - 949*y - 24*z^7 + 272*z^6 - 1182*z^5 + 2708*z^4 - 3709*z^3 + 3085*z^2 - 1450*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ 1, -1, -2 ], [ 2, -1, -1 ], [ 2, -2, -3 ] ], "source_shift": [ 1, 2, 1 ], "target_matrix": [ [ -1, -1, -2 ], [ -2, -1, -1 ], [ 1, 1, 1 ] ], "target_shift": [ 285, 437, -290 ], "collision_preimages": [ [ "-7/4", "-9/4", "3/4" ], [ "-9", "-20", "11/2" ], [ "-8", "-25", "19/2" ] ], "collision_image": [ "1141/4", "875/2", "-1161/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0071248740023293 ], "surface_features": [ 3.7142398546496214, 4.591131416423124, 3.439963935920905, 0.0, 3.808547394917643, 4980 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C005", "pair": 5, "family": "counterexample", "mapping": "F(x,y,z)=(-2*x^7 - 20*x^6*y + 20*x^6*z + 29*x^6 - 84*x^5*y^2 + 174*x^5*y*z + 243*x^5*y - 84*x^5*z^2 - 258*x^5*z - 174*x^5 - 192*x^4*y^3 + 618*x^4*y^2*z + 831*x^4*y^2 - 618*x^4*y*z^2 - 1827*x^4*y*z - 1191*x^4*y + 192*x^4*z^3 + 942*x^4*z^2 + 1323*x^4*z + 571*x^4 - 258*x^3*y^4 + 1146*x^3*y^3*z + 1485*x^3*y^3 - 1782*x^3*y^2*z^2 - 5064*x^3*y^2*z - 3192*x^3*y^2 + 1146*x^3*y*z^3 + 5415*x^3*y*z^2 + 7344*x^3*y*z + 3060*x^3*y - 258*x^3*z^4 - 1806*x^3*z^3 - 3972*x^3*z^2 - 3550*x^3*z - 1110*x^3 - 204*x^2*y^5 + 1170*x^2*y^4*z + 1464*x^2*y^4 - 2514*x^2*y^3*z^2 - 6867*x^2*y^3*z - 4191*x^2*y^3 + 2514*x^2*y^2*z^3 + 11412*x^2*y^2*z^2 + 14943*x^2*y^2*z + 6015*x^2*y^2 - 1170*x^2*y*z^4 - 7905*x^2*y*z^3 - 16779*x^2*y*z^2 - 14460*x^2*y*z - 4350*x^2*y + 204*x^2*z^5 + 1917*x^2*z^4 + 5883*x^2*z^3 + 8181*x^2*z^2 + 5286*x^2*z + 1277*x^2 - 88*x*y^6 + 624*x*y^5*z + 756*x*y^5 - 1734*x*y^4*z^2 - 4560*x*y^4*z - 2700*x*y^4 + 2398*x*y^3*z^3 + 10443*x*y^3*z^2 + 13218*x*y^3*z + 5150*x*y^3 - 1734*x*y^2*z^4 - 11268*x*y^2*z^3 - 23058*x*y^2*z^2 - 19164*x*y^2*z - 5556*x*y^2 + 624*x*y*z^5 + 5682*x*y*z^4 + 16830*x*y*z^3 + 22542*x*y*z^2 + 13986*x*y*z + 3235*x*y - 88*x*z^6 - 1068*x*z^5 - 4296*x*z^4 - 8278*x*z^3 - 8304*x*z^2 - 4142*x*z - 802*x - 16*y^7 + 136*y^6*z + 160*y^6 - 468*y^5*z^2 - 1188*y^5*z - 684*y^5 + 838*y^4*z^3 + 3504*y^4*z^2 + 4296*y^4*z + 1624*y^4 - 838*y^3*z^4 - 5221*y^3*z^3 - 10311*y^3*z^2 - 8282*y^3*z - 2320*y^3 + 468*y^2*z^5 + 4113*y^2*z^4 + 11727*y^2*z^3 + 15117*y^2*z^2 + 9024*y^2*z + 2006*y^2 - 136*y*z^6 - 1608*y*z^5 - 6252*y*z^4 - 11594*y*z^3 - 11136*y*z^2 - 5303*y*z - 983*y + 16*z^7 + 244*z^6 + 1236*z^5 + 3100*z^4 + 4304*z^3 + 3329*z^2 + 1323*z + 1, -x^7 - 10*x^6*y + 10*x^6*z + 16*x^6 - 42*x^5*y^2 + 87*x^5*y*z + 135*x^5*y - 42*x^5*z^2 - 141*x^5*z - 105*x^5 - 96*x^4*y^3 + 309*x^4*y^2*z + 465*x^4*y^2 - 309*x^4*y*z^2 - 1005*x^4*y*z - 726*x^4*y + 96*x^4*z^3 + 510*x^4*z^2 + 789*x^4*z + 369*x^4 - 129*x^3*y^4 + 573*x^3*y^3*z + 837*x^3*y^3 - 891*x^3*y^2*z^2 - 2805*x^3*y^2*z - 1965*x^3*y^2 + 573*x^3*y*z^3 + 2949*x^3*y*z^2 + 4425*x^3*y*z + 1999*x^3*y - 129*x^3*z^4 - 969*x^3*z^3 - 2340*x^3*z^2 - 2268*x^3*z - 754*x^3 - 102*x^2*y^5 + 585*x^2*y^4*z + 831*x^2*y^4 - 1257*x^2*y^3*z^2 - 3831*x^2*y^3*z - 2604*x^2*y^3 + 1257*x^2*y^2*z^3 + 6255*x^2*y^2*z^2 + 9096*x^2*y^2*z + 3969*x^2*y^2 - 585*x^2*y*z^4 - 4263*x^2*y*z^3 - 9987*x^2*y*z^2 - 9345*x^2*y*z - 2985*x^2*y + 102*x^2*z^5 + 1020*x^2*z^4 + 3423*x^2*z^3 + 5163*x^2*z^2 + 3555*x^2*z + 898*x^2 - 44*x*y^6 + 312*x*y^5*z + 432*x*y^5 - 867*x*y^4*z^2 - 2562*x*y^4*z - 1692*x*y^4 + 1199*x*y^3*z^3 + 5763*x*y^3*z^2 + 8124*x*y^3*z + 3429*x*y^3 - 867*x*y^2*z^4 - 6111*x*y^2*z^3 - 13869*x*y^2*z^2 - 12519*x*y^2*z - 3846*x*y^2 + 312*x*y*z^5 + 3036*x*y*z^4 + 9891*x*y*z^3 + 14400*x*y*z^2 + 9513*x*y*z + 2294*x*y - 44*x*z^6 - 564*x*z^5 - 2469*x*z^4 - 5157*x*z^3 - 5523*x*z^2 - 2890*x*z - 577*x - 8*y^7 + 68*y^6*z + 92*y^6 - 234*y^5*z^2 - 672*y^5*z - 432*y^5 + 419*y^4*z^3 + 1947*y^4*z^2 + 2664*y^4*z + 1090*y^4 - 419*y^3*z^4 - 2849*y^3*z^3 - 6264*y^3*z^2 - 5463*y^3*z - 1618*y^3 + 234*y^2*z^5 + 2208*y^2*z^4 + 6966*y^2*z^3 + 9768*y^2*z^2 + 6198*y^2*z + 1432*y^2 - 68*y*z^6 - 852*y*z^5 - 3627*y*z^4 - 7316*y*z^3 - 7500*y*z^2 - 3736*y*z - 712*y + 8*z^7 + 128*z^6 + 702*z^5 + 1905*z^4 + 2828*z^3 + 2302*z^2 + 946*z + 1, -2*x^7 - 20*x^6*y + 20*x^6*z + 32*x^6 - 84*x^5*y^2 + 174*x^5*y*z + 270*x^5*y - 84*x^5*z^2 - 282*x^5*z - 210*x^5 - 192*x^4*y^3 + 618*x^4*y^2*z + 930*x^4*y^2 - 618*x^4*y*z^2 - 2010*x^4*y*z - 1452*x^4*y + 192*x^4*z^3 + 1020*x^4*z^2 + 1578*x^4*z + 737*x^4 - 258*x^3*y^4 + 1146*x^3*y^3*z + 1674*x^3*y^3 - 1782*x^3*y^2*z^2 - 5610*x^3*y^2*z - 3930*x^3*y^2 + 1146*x^3*y*z^3 + 5898*x^3*y*z^2 + 8850*x^3*y*z + 3991*x^3*y - 258*x^3*z^4 - 1938*x^3*z^3 - 4680*x^3*z^2 - 4532*x^3*z - 1498*x^3 - 204*x^2*y^5 + 1170*x^2*y^4*z + 1662*x^2*y^4 - 2514*x^2*y^3*z^2 - 7662*x^2*y^3*z - 5208*x^2*y^3 + 2514*x^2*y^2*z^3 + 12510*x^2*y^2*z^2 + 18192*x^2*y^2*z + 7920*x^2*y^2 - 1170*x^2*y*z^4 - 8526*x^2*y*z^3 - 19974*x^2*y*z^2 - 18669*x^2*y*z - 5919*x^2*y + 204*x^2*z^5 + 2040*x^2*z^4 + 6846*x^2*z^3 + 10320*x^2*z^2 + 7077*x^2*z + 1763*x^2 - 88*x*y^6 + 624*x*y^5*z + 864*x*y^5 - 1734*x*y^4*z^2 - 5124*x*y^4*z - 3384*x*y^4 + 2398*x*y^3*z^3 + 11526*x*y^3*z^2 + 16248*x*y^3*z + 6838*x*y^3 - 1734*x*y^2*z^4 - 12222*x*y^2*z^3 - 27738*x*y^2*z^2 - 25002*x*y^2*z - 7608*x*y^2 + 624*x*y*z^5 + 6072*x*y*z^4 + 19782*x*y*z^3 + 28779*x*y*z^2 + 18912*x*y*z + 4480*x*y - 88*x*z^6 - 1128*x*z^5 - 4938*x*z^4 - 10310*x*z^3 - 11010*x*z^2 - 5702*x*z - 1112*x - 16*y^7 + 136*y^6*z + 184*y^6 - 468*y^5*z^2 - 1344*y^5*z - 864*y^5 + 838*y^4*z^3 + 3894*y^4*z^2 + 5328*y^4*z + 2172*y^4 - 838*y^3*z^4 - 5698*y^3*z^3 - 12528*y^3*z^2 - 10906*y^3*z - 3192*y^3 + 468*y^2*z^5 + 4416*y^2*z^4 + 13932*y^2*z^3 + 19518*y^2*z^2 + 12300*y^2*z + 2780*y^2 - 136*y*z^6 - 1704*y*z^5 - 7254*y*z^4 - 14625*y*z^3 - 14937*y*z^2 - 7340*y*z - 1360*y + 16*z^7 + 256*z^6 + 1404*z^5 + 3809*z^4 + 5643*z^3 + 4559*z^2 + 1838*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -1, -2, 1 ], [ -1, -1, 2 ], [ -1, -1, 1 ] ], "source_shift": [ 2, 1, 1 ], "target_matrix": [ [ 2, 1, 1 ], [ 1, 1, 1 ], [ 2, 2, 3 ] ], "target_shift": [ -212, -155, -295 ], "collision_preimages": [ [ "3/4", "3/4", "1/4" ], [ "-20", "13/2", "-8" ], [ "-19", "17/2", "-5" ] ], "collision_image": [ "-425/2", "-621/4", "-591/2" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.135697498000809 ], "surface_features": [ 3.589340094729066, 4.459075699189636, 3.2390490931401916, 0.0, 3.705939279742582, 4906 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C222", "pair": 222, "family": "counterexample", "mapping": "F(x,y,z)=(54*x^7 - 459*x^6*y + 297*x^6*z - 567*x^6 + 1584*x^5*y^2 - 2115*x^5*y*z + 3951*x^5*y + 693*x^5*z^2 - 2655*x^5*z + 1917*x^5 - 2860*x^4*y^3 + 5934*x^4*y^2*z - 10830*x^4*y^2 - 4017*x^4*y*z^2 + 15072*x^4*y*z - 10755*x^4*y + 889*x^4*z^3 - 5133*x^4*z^2 + 7407*x^4*z - 3291*x^4 + 2912*x^3*y^4 - 8360*x^3*y^3*z + 14920*x^3*y^3 + 8796*x^3*y^2*z^2 - 32304*x^3*y^2*z + 22752*x^3*y^2 - 4025*x^3*y*z^3 + 22791*x^3*y*z^2 - 32508*x^3*y*z + 14163*x^3*y + 677*x^3*z^4 - 5245*x^3*z^3 + 11349*x^3*z^2 - 10032*x^3*z + 3219*x^3 - 1680*x^2*y^5 + 6240*x^2*y^4*z - 10944*x^2*y^4 - 9072*x^2*y^3*z^2 + 32688*x^2*y^3*z - 22728*x^2*y^3 + 6450*x^2*y^2*z^3 - 35838*x^2*y^2*z^2 + 50496*x^2*y^2*z - 21474*x^2*y^2 - 2244*x^2*y*z^4 + 17082*x^2*y*z^3 - 36552*x^2*y*z^2 + 31647*x^2*y*z - 9867*x^2*y + 306*x^2*z^5 - 2988*x^2*z^4 + 8622*x^2*z^3 - 11370*x^2*z^2 + 7230*x^2*z - 1830*x^2 + 512*x*y^6 - 2352*x*y^5*z + 4080*x*y^5 + 4416*x*y^4*z^2 - 15696*x*y^4*z + 10800*x*y^4 - 4332*x*y^3*z^3 + 23700*x*y^3*z^2 - 33024*x*y^3*z + 13716*x*y^3 + 2340*x*y^2*z^4 - 17532*x*y^2*z^3 + 37104*x*y^2*z^2 - 31380*x*y^2*z + 9480*x*y^2 - 660*x*y*z^5 + 6348*x*y*z^4 - 18129*x*y*z^3 + 23406*x*y*z^2 - 14454*x*y*z + 3513*x*y + 76*x*z^6 - 900*x*z^5 + 3249*x*z^4 - 5682*x*z^3 + 5370*x*z^2 - 2682*x*z + 557*x - 64*y^7 + 352*y^6*z - 608*y^6 - 816*y^5*z^2 + 2880*y^5*z - 1968*y^5 + 1032*y^4*z^3 - 5592*y^4*z^2 + 7728*y^4*z - 3144*y^4 - 768*y^3*z^4 + 5688*y^3*z^3 - 11928*y^3*z^2 + 9868*y^3*z - 2892*y^3 + 336*y^2*z^5 - 3192*y^2*z^4 + 9030*y^2*z^3 - 11400*y^2*z^2 + 6828*y^2*z - 1594*y^2 - 80*y*z^6 + 936*y*z^5 - 3348*y*z^4 + 5733*y*z^3 - 5262*y*z^2 + 2525*y*z - 505*y + 8*z^7 - 112*z^6 + 486*z^5 - 1057*z^4 + 1320*z^3 - 976*z^2 + 400*z - 1, 54*x^7 - 459*x^6*y + 297*x^6*z - 567*x^6 + 1584*x^5*y^2 - 2115*x^5*y*z + 3951*x^5*y + 693*x^5*z^2 - 2655*x^5*z + 1917*x^5 - 2860*x^4*y^3 + 5934*x^4*y^2*z - 10830*x^4*y^2 - 4017*x^4*y*z^2 + 15072*x^4*y*z - 10755*x^4*y + 889*x^4*z^3 - 5133*x^4*z^2 + 7407*x^4*z - 3289*x^4 + 2912*x^3*y^4 - 8360*x^3*y^3*z + 14920*x^3*y^3 + 8796*x^3*y^2*z^2 - 32304*x^3*y^2*z + 22752*x^3*y^2 - 4025*x^3*y*z^3 + 22791*x^3*y*z^2 - 32508*x^3*y*z + 14150*x^3*y + 677*x^3*z^4 - 5245*x^3*z^3 + 11349*x^3*z^2 - 10025*x^3*z + 3202*x^3 - 1680*x^2*y^5 + 6240*x^2*y^4*z - 10944*x^2*y^4 - 9072*x^2*y^3*z^2 + 32688*x^2*y^3*z - 22728*x^2*y^3 + 6450*x^2*y^2*z^3 - 35838*x^2*y^2*z^2 + 50496*x^2*y^2*z - 21444*x^2*y^2 - 2244*x^2*y*z^4 + 17082*x^2*y*z^3 - 36552*x^2*y*z^2 + 31614*x^2*y*z - 9786*x^2*y + 306*x^2*z^5 - 2988*x^2*z^4 + 8622*x^2*z^3 - 11361*x^2*z^2 + 7185*x^2*z - 1797*x^2 + 512*x*y^6 - 2352*x*y^5*z + 4080*x*y^5 + 4416*x*y^4*z^2 - 15696*x*y^4*z + 10800*x*y^4 - 4332*x*y^3*z^3 + 23700*x*y^3*z^2 - 33024*x*y^3*z + 13688*x*y^3 + 2340*x*y^2*z^4 - 17532*x*y^2*z^3 + 37104*x*y^2*z^2 - 31332*x*y^2*z + 9360*x*y^2 - 660*x*y*z^5 + 6348*x*y*z^4 - 18129*x*y*z^3 + 23379*x*y*z^2 - 14316*x*y*z + 3414*x*y + 76*x*z^6 - 900*x*z^5 + 3249*x*z^4 - 5677*x*z^3 + 5331*x*z^2 - 2625*x*z + 536*x - 64*y^7 + 352*y^6*z - 608*y^6 - 816*y^5*z^2 + 2880*y^5*z - 1968*y^5 + 1032*y^4*z^3 - 5592*y^4*z^2 + 7728*y^4*z - 3136*y^4 - 768*y^3*z^4 + 5688*y^3*z^3 - 11928*y^3*z^2 + 9848*y^3*z - 2840*y^3 + 336*y^2*z^5 - 3192*y^2*z^4 + 9030*y^2*z^3 - 11382*y^2*z^2 + 6732*y^2*z - 1528*y^2 - 80*y*z^6 + 936*y*z^5 - 3348*y*z^4 + 5726*y*z^3 - 5205*y*z^2 + 2444*y*z - 478*y + 8*z^7 - 112*z^6 + 486*z^5 - 1056*z^4 + 1309*z^3 - 952*z^2 + 383*z - 1, -108*x^7 + 918*x^6*y - 594*x^6*z + 1080*x^6 - 3168*x^5*y^2 + 4230*x^5*y*z - 7479*x^5*y - 1386*x^5*z^2 + 5049*x^5*z - 3339*x^5 + 5720*x^4*y^3 - 11868*x^4*y^2*z + 20358*x^4*y^2 + 8034*x^4*y*z^2 - 28485*x^4*y*z + 18453*x^4*y - 1778*x^4*z^3 + 9747*x^4*z^2 - 12837*x^4*z + 5345*x^4 - 5824*x^3*y^4 + 16720*x^3*y^3*z - 27848*x^3*y^3 - 17592*x^3*y^2*z^2 + 60648*x^3*y^2*z - 38448*x^3*y^2 + 8050*x^3*y*z^3 - 43017*x^3*y*z^2 + 55512*x^3*y*z - 22591*x^3*y - 1354*x^3*z^4 + 9947*x^3*z^3 - 19575*x^3*z^2 + 16195*x^3*z - 4961*x^3 + 3360*x^2*y^5 - 12480*x^2*y^4*z + 20304*x^2*y^4 + 18144*x^2*y^3*z^2 - 60984*x^2*y^3*z + 37896*x^2*y^3 - 12900*x^2*y^2*z^3 + 67230*x^2*y^2*z^2 - 85008*x^2*y^2*z + 33690*x^2*y^2 + 4488*x^2*y*z^4 - 32211*x^2*y*z^3 + 62151*x^2*y*z^2 - 50217*x^2*y*z + 14919*x^2*y - 612*x^2*z^5 + 5661*x^2*z^4 - 14805*x^2*z^3 + 18252*x^2*z^2 - 11073*x^2*z + 2715*x^2 - 1024*x*y^6 + 4704*x*y^5*z - 7536*x*y^5 - 8832*x*y^4*z^2 + 29136*x*y^4*z - 17808*x*y^4 + 8664*x*y^3*z^3 - 44220*x*y^3*z^2 + 54912*x*y^3*z - 21220*x*y^3 - 4680*x*y^2*z^4 + 32880*x*y^2*z^3 - 62256*x*y^2*z^2 + 49032*x*y^2*z - 14112*x*y^2 + 1320*x*y*z^5 - 11964*x*y*z^4 + 30708*x*y*z^3 - 36966*x*y*z^2 + 21750*x*y*z - 5121*x*y - 152*x*z^6 + 1704*x*z^5 - 5556*x*z^4 + 9074*x*z^3 - 8178*x*z^2 + 3957*x*z - 805*x + 128*y^7 - 704*y^6*z + 1120*y^6 + 1632*y^5*z^2 - 5328*y^5*z + 3216*y^5 - 2064*y^4*z^3 + 10392*y^4*z^2 - 12720*y^4*z + 4808*y^4 + 1536*y^3*z^4 - 10620*y^3*z^3 + 19788*y^3*z^2 - 15220*y^3*z + 4252*y^3 - 672*y^2*z^5 + 5988*y^2*z^4 - 15108*y^2*z^3 + 17748*y^2*z^2 - 10128*y^2*z + 2294*y^2 + 160*y*z^6 - 1764*y*z^5 + 5652*y*z^4 - 9016*y*z^3 + 7884*y*z^2 - 3667*y*z + 717*y - 16*z^7 + 212*z^6 - 828*z^5 + 1680*z^4 - 2000*z^3 + 1433*z^2 - 575*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -1, 2, -1 ], [ -3, 2, -2 ], [ -2, 1, -1 ] ], "source_shift": [ 1, 1, 2 ], "target_matrix": [ [ -1, -1, -1 ], [ -1, -1, -2 ], [ 2, 1, 1 ] ], "target_shift": [ 72, 69, -102 ], "collision_preimages": [ [ "7/2", "-9/4", "-7" ], [ "-23/2", "7", "51/2" ], [ "-17/2", "2", "29/2" ] ], "collision_image": [ "289/4", "277/4", "-205/2" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0588038050009345 ], "surface_features": [ 4.019003026639242, 4.929459798565662, 3.727378569451489, 0.0, 4.126296189482866, 5076 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B119", "pair": 119, "family": "bijection", "mapping": "F(x,y,z)=(128*x^7 - 1344*x^6*y + 896*x^6*z - 960*x^6 + 6048*x^5*y^2 - 8064*x^5*y*z + 8576*x^5*y + 2688*x^5*z^2 - 5728*x^5*z + 3072*x^5 - 15120*x^4*y^3 + 30240*x^4*y^2*z - 31920*x^4*y^2 - 20160*x^4*y*z^2 + 42640*x^4*y*z - 22720*x^4*y + 4480*x^4*z^3 - 14240*x^4*z^2 + 15200*x^4*z - 5456*x^4 + 22680*x^3*y^4 - 60480*x^3*y^3*z + 63360*x^3*y^3 + 60480*x^3*y^2*z^2 - 126960*x^3*y^2*z + 67200*x^3*y^2 - 26880*x^3*y*z^3 + 84800*x^3*y*z^2 - 89920*x^3*y*z + 32096*x^3*y + 4480*x^3*z^4 - 18880*x^3*z^3 + 30080*x^3*z^2 - 21504*x^3*z + 5836*x^3 - 20412*x^2*y^5 + 68040*x^2*y^4*z - 70740*x^2*y^4 - 90720*x^2*y^3*z^2 + 189000*x^2*y^3*z - 99360*x^2*y^3 + 60480*x^2*y^2*z^3 - 189360*x^2*y^2*z^2 + 199440*x^2*y^2*z - 70776*x^2*y^2 - 20160*x^2*y*z^4 + 84320*x^2*y*z^3 - 133440*x^2*y*z^2 + 94848*x^2*y*z - 25612*x^2*y + 2688*x^2*z^5 - 14080*x^2*z^4 + 29760*x^2*z^3 - 31776*x^2*z^2 + 17184*x^2*z - 3774*x^2 + 10206*x*y^6 - 40824*x*y^5*z + 42120*x*y^5 + 68040*x*y^4*z^2 - 140670*x*y^4*z + 73440*x*y^4 - 60480*x*y^3*z^3 + 187920*x*y^3*z^2 - 196560*x*y^3*z + 69336*x*y^3 + 30240*x*y^2*z^4 - 125520*x*y^2*z^3 + 197280*x*y^2*z^2 - 139392*x*y^2*z + 37443*x*y^2 - 8064*x*y*z^5 + 41920*x*y*z^4 - 88000*x*y*z^3 + 93408*x*y*z^2 - 50252*x*y*z + 10979*x*y + 896*x*z^6 - 5600*x*z^5 + 14720*x*z^4 - 20864*x*z^3 + 16860*x*z^2 - 7379*x*z + 1366*x - 2187*y^7 + 10206*y^6*z - 10449*y^6 - 20412*y^5*z^2 + 41877*y^5*z - 21708*y^5 + 22680*y^4*z^3 - 69930*y^4*z^2 + 72630*y^4*z - 25461*y^4 - 15120*y^3*z^4 + 62280*y^3*z^3 - 97200*y^3*z^2 + 68256*y^3*z - 18234*y^3 + 6048*y^2*z^5 - 31200*y^2*z^4 + 65040*y^2*z^3 - 68616*y^2*z^2 + 36714*y^2*z - 7978*y^2 - 1344*y*z^6 + 8336*y*z^5 - 21760*y*z^4 + 30656*y*z^3 - 24640*y*z^2 + 10726*y*z - 1976*y + 128*z^7 - 928*z^6 + 2912*z^5 - 5136*z^4 + 5512*z^3 - 3605*z^2 + 1331*z - 1, -128*x^7 + 1344*x^6*y - 896*x^6*z + 960*x^6 - 6048*x^5*y^2 + 8064*x^5*y*z - 8576*x^5*y - 2688*x^5*z^2 + 5728*x^5*z - 3104*x^5 + 15120*x^4*y^3 - 30240*x^4*y^2*z + 31920*x^4*y^2 + 20160*x^4*y*z^2 - 42640*x^4*y*z + 22960*x^4*y - 4480*x^4*z^3 + 14240*x^4*z^2 - 15360*x^4*z + 5616*x^4 - 22680*x^3*y^4 + 60480*x^3*y^3*z - 63360*x^3*y^3 - 60480*x^3*y^2*z^2 + 126960*x^3*y^2*z - 67920*x^3*y^2 + 26880*x^3*y*z^3 - 84800*x^3*y*z^2 + 90880*x^3*y*z - 33056*x^3*y - 4480*x^3*z^4 + 18880*x^3*z^3 - 30400*x^3*z^2 + 22144*x^3*z - 6156*x^3 + 20412*x^2*y^5 - 68040*x^2*y^4*z + 70740*x^2*y^4 + 90720*x^2*y^3*z^2 - 189000*x^2*y^3*z + 100440*x^2*y^3 - 60480*x^2*y^2*z^3 + 189360*x^2*y^2*z^2 - 201600*x^2*y^2*z + 72936*x^2*y^2 + 20160*x^2*y*z^4 - 84320*x^2*y*z^3 + 134880*x^2*y*z^2 - 97728*x^2*y*z + 27052*x^2*y - 2688*x^2*z^5 + 14080*x^2*z^4 - 30080*x^2*z^3 + 32736*x^2*z^2 - 18144*x^2*z + 4098*x^2 - 10206*x*y^6 + 40824*x*y^5*z - 42120*x*y^5 - 68040*x*y^4*z^2 + 140670*x*y^4*z - 74250*x*y^4 + 60480*x*y^3*z^3 - 187920*x*y^3*z^2 + 198720*x*y^3*z - 71496*x*y^3 - 30240*x*y^2*z^4 + 125520*x*y^2*z^3 - 199440*x*y^2*z^2 + 143712*x*y^2*z - 39603*x*y^2 + 8064*x*y*z^5 - 41920*x*y*z^4 + 88960*x*y*z^3 - 96288*x*y*z^2 + 53132*x*y*z - 11951*x*y - 896*x*z^6 + 5600*x*z^5 - 14880*x*z^4 + 21504*x*z^3 - 17820*x*z^2 + 8027*x*z - 1535*x + 2187*y^7 - 10206*y^6*z + 10449*y^6 + 20412*y^5*z^2 - 41877*y^5*z + 21951*y^5 - 22680*y^4*z^3 + 69930*y^4*z^2 - 73440*y^4*z + 26271*y^4 + 15120*y^3*z^4 - 62280*y^3*z^3 + 98280*y^3*z^2 - 70416*y^3*z + 19314*y^3 - 6048*y^2*z^5 + 31200*y^2*z^4 - 65760*y^2*z^3 + 70776*y^2*z^2 - 38874*y^2*z + 8707*y^2 + 1344*y*z^6 - 8336*y*z^5 + 22000*y*z^4 - 31616*y*z^3 + 26080*y*z^2 - 11698*y*z + 2229*y - 128*z^7 + 928*z^6 - 2944*z^5 + 5296*z^4 - 5832*z^3 + 3929*z^2 - 1500*z + 1, 256*x^7 - 2688*x^6*y + 1792*x^6*z - 1920*x^6 + 12096*x^5*y^2 - 16128*x^5*y*z + 17152*x^5*y + 5376*x^5*z^2 - 11456*x^5*z + 6144*x^5 - 30240*x^4*y^3 + 60480*x^4*y^2*z - 63840*x^4*y^2 - 40320*x^4*y*z^2 + 85280*x^4*y*z - 45440*x^4*y + 8960*x^4*z^3 - 28480*x^4*z^2 + 30400*x^4*z - 10912*x^4 + 45360*x^3*y^4 - 120960*x^3*y^3*z + 126720*x^3*y^3 + 120960*x^3*y^2*z^2 - 253920*x^3*y^2*z + 134400*x^3*y^2 - 53760*x^3*y*z^3 + 169600*x^3*y*z^2 - 179840*x^3*y*z + 64192*x^3*y + 8960*x^3*z^4 - 37760*x^3*z^3 + 60160*x^3*z^2 - 43008*x^3*z + 11672*x^3 - 40824*x^2*y^5 + 136080*x^2*y^4*z - 141480*x^2*y^4 - 181440*x^2*y^3*z^2 + 378000*x^2*y^3*z - 198720*x^2*y^3 + 120960*x^2*y^2*z^3 - 378720*x^2*y^2*z^2 + 398880*x^2*y^2*z - 141552*x^2*y^2 - 40320*x^2*y*z^4 + 168640*x^2*y*z^3 - 266880*x^2*y*z^2 + 189696*x^2*y*z - 51224*x^2*y + 5376*x^2*z^5 - 28160*x^2*z^4 + 59520*x^2*z^3 - 63552*x^2*z^2 + 34368*x^2*z - 7540*x^2 + 20412*x*y^6 - 81648*x*y^5*z + 84240*x*y^5 + 136080*x*y^4*z^2 - 281340*x*y^4*z + 146880*x*y^4 - 120960*x*y^3*z^3 + 375840*x*y^3*z^2 - 393120*x*y^3*z + 138672*x*y^3 + 60480*x*y^2*z^4 - 251040*x*y^2*z^3 + 394560*x*y^2*z^2 - 278784*x*y^2*z + 74886*x*y^2 - 16128*x*y*z^5 + 83840*x*y*z^4 - 176000*x*y*z^3 + 186816*x*y*z^2 - 100504*x*y*z + 21934*x*y + 1792*x*z^6 - 11200*x*z^5 + 29440*x*z^4 - 41728*x*z^3 + 33720*x*z^2 - 14742*x*z + 2712*x - 4374*y^7 + 20412*y^6*z - 20898*y^6 - 40824*y^5*z^2 + 83754*y^5*z - 43416*y^5 + 45360*y^4*z^3 - 139860*y^4*z^2 + 145260*y^4*z - 50922*y^4 - 30240*y^3*z^4 + 124560*y^3*z^3 - 194400*y^3*z^2 + 136512*y^3*z - 36468*y^3 + 12096*y^2*z^5 - 62400*y^2*z^4 + 130080*y^2*z^3 - 137232*y^2*z^2 + 73428*y^2*z - 15938*y^2 - 2688*y*z^6 + 16672*y*z^5 - 43520*y*z^4 + 61312*y*z^3 - 49280*y*z^2 + 21428*y*z - 3926*y + 256*z^7 - 1856*z^6 + 5824*z^5 - 10272*z^4 + 11024*z^3 - 7202*z^2 + 2644*z - 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -2, 3, -2 ], [ -1, 1, -1 ], [ -2, 1, -1 ] ], "source_shift": [ 2, 1, 1 ], "target_matrix": [ [ -1, -1, -1 ], [ 1, 2, 1 ], [ -2, -2, -3 ] ], "target_shift": [ 213, -250, 417 ], "inverse_recipe": { "p": 5, "q": 2, "f": "x**5 + x**2", "g": "x**2", "h": "y*z", "polynomials": { "f": [ [ [ 5, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "g": [ [ [ 2, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0392742039985023 ], "surface_features": [ 4.758324323251653, 5.600842261159698, 4.480581786829169, 0.0, 4.854749462427717, 5355 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B104", "pair": 104, "family": "bijection", "mapping": "F(x,y,z)=(384*x^7 - 1344*x^6*y + 1344*x^6*z - 3456*x^6 + 2016*x^5*y^2 - 4032*x^5*y*z + 10464*x^5*y + 2016*x^5*z^2 - 10464*x^5*z + 13224*x^5 - 1680*x^4*y^3 + 5040*x^4*y^2*z - 13200*x^4*y^2 - 5040*x^4*y*z^2 + 26400*x^4*y*z - 33732*x^4*y + 1680*x^4*z^3 - 13200*x^4*z^2 + 33684*x^4*z - 27996*x^4 + 840*x^3*y^4 - 3360*x^3*y^3*z + 8880*x^3*y^3 + 5040*x^3*y^2*z^2 - 26640*x^3*y^2*z + 34410*x^3*y^2 - 3360*x^3*y*z^3 + 26640*x^3*y*z^2 - 68724*x^3*y*z + 57876*x^3*y + 840*x^3*z^4 - 8880*x^3*z^3 + 34314*x^3*z^2 - 57612*x^3*z + 35643*x^3 - 252*x^2*y^5 + 1260*x^2*y^4*z - 3360*x^2*y^4 - 2520*x^2*y^3*z^2 + 13440*x^2*y^3*z - 17547*x^2*y^3 + 2520*x^2*y^2*z^3 - 20160*x^2*y^2*z^2 + 52569*x^2*y^2*z - 44859*x^2*y^2 - 1260*x^2*y*z^4 + 13440*x^2*y*z^3 - 52497*x^2*y*z^2 + 89310*x^2*y*z - 56118*x^2*y + 252*x^2*z^5 - 3360*x^2*z^4 + 17475*x^2*z^3 - 44451*x^2*z^2 + 55587*x^2*z - 27491*x^2 + 42*x*y^6 - 252*x*y^5*z + 678*x*y^5 + 630*x*y^4*z^2 - 3390*x*y^4*z + 4473*x*y^4 - 840*x*y^3*z^3 + 6780*x*y^3*z^2 - 17868*x*y^3*z + 15450*x*y^3 + 630*x*y^2*z^4 - 6780*x*y^2*z^3 + 26766*x*y^2*z^2 - 46140*x*y^2*z + 29460*x*y^2 - 252*x*y*z^5 + 3390*x*y*z^4 - 17820*x*y*z^3 + 45930*x*y*z^2 - 58356*x*y*z + 29372*x*y + 42*x*z^6 - 678*x*z^5 + 4449*x*z^4 - 15240*x*z^3 + 28896*x*z^2 - 28910*x*z + 11974*x - 3*y^7 + 21*y^6*z - 57*y^6 - 63*y^5*z^2 + 342*y^5*z - 456*y^5 + 105*y^4*z^3 - 855*y^4*z^2 + 2277*y^4*z - 1995*y^4 - 105*y^3*z^4 + 1140*y^3*z^3 - 4548*y^3*z^2 + 7944*y^3*z - 5157*y^3 + 63*y^2*z^5 - 855*y^2*z^4 + 4542*y^2*z^3 - 11862*y^2*z^2 + 15321*y^2*z - 7856*y^2 - 21*y*z^6 + 342*y*z^5 - 2268*y*z^4 + 7872*y*z^3 - 15171*y*z^2 + 15460*y*z - 6531*y + 3*z^7 - 57*z^6 + 453*z^5 - 1959*z^4 + 5007*z^3 - 7604*z^2 + 6378*z - 1, 128*x^7 - 448*x^6*y + 448*x^6*z - 1152*x^6 + 672*x^5*y^2 - 1344*x^5*y*z + 3488*x^5*y + 672*x^5*z^2 - 3488*x^5*z + 4408*x^5 - 560*x^4*y^3 + 1680*x^4*y^2*z - 4400*x^4*y^2 - 1680*x^4*y*z^2 + 8800*x^4*y*z - 11244*x^4*y + 560*x^4*z^3 - 4400*x^4*z^2 + 11228*x^4*z - 9332*x^4 + 280*x^3*y^4 - 1120*x^3*y^3*z + 2960*x^3*y^3 + 1680*x^3*y^2*z^2 - 8880*x^3*y^2*z + 11470*x^3*y^2 - 1120*x^3*y*z^3 + 8880*x^3*y*z^2 - 22908*x^3*y*z + 19292*x^3*y + 280*x^3*z^4 - 2960*x^3*z^3 + 11438*x^3*z^2 - 19204*x^3*z + 11889*x^3 - 84*x^2*y^5 + 420*x^2*y^4*z - 1120*x^2*y^4 - 840*x^2*y^3*z^2 + 4480*x^2*y^3*z - 5849*x^2*y^3 + 840*x^2*y^2*z^3 - 6720*x^2*y^2*z^2 + 17523*x^2*y^2*z - 14953*x^2*y^2 - 420*x^2*y*z^4 + 4480*x^2*y*z^3 - 17499*x^2*y*z^2 + 29770*x^2*y*z - 18718*x^2*y + 84*x^2*z^5 - 1120*x^2*z^4 + 5825*x^2*z^3 - 14817*x^2*z^2 + 18541*x^2*z - 9189*x^2 + 14*x*y^6 - 84*x*y^5*z + 226*x*y^5 + 210*x*y^4*z^2 - 1130*x*y^4*z + 1491*x*y^4 - 280*x*y^3*z^3 + 2260*x*y^3*z^2 - 5956*x*y^3*z + 5150*x*y^3 + 210*x*y^2*z^4 - 2260*x*y^2*z^3 + 8922*x*y^2*z^2 - 15380*x*y^2*z + 9826*x*y^2 - 84*x*y*z^5 + 1130*x*y*z^4 - 5940*x*y*z^3 + 15310*x*y*z^2 - 19464*x*y*z + 9816*x*y + 14*x*z^6 - 226*x*z^5 + 1483*x*z^4 - 5080*x*z^3 + 9638*x*z^2 - 9662*x*z + 4017*x - y^7 + 7*y^6*z - 19*y^6 - 21*y^5*z^2 + 114*y^5*z - 152*y^5 + 35*y^4*z^3 - 285*y^4*z^2 + 759*y^4*z - 665*y^4 - 35*y^3*z^4 + 380*y^3*z^3 - 1516*y^3*z^2 + 2648*y^3*z - 1720*y^3 + 21*y^2*z^5 - 285*y^2*z^4 + 1514*y^2*z^3 - 3954*y^2*z^2 + 5110*y^2*z - 2625*y^2 - 7*y*z^6 + 114*y*z^5 - 756*y*z^4 + 2624*y*z^3 - 5060*y*z^2 + 5166*y*z - 2191*y + z^7 - 19*z^6 + 151*z^5 - 653*z^4 + 1670*z^3 - 2541*z^2 + 2139*z - 1, -128*x^7 + 448*x^6*y - 448*x^6*z + 1152*x^6 - 672*x^5*y^2 + 1344*x^5*y*z - 3488*x^5*y - 672*x^5*z^2 + 3488*x^5*z - 4408*x^5 + 560*x^4*y^3 - 1680*x^4*y^2*z + 4400*x^4*y^2 + 1680*x^4*y*z^2 - 8800*x^4*y*z + 11244*x^4*y - 560*x^4*z^3 + 4400*x^4*z^2 - 11228*x^4*z + 9332*x^4 - 280*x^3*y^4 + 1120*x^3*y^3*z - 2960*x^3*y^3 - 1680*x^3*y^2*z^2 + 8880*x^3*y^2*z - 11470*x^3*y^2 + 1120*x^3*y*z^3 - 8880*x^3*y*z^2 + 22908*x^3*y*z - 19292*x^3*y - 280*x^3*z^4 + 2960*x^3*z^3 - 11438*x^3*z^2 + 19204*x^3*z - 11881*x^3 + 84*x^2*y^5 - 420*x^2*y^4*z + 1120*x^2*y^4 + 840*x^2*y^3*z^2 - 4480*x^2*y^3*z + 5849*x^2*y^3 - 840*x^2*y^2*z^3 + 6720*x^2*y^2*z^2 - 17523*x^2*y^2*z + 14953*x^2*y^2 + 420*x^2*y*z^4 - 4480*x^2*y*z^3 + 17499*x^2*y*z^2 - 29770*x^2*y*z + 18706*x^2*y - 84*x^2*z^5 + 1120*x^2*z^4 - 5825*x^2*z^3 + 14817*x^2*z^2 - 18529*x^2*z + 9161*x^2 - 14*x*y^6 + 84*x*y^5*z - 226*x*y^5 - 210*x*y^4*z^2 + 1130*x*y^4*z - 1491*x*y^4 + 280*x*y^3*z^3 - 2260*x*y^3*z^2 + 5956*x*y^3*z - 5150*x*y^3 - 210*x*y^2*z^4 + 2260*x*y^2*z^3 - 8922*x*y^2*z^2 + 15380*x*y^2*z - 9820*x*y^2 + 84*x*y*z^5 - 1130*x*y*z^4 + 5940*x*y*z^3 - 15310*x*y*z^2 + 19452*x*y*z - 9788*x*y - 14*x*z^6 + 226*x*z^5 - 1483*x*z^4 + 5080*x*z^3 - 9632*x*z^2 + 9634*x*z - 3984*x + y^7 - 7*y^6*z + 19*y^6 + 21*y^5*z^2 - 114*y^5*z + 152*y^5 - 35*y^4*z^3 + 285*y^4*z^2 - 759*y^4*z + 665*y^4 + 35*y^3*z^4 - 380*y^3*z^3 + 1516*y^3*z^2 - 2648*y^3*z + 1719*y^3 - 21*y^2*z^5 + 285*y^2*z^4 - 1514*y^2*z^3 + 3954*y^2*z^2 - 5107*y^2*z + 2618*y^2 + 7*y*z^6 - 114*y*z^5 + 756*y*z^4 - 2624*y*z^3 + 5057*y*z^2 - 5152*y*z + 2173*y - z^7 + 19*z^6 - 151*z^5 + 653*z^4 - 1669*z^3 + 2534*z^2 - 2122*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -2, 1, -1 ], [ -1, 2, -1 ], [ -1, 1, -1 ] ], "source_shift": [ 2, 2, 1 ], "target_matrix": [ [ -3, -3, -2 ], [ -1, -2, -1 ], [ 1, 1, 1 ] ], "target_shift": [ 2287, 771, -757 ], "inverse_recipe": { "p": 3, "q": 2, "f": "x**3 + x**2", "g": "x**2 + x", "h": "y*z**2 + y + z**2 + z", "polynomials": { "f": [ [ [ 3, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "g": [ [ [ 2, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 2 ], 1 ], [ [ 0, 1, 0 ], 1 ], [ [ 0, 0, 2 ], 1 ], [ [ 0, 0, 1 ], 1 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0515464550007891 ], "surface_features": [ 3.881310936492235, 4.950900089366387, 3.4026052419199146, 0.0, 4.08987719343617, 4941 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B227", "pair": 227, "family": "bijection", "mapping": "F(x,y,z)=(-128*x^7 + 896*x^6*y - 1344*x^6*z + 544*x^6 - 2688*x^5*y^2 + 8064*x^5*y*z - 3264*x^5*y - 6048*x^5*z^2 + 4880*x^5*z - 960*x^5 + 4480*x^4*y^3 - 20160*x^4*y^2*z + 8160*x^4*y^2 + 30240*x^4*y*z^2 - 24400*x^4*y*z + 4800*x^4*y - 15120*x^4*z^3 + 18240*x^4*z^2 - 7152*x^4*z + 904*x^4 - 4480*x^3*y^4 + 26880*x^3*y^3*z - 10880*x^3*y^3 - 60480*x^3*y^2*z^2 + 48800*x^3*y^2*z - 9600*x^3*y^2 + 60480*x^3*y*z^3 - 72960*x^3*y*z^2 + 28608*x^3*y*z - 3616*x^3*y - 22680*x^3*z^4 + 36360*x^3*z^3 - 21312*x^3*z^2 + 5364*x^3*z - 488*x^3 + 2688*x^2*y^5 - 20160*x^2*y^4*z + 8160*x^2*y^4 + 60480*x^2*y^3*z^2 - 48800*x^2*y^3*z + 9600*x^2*y^3 - 90720*x^2*y^2*z^3 + 109440*x^2*y^2*z^2 - 42912*x^2*y^2*z + 5424*x^2*y^2 + 68040*x^2*y*z^4 - 109080*x^2*y*z^3 + 63936*x^2*y*z^2 - 16092*x^2*y*z + 1468*x^2*y - 20412*x^2*z^5 + 40770*x^2*z^4 - 31752*x^2*z^3 + 11934*x^2*z^2 - 2156*x^2*z + 159*x^2 - 896*x*y^6 + 8064*x*y^5*z - 3264*x*y^5 - 30240*x*y^4*z^2 + 24400*x*y^4*z - 4800*x*y^4 + 60480*x*y^3*z^3 - 72960*x*y^3*z^2 + 28608*x*y^3*z - 3616*x*y^3 - 68040*x*y^2*z^4 + 109080*x*y^2*z^3 - 63936*x*y^2*z^2 + 16092*x*y^2*z - 1472*x*y^2 + 40824*x*y*z^5 - 81540*x*y*z^4 + 63504*x*y*z^3 - 23868*x*y*z^2 + 4324*x*y*z - 323*x*y - 10206*x*z^6 + 24381*x*z^5 - 23652*x*z^4 + 11799*x*z^3 - 3174*x*z^2 + 461*x*z - 32*x + 128*y^7 - 1344*y^6*z + 544*y^6 + 6048*y^5*z^2 - 4880*y^5*z + 960*y^5 - 15120*y^4*z^3 + 18240*y^4*z^2 - 7152*y^4*z + 904*y^4 + 22680*y^3*z^4 - 36360*y^3*z^3 + 21312*y^3*z^2 - 5364*y^3*z + 492*y^3 - 20412*y^2*z^5 + 40770*y^2*z^4 - 31752*y^2*z^3 + 11934*y^2*z^2 - 2168*y^2*z + 164*y^2 + 10206*y*z^6 - 24381*y*z^5 + 23652*y*z^4 - 11799*y*z^3 + 3183*y*z^2 - 468*y*z + 32*y - 2187*z^7 + 6075*z^6 - 7047*z^5 + 4374*z^4 - 1557*z^3 + 334*z^2 - 45*z + 1, 128*x^7 - 896*x^6*y + 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4200*x*y*z^4 - 17440*x*y*z^3 + 37602*x*y*z^2 - 41966*x*y*z + 19376*x*y + 70*x*z^6 - 840*x*z^5 + 4350*x*z^4 - 12454*x*z^3 + 20733*x*z^2 - 18996*x*z + 7432*x - 10*y^7 + 70*y^6*z - 140*y^6 - 210*y^5*z^2 + 840*y^5*z - 880*y^5 + 350*y^4*z^3 - 2100*y^4*z^2 + 4390*y^4*z - 3196*y^4 - 350*y^3*z^4 + 2800*y^3*z^3 - 8760*y^3*z^2 + 12704*y^3*z - 7181*y^3 + 210*y^2*z^5 - 2100*y^2*z^4 + 8740*y^2*z^3 - 18936*y^2*z^2 + 21293*y^2*z - 9958*y^2 - 70*y*z^6 + 840*y*z^5 - 4360*y*z^4 + 12544*y*z^3 - 21043*y*z^2 + 19526*y*z - 7781*y + 10*z^7 - 140*z^6 + 870*z^5 - 3116*z^4 + 6931*z^3 - 9568*z^2 + 7538*z - 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -1, 1, -1 ], [ -1, 2, -1 ], [ -2, 3, -3 ] ], "source_shift": [ 2, 1, 2 ], "target_matrix": [ [ 1, 2, 1 ], [ -1, -1, -2 ], [ -2, -3, -2 ] ], "target_shift": [ -1299, 1244, 2588 ], "inverse_recipe": { "p": 3, "q": 4, "f": "x**3 + x", "g": "x**4 + x**2", "h": "y*z + 2*y", "polynomials": { "f": [ [ [ 3, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "g": [ [ [ 4, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 1, 0 ], 2 ] ] }, "h_multiplier": 5 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.116367075999733 ], "surface_features": [ 3.6932624665916345, 4.622897576294769, 3.322219294733919, 0.0, 3.8271351728511664, 4928 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B050", "pair": 50, "family": "bijection", "mapping": "F(x,y,z)=(-384*x^7 - 1344*x^6*y - 1344*x^6*z - 1536*x^6 - 2016*x^5*y^2 - 4032*x^5*y*z - 4608*x^5*y - 2016*x^5*z^2 - 4608*x^5*z - 2688*x^5 - 1680*x^4*y^3 - 5040*x^4*y^2*z - 5760*x^4*y^2 - 5040*x^4*y*z^2 - 11520*x^4*y*z - 6816*x^4*y - 1680*x^4*z^3 - 5760*x^4*z^2 - 6768*x^4*z - 2728*x^4 - 840*x^3*y^4 - 3360*x^3*y^3*z - 3840*x^3*y^3 - 5040*x^3*y^2*z^2 - 11520*x^3*y^2*z - 6912*x^3*y^2 - 3360*x^3*y*z^3 - 11520*x^3*y*z^2 - 13728*x^3*y*z - 5684*x^3*y - 840*x^3*z^4 - 3840*x^3*z^3 - 6816*x^3*z^2 - 5564*x^3*z - 1728*x^3 - 252*x^2*y^5 - 1260*x^2*y^4*z - 1440*x^2*y^4 - 2520*x^2*y^3*z^2 - 5760*x^2*y^3*z - 3504*x^2*y^3 - 2520*x^2*y^2*z^3 - 8640*x^2*y^2*z^2 - 10440*x^2*y^2*z - 4434*x^2*y^2 - 1260*x^2*y*z^4 - 5760*x^2*y*z^3 - 10368*x^2*y*z^2 - 8688*x^2*y*z - 2808*x^2*y - 252*x^2*z^5 - 1440*x^2*z^4 - 3432*x^2*z^3 - 4254*x^2*z^2 - 2688*x^2*z - 668*x^2 - 42*x*y^6 - 252*x*y^5*z - 288*x*y^5 - 630*x*y^4*z^2 - 1440*x*y^4*z - 888*x*y^4 - 840*x*y^3*z^3 - 2880*x*y^3*z^2 - 3528*x*y^3*z - 1535*x*y^3 - 630*x*y^2*z^4 - 2880*x*y^2*z^3 - 5256*x*y^2*z^2 - 4515*x*y^2*z - 1512*x*y^2 - 252*x*y*z^5 - 1440*x*y*z^4 - 3480*x*y*z^3 - 4425*x*y*z^2 - 2904*x*y*z - 776*x*y - 42*x*z^6 - 288*x*z^5 - 864*x*z^4 - 1445*x*z^3 - 1392*x*z^2 - 713*x*z - 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17280*x^2*y^3*z + 10512*x^2*y^3 + 7560*x^2*y^2*z^3 + 25920*x^2*y^2*z^2 + 31320*x^2*y^2*z + 13278*x^2*y^2 + 3780*x^2*y*z^4 + 17280*x^2*y*z^3 + 31104*x^2*y*z^2 + 26016*x^2*y*z + 8448*x^2*y + 756*x^2*z^5 + 4320*x^2*z^4 + 10296*x^2*z^3 + 12738*x^2*z^2 + 8088*x^2*z + 2076*x^2 + 126*x*y^6 + 756*x*y^5*z + 864*x*y^5 + 1890*x*y^4*z^2 + 4320*x*y^4*z + 2664*x*y^4 + 2520*x*y^3*z^3 + 8640*x*y^3*z^2 + 10584*x*y^3*z + 4597*x*y^3 + 1890*x*y^2*z^4 + 8640*x*y^2*z^3 + 15768*x*y^2*z^2 + 13521*x*y^2*z + 4548*x*y^2 + 756*x*y*z^5 + 4320*x*y*z^4 + 10440*x*y*z^3 + 13251*x*y*z^2 + 8736*x*y*z + 2400*x*y + 126*x*z^6 + 864*x*z^5 + 2592*x*z^4 + 4327*x*z^3 + 4188*x*z^2 + 2211*x*z + 492*x + 9*y^7 + 63*y^6*z + 72*y^6 + 189*y^5*z^2 + 432*y^5*z + 270*y^5 + 315*y^4*z^3 + 1080*y^4*z^2 + 1341*y^4*z + 596*y^4 + 315*y^3*z^4 + 1440*y^3*z^3 + 2664*y^3*z^2 + 2339*y^3*z + 812*y^3 + 189*y^2*z^5 + 1080*y^2*z^4 + 2646*y^2*z^3 + 3441*y^2*z^2 + 2346*y^2*z + 708*y^2 + 63*y*z^6 + 432*y*z^5 + 1314*y*z^4 + 2249*y*z^3 + 2256*y*z^2 + 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888*x*y^4 - 840*x*y^3*z^3 - 2880*x*y^3*z^2 - 3528*x*y^3*z - 1535*x*y^3 - 630*x*y^2*z^4 - 2880*x*y^2*z^3 - 5256*x*y^2*z^2 - 4515*x*y^2*z - 1524*x*y^2 - 252*x*y*z^5 - 1440*x*y*z^4 - 3480*x*y*z^3 - 4425*x*y*z^2 - 2928*x*y*z - 808*x*y - 42*x*z^6 - 288*x*z^5 - 864*x*z^4 - 1445*x*z^3 - 1404*x*z^2 - 745*x*z - 167*x - 3*y^7 - 21*y^6*z - 24*y^6 - 63*y^5*z^2 - 144*y^5*z - 90*y^5 - 105*y^4*z^3 - 360*y^4*z^2 - 447*y^4*z - 199*y^4 - 105*y^3*z^4 - 480*y^3*z^3 - 888*y^3*z^2 - 781*y^3*z - 272*y^3 - 63*y^2*z^5 - 360*y^2*z^4 - 882*y^2*z^3 - 1149*y^2*z^2 - 786*y^2*z - 238*y^2 - 21*y*z^6 - 144*y*z^5 - 438*y*z^4 - 751*y*z^3 - 756*y*z^2 - 434*y*z - 111*y - 3*z^7 - 24*z^6 - 87*z^5 - 184*z^4 - 242*z^3 - 199*z^2 - 94*z - 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ 2, 1, 1 ], [ 1, 2, 1 ], [ 2, 3, 2 ] ], "source_shift": [ 1, 1, 2 ], "target_matrix": [ [ -1, -1, -2 ], [ 3, 2, 3 ], [ -1, -1, -1 ] ], "target_shift": [ 10, -54, 18 ], "inverse_recipe": { "p": 4, "q": 3, "f": "x**4", "g": "x**3 + x**2", "h": "y*z", "polynomials": { "f": [ [ [ 4, 0, 0 ], 1 ] ], "g": [ [ [ 3, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ] ] }, "h_multiplier": 3 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0682888049996109 ], "surface_features": [ 3.6206868186937577, 4.614728525224293, 3.225309281725863, 0.0, 3.7874593925003635, 4910 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B019", "pair": 19, "family": "bijection", "mapping": "F(x,y,z)=(128*x^7 - 448*x^6*y + 448*x^6*z - 832*x^6 + 672*x^5*y^2 - 1344*x^5*y*z + 2528*x^5*y + 672*x^5*z^2 - 2528*x^5*z + 2376*x^5 - 560*x^4*y^3 + 1680*x^4*y^2*z - 3200*x^4*y^2 - 1680*x^4*y*z^2 + 6400*x^4*y*z - 6092*x^4*y + 560*x^4*z^3 - 3200*x^4*z^2 + 6108*x^4*z - 3900*x^4 + 280*x^3*y^4 - 1120*x^3*y^3*z + 2160*x^3*y^3 + 1680*x^3*y^2*z^2 - 6480*x^3*y^2*z + 6246*x^3*y^2 - 1120*x^3*y*z^3 + 6480*x^3*y*z^2 - 12524*x^3*y*z + 8120*x^3*y + 280*x^3*z^4 - 2160*x^3*z^3 + 6278*x^3*z^2 - 8192*x^3*z + 3976*x^3 - 84*x^2*y^5 + 420*x^2*y^4*z - 820*x^2*y^4 - 840*x^2*y^3*z^2 + 3280*x^2*y^3*z - 3201*x^2*y^3 + 840*x^2*y^2*z^3 - 4920*x^2*y^2*z^2 + 9627*x^2*y^2*z - 6339*x^2*y^2 - 420*x^2*y*z^4 + 3280*x^2*y*z^3 - 9651*x^2*y*z^2 + 12790*x^2*y*z - 6323*x^2*y + 84*x^2*z^5 - 820*x^2*z^4 + 3225*x^2*z^3 - 6451*x^2*z^2 + 6440*x^2*z - 2532*x^2 + 14*x*y^6 - 84*x*y^5*z + 166*x*y^5 + 210*x*y^4*z^2 - 830*x*y^4*z + 820*x*y^4 - 280*x*y^3*z^3 + 1660*x*y^3*z^2 - 3288*x*y^3*z + 2199*x*y^3 + 210*x*y^2*z^4 - 1660*x*y^2*z^3 + 4944*x*y^2*z^2 - 6655*x*y^2*z + 3353*x*y^2 - 84*x*y*z^5 + 830*x*y*z^4 - 3304*x*y*z^3 + 6713*x*y*z^2 - 6830*x*y*z + 2747*x*y + 14*x*z^6 - 166*x*z^5 + 828*x*z^4 - 2257*x*z^3 + 3477*x*z^2 - 2836*x*z + 942*x - y^7 + 7*y^6*z - 14*y^6 - 21*y^5*z^2 + 84*y^5*z - 84*y^5 + 35*y^4*z^3 - 210*y^4*z^2 + 421*y^4*z - 286*y^4 - 35*y^3*z^4 + 280*y^3*z^3 - 844*y^3*z^2 + 1154*y^3*z - 593*y^3 + 21*y^2*z^5 - 210*y^2*z^4 + 846*y^2*z^3 - 1746*y^2*z^2 + 1812*y^2*z - 747*y^2 - 7*y*z^6 + 84*y*z^5 - 424*y*z^4 + 1174*y*z^3 - 1845*y*z^2 + 1544*y*z - 527*y + z^7 - 14*z^6 + 85*z^5 - 296*z^4 + 626*z^3 - 798*z^2 + 553*z - 1, -256*x^7 + 896*x^6*y - 896*x^6*z + 1664*x^6 - 1344*x^5*y^2 + 2688*x^5*y*z - 5056*x^5*y - 1344*x^5*z^2 + 5056*x^5*z - 4752*x^5 + 1120*x^4*y^3 - 3360*x^4*y^2*z + 6400*x^4*y^2 + 3360*x^4*y*z^2 - 12800*x^4*y*z + 12184*x^4*y - 1120*x^4*z^3 + 6400*x^4*z^2 - 12216*x^4*z + 7800*x^4 - 560*x^3*y^4 + 2240*x^3*y^3*z - 4320*x^3*y^3 - 3360*x^3*y^2*z^2 + 12960*x^3*y^2*z - 12492*x^3*y^2 + 2240*x^3*y*z^3 - 12960*x^3*y*z^2 + 25048*x^3*y*z - 16240*x^3*y - 560*x^3*z^4 + 4320*x^3*z^3 - 12556*x^3*z^2 + 16384*x^3*z - 7936*x^3 + 168*x^2*y^5 - 840*x^2*y^4*z + 1640*x^2*y^4 + 1680*x^2*y^3*z^2 - 6560*x^2*y^3*z + 6402*x^2*y^3 - 1680*x^2*y^2*z^3 + 9840*x^2*y^2*z^2 - 19254*x^2*y^2*z + 12678*x^2*y^2 + 840*x^2*y*z^4 - 6560*x^2*y*z^3 + 19302*x^2*y*z^2 - 25580*x^2*y*z + 12622*x^2*y - 168*x^2*z^5 + 1640*x^2*z^4 - 6450*x^2*z^3 + 12902*x^2*z^2 - 12856*x^2*z + 5028*x^2 - 28*x*y^6 + 168*x*y^5*z - 332*x*y^5 - 420*x*y^4*z^2 + 1660*x*y^4*z - 1640*x*y^4 + 560*x*y^3*z^3 - 3320*x*y^3*z^2 + 6576*x*y^3*z - 4398*x*y^3 - 420*x*y^2*z^4 + 3320*x*y^2*z^3 - 9888*x*y^2*z^2 + 13310*x*y^2*z - 6694*x*y^2 + 168*x*y*z^5 - 1660*x*y*z^4 + 6608*x*y*z^3 - 13426*x*y*z^2 + 13636*x*y*z - 5458*x*y - 28*x*z^6 + 332*x*z^5 - 1656*x*z^4 + 4514*x*z^3 - 6942*x*z^2 + 5636*x*z - 1851*x + 2*y^7 - 14*y^6*z + 28*y^6 + 42*y^5*z^2 - 168*y^5*z + 168*y^5 - 70*y^4*z^3 + 420*y^4*z^2 - 842*y^4*z + 572*y^4 + 70*y^3*z^4 - 560*y^3*z^3 + 1688*y^3*z^2 - 2308*y^3*z + 1184*y^3 - 42*y^2*z^5 + 420*y^2*z^4 - 1692*y^2*z^3 + 3492*y^2*z^2 - 3618*y^2*z + 1485*y^2 + 14*y*z^6 - 168*y*z^5 + 848*y*z^4 - 2348*y*z^3 + 3684*y*z^2 - 3070*y*z + 1036*y - 2*z^7 + 28*z^6 - 170*z^5 + 592*z^4 - 1250*z^3 + 1587*z^2 - 1086*z + 1, 128*x^7 - 448*x^6*y + 448*x^6*z - 832*x^6 + 672*x^5*y^2 - 1344*x^5*y*z + 2528*x^5*y + 672*x^5*z^2 - 2528*x^5*z + 2376*x^5 - 560*x^4*y^3 + 1680*x^4*y^2*z - 3200*x^4*y^2 - 1680*x^4*y*z^2 + 6400*x^4*y*z - 6092*x^4*y + 560*x^4*z^3 - 3200*x^4*z^2 + 6108*x^4*z - 3900*x^4 + 280*x^3*y^4 - 1120*x^3*y^3*z + 2160*x^3*y^3 + 1680*x^3*y^2*z^2 - 6480*x^3*y^2*z + 6246*x^3*y^2 - 1120*x^3*y*z^3 + 6480*x^3*y*z^2 - 12524*x^3*y*z + 8120*x^3*y + 280*x^3*z^4 - 2160*x^3*z^3 + 6278*x^3*z^2 - 8192*x^3*z + 3976*x^3 - 84*x^2*y^5 + 420*x^2*y^4*z - 820*x^2*y^4 - 840*x^2*y^3*z^2 + 3280*x^2*y^3*z - 3201*x^2*y^3 + 840*x^2*y^2*z^3 - 4920*x^2*y^2*z^2 + 9627*x^2*y^2*z - 6339*x^2*y^2 - 420*x^2*y*z^4 + 3280*x^2*y*z^3 - 9651*x^2*y*z^2 + 12790*x^2*y*z - 6323*x^2*y + 84*x^2*z^5 - 820*x^2*z^4 + 3225*x^2*z^3 - 6451*x^2*z^2 + 6440*x^2*z - 2528*x^2 + 14*x*y^6 - 84*x*y^5*z + 166*x*y^5 + 210*x*y^4*z^2 - 830*x*y^4*z + 820*x*y^4 - 280*x*y^3*z^3 + 1660*x*y^3*z^2 - 3288*x*y^3*z + 2199*x*y^3 + 210*x*y^2*z^4 - 1660*x*y^2*z^3 + 4944*x*y^2*z^2 - 6655*x*y^2*z + 3353*x*y^2 - 84*x*y*z^5 + 830*x*y*z^4 - 3304*x*y*z^3 + 6713*x*y*z^2 - 6830*x*y*z + 2743*x*y + 14*x*z^6 - 166*x*z^5 + 828*x*z^4 - 2257*x*z^3 + 3477*x*z^2 - 2832*x*z + 935*x - y^7 + 7*y^6*z - 14*y^6 - 21*y^5*z^2 + 84*y^5*z - 84*y^5 + 35*y^4*z^3 - 210*y^4*z^2 + 421*y^4*z - 286*y^4 - 35*y^3*z^4 + 280*y^3*z^3 - 844*y^3*z^2 + 1154*y^3*z - 593*y^3 + 21*y^2*z^5 - 210*y^2*z^4 + 846*y^2*z^3 - 1746*y^2*z^2 + 1812*y^2*z - 746*y^2 - 7*y*z^6 + 84*y*z^5 - 424*y*z^4 + 1174*y*z^3 - 1845*y*z^2 + 1542*y*z - 523*y + z^7 - 14*z^6 + 85*z^5 - 296*z^4 + 626*z^3 - 797*z^2 + 549*z - 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -2, 1, -1 ], [ -1, 1, -1 ], [ -1, 1, -2 ] ], "source_shift": [ 1, 2, 1 ], "target_matrix": [ [ -1, -2, -1 ], [ 2, 3, 3 ], [ -1, -1, -1 ] ], "target_shift": [ 160, -309, 156 ], "inverse_recipe": { "p": 2, "q": 3, "f": "x**2 + x", "g": "x**3 + x**2 + x", "h": "y**3 + y**2*z + y*z + z**2", "polynomials": { "f": [ [ [ 2, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "g": [ [ [ 3, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "h": [ [ [ 0, 2, 1 ], 1 ], [ [ 0, 0, 2 ], 1 ], [ [ 0, 3, 0 ], 1 ], [ [ 0, 1, 1 ], 1 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 6.143368853998254 ], "surface_features": [ 3.476439118229997, 4.407900540142635, 3.1283992687178066, 0.0, 3.6048592815252656, 4831 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C171", "pair": 171, "family": "counterexample", "mapping": "F(x,y,z)=(64*x^7 - 512*x^6*y - 352*x^6*z + 800*x^6 + 1680*x^5*y^2 + 2352*x^5*y*z - 5136*x^5*y + 816*x^5*z^2 - 3648*x^5*z + 3888*x^5 - 2912*x^4*y^3 - 6240*x^4*y^2*z + 13344*x^4*y^2 - 4416*x^4*y*z^2 + 19152*x^4*y*z - 20112*x^4*y - 1032*x^4*z^3 + 6840*x^4*z^2 - 14544*x^4*z + 9872*x^4 + 2860*x^3*y^4 + 8360*x^3*y^3*z - 17848*x^3*y^3 + 9072*x^3*y^2*z^2 - 38928*x^3*y^2*z + 40488*x^3*y^2 + 4332*x^3*y*z^3 - 28140*x^3*y*z^2 + 59064*x^3*y*z - 40016*x^3*y + 768*x^3*z^4 - 6744*x^3*z^3 + 21468*x^3*z^2 - 29264*x^3*z + 14460*x^3 - 1584*x^2*y^5 - 5934*x^2*y^4*z + 12858*x^2*y^4 - 8796*x^2*y^3*z^2 + 38016*x^2*y^3*z - 39480*x^2*y^3 - 6450*x^2*y^2*z^3 + 41850*x^2*y^2*z^2 - 87324*x^2*y^2*z + 59064*x^2*y^2 - 2340*x^2*y*z^4 + 20340*x^2*y*z^3 - 64104*x^2*y*z^2 + 87156*x^2*y*z - 43272*x^2*y - 336*x^2*z^5 + 3684*x^2*z^4 - 15624*x^2*z^3 + 32052*x^2*z^2 - 31902*x^2*z + 12406*x^2 + 459*x*y^6 + 2115*x*y^5*z - 4689*x*y^5 + 4017*x*y^4*z^2 - 17688*x*y^4*z + 18531*x*y^4 + 4025*x*y^3*z^3 - 26457*x*y^3*z^2 + 55410*x*y^3*z - 37492*x*y^3 + 2244*x*y^2*z^4 - 19626*x*y^2*z^3 + 61779*x*y^2*z^2 - 83880*x*y^2*z + 41757*x*y^2 + 660*x*y*z^5 - 7224*x*y*z^4 + 30456*x*y*z^3 - 62304*x*y*z^2 + 62187*x*y*z - 24397*x*y + 80*x*z^6 - 1056*x*z^5 + 5604*x*z^4 - 15370*x*z^3 + 23076*x*z^2 - 18106*x*z + 5849*x - 54*y^7 - 297*y^6*z + 675*y^6 - 693*y^5*z^2 + 3123*y^5*z - 3321*y^5 - 889*y^4*z^3 + 5964*y^4*z^2 - 12636*y^4*z + 8581*y^4 - 677*y^3*z^4 + 6019*y^3*z^3 - 19089*y^3*z^2 + 25957*y^3*z - 12948*y^3 - 306*y^2*z^5 + 3387*y^2*z^4 - 14322*y^2*z^3 + 29283*y^2*z^2 - 29271*y^2*z + 11548*y^2 - 76*y*z^6 + 1008*y*z^5 - 5340*y*z^4 + 14611*y*z^3 - 21963*y*z^2 + 17339*y*z - 5677*y - 8*z^7 + 124*z^6 - 792*z^5 + 2722*z^4 - 5472*z^3 + 6484*z^2 - 4236*z + 1, 128*x^7 - 1024*x^6*y - 704*x^6*z + 1504*x^6 + 3360*x^5*y^2 + 4704*x^5*y*z - 9552*x^5*y + 1632*x^5*z^2 - 6816*x^5*z + 6816*x^5 - 5824*x^4*y^3 - 12480*x^4*y^2*z + 24528*x^4*y^2 - 8832*x^4*y*z^2 + 35376*x^4*y*z - 34800*x^4*y - 2064*x^4*z^3 + 12696*x^4*z^2 - 25296*x^4*z + 16360*x^4 + 5720*x^3*y^4 + 16720*x^3*y^3*z - 32408*x^3*y^3 + 18144*x^3*y^2*z^2 - 71040*x^3*y^2*z + 69072*x^3*y^2 + 8664*x^3*y*z^3 - 51612*x^3*y*z^2 + 101328*x^3*y*z - 65428*x^3*y + 1536*x^3*z^4 - 12432*x^3*z^3 + 37032*x^3*z^2 - 48088*x^3*z + 23016*x^3 - 3168*x^2*y^5 - 11868*x^2*y^4*z + 23070*x^2*y^4 - 17592*x^2*y^3*z^2 + 68544*x^2*y^3*z - 66360*x^2*y^3 - 12900*x^2*y^2*z^3 + 75834*x^2*y^2*z^2 - 147648*x^2*y^2*z + 95238*x^2*y^2 - 4680*x^2*y*z^4 + 37044*x^2*y*z^3 - 109020*x^2*y*z^2 + 141276*x^2*y*z - 68028*x^2*y - 672*x^2*z^5 + 6744*x^2*z^4 - 26724*x^2*z^3 + 52224*x^2*z^2 - 50376*x^2*z + 19238*x^2 + 918*x*y^6 + 4230*x*y^5*z - 8325*x*y^5 + 8034*x*y^4*z^2 - 31542*x*y^4*z + 30690*x*y^4 + 8050*x*y^3*z^3 - 47397*x*y^3*z^2 + 92316*x*y^3*z - 59639*x*y^3 + 4488*x*y^2*z^4 - 35328*x*y^2*z^3 + 103542*x*y^2*z^2 - 134130*x*y^2*z + 64890*x*y^2 + 1320*x*y*z^5 - 13068*x*y*z^4 + 51348*x*y*z^3 - 100152*x*y*z^2 + 97050*x*y*z - 37457*x*y + 160*x*z^6 - 1920*x*z^5 + 9504*x*z^4 - 24836*x*z^3 + 36168*x*z^2 - 27896*x*z + 8930*x - 108*y^7 - 594*y^6*z + 1188*y^6 - 1386*y^5*z^2 + 5517*y^5*z - 5427*y^5 - 1778*y^4*z^3 + 10578*y^4*z^2 - 20763*y^4*z + 13490*y^4 - 1354*y^3*z^4 + 10721*y^3*z^3 - 31545*y^3*z^2 + 40997*y^3*z - 19935*y^3 - 612*y^2*z^5 + 6060*y^2*z^4 - 23805*y^2*z^3 + 46476*y^2*z^2 - 45225*y^2*z + 17600*y^2 - 152*y*z^6 + 1812*y*z^5 - 8928*y*z^4 + 23306*y*z^3 - 34062*y*z^2 + 26497*y*z - 8605*y - 16*z^7 + 224*z^6 - 1332*z^5 + 4364*z^4 - 8520*z^3 + 9938*z^2 - 6437*z + 1, 64*x^7 - 512*x^6*y - 352*x^6*z + 800*x^6 + 1680*x^5*y^2 + 2352*x^5*y*z - 5136*x^5*y + 816*x^5*z^2 - 3648*x^5*z + 3888*x^5 - 2912*x^4*y^3 - 6240*x^4*y^2*z + 13344*x^4*y^2 - 4416*x^4*y*z^2 + 19152*x^4*y*z - 20112*x^4*y - 1032*x^4*z^3 + 6840*x^4*z^2 - 14544*x^4*z + 9880*x^4 + 2860*x^3*y^4 + 8360*x^3*y^3*z - 17848*x^3*y^3 + 9072*x^3*y^2*z^2 - 38928*x^3*y^2*z + 40488*x^3*y^2 + 4332*x^3*y*z^3 - 28140*x^3*y*z^2 + 59064*x^3*y*z - 40068*x^3*y + 768*x^3*z^4 - 6744*x^3*z^3 + 21468*x^3*z^2 - 29296*x^3*z + 14524*x^3 - 1584*x^2*y^5 - 5934*x^2*y^4*z + 12858*x^2*y^4 - 8796*x^2*y^3*z^2 + 38016*x^2*y^3*z - 39480*x^2*y^3 - 6450*x^2*y^2*z^3 + 41850*x^2*y^2*z^2 - 87324*x^2*y^2*z + 59190*x^2*y^2 - 2340*x^2*y*z^4 + 20340*x^2*y*z^3 - 64104*x^2*y*z^2 + 87312*x^2*y*z - 43548*x^2*y - 336*x^2*z^5 + 3684*x^2*z^4 - 15624*x^2*z^3 + 32100*x^2*z^2 - 32082*x^2*z + 12550*x^2 + 459*x*y^6 + 2115*x*y^5*z - 4689*x*y^5 + 4017*x*y^4*z^2 - 17688*x*y^4*z + 18531*x*y^4 + 4025*x*y^3*z^3 - 26457*x*y^3*z^2 + 55410*x*y^3*z - 37627*x*y^3 + 2244*x*y^2*z^4 - 19626*x*y^2*z^3 + 61779*x*y^2*z^2 - 84132*x*y^2*z + 42153*x*y^2 + 660*x*y*z^5 - 7224*x*y*z^4 + 30456*x*y*z^3 - 62460*x*y*z^2 + 62703*x*y*z - 24793*x*y + 80*x*z^6 - 1056*x*z^5 + 5604*x*z^4 - 15402*x*z^3 + 23244*x*z^2 - 18370*x*z + 5973*x - 54*y^7 - 297*y^6*z + 675*y^6 - 693*y^5*z^2 + 3123*y^5*z - 3321*y^5 - 889*y^4*z^3 + 5964*y^4*z^2 - 12636*y^4*z + 8635*y^4 - 677*y^3*z^4 + 6019*y^3*z^3 - 19089*y^3*z^2 + 26092*y^3*z - 13137*y^3 - 306*y^2*z^5 + 3387*y^2*z^4 - 14322*y^2*z^3 + 29409*y^2*z^2 - 29640*y^2*z + 11818*y^2 - 76*y*z^6 + 1008*y*z^5 - 5340*y*z^4 + 14663*y*z^3 - 22203*y*z^2 + 17699*y*z - 5843*y - 8*z^7 + 124*z^6 - 792*z^5 + 2730*z^4 - 5524*z^3 + 6604*z^2 - 4348*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ 2, -3, -2 ], [ 2, -1, -1 ], [ 1, -2, -1 ] ], "source_shift": [ 2, 2, 2 ], "target_matrix": [ [ 1, 1, 2 ], [ 2, 1, 1 ], [ 1, 1, 1 ] ], "target_shift": [ -1183, -1789, -1219 ], "collision_preimages": [ [ "-9/4", "5/2", "-5" ], [ "2", "-10", "35/2" ], [ "7", "-12", "53/2" ] ], "collision_image": [ "-4733/4", "-3579/2", "-4877/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 6.088664374001382 ], "surface_features": [ 4.303105840117151, 5.16922756850498, 3.9679222067305173, 0.0, 4.404291242968723, 5153 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B121", "pair": 121, "family": "bijection", "mapping": "F(x,y,z)=(384*x^7 - 1344*x^6*y + 1344*x^6*z - 2112*x^6 + 2016*x^5*y^2 - 4032*x^5*y*z + 6432*x^5*y + 2016*x^5*z^2 - 6624*x^5*z + 5064*x^5 - 1680*x^4*y^3 + 5040*x^4*y^2*z - 8160*x^4*y^2 - 5040*x^4*y*z^2 + 16800*x^4*y*z - 13044*x^4*y + 1680*x^4*z^3 - 8640*x^4*z^2 + 13764*x^4*z - 6936*x^4 + 840*x^3*y^4 - 3360*x^3*y^3*z + 5520*x^3*y^3 + 5040*x^3*y^2*z^2 - 17040*x^3*y^2*z + 13434*x^3*y^2 - 3360*x^3*y*z^3 + 17520*x^3*y*z^2 - 28332*x^3*y*z + 14520*x^3*y + 840*x^3*z^4 - 6000*x^3*z^3 + 14922*x^3*z^2 - 15648*x^3*z + 5959*x^3 - 252*x^2*y^5 + 1260*x^2*y^4*z - 2100*x^2*y^4 - 2520*x^2*y^3*z^2 + 8640*x^2*y^3*z - 6915*x^2*y^3 + 2520*x^2*y^2*z^3 - 13320*x^2*y^2*z^2 + 21861*x^2*y^2*z - 11394*x^2*y^2 - 1260*x^2*y*z^4 + 9120*x^2*y*z^3 - 23013*x^2*y*z^2 + 24540*x^2*y*z - 9525*x^2*y + 252*x^2*z^5 - 2340*x^2*z^4 + 8067*x^2*z^3 - 13194*x^2*z^2 + 10479*x^2*z - 3241*x^2 + 42*x*y^6 - 252*x*y^5*z + 426*x*y^5 + 630*x*y^4*z^2 - 2190*x*y^4*z + 1779*x*y^4 - 840*x*y^3*z^3 + 4500*x*y^3*z^2 - 7494*x*y^3*z + 3972*x*y^3 + 630*x*y^2*z^4 - 4620*x*y^2*z^3 + 11826*x*y^2*z^2 - 12822*x*y^2*z + 5076*x*y^2 - 252*x*y*z^5 + 2370*x*y*z^4 - 8286*x*y*z^3 + 13776*x*y*z^2 - 11166*x*y*z + 3526*x*y + 42*x*z^6 - 486*x*z^5 + 2175*x*z^4 - 4926*x*z^3 + 6129*x*z^2 - 3970*x*z + 1045*x - 3*y^7 + 21*y^6*z - 36*y^6 - 63*y^5*z^2 + 222*y^5*z - 183*y^5 + 105*y^4*z^3 - 570*y^4*z^2 + 963*y^4*z - 519*y^4 - 105*y^3*z^4 + 780*y^3*z^3 - 2025*y^3*z^2 + 2232*y^3*z - 902*y^3 + 63*y^2*z^5 - 600*y^2*z^4 + 2127*y^2*z^3 - 3594*y^2*z^2 + 2976*y^2*z - 961*y^2 - 21*y*z^6 + 246*y*z^5 - 1116*y*z^4 + 2568*y*z^3 - 3267*y*z^2 + 2168*y*z - 583*y + 3*z^7 - 42*z^6 + 234*z^5 - 687*z^4 + 1193*z^3 - 1222*z^2 + 672*z - 1, -256*x^7 + 896*x^6*y - 896*x^6*z + 1408*x^6 - 1344*x^5*y^2 + 2688*x^5*y*z - 4288*x^5*y - 1344*x^5*z^2 + 4416*x^5*z - 3376*x^5 + 1120*x^4*y^3 - 3360*x^4*y^2*z + 5440*x^4*y^2 + 3360*x^4*y*z^2 - 11200*x^4*y*z + 8696*x^4*y - 1120*x^4*z^3 + 5760*x^4*z^2 - 9176*x^4*z + 4624*x^4 - 560*x^3*y^4 + 2240*x^3*y^3*z - 3680*x^3*y^3 - 3360*x^3*y^2*z^2 + 11360*x^3*y^2*z - 8956*x^3*y^2 + 2240*x^3*y*z^3 - 11680*x^3*y*z^2 + 18888*x^3*y*z - 9680*x^3*y - 560*x^3*z^4 + 4000*x^3*z^3 - 9948*x^3*z^2 + 10432*x^3*z - 3970*x^3 + 168*x^2*y^5 - 840*x^2*y^4*z + 1400*x^2*y^4 + 1680*x^2*y^3*z^2 - 5760*x^2*y^3*z + 4610*x^2*y^3 - 1680*x^2*y^2*z^3 + 8880*x^2*y^2*z^2 - 14574*x^2*y^2*z + 7596*x^2*y^2 + 840*x^2*y*z^4 - 6080*x^2*y*z^3 + 15342*x^2*y*z^2 - 16360*x^2*y*z + 6346*x^2*y - 168*x^2*z^5 + 1560*x^2*z^4 - 5378*x^2*z^3 + 8796*x^2*z^2 - 6982*x^2*z + 2158*x^2 - 28*x*y^6 + 168*x*y^5*z - 284*x*y^5 - 420*x*y^4*z^2 + 1460*x*y^4*z - 1186*x*y^4 + 560*x*y^3*z^3 - 3000*x*y^3*z^2 + 4996*x*y^3*z - 2648*x*y^3 - 420*x*y^2*z^4 + 3080*x*y^2*z^3 - 7884*x*y^2*z^2 + 8548*x*y^2*z - 3382*x*y^2 + 168*x*y*z^5 - 1580*x*y*z^4 + 5524*x*y*z^3 - 9184*x*y*z^2 + 7440*x*y*z - 2348*x*y - 28*x*z^6 + 324*x*z^5 - 1450*x*z^4 + 3284*x*z^3 - 4084*x*z^2 + 2644*x*z - 697*x + 2*y^7 - 14*y^6*z + 24*y^6 + 42*y^5*z^2 - 148*y^5*z + 122*y^5 - 70*y^4*z^3 + 380*y^4*z^2 - 642*y^4*z + 346*y^4 + 70*y^3*z^4 - 520*y^3*z^3 + 1350*y^3*z^2 - 1488*y^3*z + 601*y^3 - 42*y^2*z^5 + 400*y^2*z^4 - 1418*y^2*z^3 + 2396*y^2*z^2 - 1983*y^2*z + 640*y^2 + 14*y*z^6 - 164*y*z^5 + 744*y*z^4 - 1712*y*z^3 + 2177*y*z^2 - 1444*y*z + 389*y - 2*z^7 + 28*z^6 - 156*z^5 + 458*z^4 - 795*z^3 + 814*z^2 - 449*z + 1, 128*x^7 - 448*x^6*y + 448*x^6*z - 704*x^6 + 672*x^5*y^2 - 1344*x^5*y*z + 2144*x^5*y + 672*x^5*z^2 - 2208*x^5*z + 1688*x^5 - 560*x^4*y^3 + 1680*x^4*y^2*z - 2720*x^4*y^2 - 1680*x^4*y*z^2 + 5600*x^4*y*z - 4348*x^4*y + 560*x^4*z^3 - 2880*x^4*z^2 + 4588*x^4*z - 2312*x^4 + 280*x^3*y^4 - 1120*x^3*y^3*z + 1840*x^3*y^3 + 1680*x^3*y^2*z^2 - 5680*x^3*y^2*z + 4478*x^3*y^2 - 1120*x^3*y*z^3 + 5840*x^3*y*z^2 - 9444*x^3*y*z + 4840*x^3*y + 280*x^3*z^4 - 2000*x^3*z^3 + 4974*x^3*z^2 - 5216*x^3*z + 1997*x^3 - 84*x^2*y^5 + 420*x^2*y^4*z - 700*x^2*y^4 - 840*x^2*y^3*z^2 + 2880*x^2*y^3*z - 2305*x^2*y^3 + 840*x^2*y^2*z^3 - 4440*x^2*y^2*z^2 + 7287*x^2*y^2*z - 3798*x^2*y^2 - 420*x^2*y*z^4 + 3040*x^2*y*z^3 - 7671*x^2*y*z^2 + 8180*x^2*y*z - 3191*x^2*y + 84*x^2*z^5 - 780*x^2*z^4 + 2689*x^2*z^3 - 4398*x^2*z^2 + 3509*x^2*z - 1099*x^2 + 14*x*y^6 - 84*x*y^5*z + 142*x*y^5 + 210*x*y^4*z^2 - 730*x*y^4*z + 593*x*y^4 - 280*x*y^3*z^3 + 1500*x*y^3*z^2 - 2498*x*y^3*z + 1324*x*y^3 + 210*x*y^2*z^4 - 1540*x*y^2*z^3 + 3942*x*y^2*z^2 - 4274*x*y^2*z + 1700*x*y^2 - 84*x*y*z^5 + 790*x*y*z^4 - 2762*x*y*z^3 + 4592*x*y*z^2 - 3738*x*y*z + 1194*x*y + 14*x*z^6 - 162*x*z^5 + 725*x*z^4 - 1642*x*z^3 + 2051*x*z^2 - 1342*x*z + 361*x - y^7 + 7*y^6*z - 12*y^6 - 21*y^5*z^2 + 74*y^5*z - 61*y^5 + 35*y^4*z^3 - 190*y^4*z^2 + 321*y^4*z - 173*y^4 - 35*y^3*z^4 + 260*y^3*z^3 - 675*y^3*z^2 + 744*y^3*z - 302*y^3 + 21*y^2*z^5 - 200*y^2*z^4 + 709*y^2*z^3 - 1198*y^2*z^2 + 996*y^2*z - 325*y^2 - 7*y*z^6 + 82*y*z^5 - 372*y*z^4 + 856*y*z^3 - 1093*y*z^2 + 732*y*z - 201*y + z^7 - 14*z^6 + 78*z^5 - 229*z^4 + 399*z^3 - 412*z^2 + 230*z - 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -2, 1, -1 ], [ -1, 1, -1 ], [ -1, 1, -2 ] ], "source_shift": [ 1, 2, 1 ], "target_matrix": [ [ -3, -2, -2 ], [ 2, 1, 1 ], [ -1, -2, -1 ] ], "target_shift": [ 159, -106, 57 ], "inverse_recipe": { "p": 3, "q": 2, "f": "x**3 + x**2 + x", "g": "x**2 + x", "h": "y*z**2 + y + z", "polynomials": { "f": [ [ [ 3, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "g": [ [ [ 2, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 2 ], 1 ], [ [ 0, 0, 1 ], 1 ], [ [ 0, 1, 0 ], 1 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 6.0171169669993105 ], "surface_features": [ 3.514620792990414, 4.452277233126352, 3.162862993321926, 0.0, 3.647854557254789, 4864 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B170", "pair": 170, "family": "bijection", "mapping": "F(x,y,z)=(-128*x^7 + 448*x^6*y - 448*x^6*z + 928*x^6 - 672*x^5*y^2 + 1344*x^5*y*z - 2800*x^5*y - 672*x^5*z^2 + 2800*x^5*z - 2848*x^5 + 560*x^4*y^3 - 1680*x^4*y^2*z + 3520*x^4*y^2 + 1680*x^4*y*z^2 - 7040*x^4*y*z + 7200*x^4*y - 560*x^4*z^3 + 3520*x^4*z^2 - 7200*x^4*z + 4824*x^4 - 280*x^3*y^4 + 1120*x^3*y^3*z - 2360*x^3*y^3 - 1680*x^3*y^2*z^2 + 7080*x^3*y^2*z - 7280*x^3*y^2 + 1120*x^3*y*z^3 - 7080*x^3*y*z^2 + 14560*x^3*y*z - 9812*x^3*y - 280*x^3*z^4 + 2360*x^3*z^3 - 7280*x^3*z^2 + 9812*x^3*z - 4896*x^3 + 84*x^2*y^5 - 420*x^2*y^4*z + 890*x^2*y^4 + 840*x^2*y^3*z^2 - 3560*x^2*y^3*z + 3680*x^2*y^3 - 840*x^2*y^2*z^3 + 5340*x^2*y^2*z^2 - 11040*x^2*y^2*z + 7482*x^2*y^2 + 420*x^2*y*z^4 - 3560*x^2*y*z^3 + 11040*x^2*y*z^2 - 14964*x^2*y*z + 7520*x^2*y - 84*x^2*z^5 + 890*x^2*z^4 - 3680*x^2*z^3 + 7482*x^2*z^2 - 7524*x^2*z + 2991*x^2 - 14*x*y^6 + 84*x*y^5*z - 179*x*y^5 - 210*x*y^4*z^2 + 895*x*y^4*z - 930*x*y^4 + 280*x*y^3*z^3 - 1790*x*y^3*z^2 + 3720*x*y^3*z - 2535*x*y^3 - 210*x*y^2*z^4 + 1790*x*y^2*z^3 - 5580*x*y^2*z^2 + 7605*x*y^2*z - 3848*x*y^2 + 84*x*y*z^5 - 895*x*y*z^4 + 3720*x*y*z^3 - 7605*x*y*z^2 + 7700*x*y*z - 3093*x*y - 14*x*z^6 + 179*x*z^5 - 930*x*z^4 + 2535*x*z^3 - 3852*x*z^2 + 3102*x*z - 1019*x + y^7 - 7*y^6*z + 15*y^6 + 21*y^5*z^2 - 90*y^5*z + 94*y^5 - 35*y^4*z^3 + 225*y^4*z^2 - 470*y^4*z + 322*y^4 + 35*y^3*z^4 - 300*y^3*z^3 + 940*y^3*z^2 - 1288*y^3*z + 656*y^3 - 21*y^2*z^5 + 225*y^2*z^4 - 940*y^2*z^3 + 1932*y^2*z^2 - 1969*y^2*z + 799*y^2 + 7*y*z^6 - 90*y*z^5 + 470*y*z^4 - 1288*y*z^3 + 1970*y*z^2 - 1603*y*z + 535*y - z^7 + 15*z^6 - 94*z^5 + 322*z^4 - 657*z^3 + 804*z^2 - 538*z + 1, 256*x^7 - 896*x^6*y + 896*x^6*z - 1856*x^6 + 1344*x^5*y^2 - 2688*x^5*y*z + 5600*x^5*y + 1344*x^5*z^2 - 5600*x^5*z + 5760*x^5 - 1120*x^4*y^3 + 3360*x^4*y^2*z - 7040*x^4*y^2 - 3360*x^4*y*z^2 + 14080*x^4*y*z - 14560*x^4*y + 1120*x^4*z^3 - 7040*x^4*z^2 + 14560*x^4*z - 9968*x^4 + 560*x^3*y^4 - 2240*x^3*y^3*z + 4720*x^3*y^3 + 3360*x^3*y^2*z^2 - 14160*x^3*y^2*z + 14720*x^3*y^2 - 2240*x^3*y*z^3 + 14160*x^3*y*z^2 - 29440*x^3*y*z + 20264*x^3*y + 560*x^3*z^4 - 4720*x^3*z^3 + 14720*x^3*z^2 - 20264*x^3*z + 10448*x^3 - 168*x^2*y^5 + 840*x^2*y^4*z - 1780*x^2*y^4 - 1680*x^2*y^3*z^2 + 7120*x^2*y^3*z - 7440*x^2*y^3 + 1680*x^2*y^2*z^3 - 10680*x^2*y^2*z^2 + 22320*x^2*y^2*z - 15444*x^2*y^2 - 840*x^2*y*z^4 + 7120*x^2*y*z^3 - 22320*x^2*y*z^2 + 30888*x^2*y*z - 16024*x^2*y + 168*x^2*z^5 - 1780*x^2*z^4 + 7440*x^2*z^3 - 15444*x^2*z^2 + 16032*x^2*z - 6674*x^2 + 28*x*y^6 - 168*x*y^5*z + 358*x*y^5 + 420*x*y^4*z^2 - 1790*x*y^4*z + 1880*x*y^4 - 560*x*y^3*z^3 + 3580*x*y^3*z^2 - 7520*x*y^3*z + 5230*x*y^3 + 420*x*y^2*z^4 - 3580*x*y^2*z^3 + 11280*x*y^2*z^2 - 15690*x*y^2*z + 8188*x*y^2 - 168*x*y*z^5 + 1790*x*y*z^4 - 7520*x*y*z^3 + 15690*x*y*z^2 - 16384*x*y*z + 6878*x*y + 28*x*z^6 - 358*x*z^5 + 1880*x*z^4 - 5230*x*z^3 + 8196*x*z^2 - 6896*x*z + 2419*x - 2*y^7 + 14*y^6*z - 30*y^6 - 42*y^5*z^2 + 180*y^5*z - 190*y^5 + 70*y^4*z^3 - 450*y^4*z^2 + 950*y^4*z - 664*y^4 - 70*y^3*z^4 + 600*y^3*z^3 - 1900*y^3*z^2 + 2656*y^3*z - 1394*y^3 + 42*y^2*z^5 - 450*y^2*z^4 + 1900*y^2*z^3 - 3984*y^2*z^2 + 4184*y^2*z - 1771*y^2 - 14*y*z^6 + 180*y*z^5 - 950*y*z^4 + 2656*y*z^3 - 4186*y*z^2 + 3552*y*z - 1261*y + 2*z^7 - 30*z^6 + 190*z^5 - 664*z^4 + 1396*z^3 - 1781*z^2 + 1269*z - 1, 128*x^7 - 448*x^6*y + 448*x^6*z - 928*x^6 + 672*x^5*y^2 - 1344*x^5*y*z + 2800*x^5*y + 672*x^5*z^2 - 2800*x^5*z + 2848*x^5 - 560*x^4*y^3 + 1680*x^4*y^2*z - 3520*x^4*y^2 - 1680*x^4*y*z^2 + 7040*x^4*y*z - 7200*x^4*y + 560*x^4*z^3 - 3520*x^4*z^2 + 7200*x^4*z - 4824*x^4 + 280*x^3*y^4 - 1120*x^3*y^3*z + 2360*x^3*y^3 + 1680*x^3*y^2*z^2 - 7080*x^3*y^2*z + 7280*x^3*y^2 - 1120*x^3*y*z^3 + 7080*x^3*y*z^2 - 14560*x^3*y*z + 9812*x^3*y + 280*x^3*z^4 - 2360*x^3*z^3 + 7280*x^3*z^2 - 9812*x^3*z + 4896*x^3 - 84*x^2*y^5 + 420*x^2*y^4*z - 890*x^2*y^4 - 840*x^2*y^3*z^2 + 3560*x^2*y^3*z - 3680*x^2*y^3 + 840*x^2*y^2*z^3 - 5340*x^2*y^2*z^2 + 11040*x^2*y^2*z - 7482*x^2*y^2 - 420*x^2*y*z^4 + 3560*x^2*y*z^3 - 11040*x^2*y*z^2 + 14964*x^2*y*z - 7520*x^2*y + 84*x^2*z^5 - 890*x^2*z^4 + 3680*x^2*z^3 - 7482*x^2*z^2 + 7524*x^2*z - 2995*x^2 + 14*x*y^6 - 84*x*y^5*z + 179*x*y^5 + 210*x*y^4*z^2 - 895*x*y^4*z + 930*x*y^4 - 280*x*y^3*z^3 + 1790*x*y^3*z^2 - 3720*x*y^3*z + 2535*x*y^3 + 210*x*y^2*z^4 - 1790*x*y^2*z^3 + 5580*x*y^2*z^2 - 7605*x*y^2*z + 3848*x*y^2 - 84*x*y*z^5 + 895*x*y*z^4 - 3720*x*y*z^3 + 7605*x*y*z^2 - 7700*x*y*z + 3097*x*y + 14*x*z^6 - 179*x*z^5 + 930*x*z^4 - 2535*x*z^3 + 3852*x*z^2 - 3106*x*z + 1028*x - y^7 + 7*y^6*z - 15*y^6 - 21*y^5*z^2 + 90*y^5*z - 94*y^5 + 35*y^4*z^3 - 225*y^4*z^2 + 470*y^4*z - 322*y^4 - 35*y^3*z^4 + 300*y^3*z^3 - 940*y^3*z^2 + 1288*y^3*z - 656*y^3 + 21*y^2*z^5 - 225*y^2*z^4 + 940*y^2*z^3 - 1932*y^2*z^2 + 1969*y^2*z - 800*y^2 - 7*y*z^6 + 90*y*z^5 - 470*y*z^4 + 1288*y*z^3 - 1970*y*z^2 + 1605*y*z - 540*y + z^7 - 15*z^6 + 94*z^5 - 322*z^4 + 657*z^3 - 805*z^2 + 543*z - 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -2, 1, -1 ], [ -1, 1, -1 ], [ -1, 1, -2 ] ], "source_shift": [ 2, 1, 1 ], "target_matrix": [ [ 1, 1, 1 ], [ -2, -1, -1 ], [ -1, -2, -1 ] ], "target_shift": [ -147, 384, 152 ], "inverse_recipe": { "p": 2, "q": 5, "f": "x**2", "g": "x**5 + x**3 + x**2 + x", "h": "y*z", "polynomials": { "f": [ [ [ 2, 0, 0 ], 1 ] ], "g": [ [ [ 5, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ], [ [ 3, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 5.966380489000585 ], "surface_features": [ 3.536245163446622, 4.489789788615993, 3.13640344813399, 0.0, 3.6778915116401207, 4840 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C197", "pair": 197, "family": "counterexample", "mapping": "F(x,y,z)=(24*x^7 - 124*x^6*y - 196*x^6*z + 280*x^6 + 270*x^5*y^2 + 840*x^5*y*z - 1200*x^5*y + 642*x^5*z^2 - 1848*x^5*z + 1314*x^5 - 321*x^4*y^3 - 1473*x^4*y^2*z + 2106*x^4*y^2 - 2211*x^4*y*z^2 + 6372*x^4*y*z - 4545*x^4*y - 1083*x^4*z^3 + 4722*x^4*z^2 - 6795*x^4*z + 3245*x^4 + 225*x^3*y^4 + 1353*x^3*y^3*z - 1938*x^3*y^3 + 2991*x^3*y^2*z^2 - 8640*x^3*y^2*z + 6180*x^3*y^2 + 2875*x^3*y*z^3 - 12570*x^3*y*z^2 + 18129*x^3*y*z - 8677*x^3*y + 1012*x^3*z^4 - 5956*x^3*z^3 + 12987*x^3*z^2 - 12516*x^3*z + 4533*x^3 - 93*x^2*y^5 - 687*x^2*y^4*z + 987*x^2*y^4 - 1989*x^2*y^3*z^2 + 5766*x^2*y^3*z - 4134*x^2*y^3 - 2817*x^2*y^2*z^3 + 12366*x^2*y^2*z^2 - 17862*x^2*y^2*z + 8556*x^2*y^2 - 1950*x^2*y*z^4 + 11526*x^2*y*z^3 - 25137*x^2*y*z^2 + 24198*x^2*y*z - 8757*x^2*y - 528*x^2*z^5 + 3939*x^2*z^4 - 11511*x^2*z^3 + 16665*x^2*z^2 - 12087*x^2*z + 3547*x^2 + 21*x*y^6 + 183*x*y^5*z - 264*x*y^5 + 651*x*y^4*z^2 - 1896*x*y^4*z + 1362*x*y^4 + 1209*x*y^3*z^3 - 5334*x*y^3*z^2 + 7713*x*y^3*z - 3694*x*y^3 + 1236*x*y^2*z^4 - 7344*x*y^2*z^3 + 16017*x*y^2*z^2 - 15393*x*y^2*z + 5559*x*y^2 + 660*x*y*z^5 - 4950*x*y*z^4 + 14454*x*y*z^3 - 20865*x*y*z^2 + 15069*x*y*z - 4399*x*y + 144*x*z^6 - 1308*x*z^5 + 4788*x*z^4 - 9211*x*z^3 + 9945*x*z^2 - 5773*x*z + 1422*x - 2*y^7 - 20*y^6*z + 29*y^6 - 84*y^5*z^2 + 246*y^5*z - 177*y^5 - 192*y^4*z^3 + 852*y^4*z^2 - 1233*y^4*z + 590*y^4 - 258*y^3*z^4 + 1542*y^3*z^3 - 3363*y^3*z^2 + 3225*y^3*z - 1161*y^3 - 204*y^2*z^5 + 1539*y^2*z^4 - 4491*y^2*z^3 + 6462*y^2*z^2 - 4644*y^2*z + 1348*y^2 - 88*y*z^6 + 804*y*z^5 - 2940*y*z^4 + 5633*y*z^3 - 6045*y*z^2 + 3485*y*z - 854*y - 16*z^7 + 172*z^6 - 756*z^5 + 1806*z^4 - 2568*z^3 + 2200*z^2 - 1067*z + 1, 72*x^7 - 372*x^6*y - 588*x^6*z + 804*x^6 + 810*x^5*y^2 + 2520*x^5*y*z - 3432*x^5*y + 1926*x^5*z^2 - 5268*x^5*z + 3594*x^5 - 963*x^4*y^3 - 4419*x^4*y^2*z + 5997*x^4*y^2 - 6633*x^4*y*z^2 + 18078*x^4*y*z - 12345*x^4*y - 3249*x^4*z^3 + 13341*x^4*z^2 - 18339*x^4*z + 8478*x^4 + 675*x^3*y^4 + 4059*x^3*y^3*z - 5493*x^3*y^3 + 8973*x^3*y^2*z^2 - 24390*x^3*y^2*z + 16668*x^3*y^2 + 8625*x^3*y*z^3 - 35325*x^3*y*z^2 + 48573*x^3*y*z - 22488*x^3*y + 3036*x^3*z^4 - 16656*x^3*z^3 + 34551*x^3*z^2 - 32178*x^3*z + 11379*x^3 - 279*x^2*y^5 - 2061*x^2*y^4*z + 2784*x^2*y^4 - 5967*x^2*y^3*z^2 + 16194*x^2*y^3*z - 11073*x^2*y^3 - 8451*x^2*y^2*z^3 + 34569*x^2*y^2*z^2 - 47529*x^2*y^2*z + 22008*x^2*y^2 - 5850*x^2*y*z^4 + 32064*x^2*y*z^3 - 66438*x^2*y*z^2 + 61770*x^2*y*z - 21810*x^2*y - 1584*x^2*z^5 + 10905*x^2*z^4 - 30228*x^2*z^3 + 42228*x^2*z^2 - 29850*x^2*z + 8595*x^2 + 63*x*y^6 + 549*x*y^5*z - 741*x*y^5 + 1953*x*y^4*z^2 - 5298*x*y^4*z + 3624*x*y^4 + 3627*x*y^3*z^3 - 14835*x*y^3*z^2 + 20391*x*y^3*z - 9435*x*y^3 + 3708*x*y^2*z^4 - 20328*x*y^2*z^3 + 42075*x*y^2*z^2 - 39033*x*y^2*z + 13743*x*y^2 + 1980*x*y*z^5 - 13638*x*y*z^4 + 37740*x*y*z^3 - 52545*x*y*z^2 + 36957*x*y*z - 10572*x*y + 432*x*z^6 - 3588*x*z^5 + 12432*x*z^4 - 23049*x*z^3 + 24207*x*z^2 - 13743*x*z + 3328*x - 6*y^7 - 60*y^6*z + 81*y^6 - 252*y^5*z^2 + 684*y^5*z - 468*y^5 - 576*y^4*z^3 + 2358*y^4*z^2 - 3240*y^4*z + 1497*y^4 - 774*y^3*z^4 + 4248*y^3*z^3 - 8784*y^3*z^2 + 8127*y^3*z - 2850*y^3 - 612*y^2*z^5 + 4221*y^2*z^4 - 11664*y^2*z^3 + 16179*y^2*z^2 - 11313*y^2*z + 3213*y^2 - 264*y*z^6 + 2196*y*z^5 - 7596*y*z^4 + 14019*y*z^3 - 14619*y*z^2 + 8229*y*z - 1979*y - 48*z^7 + 468*z^6 - 1944*z^5 + 4470*z^4 - 6168*z^3 + 5148*z^2 - 2444*z + 1, 48*x^7 - 248*x^6*y - 392*x^6*z + 524*x^6 + 540*x^5*y^2 + 1680*x^5*y*z - 2232*x^5*y + 1284*x^5*z^2 - 3420*x^5*z + 2280*x^5 - 642*x^4*y^3 - 2946*x^4*y^2*z + 3891*x^4*y^2 - 4422*x^4*y*z^2 + 11706*x^4*y*z - 7800*x^4*y - 2166*x^4*z^3 + 8619*x^4*z^2 - 11544*x^4*z + 5236*x^4 + 450*x^3*y^4 + 2706*x^3*y^3*z - 3555*x^3*y^3 + 5982*x^3*y^2*z^2 - 15750*x^3*y^2*z + 10488*x^3*y^2 + 5750*x^3*y*z^3 - 22755*x^3*y*z^2 + 30444*x^3*y*z - 13822*x^3*y + 2024*x^3*z^4 - 10700*x^3*z^3 + 21564*x^3*z^2 - 19682*x^3*z + 6872*x^3 - 186*x^2*y^5 - 1374*x^2*y^4*z + 1797*x^2*y^4 - 3978*x^2*y^3*z^2 + 10428*x^2*y^3*z - 6939*x^2*y^3 - 5634*x^2*y^2*z^3 + 22203*x^2*y^2*z^2 - 29667*x^2*y^2*z + 13467*x^2*y^2 - 3900*x^2*y*z^4 + 20538*x^2*y*z^3 - 41301*x^2*y*z^2 + 37626*x^2*y*z - 13122*x^2*y - 1056*x^2*z^5 + 6966*x^2*z^4 - 18717*x^2*z^3 + 25611*x^2*z^2 - 17886*x^2*z + 5123*x^2 + 42*x*y^6 + 366*x*y^5*z - 477*x*y^5 + 1302*x*y^4*z^2 - 3402*x*y^4*z + 2262*x*y^4 + 2418*x*y^3*z^3 - 9501*x*y^3*z^2 + 12678*x*y^3*z - 5750*x*y^3 + 2472*x*y^2*z^4 - 12984*x*y^2*z^3 + 26058*x*y^2*z^2 - 23688*x*y^2*z + 8244*x*y^2 + 1320*x*y*z^5 - 8688*x*y*z^4 + 23286*x*y*z^3 - 31764*x*y*z^2 + 22098*x*y*z - 6299*x*y + 288*x*z^6 - 2280*x*z^5 + 7644*x*z^4 - 13886*x*z^3 + 14442*x*z^2 - 8186*x*z + 1988*x - 4*y^7 - 40*y^6*z + 52*y^6 - 168*y^5*z^2 + 438*y^5*z - 291*y^5 - 384*y^4*z^3 + 1506*y^4*z^2 - 2007*y^4*z + 909*y^4 - 516*y^3*z^4 + 2706*y^3*z^3 - 5421*y^3*z^2 + 4916*y^3*z - 1706*y^3 - 408*y^2*z^5 + 2682*y^2*z^4 - 7173*y^2*z^3 + 9753*y^2*z^2 - 6756*y^2*z + 1916*y^2 - 176*y*z^6 + 1392*y*z^5 - 4656*y*z^4 + 8426*y*z^3 - 8718*y*z^2 + 4912*y*z - 1187*y - 32*z^7 + 296*z^6 - 1188*z^5 + 2680*z^4 - 3676*z^3 + 3080*z^2 - 1473*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ 1, -1, -2 ], [ 2, -1, -1 ], [ 3, -2, -2 ] ], "source_shift": [ 2, 1, 2 ], "target_matrix": [ [ 1, 1, 1 ], [ 3, 2, 3 ], [ 2, 1, 1 ] ], "target_shift": [ -228, -517, -312 ], "collision_preimages": [ [ "1/4", "3/4", "3/4" ], [ "-19/2", "-49/2", "8" ], [ "-7/2", "-29/2", "7" ] ], "collision_image": [ "-913/4", "-2071/4", "-625/2" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 5.810195313999429 ], "surface_features": [ 3.8946152448861167, 4.822416550298963, 3.5593679094633317, 0.0, 4.021579703301161, 5021 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B146", "pair": 146, "family": "bijection", "mapping": "F(x,y,z)=(-2*x^7 + 14*x^6*y - 14*x^6*z + 34*x^6 - 42*x^5*y^2 + 84*x^5*y*z - 208*x^5*y - 42*x^5*z^2 + 204*x^5*z - 246*x^5 + 70*x^4*y^3 - 210*x^4*y^2*z + 530*x^4*y^2 + 210*x^4*y*z^2 - 1040*x^4*y*z + 1278*x^4*y - 70*x^4*z^3 + 510*x^4*z^2 - 1232*x^4*z + 982*x^4 - 70*x^3*y^4 + 280*x^3*y^3*z - 720*x^3*y^3 - 420*x^3*y^2*z^2 + 2120*x^3*y^2*z - 2654*x^3*y^2 + 280*x^3*y*z^3 - 2080*x^3*y*z^2 + 5120*x^3*y*z - 4160*x^3*y - 70*x^3*z^4 + 680*x^3*z^3 - 2468*x^3*z^2 + 3948*x^3*z - 2328*x^3 + 42*x^2*y^5 - 210*x^2*y^4*z + 550*x^2*y^4 + 420*x^2*y^3*z^2 - 2160*x^2*y^3*z + 2754*x^2*y^3 - 420*x^2*y^2*z^3 + 3180*x^2*y^2*z^2 - 7974*x^2*y^2*z + 6602*x^2*y^2 + 210*x^2*y*z^4 - 2080*x^2*y*z^3 + 7692*x^2*y*z^2 - 12544*x^2*y*z + 7548*x^2*y - 42*x^2*z^5 + 510*x^2*z^4 - 2472*x^2*z^3 + 5952*x^2*z^2 - 7054*x^2*z + 3257*x^2 - 14*x*y^6 + 84*x*y^5*z - 224*x*y^5 - 210*x*y^4*z^2 + 1100*x*y^4*z - 1428*x*y^4 + 280*x*y^3*z^3 - 2160*x*y^3*z^2 + 5516*x*y^3*z - 4652*x*y^3 - 210*x*y^2*z^4 + 2120*x*y^2*z^3 - 7986*x*y^2*z^2 + 13272*x*y^2*z - 8148*x*y^2 + 84*x*y*z^5 - 1040*x*y*z^4 + 5136*x*y*z^3 - 12608*x*y*z^2 + 15256*x*y*z - 7194*x*y - 14*x*z^6 + 204*x*z^5 - 1238*x*z^4 + 3988*x*z^3 - 7124*x*z^2 + 6614*x*z - 2470*x + 2*y^7 - 14*y^6*z + 38*y^6 + 42*y^5*z^2 - 224*y^5*z + 296*y^5 - 70*y^4*z^3 + 550*y^4*z^2 - 1430*y^4*z + 1228*y^4 + 70*y^3*z^4 - 720*y^3*z^3 + 2762*y^3*z^2 - 4676*y^3*z + 2928*y^3 - 42*y^2*z^5 + 530*y^2*z^4 - 2666*y^2*z^3 + 6670*y^2*z^2 - 8240*y^2*z + 3969*y^2 + 14*y*z^6 - 208*y*z^5 + 1286*y*z^4 - 4224*y*z^3 + 7708*y*z^2 - 7314*y*z + 2791*y - 2*z^7 + 34*z^6 - 248*z^5 + 1002*z^4 - 2398*z^3 + 3357*z^2 - 2518*z + 1, 2*x^7 - 14*x^6*y + 14*x^6*z - 34*x^6 + 42*x^5*y^2 - 84*x^5*y*z + 208*x^5*y + 42*x^5*z^2 - 204*x^5*z + 246*x^5 - 70*x^4*y^3 + 210*x^4*y^2*z - 530*x^4*y^2 - 210*x^4*y*z^2 + 1040*x^4*y*z - 1278*x^4*y + 70*x^4*z^3 - 510*x^4*z^2 + 1232*x^4*z - 982*x^4 + 70*x^3*y^4 - 280*x^3*y^3*z + 720*x^3*y^3 + 420*x^3*y^2*z^2 - 2120*x^3*y^2*z + 2654*x^3*y^2 - 280*x^3*y*z^3 + 2080*x^3*y*z^2 - 5120*x^3*y*z + 4160*x^3*y + 70*x^3*z^4 - 680*x^3*z^3 + 2468*x^3*z^2 - 3948*x^3*z + 2330*x^3 - 42*x^2*y^5 + 210*x^2*y^4*z - 550*x^2*y^4 - 420*x^2*y^3*z^2 + 2160*x^2*y^3*z - 2754*x^2*y^3 + 420*x^2*y^2*z^3 - 3180*x^2*y^2*z^2 + 7974*x^2*y^2*z - 6602*x^2*y^2 - 210*x^2*y*z^4 + 2080*x^2*y*z^3 - 7692*x^2*y*z^2 + 12544*x^2*y*z - 7554*x^2*y + 42*x^2*z^5 - 510*x^2*z^4 + 2472*x^2*z^3 - 5952*x^2*z^2 + 7060*x^2*z - 3271*x^2 + 14*x*y^6 - 84*x*y^5*z + 224*x*y^5 + 210*x*y^4*z^2 - 1100*x*y^4*z + 1428*x*y^4 - 280*x*y^3*z^3 + 2160*x*y^3*z^2 - 5516*x*y^3*z + 4652*x*y^3 + 210*x*y^2*z^4 - 2120*x*y^2*z^3 + 7986*x*y^2*z^2 - 13272*x*y^2*z + 8154*x*y^2 - 84*x*y*z^5 + 1040*x*y*z^4 - 5136*x*y*z^3 + 12608*x*y*z^2 - 15268*x*y*z + 7222*x*y + 14*x*z^6 - 204*x*z^5 + 1238*x*z^4 - 3988*x*z^3 + 7130*x*z^2 - 6642*x*z + 2506*x - 2*y^7 + 14*y^6*z - 38*y^6 - 42*y^5*z^2 + 224*y^5*z - 296*y^5 + 70*y^4*z^3 - 550*y^4*z^2 + 1430*y^4*z - 1228*y^4 - 70*y^3*z^4 + 720*y^3*z^3 - 2762*y^3*z^2 + 4676*y^3*z - 2930*y^3 + 42*y^2*z^5 - 530*y^2*z^4 + 2666*y^2*z^3 - 6670*y^2*z^2 + 8246*y^2*z - 3983*y^2 - 14*y*z^6 + 208*y*z^5 - 1286*y*z^4 + 4224*y*z^3 - 7714*y*z^2 + 7342*y*z - 2827*y + 2*z^7 - 34*z^6 + 248*z^5 - 1002*z^4 + 2400*z^3 - 3371*z^2 + 2556*z - 1, 2*x^7 - 14*x^6*y + 14*x^6*z - 34*x^6 + 42*x^5*y^2 - 84*x^5*y*z + 208*x^5*y + 42*x^5*z^2 - 204*x^5*z + 246*x^5 - 70*x^4*y^3 + 210*x^4*y^2*z - 530*x^4*y^2 - 210*x^4*y*z^2 + 1040*x^4*y*z - 1278*x^4*y + 70*x^4*z^3 - 510*x^4*z^2 + 1232*x^4*z - 982*x^4 + 70*x^3*y^4 - 280*x^3*y^3*z + 720*x^3*y^3 + 420*x^3*y^2*z^2 - 2120*x^3*y^2*z + 2654*x^3*y^2 - 280*x^3*y*z^3 + 2080*x^3*y*z^2 - 5120*x^3*y*z + 4160*x^3*y + 70*x^3*z^4 - 680*x^3*z^3 + 2468*x^3*z^2 - 3948*x^3*z + 2330*x^3 - 42*x^2*y^5 + 210*x^2*y^4*z - 550*x^2*y^4 - 420*x^2*y^3*z^2 + 2160*x^2*y^3*z - 2754*x^2*y^3 + 420*x^2*y^2*z^3 - 3180*x^2*y^2*z^2 + 7974*x^2*y^2*z - 6602*x^2*y^2 - 210*x^2*y*z^4 + 2080*x^2*y*z^3 - 7692*x^2*y*z^2 + 12544*x^2*y*z - 7554*x^2*y + 42*x^2*z^5 - 510*x^2*z^4 + 2472*x^2*z^3 - 5952*x^2*z^2 + 7060*x^2*z - 3272*x^2 + 14*x*y^6 - 84*x*y^5*z + 224*x*y^5 + 210*x*y^4*z^2 - 1100*x*y^4*z + 1428*x*y^4 - 280*x*y^3*z^3 + 2160*x*y^3*z^2 - 5516*x*y^3*z + 4652*x*y^3 + 210*x*y^2*z^4 - 2120*x*y^2*z^3 + 7986*x*y^2*z^2 - 13272*x*y^2*z + 8154*x*y^2 - 84*x*y*z^5 + 1040*x*y*z^4 - 5136*x*y*z^3 + 12608*x*y*z^2 - 15268*x*y*z + 7224*x*y + 14*x*z^6 - 204*x*z^5 + 1238*x*z^4 - 3988*x*z^3 + 7130*x*z^2 - 6644*x*z + 2511*x - 2*y^7 + 14*y^6*z - 38*y^6 - 42*y^5*z^2 + 224*y^5*z - 296*y^5 + 70*y^4*z^3 - 550*y^4*z^2 + 1430*y^4*z - 1228*y^4 - 70*y^3*z^4 + 720*y^3*z^3 - 2762*y^3*z^2 + 4676*y^3*z - 2930*y^3 + 42*y^2*z^5 - 530*y^2*z^4 + 2666*y^2*z^3 - 6670*y^2*z^2 + 8246*y^2*z - 3984*y^2 - 14*y*z^6 + 208*y*z^5 - 1286*y*z^4 + 4224*y*z^3 - 7714*y*z^2 + 7344*y*z - 2833*y + 2*z^7 - 34*z^6 + 248*z^5 - 1002*z^4 + 2400*z^3 - 3372*z^2 + 2561*z - 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -1, 1, -1 ], [ -1, 2, -1 ], [ -1, 1, -2 ] ], "source_shift": [ 2, 1, 2 ], "target_matrix": [ [ 1, 1, 2 ], [ -1, -1, -1 ], [ -1, -2, -1 ] ], "target_shift": [ -774, 806, 811 ], "inverse_recipe": { "p": 2, "q": 3, "f": "x**2", "g": "x**3 + x**2 + x", "h": "y**2*z + z", "polynomials": { "f": [ [ [ 2, 0, 0 ], 1 ] ], "g": [ [ [ 3, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "h": [ [ [ 0, 2, 1 ], 1 ], [ [ 0, 0, 1 ], 1 ] ] }, "h_multiplier": 2 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 5.204519497001456 ], "surface_features": [ 3.3816274244431996, 4.183782151277061, 3.0090257420869104, 0.0, 3.5025132646976225, 4786 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B030", "pair": 30, "family": "bijection", "mapping": "F(x,y,z)=(1792*x^7 - 6272*x^6*y + 6272*x^6*z - 8512*x^6 + 9408*x^5*y^2 - 18816*x^5*y*z + 25760*x^5*y + 9408*x^5*z^2 - 25760*x^5*z + 17280*x^5 - 7840*x^4*y^3 + 23520*x^4*y^2*z - 32480*x^4*y^2 - 23520*x^4*y*z^2 + 64960*x^4*y*z - 44208*x^4*y + 7840*x^4*z^3 - 32480*x^4*z^2 + 44208*x^4*z - 19976*x^4 + 3920*x^3*y^4 - 15680*x^3*y^3*z + 21840*x^3*y^3 + 23520*x^3*y^2*z^2 - 65520*x^3*y^2*z + 45216*x^3*y^2 - 15680*x^3*y*z^3 + 65520*x^3*y*z^2 - 90432*x^3*y*z + 41576*x^3*y + 3920*x^3*z^4 - 21840*x^3*z^3 + 45216*x^3*z^2 - 41576*x^3*z + 14574*x^3 - 1176*x^2*y^5 + 5880*x^2*y^4*z - 8260*x^2*y^4 - 11760*x^2*y^3*z^2 + 33040*x^2*y^3*z - 23112*x^2*y^3 + 11760*x^2*y^2*z^3 - 49560*x^2*y^2*z^2 + 69336*x^2*y^2*z - 32442*x^2*y^2 - 5880*x^2*y*z^4 + 33040*x^2*y*z^3 - 69336*x^2*y*z^2 + 64884*x^2*y*z - 23310*x^2*y + 1176*x^2*z^5 - 8260*x^2*z^4 + 23112*x^2*z^3 - 32442*x^2*z^2 + 23254*x^2*z - 6936*x^2 + 196*x*y^6 - 1176*x*y^5*z + 1666*x*y^5 + 2940*x*y^4*z^2 - 8330*x*y^4*z + 5904*x*y^4 - 3920*x*y^3*z^3 + 16660*x*y^3*z^2 - 23616*x*y^3*z + 11248*x*y^3 + 2940*x*y^2*z^4 - 16660*x*y^2*z^3 + 35424*x*y^2*z^2 - 33744*x*y^2*z + 12432*x*y^2 - 1176*x*y*z^5 + 8330*x*y*z^4 - 23616*x*y*z^3 + 33744*x*y*z^2 - 24808*x*y*z + 7678*x*y + 196*x*z^6 - 1666*x*z^5 + 5904*x*z^4 - 11248*x*z^3 + 12376*x*z^2 - 7608*x*z + 2061*x - 14*y^7 + 98*y^6*z - 140*y^6 - 294*y^5*z^2 + 840*y^5*z - 603*y^5 + 490*y^4*z^3 - 2100*y^4*z^2 + 3015*y^4*z - 1462*y^4 - 490*y^3*z^4 + 2800*y^3*z^3 - 6030*y^3*z^2 + 5848*y^3*z - 2212*y^3 + 294*y^2*z^5 - 2100*y^2*z^4 + 6030*y^2*z^3 - 8772*y^2*z^2 + 6622*y^2*z - 2140*y^2 - 98*y*z^6 + 840*y*z^5 - 3015*y*z^4 + 5848*y*z^3 - 6608*y*z^2 + 4238*y*z - 1219*y + 14*z^7 - 140*z^6 + 603*z^5 - 1462*z^4 + 2198*z^3 - 2098*z^2 + 1190*z - 1, 1792*x^7 - 6272*x^6*y + 6272*x^6*z - 8512*x^6 + 9408*x^5*y^2 - 18816*x^5*y*z + 25760*x^5*y + 9408*x^5*z^2 - 25760*x^5*z + 17280*x^5 - 7840*x^4*y^3 + 23520*x^4*y^2*z - 32480*x^4*y^2 - 23520*x^4*y*z^2 + 64960*x^4*y*z - 44208*x^4*y + 7840*x^4*z^3 - 32480*x^4*z^2 + 44208*x^4*z - 19976*x^4 + 3920*x^3*y^4 - 15680*x^3*y^3*z + 21840*x^3*y^3 + 23520*x^3*y^2*z^2 - 65520*x^3*y^2*z + 45216*x^3*y^2 - 15680*x^3*y*z^3 + 65520*x^3*y*z^2 - 90432*x^3*y*z + 41576*x^3*y + 3920*x^3*z^4 - 21840*x^3*z^3 + 45216*x^3*z^2 - 41576*x^3*z + 14574*x^3 - 1176*x^2*y^5 + 5880*x^2*y^4*z - 8260*x^2*y^4 - 11760*x^2*y^3*z^2 + 33040*x^2*y^3*z - 23112*x^2*y^3 + 11760*x^2*y^2*z^3 - 49560*x^2*y^2*z^2 + 69336*x^2*y^2*z - 32442*x^2*y^2 - 5880*x^2*y*z^4 + 33040*x^2*y*z^3 - 69336*x^2*y*z^2 + 64884*x^2*y*z - 23310*x^2*y + 1176*x^2*z^5 - 8260*x^2*z^4 + 23112*x^2*z^3 - 32442*x^2*z^2 + 23254*x^2*z - 6944*x^2 + 196*x*y^6 - 1176*x*y^5*z + 1666*x*y^5 + 2940*x*y^4*z^2 - 8330*x*y^4*z + 5904*x*y^4 - 3920*x*y^3*z^3 + 16660*x*y^3*z^2 - 23616*x*y^3*z + 11248*x*y^3 + 2940*x*y^2*z^4 - 16660*x*y^2*z^3 + 35424*x*y^2*z^2 - 33744*x*y^2*z + 12432*x*y^2 - 1176*x*y*z^5 + 8330*x*y*z^4 - 23616*x*y*z^3 + 33744*x*y*z^2 - 24808*x*y*z + 7686*x*y + 196*x*z^6 - 1666*x*z^5 + 5904*x*z^4 - 11248*x*z^3 + 12376*x*z^2 - 7616*x*z + 2071*x - 14*y^7 + 98*y^6*z - 140*y^6 - 294*y^5*z^2 + 840*y^5*z - 603*y^5 + 490*y^4*z^3 - 2100*y^4*z^2 + 3015*y^4*z - 1462*y^4 - 490*y^3*z^4 + 2800*y^3*z^3 - 6030*y^3*z^2 + 5848*y^3*z - 2212*y^3 + 294*y^2*z^5 - 2100*y^2*z^4 + 6030*y^2*z^3 - 8772*y^2*z^2 + 6622*y^2*z - 2142*y^2 - 98*y*z^6 + 840*y*z^5 - 3015*y*z^4 + 5848*y*z^3 - 6608*y*z^2 + 4242*y*z - 1225*y + 14*z^7 - 140*z^6 + 603*z^5 - 1462*z^4 + 2198*z^3 - 2100*z^2 + 1196*z - 1, -1792*x^7 + 6272*x^6*y - 6272*x^6*z + 8512*x^6 - 9408*x^5*y^2 + 18816*x^5*y*z - 25760*x^5*y - 9408*x^5*z^2 + 25760*x^5*z - 17312*x^5 + 7840*x^4*y^3 - 23520*x^4*y^2*z + 32480*x^4*y^2 + 23520*x^4*y*z^2 - 64960*x^4*y*z + 44288*x^4*y - 7840*x^4*z^3 + 32480*x^4*z^2 - 44288*x^4*z + 20072*x^4 - 3920*x^3*y^4 + 15680*x^3*y^3*z - 21840*x^3*y^3 - 23520*x^3*y^2*z^2 + 65520*x^3*y^2*z - 45296*x^3*y^2 + 15680*x^3*y*z^3 - 65520*x^3*y*z^2 + 90592*x^3*y*z - 41768*x^3*y - 3920*x^3*z^4 + 21840*x^3*z^3 - 45296*x^3*z^2 + 41768*x^3*z - 14686*x^3 + 1176*x^2*y^5 - 5880*x^2*y^4*z + 8260*x^2*y^4 + 11760*x^2*y^3*z^2 - 33040*x^2*y^3*z + 23152*x^2*y^3 - 11760*x^2*y^2*z^3 + 49560*x^2*y^2*z^2 - 69456*x^2*y^2*z + 32586*x^2*y^2 + 5880*x^2*y*z^4 - 33040*x^2*y*z^3 + 69456*x^2*y*z^2 - 65172*x^2*y*z + 23478*x^2*y - 1176*x^2*z^5 + 8260*x^2*z^4 - 23152*x^2*z^3 + 32586*x^2*z^2 - 23422*x^2*z + 7008*x^2 - 196*x*y^6 + 1176*x*y^5*z - 1666*x*y^5 - 2940*x*y^4*z^2 + 8330*x*y^4*z - 5914*x*y^4 + 3920*x*y^3*z^3 - 16660*x*y^3*z^2 + 23656*x*y^3*z - 11296*x*y^3 - 2940*x*y^2*z^4 + 16660*x*y^2*z^3 - 35484*x*y^2*z^2 + 33888*x*y^2*z - 12516*x*y^2 + 1176*x*y*z^5 - 8330*x*y*z^4 + 23656*x*y*z^3 - 33888*x*y*z^2 + 24976*x*y*z - 7750*x*y - 196*x*z^6 + 1666*x*z^5 - 5914*x*z^4 + 11296*x*z^3 - 12460*x*z^2 + 7680*x*z - 2092*x + 14*y^7 - 98*y^6*z + 140*y^6 + 294*y^5*z^2 - 840*y^5*z + 604*y^5 - 490*y^4*z^3 + 2100*y^4*z^2 - 3020*y^4*z + 1468*y^4 + 490*y^3*z^4 - 2800*y^3*z^3 + 6040*y^3*z^2 - 5872*y^3*z + 2226*y^3 - 294*y^2*z^5 + 2100*y^2*z^4 - 6040*y^2*z^3 + 8808*y^2*z^2 - 6664*y^2*z + 2158*y^2 + 98*y*z^6 - 840*y*z^5 + 3020*y*z^4 - 5872*y*z^3 + 6650*y*z^2 - 4274*y*z + 1237*y - 14*z^7 + 140*z^6 - 604*z^5 + 1468*z^4 - 2212*z^3 + 2116*z^2 - 1207*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -2, 1, -1 ], [ -3, 3, -2 ], [ -1, 1, -1 ] ], "source_shift": [ 1, 2, 1 ], "target_matrix": [ [ -1, -1, -2 ], [ -1, -1, -1 ], [ 1, 2, 1 ] ], "target_shift": [ 304, 308, -312 ], "inverse_recipe": { "p": 5, "q": 2, "f": "x**5 + x**4", "g": "x**2", "h": "y*z + z**3 + z**2 + z", "polynomials": { "f": [ [ [ 5, 0, 0 ], 1 ], [ [ 4, 0, 0 ], 1 ] ], "g": [ [ [ 2, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 0, 2 ], 1 ], [ [ 0, 0, 3 ], 1 ], [ [ 0, 0, 1 ], 1 ] ] }, "h_multiplier": 14 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 5.170855353000661 ], "surface_features": [ 4.196088405005884, 4.957089847685009, 3.8943160626844384, 0.0, 4.2699904980286, 5131 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B154", "pair": 154, "family": "bijection", "mapping": "F(x,y,z)=(768*x^7 - 2688*x^6*y + 2688*x^6*z - 5376*x^6 + 4032*x^5*y^2 - 8064*x^5*y*z + 16128*x^5*y + 4032*x^5*z^2 - 16128*x^5*z + 16608*x^5 - 3360*x^4*y^3 + 10080*x^4*y^2*z - 20160*x^4*y^2 - 10080*x^4*y*z^2 + 40320*x^4*y*z - 41568*x^4*y + 3360*x^4*z^3 - 20160*x^4*z^2 + 41664*x^4*z - 29488*x^4 + 1680*x^3*y^4 - 6720*x^3*y^3*z + 13440*x^3*y^3 + 10080*x^3*y^2*z^2 - 40320*x^3*y^2*z + 41616*x^3*y^2 - 6720*x^3*y*z^3 + 40320*x^3*y*z^2 - 83424*x^3*y*z + 59216*x^3*y + 1680*x^3*z^4 - 13440*x^3*z^3 + 41808*x^3*z^2 - 59648*x^3*z + 32520*x^3 - 504*x^2*y^5 + 2520*x^2*y^4*z - 5040*x^2*y^4 - 5040*x^2*y^3*z^2 + 20160*x^2*y^3*z - 20832*x^2*y^3 + 5040*x^2*y^2*z^3 - 30240*x^2*y^2*z^2 + 62640*x^2*y^2*z - 44592*x^2*y^2 - 2520*x^2*y*z^4 + 20160*x^2*y*z^3 - 62784*x^2*y*z^2 + 89832*x^2*y*z - 49224*x^2*y + 504*x^2*z^5 - 5040*x^2*z^4 + 20976*x^2*z^3 - 45240*x^2*z^2 + 49968*x^2*z - 22264*x^2 + 84*x*y^6 - 504*x*y^5*z + 1008*x*y^5 + 1260*x*y^4*z^2 - 5040*x*y^4*z + 5214*x*y^4 - 1680*x*y^3*z^3 + 10080*x*y^3*z^2 - 20904*x*y^3*z + 14924*x*y^3 + 1260*x*y^2*z^4 - 10080*x*y^2*z^3 + 31428*x*y^2*z^2 - 45096*x*y^2*z + 24834*x*y^2 - 504*x*y*z^5 + 5040*x*y*z^4 - 21000*x*y*z^3 + 45420*x*y*z^2 - 50412*x*y*z + 22648*x*y + 84*x*z^6 - 1008*x*z^5 + 5262*x*z^4 - 15248*x*z^3 + 25578*x*z^2 - 23254*x*z + 8752*x - 6*y^7 + 42*y^6*z - 84*y^6 - 126*y^5*z^2 + 504*y^5*z - 522*y^5 + 210*y^4*z^3 - 1260*y^4*z^2 + 2616*y^4*z - 1873*y^4 - 210*y^3*z^4 + 1680*y^3*z^3 - 5244*y^3*z^2 + 7546*y^3*z - 4176*y^3 + 126*y^2*z^5 - 1260*y^2*z^4 + 5256*y^2*z^3 - 11400*y^2*z^2 + 12714*y^2*z - 5761*y^2 - 42*y*z^6 + 504*y*z^5 - 2634*y*z^4 + 7654*y*z^3 - 12900*y*z^2 + 11834*y*z - 4511*y + 6*z^7 - 84*z^6 + 528*z^5 - 1927*z^4 + 4362*z^3 - 6079*z^2 + 4708*z - 1, -768*x^7 + 2688*x^6*y - 2688*x^6*z + 5376*x^6 - 4032*x^5*y^2 + 8064*x^5*y*z - 16128*x^5*y - 4032*x^5*z^2 + 16128*x^5*z - 16608*x^5 + 3360*x^4*y^3 - 10080*x^4*y^2*z + 20160*x^4*y^2 + 10080*x^4*y*z^2 - 40320*x^4*y*z + 41568*x^4*y - 3360*x^4*z^3 + 20160*x^4*z^2 - 41664*x^4*z + 29504*x^4 - 1680*x^3*y^4 + 6720*x^3*y^3*z - 13440*x^3*y^3 - 10080*x^3*y^2*z^2 + 40320*x^3*y^2*z - 41616*x^3*y^2 + 6720*x^3*y*z^3 - 40320*x^3*y*z^2 + 83424*x^3*y*z - 59248*x^3*y - 1680*x^3*z^4 + 13440*x^3*z^3 - 41808*x^3*z^2 + 59680*x^3*z - 32584*x^3 + 504*x^2*y^5 - 2520*x^2*y^4*z + 5040*x^2*y^4 + 5040*x^2*y^3*z^2 - 20160*x^2*y^3*z + 20832*x^2*y^3 - 5040*x^2*y^2*z^3 + 30240*x^2*y^2*z^2 - 62640*x^2*y^2*z + 44616*x^2*y^2 + 2520*x^2*y*z^4 - 20160*x^2*y*z^3 + 62784*x^2*y*z^2 - 89880*x^2*y*z + 49320*x^2*y - 504*x^2*z^5 + 5040*x^2*z^4 - 20976*x^2*z^3 + 45264*x^2*z^2 - 50064*x^2*z + 22364*x^2 - 84*x*y^6 + 504*x*y^5*z - 1008*x*y^5 - 1260*x*y^4*z^2 + 5040*x*y^4*z - 5214*x*y^4 + 1680*x*y^3*z^3 - 10080*x*y^3*z^2 + 20904*x*y^3*z - 14932*x*y^3 - 1260*x*y^2*z^4 + 10080*x*y^2*z^3 - 31428*x*y^2*z^2 + 45120*x*y^2*z - 24882*x*y^2 + 504*x*y*z^5 - 5040*x*y*z^4 + 21000*x*y*z^3 - 45444*x*y*z^2 + 50508*x*y*z - 22748*x*y - 84*x*z^6 + 1008*x*z^5 - 5262*x*z^4 + 15256*x*z^3 - 25626*x*z^2 + 23354*x*z - 8826*x + 6*y^7 - 42*y^6*z + 84*y^6 + 126*y^5*z^2 - 504*y^5*z + 522*y^5 - 210*y^4*z^3 + 1260*y^4*z^2 - 2616*y^4*z + 1874*y^4 + 210*y^3*z^4 - 1680*y^3*z^3 + 5244*y^3*z^2 - 7550*y^3*z + 4184*y^3 - 126*y^2*z^5 + 1260*y^2*z^4 - 5256*y^2*z^3 + 11406*y^2*z^2 - 12738*y^2*z + 5786*y^2 + 42*y*z^6 - 504*y*z^5 + 2634*y*z^4 - 7658*y*z^3 + 12924*y*z^2 - 11884*y*z + 4549*y - 6*z^7 + 84*z^6 - 528*z^5 + 1928*z^4 - 4370*z^3 + 6104*z^2 - 4747*z + 1, 768*x^7 - 2688*x^6*y + 2688*x^6*z - 5376*x^6 + 4032*x^5*y^2 - 8064*x^5*y*z + 16128*x^5*y + 4032*x^5*z^2 - 16128*x^5*z + 16608*x^5 - 3360*x^4*y^3 + 10080*x^4*y^2*z - 20160*x^4*y^2 - 10080*x^4*y*z^2 + 40320*x^4*y*z - 41568*x^4*y + 3360*x^4*z^3 - 20160*x^4*z^2 + 41664*x^4*z - 29488*x^4 + 1680*x^3*y^4 - 6720*x^3*y^3*z + 13440*x^3*y^3 + 10080*x^3*y^2*z^2 - 40320*x^3*y^2*z + 41616*x^3*y^2 - 6720*x^3*y*z^3 + 40320*x^3*y*z^2 - 83424*x^3*y*z + 59216*x^3*y + 1680*x^3*z^4 - 13440*x^3*z^3 + 41808*x^3*z^2 - 59648*x^3*z + 32504*x^3 - 504*x^2*y^5 + 2520*x^2*y^4*z - 5040*x^2*y^4 - 5040*x^2*y^3*z^2 + 20160*x^2*y^3*z - 20832*x^2*y^3 + 5040*x^2*y^2*z^3 - 30240*x^2*y^2*z^2 + 62640*x^2*y^2*z - 44592*x^2*y^2 - 2520*x^2*y*z^4 + 20160*x^2*y*z^3 - 62784*x^2*y*z^2 + 89832*x^2*y*z - 49200*x^2*y + 504*x^2*z^5 - 5040*x^2*z^4 + 20976*x^2*z^3 - 45240*x^2*z^2 + 49944*x^2*z - 22216*x^2 + 84*x*y^6 - 504*x*y^5*z + 1008*x*y^5 + 1260*x*y^4*z^2 - 5040*x*y^4*z + 5214*x*y^4 - 1680*x*y^3*z^3 + 10080*x*y^3*z^2 - 20904*x*y^3*z + 14924*x*y^3 + 1260*x*y^2*z^4 - 10080*x*y^2*z^3 + 31428*x*y^2*z^2 - 45096*x*y^2*z + 24822*x*y^2 - 504*x*y*z^5 + 5040*x*y*z^4 - 21000*x*y*z^3 + 45420*x*y*z^2 - 50388*x*y*z + 22600*x*y + 84*x*z^6 - 1008*x*z^5 + 5262*x*z^4 - 15248*x*z^3 + 25566*x*z^2 - 23206*x*z + 8698*x - 6*y^7 + 42*y^6*z - 84*y^6 - 126*y^5*z^2 + 504*y^5*z - 522*y^5 + 210*y^4*z^3 - 1260*y^4*z^2 + 2616*y^4*z - 1873*y^4 - 210*y^3*z^4 + 1680*y^3*z^3 - 5244*y^3*z^2 + 7546*y^3*z - 4174*y^3 + 126*y^2*z^5 - 1260*y^2*z^4 + 5256*y^2*z^3 - 11400*y^2*z^2 + 12708*y^2*z - 5749*y^2 - 42*y*z^6 + 504*y*z^5 - 2634*y*z^4 + 7654*y*z^3 - 12894*y*z^2 + 11810*y*z - 4483*y + 6*z^7 - 84*z^6 + 528*z^5 - 1927*z^4 + 4360*z^3 - 6067*z^2 + 4678*z - 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -2, 1, -1 ], [ -2, 2, -3 ], [ -1, 1, -2 ] ], "source_shift": [ 2, 1, 2 ], "target_matrix": [ [ -1, -1, -1 ], [ 1, 2, 1 ], [ -1, -1, -2 ] ], "target_shift": [ 1510, -1531, 1486 ], "inverse_recipe": { "p": 4, "q": 3, "f": "x**4 + x**2", "g": "x**3 + x", "h": "y*z", "polynomials": { "f": [ [ [ 4, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "g": [ [ [ 3, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ] ] }, "h_multiplier": 6 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 5.090848244999506 ], "surface_features": [ 4.183932803319563, 4.953663063747091, 3.7727982887371274, 0.0, 4.2774751917799065, 5104 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B045", "pair": 45, "family": "bijection", "mapping": "F(x,y,z)=(28*x^7 - 196*x^6*y + 196*x^6*z - 420*x^6 + 588*x^5*y^2 - 1176*x^5*y*z + 2520*x^5*y + 588*x^5*z^2 - 2520*x^5*z + 2716*x^5 - 980*x^4*y^3 + 2940*x^4*y^2*z - 6300*x^4*y^2 - 2940*x^4*y*z^2 + 12600*x^4*y*z - 13580*x^4*y + 980*x^4*z^3 - 6300*x^4*z^2 + 13608*x^4*z - 9914*x^4 + 980*x^3*y^4 - 3920*x^3*y^3*z + 8400*x^3*y^3 + 5880*x^3*y^2*z^2 - 25200*x^3*y^2*z + 27160*x^3*y^2 - 3920*x^3*y*z^3 + 25200*x^3*y*z^2 - 54432*x^3*y*z + 39684*x^3*y + 980*x^3*z^4 - 8400*x^3*z^3 + 27272*x^3*z^2 - 39936*x^3*z + 22160*x^3 - 588*x^2*y^5 + 2940*x^2*y^4*z - 6300*x^2*y^4 - 5880*x^2*y^3*z^2 + 25200*x^2*y^3*z - 27160*x^2*y^3 + 5880*x^2*y^2*z^3 - 37800*x^2*y^2*z^2 + 81648*x^2*y^2*z - 59568*x^2*y^2 - 2940*x^2*y*z^4 + 25200*x^2*y*z^3 - 81816*x^2*y*z^2 + 119892*x^2*y*z - 66648*x^2*y + 588*x^2*z^5 - 6300*x^2*z^4 + 27328*x^2*z^3 - 60324*x^2*z^2 + 67488*x^2*z - 30404*x^2 + 196*x*y^6 - 1176*x*y^5*z + 2520*x*y^5 + 2940*x*y^4*z^2 - 12600*x*y^4*z + 13580*x*y^4 - 3920*x*y^3*z^3 + 25200*x*y^3*z^2 - 54432*x*y^3*z + 39740*x*y^3 + 2940*x*y^2*z^4 - 25200*x*y^2*z^3 + 81816*x*y^2*z^2 - 119976*x*y^2*z + 66816*x*y^2 - 1176*x*y*z^5 + 12600*x*y*z^4 - 54656*x*y*z^3 + 120732*x*y*z^2 - 135312*x*y*z + 61172*x*y + 196*x*z^6 - 2520*x*z^5 + 13692*x*z^4 - 40496*x*z^3 + 68496*x*z^2 - 62488*x*z + 23734*x - 28*y^7 + 196*y^6*z - 420*y^6 - 588*y^5*z^2 + 2520*y^5*z - 2716*y^5 + 980*y^4*z^3 - 6300*y^4*z^2 + 13608*y^4*z - 9942*y^4 - 980*y^3*z^4 + 8400*y^3*z^3 - 27272*y^3*z^2 + 40020*y^3*z - 22328*y^3 + 588*y^2*z^5 - 6300*y^2*z^4 + 27328*y^2*z^3 - 60408*y^2*z^2 + 67824*y^2*z - 30768*y^2 - 196*y*z^6 + 2520*y*z^5 - 13692*y*z^4 + 40524*y*z^3 - 68664*y*z^2 + 62880*y*z - 24016*y + 28*z^7 - 420*z^6 + 2744*z^5 - 10194*z^4 + 23168*z^3 - 32112*z^2 + 24822*z - 1, -14*x^7 + 98*x^6*y - 98*x^6*z + 210*x^6 - 294*x^5*y^2 + 588*x^5*y*z - 1260*x^5*y - 294*x^5*z^2 + 1260*x^5*z - 1358*x^5 + 490*x^4*y^3 - 1470*x^4*y^2*z + 3150*x^4*y^2 + 1470*x^4*y*z^2 - 6300*x^4*y*z + 6790*x^4*y - 490*x^4*z^3 + 3150*x^4*z^2 - 6804*x^4*z + 4958*x^4 - 490*x^3*y^4 + 1960*x^3*y^3*z - 4200*x^3*y^3 - 2940*x^3*y^2*z^2 + 12600*x^3*y^2*z - 13580*x^3*y^2 + 1960*x^3*y*z^3 - 12600*x^3*y*z^2 + 27216*x^3*y*z - 19846*x^3*y - 490*x^3*z^4 + 4200*x^3*z^3 - 13636*x^3*z^2 + 19972*x^3*z - 11090*x^3 + 294*x^2*y^5 - 1470*x^2*y^4*z + 3150*x^2*y^4 + 2940*x^2*y^3*z^2 - 12600*x^2*y^3*z + 13580*x^2*y^3 - 2940*x^2*y^2*z^3 + 18900*x^2*y^2*z^2 - 40824*x^2*y^2*z + 29790*x^2*y^2 + 1470*x^2*y*z^4 - 12600*x^2*y*z^3 + 40908*x^2*y*z^2 - 59958*x^2*y*z + 33354*x^2*y - 294*x^2*z^5 + 3150*x^2*z^4 - 13664*x^2*z^3 + 30168*x^2*z^2 - 33774*x^2*z + 15238*x^2 - 98*x*y^6 + 588*x*y^5*z - 1260*x*y^5 - 1470*x*y^4*z^2 + 6300*x*y^4*z - 6790*x*y^4 + 1960*x*y^3*z^3 - 12600*x*y^3*z^2 + 27216*x*y^3*z - 19874*x*y^3 - 1470*x*y^2*z^4 + 12600*x*y^2*z^3 - 40908*x*y^2*z^2 + 60000*x*y^2*z - 33438*x*y^2 + 588*x*y*z^5 - 6300*x*y*z^4 + 27328*x*y*z^3 - 60378*x*y*z^2 + 67716*x*y*z - 30658*x*y - 98*x*z^6 + 1260*x*z^5 - 6846*x*z^4 + 20252*x*z^3 - 34278*x*z^2 + 31316*x*z - 11927*x + 14*y^7 - 98*y^6*z + 210*y^6 + 294*y^5*z^2 - 1260*y^5*z + 1358*y^5 - 490*y^4*z^3 + 3150*y^4*z^2 - 6804*y^4*z + 4972*y^4 + 490*y^3*z^4 - 4200*y^3*z^3 + 13636*y^3*z^2 - 20014*y^3*z + 11174*y^3 - 294*y^2*z^5 + 3150*y^2*z^4 - 13664*y^2*z^3 + 30210*y^2*z^2 - 33942*y^2*z + 15420*y^2 + 98*y*z^6 - 1260*y*z^5 + 6846*y*z^4 - 20266*y*z^3 + 34362*y*z^2 - 31512*y*z + 12069*y - 14*z^7 + 210*z^6 - 1372*z^5 + 5098*z^4 - 11594*z^3 + 16092*z^2 - 12473*z + 1, 14*x^7 - 98*x^6*y + 98*x^6*z - 210*x^6 + 294*x^5*y^2 - 588*x^5*y*z + 1260*x^5*y + 294*x^5*z^2 - 1260*x^5*z + 1358*x^5 - 490*x^4*y^3 + 1470*x^4*y^2*z - 3150*x^4*y^2 - 1470*x^4*y*z^2 + 6300*x^4*y*z - 6790*x^4*y + 490*x^4*z^3 - 3150*x^4*z^2 + 6804*x^4*z - 4957*x^4 + 490*x^3*y^4 - 1960*x^3*y^3*z + 4200*x^3*y^3 + 2940*x^3*y^2*z^2 - 12600*x^3*y^2*z + 13580*x^3*y^2 - 1960*x^3*y*z^3 + 12600*x^3*y*z^2 - 27216*x^3*y*z + 19842*x^3*y + 490*x^3*z^4 - 4200*x^3*z^3 + 13636*x^3*z^2 - 19968*x^3*z + 11079*x^3 - 294*x^2*y^5 + 1470*x^2*y^4*z - 3150*x^2*y^4 - 2940*x^2*y^3*z^2 + 12600*x^2*y^3*z - 13580*x^2*y^3 + 2940*x^2*y^2*z^3 - 18900*x^2*y^2*z^2 + 40824*x^2*y^2*z - 29784*x^2*y^2 - 1470*x^2*y*z^4 + 12600*x^2*y*z^3 - 40908*x^2*y*z^2 + 59946*x^2*y*z - 33321*x^2*y + 294*x^2*z^5 - 3150*x^2*z^4 + 13664*x^2*z^3 - 30162*x^2*z^2 + 33741*x^2*z - 15196*x^2 + 98*x*y^6 - 588*x*y^5*z + 1260*x*y^5 + 1470*x*y^4*z^2 - 6300*x*y^4*z + 6790*x*y^4 - 1960*x*y^3*z^3 + 12600*x*y^3*z^2 - 27216*x*y^3*z + 19870*x*y^3 + 1470*x*y^2*z^4 - 12600*x*y^2*z^3 + 40908*x*y^2*z^2 - 59988*x*y^2*z + 33405*x*y^2 - 588*x*y*z^5 + 6300*x*y*z^4 - 27328*x*y*z^3 + 60366*x*y*z^2 - 67650*x*y*z + 30574*x*y + 98*x*z^6 - 1260*x*z^5 + 6846*x*z^4 - 20248*x*z^3 + 34245*x*z^2 - 31232*x*z + 11854*x - 14*y^7 + 98*y^6*z - 210*y^6 - 294*y^5*z^2 + 1260*y^5*z - 1358*y^5 + 490*y^4*z^3 - 3150*y^4*z^2 + 6804*y^4*z - 4971*y^4 - 490*y^3*z^4 + 4200*y^3*z^3 - 13636*y^3*z^2 + 20010*y^3*z - 11163*y^3 + 294*y^2*z^5 - 3150*y^2*z^4 + 13664*y^2*z^3 - 30204*y^2*z^2 + 33909*y^2*z - 15378*y^2 - 98*y*z^6 + 1260*y*z^5 - 6846*y*z^4 + 20262*y*z^3 - 34329*y*z^2 + 31428*y*z - 11995*y + 14*z^7 - 210*z^6 + 1372*z^5 - 5097*z^4 + 11583*z^3 - 16050*z^2 + 12397*z - 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -1, 1, -1 ], [ -2, 3, -3 ], [ -1, 1, -2 ] ], "source_shift": [ 2, 1, 1 ], "target_matrix": [ [ -2, -2, -3 ], [ 1, 2, 1 ], [ -1, -1, -2 ] ], "target_shift": [ 8067, -4069, 4024 ], "inverse_recipe": { "p": 4, "q": 3, "f": "x**4 + x**3 + x", "g": "x**3", "h": "y*z + y + 2*z", "polynomials": { "f": [ [ [ 4, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ], [ [ 3, 0, 0 ], 1 ] ], "g": [ [ [ 3, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 0, 1 ], 2 ], [ [ 0, 1, 0 ], 1 ] ] }, "h_multiplier": 14 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0970263079980214 ], "surface_features": [ 4.192302868771843, 5.131336313247189, 3.7993405494535817, 0.0, 4.331590804234297, 5103 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C223", "pair": 223, "family": "counterexample", "mapping": "F(x,y,z)=(-8*x^7 + 80*x^6*y - 68*x^6*z + 112*x^6 - 336*x^5*y^2 + 588*x^5*y*z - 936*x^5*y - 246*x^5*z^2 + 852*x^5*z - 486*x^5 + 768*x^4*y^3 - 2076*x^4*y^2*z + 3192*x^4*y^2 + 1788*x^4*y*z^2 - 5988*x^4*y*z + 3348*x^4*y - 491*x^4*z^3 + 2688*x^4*z^2 - 3123*x^4*z + 1056*x^4 - 1032*x^3*y^4 + 3828*x^3*y^3*z - 5688*x^3*y^3 - 5094*x^3*y^2*z^2 + 16476*x^3*y^2*z - 9030*x^3*y^2 + 2879*x^3*y*z^3 - 15258*x^3*y*z^2 + 17379*x^3*y*z - 5726*x^3*y - 584*x^3*z^4 + 4501*x^3*z^3 - 7989*x^3*z^2 + 5493*x^3*z - 1309*x^3 + 816*x^2*y^5 - 3888*x^2*y^4*z + 5592*x^2*y^4 + 7104*x^2*y^3*z^2 - 22188*x^2*y^3*z + 11928*x^2*y^3 - 6204*x^2*y^2*z^3 + 31770*x^2*y^2*z^2 - 35472*x^2*y^2*z + 11382*x^2*y^2 + 2589*x^2*y*z^4 - 19353*x^2*y*z^3 + 33684*x^2*y*z^2 - 22569*x^2*y*z + 5205*x^2*y - 414*x^2*z^5 + 4218*x^2*z^4 - 10167*x^2*z^3 + 10662*x^2*z^2 - 5172*x^2*z + 952*x^2 - 352*x*y^6 + 2064*x*y^5*z - 2880*x*y^5 - 4848*x*y^4*z^2 + 14640*x*y^4*z - 7728*x*y^4 + 5816*x*y^3*z^3 - 28752*x*y^3*z^2 + 31488*x*y^3*z - 9848*x*y^3 - 3750*x*y^2*z^4 + 27120*x*y^2*z^3 - 46272*x*y^2*z^2 + 30180*x*y^2*z - 6732*x*y^2 + 1233*x*y*z^5 - 12216*x*y*z^4 + 28884*x*y*z^3 - 29526*x*y*z^2 + 13860*x*y*z - 2444*x*y - 162*x*z^6 + 2097*x*z^5 - 6435*x*z^4 + 9149*x*z^3 - 6777*x*z^2 + 2547*x*z - 383*x + 64*y^7 - 448*y^6*z + 608*y^6 + 1296*y^5*z^2 - 3792*y^5*z + 1968*y^5 - 2000*y^4*z^3 + 9552*y^4*z^2 - 10272*y^4*z + 3136*y^4 + 1772*y^3*z^4 - 12376*y^3*z^3 + 20712*y^3*z^2 - 13160*y^3*z + 2840*y^3 - 900*y^2*z^5 + 8646*y^2*z^4 - 20040*y^2*z^3 + 19932*y^2*z^2 - 9048*y^2*z + 1528*y^2 + 243*y*z^6 - 3069*y*z^5 + 9243*y*z^4 - 12818*y*z^3 + 9186*y*z^2 - 3306*y*z + 478*y - 27*z^7 + 432*z^6 - 1620*z^5 + 2927*z^4 - 2944*z^3 + 1695*z^2 - 518*z + 1, 16*x^7 - 160*x^6*y + 136*x^6*z - 212*x^6 + 672*x^5*y^2 - 1176*x^5*y*z + 1764*x^5*y + 492*x^5*z^2 - 1620*x^5*z + 828*x^5 - 1536*x^4*y^3 + 4152*x^4*y^2*z - 5988*x^4*y^2 - 3576*x^4*y*z^2 + 11340*x^4*y*z - 5652*x^4*y + 982*x^4*z^3 - 5133*x^4*z^2 + 5352*x^4*z - 1680*x^4 + 2064*x^3*y^4 - 7656*x^3*y^3*z + 10620*x^3*y^3 + 10188*x^3*y^2*z^2 - 31068*x^3*y^2*z + 15108*x^3*y^2 - 5758*x^3*y*z^3 + 29031*x^3*y*z^2 - 29508*x^3*y*z + 9016*x^3*y + 1168*x^3*z^4 - 8630*x^3*z^3 + 13773*x^3*z^2 - 8790*x^3*z + 2000*x^3 - 1632*x^2*y^5 + 7776*x^2*y^4*z - 10392*x^2*y^4 - 14208*x^2*y^3*z^2 + 41652*x^2*y^3*z - 19788*x^2*y^3 + 12408*x^2*y^2*z^3 - 60210*x^2*y^2*z^2 + 59676*x^2*y^2*z - 17748*x^2*y^2 - 5178*x^2*y*z^4 + 36987*x^2*y*z^3 - 57531*x^2*y*z^2 + 35730*x^2*y*z - 7884*x^2*y + 828*x^2*z^5 - 8118*x^2*z^4 + 17634*x^2*z^3 - 17166*x^2*z^2 + 7938*x^2*z - 1433*x^2 + 704*x*y^6 - 4128*x*y^5*z + 5328*x*y^5 + 9696*x*y^4*z^2 - 27360*x*y^4*z + 12720*x*y^4 - 11632*x*y^3*z^3 + 54264*x*y^3*z^2 - 52512*x*y^3*z + 15220*x*y^3 + 7500*x*y^2*z^4 - 51648*x*y^2*z^3 + 78288*x*y^2*z^2 - 47292*x*y^2*z + 10128*x*y^2 - 2466*x*y*z^5 + 23445*x*y*z^4 - 49632*x*y*z^3 + 47013*x*y*z^2 - 21066*x*y*z + 3667*x*y + 324*x*z^6 - 4050*x*z^5 + 11229*x*z^4 - 14824*x*z^3 + 10455*x*z^2 - 3840*x*z + 575*x - 128*y^7 + 896*y^6*z - 1120*y^6 - 2592*y^5*z^2 + 7056*y^5*z - 3216*y^5 + 4000*y^4*z^3 - 17952*y^4*z^2 + 16992*y^4*z - 4808*y^4 - 3544*y^3*z^4 + 23480*y^3*z^3 - 34728*y^3*z^2 + 20428*y^3*z - 4252*y^3 + 1800*y^2*z^5 - 16542*y^2*z^4 + 34104*y^2*z^3 - 31398*y^2*z^2 + 13644*y^2*z - 2294*y^2 - 486*y*z^6 + 5913*y*z^5 - 15981*y*z^4 + 20533*y*z^3 - 14019*y*z^2 + 4959*y*z - 717*y + 54*z^7 - 837*z^6 + 2844*z^5 - 4774*z^4 + 4568*z^3 - 2562*z^2 + 778*z - 1, 8*x^7 - 80*x^6*y + 68*x^6*z - 112*x^6 + 336*x^5*y^2 - 588*x^5*y*z + 936*x^5*y + 246*x^5*z^2 - 852*x^5*z + 486*x^5 - 768*x^4*y^3 + 2076*x^4*y^2*z - 3192*x^4*y^2 - 1788*x^4*y*z^2 + 5988*x^4*y*z - 3348*x^4*y + 491*x^4*z^3 - 2688*x^4*z^2 + 3123*x^4*z - 1057*x^4 + 1032*x^3*y^4 - 3828*x^3*y^3*z + 5688*x^3*y^3 + 5094*x^3*y^2*z^2 - 16476*x^3*y^2*z + 9030*x^3*y^2 - 2879*x^3*y*z^3 + 15258*x^3*y*z^2 - 17379*x^3*y*z + 5733*x^3*y + 584*x^3*z^4 - 4501*x^3*z^3 + 7989*x^3*z^2 - 5497*x^3*z + 1320*x^3 - 816*x^2*y^5 + 3888*x^2*y^4*z - 5592*x^2*y^4 - 7104*x^2*y^3*z^2 + 22188*x^2*y^3*z - 11928*x^2*y^3 + 6204*x^2*y^2*z^3 - 31770*x^2*y^2*z^2 + 35472*x^2*y^2*z - 11400*x^2*y^2 - 2589*x^2*y*z^4 + 19353*x^2*y*z^3 - 33684*x^2*y*z^2 + 22590*x^2*y*z - 5262*x^2*y + 414*x^2*z^5 - 4218*x^2*z^4 + 10167*x^2*z^3 - 10668*x^2*z^2 + 5208*x^2*z - 976*x^2 + 352*x*y^6 - 2064*x*y^5*z + 2880*x*y^5 + 4848*x*y^4*z^2 - 14640*x*y^4*z + 7728*x*y^4 - 5816*x*y^3*z^3 + 28752*x*y^3*z^2 - 31488*x*y^3*z + 9868*x*y^3 + 3750*x*y^2*z^4 - 27120*x*y^2*z^3 + 46272*x*y^2*z^2 - 30216*x*y^2*z + 6828*x*y^2 - 1233*x*y*z^5 + 12216*x*y*z^4 - 28884*x*y*z^3 + 29547*x*y*z^2 - 13986*x*y*z + 2525*x*y + 162*x*z^6 - 2097*x*z^5 + 6435*x*z^4 - 9153*x*z^3 + 6816*x*z^2 - 2601*x*z + 400*x - 64*y^7 + 448*y^6*z - 608*y^6 - 1296*y^5*z^2 + 3792*y^5*z - 1968*y^5 + 2000*y^4*z^3 - 9552*y^4*z^2 + 10272*y^4*z - 3144*y^4 - 1772*y^3*z^4 + 12376*y^3*z^3 - 20712*y^3*z^2 + 13180*y^3*z - 2892*y^3 + 900*y^2*z^5 - 8646*y^2*z^4 + 20040*y^2*z^3 - 19950*y^2*z^2 + 9156*y^2*z - 1594*y^2 - 243*y*z^6 + 3069*y*z^5 - 9243*y*z^4 + 12825*y*z^3 - 9255*y*z^2 + 3399*y*z - 505*y + 27*z^7 - 432*z^6 + 1620*z^5 - 2928*z^4 + 2958*z^3 - 1725*z^2 + 538*z - 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -1, 2, -1 ], [ -2, 2, -3 ], [ -1, 1, -1 ] ], "source_shift": [ 1, 1, 2 ], "target_matrix": [ [ 1, 1, 2 ], [ -2, -1, -1 ], [ -1, -1, -1 ] ], "target_shift": [ -69, 102, 72 ], "collision_preimages": [ [ "7", "5/4", "-7/2" ], [ "-41/2", "-9/2", "23/2" ], [ "-39/2", "-13/2", "17/2" ] ], "collision_image": [ "-277/4", "205/2", "289/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0573795950003841 ], "surface_features": [ 3.976476075777356, 4.893695198415863, 3.6837672614253116, 0.0, 4.089460841662666, 5059 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C210", "pair": 210, "family": "counterexample", "mapping": "F(x,y,z)=(-2*x^7 + 21*x^6*y - 15*x^6*z + 35*x^6 - 93*x^5*y^2 + 135*x^5*y*z - 315*x^5*y - 48*x^5*z^2 + 225*x^5*z - 240*x^5 + 225*x^4*y^3 - 498*x^4*y^2*z + 1164*x^4*y^2 + 360*x^4*y*z^2 - 1686*x^4*y*z + 1824*x^4*y - 85*x^4*z^3 + 600*x^4*z^2 - 1284*x^4*z + 863*x^4 - 321*x^3*y^4 + 963*x^3*y^3*z - 2259*x^3*y^3 - 1062*x^3*y^2*z^2 + 4977*x^3*y^2*z - 5463*x^3*y^2 + 510*x^3*y*z^3 - 3594*x^3*y*z^2 + 7797*x^3*y*z - 5341*x^3*y - 90*x^3*z^4 + 850*x^3*z^3 - 2736*x^3*z^2 + 3694*x^3*z - 1794*x^3 + 270*x^2*y^5 - 1029*x^2*y^4*z + 2427*x^2*y^4 + 1539*x^2*y^3*z^2 - 7230*x^2*y^3*z + 8052*x^2*y^3 - 1128*x^2*y^2*z^3 + 7947*x^2*y^2*z^2 - 17481*x^2*y^2*z + 12216*x^2*y^2 + 405*x^2*y*z^4 - 3816*x^2*y*z^3 + 12447*x^2*y*z^2 - 17121*x^2*y*z + 8493*x^2*y - 57*x^2*z^5 + 675*x^2*z^4 - 2904*x^2*z^3 + 5904*x^2*z^2 - 5772*x^2*z + 2179*x^2 - 124*x*y^6 + 576*x*y^5*z - 1368*x*y^5 - 1095*x*y^4*z^2 + 5166*x*y^4*z - 5835*x*y^4 + 1089*x*y^3*z^3 - 7683*x*y^3*z^2 + 17133*x*y^3*z - 12220*x*y^3 - 597*x*y^2*z^4 + 5619*x*y^2*z^3 - 18573*x*y^2*z^2 + 26046*x*y^2*z - 13218*x*y^2 + 171*x*y*z^5 - 2019*x*y*z^4 + 8799*x*y*z^3 - 18219*x*y*z^2 + 18189*x*y*z - 7024*x*y - 20*x*z^6 + 285*x*z^5 - 1536*x*z^4 + 4178*x*z^3 - 6162*x*z^2 + 4697*x*z - 1437*x + 24*y^7 - 132*y^6*z + 316*y^6 + 306*y^5*z^2 - 1452*y^5*z + 1662*y^5 - 387*y^4*z^3 + 2739*y^4*z^2 - 6189*y^4*z + 4502*y^4 + 288*y^3*z^4 - 2712*y^3*z^3 + 9081*y^3*z^2 - 12986*y^3*z + 6753*y^3 - 126*y^2*z^5 + 1485*y^2*z^4 - 6555*y^2*z^3 + 13830*y^2*z^2 - 14115*y^2*z + 5593*y^2 + 30*y*z^6 - 426*y*z^5 + 2325*y*z^4 - 6439*y*z^3 + 9696*y*z^2 - 7561*y*z + 2360*y - 3*z^7 + 50*z^6 - 324*z^5 + 1105*z^4 - 2184*z^3 + 2518*z^2 - 1564*z + 1, 2*x^7 - 21*x^6*y + 15*x^6*z - 29*x^6 + 93*x^5*y^2 - 135*x^5*y*z + 264*x^5*y + 48*x^5*z^2 - 186*x^5*z + 177*x^5 - 225*x^4*y^3 + 498*x^4*y^2*z - 987*x^4*y^2 - 360*x^4*y*z^2 + 1410*x^4*y*z - 1362*x^4*y + 85*x^4*z^3 - 495*x^4*z^2 + 945*x^4*z - 588*x^4 + 321*x^3*y^4 - 963*x^3*y^3*z + 1938*x^3*y^3 + 1062*x^3*y^2*z^2 - 4212*x^3*y^2*z + 4134*x^3*y^2 - 510*x^3*y*z^3 + 3000*x^3*y*z^2 - 5811*x^3*y*z + 3685*x^3*y + 90*x^3*z^4 - 700*x^3*z^3 + 2010*x^3*z^2 - 2512*x^3*z + 1144*x^3 - 270*x^2*y^5 + 1029*x^2*y^4*z - 2106*x^2*y^4 - 1539*x^2*y^3*z^2 + 6192*x^2*y^3*z - 6180*x^2*y^3 + 1128*x^2*y^2*z^3 - 6714*x^2*y^2*z^2 + 13206*x^2*y^2*z - 8541*x^2*y^2 - 405*x^2*y*z^4 + 3180*x^2*y*z^3 - 9261*x^2*y*z^2 + 11793*x^2*y*z - 5499*x^2*y + 57*x^2*z^5 - 555*x^2*z^4 + 2130*x^2*z^3 - 4008*x^2*z^2 + 3672*x^2*z - 1297*x^2 + 124*x*y^6 - 576*x*y^5*z + 1200*x*y^5 + 1095*x*y^4*z^2 - 4476*x*y^4*z + 4545*x*y^4 - 1089*x*y^3*z^3 + 6570*x*y^3*z^2 - 13131*x*y^3*z + 8666*x*y^3 + 597*x*y^2*z^4 - 4740*x*y^2*z^3 + 14010*x*y^2*z^2 - 18186*x*y^2*z + 8688*x*y^2 - 171*x*y*z^5 + 1680*x*y*z^4 - 6537*x*y*z^3 + 12531*x*y*z^2 - 11757*x*y*z + 4273*x*y + 20*x*z^6 - 234*x*z^5 + 1125*x*z^4 - 2832*x*z^3 + 3912*x*z^2 - 2786*x*z + 792*x - 24*y^7 + 132*y^6*z - 280*y^6 - 306*y^5*z^2 + 1272*y^5*z - 1314*y^5 + 387*y^4*z^3 - 2370*y^4*z^2 + 4815*y^4*z - 3242*y^4 - 288*y^3*z^4 + 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"Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -1, 1, -1 ], [ -1, 2, -1 ], [ -2, 3, -3 ] ], "source_shift": [ 2, 1, 2 ], "target_matrix": [ [ 1, 2, 1 ], [ -1, -1, -2 ], [ -2, -3, -2 ] ], "target_shift": [ -397, 204, 626 ], "collision_preimages": [ [ "15/4", "1", "-3/4" ], [ "15/2", "-3/2", "-8" ], [ "27/2", "7/2", "-7" ] ], "collision_image": [ "-1589/4", "817/4", "1253/2" ] }, "probe_target": "No", "jev_answers": [ "Maybe" ], "seconds": [ 1.0824283420006395 ], "surface_features": [ 3.6703793585142286, 4.646060468555641, 3.3093107157881754, 0.0, 3.8138608946415733, 4921 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B181", "pair": 181, "family": "bijection", "mapping": "F(x,y,z)=(-18*x^7 - 126*x^6*y - 126*x^6*z - 252*x^6 - 378*x^5*y^2 - 756*x^5*y*z - 1512*x^5*y - 378*x^5*z^2 - 1512*x^5*z - 1530*x^5 - 630*x^4*y^3 - 1890*x^4*y^2*z - 3780*x^4*y^2 - 1890*x^4*y*z^2 - 7560*x^4*y*z - 7668*x^4*y - 630*x^4*z^3 - 3780*x^4*z^2 - 7650*x^4*z - 5270*x^4 - 630*x^3*y^4 - 2520*x^3*y^3*z - 5040*x^3*y^3 - 3780*x^3*y^2*z^2 - 15120*x^3*y^2*z - 15372*x^3*y^2 - 2520*x^3*y*z^3 - 15120*x^3*y*z^2 - 30672*x^3*y*z - 21224*x^3*y - 630*x^3*z^4 - 5040*x^3*z^3 - 15300*x^3*z^2 - 21098*x^3*z - 11166*x^3 - 378*x^2*y^5 - 1890*x^2*y^4*z - 3780*x^2*y^4 - 3780*x^2*y^3*z^2 - 15120*x^2*y^3*z - 15408*x^2*y^3 - 3780*x^2*y^2*z^3 - 22680*x^2*y^2*z^2 - 46116*x^2*y^2*z - 32052*x^2*y^2 - 1890*x^2*y*z^4 - 15120*x^2*y*z^3 - 46008*x^2*y*z^2 - 63726*x^2*y*z - 33930*x^2*y - 378*x^2*z^5 - 3780*x^2*z^4 - 15300*x^2*z^3 - 31674*x^2*z^2 - 33606*x^2*z - 14586*x^2 - 126*x*y^6 - 756*x*y^5*z - 1512*x*y^5 - 1890*x*y^4*z^2 - 7560*x*y^4*z - 7722*x*y^4 - 2520*x*y^3*z^3 - 15120*x*y^3*z^2 - 30816*x*y^3*z - 21512*x*y^3 - 1890*x*y^2*z^4 - 15120*x*y^2*z^3 - 46116*x*y^2*z^2 - 64158*x*y^2*z - 34362*x*y^2 - 756*x*y*z^5 - 7560*x*y*z^4 - 30672*x*y*z^3 - 63780*x*y*z^2 - 68076*x*y*z - 29802*x*y - 126*x*z^6 - 1512*x*z^5 - 7650*x*z^4 - 21134*x*z^3 - 33714*x*z^2 - 29406*x*z - 10869*x - 18*y^7 - 126*y^6*z - 252*y^6 - 378*y^5*z^2 - 1512*y^5*z - 1548*y^5 - 630*y^4*z^3 - 3780*y^4*z^2 - 7722*y^4*z - 5414*y^4 - 630*y^3*z^4 - 5040*y^3*z^3 - 15408*y^3*z^2 - 21530*y^3*z - 11598*y^3 - 378*y^2*z^5 - 3780*y^2*z^4 - 15372*y^2*z^3 - 32106*y^2*z^2 - 34470*y^2*z - 15216*y^2 - 126*y*z^6 - 1512*y*z^5 - 7668*y*z^4 - 21278*y*z^3 - 34146*y*z^2 - 30054*y*z - 11231*y - 18*z^7 - 252*z^6 - 1530*z^5 - 5288*z^4 - 11274*z^3 - 14820*z^2 - 11045*z - 1, 6*x^7 + 42*x^6*y + 42*x^6*z + 84*x^6 + 126*x^5*y^2 + 252*x^5*y*z + 504*x^5*y + 126*x^5*z^2 + 504*x^5*z + 510*x^5 + 210*x^4*y^3 + 630*x^4*y^2*z + 1260*x^4*y^2 + 630*x^4*y*z^2 + 2520*x^4*y*z + 2556*x^4*y + 210*x^4*z^3 + 1260*x^4*z^2 + 2550*x^4*z + 1756*x^4 + 210*x^3*y^4 + 840*x^3*y^3*z + 1680*x^3*y^3 + 1260*x^3*y^2*z^2 + 5040*x^3*y^2*z + 5124*x^3*y^2 + 840*x^3*y*z^3 + 5040*x^3*y*z^2 + 10224*x^3*y*z + 7072*x^3*y + 210*x^3*z^4 + 1680*x^3*z^3 + 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- 4800*x^2 - 42*x*y^6 - 252*x*y^5*z - 504*x*y^5 - 630*x*y^4*z^2 - 2520*x*y^4*z - 2574*x*y^4 - 840*x*y^3*z^3 - 5040*x*y^3*z^2 - 10272*x*y^3*z - 7160*x*y^3 - 630*x*y^2*z^4 - 5040*x*y^2*z^3 - 15372*x*y^2*z^2 - 21354*x*y^2*z - 11391*x*y^2 - 252*x*y*z^5 - 2520*x*y*z^4 - 10224*x*y*z^3 - 21228*x*y*z^2 - 22566*x*y*z - 9810*x*y - 42*x*z^6 - 504*x*z^5 - 2550*x*z^4 - 7034*x*z^3 - 11175*x*z^2 - 9678*x*z - 3534*x - 6*y^7 - 42*y^6*z - 84*y^6 - 126*y^5*z^2 - 504*y^5*z - 516*y^5 - 210*y^4*z^3 - 1260*y^4*z^2 - 2574*y^4*z - 1802*y^4 - 210*y^3*z^4 - 1680*y^3*z^3 - 5136*y^3*z^2 - 7166*y^3*z - 3845*y^3 - 126*y^2*z^5 - 1260*y^2*z^4 - 5124*y^2*z^3 - 10686*y^2*z^2 - 11427*y^2*z - 5010*y^2 - 42*y*z^6 - 504*y*z^5 - 2556*y*z^4 - 7082*y*z^3 - 11319*y*z^2 - 9894*y*z - 3655*y - 6*z^7 - 84*z^6 - 510*z^5 - 1760*z^4 - 3737*z^3 - 4878*z^2 - 3590*z - 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ 1, 1, 1 ], [ 1, 2, 1 ], [ 2, 2, 3 ] ], "source_shift": [ 2, 2, 1 ], "target_matrix": [ [ -3, -2, -2 ], [ 1, 1, 1 ], [ -1, -1, -2 ] ], "target_shift": [ 3549, -1173, 1135 ], "inverse_recipe": { "p": 3, "q": 4, "f": "x**3", "g": "x**4 + x", "h": "y*z + y", "polynomials": { "f": [ [ [ 3, 0, 0 ], 1 ] ], "g": [ [ [ 4, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 1, 0 ], 1 ] ] }, "h_multiplier": 6 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 0.9917154540016782 ], "surface_features": [ 3.841564887796166, 4.832994029612327, 3.4075608494863627, 0.0, 4.023662907688221, 4977 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B098", "pair": 98, "family": "bijection", "mapping": "F(x,y,z)=(20*x^7 - 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268*y^6 - 612*y^5*z^2 - 2184*y^5*z - 1644*y^5 + 1354*y^4*z^3 + 7347*y^4*z^2 + 11232*y^4*z + 5223*y^4 - 1778*y^3*z^4 - 13057*y^3*z^3 - 30441*y^3*z^2 - 28781*y^3*z - 9547*y^3 + 1386*y^2*z^5 + 12927*y^2*z^4 + 40896*y^2*z^3 + 59025*y^2*z^2 + 39840*y^2*z + 10151*y^2 - 594*y*z^6 - 6759*y*z^5 - 27225*y*z^4 - 53373*y*z^3 - 55041*y*z^2 - 28551*y*z - 5831*y + 108*z^7 + 1458*z^6 + 7182*z^5 + 17946*z^4 + 25164*z^3 + 19956*z^2 + 8296*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -2, -1, 1 ], [ -2, -2, 3 ], [ -1, -1, 2 ] ], "source_shift": [ 2, 2, 1 ], "target_matrix": [ [ 1, 2, 1 ], [ 1, 1, 1 ], [ 2, 1, 1 ] ], "target_shift": [ -1488, -952, -1397 ], "collision_preimages": [ [ "5/4", "-1", "-1/2" ], [ "-8", "63/2", "29/2" ], [ "-3", "41/2", "23/2" ] ], "collision_image": [ "-5953/4", "-3809/4", "-2795/2" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": 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4560*y^3*z^3 - 5259*y^3*z^2 + 2994*y^3*z - 709*y^3 - 2016*y^2*z^5 + 6720*y^2*z^4 - 10146*y^2*z^3 + 8484*y^2*z^2 - 3912*y^2*z + 808*y^2 - 1344*y*z^6 + 5280*y*z^5 - 9780*y*z^4 + 10680*y*z^3 - 7200*y*z^2 + 2872*y*z - 543*y - 384*z^7 + 1728*z^6 - 3768*z^5 + 5040*z^4 - 4418*z^3 + 2566*z^2 - 922*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ 1, -1, -2 ], [ 2, -1, -1 ], [ 2, -2, -3 ] ], "source_shift": [ 1, 2, 1 ], "target_matrix": [ [ -1, -1, -2 ], [ -2, -1, -1 ], [ 1, 1, 1 ] ], "target_shift": [ 166, 342, -170 ], "inverse_recipe": { "p": 2, "q": 3, "f": "x**2", "g": "x**3", "h": "y**3 + y**2*z + y**2 + y", "polynomials": { "f": [ [ [ 2, 0, 0 ], 1 ] ], "g": [ [ [ 3, 0, 0 ], 1 ] ], "h": [ [ [ 0, 2, 1 ], 1 ], [ [ 0, 3, 0 ], 1 ], [ [ 0, 2, 0 ], 1 ], [ [ 0, 1, 0 ], 1 ] ] }, "h_multiplier": 3 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0133351670010597 ], "surface_features": [ 3.7227990820503503, 4.636688447953283, 3.4294292643817874, 0.0, 3.8272815658676933, 4968 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B156", "pair": 156, "family": "bijection", "mapping": "F(x,y,z)=(54*x^7 + 342*x^6*y + 342*x^6*z + 738*x^6 + 924*x^5*y^2 + 1848*x^5*y*z + 3984*x^5*y + 924*x^5*z^2 + 3984*x^5*z + 4328*x^5 + 1380*x^4*y^3 + 4140*x^4*y^2*z + 8916*x^4*y^2 + 4140*x^4*y*z^2 + 17832*x^4*y*z + 19348*x^4*y + 1380*x^4*z^3 + 8916*x^4*z^2 + 19348*x^4*z + 14074*x^4 + 1230*x^3*y^4 + 4920*x^3*y^3*z + 10584*x^3*y^3 + 7380*x^3*y^2*z^2 + 31752*x^3*y^2*z + 34406*x^3*y^2 + 4920*x^3*y*z^3 + 31752*x^3*y*z^2 + 68812*x^3*y*z + 49984*x^3*y + 1230*x^3*z^4 + 10584*x^3*z^3 + 34406*x^3*z^2 + 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10240*x*y^2*z^3 - 22908*x*y^2*z^2 + 22400*x*y^2*z - 7896*x*y^2 + 336*x*y*z^5 - 2680*x*y*z^4 + 8408*x*y*z^3 - 13136*x*y*z^2 + 9968*x*y*z - 2844*x*y - 28*x*z^6 + 280*x*z^5 - 1152*x*z^4 + 2552*x*z^3 - 3154*x*z^2 + 1962*x*z - 465*x + 512*y^7 - 1792*y^6*z + 2304*y^6 + 2688*y^5*z^2 - 7296*y^5*z + 4768*y^5 - 2240*y^4*z^3 + 9600*y^4*z^2 - 13296*y^4*z + 5952*y^4 + 1120*y^3*z^4 - 6720*y^3*z^3 + 14744*y^3*z^2 - 14048*y^3*z + 4808*y^3 - 336*y^2*z^5 + 2640*y^2*z^4 - 8132*y^2*z^3 + 12384*y^2*z^2 - 9100*y^2*z + 2512*y^2 + 56*y*z^6 - 552*y*z^5 + 2232*y*z^4 - 4824*y*z^3 + 5740*y*z^2 - 3428*y*z + 785*y - 4*z^7 + 48*z^6 - 244*z^5 + 700*z^4 - 1214*z^3 + 1197*z^2 - 596*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -1, 2, -1 ], [ -1, 1, -2 ], [ -2, 2, -3 ] ], "source_shift": [ 1, 1, 2 ], "target_matrix": [ [ -1, -2, -1 ], [ -1, -1, -1 ], [ 2, 1, 1 ] ], "target_shift": [ 54, 52, -113 ], "inverse_recipe": { "p": 2, "q": 3, "f": "x**2", "g": "x**3 + x**2 + x", "h": "y**2*z + 2*y**2 + y", "polynomials": { "f": [ [ [ 2, 0, 0 ], 1 ] ], "g": [ [ [ 3, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "h": [ [ [ 0, 2, 1 ], 1 ], [ [ 0, 2, 0 ], 2 ], [ [ 0, 1, 0 ], 1 ] ] }, "h_multiplier": 2 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.1011967099984759 ], "surface_features": [ 3.520838930864233, 4.440909082065218, 3.225309281725863, 0.0, 3.629127744345644, 4892 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C009", "pair": 9, "family": "counterexample", "mapping": "F(x,y,z)=(-27*x^7 + 243*x^6*y + 162*x^6*z - 243*x^6 - 900*x^5*y^2 - 1233*x^5*y*z + 1827*x^5*y - 414*x^5*z^2 + 1233*x^5*z - 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evaluated to the same image.", "certificate_data": { "source_matrix": [ [ 3, -2, -2 ], [ 1, -2, -1 ], [ 1, -1, -1 ] ], "source_shift": [ 1, 1, 1 ], "target_matrix": [ [ -1, -1, -2 ], [ 1, 1, 1 ], [ -1, -2, -1 ] ], "target_shift": [ 51, -53, 87 ], "collision_preimages": [ [ "3/2", "-1/4", "3" ], [ "-11", "8", "-49/2" ], [ "-13", "5", "-47/2" ] ], "collision_image": [ "205/4", "-213/4", "349/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0263086310005747 ], "surface_features": [ 3.773972641518274, 4.636688447953283, 3.494154594018443, 0.0, 3.8583484843154765, 4997 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C022", "pair": 22, "family": "counterexample", "mapping": "F(x,y,z)=(4*x^7 + 26*x^6*y + 38*x^6*z + 64*x^6 + 72*x^5*y^2 + 210*x^5*y*z + 357*x^5*y + 150*x^5*z^2 + 513*x^5*z + 411*x^5 + 110*x^4*y^3 + 480*x^4*y^2*z + 825*x^4*y^2 + 684*x^4*y*z^2 + 2370*x^4*y*z + 1920*x^4*y + 318*x^4*z^3 + 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770*y^5*z^2 + 1540*y^5*z - 2031*y^5 + 1150*y^4*z^3 - 3450*y^4*z^2 + 7700*y^4*z - 5400*y^4 - 1025*y^3*z^4 + 4100*y^3*z^3 - 11895*y^3*z^2 + 15590*y^3*z - 6906*y^3 + 545*y^2*z^5 - 2725*y^2*z^4 + 9345*y^2*z^3 - 17135*y^2*z^2 + 14683*y^2*z - 4668*y^2 - 160*y*z^6 + 960*y*z^5 - 3725*y*z^4 + 8500*y*z^3 - 10533*y*z^2 + 6566*y*z - 1615*y + 20*z^7 - 140*z^6 + 601*z^5 - 1605*z^4 + 2551*z^3 - 2333*z^2 + 1133*z - 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -1, 1, -1 ], [ -3, 3, -2 ], [ -1, 2, -1 ] ], "source_shift": [ 1, 1, 1 ], "target_matrix": [ [ 2, 1, 1 ], [ -1, -1, -1 ], [ -1, -1, -2 ] ], "target_shift": [ -460, 229, 225 ], "inverse_recipe": { "p": 5, "q": 1, "f": "x**5 + x**3", "g": "x", "h": "y*z**2 + z**5 + z", "polynomials": { "f": [ [ [ 5, 0, 0 ], 1 ], [ [ 3, 0, 0 ], 1 ] ], "g": [ [ [ 1, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 2 ], 1 ], [ [ 0, 0, 1 ], 1 ], [ [ 0, 0, 5 ], 1 ] ] }, "h_multiplier": 5 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.069863990000158 ], "surface_features": [ 3.915080314543807, 4.835563751669097, 3.646060468555641, 0.0, 4.015015414183738, 5057 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B197", "pair": 197, "family": "bijection", "mapping": "F(x,y,z)=(3*x^7 - 20*x^6*y - 39*x^6*z + 37*x^6 + 57*x^5*y^2 + 222*x^5*y*z - 210*x^5*y + 216*x^5*z^2 - 408*x^5*z + 195*x^5 - 90*x^4*y^3 - 525*x^4*y^2*z + 495*x^4*y^2 - 1020*x^4*y*z^2 + 1920*x^4*y*z - 914*x^4*y - 660*x^4*z^3 + 1860*x^4*z^2 - 1767*x^4*z + 568*x^4 + 85*x^3*y^4 + 660*x^3*y^3*z - 620*x^3*y^3 + 1920*x^3*y^2*z^2 - 3600*x^3*y^2*z + 1706*x^3*y^2 + 2480*x^3*y*z^3 - 6960*x^3*y*z^2 + 6580*x^3*y*z - 2103*x^3*y + 1200*x^3*z^4 - 4480*x^3*z^3 + 6336*x^3*z^2 - 4037*x^3*z + 979*x^3 - 48*x^2*y^5 - 465*x^2*y^4*z + 435*x^2*y^4 - 1800*x^2*y^3*z^2 + 3360*x^2*y^3*z - 1584*x^2*y^3 - 3480*x^2*y^2*z^3 + 9720*x^2*y^2*z^2 - 9138*x^2*y^2*z + 2901*x^2*y^2 - 3360*x^2*y*z^4 + 12480*x^2*y*z^3 - 17544*x^2*y*z^2 + 11097*x^2*y*z - 2667*x^2*y - 1296*x^2*z^5 + 6000*x^2*z^4 - 11208*x^2*z^3 + 10590*x^2*z^2 - 5064*x^2*z + 987*x^2 + 15*x*y^6 + 174*x*y^5*z - 162*x*y^5 + 840*x*y^4*z^2 - 1560*x*y^4*z + 731*x*y^4 + 2160*x*y^3*z^3 - 6000*x*y^3*z^2 + 5604*x*y^3*z - 1765*x*y^3 + 3120*x*y^2*z^4 - 11520*x*y^2*z^3 + 16080*x*y^2*z^2 - 10083*x*y^2*z + 2397*x*y^2 + 2400*x*y*z^5 - 11040*x*y*z^4 + 20464*x*y*z^3 - 19152*x*y*z^2 + 9048*x*y*z - 1732*x*y + 768*x*z^6 - 4224*x*z^5 + 9744*x*z^4 - 12092*x*z^3 + 8508*x*z^2 - 3219*x*z + 513*x - 2*y^7 - 27*y^6*z + 25*y^6 - 156*y^5*z^2 + 288*y^5*z - 134*y^5 - 500*y^4*z^3 + 1380*y^4*z^2 - 1279*y^4*z + 399*y^4 - 960*y^3*z^4 + 3520*y^3*z^3 - 4872*y^3*z^2 + 3023*y^3*z - 709*y^3 - 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+ 4800*x*y*z^5 - 23040*x*y*z^4 + 44768*x*y*z^3 - 44160*x*y*z^2 + 22152*x*y*z - 4544*x*y + 1536*x*z^6 - 8832*x*z^5 + 21408*x*z^4 - 28088*x*z^3 + 21072*x*z^2 - 8598*x*z + 1501*x - 4*y^7 - 54*y^6*z + 52*y^6 - 312*y^5*z^2 + 600*y^5*z - 292*y^5 - 1000*y^4*z^3 + 2880*y^4*z^2 - 2798*y^4*z + 920*y^4 - 1920*y^3*z^4 + 7360*y^3*z^3 - 10704*y^3*z^2 + 7022*y^3*z - 1756*y^3 - 2208*y^2*z^5 + 10560*y^2*z^4 - 20432*y^2*z^3 + 20052*y^2*z^2 - 9996*y^2*z + 2032*y^2 - 1408*y*z^6 + 8064*y*z^5 - 19456*y*z^4 + 25384*y*z^3 - 18912*y*z^2 + 7644*y*z - 1316*y - 384*z^7 + 2560*z^6 - 7392*z^5 + 12016*z^4 - 11888*z^3 + 7164*z^2 - 2443*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ 1, -1, -2 ], [ 2, -1, -1 ], [ 3, -2, -2 ] ], "source_shift": [ 2, 1, 2 ], "target_matrix": [ [ 1, 1, 1 ], [ 3, 2, 3 ], [ 2, 1, 1 ] ], "target_shift": [ -94, -281, -366 ], "inverse_recipe": { "p": 1, "q": 6, "f": "x", "g": "x**6 + x**4 + x**3", "h": "y*z", "polynomials": { "f": [ [ [ 1, 0, 0 ], 1 ] ], "g": [ [ [ 6, 0, 0 ], 1 ], [ [ 3, 0, 0 ], 1 ], [ [ 4, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0615770259973942 ], "surface_features": [ 3.874618862015892, 4.788111781854241, 3.5200903281128424, 0.0, 4.008430172718923, 4999 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B133", "pair": 133, "family": "bijection", "mapping": "F(x,y,z)=(384*x^7 + 1344*x^6*y - 1344*x^6*z - 3072*x^6 + 2016*x^5*y^2 - 4032*x^5*y*z - 9504*x^5*y + 2016*x^5*z^2 + 9312*x^5*z + 10824*x^5 + 1680*x^4*y^3 - 5040*x^4*y^2*z - 12240*x^4*y^2 + 5040*x^4*y*z^2 + 24000*x^4*y*z + 28740*x^4*y - 1680*x^4*z^3 - 11760*x^4*z^2 - 27636*x^4*z - 21708*x^4 + 840*x^3*y^4 - 3360*x^3*y^3*z - 8400*x^3*y^3 + 5040*x^3*y^2*z^2 + 24720*x^3*y^2*z + 30474*x^3*y^2 - 3360*x^3*y*z^3 - 24240*x^3*y*z^2 - 58668*x^3*y*z - 47484*x^3*y + 840*x^3*z^4 + 7920*x^3*z^3 + 28218*x^3*z^2 + 44844*x^3*z + 26689*x^3 + 252*x^2*y^5 - 1260*x^2*y^4*z - 3240*x^2*y^4 + 2520*x^2*y^3*z^2 + 12720*x^2*y^3*z + 16131*x^2*y^3 - 2520*x^2*y^2*z^3 - 18720*x^2*y^2*z^2 - 46629*x^2*y^2*z - 38871*x^2*y^2 + 1260*x^2*y*z^4 + 12240*x^2*y*z^3 + 44901*x^2*y*z^2 + 73518*x^2*y*z + 45114*x^2*y - 252*x^2*z^5 - 3000*x^2*z^4 - 14403*x^2*z^3 - 34731*x^2*z^2 - 41856*x^2*z - 20069*x^2 + 42*x*y^6 - 252*x*y^5*z - 666*x*y^5 + 630*x*y^4*z^2 + 3270*x*y^4*z + 4263*x*y^4 - 840*x*y^3*z^3 - 6420*x*y^3*z^2 - 16446*x*y^3*z - 14112*x*y^3 + 630*x*y^2*z^4 + 6300*x*y^2*z^3 + 23778*x*y^2*z^2 + 40092*x*y^2*z + 25374*x*y^2 - 252*x*y*z^5 - 3090*x*y*z^4 - 15270*x*y*z^3 - 37932*x*y*z^2 - 47148*x*y*z - 23336*x*y + 42*x*z^6 + 606*x*z^5 + 3675*x*z^4 + 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2180*x*y^4*z + 2842*x*y^4 - 560*x*y^3*z^3 - 4280*x*y^3*z^2 - 10964*x*y^3*z - 9408*x*y^3 + 420*x*y^2*z^4 + 4200*x*y^2*z^3 + 15852*x*y^2*z^2 + 26728*x*y^2*z + 16920*x*y^2 - 168*x*y*z^5 - 2060*x*y*z^4 - 10180*x*y*z^3 - 25288*x*y*z^2 - 31440*x*y*z - 15572*x*y + 28*x*z^6 + 404*x*z^5 + 2450*x*z^4 + 7968*x*z^3 + 14590*x*z^2 + 14192*x*z + 5703*x + 2*y^7 - 14*y^6*z - 38*y^6 + 42*y^5*z^2 + 224*y^5*z + 300*y^5 - 70*y^4*z^3 - 550*y^4*z^2 - 1448*y^4*z - 1278*y^4 + 70*y^3*z^4 + 720*y^3*z^3 + 2794*y^3*z^2 + 4848*y^3*z + 3166*y^3 - 42*y^2*z^5 - 530*y^2*z^4 - 2694*y^2*z^3 - 6890*y^2*z^2 - 8838*y^2*z - 4523*y^2 + 14*y*z^6 + 208*y*z^5 + 1298*y*z^4 + 4348*y*z^3 + 8214*y*z^2 + 8254*y*z + 3428*y - 2*z^7 - 34*z^6 - 250*z^5 - 1028*z^4 - 2542*z^3 - 3761*z^2 - 3067*z - 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -2, -1, 1 ], [ -1, -2, 1 ], [ -1, -1, 1 ] ], "source_shift": [ 2, 2, 1 ], "target_matrix": [ [ -3, -3, -2 ], [ 1, 1, 1 ], [ -2, -1, -1 ] ], "target_shift": [ 1583, -517, 1059 ], "inverse_recipe": { "p": 2, "q": 3, "f": "x**2", "g": "x**3 + x**2 + x", "h": "y**2*z", "polynomials": { "f": [ [ [ 2, 0, 0 ], 1 ] ], "g": [ [ [ 3, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "h": [ [ [ 0, 2, 1 ], 1 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0268093520026014 ], "surface_features": [ 3.885863428119109, 4.8663936838952155, 3.481943999280067, 0.0, 4.04700060562269, 4968 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B091", "pair": 91, "family": "bijection", "mapping": "F(x,y,z)=(-384*x^7 + 4032*x^6*y - 2688*x^6*z + 1536*x^6 - 18144*x^5*y^2 + 24192*x^5*y*z - 13824*x^5*y - 8064*x^5*z^2 + 9216*x^5*z - 2736*x^5 + 45360*x^4*y^3 - 90720*x^4*y^2*z + 51840*x^4*y^2 + 60480*x^4*y*z^2 - 69120*x^4*y*z + 20544*x^4*y - 13440*x^4*z^3 + 23040*x^4*z^2 - 13680*x^4*z + 2912*x^4 - 68040*x^3*y^4 + 181440*x^3*y^3*z - 103680*x^3*y^3 - 181440*x^3*y^2*z^2 + 207360*x^3*y^2*z - 61704*x^3*y^2 + 80640*x^3*y*z^3 - 138240*x^3*y*z^2 + 82176*x^3*y*z - 17520*x^3*y - 13440*x^3*z^4 + 30720*x^3*z^3 - 27360*x^3*z^2 + 11672*x^3*z - 2072*x^3 + 61236*x^2*y^5 - 204120*x^2*y^4*z + 116640*x^2*y^4 + 272160*x^2*y^3*z^2 - 311040*x^2*y^3*z + 92664*x^2*y^3 - 181440*x^2*y^2*z^3 + 311040*x^2*y^2*z^2 - 185112*x^2*y^2*z + 39528*x^2*y^2 + 60480*x^2*y*z^4 - 138240*x^2*y*z^3 + 123264*x^2*y*z^2 - 52668*x^2*y*z + 9360*x^2*y - 8064*x^2*z^5 + 23040*x^2*z^4 - 27360*x^2*z^3 + 17544*x^2*z^2 - 6252*x^2*z + 999*x^2 - 30618*x*y^6 + 122472*x*y^5*z - 69984*x*y^5 - 204120*x*y^4*z^2 + 233280*x*y^4*z - 69579*x*y^4 + 181440*x*y^3*z^3 - 311040*x*y^3*z^2 + 185328*x*y^3*z - 39636*x*y^3 - 90720*x*y^2*z^4 + 207360*x*y^2*z^3 - 185112*x*y^2*z^2 + 79218*x*y^2*z - 14094*x*y^2 + 24192*x*y*z^5 - 69120*x*y*z^4 + 82176*x*y*z^3 - 52776*x*y*z^2 + 18828*x*y*z - 3009*x*y - 2688*x*z^6 + 9216*x*z^5 - 13680*x*z^4 + 11720*x*z^3 - 6288*x*z^2 + 2025*x*z - 331*x + 6561*y^7 - 30618*y^6*z + 17496*y^6 + 61236*y^5*z^2 - 69984*y^5*z + 20898*y^5 - 68040*y^4*z^3 + 116640*y^4*z^2 - 69579*y^4*z + 14904*y^4 + 45360*y^3*z^4 - 103680*y^3*z^3 + 92664*y^3*z^2 - 39717*y^3*z + 7074*y^3 - 18144*y^2*z^5 + 51840*y^2*z^4 - 61704*y^2*z^3 + 39690*y^2*z^2 - 14175*y^2*z + 2265*y^2 + 4032*y*z^6 - 13824*y*z^5 + 20544*y*z^4 - 17628*y*z^3 + 9468*y*z^2 - 3051*y*z + 503*y - 384*z^7 + 1536*z^6 - 2736*z^5 + 2936*z^4 - 2108*z^3 + 1026*z^2 - 339*z + 1, 128*x^7 - 1344*x^6*y + 896*x^6*z - 512*x^6 + 6048*x^5*y^2 - 8064*x^5*y*z + 4608*x^5*y + 2688*x^5*z^2 - 3072*x^5*z + 912*x^5 - 15120*x^4*y^3 + 30240*x^4*y^2*z - 17280*x^4*y^2 - 20160*x^4*y*z^2 + 23040*x^4*y*z - 6848*x^4*y + 4480*x^4*z^3 - 7680*x^4*z^2 + 4560*x^4*z - 960*x^4 + 22680*x^3*y^4 - 60480*x^3*y^3*z + 34560*x^3*y^3 + 60480*x^3*y^2*z^2 - 69120*x^3*y^2*z + 20568*x^3*y^2 - 26880*x^3*y*z^3 + 46080*x^3*y*z^2 - 27392*x^3*y*z + 5776*x^3*y + 4480*x^3*z^4 - 10240*x^3*z^3 + 9120*x^3*z^2 - 3848*x^3*z + 664*x^3 - 20412*x^2*y^5 + 68040*x^2*y^4*z - 38880*x^2*y^4 - 90720*x^2*y^3*z^2 + 103680*x^2*y^3*z - 30888*x^2*y^3 + 60480*x^2*y^2*z^3 - 103680*x^2*y^2*z^2 + 61704*x^2*y^2*z - 13032*x^2*y^2 - 20160*x^2*y*z^4 + 46080*x^2*y*z^3 - 41088*x^2*y*z^2 + 17364*x^2*y*z - 3000*x^2*y + 2688*x^2*z^5 - 7680*x^2*z^4 + 9120*x^2*z^3 - 5784*x^2*z^2 + 2004*x^2*z - 309*x^2 + 10206*x*y^6 - 40824*x*y^5*z + 23328*x*y^5 + 68040*x*y^4*z^2 - 77760*x*y^4*z + 23193*x*y^4 - 60480*x*y^3*z^3 + 103680*x*y^3*z^2 - 61776*x*y^3*z + 13068*x*y^3 + 30240*x*y^2*z^4 - 69120*x*y^2*z^3 + 61704*x*y^2*z^2 - 26118*x*y^2*z + 4518*x*y^2 - 8064*x*y*z^5 + 23040*x*y*z^4 - 27392*x*y*z^3 + 17400*x*y*z^2 - 6036*x*y*z + 931*x*y + 896*x*z^6 - 3072*x*z^5 + 4560*x*z^4 - 3864*x*z^3 + 2016*x*z^2 - 627*x*z + 99*x - 2187*y^7 + 10206*y^6*z - 5832*y^6 - 20412*y^5*z^2 + 23328*y^5*z - 6966*y^5 + 22680*y^4*z^3 - 38880*y^4*z^2 + 23193*y^4*z - 4914*y^4 - 15120*y^3*z^4 + 34560*y^3*z^3 - 30888*y^3*z^2 + 13095*y^3*z - 2268*y^3 + 6048*y^2*z^5 - 17280*y^2*z^4 + 20568*y^2*z^3 - 13086*y^2*z^2 + 4545*y^2*z - 701*y^2 - 1344*y*z^6 + 4608*y*z^5 - 6848*y*z^4 + 5812*y*z^3 - 3036*y*z^2 + 945*y*z - 151*y + 128*z^7 - 512*z^6 + 912*z^5 - 968*z^4 + 676*z^3 - 318*z^2 + 101*z - 1, 256*x^7 - 2688*x^6*y + 1792*x^6*z - 1024*x^6 + 12096*x^5*y^2 - 16128*x^5*y*z + 9216*x^5*y + 5376*x^5*z^2 - 6144*x^5*z + 1824*x^5 - 30240*x^4*y^3 + 60480*x^4*y^2*z - 34560*x^4*y^2 - 40320*x^4*y*z^2 + 46080*x^4*y*z - 13696*x^4*y + 8960*x^4*z^3 - 15360*x^4*z^2 + 9120*x^4*z - 1952*x^4 + 45360*x^3*y^4 - 120960*x^3*y^3*z + 69120*x^3*y^3 + 120960*x^3*y^2*z^2 - 138240*x^3*y^2*z + 41136*x^3*y^2 - 53760*x^3*y*z^3 + 92160*x^3*y*z^2 - 54784*x^3*y*z + 11744*x^3*y + 8960*x^3*z^4 - 20480*x^3*z^3 + 18240*x^3*z^2 - 7824*x^3*z + 1400*x^3 - 40824*x^2*y^5 + 136080*x^2*y^4*z - 77760*x^2*y^4 - 181440*x^2*y^3*z^2 + 207360*x^2*y^3*z - 61776*x^2*y^3 + 120960*x^2*y^2*z^3 - 207360*x^2*y^2*z^2 + 123408*x^2*y^2*z - 26496*x^2*y^2 - 40320*x^2*y*z^4 + 92160*x^2*y*z^3 - 82176*x^2*y*z^2 + 35304*x^2*y*z - 6324*x^2*y + 5376*x^2*z^5 - 15360*x^2*z^4 + 18240*x^2*z^3 - 11760*x^2*z^2 + 4224*x^2*z - 678*x^2 + 20412*x*y^6 - 81648*x*y^5*z + 46656*x*y^5 + 136080*x*y^4*z^2 - 155520*x*y^4*z + 46386*x*y^4 - 120960*x*y^3*z^3 + 207360*x*y^3*z^2 - 123552*x*y^3*z + 26568*x*y^3 + 60480*x*y^2*z^4 - 138240*x*y^2*z^3 + 123408*x*y^2*z^2 - 53100*x*y^2*z + 9522*x*y^2 - 16128*x*y*z^5 + 46080*x*y*z^4 - 54784*x*y*z^3 + 35376*x*y*z^2 - 12720*x*y*z + 2042*x*y + 1792*x*z^6 - 6144*x*z^5 + 9120*x*z^4 - 7856*x*z^3 + 4248*x*z^2 - 1374*x*z + 223*x - 4374*y^7 + 20412*y^6*z - 11664*y^6 - 40824*y^5*z^2 + 46656*y^5*z - 13932*y^5 + 45360*y^4*z^3 - 77760*y^4*z^2 + 46386*y^4*z - 9990*y^4 - 30240*y^3*z^4 + 69120*y^3*z^3 - 61776*y^3*z^2 + 26622*y^3*z - 4779*y^3 + 12096*y^2*z^5 - 34560*y^2*z^4 + 41136*y^2*z^3 - 26604*y^2*z^2 + 9576*y^2*z - 1537*y^2 - 2688*y*z^6 + 9216*y*z^5 - 13696*y*z^4 + 11816*y*z^3 - 6396*y*z^2 + 2070*y*z - 338*y + 256*z^7 - 1024*z^6 + 1824*z^5 - 1968*z^4 + 1424*z^3 - 696*z^2 + 229*z - 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -2, 3, -2 ], [ -1, 2, -1 ], [ -1, 1, -2 ] ], "source_shift": [ 1, 2, 2 ], "target_matrix": [ [ 3, 3, 2 ], [ -1, -1, -1 ], [ -2, -1, -1 ] ], "target_shift": [ -66, 18, 43 ], "inverse_recipe": { "p": 3, "q": 4, "f": "x**3 + x", "g": "x**4 + x**3 + x", "h": "y*z + y", "polynomials": { "f": [ [ [ 3, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "g": [ [ [ 4, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ], [ [ 3, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 1, 0 ], 1 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.1099387209978886 ], "surface_features": [ 4.60673247493022, 5.492816243246181, 4.243831461981921, 0.0, 4.745336320488149, 5266 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C059", "pair": 59, "family": "counterexample", "mapping": "F(x,y,z)=(4*x^7 - 38*x^6*y - 26*x^6*z + 38*x^6 + 150*x^5*y^2 + 210*x^5*y*z - 309*x^5*y + 72*x^5*z^2 - 213*x^5*z + 162*x^5 - 318*x^4*y^3 - 684*x^4*y^2*z + 1014*x^4*y^2 - 480*x^4*y*z^2 + 1431*x^4*y*z - 1092*x^4*y - 110*x^4*z^3 + 495*x^4*z^2 - 759*x^4*z + 398*x^4 + 390*x^3*y^4 + 1146*x^3*y^3*z - 1713*x^3*y^3 + 1236*x^3*y^2*z^2 - 3717*x^3*y^2*z + 2844*x^3*y^2 + 580*x^3*y*z^3 - 2634*x^3*y*z^2 + 4056*x^3*y*z - 2125*x^3*y + 100*x^3*z^4 - 610*x^3*z^3 + 1416*x^3*z^2 - 1493*x^3*z + 608*x^3 - 276*x^2*y^5 - 1038*x^2*y^4*z + 1566*x^2*y^4 - 1530*x^2*y^3*z^2 + 4647*x^2*y^3*z - 3564*x^2*y^3 - 1104*x^2*y^2*z^3 + 5067*x^2*y^2*z^2 - 7833*x^2*y^2*z + 4092*x^2*y^2 - 390*x^2*y*z^4 + 2406*x^2*y*z^3 - 5616*x^2*y*z^2 + 5919*x^2*y*z - 2394*x^2*y - 54*x^2*z^5 + 420*x^2*z^4 - 1314*x^2*z^3 + 2091*x^2*z^2 - 1707*x^2*z + 578*x^2 + 104*x*y^6 + 480*x*y^5*z - 732*x*y^5 + 906*x*y^4*z^2 - 2784*x*y^4*z + 2142*x*y^4 + 894*x*y^3*z^3 - 4155*x*y^3*z^2 + 6450*x*y^3*z - 3355*x*y^3 + 486*x*y^2*z^4 - 3039*x*y^2*z^3 + 7134*x*y^2*z^2 - 7503*x*y^2*z + 3000*x*y^2 + 138*x*y*z^5 - 1089*x*y*z^4 + 3432*x*y*z^3 - 5463*x*y*z^2 + 4428*x*y*z - 1477*x*y + 16*x*z^6 - 153*x*z^5 + 606*x*z^4 - 1295*x*z^3 + 1590*x*z^2 - 1075*x*z + 318*x - 16*y^7 - 88*y^6*z + 136*y^6 - 204*y^5*z^2 + 636*y^5*z - 492*y^5 - 258*y^4*z^3 + 1218*y^4*z^2 - 1902*y^4*z + 986*y^4 - 192*y^3*z^4 + 1221*y^3*z^3 - 2886*y^3*z^2 + 3027*y^3*z - 1194*y^3 - 84*y^2*z^5 + 675*y^2*z^4 - 2145*y^2*z^3 + 3411*y^2*z^2 - 2733*y^2*z + 890*y^2 - 20*y*z^6 + 195*y*z^5 - 780*y*z^4 + 1669*y*z^3 - 2034*y*z^2 + 1351*y*z - 388*y - 2*z^7 + 23*z^6 - 111*z^5 + 299*z^4 - 491*z^3 + 497*z^2 - 291*z + 1, -2*x^7 + 19*x^6*y + 13*x^6*z - 28*x^6 - 75*x^5*y^2 - 105*x^5*y*z + 231*x^5*y - 36*x^5*z^2 + 156*x^5*z - 144*x^5 + 159*x^4*y^3 + 342*x^4*y^2*z - 768*x^4*y^2 + 240*x^4*y*z^2 - 1062*x^4*y*z + 987*x^4*y + 55*x^4*z^3 - 360*x^4*z^2 + 672*x^4*z - 396*x^4 - 195*x^3*y^4 - 573*x^3*y^3*z + 1311*x^3*y^3 - 618*x^3*y^2*z^2 + 2790*x^3*y^2*z - 2610*x^3*y^2 - 290*x^3*y*z^3 + 1938*x^3*y*z^2 - 3648*x^3*y*z + 2154*x^3*y - 50*x^3*z^4 + 440*x^3*z^3 - 1248*x^3*z^2 + 1481*x^3*z - 650*x^3 + 138*x^2*y^5 + 519*x^2*y^4*z - 1206*x^2*y^4 + 765*x^2*y^3*z^2 - 3516*x^2*y^3*z + 3312*x^2*y^3 + 552*x^2*y^2*z^3 - 3762*x^2*y^2*z^2 + 7143*x^2*y^2*z - 4218*x^2*y^2 + 195*x^2*y*z^4 - 1752*x^2*y*z^3 + 5022*x^2*y*z^2 - 5976*x^2*y*z + 2613*x^2*y + 27*x^2*z^5 - 300*x^2*z^4 + 1152*x^2*z^3 - 2067*x^2*z^2 + 1821*x^2*z - 649*x^2 - 52*x*y^6 - 240*x*y^5*z + 564*x*y^5 - 453*x*y^4*z^2 + 2112*x*y^4*z - 2007*x*y^4 - 447*x*y^3*z^3 + 3099*x*y^3*z^2 - 5943*x*y^3*z + 3504*x*y^3 - 243*x*y^2*z^4 + 2226*x*y^2*z^3 - 6456*x*y^2*z^2 + 7692*x*y^2*z - 3336*x*y^2 - 69*x*y*z^5 + 783*x*y*z^4 - 3048*x*y*z^3 + 5490*x*y*z^2 - 4818*x*y*z + 1700*x*y - 8*x*z^6 + 108*x*z^5 - 528*x*z^4 + 1275*x*z^3 - 1692*x*z^2 + 1205*x*z - 366*x + 8*y^7 + 44*y^6*z - 104*y^6 + 102*y^5*z^2 - 480*y^5*z + 462*y^5 + 129*y^4*z^3 - 906*y^4*z^2 + 1761*y^4*z - 1038*y^4 + 96*y^3*z^4 - 894*y^3*z^3 + 2631*y^3*z^2 - 3137*y^3*z + 1346*y^3 + 42*y^2*z^5 - 486*y^2*z^4 + 1923*y^2*z^3 - 3474*y^2*z^2 + 3024*y^2*z - 1045*y^2 + 10*y*z^6 - 138*y*z^5 + 687*y*z^4 - 1668*y*z^3 + 2205*y*z^2 - 1550*y*z + 461*y + z^7 - 16*z^6 + 96*z^5 - 293*z^4 + 521*z^3 - 556*z^2 + 335*z - 1, -6*x^7 + 57*x^6*y + 39*x^6*z - 60*x^6 - 225*x^5*y^2 - 315*x^5*y*z + 489*x^5*y - 108*x^5*z^2 + 336*x^5*z - 264*x^5 + 477*x^4*y^3 + 1026*x^4*y^2*z - 1608*x^4*y^2 + 720*x^4*y*z^2 - 2262*x^4*y*z + 1785*x^4*y + 165*x^4*z^3 - 780*x^4*z^2 + 1236*x^4*z - 662*x^4 - 585*x^3*y^4 - 1719*x^3*y^3*z + 2721*x^3*y^3 - 1854*x^3*y^2*z^2 + 5886*x^3*y^2*z - 4662*x^3*y^2 - 870*x^3*y*z^3 + 4158*x^3*y*z^2 - 6624*x^3*y*z + 3547*x^3*y - 150*x^3*z^4 + 960*x^3*z^3 - 2304*x^3*z^2 + 2482*x^3*z - 1024*x^3 + 414*x^2*y^5 + 1557*x^2*y^4*z - 2490*x^2*y^4 + 2295*x^2*y^3*z^2 - 7368*x^2*y^3*z + 5856*x^2*y^3 + 1656*x^2*y^2*z^3 - 8010*x^2*y^2*z^2 + 12825*x^2*y^2*z - 6852*x^2*y^2 + 585*x^2*y*z^4 - 3792*x^2*y*z^3 + 9162*x^2*y*z^2 - 9873*x^2*y*z + 4047*x^2*y + 81*x^2*z^5 - 660*x^2*z^4 + 2136*x^2*z^3 - 3474*x^2*z^2 + 2874*x^2*z - 981*x^2 - 156*x*y^6 - 720*x*y^5*z + 1164*x*y^5 - 1359*x*y^4*z^2 + 4416*x*y^4*z - 3525*x*y^4 - 1341*x*y^3*z^3 + 6573*x*y^3*z^2 - 10581*x*y^3*z + 5632*x*y^3 - 729*x*y^2*z^4 + 4794*x*y^2*z^3 - 11664*x*y^2*z^2 + 12552*x*y^2*z - 5088*x*y^2 - 207*x*y*z^5 + 1713*x*y*z^4 - 5592*x*y*z^3 + 9105*x*y*z^2 - 7482*x*y*z + 2517*x*y - 24*x*z^6 + 240*x*z^5 - 984*x*z^4 + 2150*x*z^3 - 2676*x*z^2 + 1824*x*z - 542*x + 24*y^7 + 132*y^6*z - 216*y^6 + 306*y^5*z^2 - 1008*y^5*z + 810*y^5 + 387*y^4*z^3 - 1926*y^4*z^2 + 3123*y^4*z - 1658*y^4 + 288*y^3*z^4 - 1926*y^3*z^3 + 4725*y^3*z^2 - 5075*y^3*z + 2030*y^3 + 126*y^2*z^5 - 1062*y^2*z^4 + 3501*y^2*z^3 - 5700*y^2*z^2 + 4632*y^2*z - 1521*y^2 + 30*y*z^6 - 306*y*z^5 + 1269*y*z^4 - 2779*y*z^3 + 3435*y*z^2 - 2301*y*z + 665*y + 3*z^7 - 36*z^6 + 180*z^5 - 496*z^4 + 826*z^3 - 843*z^2 + 496*z - 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ 1, -2, -1 ], [ 1, -1, -1 ], [ 2, -1, -1 ] ], "source_shift": [ 1, 1, 1 ], "target_matrix": [ [ 2, 1, 1 ], [ -1, -2, -1 ], [ -3, -2, -2 ] ], "target_shift": [ -75, 87, 129 ], "collision_preimages": [ [ "-1/4", "0", "3/4" ], [ "8", "-5/2", "13" ], [ "5", "5/2", "2" ] ], "collision_image": [ "-151/2", "349/4", "519/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 0.9744438000016089 ], "surface_features": [ 3.2533231946838694, 4.108057373783854, 2.9923325590474645, 0.0, 3.346376250593239, 4810 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B229", "pair": 229, "family": "bijection", "mapping": "F(x,y,z)=(-2*x^7 + 14*x^6*y - 28*x^6*z + 16*x^6 - 42*x^5*y^2 + 168*x^5*y*z - 96*x^5*y - 168*x^5*z^2 + 192*x^5*z - 58*x^5 + 70*x^4*y^3 - 420*x^4*y^2*z + 240*x^4*y^2 + 840*x^4*y*z^2 - 960*x^4*y*z + 292*x^4*y - 560*x^4*z^3 + 960*x^4*z^2 - 578*x^4*z + 131*x^4 - 70*x^3*y^4 + 560*x^3*y^3*z - 320*x^3*y^3 - 1680*x^3*y^2*z^2 + 1920*x^3*y^2*z - 588*x^3*y^2 + 2240*x^3*y*z^3 - 3840*x^3*y*z^2 + 2328*x^3*y*z - 536*x^3*y - 1120*x^3*z^4 + 2560*x^3*z^3 - 2304*x^3*z^2 + 1034*x^3*z - 202*x^3 + 42*x^2*y^5 - 420*x^2*y^4*z + 240*x^2*y^4 + 1680*x^2*y^3*z^2 - 1920*x^2*y^3*z + 592*x^2*y^3 - 3360*x^2*y^2*z^3 + 5760*x^2*y^2*z^2 - 3516*x^2*y^2*z + 822*x^2*y^2 + 3360*x^2*y*z^4 - 7680*x^2*y*z^3 + 6960*x^2*y*z^2 - 3174*x^2*y*z + 632*x^2*y - 1344*x^2*z^5 + 3840*x^2*z^4 - 4592*x^2*z^3 + 3060*x^2*z^2 - 1180*x^2*z + 211*x^2 - 14*x*y^6 + 168*x*y^5*z - 96*x*y^5 - 840*x*y^4*z^2 + 960*x*y^4*z - 298*x*y^4 + 2240*x*y^3*z^3 - 3840*x*y^3*z^2 + 2360*x*y^3*z - 560*x*y^3 - 3360*x*y^2*z^4 + 7680*x*y^2*z^3 - 7008*x*y^2*z^2 + 3246*x*y^2*z - 658*x*y^2 + 2688*x*y*z^5 - 7680*x*y*z^4 + 9248*x*y*z^3 - 6264*x*y*z^2 + 2464*x*y*z - 454*x*y - 896*x*z^6 + 3072*x*z^5 - 4576*x*z^4 + 4024*x*z^3 - 2296*x*z^2 + 804*x*z - 145*x + 2*y^7 - 28*y^6*z + 16*y^6 + 168*y^5*z^2 - 192*y^5*z + 60*y^5 - 560*y^4*z^3 + 960*y^4*z^2 - 594*y^4*z + 143*y^4 + 1120*y^3*z^4 - 2560*y^3*z^3 + 2352*y^3*z^2 - 1106*y^3*z + 228*y^3 - 1344*y^2*z^5 + 3840*y^2*z^4 - 4656*y^2*z^3 + 3204*y^2*z^2 - 1284*y^2*z + 245*y^2 + 896*y*z^6 - 3072*y*z^5 + 4608*y*z^4 - 4120*y*z^3 + 2400*y*z^2 - 862*y*z + 167*y - 256*z^7 + 1024*z^6 - 1824*z^5 + 1984*z^4 - 1488*z^3 + 768*z^2 - 261*z + 1, -4*x^7 + 28*x^6*y - 56*x^6*z + 32*x^6 - 84*x^5*y^2 + 336*x^5*y*z - 192*x^5*y - 336*x^5*z^2 + 384*x^5*z - 116*x^5 + 140*x^4*y^3 - 840*x^4*y^2*z + 480*x^4*y^2 + 1680*x^4*y*z^2 - 1920*x^4*y*z + 584*x^4*y - 1120*x^4*z^3 + 1920*x^4*z^2 - 1156*x^4*z + 261*x^4 - 140*x^3*y^4 + 1120*x^3*y^3*z - 640*x^3*y^3 - 3360*x^3*y^2*z^2 + 3840*x^3*y^2*z - 1176*x^3*y^2 + 4480*x^3*y*z^3 - 7680*x^3*y*z^2 + 4656*x^3*y*z - 1068*x^3*y - 2240*x^3*z^4 + 5120*x^3*z^3 - 4608*x^3*z^2 + 2060*x^3*z - 402*x^3 + 84*x^2*y^5 - 840*x^2*y^4*z + 480*x^2*y^4 + 3360*x^2*y^3*z^2 - 3840*x^2*y^3*z + 1184*x^2*y^3 - 6720*x^2*y^2*z^3 + 11520*x^2*y^2*z^2 - 7032*x^2*y^2*z + 1638*x^2*y^2 + 6720*x^2*y*z^4 - 15360*x^2*y*z^3 + 13920*x^2*y*z^2 - 6324*x^2*y*z + 1258*x^2*y - 2688*x^2*z^5 + 7680*x^2*z^4 - 9184*x^2*z^3 + 6096*x^2*z^2 - 2348*x^2*z + 421*x^2 - 28*x*y^6 + 336*x*y^5*z - 192*x*y^5 - 1680*x*y^4*z^2 + 1920*x*y^4*z - 596*x*y^4 + 4480*x*y^3*z^3 - 7680*x*y^3*z^2 + 4720*x*y^3*z - 1116*x*y^3 - 6720*x*y^2*z^4 + 15360*x*y^2*z^3 - 14016*x*y^2*z^2 + 6468*x*y^2*z - 1310*x*y^2 + 5376*x*y*z^5 - 15360*x*y*z^4 + 18496*x*y*z^3 - 12480*x*y*z^2 + 4904*x*y*z - 906*x*y - 1792*x*z^6 + 6144*x*z^5 - 9152*x*z^4 + 8016*x*z^3 - 4568*x*z^2 + 1604*x*z - 290*x + 4*y^7 - 56*y^6*z + 32*y^6 + 336*y^5*z^2 - 384*y^5*z + 120*y^5 - 1120*y^4*z^3 + 1920*y^4*z^2 - 1188*y^4*z + 285*y^4 + 2240*y^3*z^4 - 5120*y^3*z^3 + 4704*y^3*z^2 - 2204*y^3*z + 454*y^3 - 2688*y^2*z^5 + 7680*y^2*z^4 - 9312*y^2*z^3 + 6384*y^2*z^2 - 2556*y^2*z + 489*y^2 + 1792*y*z^6 - 6144*y*z^5 + 9216*y*z^4 - 8208*y*z^3 + 4776*y*z^2 - 1720*y*z + 336*y - 512*z^7 + 2048*z^6 - 3648*z^5 + 3952*z^4 - 2960*z^3 + 1532*z^2 - 521*z + 1, -2*x^7 + 14*x^6*y - 28*x^6*z + 16*x^6 - 42*x^5*y^2 + 168*x^5*y*z - 96*x^5*y - 168*x^5*z^2 + 192*x^5*z - 58*x^5 + 70*x^4*y^3 - 420*x^4*y^2*z + 240*x^4*y^2 + 840*x^4*y*z^2 - 960*x^4*y*z + 292*x^4*y - 560*x^4*z^3 + 960*x^4*z^2 - 578*x^4*z + 132*x^4 - 70*x^3*y^4 + 560*x^3*y^3*z - 320*x^3*y^3 - 1680*x^3*y^2*z^2 + 1920*x^3*y^2*z - 588*x^3*y^2 + 2240*x^3*y*z^3 - 3840*x^3*y*z^2 + 2328*x^3*y*z - 540*x^3*y - 1120*x^3*z^4 + 2560*x^3*z^3 - 2304*x^3*z^2 + 1042*x^3*z - 206*x^3 + 42*x^2*y^5 - 420*x^2*y^4*z + 240*x^2*y^4 + 1680*x^2*y^3*z^2 - 1920*x^2*y^3*z + 592*x^2*y^3 - 3360*x^2*y^2*z^3 + 5760*x^2*y^2*z^2 - 3516*x^2*y^2*z + 828*x^2*y^2 + 3360*x^2*y*z^4 - 7680*x^2*y*z^3 + 6960*x^2*y*z^2 - 3198*x^2*y*z + 644*x^2*y - 1344*x^2*z^5 + 3840*x^2*z^4 - 4592*x^2*z^3 + 3084*x^2*z^2 - 1204*x^2*z + 218*x^2 - 14*x*y^6 + 168*x*y^5*z - 96*x*y^5 - 840*x*y^4*z^2 + 960*x*y^4*z - 298*x*y^4 + 2240*x*y^3*z^3 - 3840*x*y^3*z^2 + 2360*x*y^3*z - 564*x*y^3 - 3360*x*y^2*z^4 + 7680*x*y^2*z^3 - 7008*x*y^2*z^2 + 3270*x*y^2*z - 670*x*y^2 + 2688*x*y*z^5 - 7680*x*y*z^4 + 9248*x*y*z^3 - 6312*x*y*z^2 + 2512*x*y*z - 468*x*y - 896*x*z^6 + 3072*x*z^5 - 4576*x*z^4 + 4056*x*z^3 - 2344*x*z^2 + 832*x*z - 153*x + 2*y^7 - 28*y^6*z + 16*y^6 + 168*y^5*z^2 - 192*y^5*z + 60*y^5 - 560*y^4*z^3 + 960*y^4*z^2 - 594*y^4*z + 144*y^4 + 1120*y^3*z^4 - 2560*y^3*z^3 + 2352*y^3*z^2 - 1114*y^3*z + 232*y^3 - 1344*y^2*z^5 + 3840*y^2*z^4 - 4656*y^2*z^3 + 3228*y^2*z^2 - 1308*y^2*z + 252*y^2 + 896*y*z^6 - 3072*y*z^5 + 4608*y*z^4 - 4152*y*z^3 + 2448*y*z^2 - 890*y*z + 175*y - 256*z^7 + 1024*z^6 - 1824*z^5 + 2000*z^4 - 1520*z^3 + 796*z^2 - 276*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -1, 1, -2 ], [ -1, 1, -1 ], [ -1, 2, -1 ] ], "source_shift": [ 1, 2, 2 ], "target_matrix": [ [ 1, 1, 1 ], [ 2, 1, 1 ], [ 1, 2, 1 ] ], "target_shift": [ -53, -108, -58 ], "inverse_recipe": { "p": 4, "q": 3, "f": "x**4 + x**2 + x", "g": "x**3", "h": "y*z + z**2 + z", "polynomials": { "f": [ [ [ 4, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "g": [ [ [ 3, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 0, 1 ], 1 ], [ [ 0, 0, 2 ], 1 ] ] }, "h_multiplier": 2 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "Maybe" ], "seconds": [ 1.0156197360010992 ], "surface_features": [ 3.3636490573235838, 4.267077816740435, 3.0492180226701815, 0.0, 3.4662929213921916, 4834 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B220", "pair": 220, "family": "bijection", "mapping": "F(x,y,z)=(4*x^7 + 28*x^6*y - 28*x^6*z - 60*x^6 + 84*x^5*y^2 - 168*x^5*y*z - 360*x^5*y + 84*x^5*z^2 + 360*x^5*z + 392*x^5 + 140*x^4*y^3 - 420*x^4*y^2*z - 900*x^4*y^2 + 420*x^4*y*z^2 + 1800*x^4*y*z + 1960*x^4*y - 140*x^4*z^3 - 900*x^4*z^2 - 1964*x^4*z - 1446*x^4 + 140*x^3*y^4 - 560*x^3*y^3*z - 1200*x^3*y^3 + 840*x^3*y^2*z^2 + 3600*x^3*y^2*z + 3920*x^3*y^2 - 560*x^3*y*z^3 - 3600*x^3*y*z^2 - 7856*x^3*y*z - 5780*x^3*y + 140*x^3*z^4 + 1200*x^3*z^3 + 3936*x^3*z^2 + 5812*x^3*z + 3257*x^3 + 84*x^2*y^5 - 420*x^2*y^4*z - 900*x^2*y^4 + 840*x^2*y^3*z^2 + 3600*x^2*y^3*z + 3920*x^2*y^3 - 840*x^2*y^2*z^3 - 5400*x^2*y^2*z^2 - 11784*x^2*y^2*z - 8664*x^2*y^2 + 420*x^2*y*z^4 + 3600*x^2*y*z^3 + 11808*x^2*y*z^2 + 17424*x^2*y*z + 9743*x^2*y - 84*x^2*z^5 - 900*x^2*z^4 - 3944*x^2*z^3 - 8760*x^2*z^2 - 9839*x^2*z - 4507*x^2 + 28*x*y^6 - 168*x*y^5*z - 360*x*y^5 + 420*x*y^4*z^2 + 1800*x*y^4*z + 1960*x*y^4 - 560*x*y^3*z^3 - 3600*x*y^3*z^2 - 7856*x*y^3*z - 5772*x*y^3 + 420*x*y^2*z^4 + 3600*x*y^2*z^3 + 11808*x*y^2*z^2 + 17412*x*y^2*z + 9715*x*y^2 - 168*x*y*z^5 - 1800*x*y*z^4 - 7888*x*y*z^3 - 17508*x*y*z^2 - 19622*x*y*z - 8942*x*y + 28*x*z^6 + 360*x*z^5 + 1976*x*z^4 + 5868*x*z^3 + 9907*x*z^2 + 9082*x*z + 3562*x + 4*y^7 - 28*y^6*z - 60*y^6 + 84*y^5*z^2 + 360*y^5*z + 392*y^5 - 140*y^4*z^3 - 900*y^4*z^2 - 1964*y^4*z - 1442*y^4 + 140*y^3*z^4 + 1200*y^3*z^3 + 3936*y^3*z^2 + 5800*y^3*z + 3229*y^3 - 84*y^2*z^5 - 900*y^2*z^4 - 3944*y^2*z^3 - 8748*y^2*z^2 - 9783*y^2*z - 4435*y^2 + 28*y*z^6 + 360*y*z^5 + 1976*y*z^4 + 5864*y*z^3 + 9879*y*z^2 + 9006*y*z + 3504*y - 4*z^7 - 60*z^6 - 396*z^5 - 1474*z^4 - 3325*z^3 - 4571*z^2 - 3589*z - 1, 2*x^7 + 14*x^6*y - 14*x^6*z - 30*x^6 + 42*x^5*y^2 - 84*x^5*y*z - 180*x^5*y + 42*x^5*z^2 + 180*x^5*z + 196*x^5 + 70*x^4*y^3 - 210*x^4*y^2*z - 450*x^4*y^2 + 210*x^4*y*z^2 + 900*x^4*y*z + 980*x^4*y - 70*x^4*z^3 - 450*x^4*z^2 - 982*x^4*z - 722*x^4 + 70*x^3*y^4 - 280*x^3*y^3*z - 600*x^3*y^3 + 420*x^3*y^2*z^2 + 1800*x^3*y^2*z + 1960*x^3*y^2 - 280*x^3*y*z^3 - 1800*x^3*y*z^2 - 3928*x^3*y*z - 2886*x^3*y + 70*x^3*z^4 + 600*x^3*z^3 + 1968*x^3*z^2 + 2902*x^3*z + 1621*x^3 + 42*x^2*y^5 - 210*x^2*y^4*z - 450*x^2*y^4 + 420*x^2*y^3*z^2 + 1800*x^2*y^3*z + 1960*x^2*y^3 - 420*x^2*y^2*z^3 - 2700*x^2*y^2*z^2 - 5892*x^2*y^2*z - 4326*x^2*y^2 + 210*x^2*y*z^4 + 1800*x^2*y*z^3 + 5904*x^2*y*z^2 + 8700*x^2*y*z + 4849*x^2*y - 42*x^2*z^5 - 450*x^2*z^4 - 1972*x^2*z^3 - 4374*x^2*z^2 - 4897*x^2*z - 2233*x^2 + 14*x*y^6 - 84*x*y^5*z - 180*x*y^5 + 210*x*y^4*z^2 + 900*x*y^4*z + 980*x*y^4 - 280*x*y^3*z^3 - 1800*x*y^3*z^2 - 3928*x*y^3*z - 2882*x*y^3 + 210*x*y^2*z^4 + 1800*x*y^2*z^3 + 5904*x*y^2*z^2 + 8694*x*y^2*z + 4835*x*y^2 - 84*x*y*z^5 - 900*x*y*z^4 - 3944*x*y*z^3 - 8742*x*y*z^2 - 9766*x*y*z - 4430*x*y + 14*x*z^6 + 180*x*z^5 + 988*x*z^4 + 2930*x*z^3 + 4931*x*z^2 + 4500*x*z + 1755*x + 2*y^7 - 14*y^6*z - 30*y^6 + 42*y^5*z^2 + 180*y^5*z + 196*y^5 - 70*y^4*z^3 - 450*y^4*z^2 - 982*y^4*z - 720*y^4 + 70*y^3*z^4 + 600*y^3*z^3 + 1968*y^3*z^2 + 2896*y^3*z + 1607*y^3 - 42*y^2*z^5 - 450*y^2*z^4 - 1972*y^2*z^3 - 4368*y^2*z^2 - 4869*y^2*z - 2197*y^2 + 14*y*z^6 + 180*y*z^5 + 988*y*z^4 + 2928*y*z^3 + 4917*y*z^2 + 4462*y*z + 1727*y - 2*z^7 - 30*z^6 - 198*z^5 - 736*z^4 - 1655*z^3 - 2265*z^2 - 1770*z - 1, -2*x^7 - 14*x^6*y + 14*x^6*z + 30*x^6 - 42*x^5*y^2 + 84*x^5*y*z + 180*x^5*y - 42*x^5*z^2 - 180*x^5*z - 196*x^5 - 70*x^4*y^3 + 210*x^4*y^2*z + 450*x^4*y^2 - 210*x^4*y*z^2 - 900*x^4*y*z - 980*x^4*y + 70*x^4*z^3 + 450*x^4*z^2 + 982*x^4*z + 720*x^4 - 70*x^3*y^4 + 280*x^3*y^3*z + 600*x^3*y^3 - 420*x^3*y^2*z^2 - 1800*x^3*y^2*z - 1960*x^3*y^2 + 280*x^3*y*z^3 + 1800*x^3*y*z^2 + 3928*x^3*y*z + 2878*x^3*y - 70*x^3*z^4 - 600*x^3*z^3 - 1968*x^3*z^2 - 2894*x^3*z - 1605*x^3 - 42*x^2*y^5 + 210*x^2*y^4*z + 450*x^2*y^4 - 420*x^2*y^3*z^2 - 1800*x^2*y^3*z - 1960*x^2*y^3 + 420*x^2*y^2*z^3 + 2700*x^2*y^2*z^2 + 5892*x^2*y^2*z + 4314*x^2*y^2 - 210*x^2*y*z^4 - 1800*x^2*y*z^3 - 5904*x^2*y*z^2 - 8676*x^2*y*z - 4801*x^2*y + 42*x^2*z^5 + 450*x^2*z^4 + 1972*x^2*z^3 + 4362*x^2*z^2 + 4849*x^2*z + 2185*x^2 - 14*x*y^6 + 84*x*y^5*z + 180*x*y^5 - 210*x*y^4*z^2 - 900*x*y^4*z - 980*x*y^4 + 280*x*y^3*z^3 + 1800*x*y^3*z^2 + 3928*x*y^3*z + 2874*x*y^3 - 210*x*y^2*z^4 - 1800*x*y^2*z^3 - 5904*x*y^2*z^2 - 8670*x*y^2*z - 4787*x*y^2 + 84*x*y*z^5 + 900*x*y*z^4 + 3944*x*y*z^3 + 8718*x*y*z^2 + 9670*x*y*z + 4334*x*y - 14*x*z^6 - 180*x*z^5 - 988*x*z^4 - 2922*x*z^3 - 4883*x*z^2 - 4404*x*z - 1685*x - 2*y^7 + 14*y^6*z + 30*y^6 - 42*y^5*z^2 - 180*y^5*z - 196*y^5 + 70*y^4*z^3 + 450*y^4*z^2 + 982*y^4*z + 718*y^4 - 70*y^3*z^4 - 600*y^3*z^3 - 1968*y^3*z^2 - 2888*y^3*z - 1591*y^3 + 42*y^2*z^5 + 450*y^2*z^4 + 1972*y^2*z^3 + 4356*y^2*z^2 + 4821*y^2*z + 2149*y^2 - 14*y*z^6 - 180*y*z^5 - 988*y*z^4 - 2920*y*z^3 - 4869*y*z^2 - 4366*y*z - 1659*y + 2*z^7 + 30*z^6 + 198*z^5 + 734*z^4 + 1639*z^3 + 2217*z^2 + 1702*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -1, -1, 1 ], [ -1, -1, 2 ], [ -2, -1, 1 ] ], "source_shift": [ 2, 1, 1 ], "target_matrix": [ [ -2, -1, -1 ], [ -1, -1, -1 ], [ 1, 1, 2 ] ], "target_shift": [ 1240, 608, -570 ], "inverse_recipe": { "p": 3, "q": 4, "f": "x**3 + x**2 + x", "g": "x**4 + x", "h": "y*z + 2*y", "polynomials": { "f": [ [ [ 3, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "g": [ [ [ 4, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 1, 0 ], 2 ] ] }, "h_multiplier": 2 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0791211210016627 ], "surface_features": [ 3.3513033889619237, 4.292743271377071, 2.9542425094393248, 0.0, 3.492322787426559, 4772 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C216", "pair": 216, "family": "counterexample", "mapping": "F(x,y,z)=(3*x^7 - 30*x^6*y - 30*x^6*z + 36*x^6 + 126*x^5*y^2 + 261*x^5*y*z - 306*x^5*y + 126*x^5*z^2 - 318*x^5*z + 180*x^5 - 288*x^4*y^3 - 927*x^4*y^2*z + 1062*x^4*y^2 - 927*x^4*y*z^2 + 2283*x^4*y*z - 1269*x^4*y - 288*x^4*z^3 + 1155*x^4*z^2 - 1347*x^4*z + 496*x^4 + 387*x^3*y^4 + 1719*x^3*y^3*z - 1926*x^3*y^3 + 2673*x^3*y^2*z^2 - 6420*x^3*y^2*z + 3501*x^3*y^2 + 1719*x^3*y*z^3 - 6726*x^3*y*z^2 + 7710*x^3*y*z - 2779*x^3*y + 387*x^3*z^4 - 2208*x^3*z^3 + 3981*x^3*z^2 - 3007*x^3*z + 826*x^3 - 306*x^2*y^5 - 1755*x^2*y^4*z + 1926*x^2*y^4 - 3771*x^2*y^3*z^2 + 8835*x^2*y^3*z - 4725*x^2*y^3 - 3771*x^2*y^2*z^3 + 14373*x^2*y^2*z^2 - 16173*x^2*y^2*z + 5700*x^2*y^2 - 1755*x^2*y*z^4 + 9783*x^2*y*z^3 - 17361*x^2*y*z^2 + 12843*x^2*y*z - 3435*x^2*y - 306*x^2*z^5 + 2343*x^2*z^4 - 5805*x^2*z^3 + 6747*x^2*z^2 - 3801*x^2*z + 843*x^2 + 132*x*y^6 + 936*x*y^5*z - 1008*x*y^5 + 2601*x*y^4*z^2 - 5952*x*y^4*z + 3123*x*y^4 + 3597*x*y^3*z^3 - 13344*x*y^3*z^2 + 14727*x*y^3*z - 5075*x*y^3 + 2601*x*y^2*z^4 - 14130*x*y^2*z^3 + 24642*x*y^2*z^2 - 17823*x*y^2*z + 4632*x*y^2 + 936*x*y*z^5 - 7026*x*y*z^4 + 17163*x*y*z^3 - 19554*x*y*z^2 + 10734*x*y*z - 2301*x*y + 132*x*z^6 - 1308*x*z^5 + 4173*x*z^4 - 6634*x*z^3 + 5748*x*z^2 - 2619*x*z + 496*x - 24*y^7 - 204*y^6*z + 216*y^6 - 702*y^5*z^2 + 1572*y^5*z - 810*y^5 - 1257*y^4*z^3 + 4542*y^4*z^2 - 4917*y^4*z + 1658*y^4 - 1257*y^3*z^4 + 6639*y^3*z^3 - 11364*y^3*z^2 + 8036*y^3*z - 2030*y^3 - 702*y^2*z^5 + 5151*y^2*z^4 - 12384*y^2*z^3 + 13788*y^2*z^2 - 7350*y^2*z + 1521*y^2 - 204*y*z^6 + 1992*y*z^5 - 6279*y*z^4 + 9805*y*z^3 - 8289*y*z^2 + 3645*y*z - 665*y - 24*z^7 + 300*z^6 - 1182*z^5 + 2407*z^4 - 2851*z^3 + 2007*z^2 - 777*z + 1, x^7 - 10*x^6*y - 10*x^6*z + 13*x^6 + 42*x^5*y^2 + 87*x^5*y*z - 111*x^5*y + 42*x^5*z^2 - 114*x^5*z + 69*x^5 - 96*x^4*y^3 - 309*x^4*y^2*z + 387*x^4*y^2 - 309*x^4*y*z^2 + 822*x^4*y*z - 489*x^4*y - 96*x^4*z^3 + 411*x^4*z^2 - 513*x^4*z + 196*x^4 + 129*x^3*y^4 + 573*x^3*y^3*z - 705*x^3*y^3 + 891*x^3*y^2*z^2 - 2322*x^3*y^2*z + 1356*x^3*y^2 + 573*x^3*y*z^3 - 2403*x^3*y*z^2 + 2952*x^3*y*z - 1103*x^3*y + 129*x^3*z^4 - 780*x^3*z^3 + 1506*x^3*z^2 - 1183*x^3*z + 328*x^3 - 102*x^2*y^5 - 585*x^2*y^4*z + 708*x^2*y^4 - 1257*x^2*y^3*z^2 + 3210*x^2*y^3*z - 1839*x^2*y^3 - 1257*x^2*y^2*z^3 + 5157*x^2*y^2*z^2 - 6225*x^2*y^2*z + 2271*x^2*y^2 - 585*x^2*y*z^4 + 3468*x^2*y*z^3 - 6603*x^2*y*z^2 + 5079*x^2*y*z - 1365*x^2*y - 102*x^2*z^5 + 822*x^2*z^4 - 2181*x^2*z^3 + 2640*x^2*z^2 - 1509*x^2*z + 331*x^2 + 44*x*y^6 + 312*x*y^5*z - 372*x*y^5 + 867*x*y^4*z^2 - 2172*x*y^4*z + 1221*x*y^4 + 1199*x*y^3*z^3 - 4809*x*y^3*z^2 + 5697*x*y^3*z - 2028*x*y^3 + 867*x*y^2*z^4 - 5028*x*y^2*z^3 + 9423*x*y^2*z^2 - 7083*x*y^2*z + 1839*x*y^2 + 312*x*y*z^5 - 2472*x*y*z^4 + 6483*x*y*z^3 - 7695*x*y*z^2 + 4272*x*y*z - 899*x*y + 44*x*z^6 - 456*x*z^5 + 1557*x*z^4 - 2580*x*z^3 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5625*x^2*y^3*z - 2886*x^2*y^3 - 2514*x^2*y^2*z^3 + 9216*x^2*y^2*z^2 - 9948*x^2*y^2*z + 3411*x^2*y^2 - 1170*x^2*y*z^4 + 6315*x^2*y*z^3 - 10758*x^2*y*z^2 + 7743*x^2*y*z - 2034*x^2*y - 204*x^2*z^5 + 1521*x^2*z^4 - 3624*x^2*z^3 + 4101*x^2*z^2 - 2268*x^2*z + 497*x^2 + 88*x*y^6 + 624*x*y^5*z - 636*x*y^5 + 1734*x*y^4*z^2 - 3780*x*y^4*z + 1902*x*y^4 + 2398*x*y^3*z^3 - 8535*x*y^3*z^2 + 9030*x*y^3*z - 3027*x*y^3 + 1734*x*y^2*z^4 - 9102*x*y^2*z^3 + 15219*x*y^2*z^2 - 10704*x*y^2*z + 2733*x*y^2 + 624*x*y*z^5 - 4554*x*y*z^4 + 10680*x*y*z^3 - 11838*x*y*z^2 + 6378*x*y*z - 1351*x*y + 88*x*z^6 - 852*x*z^5 + 2616*x*z^4 - 4050*x*z^3 + 3444*x*z^2 - 1550*x*z + 291*x - 16*y^7 - 136*y^6*z + 136*y^6 - 468*y^5*z^2 + 996*y^5*z - 492*y^5 - 838*y^4*z^3 + 2898*y^4*z^2 - 3006*y^4*z + 986*y^4 - 838*y^3*z^4 + 4267*y^3*z^3 - 6996*y^3*z^2 + 4809*y^3*z - 1194*y^3 - 468*y^2*z^5 + 3333*y^2*z^4 - 7680*y^2*z^3 + 8313*y^2*z^2 - 4350*y^2*z + 890*y^2 - 136*y*z^6 + 1296*y*z^5 - 3924*y*z^4 + 5962*y*z^3 - 4944*y*z^2 + 2147*y*z - 388*y - 16*z^7 + 196*z^6 - 744*z^5 + 1476*z^4 - 1716*z^3 + 1193*z^2 - 458*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ 1, -2, -1 ], [ 1, -1, -2 ], [ 1, -1, -1 ] ], "source_shift": [ 1, 1, 1 ], "target_matrix": [ [ 3, 2, 2 ], [ 1, 1, 2 ], [ 2, 1, 1 ] ], "target_shift": [ -129, -51, -75 ], "collision_preimages": [ [ "-7/4", "-1/4", "-1/4" ], [ "19", "11/2", "8" ], [ "18", "15/2", "5" ] ], "collision_image": [ "-519/4", "-205/4", "-151/2" ] }, "probe_target": "No", "jev_answers": [ "Maybe" ], "seconds": [ 1.0456951339983789 ], "surface_features": [ 3.446340374874711, 4.391675953237296, 3.1230345297535065, 0.0, 3.5703025727318782, 4865 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B205", "pair": 205, "family": "bijection", "mapping": "F(x,y,z)=(x^7 - 7*x^6*y - 14*x^6*z + 15*x^6 + 21*x^5*y^2 + 84*x^5*y*z - 90*x^5*y + 84*x^5*z^2 - 180*x^5*z + 99*x^5 - 35*x^4*y^3 - 210*x^4*y^2*z + 225*x^4*y^2 - 420*x^4*y*z^2 + 900*x^4*y*z - 495*x^4*y - 280*x^4*z^3 + 900*x^4*z^2 - 989*x^4*z + 371*x^4 + 35*x^3*y^4 + 280*x^3*y^3*z - 300*x^3*y^3 + 840*x^3*y^2*z^2 - 1800*x^3*y^2*z + 990*x^3*y^2 + 1120*x^3*y*z^3 - 3600*x^3*y*z^2 + 3956*x^3*y*z - 1485*x^3*y + 560*x^3*z^4 - 2400*x^3*z^3 + 3952*x^3*z^2 - 2959*x^3*z + 853*x^3 - 21*x^2*y^5 - 210*x^2*y^4*z + 225*x^2*y^4 - 840*x^2*y^3*z^2 + 1800*x^2*y^3*z - 990*x^2*y^3 - 1680*x^2*y^2*z^3 + 5400*x^2*y^2*z^2 - 5934*x^2*y^2*z + 2229*x^2*y^2 - 1680*x^2*y*z^4 + 7200*x^2*y*z^3 - 11856*x^2*y*z^2 + 8883*x^2*y*z - 2566*x^2*y - 672*x^2*z^5 + 3600*x^2*z^4 - 7896*x^2*z^3 + 8850*x^2*z^2 - 5086*x^2*z + 1209*x^2 + 7*x*y^6 + 84*x*y^5*z - 90*x*y^5 + 420*x*y^4*z^2 - 900*x*y^4*z + 495*x*y^4 + 1120*x*y^3*z^3 - 3600*x*y^3*z^2 + 3956*x*y^3*z - 1487*x*y^3 + 1680*x*y^2*z^4 - 7200*x*y^2*z^3 + 11856*x*y^2*z^2 - 8889*x*y^2*z + 2573*x*y^2 + 1344*x*y*z^5 - 7200*x*y*z^4 + 15792*x*y*z^3 - 17712*x*y*z^2 + 10200*x*y*z - 2436*x*y + 448*x*z^6 - 2880*x*z^5 + 7888*x*z^4 - 11764*x*z^3 + 10108*x*z^2 - 4780*x*z + 983*x - y^7 - 14*y^6*z + 15*y^6 - 84*y^5*z^2 + 180*y^5*z - 99*y^5 - 280*y^4*z^3 + 900*y^4*z^2 - 989*y^4*z + 372*y^4 - 560*y^3*z^4 + 2400*y^3*z^3 - 3952*y^3*z^2 + 2965*y^3*z - 860*y^3 - 672*y^2*z^5 + 3600*y^2*z^4 - 7896*y^2*z^3 + 8862*y^2*z^2 - 5114*y^2*z + 1227*y^2 - 448*y*z^6 + 2880*y*z^5 - 7888*y*z^4 + 11772*y*z^3 - 10136*y*z^2 + 4815*y*z - 997*y - 128*z^7 + 960*z^6 - 3152*z^5 + 5864*z^4 - 6696*z^3 + 4725*z^2 - 1927*z + 1, -2*x^7 + 14*x^6*y + 28*x^6*z - 30*x^6 - 42*x^5*y^2 - 168*x^5*y*z + 180*x^5*y - 168*x^5*z^2 + 360*x^5*z - 198*x^5 + 70*x^4*y^3 + 420*x^4*y^2*z - 450*x^4*y^2 + 840*x^4*y*z^2 - 1800*x^4*y*z + 990*x^4*y + 560*x^4*z^3 - 1800*x^4*z^2 + 1978*x^4*z - 744*x^4 - 70*x^3*y^4 - 560*x^3*y^3*z + 600*x^3*y^3 - 1680*x^3*y^2*z^2 + 3600*x^3*y^2*z - 1980*x^3*y^2 - 2240*x^3*y*z^3 + 7200*x^3*y*z^2 - 7912*x^3*y*z + 2978*x^3*y - 1120*x^3*z^4 + 4800*x^3*z^3 - 7904*x^3*z^2 + 5934*x^3*z - 1721*x^3 + 42*x^2*y^5 + 420*x^2*y^4*z - 450*x^2*y^4 + 1680*x^2*y^3*z^2 - 3600*x^2*y^3*z + 1980*x^2*y^3 + 3360*x^2*y^2*z^3 - 10800*x^2*y^2*z^2 + 11868*x^2*y^2*z - 4470*x^2*y^2 + 3360*x^2*y*z^4 - 14400*x^2*y*z^3 + 23712*x^2*y*z^2 - 17814*x^2*y*z + 5177*x^2*y + 1344*x^2*z^5 - 7200*x^2*z^4 + 15792*x^2*z^3 - 17748*x^2*z^2 + 10262*x^2*z - 2461*x^2 - 14*x*y^6 - 168*x*y^5*z + 180*x*y^5 - 840*x*y^4*z^2 + 1800*x*y^4*z - 990*x*y^4 - 2240*x*y^3*z^3 + 7200*x*y^3*z^2 - 7912*x*y^3*z + 2982*x*y^3 - 3360*x*y^2*z^4 + 14400*x*y^2*z^3 - 23712*x*y^2*z^2 + 17826*x*y^2*z - 5191*x*y^2 - 2688*x*y*z^5 + 14400*x*y*z^4 - 31584*x*y*z^3 + 35520*x*y*z^2 - 20580*x*y*z + 4958*x*y - 896*x*z^6 + 5760*x*z^5 - 15776*x*z^4 + 23592*x*z^3 - 20396*x*z^2 + 9732*x*z - 2024*x + 2*y^7 + 28*y^6*z - 30*y^6 + 168*y^5*z^2 - 360*y^5*z + 198*y^5 + 560*y^4*z^3 - 1800*y^4*z^2 + 1978*y^4*z - 746*y^4 + 1120*y^3*z^4 - 4800*y^3*z^3 + 7904*y^3*z^2 - 5946*y^3*z + 1735*y^3 + 1344*y^2*z^5 - 7200*y^2*z^4 + 15792*y^2*z^3 - 17772*y^2*z^2 + 10318*y^2*z - 2497*y^2 + 896*y*z^6 - 5760*y*z^5 + 15776*y*z^4 - 23608*y*z^3 + 20452*y*z^2 - 9802*y*z + 2054*y + 256*z^7 - 1920*z^6 + 6304*z^5 - 11760*z^4 + 13512*z^3 - 9622*z^2 + 3969*z - 1, -x^7 + 7*x^6*y + 14*x^6*z - 15*x^6 - 21*x^5*y^2 - 84*x^5*y*z + 90*x^5*y - 84*x^5*z^2 + 180*x^5*z - 99*x^5 + 35*x^4*y^3 + 210*x^4*y^2*z - 225*x^4*y^2 + 420*x^4*y*z^2 - 900*x^4*y*z + 495*x^4*y + 280*x^4*z^3 - 900*x^4*z^2 + 989*x^4*z - 371*x^4 - 35*x^3*y^4 - 280*x^3*y^3*z + 300*x^3*y^3 - 840*x^3*y^2*z^2 + 1800*x^3*y^2*z - 990*x^3*y^2 - 1120*x^3*y*z^3 + 3600*x^3*y*z^2 - 3956*x^3*y*z + 1485*x^3*y - 560*x^3*z^4 + 2400*x^3*z^3 - 3952*x^3*z^2 + 2959*x^3*z - 854*x^3 + 21*x^2*y^5 + 210*x^2*y^4*z - 225*x^2*y^4 + 840*x^2*y^3*z^2 - 1800*x^2*y^3*z + 990*x^2*y^3 + 1680*x^2*y^2*z^3 - 5400*x^2*y^2*z^2 + 5934*x^2*y^2*z - 2229*x^2*y^2 + 1680*x^2*y*z^4 - 7200*x^2*y*z^3 + 11856*x^2*y*z^2 - 8883*x^2*y*z + 2569*x^2*y + 672*x^2*z^5 - 3600*x^2*z^4 + 7896*x^2*z^3 - 8850*x^2*z^2 + 5092*x^2*z - 1216*x^2 - 7*x*y^6 - 84*x*y^5*z + 90*x*y^5 - 420*x*y^4*z^2 + 900*x*y^4*z - 495*x*y^4 - 1120*x*y^3*z^3 + 3600*x*y^3*z^2 - 3956*x*y^3*z + 1487*x*y^3 - 1680*x*y^2*z^4 + 7200*x*y^2*z^3 - 11856*x*y^2*z^2 + 8889*x*y^2*z - 2576*x*y^2 - 1344*x*y*z^5 + 7200*x*y*z^4 - 15792*x*y*z^3 + 17712*x*y*z^2 - 10212*x*y*z + 2450*x*y - 448*x*z^6 + 2880*x*z^5 - 7888*x*z^4 + 11764*x*z^3 - 10120*x*z^2 + 4808*x*z - 1001*x + y^7 + 14*y^6*z - 15*y^6 + 84*y^5*z^2 - 180*y^5*z + 99*y^5 + 280*y^4*z^3 - 900*y^4*z^2 + 989*y^4*z - 372*y^4 + 560*y^3*z^4 - 2400*y^3*z^3 + 3952*y^3*z^2 - 2965*y^3*z + 861*y^3 + 672*y^2*z^5 - 3600*y^2*z^4 + 7896*y^2*z^3 - 8862*y^2*z^2 + 5120*y^2*z - 1234*y^2 + 448*y*z^6 - 2880*y*z^5 + 7888*y*z^4 - 11772*y*z^3 + 10148*y*z^2 - 4843*y*z + 1015*y + 128*z^7 - 960*z^6 + 3152*z^5 - 5864*z^4 + 6704*z^3 - 4753*z^2 + 1962*z - 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ 1, -1, -2 ], [ 1, -1, -1 ], [ 1, -2, -1 ] ], "source_shift": [ 2, 2, 2 ], "target_matrix": [ [ 1, 1, 1 ], [ -2, -1, -1 ], [ -1, -2, -1 ] ], "target_shift": [ -353, 739, 369 ], "inverse_recipe": { "p": 3, "q": 4, "f": "x**3 + x**2 + x", "g": "x**4 + x**2 + x", "h": "y*z", "polynomials": { "f": [ [ [ 3, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "g": [ [ [ 4, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.058614593999664 ], "surface_features": [ 3.591509499984923, 4.550472957106564, 3.225309281725863, 0.0, 3.732660163085373, 4860 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C142", "pair": 142, "family": "counterexample", "mapping": "F(x,y,z)=(-24*x^7 - 132*x^6*y - 124*x^6*z - 200*x^6 - 306*x^5*y^2 - 576*x^5*y*z - 912*x^5*y - 270*x^5*z^2 - 864*x^5*z - 690*x^5 - 387*x^4*y^3 - 1095*x^4*y^2*z - 1710*x^4*y^2 - 1029*x^4*y*z^2 - 3240*x^4*y*z - 2559*x^4*y - 321*x^4*z^3 - 1530*x^4*z^2 - 2433*x^4*z - 1301*x^4 - 288*x^3*y^4 - 1089*x^3*y^3*z - 1686*x^3*y^3 - 1539*x^3*y^2*z^2 - 4794*x^3*y^2*z - 3747*x^3*y^2 - 963*x^3*y*z^3 - 4530*x^3*y*z^2 - 7119*x^3*y*z - 3770*x^3*y - 225*x^3*z^4 - 1422*x^3*z^3 - 3372*x^3*z^2 - 3585*x^3*z - 1459*x^3 - 126*x^2*y^5 - 597*x^2*y^4*z - 921*x^2*y^4 - 1128*x^2*y^3*z^2 - 3495*x^2*y^3*z - 2706*x^2*y^3 - 1062*x^2*y^2*z^3 - 4959*x^2*y^2*z^2 - 7710*x^2*y^2*z - 4044*x^2*y^2 - 498*x^2*y*z^4 - 3117*x^2*y*z^3 - 7302*x^2*y*z^2 - 7680*x^2*y*z - 3096*x^2*y - 93*x^2*z^5 - 732*x^2*z^4 - 2298*x^2*z^3 - 3636*x^2*z^2 - 2934*x^2*z - 976*x^2 - 30*x*y^6 - 171*x*y^5*z - 264*x*y^5 - 405*x*y^4*z^2 - 1254*x*y^4*z - 963*x*y^4 - 510*x*y^3*z^3 - 2376*x*y^3*z^2 - 3660*x*y^3*z - 1903*x*y^3 - 360*x*y^2*z^4 - 2244*x*y^2*z^3 - 5202*x*y^2*z^2 - 5418*x*y^2*z - 2163*x*y^2 - 135*x*y*z^5 - 1056*x*y*z^4 - 3276*x*y*z^3 - 5127*x*y*z^2 - 4092*x*y*z - 1345*x*y - 21*x*z^6 - 198*x*z^5 - 771*x*z^4 - 1612*x*z^3 - 1929*x*z^2 - 1264*x*z - 357*x - 3*y^7 - 20*y^6*z - 31*y^6 - 57*y^5*z^2 - 177*y^5*z - 135*y^5 - 90*y^4*z^3 - 420*y^4*z^2 - 642*y^4*z - 331*y^4 - 85*y^3*z^4 - 530*y^3*z^3 - 1218*y^3*z^2 - 1257*y^3*z - 497*y^3 - 48*y^2*z^5 - 375*y^2*z^4 - 1152*y^2*z^3 - 1785*y^2*z^2 - 1410*y^2*z - 460*y^2 - 15*y*z^6 - 141*y*z^5 - 543*y*z^4 - 1123*y*z^3 - 1329*y*z^2 - 863*y*z - 243*y - 2*z^7 - 22*z^6 - 102*z^5 - 264*z^4 - 416*z^3 - 403*z^2 - 224*z - 1, -48*x^7 - 264*x^6*y - 248*x^6*z - 364*x^6 - 612*x^5*y^2 - 1152*x^5*y*z - 1644*x^5*y - 540*x^5*z^2 - 1560*x^5*z - 1152*x^5 - 774*x^4*y^3 - 2190*x^4*y^2*z - 3051*x^4*y^2 - 2058*x^4*y*z^2 - 5790*x^4*y*z - 4224*x^4*y - 642*x^4*z^3 - 2739*x^4*z^2 - 4020*x^4*z - 2044*x^4 - 576*x^3*y^4 - 2178*x^3*y^3*z - 2976*x^3*y^3 - 3078*x^3*y^2*z^2 - 8475*x^3*y^2*z - 6114*x^3*y^2 - 1926*x^3*y*z^3 - 8022*x^3*y*z^2 - 11628*x^3*y*z - 5854*x^3*y - 450*x^3*z^4 - 2523*x^3*z^3 - 5514*x^3*z^2 - 5570*x^3*z - 2196*x^3 - 252*x^2*y^5 - 1194*x^2*y^4*z - 1608*x^2*y^4 - 2256*x^2*y^3*z^2 - 6111*x^2*y^3*z - 4365*x^2*y^3 - 2124*x^2*y^2*z^3 - 8685*x^2*y^2*z^2 - 12450*x^2*y^2*z - 6207*x^2*y^2 - 996*x^2*y*z^4 - 5469*x^2*y*z^3 - 11805*x^2*y*z^2 - 11796*x^2*y*z - 4608*x^2*y - 186*x^2*z^5 - 1287*x^2*z^4 - 3720*x^2*z^3 - 5589*x^2*z^2 - 4368*x^2*z - 1427*x^2 - 60*x*y^6 - 342*x*y^5*z - 456*x*y^5 - 810*x*y^4*z^2 - 2169*x*y^4*z - 1536*x*y^4 - 1020*x*y^3*z^3 - 4116*x*y^3*z^2 - 5844*x*y^3*z - 2888*x*y^3 - 720*x*y^2*z^4 - 3894*x*y^2*z^3 - 8316*x*y^2*z^2 - 8229*x*y^2*z - 3186*x*y^2 - 270*x*y*z^5 - 1836*x*y*z^4 - 5244*x*y*z^3 - 7794*x*y*z^2 - 6030*x*y*z - 1946*x*y - 42*x*z^6 - 345*x*z^5 - 1236*x*z^4 - 2453*x*z^3 - 2844*x*z^2 - 1829*x*z - 512*x - 6*y^7 - 40*y^6*z - 53*y^6 - 114*y^5*z^2 - 303*y^5*z - 213*y^5 - 180*y^4*z^3 - 720*y^4*z^2 - 1014*y^4*z - 497*y^4 - 170*y^3*z^4 - 910*y^3*z^3 - 1926*y^3*z^2 - 1889*y^3*z - 725*y^3 - 96*y^2*z^5 - 645*y^2*z^4 - 1824*y^2*z^3 - 2685*y^2*z^2 - 2058*y^2*z - 659*y^2 - 30*y*z^6 - 243*y*z^5 - 861*y*z^4 - 1691*y*z^3 - 1941*y*z^2 - 1237*y*z - 345*y - 4*z^7 - 38*z^6 - 162*z^5 - 398*z^4 - 608*z^3 - 578*z^2 - 318*z - 1, 24*x^7 + 132*x^6*y + 124*x^6*z + 236*x^6 + 306*x^5*y^2 + 576*x^5*y*z + 1092*x^5*y + 270*x^5*z^2 + 1032*x^5*z + 918*x^5 + 387*x^4*y^3 + 1095*x^4*y^2*z + 2079*x^4*y^2 + 1029*x^4*y*z^2 + 3930*x^4*y*z + 3453*x^4*y + 321*x^4*z^3 + 1851*x^4*z^2 + 3279*x^4*z + 1862*x^4 + 288*x^3*y^4 + 1089*x^3*y^3*z + 2082*x^3*y^3 + 1539*x^3*y^2*z^2 + 5907*x^3*y^2*z + 5127*x^3*y^2 + 963*x^3*y*z^3 + 5568*x^3*y*z^2 + 9729*x^3*y*z + 5468*x^3*y + 225*x^3*z^4 + 1743*x^3*z^3 + 4602*x^3*z^2 + 5196*x^3*z + 2197*x^3 + 126*x^2*y^5 + 597*x^2*y^4*z + 1155*x^2*y^4 + 1128*x^2*y^3*z^2 + 4374*x^2*y^3*z + 3753*x^2*y^3 + 1062*x^2*y^2*z^3 + 6192*x^2*y^2*z^2 + 10680*x^2*y^2*z + 5943*x^2*y^2 + 498*x^2*y*z^4 + 3882*x^2*y*z^3 + 10101*x^2*y*z^2 + 11277*x^2*y*z + 4725*x^2*y + 93*x^2*z^5 + 909*x^2*z^4 + 3174*x^2*z^3 + 5334*x^2*z^2 + 4476*x^2*z + 1528*x^2 + 30*x*y^6 + 171*x*y^5*z + 336*x*y^5 + 405*x*y^4*z^2 + 1593*x*y^4*z + 1353*x*y^4 + 510*x*y^3*z^3 + 3012*x*y^3*z^2 + 5136*x*y^3*z + 2833*x*y^3 + 360*x*y^2*z^4 + 2838*x*y^2*z^3 + 7290*x*y^2*z^2 + 8058*x*y^2*z + 3345*x*y^2 + 135*x*y*z^5 + 1332*x*y*z^4 + 4584*x*y*z^3 + 7617*x*y*z^2 + 6324*x*y*z + 2137*x*y + 21*x*z^6 + 249*x*z^5 + 1077*x*z^4 + 2392*x*z^3 + 2979*x*z^2 + 2008*x*z + 575*x + 3*y^7 + 20*y^6*z + 40*y^6 + 57*y^5*z^2 + 228*y^5*z + 192*y^5 + 90*y^4*z^3 + 540*y^4*z^2 + 912*y^4*z + 499*y^4 + 85*y^3*z^4 + 680*y^3*z^3 + 1728*y^3*z^2 + 1893*y^3*z + 779*y^3 + 48*y^2*z^5 + 480*y^2*z^4 + 1632*y^2*z^3 + 2685*y^2*z^2 + 2208*y^2*z + 742*y^2 + 15*y*z^6 + 180*y*z^5 + 768*y*z^4 + 1687*y*z^3 + 2079*y*z^2 + 1391*y*z + 397*y + 2*z^7 + 28*z^6 + 144*z^5 + 396*z^4 + 650*z^3 + 649*z^2 + 366*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ 1, 1, 1 ], [ 2, 1, 1 ], [ 3, 3, 2 ] ], "source_shift": [ 1, 1, 1 ], "target_matrix": [ [ -1, -1, -1 ], [ -2, -1, -1 ], [ 1, 2, 1 ] ], "target_shift": [ 53, 75, -87 ], "collision_preimages": [ [ "0", "3/4", "-7/4" ], [ "-5/2", "8", "-11/2" ], [ "5/2", "7", "-23/2" ] ], "collision_image": [ "213/4", "151/2", "-349/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0817552939988673 ], "surface_features": [ 3.3084986063814927, 4.095169351431755, 3.038818787373656, 0.0, 3.3856733807321846, 4837 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C227", "pair": 227, "family": "counterexample", "mapping": "F(x,y,z)=(-8*x^7 + 64*x^6*y - 68*x^6*z + 80*x^6 - 216*x^5*y^2 + 468*x^5*y*z - 540*x^5*y - 246*x^5*z^2 + 588*x^5*z - 318*x^5 + 400*x^4*y^3 - 1320*x^4*y^2*z + 1500*x^4*y^2 + 1416*x^4*y*z^2 - 3324*x^4*y*z + 1752*x^4*y - 491*x^4*z^3 + 1788*x^4*z^2 - 1959*x^4*z + 672*x^4 - 440*x^3*y^4 + 1960*x^3*y^3*z - 2200*x^3*y^3 - 3204*x^3*y^2*z^2 + 7416*x^3*y^2*z - 3828*x^3*y^2 + 2269*x^3*y*z^3 - 8127*x^3*y*z^2 + 8679*x^3*y*z - 2917*x^3*y - 584*x^3*z^4 + 2879*x^3*z^3 - 4791*x^3*z^2 + 3328*x^3*z - 836*x^3 + 288*x^2*y^5 - 1620*x^2*y^4*z + 1800*x^2*y^4 + 3576*x^2*y^3*z^2 - 8184*x^2*y^3*z + 4152*x^2*y^3 - 3861*x^2*y^2*z^3 + 13653*x^2*y^2*z^2 - 14283*x^2*y^2*z + 4719*x^2*y^2 + 2031*x^2*y*z^4 - 9864*x^2*y*z^3 + 16005*x^2*y*z^2 - 10887*x^2*y*z + 2697*x^2*y - 414*x^2*z^5 + 2589*x^2*z^4 - 5814*x^2*z^3 + 6129*x^2*z^2 - 3117*x^2*z + 631*x^2 - 104*x*y^6 + 708*x*y^5*z - 780*x*y^5 - 1974*x*y^4*z^2 + 4476*x*y^4*z - 2238*x*y^4 + 2879*x*y^3*z^3 - 10077*x*y^3*z^2 + 10365*x*y^3*z - 3375*x*y^3 - 2310*x*y^2*z^4 + 11091*x*y^2*z^3 - 17637*x*y^2*z^2 + 11790*x*y^2*z - 2886*x*y^2 + 963*x*y*z^5 - 5943*x*y*z^4 + 13020*x*y*z^3 - 13434*x*y*z^2 + 6738*x*y*z - 1352*x*y - 162*x*z^6 + 1233*x*z^5 - 3501*x*z^4 + 4974*x*z^3 - 3837*x*z^2 + 1574*x*z - 277*x + 16*y^7 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"target_matrix": [ [ 1, 1, 1 ], [ -1, -2, -1 ], [ -2, -2, -3 ] ], "target_shift": [ -53, 87, 105 ], "collision_preimages": [ [ "7/4", "-1/4", "-1" ], [ "31/2", "8", "-5" ], [ "3/2", "5", "3" ] ], "collision_image": [ "-213/4", "349/4", "211/2" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.09810435899999 ], "surface_features": [ 3.6888534549524765, 4.547454710572725, 3.436162647040756, 0.0, 3.7801417759125644, 4972 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C209", "pair": 209, "family": "counterexample", "mapping": "F(x,y,z)=(-8*x^7 - 52*x^6*y - 44*x^6*z - 136*x^6 - 138*x^5*y^2 - 240*x^5*y*z - 672*x^5*y - 102*x^5*z^2 - 600*x^5*z - 774*x^5 - 195*x^4*y^3 - 519*x^4*y^2*z - 1350*x^4*y^2 - 453*x^4*y*z^2 - 2436*x^4*y*z - 3003*x^4*y - 129*x^4*z^3 - 1086*x^4*z^2 - 2733*x^4*z - 2174*x^4 - 159*x^3*y^4 - 573*x^3*y^3*z - 1410*x^3*y^3 - 765*x^3*y^2*z^2 - 3852*x^3*y^2*z - 4572*x^3*y^2 - 447*x^3*y*z^3 - 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1032*y^4 + 1920*y^3*z^4 + 7680*y^3*z^3 + 11520*y^3*z^2 + 7872*y^3*z + 2112*y^3 + 2208*y^2*z^5 + 11040*y^2*z^4 + 22080*y^2*z^3 + 22512*y^2*z^2 + 11904*y^2*z + 2652*y^2 + 1408*y*z^6 + 8448*y*z^5 + 21120*y*z^4 + 28592*y*z^3 + 22416*y*z^2 + 9782*y*z + 1882*y + 384*z^7 + 2688*z^6 + 8064*z^5 + 13602*z^4 + 14088*z^3 + 9066*z^2 + 3403*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ 1, 1, 2 ], [ 2, 1, 1 ], [ 2, 2, 3 ] ], "source_shift": [ 2, 1, 2 ], "target_matrix": [ [ -1, -2, -1 ], [ -1, -1, -1 ], [ 2, 1, 1 ] ], "target_shift": [ 227, 224, -582 ], "inverse_recipe": { "p": 1, "q": 6, "f": "x", "g": "x**6 + x", "h": "y**4 + y*z + y + z", "polynomials": { "f": [ [ [ 1, 0, 0 ], 1 ] ], "g": [ [ [ 6, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 4, 0 ], 1 ], [ [ 0, 0, 1 ], 1 ], [ [ 0, 1, 0 ], 1 ] ] }, 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390*y^5*z + 291*y^5 + 288*y^4*z^3 + 1389*y^4*z^2 + 2121*y^4*z + 1050*y^4 + 387*y^3*z^4 + 2604*y^3*z^3 + 6111*y^3*z^2 + 6153*y^3*z + 2283*y^3 + 306*y^2*z^5 + 2712*y^2*z^4 + 8697*y^2*z^3 + 13347*y^2*z^2 + 10041*y^2*z + 3003*y^2 + 132*y*z^6 + 1488*y*z^5 + 6108*y*z^4 + 12684*y*z^3 + 14496*y*z^2 + 8784*y*z + 2231*y + 24*z^7 + 336*z^6 + 1692*z^5 + 4449*z^4 + 6852*z^3 + 6300*z^2 + 3241*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ 1, 1, 1 ], [ 1, 1, 2 ], [ 2, 1, 1 ] ], "source_shift": [ 1, 2, 1 ], "target_matrix": [ [ -1, -1, -1 ], [ -1, -1, -2 ], [ 3, 2, 3 ] ], "target_shift": [ 290, 285, -723 ], "collision_preimages": [ [ "-1/4", "1/4", "-1" ], [ "11/2", "-2", "-7/2" ], [ "15/2", "-11", "3/2" ] ], "collision_image": [ "1161/4", "1141/4", "-2895/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0816950580010598 ], "surface_features": [ 3.447622667373501, 4.458260037360323, 3.070037866607755, 0.0, 3.6128129182783106, 4832 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C126", "pair": 126, "family": "counterexample", "mapping": "F(x,y,z)=(x^7 - 10*x^6*y - 8*x^6*z + 16*x^6 + 42*x^5*y^2 + 69*x^5*y*z - 138*x^5*y + 27*x^5*z^2 - 108*x^5*z + 96*x^5 - 96*x^4*y^3 - 243*x^4*y^2*z + 489*x^4*y^2 - 195*x^4*y*z^2 + 777*x^4*y*z - 705*x^4*y - 50*x^4*z^3 + 300*x^4*z^2 - 528*x^4*z + 293*x^4 + 129*x^3*y^4 + 447*x^3*y^3*z - 912*x^3*y^3 + 552*x^3*y^2*z^2 - 2202*x^3*y^2*z + 2043*x^3*y^2 + 290*x^3*y*z^3 - 1728*x^3*y*z^2 + 3096*x^3*y*z - 1745*x^3*y + 55*x^3*z^4 - 440*x^3*z^3 + 1152*x^3*z^2 - 1275*x^3*z + 521*x^3 - 102*x^2*y^5 - 453*x^2*y^4*z + 945*x^2*y^4 - 765*x^2*y^3*z^2 + 3075*x^2*y^3*z - 2919*x^2*y^3 - 618*x^2*y^2*z^3 + 3672*x^2*y^2*z^2 - 6708*x^2*y^2*z + 3843*x^2*y^2 - 240*x^2*y*z^4 + 1902*x^2*y*z^3 - 5058*x^2*y*z^2 + 5676*x^2*y*z - 2355*x^2*y - 36*x^2*z^5 + 360*x^2*z^4 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1971*x*z^3 + 2307*x*z^2 - 1465*x*z + 399*x - 16*y^7 - 104*y^6*z + 208*y^6 - 276*y^5*z^2 + 1008*y^5*z - 876*y^5 - 390*y^4*z^3 + 2004*y^4*z^2 - 3366*y^4*z + 1861*y^4 - 318*y^3*z^4 + 2088*y^3*z^3 - 5112*y^3*z^2 + 5533*y^3*z - 2260*y^3 - 150*y^2*z^5 + 1200*y^2*z^4 - 3828*y^2*z^3 + 6099*y^2*z^2 - 4902*y^2*z + 1607*y^2 - 38*y*z^6 + 360*y*z^5 - 1410*y*z^4 + 2949*y*z^3 - 3504*y*z^2 + 2261*y*z - 624*y - 4*z^7 + 44*z^6 - 204*z^5 + 526*z^4 - 822*z^3 + 788*z^2 - 436*z + 1, x^7 - 10*x^6*y - 8*x^6*z + 13*x^6 + 42*x^5*y^2 + 69*x^5*y*z - 114*x^5*y + 27*x^5*z^2 - 87*x^5*z + 69*x^5 - 96*x^4*y^3 - 243*x^4*y^2*z + 411*x^4*y^2 - 195*x^4*y*z^2 + 636*x^4*y*z - 513*x^4*y - 50*x^4*z^3 + 240*x^4*z^2 - 378*x^4*z + 197*x^4 + 129*x^3*y^4 + 447*x^3*y^3*z - 780*x^3*y^3 + 552*x^3*y^2*z^2 - 1833*x^3*y^2*z + 1506*x^3*y^2 + 290*x^3*y*z^3 - 1404*x^3*y*z^2 + 2244*x^3*y*z - 1187*x^3*y + 55*x^3*z^4 - 350*x^3*z^3 + 822*x^3*z^2 - 855*x^3*z + 335*x^3 - 102*x^2*y^5 - 453*x^2*y^4*z + 822*x^2*y^4 - 765*x^2*y^3*z^2 + 2604*x^2*y^3*z - 2181*x^2*y^3 - 618*x^2*y^2*z^3 + 3033*x^2*y^2*z^2 - 4926*x^2*y^2*z + 2646*x^2*y^2 - 240*x^2*y*z^4 + 1536*x^2*y*z^3 - 3654*x^2*y*z^2 + 3852*x^2*y*z - 1533*x^2*y - 36*x^2*z^5 + 285*x^2*z^4 - 888*x^2*z^3 + 1383*x^2*z^2 - 1086*x^2*z + 346*x^2 + 44*x*y^6 + 240*x*y^5*z - 456*x*y^5 + 519*x*y^4*z^2 - 1824*x*y^4*z + 1557*x*y^4 + 573*x*y^3*z^3 - 2868*x*y^3*z^2 + 4737*x*y^3*z - 2584*x*y^3 + 342*x*y^2*z^4 - 2211*x*y^2*z^3 + 5334*x*y^2*z^2 - 5700*x*y^2*z + 2304*x*y^2 + 105*x*y*z^5 - 834*x*y*z^4 + 2628*x*y*z^3 - 4143*x*y*z^2 + 3300*x*y*z - 1069*x*y + 13*x*z^6 - 123*x*z^5 + 477*x*z^4 - 989*x*z^3 + 1167*x*z^2 - 749*x*z + 205*x - 8*y^7 - 52*y^6*z + 104*y^6 - 138*y^5*z^2 + 504*y^5*z - 438*y^5 - 195*y^4*z^3 + 1002*y^4*z^2 - 1683*y^4*z + 931*y^4 - 159*y^3*z^4 + 1044*y^3*z^3 - 2556*y^3*z^2 + 2769*y^3*z - 1135*y^3 - 75*y^2*z^5 + 600*y^2*z^4 - 1914*y^2*z^3 + 3054*y^2*z^2 - 2466*y^2*z + 814*y^2 - 19*y*z^6 + 180*y*z^5 - 705*y*z^4 + 1478*y*z^3 - 1767*y*z^2 + 1150*y*z - 319*y - 2*z^7 + 22*z^6 - 102*z^5 + 264*z^4 - 416*z^3 + 403*z^2 - 224*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ 1, -1, -1 ], [ 1, -2, -1 ], [ 1, -1, -2 ] ], "source_shift": [ 1, 1, 1 ], "target_matrix": [ [ 1, 2, 1 ], [ 2, 2, 3 ], [ 1, 1, 1 ] ], "target_shift": [ -87, -105, -53 ], "collision_preimages": [ [ "-3/4", "0", "1/4" ], [ "-3", "5/2", "-11/2" ], [ "-12", "-5/2", "-15/2" ] ], "collision_image": [ "-349/4", "-211/2", "-213/4" ] }, "probe_target": "No", "jev_answers": [ "Maybe" ], "seconds": [ 1.0523697179996816 ], "surface_features": [ 3.180046186640756, 4.0563330349511615, 2.8573324964312685, 0.0, 3.286983524765943, 4768 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C237", "pair": 237, "family": "counterexample", "mapping": "F(x,y,z)=(-2*x^7 - 20*x^6*y - 15*x^6*z - 28*x^6 - 84*x^5*y^2 - 129*x^5*y*z 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12729*y^2*z^2 + 11154*y^2*z + 3794*y^2 + 58*y*z^6 + 597*y*z^5 + 2595*y*z^4 + 6048*y*z^3 + 7884*y*z^2 + 5357*y*z + 1423*y + 6*z^7 + 71*z^6 + 366*z^5 + 1055*z^4 + 1818*z^3 + 1847*z^2 + 1002*z + 1, -2*x^7 - 20*x^6*y - 15*x^6*z - 34*x^6 - 84*x^5*y^2 - 129*x^5*y*z - 291*x^5*y - 48*x^5*z^2 - 219*x^5*z - 225*x^5 - 192*x^4*y^3 - 453*x^4*y^2*z - 1020*x^4*y^2 - 345*x^4*y*z^2 - 1566*x^4*y*z - 1629*x^4*y - 85*x^4*z^3 - 585*x^4*z^2 - 1209*x^4*z - 779*x^4 - 258*x^3*y^4 - 831*x^3*y^3*z - 1875*x^3*y^3 - 972*x^3*y^2*z^2 - 4401*x^3*y^2*z - 4641*x^3*y^2 - 490*x^3*y*z^3 - 3354*x^3*y*z^2 - 7017*x^3*y*z - 4598*x^3*y - 90*x^3*z^4 - 830*x^3*z^3 - 2586*x^3*z^2 - 3358*x^3*z - 1552*x^3 - 204*x^2*y^5 - 840*x^2*y^4*z - 1908*x^2*y^4 - 1341*x^2*y^3*z^2 - 6078*x^2*y^3*z - 6507*x^2*y^3 - 1038*x^2*y^2*z^3 - 7083*x^2*y^2*z^2 - 15015*x^2*y^2*z - 10026*x^2*y^2 - 390*x^2*y*z^4 - 3576*x^2*y*z^3 - 11277*x^2*y*z^2 - 14892*x^2*y*z - 7005*x^2*y - 57*x^2*z^5 - 660*x^2*z^4 - 2754*x^2*z^3 - 5400*x^2*z^2 - 5046*x^2*z - 1789*x^2 - 88*x*y^6 - 444*x*y^5*z - 1020*x*y^5 - 906*x*y^4*z^2 - 4128*x*y^4*z - 4494*x*y^4 - 957*x*y^3*z^3 - 6531*x*y^3*z^2 - 14043*x*y^3*z - 9577*x*y^3 - 552*x*y^2*z^4 - 5043*x*y^2*z^3 - 16107*x*y^2*z^2 - 21666*x*y^2*z - 10401*x*y^2 - 165*x*y*z^5 - 1899*x*y*z^4 - 8019*x*y*z^3 - 15990*x*y*z^2 - 15213*x*y*z - 5482*x*y - 20*x*z^6 - 279*x*z^5 - 1461*x*z^4 - 3842*x*z^3 - 5436*x*z^2 - 3917*x*z - 1098*x - 16*y^7 - 96*y^6*z - 224*y^6 - 240*y^5*z^2 - 1104*y^5*z - 1224*y^5 - 324*y^4*z^3 - 2220*y^4*z^2 - 4848*y^4*z - 3382*y^4 - 255*y^3*z^4 - 2328*y^3*z^3 - 7536*y^3*z^2 - 10343*y^3*z - 5090*y^3 - 117*y^2*z^5 - 1341*y^2*z^4 - 5733*y^2*z^3 - 11640*y^2*z^2 - 11298*y^2*z - 4159*y^2 - 29*y*z^6 - 402*y*z^5 - 2130*y*z^4 - 5696*y*z^3 - 8208*y*z^2 - 6019*y*z - 1696*y - 3*z^7 - 49*z^6 - 309*z^5 - 1021*z^4 - 1942*z^3 - 2128*z^2 - 1225*z - 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ 1, 1, 1 ], [ 1, 2, 1 ], [ 2, 2, 3 ] ], "source_shift": [ 2, 1, 1 ], "target_matrix": [ [ -1, -1, -1 ], [ 2, 1, 1 ], [ -1, -2, -1 ] ], "target_shift": [ 155, -212, 270 ], "collision_preimages": [ [ "-23/4", "1", "11/4" ], [ "-7", "-3/2", "15/2" ], [ "-18", "7/2", "23/2" ] ], "collision_image": [ "621/4", "-425/2", "1081/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 0.9770557270021527 ], "surface_features": [ 3.525080816694998, 4.375627040254021, 3.196866747249239, 0.0, 3.6287386246622972, 4883 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C007", "pair": 7, "family": "counterexample", "mapping": "F(x,y,z)=(2*x^7 - 21*x^6*y - 20*x^6*z + 29*x^6 + 93*x^5*y^2 + 183*x^5*y*z - 258*x^5*y + 84*x^5*z^2 - 252*x^5*z + 177*x^5 - 225*x^4*y^3 - 687*x^4*y^2*z + 942*x^4*y^2 - 651*x^4*y*z^2 + 1896*x^4*y*z - 1293*x^4*y - 192*x^4*z^3 + 897*x^4*z^2 - 1302*x^4*z + 590*x^4 + 321*x^3*y^4 + 1353*x^3*y^3*z - 1806*x^3*y^3 + 1989*x^3*y^2*z^2 - 5622*x^3*y^2*z + 3720*x^3*y^2 + 1209*x^3*y*z^3 - 5478*x^3*y*z^2 + 7719*x^3*y*z - 3386*x^3*y + 258*x^3*z^4 - 1674*x^3*z^3 + 3771*x^3*z^2 - 3533*x^3*z + 1161*x^3 - 270*x^2*y^5 - 1473*x^2*y^4*z + 1917*x^2*y^4 - 2991*x^2*y^3*z^2 + 8208*x^2*y^3*z - 5268*x^2*y^3 - 2817*x^2*y^2*z^3 + 12366*x^2*y^2*z^2 - 16899*x^2*y^2*z + 7170*x^2*y^2 - 1236*x^2*y*z^4 + 7776*x^2*y*z^3 - 17007*x^2*y*z^2 + 15426*x^2*y*z - 4890*x^2*y - 204*x^2*z^5 + 1728*x^2*z^4 - 5376*x^2*z^3 + 7815*x^2*z^2 - 5313*x^2*z + 1348*x^2 + 124*x*y^6 + 840*x*y^5*z - 1068*x*y^5 + 2211*x*y^4*z^2 - 5898*x*y^4*z + 3672*x*y^4 + 2875*x*y^3*z^3 - 12222*x*y^3*z^2 + 16188*x*y^3*z - 6641*x*y^3 + 1950*x*y^2*z^4 - 11874*x*y^2*z^3 + 25182*x*y^2*z^2 - 22089*x*y^2*z + 6750*x*y^2 + 660*x*y*z^5 - 5424*x*y*z^4 + 16389*x*y*z^3 - 23073*x*y*z^2 + 15135*x*y*z - 3689*x*y + 88*x*z^6 - 936*x*z^5 + 3774*x*z^4 - 7570*x*z^3 + 7986*x*z^2 - 4195*x*z + 854*x - 24*y^7 - 196*y^6*z + 244*y^6 - 642*y^5*z^2 + 1668*y^5*z - 1008*y^5 - 1083*y^4*z^3 + 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3900*x*y^2*z^4 - 21582*x*y^2*z^3 + 41481*x*y^2*z^2 - 33636*x*y^2*z + 9720*x*y^2 + 1320*x*y*z^5 - 10020*x*y*z^4 + 27660*x*y*z^3 - 35988*x*y*z^2 + 22218*x*y*z - 5197*x*y + 176*x*z^6 - 1752*x*z^5 + 6516*x*z^4 - 12112*x*z^3 + 12000*x*z^2 - 6014*x*z + 1187*x - 48*y^7 - 392*y^6*z + 416*y^6 - 1284*y^5*z^2 + 2892*y^5*z - 1572*y^5 - 2166*y^4*z^3 + 7884*y^4*z^2 - 9108*y^4*z + 3352*y^4 - 2024*y^3*z^4 + 10739*y^3*z^3 - 19929*y^3*z^2 + 15626*y^3*z - 4360*y^3 - 1056*y^2*z^5 + 7710*y^2*z^4 - 20532*y^2*z^3 + 25809*y^2*z^2 - 15396*y^2*z + 3470*y^2 - 288*y*z^6 + 2772*y*z^5 - 9972*y*z^4 + 17912*y*z^3 - 17112*y*z^2 + 8261*y*z - 1573*y - 32*z^7 + 392*z^6 - 1824*z^5 + 4396*z^4 - 6008*z^3 + 4658*z^2 - 1888*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ 1, -2, -1 ], [ 1, -1, -2 ], [ 2, -3, -2 ] ], "source_shift": [ 2, 1, 2 ], "target_matrix": [ [ 1, 1, 1 ], [ 1, 1, 2 ], [ 2, 1, 1 ] ], "target_shift": [ -228, -204, -312 ], "collision_preimages": [ [ "9/4", "7/4", "3/4" ], [ "20", "13/2", "8" ], [ "25", "21/2", "7" ] ], "collision_image": [ "-913/4", "-817/4", "-625/2" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.192538424002123 ], "surface_features": [ 3.7331091752941448, 4.617849217563845, 3.4497868469857735, 0.0, 3.8327133701174865, 4983 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C234", "pair": 234, "family": "counterexample", "mapping": "F(x,y,z)=(-16*x^7 + 136*x^6*y - 160*x^6*z + 200*x^6 - 492*x^5*y^2 + 1176*x^5*y*z - 1464*x^5*y - 672*x^5*z^2 + 1728*x^5*z - 1008*x^5 + 982*x^4*y^3 - 3576*x^4*y^2*z + 4434*x^4*y^2 + 4152*x^4*y*z^2 - 10632*x^4*y*z + 6168*x^4*y - 1536*x^4*z^3 + 6120*x^4*z^2 - 7320*x^4*z + 2676*x^4 - 1168*x^3*y^4 + 5758*x^3*y^3*z - 7112*x^3*y^3 - 10188*x^3*y^2*z^2 + 25980*x^3*y^2*z - 14988*x^3*y^2 + 7656*x^3*y*z^3 - 30384*x^3*y*z^2 + 36168*x^3*y*z - 13106*x^3*y - 2064*x^3*z^4 + 11376*x^3*z^3 - 20928*x^3*z^2 + 15684*x^3*z - 4112*x^3 + 828*x^2*y^5 - 5178*x^2*y^4*z + 6372*x^2*y^4 + 12408*x^2*y^3*z^2 - 31512*x^2*y^3*z + 18078*x^2*y^3 - 14208*x^2*y^2*z^3 + 56160*x^2*y^2*z^2 - 66522*x^2*y^2*z + 23886*x^2*y^2 + 7776*x^2*y*z^4 - 42720*x^2*y*z^3 + 78288*x^2*y*z^2 - 58212*x^2*y*z + 15060*x^2*y - 1632*x^2*z^5 + 11712*x^2*z^4 - 29448*x^2*z^3 + 33888*x^2*z^2 - 18276*x^2*z + 3728*x^2 - 324*x*y^6 + 2466*x*y^5*z - 3024*x*y^5 - 7500*x*y^4*z^2 + 18972*x*y^4*z - 10824*x*y^4 + 11632*x*y^3*z^3 - 45792*x*y^3*z^2 + 53970*x*y^3*z - 19199*x*y^3 - 9696*x*y^2*z^4 + 53088*x*y^2*z^3 - 96900*x*y^2*z^2 + 71460*x*y^2*z - 18225*x*y^2 + 4128*x*y*z^5 - 29568*x*y*z^4 + 74136*x*y*z^3 - 84732*x*y*z^2 + 45174*x*y*z - 9046*x*y - 704*x*z^6 + 6336*x*z^5 - 20400*x*z^4 + 32000*x*z^3 - 26544*x*z^2 + 11192*x*z - 1900*x + 54*y^7 - 486*y^6*z + 594*y^6 + 1800*y^5*z^2 - 4536*y^5*z + 2574*y^5 - 3544*y^4*z^3 + 13896*y^4*z^2 - 16296*y^4*z + 5743*y^4 + 4000*y^3*z^4 - 21824*y^3*z^3 + 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"target_shift": [ -433, 228, 397 ], "collision_preimages": [ [ "6", "5/2", "-5/4" ], [ "-43/2", "-10", "7" ], [ "-41/2", "-12", "4" ] ], "collision_image": [ "-867/2", "913/4", "1589/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.1204929410014302 ], "surface_features": [ 4.092866982957736, 4.986323777050766, 3.7829024059746446, 0.0, 4.201876275950509, 5098 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C231", "pair": 231, "family": "counterexample", "mapping": "F(x,y,z)=(-72*x^7 - 588*x^6*y - 372*x^6*z - 852*x^6 - 1926*x^5*y^2 - 2520*x^5*y*z - 5712*x^5*y - 810*x^5*z^2 - 3696*x^5*z - 4188*x^5 - 3249*x^4*y^3 - 6633*x^4*y^2*z - 14937*x^4*y^2 - 4419*x^4*y*z^2 - 19998*x^4*y*z - 22644*x^4*y - 963*x^4*z^3 - 6573*x^4*z^2 - 14916*x^4*z - 11274*x^4 - 3036*x^3*y^4 - 8625*x^3*y^3*z - 19479*x^3*y^3 - 8973*x^3*y^2*z^2 - 40521*x^3*y^2*z - 46167*x^3*y^2 - 4059*x^3*y*z^3 - 27561*x^3*y*z^2 - 62706*x^3*y*z - 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- 36786*x*y^2*z - 22272*x*y^2 - 183*x*y*z^5 - 2214*x*y*z^4 - 10650*x*y*z^3 - 25551*x*y*z^2 - 30591*x*y*z - 14602*x*y - 21*x*z^6 - 306*x*z^5 - 1836*x*z^4 - 5837*x*z^3 - 10389*x*z^2 - 9818*x*z - 3842*x - 16*y^7 - 88*y^6*z - 224*y^6 - 204*y^5*z^2 - 1020*y^5*z - 1284*y^5 - 258*y^4*z^3 - 1908*y^4*z^2 - 4740*y^4*z - 3952*y^4 - 192*y^3*z^4 - 1875*y^3*z^3 - 6909*y^3*z^2 - 11380*y^3*z - 7070*y^3 - 84*y^2*z^5 - 1020*y^2*z^4 - 4968*y^2*z^3 - 12141*y^2*z^2 - 14892*y^2*z - 7336*y^2 - 20*y*z^6 - 291*y*z^5 - 1761*y*z^4 - 5687*y*z^3 - 10344*y*z^2 - 10045*y*z - 4065*y - 2*z^7 - 34*z^6 - 246*z^5 - 986*z^4 - 2368*z^3 - 3409*z^2 - 2722*z - 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ 2, 1, 1 ], [ 1, 2, 1 ], [ 3, 2, 2 ] ], "source_shift": [ 2, 2, 1 ], "target_matrix": [ [ -3, -2, -2 ], [ 1, 1, 1 ], [ -1, -1, -2 ] ], "target_shift": [ 2350, -952, 924 ], "collision_preimages": [ [ "-11/4", "-11/4", "25/4" ], [ "-15/2", "-10", "24" ], [ "-23/2", "-9", "29" ] ], "collision_image": [ "9403/4", "-3809/4", "3697/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0587790210011008 ], "surface_features": [ 4.027447692425845, 4.994910766603217, 3.6757783416740852, 0.0, 4.194745599183956, 5043 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B125", "pair": 125, "family": "bijection", "mapping": "F(x,y,z)=(-8*x^7 + 56*x^6*y - 56*x^6*z + 80*x^6 - 168*x^5*y^2 + 336*x^5*y*z - 488*x^5*y - 168*x^5*z^2 + 480*x^5*z - 334*x^5 + 280*x^4*y^3 - 840*x^4*y^2*z + 1240*x^4*y^2 + 840*x^4*y*z^2 - 2440*x^4*y*z + 1718*x^4*y - 280*x^4*z^3 + 1200*x^4*z^2 - 1670*x^4*z + 764*x^4 - 280*x^3*y^4 + 1120*x^3*y^3*z - 1680*x^3*y^3 - 1680*x^3*y^2*z^2 + 4960*x^3*y^2*z - 3532*x^3*y^2 + 1120*x^3*y*z^3 - 4880*x^3*y*z^2 + 6872*x^3*y*z - 3168*x^3*y - 280*x^3*z^4 + 1600*x^3*z^3 - 3340*x^3*z^2 + 3056*x^3*z - 1076*x^3 + 168*x^2*y^5 - 840*x^2*y^4*z + 1280*x^2*y^4 + 1680*x^2*y^3*z^2 - 5040*x^2*y^3*z + 3628*x^2*y^3 - 1680*x^2*y^2*z^3 + 7440*x^2*y^2*z^2 - 10596*x^2*y^2*z + 4920*x^2*y^2 + 840*x^2*y*z^4 - 4880*x^2*y*z^3 + 10308*x^2*y*z^2 - 9504*x^2*y*z + 3364*x^2*y - 168*x^2*z^5 + 1200*x^2*z^4 - 3340*x^2*z^3 + 4584*x^2*z^2 - 3236*x^2*z + 1008*x^2 - 56*x*y^6 + 336*x*y^5*z - 520*x*y^5 - 840*x*y^4*z^2 + 2560*x*y^4*z - 1862*x*y^4 + 1120*x*y^3*z^3 - 5040*x*y^3*z^2 + 7256*x*y^3*z - 3392*x*y^3 - 840*x*y^2*z^4 + 4960*x*y^2*z^3 - 10596*x*y^2*z^2 + 9840*x*y^2*z - 3500*x*y^2 + 336*x*y*z^5 - 2440*x*y*z^4 + 6872*x*y*z^3 - 9504*x*y*z^2 + 6744*x*y*z - 2128*x*y - 56*x*z^6 + 480*x*z^5 - 1670*x*z^4 + 3056*x*z^3 - 3244*x*z^2 + 2048*x*z - 621*x + 8*y^7 - 56*y^6*z + 88*y^6 + 168*y^5*z^2 - 520*y^5*z + 382*y^5 - 280*y^4*z^3 + 1280*y^4*z^2 - 1862*y^4*z + 876*y^4 + 280*y^3*z^4 - 1680*y^3*z^3 + 3628*y^3*z^2 - 3392*y^3*z + 1212*y^3 - 168*y^2*z^5 + 1240*y^2*z^4 - 3532*y^2*z^3 + 4920*y^2*z^2 - 3508*y^2*z + 1120*y^2 + 56*y*z^6 - 488*y*z^5 + 1718*y*z^4 - 3168*y*z^3 + 3380*y*z^2 - 2168*y*z + 671*y - 8*z^7 + 80*z^6 - 334*z^5 + 764*z^4 - 1084*z^3 + 1040*z^2 - 651*z + 1, 24*x^7 - 168*x^6*y + 168*x^6*z - 240*x^6 + 504*x^5*y^2 - 1008*x^5*y*z + 1464*x^5*y + 504*x^5*z^2 - 1440*x^5*z + 1004*x^5 - 840*x^4*y^3 + 2520*x^4*y^2*z - 3720*x^4*y^2 - 2520*x^4*y*z^2 + 7320*x^4*y*z - 5164*x^4*y + 840*x^4*z^3 - 3600*x^4*z^2 + 5020*x^4*z - 2304*x^4 + 840*x^3*y^4 - 3360*x^3*y^3*z + 5040*x^3*y^3 + 5040*x^3*y^2*z^2 - 14880*x^3*y^2*z + 10616*x^3*y^2 - 3360*x^3*y*z^3 + 14640*x^3*y*z^2 - 20656*x^3*y*z + 9552*x^3*y + 840*x^3*z^4 - 4800*x^3*z^3 + 10040*x^3*z^2 - 9216*x^3*z + 3256*x^3 - 504*x^2*y^5 + 2520*x^2*y^4*z - 3840*x^2*y^4 - 5040*x^2*y^3*z^2 + 15120*x^2*y^3*z - 10904*x^2*y^3 + 5040*x^2*y^2*z^3 - 22320*x^2*y^2*z^2 + 31848*x^2*y^2*z - 14832*x^2*y^2 - 2520*x^2*y*z^4 + 14640*x^2*y*z^3 - 30984*x^2*y*z^2 + 28656*x^2*y*z - 10176*x^2*y + 504*x^2*z^5 - 3600*x^2*z^4 + 10040*x^2*z^3 - 13824*x^2*z^2 + 9792*x^2*z - 3054*x^2 + 168*x*y^6 - 1008*x*y^5*z + 1560*x*y^5 + 2520*x*y^4*z^2 - 7680*x*y^4*z + 5596*x*y^4 - 3360*x*y^3*z^3 + 15120*x*y^3*z^2 - 21808*x*y^3*z + 10224*x*y^3 + 2520*x*y^2*z^4 - 14880*x*y^2*z^3 + 31848*x*y^2*z^2 - 29664*x*y^2*z + 10584*x*y^2 - 1008*x*y*z^5 + 7320*x*y*z^4 - 20656*x*y*z^3 + 28656*x*y*z^2 - 20400*x*y*z + 6444*x*y + 168*x*z^6 - 1440*x*z^5 + 5020*x*z^4 - 9216*x*z^3 + 9816*x*z^2 - 6204*x*z + 1875*x - 24*y^7 + 168*y^6*z - 264*y^6 - 504*y^5*z^2 + 1560*y^5*z - 1148*y^5 + 840*y^4*z^3 - 3840*y^4*z^2 + 5596*y^4*z - 2640*y^4 - 840*y^3*z^4 + 5040*y^3*z^3 - 10904*y^3*z^2 + 10224*y^3*z - 3664*y^3 + 504*y^2*z^5 - 3720*y^2*z^4 + 10616*y^2*z^3 - 14832*y^2*z^2 + 10608*y^2*z - 3390*y^2 - 168*y*z^6 + 1464*y*z^5 - 5164*y*z^4 + 9552*y*z^3 - 10224*y*z^2 + 6564*y*z - 2021*y + 24*z^7 - 240*z^6 + 1004*z^5 - 2304*z^4 + 3280*z^3 - 3150*z^2 + 1967*z - 1, 16*x^7 - 112*x^6*y + 112*x^6*z - 160*x^6 + 336*x^5*y^2 - 672*x^5*y*z + 976*x^5*y + 336*x^5*z^2 - 960*x^5*z + 670*x^5 - 560*x^4*y^3 + 1680*x^4*y^2*z - 2480*x^4*y^2 - 1680*x^4*y*z^2 + 4880*x^4*y*z - 3446*x^4*y + 560*x^4*z^3 - 2400*x^4*z^2 + 3350*x^4*z - 1540*x^4 + 560*x^3*y^4 - 2240*x^3*y^3*z + 3360*x^3*y^3 + 3360*x^3*y^2*z^2 - 9920*x^3*y^2*z + 7084*x^3*y^2 - 2240*x^3*y*z^3 + 9760*x^3*y*z^2 - 13784*x^3*y*z + 6384*x^3*y + 560*x^3*z^4 - 3200*x^3*z^3 + 6700*x^3*z^2 - 6160*x^3*z + 2180*x^3 - 336*x^2*y^5 + 1680*x^2*y^4*z - 2560*x^2*y^4 - 3360*x^2*y^3*z^2 + 10080*x^2*y^3*z - 7276*x^2*y^3 + 3360*x^2*y^2*z^3 - 14880*x^2*y^2*z^2 + 21252*x^2*y^2*z - 9912*x^2*y^2 - 1680*x^2*y*z^4 + 9760*x^2*y*z^3 - 20676*x^2*y*z^2 + 19152*x^2*y*z - 6812*x^2*y + 336*x^2*z^5 - 2400*x^2*z^4 + 6700*x^2*z^3 - 9240*x^2*z^2 + 6556*x^2*z - 2047*x^2 + 112*x*y^6 - 672*x*y^5*z + 1040*x*y^5 + 1680*x*y^4*z^2 - 5120*x*y^4*z + 3734*x*y^4 - 2240*x*y^3*z^3 + 10080*x*y^3*z^2 - 14552*x*y^3*z + 6832*x*y^3 + 1680*x*y^2*z^4 - 9920*x*y^2*z^3 + 21252*x*y^2*z^2 - 19824*x*y^2*z + 7084*x*y^2 - 672*x*y*z^5 + 4880*x*y*z^4 - 13784*x*y*z^3 + 19152*x*y*z^2 - 13656*x*y*z + 4318*x*y + 112*x*z^6 - 960*x*z^5 + 3350*x*z^4 - 6160*x*z^3 + 6572*x*z^2 - 4158*x*z + 1258*x - 16*y^7 + 112*y^6*z - 176*y^6 - 336*y^5*z^2 + 1040*y^5*z - 766*y^5 + 560*y^4*z^3 - 2560*y^4*z^2 + 3734*y^4*z - 1764*y^4 - 560*y^3*z^4 + 3360*y^3*z^3 - 7276*y^3*z^2 + 6832*y^3*z - 2452*y^3 + 336*y^2*z^5 - 2480*y^2*z^4 + 7084*y^2*z^3 - 9912*y^2*z^2 + 7100*y^2*z - 2271*y^2 - 112*y*z^6 + 976*y*z^5 - 3446*y*z^4 + 6384*y*z^3 - 6844*y*z^2 + 4398*y*z - 1355*y + 16*z^7 - 160*z^6 + 670*z^5 - 1540*z^4 + 2196*z^3 - 2111*z^2 + 1320*z - 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -1, 1, -1 ], [ -1, 2, -1 ], [ -2, 2, -3 ] ], "source_shift": [ 1, 2, 1 ], "target_matrix": [ [ 1, 2, 1 ], [ -3, -2, -2 ], [ -2, -1, -1 ] ], "target_shift": [ -190, 566, 379 ], "inverse_recipe": { "p": 2, "q": 5, "f": "x**2 + x", "g": "x**5 + x**4 + x**2 + x", "h": "y*z + y", "polynomials": { "f": [ [ [ 2, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "g": [ [ [ 5, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ], [ [ 4, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 1, 0 ], 1 ] ] }, "h_multiplier": 8 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0584055349972914 ], "surface_features": [ 3.653798415930882, 4.503082164576042, 3.383815365980431, 0.0, 3.762591742463192, 4959 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B185", "pair": 185, "family": "bijection", "mapping": "F(x,y,z)=(384*x^7 + 1344*x^6*y - 1344*x^6*z - 2880*x^6 + 2016*x^5*y^2 - 4032*x^5*y*z - 8640*x^5*y + 2016*x^5*z^2 + 8640*x^5*z + 9456*x^5 + 1680*x^4*y^3 - 5040*x^4*y^2*z - 10800*x^4*y^2 + 5040*x^4*y*z^2 + 21600*x^4*y*z + 23664*x^4*y - 1680*x^4*z^3 - 10800*x^4*z^2 - 23664*x^4*z - 17648*x^4 + 840*x^3*y^4 - 3360*x^3*y^3*z - 7200*x^3*y^3 + 5040*x^3*y^2*z^2 + 21600*x^3*y^2*z + 23688*x^3*y^2 - 3360*x^3*y*z^3 - 21600*x^3*y*z^2 - 47376*x^3*y*z - 35416*x^3*y + 840*x^3*z^4 + 7200*x^3*z^3 + 23688*x^3*z^2 + 35440*x^3*z + 20260*x^3 + 252*x^2*y^5 - 1260*x^2*y^4*z - 2700*x^2*y^4 + 2520*x^2*y^3*z^2 + 10800*x^2*y^3*z + 11856*x^2*y^3 - 2520*x^2*y^2*z^3 - 16200*x^2*y^2*z^2 - 35568*x^2*y^2*z - 26652*x^2*y^2 + 1260*x^2*y*z^4 + 10800*x^2*y*z^3 + 35568*x^2*y*z^2 + 53340*x^2*y*z + 30612*x^2*y - 252*x^2*z^5 - 2700*x^2*z^4 - 11856*x^2*z^3 - 26688*x^2*z^2 - 30684*x^2*z - 14339*x^2 + 42*x*y^6 - 252*x*y^5*z - 540*x*y^5 + 630*x*y^4*z^2 + 2700*x*y^4*z + 2967*x*y^4 - 840*x*y^3*z^3 - 5400*x*y^3*z^2 - 11868*x*y^3*z - 8914*x*y^3 + 630*x*y^2*z^4 + 5400*x*y^2*z^3 + 17802*x*y^2*z^2 + 26760*x*y^2*z + 15417*x*y^2 - 252*x*y*z^5 - 2700*x*y*z^4 - 11868*x*y*z^3 - 26778*x*y*z^2 - 30906*x*y*z - 14528*x*y + 42*x*z^6 + 540*x*z^5 + 2967*x*z^4 + 8932*x*z^3 + 15489*x*z^2 + 14609*x*z + 5820*x + 3*y^7 - 21*y^6*z - 45*y^6 + 63*y^5*z^2 + 270*y^5*z + 297*y^5 - 105*y^4*z^3 - 675*y^4*z^2 - 1485*y^4*z - 1118*y^4 + 105*y^3*z^4 + 900*y^3*z^3 + 2970*y^3*z^2 + 4475*y^3*z + 2588*y^3 - 63*y^2*z^5 - 675*y^2*z^4 - 2970*y^2*z^3 - 6717*y^2*z^2 - 7782*y^2*z - 3680*y^2 + 21*y*z^6 + 270*y*z^5 + 1485*y*z^4 + 4481*y*z^3 + 7800*y*z^2 + 7402*y*z + 2975*y - 3*z^7 - 45*z^6 - 297*z^5 - 1121*z^4 - 2606*z^3 - 3722*z^2 - 3007*z - 1, -128*x^7 - 448*x^6*y + 448*x^6*z + 960*x^6 - 672*x^5*y^2 + 1344*x^5*y*z + 2880*x^5*y - 672*x^5*z^2 - 2880*x^5*z - 3152*x^5 - 560*x^4*y^3 + 1680*x^4*y^2*z + 3600*x^4*y^2 - 1680*x^4*y*z^2 - 7200*x^4*y*z - 7888*x^4*y + 560*x^4*z^3 + 3600*x^4*z^2 + 7888*x^4*z + 5872*x^4 - 280*x^3*y^4 + 1120*x^3*y^3*z + 2400*x^3*y^3 - 1680*x^3*y^2*z^2 - 7200*x^3*y^2*z - 7896*x^3*y^2 + 1120*x^3*y*z^3 + 7200*x^3*y*z^2 + 15792*x^3*y*z + 11784*x^3*y - 280*x^3*z^4 - 2400*x^3*z^3 - 7896*x^3*z^2 - 11792*x^3*z - 6716*x^3 - 84*x^2*y^5 + 420*x^2*y^4*z + 900*x^2*y^4 - 840*x^2*y^3*z^2 - 3600*x^2*y^3*z - 3952*x^2*y^3 + 840*x^2*y^2*z^3 + 5400*x^2*y^2*z^2 + 11856*x^2*y^2*z + 8868*x^2*y^2 - 420*x^2*y*z^4 - 3600*x^2*y*z^3 - 11856*x^2*y*z^2 - 17748*x^2*y*z - 10148*x^2*y + 84*x^2*z^5 + 900*x^2*z^4 + 3952*x^2*z^3 + 8880*x^2*z^2 + 10172*x^2*z + 4729*x^2 - 14*x*y^6 + 84*x*y^5*z + 180*x*y^5 - 210*x*y^4*z^2 - 900*x*y^4*z - 989*x*y^4 + 280*x*y^3*z^3 + 1800*x*y^3*z^2 + 3956*x*y^3*z + 2966*x*y^3 - 210*x*y^2*z^4 - 1800*x*y^2*z^3 - 5934*x*y^2*z^2 - 8904*x*y^2*z - 5111*x*y^2 + 84*x*y*z^5 + 900*x*y*z^4 + 3956*x*y*z^3 + 8910*x*y*z^2 + 10246*x*y*z + 4792*x*y - 14*x*z^6 - 180*x*z^5 - 989*x*z^4 - 2972*x*z^3 - 5135*x*z^2 - 4819*x*z - 1910*x - y^7 + 7*y^6*z + 15*y^6 - 21*y^5*z^2 - 90*y^5*z - 99*y^5 + 35*y^4*z^3 + 225*y^4*z^2 + 495*y^4*z + 372*y^4 - 35*y^3*z^4 - 300*y^3*z^3 - 990*y^3*z^2 - 1489*y^3*z - 858*y^3 + 21*y^2*z^5 + 225*y^2*z^4 + 990*y^2*z^3 + 2235*y^2*z^2 + 2580*y^2*z + 1214*y^2 - 7*y*z^6 - 90*y*z^5 - 495*y*z^4 - 1491*y*z^3 - 2586*y*z^2 - 2442*y*z - 977*y + z^7 + 15*z^6 + 99*z^5 + 373*z^4 + 864*z^3 + 1228*z^2 + 987*z + 1, 128*x^7 + 448*x^6*y - 448*x^6*z - 960*x^6 + 672*x^5*y^2 - 1344*x^5*y*z - 2880*x^5*y + 672*x^5*z^2 + 2880*x^5*z + 3152*x^5 + 560*x^4*y^3 - 1680*x^4*y^2*z - 3600*x^4*y^2 + 1680*x^4*y*z^2 + 7200*x^4*y*z + 7888*x^4*y - 560*x^4*z^3 - 3600*x^4*z^2 - 7888*x^4*z - 5872*x^4 + 280*x^3*y^4 - 1120*x^3*y^3*z - 2400*x^3*y^3 + 1680*x^3*y^2*z^2 + 7200*x^3*y^2*z + 7896*x^3*y^2 - 1120*x^3*y*z^3 - 7200*x^3*y*z^2 - 15792*x^3*y*z - 11784*x^3*y + 280*x^3*z^4 + 2400*x^3*z^3 + 7896*x^3*z^2 + 11792*x^3*z + 6708*x^3 + 84*x^2*y^5 - 420*x^2*y^4*z - 900*x^2*y^4 + 840*x^2*y^3*z^2 + 3600*x^2*y^3*z + 3952*x^2*y^3 - 840*x^2*y^2*z^3 - 5400*x^2*y^2*z^2 - 11856*x^2*y^2*z - 8868*x^2*y^2 + 420*x^2*y*z^4 + 3600*x^2*y*z^3 + 11856*x^2*y*z^2 + 17748*x^2*y*z + 10136*x^2*y - 84*x^2*z^5 - 900*x^2*z^4 - 3952*x^2*z^3 - 8880*x^2*z^2 - 10160*x^2*z - 4705*x^2 + 14*x*y^6 - 84*x*y^5*z - 180*x*y^5 + 210*x*y^4*z^2 + 900*x*y^4*z + 989*x*y^4 - 280*x*y^3*z^3 - 1800*x*y^3*z^2 - 3956*x*y^3*z - 2966*x*y^3 + 210*x*y^2*z^4 + 1800*x*y^2*z^3 + 5934*x*y^2*z^2 + 8904*x*y^2*z + 5105*x*y^2 - 84*x*y*z^5 - 900*x*y*z^4 - 3956*x*y*z^3 - 8910*x*y*z^2 - 10234*x*y*z - 4768*x*y + 14*x*z^6 + 180*x*z^5 + 989*x*z^4 + 2972*x*z^3 + 5129*x*z^2 + 4795*x*z + 1883*x + y^7 - 7*y^6*z - 15*y^6 + 21*y^5*z^2 + 90*y^5*z + 99*y^5 - 35*y^4*z^3 - 225*y^4*z^2 - 495*y^4*z - 372*y^4 + 35*y^3*z^4 + 300*y^3*z^3 + 990*y^3*z^2 + 1489*y^3*z + 857*y^3 - 21*y^2*z^5 - 225*y^2*z^4 - 990*y^2*z^3 - 2235*y^2*z^2 - 2577*y^2*z - 1208*y^2 + 7*y*z^6 + 90*y*z^5 + 495*y*z^4 + 1491*y*z^3 + 2583*y*z^2 + 2430*y*z + 963*y - z^7 - 15*z^6 - 99*z^5 - 373*z^4 - 863*z^3 - 1222*z^2 - 973*z - 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -2, -1, 1 ], [ -1, -1, 1 ], [ -1, -1, 2 ] ], "source_shift": [ 2, 2, 2 ], "target_matrix": [ [ -3, -2, -2 ], [ 1, 2, 1 ], [ -1, -1, -1 ] ], "target_shift": [ 1053, -345, 333 ], "inverse_recipe": { "p": 3, "q": 4, "f": "x**3 + x", "g": "x**4 + x**3 + x**2 + x", "h": "y*z", "polynomials": { "f": [ [ [ 3, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "g": [ [ [ 4, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ], [ [ 3, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.174118456998258 ], "surface_features": [ 3.6886061766183125, 4.727053011353857, 3.2405492482825995, 0.0, 3.880825206105286, 4878 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C191", "pair": 191, "family": "counterexample", "mapping": "F(x,y,z)=(-8*x^7 - 68*x^6*y + 44*x^6*z + 124*x^6 - 234*x^5*y^2 + 312*x^5*y*z + 888*x^5*y - 102*x^5*z^2 - 576*x^5*z - 792*x^5 - 419*x^4*y^3 + 867*x^4*y^2*z + 2511*x^4*y^2 - 585*x^4*y*z^2 - 3342*x^4*y*z - 4668*x^4*y + 129*x^4*z^3 + 1095*x^4*z^2 + 3024*x^4*z + 2722*x^4 - 419*x^3*y^4 + 1199*x^3*y^3*z + 3571*x^3*y^3 - 1257*x^3*y^2*z^2 - 7323*x^3*y^2*z - 10434*x^3*y^2 + 573*x^3*y*z^3 + 4929*x^3*y*z^2 + 13836*x^3*y*z + 12687*x^3*y - 96*x^3*z^4 - 1089*x^3*z^3 - 4530*x^3*z^2 - 8205*x^3*z - 5472*x^3 - 234*x^2*y^5 + 867*x^2*y^4*z + 2685*x^2*y^4 - 1257*x^2*y^3*z^2 - 7542*x^2*y^3*z - 10989*x^2*y^3 + 891*x^2*y^2*z^3 + 7830*x^2*y^2*z^2 + 22410*x^2*y^2*z + 20928*x^2*y^2 - 309*x^2*y*z^4 - 3558*x^2*y*z^3 - 15045*x^2*y*z^2 - 27735*x^2*y*z - 18867*x^2*y + 42*x^2*z^5 + 597*x^2*z^4 + 3324*x^2*z^3 + 9081*x^2*z^2 + 12207*x^2*z + 6484*x^2 - 68*x*y^6 + 312*x*y^5*z + 1008*x*y^5 - 585*x*y^4*z^2 - 3642*x*y^4*z - 5412*x*y^4 + 573*x*y^3*z^3 + 5187*x*y^3*z^2 + 15138*x*y^3*z + 14360*x*y^3 - 309*x*y^2*z^4 - 3639*x*y^2*z^3 - 15666*x*y^2*z^2 - 29346*x*y^2*z - 20307*x*y^2 + 87*x*y*z^5 + 1257*x*y*z^4 + 7110*x*y*z^3 + 19737*x*y*z^2 + 27009*x*y*z + 14647*x*y - 10*x*z^6 - 171*x*z^5 - 1194*x*z^4 - 4369*x*z^3 - 8868*x*z^2 - 9509*x*z - 4236*x - 8*y^7 + 44*y^6*z + 148*y^6 - 102*y^5*z^2 - 660*y^5*z - 996*y^5 + 129*y^4*z^3 + 1209*y^4*z^2 + 3588*y^4*z + 3438*y^4 - 96*y^3*z^4 - 1164*y^3*z^3 - 5097*y^3*z^2 - 9666*y^3*z - 6762*y^3 + 42*y^2*z^5 + 621*y^2*z^4 + 3570*y^2*z^3 + 10044*y^2*z^2 + 13929*y^2*z + 7669*y^2 - 10*y*z^6 - 174*y*z^5 - 1233*y*z^4 - 4575*y*z^3 - 9426*y*z^2 - 10282*y*z - 4667*y + z^7 + 20*z^6 + 168*z^5 + 771*z^4 + 2097*z^3 + 3397*z^2 + 3054*z + 1, -8*x^7 - 68*x^6*y + 44*x^6*z + 124*x^6 - 234*x^5*y^2 + 312*x^5*y*z + 888*x^5*y - 102*x^5*z^2 - 576*x^5*z - 792*x^5 - 419*x^4*y^3 + 867*x^4*y^2*z + 2511*x^4*y^2 - 585*x^4*y*z^2 - 3342*x^4*y*z - 4668*x^4*y + 129*x^4*z^3 + 1095*x^4*z^2 + 3024*x^4*z + 2730*x^4 - 419*x^3*y^4 + 1199*x^3*y^3*z + 3571*x^3*y^3 - 1257*x^3*y^2*z^2 - 7323*x^3*y^2*z - 10434*x^3*y^2 + 573*x^3*y*z^3 + 4929*x^3*y*z^2 + 13836*x^3*y*z + 12707*x^3*y - 96*x^3*z^4 - 1089*x^3*z^3 - 4530*x^3*z^2 - 8225*x^3*z - 5524*x^3 - 234*x^2*y^5 + 867*x^2*y^4*z + 2685*x^2*y^4 - 1257*x^2*y^3*z^2 - 7542*x^2*y^3*z - 10989*x^2*y^3 + 891*x^2*y^2*z^3 + 7830*x^2*y^2*z^2 + 22410*x^2*y^2*z + 20946*x^2*y^2 - 309*x^2*y*z^4 - 3558*x^2*y*z^3 - 15045*x^2*y*z^2 - 27771*x^2*y*z - 18975*x^2*y + 42*x^2*z^5 + 597*x^2*z^4 + 3324*x^2*z^3 + 9099*x^2*z^2 + 12303*x^2*z + 6604*x^2 - 68*x*y^6 + 312*x*y^5*z + 1008*x*y^5 - 585*x*y^4*z^2 - 3642*x*y^4*z - 5412*x*y^4 + 573*x*y^3*z^3 + 5187*x*y^3*z^2 + 15138*x*y^3*z + 14367*x*y^3 - 309*x*y^2*z^4 - 3639*x*y^2*z^3 - 15666*x*y^2*z^2 - 29367*x*y^2*z - 20376*x*y^2 + 87*x*y*z^5 + 1257*x*y*z^4 + 7110*x*y*z^3 + 19758*x*y*z^2 + 27135*x*y*z + 14815*x*y - 10*x*z^6 - 171*x*z^5 - 1194*x*z^4 - 4376*x*z^3 - 8925*x*z^2 - 9653*x*z - 4348*x - 8*y^7 + 44*y^6*z + 148*y^6 - 102*y^5*z^2 - 660*y^5*z - 996*y^5 + 129*y^4*z^3 + 1209*y^4*z^2 + 3588*y^4*z + 3439*y^4 - 96*y^3*z^4 - 1164*y^3*z^3 - 5097*y^3*z^2 - 9670*y^3*z - 6776*y^3 + 42*y^2*z^5 + 621*y^2*z^4 + 3570*y^2*z^3 + 10050*y^2*z^2 + 13968*y^2*z + 7723*y^2 - 10*y*z^6 - 174*y*z^5 - 1233*y*z^4 - 4579*y*z^3 - 9462*y*z^2 - 10378*y*z - 4745*y + z^7 + 20*z^6 + 168*z^5 + 772*z^4 + 2108*z^3 + 3439*z^2 + 3120*z + 1, -8*x^7 - 68*x^6*y + 44*x^6*z + 148*x^6 - 234*x^5*y^2 + 312*x^5*y*z + 1044*x^5*y - 102*x^5*z^2 - 684*x^5*z - 1044*x^5 - 419*x^4*y^3 + 867*x^4*y^2*z + 2901*x^4*y^2 - 585*x^4*y*z^2 - 3906*x^4*y*z - 6048*x^4*y + 129*x^4*z^3 + 1293*x^4*z^2 + 3960*x^4*z + 3834*x^4 - 419*x^3*y^4 + 1199*x^3*y^3*z + 4048*x^3*y^3 - 1257*x^3*y^2*z^2 - 8406*x^3*y^2*z - 13257*x^3*y^2 + 573*x^3*y*z^3 + 5724*x^3*y*z^2 + 17790*x^3*y*z + 17531*x^3*y - 96*x^3*z^4 - 1278*x^3*z^3 - 5889*x^3*z^2 - 11465*x^3*z - 8104*x^3 - 234*x^2*y^5 + 867*x^2*y^4*z + 2988*x^2*y^4 - 1257*x^2*y^3*z^2 - 8496*x^2*y^3*z - 13671*x^2*y^3 + 891*x^2*y^2*z^3 + 8928*x^2*y^2*z^2 + 28233*x^2*y^2*z + 28326*x^2*y^2 - 309*x^2*y*z^4 - 4104*x^2*y*z^3 - 19185*x^2*y*z^2 - 37983*x^2*y*z - 27315*x^2*y + 42*x^2*z^5 + 696*x^2*z^4 + 4287*x^2*z^3 + 12579*x^2*z^2 + 17895*x^2*z + 9994*x^2 - 68*x*y^6 + 312*x*y^5*z + 1104*x*y^5 - 585*x*y^4*z^2 - 4032*x*y^4*z - 6588*x*y^4 + 573*x*y^3*z^3 + 5808*x*y^3*z^2 + 18666*x*y^3*z + 19005*x*y^3 - 309*x*y^2*z^4 - 4122*x*y^2*z^3 - 19563*x*y^2*z^2 - 39327*x*y^2*z - 28716*x*y^2 + 87*x*y*z^5 + 1440*x*y*z^4 + 8988*x*y*z^3 + 26772*x*y*z^2 + 38679*x*y*z + 21952*x*y - 10*x*z^6 - 198*x*z^5 - 1527*x*z^4 - 5996*x*z^3 - 12861*x*z^2 - 14462*x*z - 6719*x - 8*y^7 + 44*y^6*z + 160*y^6 - 102*y^5*z^2 - 720*y^5*z - 1188*y^5 + 129*y^4*z^3 + 1332*y^4*z^2 + 4332*y^4*z + 4447*y^4 - 96*y^3*z^4 - 1296*y^3*z^3 - 6231*y^3*z^2 - 12664*y^3*z - 9332*y^3 + 42*y^2*z^5 + 699*y^2*z^4 + 4419*y^2*z^3 + 13326*y^2*z^2 + 19476*y^2*z + 11182*y^2 - 10*y*z^6 - 198*y*z^5 - 1545*y*z^4 - 6145*y*z^3 - 13350*y*z^2 - 15208*y*z - 7159*y + z^7 + 23*z^6 + 213*z^5 + 1048*z^4 + 3008*z^3 + 5098*z^2 + 4765*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -2, -1, 1 ], [ -1, -2, 1 ], [ -1, -1, 1 ] ], "source_shift": [ 2, 2, 2 ], "target_matrix": [ [ 1, 1, 2 ], [ 1, 1, 1 ], [ 1, 2, 1 ] ], "target_shift": [ -1183, -1219, -1905 ], "collision_preimages": [ [ "-1/4", "-1/4", "-11/4" ], [ "11/2", "8", "18" ], [ "15/2", "5", "17" ] ], "collision_image": [ "-4733/4", "-4877/4", "-7621/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.096346234997327 ], "surface_features": [ 3.740218778188363, 4.59469081818349, 3.3998467127129226, 0.0, 3.8724040679259666, 4913 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B214", "pair": 214, "family": "bijection", "mapping": "F(x,y,z)=(2*x^7 - 14*x^6*y + 14*x^6*z - 38*x^6 + 42*x^5*y^2 - 84*x^5*y*z + 232*x^5*y + 42*x^5*z^2 - 228*x^5*z + 308*x^5 - 70*x^4*y^3 + 210*x^4*y^2*z - 590*x^4*y^2 - 210*x^4*y*z^2 + 1160*x^4*y*z - 1592*x^4*y + 70*x^4*z^3 - 570*x^4*z^2 + 1542*x^4*z - 1384*x^4 + 70*x^3*y^4 - 280*x^3*y^3*z + 800*x^3*y^3 + 420*x^3*y^2*z^2 - 2360*x^3*y^2*z + 3290*x^3*y^2 - 280*x^3*y*z^3 + 2320*x^3*y*z^2 - 6376*x^3*y*z + 5808*x^3*y + 70*x^3*z^4 - 760*x^3*z^3 + 3088*x^3*z^2 - 5560*x^3*z + 3726*x^3 - 42*x^2*y^5 + 210*x^2*y^4*z - 610*x^2*y^4 - 420*x^2*y^3*z^2 + 2400*x^2*y^3*z - 3398*x^2*y^3 + 420*x^2*y^2*z^3 - 3540*x^2*y^2*z^2 + 9882*x^2*y^2*z - 9134*x^2*y^2 - 210*x^2*y*z^4 + 2320*x^2*y*z^3 - 9576*x^2*y*z^2 + 17500*x^2*y*z - 11898*x^2*y + 42*x^2*z^5 - 570*x^2*z^4 + 3092*x^2*z^3 - 8376*x^2*z^2 + 11282*x^2*z - 6004*x^2 + 14*x*y^6 - 84*x*y^5*z + 248*x*y^5 + 210*x*y^4*z^2 - 1220*x*y^4*z + 1754*x*y^4 - 280*x*y^3*z^3 + 2400*x*y^3*z^2 - 6804*x*y^3*z + 6380*x*y^3 + 210*x*y^2*z^4 - 2360*x*y^2*z^3 + 9894*x*y^2*z^2 - 18348*x*y^2*z + 12654*x*y^2 - 84*x*y*z^5 + 1160*x*y*z^4 - 6392*x*y*z^3 + 17576*x*y*z^2 - 24028*x*y*z + 12968*x*y + 14*x*z^6 - 228*x*z^5 + 1548*x*z^4 - 5608*x*z^3 + 11386*x*z^2 - 12200*x*z + 5353*x - 2*y^7 + 14*y^6*z - 42*y^6 - 42*y^5*z^2 + 248*y^5*z - 362*y^5 + 70*y^4*z^3 - 610*y^4*z^2 + 1756*y^4*z - 1670*y^4 - 70*y^3*z^4 + 800*y^3*z^3 - 3406*y^3*z^2 + 6408*y^3*z - 4482*y^3 + 42*y^2*z^5 - 590*y^2*z^4 + 3302*y^2*z^3 - 9214*y^2*z^2 + 12784*y^2*z - 6996*y^2 - 14*y*z^6 + 232*y*z^5 - 1600*y*z^4 + 5884*y*z^3 - 12130*y*z^2 + 13192*y*z - 5867*y + 2*z^7 - 38*z^6 + 310*z^5 - 1408*z^4 + 3830*z^3 - 6196*z^2 + 5479*z - 1, 2*x^7 - 14*x^6*y + 14*x^6*z - 38*x^6 + 42*x^5*y^2 - 84*x^5*y*z + 232*x^5*y + 42*x^5*z^2 - 228*x^5*z + 308*x^5 - 70*x^4*y^3 + 210*x^4*y^2*z - 590*x^4*y^2 - 210*x^4*y*z^2 + 1160*x^4*y*z - 1592*x^4*y + 70*x^4*z^3 - 570*x^4*z^2 + 1542*x^4*z - 1384*x^4 + 70*x^3*y^4 - 280*x^3*y^3*z + 800*x^3*y^3 + 420*x^3*y^2*z^2 - 2360*x^3*y^2*z + 3290*x^3*y^2 - 280*x^3*y*z^3 + 2320*x^3*y*z^2 - 6376*x^3*y*z + 5808*x^3*y + 70*x^3*z^4 - 760*x^3*z^3 + 3088*x^3*z^2 - 5560*x^3*z + 3724*x^3 - 42*x^2*y^5 + 210*x^2*y^4*z - 610*x^2*y^4 - 420*x^2*y^3*z^2 + 2400*x^2*y^3*z - 3398*x^2*y^3 + 420*x^2*y^2*z^3 - 3540*x^2*y^2*z^2 + 9882*x^2*y^2*z - 9134*x^2*y^2 - 210*x^2*y*z^4 + 2320*x^2*y*z^3 - 9576*x^2*y*z^2 + 17500*x^2*y*z - 11892*x^2*y + 42*x^2*z^5 - 570*x^2*z^4 + 3092*x^2*z^3 - 8376*x^2*z^2 + 11276*x^2*z - 5989*x^2 + 14*x*y^6 - 84*x*y^5*z + 248*x*y^5 + 210*x*y^4*z^2 - 1220*x*y^4*z + 1754*x*y^4 - 280*x*y^3*z^3 + 2400*x*y^3*z^2 - 6804*x*y^3*z + 6380*x*y^3 + 210*x*y^2*z^4 - 2360*x*y^2*z^3 + 9894*x*y^2*z^2 - 18348*x*y^2*z + 12648*x*y^2 - 84*x*y*z^5 + 1160*x*y*z^4 - 6392*x*y*z^3 + 17576*x*y*z^2 - 24016*x*y*z + 12938*x*y + 14*x*z^6 - 228*x*z^5 + 1548*x*z^4 - 5608*x*z^3 + 11380*x*z^2 - 12170*x*z + 5311*x - 2*y^7 + 14*y^6*z - 42*y^6 - 42*y^5*z^2 + 248*y^5*z - 362*y^5 + 70*y^4*z^3 - 610*y^4*z^2 + 1756*y^4*z - 1670*y^4 - 70*y^3*z^4 + 800*y^3*z^3 - 3406*y^3*z^2 + 6408*y^3*z - 4480*y^3 + 42*y^2*z^5 - 590*y^2*z^4 + 3302*y^2*z^3 - 9214*y^2*z^2 + 12778*y^2*z - 6981*y^2 - 14*y*z^6 + 232*y*z^5 - 1600*y*z^4 + 5884*y*z^3 - 12124*y*z^2 + 13162*y*z - 5824*y + 2*z^7 - 38*z^6 + 310*z^5 - 1408*z^4 + 3828*z^3 - 6181*z^2 + 5435*z - 1, -2*x^7 + 14*x^6*y - 14*x^6*z + 38*x^6 - 42*x^5*y^2 + 84*x^5*y*z - 232*x^5*y - 42*x^5*z^2 + 228*x^5*z - 308*x^5 + 70*x^4*y^3 - 210*x^4*y^2*z + 590*x^4*y^2 + 210*x^4*y*z^2 - 1160*x^4*y*z + 1592*x^4*y - 70*x^4*z^3 + 570*x^4*z^2 - 1542*x^4*z + 1384*x^4 - 70*x^3*y^4 + 280*x^3*y^3*z - 800*x^3*y^3 - 420*x^3*y^2*z^2 + 2360*x^3*y^2*z - 3290*x^3*y^2 + 280*x^3*y*z^3 - 2320*x^3*y*z^2 + 6376*x^3*y*z - 5808*x^3*y - 70*x^3*z^4 + 760*x^3*z^3 - 3088*x^3*z^2 + 5560*x^3*z - 3726*x^3 + 42*x^2*y^5 - 210*x^2*y^4*z + 610*x^2*y^4 + 420*x^2*y^3*z^2 - 2400*x^2*y^3*z + 3398*x^2*y^3 - 420*x^2*y^2*z^3 + 3540*x^2*y^2*z^2 - 9882*x^2*y^2*z + 9134*x^2*y^2 + 210*x^2*y*z^4 - 2320*x^2*y*z^3 + 9576*x^2*y*z^2 - 17500*x^2*y*z + 11898*x^2*y - 42*x^2*z^5 + 570*x^2*z^4 - 3092*x^2*z^3 + 8376*x^2*z^2 - 11282*x^2*z + 6003*x^2 - 14*x*y^6 + 84*x*y^5*z - 248*x*y^5 - 210*x*y^4*z^2 + 1220*x*y^4*z - 1754*x*y^4 + 280*x*y^3*z^3 - 2400*x*y^3*z^2 + 6804*x*y^3*z - 6380*x*y^3 - 210*x*y^2*z^4 + 2360*x*y^2*z^3 - 9894*x*y^2*z^2 + 18348*x*y^2*z - 12654*x*y^2 + 84*x*y*z^5 - 1160*x*y*z^4 + 6392*x*y*z^3 - 17576*x*y*z^2 + 24028*x*y*z - 12966*x*y - 14*x*z^6 + 228*x*z^5 - 1548*x*z^4 + 5608*x*z^3 - 11386*x*z^2 + 12198*x*z - 5347*x + 2*y^7 - 14*y^6*z + 42*y^6 + 42*y^5*z^2 - 248*y^5*z + 362*y^5 - 70*y^4*z^3 + 610*y^4*z^2 - 1756*y^4*z + 1670*y^4 + 70*y^3*z^4 - 800*y^3*z^3 + 3406*y^3*z^2 - 6408*y^3*z + 4482*y^3 - 42*y^2*z^5 + 590*y^2*z^4 - 3302*y^2*z^3 + 9214*y^2*z^2 - 12784*y^2*z + 6995*y^2 + 14*y*z^6 - 232*y*z^5 + 1600*y*z^4 - 5884*y*z^3 + 12130*y*z^2 - 13190*y*z + 5860*y - 2*z^7 + 38*z^6 - 310*z^5 + 1408*z^4 - 3830*z^3 + 6195*z^2 - 5473*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -1, 1, -1 ], [ -1, 2, -1 ], [ -1, 1, -2 ] ], "source_shift": [ 2, 2, 1 ], "target_matrix": [ [ -1, -2, -1 ], [ -1, -1, -2 ], [ 1, 1, 1 ] ], "target_shift": [ 2035, 1997, -2027 ], "inverse_recipe": { "p": 2, "q": 3, "f": "x**2 + x", "g": "x**3 + x**2 + x", "h": "y**2*z + y**2", "polynomials": { "f": [ [ [ 2, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "g": [ [ [ 3, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "h": [ [ [ 0, 2, 1 ], 1 ], [ [ 0, 2, 0 ], 1 ] ] }, "h_multiplier": 2 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0897828699999081 ], "surface_features": [ 3.5387817275809823, 4.380717623275317, 3.0755469613925306, 0.0, 3.678289205797066, 4816 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B022", "pair": 22, "family": "bijection", "mapping": "F(x,y,z)=(10*x^7 + 70*x^6*y + 70*x^6*z + 130*x^6 + 210*x^5*y^2 + 420*x^5*y*z + 760*x^5*y + 210*x^5*z^2 + 760*x^5*z + 740*x^5 + 350*x^4*y^3 + 1050*x^4*y^2*z + 1850*x^4*y^2 + 1050*x^4*y*z^2 + 3700*x^4*y*z + 3510*x^4*y + 350*x^4*z^3 + 1850*x^4*z^2 + 3520*x^4*z + 2370*x^4 + 350*x^3*y^4 + 1400*x^3*y^3*z + 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35*x^6*z + 65*x^6 + 105*x^5*y^2 + 210*x^5*y*z + 380*x^5*y + 105*x^5*z^2 + 380*x^5*z + 370*x^5 + 175*x^4*y^3 + 525*x^4*y^2*z + 925*x^4*y^2 + 525*x^4*y*z^2 + 1850*x^4*y*z + 1755*x^4*y + 175*x^4*z^3 + 925*x^4*z^2 + 1760*x^4*z + 1185*x^4 + 175*x^3*y^4 + 700*x^3*y^3*z + 1200*x^3*y^3 + 1050*x^3*y^2*z^2 + 3600*x^3*y^2*z + 3325*x^3*y^2 + 700*x^3*y*z^3 + 3600*x^3*y*z^2 + 6670*x^3*y*z + 4360*x^3*y + 175*x^3*z^4 + 1200*x^3*z^3 + 3345*x^3*z^2 + 4400*x^3*z + 2276*x^3 + 105*x^2*y^5 + 525*x^2*y^4*z + 875*x^2*y^4 + 1050*x^2*y^3*z^2 + 3500*x^2*y^3*z + 3145*x^2*y^3 + 1050*x^2*y^2*z^3 + 5250*x^2*y^2*z^2 + 9465*x^2*y^2*z + 6005*x^2*y^2 + 525*x^2*y*z^4 + 3500*x^2*y*z^3 + 9495*x^2*y*z^2 + 12120*x^2*y*z + 6058*x^2*y + 105*x^2*z^5 + 875*x^2*z^4 + 3175*x^2*z^3 + 6115*x^2*z^2 + 6178*x^2*z + 2597*x^2 + 35*x*y^6 + 210*x*y^5*z + 340*x*y^5 + 525*x*y^4*z^2 + 1700*x*y^4*z + 1485*x*y^4 + 700*x*y^3*z^3 + 3400*x*y^3*z^2 + 5960*x*y^3*z + 3670*x*y^3 + 525*x*y^2*z^4 + 3400*x*y^2*z^3 + 8970*x*y^2*z^2 + 11110*x*y^2*z + 5378*x*y^2 + 210*x*y*z^5 + 1700*x*y*z^4 + 6000*x*y*z^3 + 11210*x*y*z^2 + 10956*x*y*z + 4434*x*y + 35*x*z^6 + 340*x*z^5 + 1505*x*z^4 + 3770*x*z^3 + 5578*x*z^2 + 4594*x*z + 1629*x + 5*y^7 + 35*y^6*z + 55*y^6 + 105*y^5*z^2 + 330*y^5*z + 280*y^5 + 175*y^4*z^3 + 825*y^4*z^2 + 1405*y^4*z + 840*y^4 + 175*y^3*z^4 + 1100*y^3*z^3 + 2820*y^3*z^2 + 3390*y^3*z + 1591*y^3 + 105*y^2*z^5 + 825*y^2*z^4 + 2830*y^2*z^3 + 5130*y^2*z^2 + 4858*y^2*z + 1902*y^2 + 35*y*z^6 + 330*y*z^5 + 1420*y*z^4 + 3450*y*z^3 + 4943*y*z^2 + 3924*y*z + 1336*y + 5*z^7 + 55*z^6 + 285*z^5 + 870*z^4 + 1676*z^3 + 2022*z^2 + 1412*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ 1, 1, 1 ], [ 1, 1, 2 ], [ 2, 1, 1 ] ], "source_shift": [ 1, 2, 2 ], "target_matrix": [ [ 2, 3, 2 ], [ 1, 1, 2 ], [ 1, 1, 1 ] ], "target_shift": [ -871, -427, -433 ], "inverse_recipe": { "p": 3, "q": 2, "f": "x**3 + x**2", "g": "x**2", "h": "y*z**2 + 2*y*z + z**3", "polynomials": { "f": [ [ [ 3, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "g": [ [ [ 2, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 2 ], 1 ], [ [ 0, 0, 3 ], 1 ], [ [ 0, 1, 1 ], 2 ] ] }, "h_multiplier": 5 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0999420859989186 ], "surface_features": [ 3.4927662075098582, 4.3845326154942486, 3.189770956346874, 0.0, 3.60538947522543, 4882 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C015", "pair": 15, "family": "counterexample", "mapping": "F(x,y,z)=(-2*x^7 + 21*x^6*y + 15*x^6*z - 29*x^6 - 93*x^5*y^2 - 135*x^5*y*z + 258*x^5*y - 48*x^5*z^2 + 186*x^5*z - 177*x^5 + 225*x^4*y^3 + 498*x^4*y^2*z - 942*x^4*y^2 + 360*x^4*y*z^2 - 1377*x^4*y*z + 1305*x^4*y + 85*x^4*z^3 - 495*x^4*z^2 + 942*x^4*z - 602*x^4 - 321*x^3*y^4 - 963*x^3*y^3*z + 1806*x^3*y^3 - 1062*x^3*y^2*z^2 + 4014*x^3*y^2*z - 3792*x^3*y^2 - 510*x^3*y*z^3 + 2928*x^3*y*z^2 - 5541*x^3*y*z + 3538*x^3*y - 90*x^3*z^4 + 700*x^3*z^3 - 1998*x^3*z^2 + 2551*x^3*z - 1241*x^3 + 270*x^2*y^5 + 1029*x^2*y^4*z - 1917*x^2*y^4 + 1539*x^2*y^3*z^2 - 5757*x^2*y^3*z + 5424*x^2*y^3 + 1128*x^2*y^2*z^3 - 6390*x^2*y^2*z^2 + 12036*x^2*y^2*z - 7674*x^2*y^2 + 405*x^2*y*z^4 - 3102*x^2*y*z^3 + 8793*x^2*y*z^2 - 11202*x^2*y*z + 5448*x^2*y + 57*x^2*z^5 - 555*x^2*z^4 + 2112*x^2*z^3 - 4041*x^2*z^2 + 3927*x^2*z - 1558*x^2 - 124*x*y^6 - 576*x*y^5*z + 1068*x*y^5 - 1095*x*y^4*z^2 + 4062*x*y^4*z - 3816*x*y^4 - 1089*x*y^3*z^3 + 6096*x*y^3*z^2 - 11436*x*y^3*z + 7271*x*y^3 - 597*x*y^2*z^4 + 4506*x*y^2*z^3 - 12696*x*y^2*z^2 + 16131*x*y^2*z - 7830*x*y^2 - 171*x*y*z^5 + 1638*x*y*z^4 - 6183*x*y*z^3 + 11790*x*y*z^2 - 11439*x*y*z + 4537*x*y - 20*x*z^6 + 234*x*z^5 - 1113*x*z^4 + 2837*x*z^3 - 4131*x*z^2 + 3275*x*z - 1108*x + 24*y^7 + 132*y^6*z - 244*y^6 + 306*y^5*z^2 - 1128*y^5*z + 1056*y^5 + 387*y^4*z^3 - 2145*y^4*z^2 + 4008*y^4*z - 2537*y^4 + 288*y^3*z^4 - 2145*y^3*z^3 + 6012*y^3*z^2 - 7610*y^3*z + 3677*y^3 + 126*y^2*z^5 - 1188*y^2*z^4 + 4452*y^2*z^3 - 8457*y^2*z^2 + 8181*y^2*z - 3232*y^2 + 30*y*z^6 - 345*y*z^5 + 1626*y*z^4 - 4126*y*z^3 + 5991*y*z^2 - 4741*y*z + 1601*y + 3*z^7 - 41*z^6 + 234*z^5 - 745*z^4 + 1445*z^3 - 1717*z^2 + 1161*z - 1, -4*x^7 + 42*x^6*y + 30*x^6*z - 52*x^6 - 186*x^5*y^2 - 270*x^5*y*z + 465*x^5*y - 96*x^5*z^2 + 333*x^5*z - 297*x^5 + 450*x^4*y^3 + 996*x^4*y^2*z - 1707*x^4*y^2 + 720*x^4*y*z^2 - 2478*x^4*y*z + 2205*x^4*y + 170*x^4*z^3 - 885*x^4*z^2 + 1578*x^4*z - 966*x^4 - 642*x^3*y^4 - 1926*x^3*y^3*z + 3291*x^3*y^3 - 2124*x^3*y^2*z^2 + 7263*x^3*y^2*z - 6453*x^3*y^2 - 1020*x^3*y*z^3 + 5262*x^3*y*z^2 - 9348*x^3*y*z + 5720*x^3*y - 180*x^3*z^4 + 1250*x^3*z^3 - 3342*x^3*z^2 + 4088*x^3*z - 1932*x^3 + 540*x^2*y^5 + 2058*x^2*y^4*z - 3513*x^2*y^4 + 3078*x^2*y^3*z^2 - 10476*x^2*y^3*z + 9297*x^2*y^3 + 2256*x^2*y^2*z^3 - 11547*x^2*y^2*z^2 + 20454*x^2*y^2*z - 12504*x^2*y^2 + 810*x^2*y*z^4 - 5568*x^2*y*z^3 + 14814*x^2*y*z^2 - 18090*x^2*y*z + 8550*x^2*y + 114*x^2*z^5 - 990*x^2*z^4 + 3528*x^2*z^3 - 6468*x^2*z^2 + 6108*x^2*z - 2378*x^2 - 248*x*y^6 - 1152*x*y^5*z + 1968*x*y^5 - 2190*x*y^4*z^2 + 7434*x*y^4*z - 6588*x*y^4 - 2178*x*y^3*z^3 + 11079*x*y^3*z^2 - 19578*x*y^3*z + 11943*x*y^3 - 1194*x*y^2*z^4 + 8133*x*y^2*z^3 - 21549*x*y^2*z^2 + 26259*x*y^2*z - 12393*x*y^2 - 342*x*y*z^5 + 2937*x*y*z^4 - 10404*x*y*z^3 + 19020*x*y*z^2 - 17940*x*y*z + 6986*x*y - 40*x*z^6 + 417*x*z^5 - 1857*x*z^4 + 4536*x*z^3 - 6420*x*z^2 + 4996*x*z - 1672*x + 48*y^7 + 264*y^6*z - 452*y^6 + 612*y^5*z^2 - 2076*y^5*z + 1836*y^5 + 774*y^4*z^3 - 3921*y^4*z^2 + 6912*y^4*z - 4201*y^4 + 576*y^3*z^4 - 3894*y^3*z^3 + 10281*y^3*z^2 - 12490*y^3*z + 5871*y^3 + 252*y^2*z^5 - 2142*y^2*z^4 + 7548*y^2*z^3 - 13755*y^2*z^2 + 12942*y^2*z - 5024*y^2 + 60*y*z^6 - 618*y*z^5 + 2733*y*z^4 - 6650*y*z^3 + 9390*y*z^2 - 7298*y*z + 2442*y + 6*z^7 - 73*z^6 + 390*z^5 - 1190*z^4 + 2244*z^3 - 2618*z^2 + 1752*z - 1, 2*x^7 - 21*x^6*y - 15*x^6*z + 35*x^6 + 93*x^5*y^2 + 135*x^5*y*z - 309*x^5*y + 48*x^5*z^2 - 225*x^5*z + 234*x^5 - 225*x^4*y^3 - 498*x^4*y^2*z + 1119*x^4*y^2 - 360*x^4*y*z^2 + 1653*x^4*y*z - 1710*x^4*y - 85*x^4*z^3 + 600*x^4*z^2 - 1248*x^4*z + 842*x^4 + 321*x^3*y^4 + 963*x^3*y^3*z - 2127*x^3*y^3 + 1062*x^3*y^2*z^2 - 4779*x^3*y^2*z + 4923*x^3*y^2 + 510*x^3*y*z^3 - 3522*x^3*y*z^2 + 7275*x^3*y*z - 4903*x^3*y + 90*x^3*z^4 - 850*x^3*z^3 + 2652*x^3*z^2 - 3574*x^3*z + 1802*x^3 - 270*x^2*y^5 - 1029*x^2*y^4*z + 2238*x^2*y^4 - 1539*x^2*y^3*z^2 + 6795*x^2*y^3*z - 6975*x^2*y^3 - 1128*x^2*y^2*z^3 + 7623*x^2*y^2*z^2 - 15654*x^2*y^2*z + 10533*x^2*y^2 - 405*x^2*y*z^4 + 3738*x^2*y*z^3 - 11565*x^2*y*z^2 + 15546*x^2*y*z - 7833*x^2*y - 57*x^2*z^5 + 675*x^2*z^4 - 2808*x^2*z^3 + 5670*x^2*z^2 - 5709*x^2*z + 2320*x^2 + 124*x*y^6 + 576*x*y^5*z - 1236*x*y^5 + 1095*x*y^4*z^2 - 4752*x*y^4*z + 4860*x*y^4 + 1089*x*y^3*z^3 - 7209*x*y^3*z^2 + 14730*x*y^3*z - 9881*x*y^3 + 597*x*y^2*z^4 - 5385*x*y^2*z^3 + 16539*x*y^2*z^2 - 22167*x*y^2*z + 11142*x*y^2 + 171*x*y*z^5 - 1977*x*y*z^4 + 8145*x*y*z^3 - 16383*x*y*z^2 + 16461*x*y*z - 6682*x*y + 20*x*z^6 - 285*x*z^5 + 1482*x*z^4 - 3986*x*z^3 + 6012*x*z^2 - 4880*x*z + 1673*x - 24*y^7 - 132*y^6*z + 280*y^6 - 306*y^5*z^2 + 1308*y^5*z - 1332*y^5 - 387*y^4*z^3 + 2514*y^4*z^2 - 5112*y^4*z + 3413*y^4 - 288*y^3*z^4 + 2541*y^3*z^3 - 7755*y^3*z^2 + 10352*y^3*z - 5177*y^3 - 126*y^2*z^5 + 1422*y^2*z^4 - 5808*y^2*z^3 + 11634*y^2*z^2 - 11649*y^2*z + 4705*y^2 - 30*y*z^6 + 417*y*z^5 - 2145*y*z^4 + 5740*y*z^3 - 8628*y*z^2 + 6985*y*z - 2386*y - 3*z^7 + 50*z^6 - 312*z^5 + 1048*z^4 - 2105*z^3 + 2560*z^2 - 1753*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ 1, -1, -1 ], [ 1, -2, -1 ], [ 2, -3, -3 ] ], "source_shift": [ 1, 2, 2 ], "target_matrix": [ [ -1, -1, -1 ], [ -2, -1, -1 ], [ 1, 2, 1 ] ], "target_shift": [ 343, 517, -519 ], "collision_preimages": [ [ "-3/4", "1", "-3/4" ], [ "-9/2", "7/2", "-8" ], [ "-21/2", "-3/2", "-7" ] ], "collision_image": [ "1373/4", "1035/2", "-2077/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 6.040965873999085 ], "surface_features": [ 3.566179772324682, 4.419278183187153, 3.256236533205923, 0.0, 3.660857012745842, 4907 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C073", "pair": 73, "family": "counterexample", "mapping": "F(x,y,z)=(-2*x^7 - 16*x^6*y - 20*x^6*z - 35*x^6 - 54*x^5*y^2 - 138*x^5*y*z - 237*x^5*y - 84*x^5*z^2 - 297*x^5*z - 246*x^5 - 100*x^4*y^3 - 390*x^4*y^2*z - 660*x^4*y^2 - 486*x^4*y*z^2 - 1689*x^4*y*z - 1374*x^4*y - 192*x^4*z^3 - 1029*x^4*z^2 - 1713*x^4*z - 906*x^4 - 110*x^3*y^4 - 580*x^3*y^3*z - 970*x^3*y^3 - 1104*x^3*y^2*z^2 - 3786*x^3*y^2*z - 3036*x^3*y^2 - 894*x^3*y*z^3 - 4719*x^3*y*z^2 - 7716*x^3*y*z - 4027*x^3*y - 258*x^3*z^4 - 1863*x^3*z^3 - 4668*x^3*z^2 - 4943*x^3*z - 1916*x^3 - 72*x^2*y^5 - 480*x^2*y^4*z - 795*x^2*y^4 - 1236*x^2*y^3*z^2 - 4194*x^2*y^3*z - 3324*x^2*y^3 - 1530*x^2*y^2*z^3 - 7983*x^2*y^2*z^2 - 12870*x^2*y^2*z - 6645*x^2*y^2 - 906*x^2*y*z^4 - 6459*x^2*y*z^3 - 15900*x^2*y*z^2 - 16617*x^2*y*z - 6384*x^2*y - 204*x^2*z^5 - 1860*x^2*z^4 - 6225*x^2*z^3 - 9879*x^2*z^2 - 7641*x^2*z - 2348*x^2 - 26*x*y^6 - 210*x*y^5*z - 345*x*y^5 - 684*x*y^4*z^2 - 2301*x*y^4*z - 1806*x*y^4 - 1146*x*y^3*z^3 - 5925*x*y^3*z^2 - 9444*x*y^3*z - 4833*x*y^3 - 1038*x*y^2*z^4 - 7329*x*y^2*z^3 - 17796*x*y^2*z^2 - 18405*x*y^2*z - 7020*x*y^2 - 480*x*y*z^5 - 4332*x*y*z^4 - 14250*x*y*z^3 - 22326*x*y*z^2 - 17124*x*y*z - 5239*x*y - 88*x*z^6 - 972*x*z^5 - 4068*x*z^4 - 8586*x*z^3 - 9912*x*z^2 - 6049*x*z - 1548*x - 4*y^7 - 38*y^6*z - 62*y^6 - 150*y^5*z^2 - 501*y^5*z - 390*y^5 - 318*y^4*z^3 - 1632*y^4*z^2 - 2577*y^4*z - 1309*y^4 - 390*y^3*z^4 - 2733*y^3*z^3 - 6564*y^3*z^2 - 6731*y^3*z - 2552*y^3 - 276*y^2*z^5 - 2472*y^2*z^4 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1450*x^2 + 13*x*y^6 + 105*x*y^5*z + 189*x*y^5 + 342*x*y^4*z^2 + 1266*x*y^4*z + 1041*x*y^4 + 573*x*y^3*z^3 + 3273*x*y^3*z^2 + 5469*x*y^3*z + 2874*x*y^3 + 519*x*y^2*z^4 + 4062*x*y^2*z^3 + 10347*x*y^2*z^2 + 10992*x*y^2*z + 4266*x*y^2 + 240*x*y*z^5 + 2406*x*y*z^4 + 8310*x*y*z^3 + 13380*x*y*z^2 + 10449*x*y*z + 3239*x*y + 44*x*z^6 + 540*x*z^5 + 2376*x*z^4 + 5157*x*z^3 + 6066*x*z^2 + 3755*x*z + 971*x + 2*y^7 + 19*y^6*z + 34*y^6 + 75*y^5*z^2 + 276*y^5*z + 225*y^5 + 159*y^4*z^3 + 903*y^4*z^2 + 1494*y^4*z + 779*y^4 + 195*y^3*z^4 + 1518*y^3*z^3 + 3822*y^3*z^2 + 4024*y^3*z + 1552*y^3 + 138*y^2*z^5 + 1377*y^2*z^4 + 4689*y^2*z^3 + 7470*y^2*z^2 + 5793*y^2*z + 1789*y^2 + 52*y*z^6 + 636*y*z^5 + 2748*y*z^4 + 5885*y*z^3 + 6864*y*z^2 + 4235*y*z + 1098*y + 8*z^7 + 116*z^6 + 612*z^5 + 1654*z^4 + 2578*z^3 + 2374*z^2 + 1217*z + 1, -x^7 - 8*x^6*y - 10*x^6*z - 16*x^6 - 27*x^5*y^2 - 69*x^5*y*z - 108*x^5*y - 42*x^5*z^2 - 135*x^5*z - 105*x^5 - 50*x^4*y^3 - 195*x^4*y^2*z - 300*x^4*y^2 - 243*x^4*y*z^2 - 765*x^4*y*z 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19*y^6*z - 28*y^6 - 75*y^5*z^2 - 225*y^5*z - 165*y^5 - 159*y^4*z^3 - 729*y^4*z^2 - 1083*y^4*z - 528*y^4 - 195*y^3*z^4 - 1215*y^3*z^3 - 2742*y^3*z^2 - 2694*y^3*z - 984*y^3 - 138*y^2*z^5 - 1095*y^2*z^4 - 3336*y^2*z^3 - 4947*y^2*z^2 - 3615*y^2*z - 1057*y^2 - 52*y*z^6 - 504*y*z^5 - 1944*y*z^4 - 3865*y*z^3 - 4224*y*z^2 - 2444*y*z - 600*y - 8*z^7 - 92*z^6 - 432*z^5 - 1082*z^4 - 1574*z^3 - 1348*z^2 - 648*z - 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ 1, 1, 2 ], [ 1, 1, 1 ], [ 1, 2, 1 ] ], "source_shift": [ 2, 1, 1 ], "target_matrix": [ [ -2, -3, -2 ], [ 1, 2, 1 ], [ -1, -1, -2 ] ], "target_shift": [ 426, -270, 139 ], "collision_preimages": [ [ "1/4", "-1/4", "-1" ], [ "-12", "8", "3/2" ], [ "-1", "5", "-7/2" ] ], "collision_image": [ "853/2", "-1081/4", "557/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 6.0022512239993375 ], 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7152*x^3*y^2*z^2 - 31488*x^3*y^2*z - 32934*x^3*y^2 + 3920*x^3*y*z^3 + 25584*x^3*y*z^2 + 52992*x^3*y*z + 35430*x^3*y - 800*x^3*z^4 - 6880*x^3*z^3 - 21168*x^3*z^2 - 28072*x^3*z - 13696*x^3 - 1098*x^2*y^5 + 4620*x^2*y^4*z + 10458*x^2*y^4 - 7722*x^2*y^3*z^2 - 34590*x^2*y^3*z - 36852*x^2*y^3 + 6408*x^2*y^2*z^3 + 42588*x^2*y^2*z^2 + 89904*x^2*y^2*z + 61218*x^2*y^2 - 2640*x^2*y*z^4 - 23136*x^2*y*z^3 - 72576*x^2*y*z^2 - 98046*x^2*y*z - 48672*x^2*y + 432*x^2*z^5 + 4680*x^2*z^4 + 19392*x^2*z^3 + 38976*x^2*z^2 + 38388*x^2*z + 14936*x^2 - 378*x*y^6 + 1926*x*y^5*z + 4428*x*y^5 - 4062*x*y^4*z^2 - 18504*x*y^4*z - 20094*x*y^4 + 4538*x*y^3*z^3 + 30708*x*y^3*z^2 + 66132*x*y^3*z + 45930*x*y^3 - 2832*x*y^2*z^4 - 25296*x*y^2*z^3 - 81006*x*y^2*z^2 - 111666*x*y^2*z - 56481*x*y^2 + 936*x*y*z^5 + 10344*x*y*z^4 + 43776*x*y*z^3 + 89802*x*y*z^2 + 90138*x*y*z + 35692*x*y - 128*x*z^6 - 1680*x*z^5 - 8808*x*z^4 - 23896*x*z^3 - 35688*x*z^2 - 28024*x*z - 9110*x - 54*y^7 + 324*y^6*z + 756*y^6 - 828*y^5*z^2 - 3834*y^5*z - 4248*y^5 + 1168*y^4*z^3 + 8046*y^4*z^2 + 17700*y^4*z + 12565*y^4 - 982*y^3*z^4 - 8942*y^3*z^3 - 29280*y^3*z^2 - 41279*y^3*z - 21309*y^3 + 492*y^2*z^5 + 5550*y^2*z^4 + 24036*y^2*z^3 + 50448*y^2*z^2 + 51705*y^2*z + 20864*y^2 - 136*y*z^6 - 1824*y*z^5 - 9792*y*z^4 - 27186*y*z^3 - 41466*y*z^2 - 33190*y*z - 10994*y + 16*z^7 + 248*z^6 + 1584*z^5 + 5452*z^4 + 10996*z^3 + 13088*z^2 + 8584*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -2, -3, 2 ], [ -1, -1, 1 ], [ -2, -1, 1 ] ], "source_shift": [ 2, 2, 2 ], "target_matrix": [ [ 1, 2, 1 ], [ 1, 1, 2 ], [ 2, 2, 3 ] ], "target_shift": [ -1905, -1183, -2403 ], "collision_preimages": [ [ "1/4", "-2", "-15/4" ], [ "-8", "-6", "-35/2" ], [ "-5", "2", "-7/2" ] ], "collision_image": [ "-7621/4", "-4733/4", "-4807/2" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 5.9814212919991405 ], 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855*x^5*z + 783*x^5 + 165*x^4*y^3 - 720*x^4*y^2*z - 1350*x^4*y^2 + 1026*x^4*y*z^2 + 3942*x^4*y*z + 3636*x^4*y - 477*x^4*z^3 - 2826*x^4*z^2 - 5337*x^4*z - 3245*x^4 + 150*x^3*y^4 - 870*x^3*y^3*z - 1650*x^3*y^3 + 1854*x^3*y^2*z^2 + 7218*x^3*y^2*z + 6714*x^3*y^2 - 1719*x^3*y*z^3 - 10341*x^3*y*z^2 - 19710*x^3*y*z - 12052*x^3*y + 585*x^3*z^4 + 4851*x^3*z^3 + 14229*x^3*z^2 + 17803*x^3*z + 8096*x^3 + 81*x^2*y^5 - 585*x^2*y^4*z - 1125*x^2*y^4 + 1656*x^2*y^3*z^2 + 6552*x^2*y^3*z + 6156*x^2*y^3 - 2295*x^2*y^2*z^3 - 14067*x^2*y^2*z^2 - 27108*x^2*y^2*z - 16686*x^2*y^2 + 1557*x^2*y*z^4 + 13194*x^2*y*z^3 + 39168*x^2*y*z^2 + 49347*x^2*y*z + 22527*x^2*y - 414*x^2*z^5 - 4563*x^2*z^4 - 18567*x^2*z^3 - 35928*x^2*z^2 - 33537*x^2*z - 12180*x^2 + 24*x*y^6 - 207*x*y^5*z - 405*x*y^5 + 729*x*y^4*z^2 + 2943*x*y^4*z + 2799*x*y^4 - 1341*x*y^3*z^3 - 8415*x*y^3*z^2 - 16434*x*y^3*z - 10196*x*y^3 + 1359*x*y^2*z^4 + 11835*x*y^2*z^3 + 35649*x*y^2*z^2 + 45285*x*y^2*z + 20766*x*y^2 - 720*x*y*z^5 - 8190*x*y*z^4 - 33858*x*y*z^3 - 66084*x*y*z^2 - 61959*x*y*z - 22539*x*y + 156*x*z^6 + 2232*x*z^5 + 11880*x*z^4 + 31676*x*z^3 + 45549*x*z^2 + 33861*x*z + 10235*x + 3*y^7 - 30*y^6*z - 60*y^6 + 126*y^5*z^2 + 522*y^5*z + 504*y^5 - 288*y^4*z^3 - 1863*y^4*z^2 - 3699*y^4*z - 2317*y^4 + 387*y^3*z^4 + 3492*y^3*z^3 + 10710*y^3*z^2 + 13741*y^3*z + 6335*y^3 - 306*y^2*z^5 - 3627*y^2*z^4 - 15291*y^2*z^3 - 30156*y^2*z^2 - 28422*y^2*z - 10359*y^2 + 132*y*z^6 + 1980*y*z^5 + 10764*y*z^4 + 29014*y*z^3 + 41943*y*z^2 + 31230*y*z + 9426*y - 24*z^7 - 444*z^6 - 2988*z^5 - 10318*z^4 - 20342*z^3 - 23223*z^2 - 14313*z - 1, -2*x^7 - 13*x^6*y + 19*x^6*z + 41*x^6 - 36*x^5*y^2 + 105*x^5*y*z + 228*x^5*y - 75*x^5*z^2 - 330*x^5*z - 336*x^5 - 55*x^4*y^3 + 240*x^4*y^2*z + 525*x^4*y^2 - 342*x^4*y*z^2 - 1518*x^4*y*z - 1557*x^4*y + 159*x^4*z^3 + 1077*x^4*z^2 + 2259*x^4*z + 1483*x^4 - 50*x^3*y^4 + 290*x^3*y^3*z + 640*x^3*y^3 - 618*x^3*y^2*z^2 - 2772*x^3*y^2*z - 2868*x^3*y^2 + 573*x^3*y*z^3 + 3930*x^3*y*z^2 + 8322*x^3*y*z + 5497*x^3*y - 195*x^3*z^4 - 1824*x^3*z^3 - 5934*x^3*z^2 - 8026*x^3*z - 3869*x^3 - 27*x^2*y^5 + 195*x^2*y^4*z + 435*x^2*y^4 - 552*x^2*y^3*z^2 - 2508*x^2*y^3*z - 2622*x^2*y^3 + 765*x^2*y^2*z^3 + 5328*x^2*y^2*z^2 + 11412*x^2*y^2*z + 7593*x^2*y^2 - 519*x^2*y*z^4 - 4944*x^2*y*z^3 - 16287*x^2*y*z^2 - 22197*x^2*y*z - 10746*x^2*y + 138*x^2*z^5 + 1692*x^2*z^4 + 7623*x^2*z^3 + 15963*x^2*z^2 + 15819*x^2*z + 6016*x^2 - 8*x*y^6 + 69*x*y^5*z + 156*x*y^5 - 243*x*y^4*z^2 - 1122*x*y^4*z - 1188*x*y^4 + 447*x*y^3*z^3 + 3174*x*y^3*z^2 + 6894*x*y^3*z + 4627*x*y^3 - 453*x*y^2*z^4 - 4416*x*y^2*z^3 - 14772*x*y^2*z^2 - 20316*x*y^2*z - 9885*x*y^2 + 240*x*y*z^5 + 3024*x*y*z^4 + 13854*x*y*z^3 + 29289*x*y*z^2 + 29169*x*y*z + 11114*x*y - 52*x*z^6 - 816*x*z^5 - 4800*x*z^4 - 13863*x*z^3 - 21192*x*z^2 - 16516*x*z - 5182*x - y^7 + 10*y^6*z + 23*y^6 - 42*y^5*z^2 - 198*y^5*z - 213*y^5 + 96*y^4*z^3 + 699*y^4*z^2 + 1545*y^4*z + 1048*y^4 - 129*y^3*z^4 - 1296*y^3*z^3 - 4419*y^3*z^2 - 6145*y^3*z - 3008*y^3 + 102*y^2*z^5 + 1332*y^2*z^4 + 6231*y^2*z^3 + 13326*y^2*z^2 + 13350*y^2*z + 5098*y^2 - 44*y*z^6 - 720*y*z^5 - 4332*y*z^4 - 12664*y*z^3 - 19476*y*z^2 - 15208*y*z - 4765*y + 8*z^7 + 160*z^6 + 1188*z^5 + 4447*z^4 + 9332*z^3 + 11182*z^2 + 7159*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -1, -1, 1 ], [ -1, -1, 2 ], [ -2, -1, 1 ] ], "source_shift": [ 2, 2, 2 ], "target_matrix": [ [ -1, -1, -1 ], [ -3, -3, -2 ], [ 1, 2, 1 ] ], "target_shift": [ 1219, 3695, -1905 ], "collision_preimages": [ [ "1/4", "7/4", "0" ], [ "-11/2", "4", "-5/2" ], [ "-15/2", "13", "5/2" ] ], "collision_image": [ "4877/4", "14783/4", "-7621/4" ] }, "probe_target": "No", "jev_answers": [ "Maybe" ], "seconds": [ 6.023907866998343 ], "surface_features": [ 3.79204267384391, 4.820096322528907, 3.2836403888003676, 0.0, 3.9892349771380076, 4914 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C079", "pair": 79, "family": "counterexample", "mapping": "F(x,y,z)=(x^7 + 10*x^6*y - 10*x^6*z - 14*x^6 + 42*x^5*y^2 - 87*x^5*y*z - 120*x^5*y + 42*x^5*z^2 + 123*x^5*z + 78*x^5 + 96*x^4*y^3 - 309*x^4*y^2*z - 420*x^4*y^2 + 309*x^4*y*z^2 + 891*x^4*y*z + 558*x^4*y - 96*x^4*z^3 - 444*x^4*z^2 - 579*x^4*z - 230*x^4 + 129*x^3*y^4 - 573*x^3*y^3*z - 768*x^3*y^3 + 891*x^3*y^2*z^2 + 2529*x^3*y^2*z + 1563*x^3*y^2 - 573*x^3*y*z^3 - 2610*x^3*y*z^2 - 3366*x^3*y*z - 1314*x^3*y + 129*x^3*z^4 + 843*x^3*z^3 + 1695*x^3*z^2 + 1379*x^3*z + 402*x^3 + 102*x^2*y^5 - 585*x^2*y^4*z - 774*x^2*y^4 + 1257*x^2*y^3*z^2 + 3513*x^2*y^3*z + 2142*x^2*y^3 - 1257*x^2*y^2*z^3 - 5634*x^2*y^2*z^2 - 7179*x^2*y^2*z - 2751*x^2*y^2 + 585*x^2*y*z^4 + 3771*x^2*y*z^3 + 7512*x^2*y*z^2 + 6012*x^2*y*z + 1713*x^2*y - 102*x^2*z^5 - 888*x^2*z^4 - 2445*x^2*z^3 - 3054*x^2*z^2 - 1824*x^2*z - 427*x^2 + 44*x*y^6 - 312*x*y^5*z - 408*x*y^5 + 867*x*y^4*z^2 + 2388*x*y^4*z + 1437*x*y^4 - 1199*x*y^3*z^3 - 5286*x*y^3*z^2 - 6651*x*y^3*z - 2501*x*y^3 + 867*x*y^2*z^4 + 5505*x*y^2*z^3 + 10854*x*y^2*z^2 + 8532*x*y^2*z + 2370*x*y^2 - 312*x*y*z^5 - 2688*x*y*z^4 - 7347*x*y*z^3 - 9039*x*y*z^2 - 5286*x*y*z - 1202*x*y + 44*x*z^6 + 492*x*z^5 + 1737*x*z^4 + 2959*x*z^3 + 2712*x*z^2 + 1303*x*z + 262*x + 8*y^7 - 68*y^6*z - 88*y^6 + 234*y^5*z^2 + 636*y^5*z + 378*y^5 - 419*y^4*z^3 - 1818*y^4*z^2 - 2259*y^4*z - 834*y^4 + 419*y^3*z^4 + 2617*y^3*z^3 + 5103*y^3*z^2 + 3938*y^3*z + 1065*y^3 - 234*y^2*z^5 - 1992*y^2*z^4 - 5400*y^2*z^3 - 6528*y^2*z^2 - 3726*y^2*z - 820*y^2 + 68*y*z^6 + 756*y*z^5 + 2655*y*z^4 + 4464*y*z^3 + 4014*y*z^2 + 1876*y*z + 367*y - 8*z^7 - 112*z^6 - 486*z^5 - 1057*z^4 - 1320*z^3 - 976*z^2 - 400*z - 1, 2*x^7 + 20*x^6*y - 20*x^6*z - 31*x^6 + 84*x^5*y^2 - 174*x^5*y*z - 267*x^5*y + 84*x^5*z^2 + 270*x^5*z + 186*x^5 + 192*x^4*y^3 - 618*x^4*y^2*z - 939*x^4*y^2 + 618*x^4*y*z^2 + 1965*x^4*y*z + 1338*x^4*y - 192*x^4*z^3 - 966*x^4*z^2 - 1371*x^4*z - 571*x^4 + 258*x^3*y^4 - 1146*x^3*y^3*z - 1725*x^3*y^3 + 1782*x^3*y^2*z^2 + 5604*x^3*y^2*z + 3768*x^3*y^2 - 1146*x^3*y*z^3 - 5703*x^3*y*z^2 - 8016*x^3*y*z - 3282*x^3*y + 258*x^3*z^4 + 1818*x^3*z^3 + 3984*x^3*z^2 + 3400*x^3*z + 1026*x^3 + 204*x^2*y^5 - 1170*x^2*y^4*z - 1746*x^2*y^4 + 2514*x^2*y^3*z^2 + 7821*x^2*y^3*z + 5190*x^2*y^3 - 2514*x^2*y^2*z^3 - 12366*x^2*y^2*z^2 - 17193*x^2*y^2*z - 6912*x^2*y^2 + 1170*x^2*y*z^4 + 8163*x^2*y*z^3 + 17760*x^2*y*z^2 + 14916*x^2*y*z + 4404*x^2*y - 204*x^2*z^5 - 1899*x^2*z^4 - 5703*x^2*z^3 - 7476*x^2*z^2 - 4620*x^2*z - 1112*x^2 + 88*x*y^6 - 624*x*y^5*z - 924*x*y^5 + 1734*x*y^4*z^2 + 5340*x*y^4*z + 3498*x*y^4 - 2398*x*y^3*z^3 - 11655*x*y^3*z^2 - 16014*x*y^3*z - 6319*x*y^3 + 1734*x*y^2*z^4 + 11964*x*y^2*z^3 + 25812*x*y^2*z^2 + 21300*x*y^2*z + 6138*x*y^2 - 624*x*y*z^5 - 5766*x*y*z^4 - 17238*x*y*z^3 - 22269*x*y*z^2 - 13488*x*y*z - 3160*x*y + 88*x*z^6 + 1044*x*z^5 + 4020*x*z^4 + 7190*x*z^3 + 6816*x*z^2 + 3362*x*z + 690*x + 16*y^7 - 136*y^6*z - 200*y^6 + 468*y^5*z^2 + 1428*y^5*z + 924*y^5 - 838*y^4*z^3 - 4026*y^4*z^2 - 5466*y^4*z - 2118*y^4 + 838*y^3*z^4 + 5711*y^3*z^3 + 12204*y^3*z^2 + 9889*y^3*z + 2778*y^3 - 468*y^2*z^5 - 4287*y^2*z^4 - 12747*y^2*z^3 - 16188*y^2*z^2 - 9579*y^2*z - 2177*y^2 + 136*y*z^6 + 1608*y*z^5 + 6180*y*z^4 + 10917*y*z^3 + 10164*y*z^2 + 4886*y*z + 978*y - 16*z^7 - 236*z^6 - 1116*z^5 - 2549*z^4 - 3291*z^3 - 2495*z^2 - 1042*z - 1, -x^7 - 10*x^6*y + 10*x^6*z + 14*x^6 - 42*x^5*y^2 + 87*x^5*y*z + 120*x^5*y - 42*x^5*z^2 - 123*x^5*z - 78*x^5 - 96*x^4*y^3 + 309*x^4*y^2*z + 420*x^4*y^2 - 309*x^4*y*z^2 - 891*x^4*y*z - 558*x^4*y + 96*x^4*z^3 + 444*x^4*z^2 + 579*x^4*z + 229*x^4 - 129*x^3*y^4 + 573*x^3*y^3*z + 768*x^3*y^3 - 891*x^3*y^2*z^2 - 2529*x^3*y^2*z - 1563*x^3*y^2 + 573*x^3*y*z^3 + 2610*x^3*y*z^2 + 3366*x^3*y*z + 1307*x^3*y - 129*x^3*z^4 - 843*x^3*z^3 - 1695*x^3*z^2 - 1375*x^3*z - 394*x^3 - 102*x^2*y^5 + 585*x^2*y^4*z + 774*x^2*y^4 - 1257*x^2*y^3*z^2 - 3513*x^2*y^3*z - 2142*x^2*y^3 + 1257*x^2*y^2*z^3 + 5634*x^2*y^2*z^2 + 7179*x^2*y^2*z + 2733*x^2*y^2 - 585*x^2*y*z^4 - 3771*x^2*y*z^3 - 7512*x^2*y*z^2 - 5991*x^2*y*z - 1671*x^2*y + 102*x^2*z^5 + 888*x^2*z^4 + 2445*x^2*z^3 + 3048*x^2*z^2 + 1797*x^2*z + 409*x^2 - 44*x*y^6 + 312*x*y^5*z + 408*x*y^5 - 867*x*y^4*z^2 - 2388*x*y^4*z - 1437*x*y^4 + 1199*x*y^3*z^3 + 5286*x*y^3*z^2 + 6651*x*y^3*z + 2481*x*y^3 - 867*x*y^2*z^4 - 5505*x*y^2*z^3 - 10854*x*y^2*z^2 - 8496*x*y^2*z - 2298*x*y^2 + 312*x*y*z^5 + 2688*x*y*z^4 + 7347*x*y*z^3 + 9018*x*y*z^2 + 5190*x*y*z + 1139*x*y - 44*x*z^6 - 492*x*z^5 - 1737*x*z^4 - 2955*x*z^3 - 2682*x*z^2 - 1261*x*z - 248*x - 8*y^7 + 68*y^6*z + 88*y^6 - 234*y^5*z^2 - 636*y^5*z - 378*y^5 + 419*y^4*z^3 + 1818*y^4*z^2 + 2259*y^4*z + 826*y^4 - 419*y^3*z^4 - 2617*y^3*z^3 - 5103*y^3*z^2 - 3918*y^3*z - 1025*y^3 + 234*y^2*z^5 + 1992*y^2*z^4 + 5400*y^2*z^3 + 6510*y^2*z^2 + 3642*y^2*z + 766*y^2 - 68*y*z^6 - 756*y*z^5 - 2655*y*z^4 - 4457*y*z^3 - 3960*y*z^2 - 1801*y*z - 343*y + 8*z^7 + 112*z^6 + 486*z^5 + 1056*z^4 + 1309*z^3 + 952*z^2 + 383*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -1, -2, 1 ], [ -1, -1, 2 ], [ -1, -1, 1 ] ], "source_shift": [ 1, 1, 2 ], "target_matrix": [ [ -1, -1, -1 ], [ -2, -3, -2 ], [ 1, 1, 2 ] ], "target_shift": [ 72, 191, -69 ], "collision_preimages": [ [ "19/4", "-5/4", "5/4" ], [ "-16", "9/2", "-7" ], [ "-15", "13/2", "-4" ] ], "collision_image": [ "289/4", "383/2", "-277/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 5.123382333000336 ], "surface_features": [ 3.4229216253854835, 4.411821656435087, 3.0637085593914173, 0.0, 3.576881713527729, 4837 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C026", "pair": 26, "family": "counterexample", "mapping": "F(x,y,z)=(6*x^7 + 60*x^6*y + 40*x^6*z + 71*x^6 + 252*x^5*y^2 + 342*x^5*y*z + 597*x^5*y + 114*x^5*z^2 + 405*x^5*z + 366*x^5 + 576*x^4*y^3 + 1194*x^4*y^2*z + 2049*x^4*y^2 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3966*x*y^4*z + 3660*x*y^4 + 963*x*y^3*z^3 + 5607*x*y^3*z^2 + 10512*x*y^3*z + 6313*x*y^3 + 498*x*y^2*z^4 + 3900*x*y^2*z^3 + 11145*x*y^2*z^2 + 13698*x*y^2*z + 6048*x*y^2 + 135*x*y*z^5 + 1335*x*y*z^4 + 5169*x*y*z^3 + 9744*x*y*z^2 + 8871*x*y*z + 3086*x*y + 15*x*z^6 + 180*x*z^5 + 885*x*z^4 + 2272*x*z^3 + 3192*x*z^2 + 2306*x*z + 665*x + 24*y^7 + 124*y^6*z + 236*y^6 + 270*y^5*z^2 + 1032*y^5*z + 936*y^5 + 321*y^4*z^3 + 1851*y^4*z^2 + 3408*y^4*z + 1994*y^4 + 225*y^3*z^4 + 1743*y^3*z^3 + 4890*y^3*z^2 + 5857*y^3*z + 2498*y^3 + 93*y^2*z^5 + 909*y^2*z^4 + 3456*y^2*z^3 + 6351*y^2*z^2 + 5586*y^2*z + 1864*y^2 + 21*y*z^6 + 249*y*z^5 + 1203*y*z^4 + 3014*y*z^3 + 4095*y*z^2 + 2828*y*z + 778*y + 2*z^7 + 28*z^6 + 165*z^5 + 528*z^4 + 984*z^3 + 1057*z^2 + 600*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ 1, 2, 1 ], [ 1, 1, 1 ], [ 3, 3, 2 ] ], "source_shift": [ 2, 1, 1 ], "target_matrix": [ [ 2, 1, 1 ], [ 2, 2, 3 ], [ 1, 1, 2 ] ], "target_shift": [ -212, -295, -139 ], "collision_preimages": [ [ "7/4", "-1", "-7/4" ], [ "9", "3/2", "-13" ], [ "8", "-7/2", "-4" ] ], "collision_image": [ "-425/2", "-591/2", "-557/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 5.106828021002002 ], "surface_features": [ 3.5684830114359776, 4.439206335237023, 3.2805783703680764, 0.0, 3.6687175468064055, 4907 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B000", "pair": 0, "family": "bijection", "mapping": "F(x,y,z)=(256*x^7 + 896*x^6*y - 896*x^6*z - 1280*x^6 + 1344*x^5*y^2 - 2688*x^5*y*z - 3840*x^5*y + 1344*x^5*z^2 + 3840*x^5*z + 2784*x^5 + 1120*x^4*y^3 - 3360*x^4*y^2*z - 4800*x^4*y^2 + 3360*x^4*y*z^2 + 9600*x^4*y*z + 6976*x^4*y - 1120*x^4*z^3 - 4800*x^4*z^2 - 7008*x^4*z - 3488*x^4 + 560*x^3*y^4 - 2240*x^3*y^3*z - 3200*x^3*y^3 + 3360*x^3*y^2*z^2 + 9600*x^3*y^2*z + 6992*x^3*y^2 - 2240*x^3*y*z^3 - 9600*x^3*y*z^2 - 14048*x^3*y*z - 7040*x^3*y + 560*x^3*z^4 + 3200*x^3*z^3 + 7056*x^3*z^2 + 7152*x^3*z + 2752*x^3 + 168*x^2*y^5 - 840*x^2*y^4*z - 1200*x^2*y^4 + 1680*x^2*y^3*z^2 + 4800*x^2*y^3*z + 3504*x^2*y^3 - 1680*x^2*y^2*z^3 - 7200*x^2*y^2*z^2 - 10560*x^2*y^2*z - 5328*x^2*y^2 + 840*x^2*y*z^4 + 4800*x^2*y*z^3 + 10608*x^2*y*z^2 + 10824*x^2*y*z + 4216*x^2*y - 168*x^2*z^5 - 1200*x^2*z^4 - 3552*x^2*z^3 - 5496*x^2*z^2 - 4360*x^2*z - 1380*x^2 + 28*x*y^6 - 168*x*y^5*z - 240*x*y^5 + 420*x*y^4*z^2 + 1200*x*y^4*z + 878*x*y^4 - 560*x*y^3*z^3 - 2400*x*y^3*z^2 - 3528*x*y^3*z - 1792*x*y^3 + 420*x*y^2*z^4 + 2400*x*y^2*z^3 + 5316*x*y^2*z^2 + 5460*x*y^2*z + 2152*x*y^2 - 168*x*y*z^5 - 1200*x*y*z^4 - 3560*x*y*z^3 - 5544*x*y*z^2 - 4448*x*y*z - 1432*x*y + 28*x*z^6 + 240*x*z^5 + 894*x*z^4 + 1876*x*z^3 + 2296*x*z^2 + 1514*x*z + 414*x + 2*y^7 - 14*y^6*z - 20*y^6 + 42*y^5*z^2 + 120*y^5*z + 88*y^5 - 70*y^4*z^3 - 300*y^4*z^2 - 442*y^4*z - 226*y^4 + 70*y^3*z^4 + 400*y^3*z^3 + 888*y^3*z^2 + 918*y^3*z + 366*y^3 - 42*y^2*z^5 - 300*y^2*z^4 - 892*y^2*z^3 - 1398*y^2*z^2 - 1134*y^2*z - 372*y^2 + 14*y*z^6 + 120*y*z^5 + 448*y*z^4 + 946*y*z^3 + 1170*y*z^2 + 788*y*z + 222*y - 2*z^7 - 20*z^6 - 90*z^5 - 240*z^4 - 402*z^3 - 418*z^2 - 242*z - 1, -128*x^7 - 448*x^6*y + 448*x^6*z + 640*x^6 - 672*x^5*y^2 + 1344*x^5*y*z + 1920*x^5*y - 672*x^5*z^2 - 1920*x^5*z - 1392*x^5 - 560*x^4*y^3 + 1680*x^4*y^2*z + 2400*x^4*y^2 - 1680*x^4*y*z^2 - 4800*x^4*y*z - 3488*x^4*y + 560*x^4*z^3 + 2400*x^4*z^2 + 3504*x^4*z + 1744*x^4 - 280*x^3*y^4 + 1120*x^3*y^3*z + 1600*x^3*y^3 - 1680*x^3*y^2*z^2 - 4800*x^3*y^2*z - 3496*x^3*y^2 + 1120*x^3*y*z^3 + 4800*x^3*y*z^2 + 7024*x^3*y*z + 3520*x^3*y - 280*x^3*z^4 - 1600*x^3*z^3 - 3528*x^3*z^2 - 3576*x^3*z - 1384*x^3 - 84*x^2*y^5 + 420*x^2*y^4*z + 600*x^2*y^4 - 840*x^2*y^3*z^2 - 2400*x^2*y^3*z - 1752*x^2*y^3 + 840*x^2*y^2*z^3 + 3600*x^2*y^2*z^2 + 5280*x^2*y^2*z + 2664*x^2*y^2 - 420*x^2*y*z^4 - 2400*x^2*y*z^3 - 5304*x^2*y*z^2 - 5412*x^2*y*z - 2120*x^2*y + 84*x^2*z^5 + 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813*x*z + 237*x + y^7 - 7*y^6*z - 10*y^6 + 21*y^5*z^2 + 60*y^5*z + 44*y^5 - 35*y^4*z^3 - 150*y^4*z^2 - 221*y^4*z - 114*y^4 + 35*y^3*z^4 + 200*y^3*z^3 + 444*y^3*z^2 + 463*y^3*z + 189*y^3 - 21*y^2*z^5 - 150*y^2*z^4 - 446*y^2*z^3 - 705*y^2*z^2 - 585*y^2*z - 200*y^2 + 7*y*z^6 + 60*y*z^5 + 224*y*z^4 + 477*y*z^3 + 603*y*z^2 + 422*y*z + 127*y - z^7 - 10*z^6 - 45*z^5 - 121*z^4 - 207*z^3 - 223*z^2 - 138*z - 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -2, -1, 1 ], [ -1, -1, 1 ], [ -1, -1, 2 ] ], "source_shift": [ 1, 1, 2 ], "target_matrix": [ [ -2, -2, -3 ], [ 1, 1, 1 ], [ -1, -2, -1 ] ], "target_shift": [ 57, -32, 36 ], "inverse_recipe": { "p": 4, "q": 3, "f": "x**4 + x**3 + x**2", "g": "x**3 + x**2", "h": "y*z + z**2 + z", "polynomials": { "f": [ [ [ 4, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ], [ [ 3, 0, 0 ], 1 ] ], "g": [ [ [ 3, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 0, 1 ], 1 ], [ [ 0, 0, 2 ], 1 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 5.057554360002541 ], "surface_features": [ 3.211812948968892, 4.147614498562027, 2.8923729073984363, 0.0, 3.3257306781300855, 4773 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B033", "pair": 33, "family": "bijection", "mapping": "F(x,y,z)=(-3*x^7 - 21*x^6*y + 21*x^6*z + 42*x^6 - 63*x^5*y^2 + 126*x^5*y*z + 252*x^5*y - 63*x^5*z^2 - 252*x^5*z - 261*x^5 - 105*x^4*y^3 + 315*x^4*y^2*z + 630*x^4*y^2 - 315*x^4*y*z^2 - 1260*x^4*y*z - 1302*x^4*y + 105*x^4*z^3 + 630*x^4*z^2 + 1302*x^4*z + 922*x^4 - 105*x^3*y^4 + 420*x^3*y^3*z + 840*x^3*y^3 - 630*x^3*y^2*z^2 - 2520*x^3*y^2*z - 2598*x^3*y^2 + 420*x^3*y*z^3 + 2520*x^3*y*z^2 + 5196*x^3*y*z + 3667*x^3*y - 105*x^3*z^4 - 840*x^3*z^3 - 2598*x^3*z^2 - 3664*x^3*z - 1983*x^3 - 63*x^2*y^5 + 315*x^2*y^4*z + 630*x^2*y^4 - 630*x^2*y^3*z^2 - 2520*x^2*y^3*z - 2592*x^2*y^3 + 630*x^2*y^2*z^3 + 3780*x^2*y^2*z^2 + 7776*x^2*y^2*z + 5469*x^2*y^2 - 315*x^2*y*z^4 - 2520*x^2*y*z^3 - 7776*x^2*y*z^2 - 10929*x^2*y*z - 5892*x^2*y + 63*x^2*z^5 + 630*x^2*z^4 + 2592*x^2*z^3 + 5460*x^2*z^2 + 5874*x^2*z + 2581*x^2 - 21*x*y^6 + 126*x*y^5*z + 252*x*y^5 - 315*x*y^4*z^2 - 1260*x*y^4*z - 1293*x*y^4 + 420*x*y^3*z^3 + 2520*x*y^3*z^2 + 5172*x*y^3*z + 3625*x*y^3 - 315*x*y^2*z^4 - 2520*x*y^2*z^3 - 7758*x*y^2*z^2 - 10866*x*y^2*z - 5835*x*y^2 + 126*x*y*z^5 + 1260*x*y*z^4 + 5172*x*y*z^3 + 10857*x*y*z^2 + 11634*x*y*z + 5093*x*y - 21*x*z^6 - 252*x*z^5 - 1293*x*z^4 - 3616*x*z^3 - 5799*x*z^2 - 5051*x*z - 1864*x - 3*y^7 + 21*y^6*z + 42*y^6 - 63*y^5*z^2 - 252*y^5*z - 258*y^5 + 105*y^4*z^3 + 630*y^4*z^2 + 1290*y^4*z + 901*y^4 - 105*y^3*z^4 - 840*y^3*z^3 - 2580*y^3*z^2 - 3601*y^3*z - 1926*y^3 + 63*y^2*z^5 + 630*y^2*z^4 + 2580*y^2*z^3 + 5397*y^2*z^2 + 5760*y^2*z + 2509*y^2 - 21*y*z^6 - 252*y*z^5 - 1290*y*z^4 - 3595*y*z^3 - 5742*y*z^2 - 4979*y*z - 1822*y + 3*z^7 + 42*z^6 + 258*z^5 + 898*z^4 + 1908*z^3 + 2470*z^2 + 1797*z + 1, 3*x^7 + 21*x^6*y - 21*x^6*z - 42*x^6 + 63*x^5*y^2 - 126*x^5*y*z - 252*x^5*y + 63*x^5*z^2 + 252*x^5*z + 261*x^5 + 105*x^4*y^3 - 315*x^4*y^2*z - 630*x^4*y^2 + 315*x^4*y*z^2 + 1260*x^4*y*z + 1302*x^4*y - 105*x^4*z^3 - 630*x^4*z^2 - 1302*x^4*z - 922*x^4 + 105*x^3*y^4 - 420*x^3*y^3*z - 840*x^3*y^3 + 630*x^3*y^2*z^2 + 2520*x^3*y^2*z + 2598*x^3*y^2 - 420*x^3*y*z^3 - 2520*x^3*y*z^2 - 5196*x^3*y*z - 3667*x^3*y + 105*x^3*z^4 + 840*x^3*z^3 + 2598*x^3*z^2 + 3664*x^3*z + 1984*x^3 + 63*x^2*y^5 - 315*x^2*y^4*z - 630*x^2*y^4 + 630*x^2*y^3*z^2 + 2520*x^2*y^3*z + 2592*x^2*y^3 - 630*x^2*y^2*z^3 - 3780*x^2*y^2*z^2 - 7776*x^2*y^2*z - 5469*x^2*y^2 + 315*x^2*y*z^4 + 2520*x^2*y*z^3 + 7776*x^2*y*z^2 + 10929*x^2*y*z + 5895*x^2*y - 63*x^2*z^5 - 630*x^2*z^4 - 2592*x^2*z^3 - 5460*x^2*z^2 - 5877*x^2*z - 2587*x^2 + 21*x*y^6 - 126*x*y^5*z - 252*x*y^5 + 315*x*y^4*z^2 + 1260*x*y^4*z + 1293*x*y^4 - 420*x*y^3*z^3 - 2520*x*y^3*z^2 - 5172*x*y^3*z - 3625*x*y^3 + 315*x*y^2*z^4 + 2520*x*y^2*z^3 + 7758*x*y^2*z^2 + 10866*x*y^2*z + 5838*x*y^2 - 126*x*y*z^5 - 1260*x*y*z^4 - 5172*x*y*z^3 - 10857*x*y*z^2 - 11640*x*y*z - 5105*x*y + 21*x*z^6 + 252*x*z^5 + 1293*x*z^4 + 3616*x*z^3 + 5802*x*z^2 + 5063*x*z + 1878*x + 3*y^7 - 21*y^6*z - 42*y^6 + 63*y^5*z^2 + 252*y^5*z + 258*y^5 - 105*y^4*z^3 - 630*y^4*z^2 - 1290*y^4*z - 901*y^4 + 105*y^3*z^4 + 840*y^3*z^3 + 2580*y^3*z^2 + 3601*y^3*z + 1927*y^3 - 63*y^2*z^5 - 630*y^2*z^4 - 2580*y^2*z^3 - 5397*y^2*z^2 - 5763*y^2*z - 2515*y^2 + 21*y*z^6 + 252*y*z^5 + 1290*y*z^4 + 3595*y*z^3 + 5745*y*z^2 + 4991*y*z + 1835*y - 3*z^7 - 42*z^6 - 258*z^5 - 898*z^4 - 1909*z^3 - 2476*z^2 - 1810*z - 1, 6*x^7 + 42*x^6*y - 42*x^6*z - 84*x^6 + 126*x^5*y^2 - 252*x^5*y*z - 504*x^5*y + 126*x^5*z^2 + 504*x^5*z + 522*x^5 + 210*x^4*y^3 - 630*x^4*y^2*z 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504*x*z^5 + 2586*x*z^4 + 7224*x*z^3 + 11550*x*z^2 + 10002*x*z + 3654*x + 6*y^7 - 42*y^6*z - 84*y^6 + 126*y^5*z^2 + 504*y^5*z + 516*y^5 - 210*y^4*z^3 - 1260*y^4*z^2 - 2580*y^4*z - 1800*y^4 + 210*y^3*z^4 + 1680*y^3*z^3 + 5160*y^3*z^2 + 7194*y^3*z + 3836*y^3 - 126*y^2*z^5 - 1260*y^2*z^4 - 5160*y^2*z^3 - 10782*y^2*z^2 - 11472*y^2*z - 4968*y^2 + 42*y*z^6 + 504*y*z^5 + 2580*y*z^4 + 7182*y*z^3 + 11436*y*z^2 + 9858*y*z + 3568*y - 6*z^7 - 84*z^6 - 516*z^5 - 1794*z^4 - 3800*z^3 - 4890*z^2 - 3520*z - 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -1, -1, 1 ], [ -2, -1, 1 ], [ -1, -2, 1 ] ], "source_shift": [ 2, 1, 2 ], "target_matrix": [ [ 1, 1, 1 ], [ -1, -2, -1 ], [ -2, -2, -3 ] ], "target_shift": [ -560, 569, 1077 ], "inverse_recipe": { "p": 3, "q": 4, "f": "x**3", "g": "x**4 + x**2", "h": "y*z", "polynomials": { "f": [ [ [ 3, 0, 0 ], 1 ] ], "g": [ [ [ 4, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ] ] }, "h_multiplier": 3 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 5.017758671001502 ], "surface_features": [ 3.4502764861471396, 4.364963519827019, 3.100370545117563, 0.0, 3.582301211593305, 4845 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C172", "pair": 172, "family": "counterexample", "mapping": "F(x,y,z)=(-3*x^7 - 24*x^6*y - 30*x^6*z - 48*x^6 - 81*x^5*y^2 - 207*x^5*y*z - 324*x^5*y - 126*x^5*z^2 - 405*x^5*z - 315*x^5 - 150*x^4*y^3 - 585*x^4*y^2*z - 900*x^4*y^2 - 729*x^4*y*z^2 - 2295*x^4*y*z - 1755*x^4*y - 288*x^4*z^3 - 1395*x^4*z^2 - 2178*x^4*z - 1108*x^4 - 165*x^3*y^4 - 870*x^3*y^3*z - 1320*x^3*y^3 - 1656*x^3*y^2*z^2 - 5130*x^3*y^2*z - 3870*x^3*y^2 - 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20576*x*z^4 - 16576*x*z^3 - 8392*x*z^2 - 2664*x*z - 410*x - 36*y^7 - 432*y^6*z - 228*y^6 - 2196*y^5*z^2 - 2328*y^5*z - 653*y^5 - 6120*y^4*z^3 - 9780*y^4*z^2 - 5498*y^4*z - 1117*y^4 - 10080*y^3*z^4 - 21600*y^3*z^3 - 18260*y^3*z^2 - 7400*y^3*z - 1327*y^3 - 9792*y^2*z^5 - 26400*y^2*z^4 - 29848*y^2*z^3 - 18096*y^2*z^2 - 6318*y^2*z - 1078*y^2 - 5184*y*z^6 - 16896*y*z^5 - 23968*y*z^4 - 19328*y*z^3 - 9888*y*z^2 - 3208*y*z - 504*y - 1152*z^7 - 4416*z^6 - 7552*z^5 - 7600*z^4 - 5084*z^3 - 2368*z^2 - 709*z - 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ 1, 1, 2 ], [ 2, 2, 3 ], [ 1, 2, 1 ] ], "source_shift": [ 1, 2, 1 ], "target_matrix": [ [ -2, -1, -1 ], [ 1, 2, 1 ], [ -1, -1, -1 ] ], "target_shift": [ 195, -104, 98 ], "inverse_recipe": { "p": 5, "q": 1, "f": "x**5 + x**3 + x**2 + x", "g": "x", "h": "y*z**2", "polynomials": { "f": [ [ [ 5, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ], [ [ 3, 0, 0 ], 1 ] ], "g": [ [ [ 1, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 2 ], 1 ] ] }, "h_multiplier": 4 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 0.986305316000653 ], "surface_features": [ 4.165929760446123, 5.055378331375, 3.8549130223078554, 0.0, 4.26955193863053, 5137 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C103", "pair": 103, "family": "counterexample", "mapping": "F(x,y,z)=(128*x^7 - 704*x^6*y + 1024*x^6*z - 1504*x^6 + 1632*x^5*y^2 - 4704*x^5*y*z + 7152*x^5*y + 3360*x^5*z^2 - 10224*x^5*z + 6816*x^5 - 2064*x^4*y^3 + 8832*x^4*y^2*z - 13944*x^4*y^2 - 12480*x^4*y*z^2 + 39504*x^4*y*z - 27072*x^4*y + 5824*x^4*z^3 - 27792*x^4*z^2 + 38352*x^4*z - 16360*x^4 + 1536*x^3*y^4 - 8664*x^3*y^3*z + 14244*x^3*y^3 + 18144*x^3*y^2*z^2 - 59916*x^3*y^2*z + 42312*x^3*y^2 - 16720*x^3*y*z^3 + 83400*x^3*y*z^2 - 118800*x^3*y*z + 52044*x^3*y + 5720*x^3*z^4 - 38408*x^3*z^3 + 82896*x^3*z^2 - 73340*x^3*z + 23016*x^3 - 672*x^2*y^5 + 4680*x^2*y^4*z - 8028*x^2*y^4 - 12900*x^2*y^3*z^2 + 44520*x^2*y^3*z - 32472*x^2*y^3 + 17592*x^2*y^2*z^3 - 91854*x^2*y^2*z^2 + 135372*x^2*y^2*z - 61050*x^2*y^2 - 11868*x^2*y*z^4 + 83544*x^2*y*z^3 - 186888*x^2*y*z^2 + 170496*x^2*y*z - 54996*x^2*y + 3168*x^2*z^5 - 28254*x^2*z^4 + 85416*x^2*z^3 - 118374*x^2*z^2 + 77268*x^2*z - 19238*x^2 + 160*x*y^6 - 1320*x*y^5*z + 2364*x*y^5 + 4488*x*y^4*z^2 - 16188*x*y^4*z + 12216*x*y^4 - 8050*x*y^3*z^3 + 43965*x*y^3*z^2 - 67140*x*y^3*z + 31237*x*y^3 + 8034*x*y^2*z^4 - 59181*x*y^2*z^3 + 137382*x*y^2*z^2 - 129507*x*y^2*z + 43044*x*y^2 - 4230*x*y*z^5 + 39471*x*y*z^4 - 123996*x*y*z^3 + 177855*x*y*z^2 - 119826*x*y*z + 30721*x*y + 918*x*z^6 - 10431*x*z^5 + 41634*x*z^4 - 80877*x*z^3 + 82938*x*z^2 - 43107*x*z + 8930*x - 16*y^7 + 152*y^6*z - 284*y^6 - 612*y^5*z^2 + 2304*y^5*z - 1800*y^5 + 1354*y^4*z^3 - 7719*y^4*z^2 + 12216*y^4*z - 5871*y^4 - 1778*y^3*z^4 + 13667*y^3*z^3 - 32904*y^3*z^2 + 32081*y^3*z - 11010*y^3 + 1386*y^2*z^5 - 13485*y^2*z^4 + 43953*y^2*z^3 - 65277*y^2*z^2 + 45483*y^2*z - 12041*y^2 - 594*y*z^6 + 7029*y*z^5 - 29106*y*z^4 + 58593*y*z^3 - 62238*y*z^2 + 33465*y*z - 7154*y + 108*z^7 - 1512*z^6 + 7641*z^5 - 19566*z^4 + 28197*z^3 - 23124*z^2 + 10039*z - 1, -64*x^7 + 352*x^6*y - 512*x^6*z + 896*x^6 - 816*x^5*y^2 + 2352*x^5*y*z - 4224*x^5*y - 1680*x^5*z^2 + 6048*x^5*z - 4848*x^5 + 1032*x^4*y^3 - 4416*x^4*y^2*z + 8160*x^4*y^2 + 6240*x^4*y*z^2 - 23136*x^4*y*z + 19008*x^4*y - 2912*x^4*z^3 + 16272*x^4*z^2 - 26880*x^4*z + 13288*x^4 - 768*x^3*y^4 + 4332*x^3*y^3*z - 8256*x^3*y^3 - 9072*x^3*y^2*z^2 + 34728*x^3*y^2*z - 29292*x^3*y^2 + 8360*x^3*y*z^3 - 48288*x^3*y*z^2 + 82008*x^3*y*z - 41636*x^3*y - 2860*x^3*z^4 + 22192*x^3*z^3 - 57000*x^3*z^2 + 58436*x^3*z - 20620*x^3 + 336*x^2*y^5 - 2340*x^2*y^4*z + 4608*x^2*y^4 + 6450*x^2*y^3*z^2 - 25536*x^2*y^3*z + 22140*x^2*y^3 - 8796*x^2*y^2*z^3 + 52596*x^2*y^2*z^2 - 91932*x^2*y^2*z + 48030*x^2*y^2 + 5934*x^2*y*z^4 - 47712*x^2*y*z^3 + 126276*x^2*y*z^2 - 133428*x^2*y*z + 48462*x^2*y - 1584*x^2*z^5 + 16080*x^2*z^4 - 57360*x^2*z^3 + 92058*x^2*z^2 - 67716*x^2*z + 18556*x^2 - 80*x*y^6 + 660*x*y^5*z - 1344*x*y^5 - 2244*x*y^4*z^2 + 9192*x*y^4*z - 8196*x*y^4 + 4025*x*y^3*z^3 - 24912*x*y^3*z^2 + 44820*x*y^3*z - 24129*x*y^3 - 4017*x*y^2*z^4 + 33438*x*y^2*z^3 - 91155*x*y^2*z^2 + 99381*x*y^2*z - 37236*x*y^2 + 2115*x*y*z^5 - 22224*x*y*z^4 + 81690*x*y*z^3 - 135441*x*y*z^2 + 102951*x*y*z - 29108*x*y - 459*x*z^6 + 5850*x*z^5 - 27207*x*z^4 + 61051*x*z^3 - 70701*x*z^2 + 40578*x*z - 9113*x + 8*y^7 - 76*y^6*z + 160*y^6 + 306*y^5*z^2 - 1296*y^5*z + 1188*y^5 - 677*y^4*z^3 + 4332*y^4*z^2 - 8016*y^4*z + 4447*y^4 + 889*y^3*z^4 - 7648*y^3*z^3 + 21447*y^3*z^2 - 24111*y^3*z + 9332*y^3 - 693*y^2*z^5 + 7521*y^2*z^4 - 28434*y^2*z^3 + 48627*y^2*z^2 - 38232*y^2*z + 11182*y^2 + 297*y*z^6 - 3906*y*z^5 + 18675*y*z^4 - 43216*y*z^3 + 51825*y*z^2 - 30828*y*z + 7159*y - 54*z^7 + 837*z^6 - 4860*z^5 + 14273*z^4 - 23231*z^3 + 21108*z^2 - 9970*z + 1, 128*x^7 - 704*x^6*y + 1024*x^6*z - 1696*x^6 + 1632*x^5*y^2 - 4704*x^5*y*z + 8016*x^5*y + 3360*x^5*z^2 - 11472*x^5*z + 8736*x^5 - 2064*x^4*y^3 + 8832*x^4*y^2*z - 15528*x^4*y^2 - 12480*x^4*y*z^2 + 44016*x^4*y*z - 34368*x^4*y + 5824*x^4*z^3 - 30960*x^4*z^2 + 48624*x^4*z - 23168*x^4 + 1536*x^3*y^4 - 8664*x^3*y^3*z + 15756*x^3*y^3 + 18144*x^3*y^2*z^2 - 66276*x^3*y^2*z + 53160*x^3*y^2 - 16720*x^3*y*z^3 + 92184*x^3*y*z^2 - 148944*x^3*y*z + 72856*x^3*y + 5720*x^3*z^4 - 42392*x^3*z^3 + 103632*x^3*z^2 - 102352*x^3*z + 35144*x^3 - 672*x^2*y^5 + 4680*x^2*y^4*z - 8820*x^2*y^4 - 12900*x^2*y^3*z^2 + 48888*x^2*y^3*z - 40344*x^2*y^3 + 17592*x^2*y^2*z^3 - 100746*x^2*y^2*z^2 + 167700*x^2*y^2*z - 84384*x^2*y^2 - 11868*x^2*y*z^4 + 91464*x^2*y*z^3 - 230664*x^2*y*z^2 + 234720*x^2*y*z - 82908*x^2*y + 3168*x^2*z^5 - 30858*x^2*z^4 + 104952*x^2*z^3 - 162192*x^2*z^2 + 115992*x^2*z - 31106*x^2 + 160*x*y^6 - 1320*x*y^5*z + 2580*x*y^5 + 4488*x*y^4*z^2 - 17652*x*y^4*z + 15000*x*y^4 - 8050*x*y^3*z^3 + 47871*x*y^3*z^2 - 82140*x*y^3*z + 42582*x*y^3 + 8034*x*y^2*z^4 - 64311*x*y^2*z^3 + 167334*x*y^2*z^2 - 175668*x*y^2*z + 63972*x*y^2 - 4230*x*y*z^5 + 42789*x*y*z^4 - 150252*x*y*z^3 + 239862*x*y*z^2 - 177150*x*y*z + 48991*x*y + 918*x*z^6 - 11277*x*z^5 + 50154*x*z^4 - 108356*x*z^3 + 121878*x*z^2 - 68397*x*z + 15086*x - 16*y^7 + 152*y^6*z - 308*y^6 - 612*y^5*z^2 + 2496*y^5*z - 2184*y^5 + 1354*y^4*z^3 - 8349*y^4*z^2 + 14760*y^4*z - 7886*y^4 - 1778*y^3*z^4 + 14753*y^3*z^3 - 39564*y^3*z^2 + 42840*y^3*z - 16108*y^3 + 1386*y^2*z^5 - 14523*y^2*z^4 + 52563*y^2*z^3 - 86592*y^2*z^2 + 66120*y^2*z - 18905*y^2 - 594*y*z^6 + 7551*y*z^5 - 34602*y*z^4 + 77150*y*z^3 - 89826*y*z^2 + 52215*y*z - 11904*y + 108*z^7 - 1620*z^6 + 9027*z^5 - 25552*z^4 + 40366*z^3 - 35826*z^2 + 16607*z - 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -2, 1, -1 ], [ -2, 2, -3 ], [ -1, 1, -2 ] ], "source_shift": [ 2, 2, 2 ], "target_matrix": [ [ -2, -1, -1 ], [ 1, 2, 1 ], [ -2, -3, -2 ] ], "target_shift": [ 1789, -1905, 3125 ], "collision_preimages": [ [ "9/4", "5", "5/2" ], [ "-7", "-55/2", "-25/2" ], [ "-2", "-33/2", "-19/2" ] ], "collision_image": [ "3579/2", "-7621/4", "6251/2" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 0.9735609780000232 ], "surface_features": [ 4.5259613672932435, 5.379961450562672, 4.1773344555057, 0.0, 4.639644570424524, 5226 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B049", "pair": 49, "family": "bijection", "mapping": "F(x,y,z)=(-2*x^7 - 27*x^6*y + 15*x^6*z + 22*x^6 - 156*x^5*y^2 + 174*x^5*y*z + 249*x^5*y - 48*x^5*z^2 - 141*x^5*z - 102*x^5 - 500*x^4*y^3 + 840*x^4*y^2*z + 1173*x^4*y^2 - 465*x^4*y*z^2 - 1332*x^4*y*z - 939*x^4*y + 85*x^4*z^3 + 375*x^4*z^2 + 543*x^4*z + 258*x^4 - 960*x^3*y^4 + 2160*x^3*y^3*z + 2944*x^3*y^3 - 1800*x^3*y^2*z^2 - 5028*x^3*y^2*z - 3456*x^3*y^2 + 660*x^3*y*z^3 + 2838*x^3*y*z^2 + 4002*x^3*y*z + 1851*x^3*y - 90*x^3*z^4 - 530*x^3*z^3 - 1152*x^3*z^2 - 1095*x^3*z - 392*x^3 - 1104*x^2*y^5 + 3120*x^2*y^4*z + 4152*x^2*y^4 - 3480*x^2*y^3*z^2 - 9480*x^2*y^3*z - 6357*x^2*y^3 + 1920*x^2*y^2*z^3 + 8046*x^2*y^2*z^2 + 11055*x^2*y^2*z + 4983*x^2*y^2 - 525*x^2*y*z^4 - 3012*x^2*y*z^3 - 6372*x^2*y*z^2 - 5892*x^2*y*z - 2046*x^2*y + 57*x^2*z^5 + 420*x^2*z^4 + 1218*x^2*z^3 + 1737*x^2*z^2 + 1254*x^2*z + 372*x^2 - 704*x*y^6 + 2400*x*y^5*z + 3120*x*y^5 - 3360*x*y^4*z^2 - 8928*x*y^4*z - 5844*x*y^4 + 2480*x*y^3*z^3 + 10128*x*y^3*z^2 + 13566*x*y^3*z + 5965*x*y^3 - 1020*x*y^2*z^4 - 5700*x*y^2*z^3 - 11742*x*y^2*z^2 - 10575*x*y^2*z - 3570*x*y^2 + 222*x*y*z^5 + 1593*x*y*z^4 + 4494*x*y*z^3 + 6231*x*y*z^2 + 4356*x*y*z + 1244*x*y - 20*x*z^6 - 177*x*z^5 - 642*x*z^4 - 1221*x*z^3 - 1338*x*z^2 - 820*x*z - 217*x - 192*y^7 + 768*y^6*z + 976*y^6 - 1296*y^5*z^2 - 3360*y^5*z - 2148*y^5 + 1200*y^4*z^3 + 4776*y^4*z^2 + 6240*y^4*z + 2679*y^4 - 660*y^3*z^4 - 3592*y^3*z^3 - 7209*y^3*z^2 - 6330*y^3*z - 2081*y^3 + 216*y^2*z^5 + 1509*y^2*z^4 + 4143*y^2*z^3 + 5592*y^2*z^2 + 3795*y^2*z + 1046*y^2 - 39*y*z^6 - 336*y*z^5 - 1185*y*z^4 - 2190*y*z^3 - 2319*y*z^2 - 1364*y*z - 348*y + 3*z^7 + 31*z^6 + 135*z^5 + 321*z^4 + 477*z^3 + 458*z^2 + 259*z + 1, 4*x^7 + 54*x^6*y - 30*x^6*z - 44*x^6 + 312*x^5*y^2 - 348*x^5*y*z - 498*x^5*y + 96*x^5*z^2 + 282*x^5*z + 204*x^5 + 1000*x^4*y^3 - 1680*x^4*y^2*z - 2346*x^4*y^2 + 930*x^4*y*z^2 + 2664*x^4*y*z + 1878*x^4*y - 170*x^4*z^3 - 750*x^4*z^2 - 1086*x^4*z - 516*x^4 + 1920*x^3*y^4 - 4320*x^3*y^3*z - 5888*x^3*y^3 + 3600*x^3*y^2*z^2 + 10056*x^3*y^2*z + 6912*x^3*y^2 - 1320*x^3*y*z^3 - 5676*x^3*y*z^2 - 8004*x^3*y*z - 3702*x^3*y + 180*x^3*z^4 + 1060*x^3*z^3 + 2304*x^3*z^2 + 2190*x^3*z + 784*x^3 + 2208*x^2*y^5 - 6240*x^2*y^4*z - 8304*x^2*y^4 + 6960*x^2*y^3*z^2 + 18960*x^2*y^3*z + 12714*x^2*y^3 - 3840*x^2*y^2*z^3 - 16092*x^2*y^2*z^2 - 22110*x^2*y^2*z - 9966*x^2*y^2 + 1050*x^2*y*z^4 + 6024*x^2*y*z^3 + 12744*x^2*y*z^2 + 11784*x^2*y*z + 4092*x^2*y - 114*x^2*z^5 - 840*x^2*z^4 - 2436*x^2*z^3 - 3474*x^2*z^2 - 2508*x^2*z - 750*x^2 + 1408*x*y^6 - 4800*x*y^5*z - 6240*x*y^5 + 6720*x*y^4*z^2 + 17856*x*y^4*z + 11688*x*y^4 - 4960*x*y^3*z^3 - 20256*x*y^3*z^2 - 27132*x*y^3*z - 11930*x*y^3 + 2040*x*y^2*z^4 + 11400*x*y^2*z^3 + 23484*x*y^2*z^2 + 21150*x*y^2*z + 7140*x*y^2 - 444*x*y*z^5 - 3186*x*y*z^4 - 8988*x*y*z^3 - 12462*x*y*z^2 - 8712*x*y*z - 2512*x*y + 40*x*z^6 + 354*x*z^5 + 1284*x*z^4 + 2442*x*z^3 + 2676*x*z^2 + 1652*x*z + 450*x + 384*y^7 - 1536*y^6*z - 1952*y^6 + 2592*y^5*z^2 + 6720*y^5*z + 4296*y^5 - 2400*y^4*z^3 - 9552*y^4*z^2 - 12480*y^4*z - 5358*y^4 + 1320*y^3*z^4 + 7184*y^3*z^3 + 14418*y^3*z^2 + 12660*y^3*z + 4162*y^3 - 432*y^2*z^5 - 3018*y^2*z^4 - 8286*y^2*z^3 - 11184*y^2*z^2 - 7590*y^2*z - 2116*y^2 + 78*y*z^6 + 672*y*z^5 + 2370*y*z^4 + 4380*y*z^3 + 4638*y*z^2 + 2752*y*z + 723*y - 6*z^7 - 62*z^6 - 270*z^5 - 642*z^4 - 954*z^3 - 922*z^2 - 533*z - 1, 2*x^7 + 27*x^6*y - 15*x^6*z - 22*x^6 + 156*x^5*y^2 - 174*x^5*y*z - 249*x^5*y + 48*x^5*z^2 + 141*x^5*z + 102*x^5 + 500*x^4*y^3 - 840*x^4*y^2*z - 1173*x^4*y^2 + 465*x^4*y*z^2 + 1332*x^4*y*z + 939*x^4*y - 85*x^4*z^3 - 375*x^4*z^2 - 543*x^4*z - 258*x^4 + 960*x^3*y^4 - 2160*x^3*y^3*z - 2944*x^3*y^3 + 1800*x^3*y^2*z^2 + 5028*x^3*y^2*z + 3456*x^3*y^2 - 660*x^3*y*z^3 - 2838*x^3*y*z^2 - 4002*x^3*y*z - 1851*x^3*y + 90*x^3*z^4 + 530*x^3*z^3 + 1152*x^3*z^2 + 1095*x^3*z + 392*x^3 + 1104*x^2*y^5 - 3120*x^2*y^4*z - 4152*x^2*y^4 + 3480*x^2*y^3*z^2 + 9480*x^2*y^3*z + 6357*x^2*y^3 - 1920*x^2*y^2*z^3 - 8046*x^2*y^2*z^2 - 11055*x^2*y^2*z - 4983*x^2*y^2 + 525*x^2*y*z^4 + 3012*x^2*y*z^3 + 6372*x^2*y*z^2 + 5892*x^2*y*z + 2046*x^2*y - 57*x^2*z^5 - 420*x^2*z^4 - 1218*x^2*z^3 - 1737*x^2*z^2 - 1254*x^2*z - 374*x^2 + 704*x*y^6 - 2400*x*y^5*z - 3120*x*y^5 + 3360*x*y^4*z^2 + 8928*x*y^4*z + 5844*x*y^4 - 2480*x*y^3*z^3 - 10128*x*y^3*z^2 - 13566*x*y^3*z - 5965*x*y^3 + 1020*x*y^2*z^4 + 5700*x*y^2*z^3 + 11742*x*y^2*z^2 + 10575*x*y^2*z + 3570*x*y^2 - 222*x*y*z^5 - 1593*x*y*z^4 - 4494*x*y*z^3 - 6231*x*y*z^2 - 4356*x*y*z - 1252*x*y + 20*x*z^6 + 177*x*z^5 + 642*x*z^4 + 1221*x*z^3 + 1338*x*z^2 + 824*x*z + 223*x + 192*y^7 - 768*y^6*z - 976*y^6 + 1296*y^5*z^2 + 3360*y^5*z + 2148*y^5 - 1200*y^4*z^3 - 4776*y^4*z^2 - 6240*y^4*z - 2679*y^4 + 660*y^3*z^4 + 3592*y^3*z^3 + 7209*y^3*z^2 + 6330*y^3*z + 2081*y^3 - 216*y^2*z^5 - 1509*y^2*z^4 - 4143*y^2*z^3 - 5592*y^2*z^2 - 3795*y^2*z - 1054*y^2 + 39*y*z^6 + 336*y*z^5 + 1185*y*z^4 + 2190*y*z^3 + 2319*y*z^2 + 1372*y*z + 358*y - 3*z^7 - 31*z^6 - 135*z^5 - 321*z^4 - 477*z^3 - 460*z^2 - 265*z - 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -1, -2, 1 ], [ -1, -1, 2 ], [ -1, -1, 1 ] ], "source_shift": [ 1, 2, 1 ], "target_matrix": [ [ 1, 1, 2 ], [ -2, -1, -1 ], [ -1, -1, -1 ] ], "target_shift": [ -63, 136, 67 ], "inverse_recipe": { "p": 1, "q": 2, "f": "x", "g": "x**2", "h": "y**3 + y**2 + y*z**3 + z**3", "polynomials": { "f": [ [ [ 1, 0, 0 ], 1 ] ], "g": [ [ [ 2, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 3 ], 1 ], [ [ 0, 0, 3 ], 1 ], [ [ 0, 3, 0 ], 1 ], [ [ 0, 2, 0 ], 1 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0353659369975503 ], "surface_features": [ 3.5141177462115025, 4.433481808392984, 3.2022157758011316, 0.0, 3.6245796778157082, 4896 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B129", "pair": 129, "family": "bijection", "mapping": "F(x,y,z)=(128*x^7 - 1344*x^6*y + 1344*x^6*z - 480*x^6 + 6048*x^5*y^2 - 12096*x^5*y*z + 4304*x^5*y + 6048*x^5*z^2 - 4304*x^5*z + 816*x^5 - 15120*x^4*y^3 + 45360*x^4*y^2*z - 16080*x^4*y^2 - 45360*x^4*y*z^2 + 32160*x^4*y*z - 6080*x^4*y + 15120*x^4*z^3 - 16080*x^4*z^2 + 6080*x^4*z - 840*x^4 + 22680*x^3*y^4 - 90720*x^3*y^3*z + 32040*x^3*y^3 + 136080*x^3*y^2*z^2 - 96120*x^3*y^2*z + 18120*x^3*y^2 - 90720*x^3*y*z^3 + 96120*x^3*y*z^2 - 36240*x^3*y*z + 4996*x^3*y + 22680*x^3*z^4 - 32040*x^3*z^3 + 18120*x^3*z^2 - 4996*x^3*z + 592*x^3 - 20412*x^2*y^5 + 102060*x^2*y^4*z - 35910*x^2*y^4 - 204120*x^2*y^3*z^2 + 143640*x^2*y^3*z - 27000*x^2*y^3 + 204120*x^2*y^2*z^3 - 215460*x^2*y^2*z^2 + 81000*x^2*y^2*z - 11142*x^2*y^2 - 102060*x^2*y*z^4 + 143640*x^2*y*z^3 - 81000*x^2*y*z^2 + 22284*x^2*y*z - 2632*x^2*y + 20412*x^2*z^5 - 35910*x^2*z^4 + 27000*x^2*z^3 - 11142*x^2*z^2 + 2628*x^2*z - 317*x^2 + 10206*x*y^6 - 61236*x*y^5*z + 21465*x*y^5 + 153090*x*y^4*z^2 - 107325*x*y^4*z + 20115*x*y^4 - 204120*x*y^3*z^3 + 214650*x*y^3*z^2 - 80460*x*y^3*z + 11043*x*y^3 + 153090*x*y^2*z^4 - 214650*x*y^2*z^3 + 120690*x*y^2*z^2 - 33129*x*y^2*z + 3900*x*y^2 - 61236*x*y*z^5 + 107325*x*y*z^4 - 80460*x*y*z^3 + 33129*x*y*z^2 - 7788*x*y*z + 934*x*y + 10206*x*z^6 - 21465*x*z^5 + 20115*x*z^4 - 11043*x*z^3 + 3888*x*z^2 - 929*x*z + 125*x - 2187*y^7 + 15309*y^6*z - 5346*y^6 - 45927*y^5*z^2 + 32076*y^5*z - 5994*y^5 + 76545*y^4*z^3 - 80190*y^4*z^2 + 29970*y^4*z - 4104*y^4 - 76545*y^3*z^4 + 106920*y^3*z^3 - 59940*y^3*z^2 + 16416*y^3*z - 1926*y^3 + 45927*y^2*z^5 - 80190*y^2*z^4 + 59940*y^2*z^3 - 24624*y^2*z^2 + 5769*y^2*z - 688*y^2 - 15309*y*z^6 + 32076*y*z^5 - 29970*y*z^4 + 16416*y*z^3 - 5760*y*z^2 + 1369*y*z - 180*y + 2187*z^7 - 5346*z^6 + 5994*z^5 - 4104*z^4 + 1917*z^3 - 681*z^2 + 178*z - 1, 128*x^7 - 1344*x^6*y + 1344*x^6*z - 480*x^6 + 6048*x^5*y^2 - 12096*x^5*y*z + 4304*x^5*y + 6048*x^5*z^2 - 4304*x^5*z + 752*x^5 - 15120*x^4*y^3 + 45360*x^4*y^2*z - 16080*x^4*y^2 - 45360*x^4*y*z^2 + 32160*x^4*y*z - 5600*x^4*y + 15120*x^4*z^3 - 16080*x^4*z^2 + 5600*x^4*z - 680*x^4 + 22680*x^3*y^4 - 90720*x^3*y^3*z + 32040*x^3*y^3 + 136080*x^3*y^2*z^2 - 96120*x^3*y^2*z + 16680*x^3*y^2 - 90720*x^3*y*z^3 + 96120*x^3*y*z^2 - 33360*x^3*y*z + 4036*x^3*y + 22680*x^3*z^4 - 32040*x^3*z^3 + 16680*x^3*z^2 - 4036*x^3*z + 416*x^3 - 20412*x^2*y^5 + 102060*x^2*y^4*z - 35910*x^2*y^4 - 204120*x^2*y^3*z^2 + 143640*x^2*y^3*z - 24840*x^2*y^3 + 204120*x^2*y^2*z^3 - 215460*x^2*y^2*z^2 + 74520*x^2*y^2*z - 8982*x^2*y^2 - 102060*x^2*y*z^4 + 143640*x^2*y*z^3 - 74520*x^2*y*z^2 + 17964*x^2*y*z - 1840*x^2*y + 20412*x^2*z^5 - 35910*x^2*z^4 + 24840*x^2*z^3 - 8982*x^2*z^2 + 1836*x^2*z - 213*x^2 + 10206*x*y^6 - 61236*x*y^5*z + 21465*x*y^5 + 153090*x*y^4*z^2 - 107325*x*y^4*z + 18495*x*y^4 - 204120*x*y^3*z^3 + 214650*x*y^3*z^2 - 73980*x*y^3*z + 8883*x*y^3 + 153090*x*y^2*z^4 - 214650*x*y^2*z^3 + 110970*x*y^2*z^2 - 26649*x*y^2*z + 2712*x*y^2 - 61236*x*y*z^5 + 107325*x*y*z^4 - 73980*x*y*z^3 + 26649*x*y*z^2 - 5412*x*y*z + 622*x*y + 10206*x*z^6 - 21465*x*z^5 + 18495*x*z^4 - 8883*x*z^3 + 2700*x*z^2 - 617*x*z + 87*x - 2187*y^7 + 15309*y^6*z - 5346*y^6 - 45927*y^5*z^2 + 32076*y^5*z - 5508*y^5 + 76545*y^4*z^3 - 80190*y^4*z^2 + 27540*y^4*z - 3294*y^4 - 76545*y^3*z^4 + 106920*y^3*z^3 - 55080*y^3*z^2 + 13176*y^3*z - 1332*y^3 + 45927*y^2*z^5 - 80190*y^2*z^4 + 55080*y^2*z^3 - 19764*y^2*z^2 + 3987*y^2*z - 454*y^2 - 15309*y*z^6 + 32076*y*z^5 - 27540*y*z^4 + 13176*y*z^3 - 3978*y*z^2 + 901*y*z - 122*y + 2187*z^7 - 5346*z^6 + 5508*z^5 - 3294*z^4 + 1323*z^3 - 447*z^2 + 122*z - 1, -384*x^7 + 4032*x^6*y - 4032*x^6*z + 1440*x^6 - 18144*x^5*y^2 + 36288*x^5*y*z - 12912*x^5*y - 18144*x^5*z^2 + 12912*x^5*z - 2448*x^5 + 45360*x^4*y^3 - 136080*x^4*y^2*z + 48240*x^4*y^2 + 136080*x^4*y*z^2 - 96480*x^4*y*z + 18240*x^4*y - 45360*x^4*z^3 + 48240*x^4*z^2 - 18240*x^4*z + 2520*x^4 - 68040*x^3*y^4 + 272160*x^3*y^3*z - 96120*x^3*y^3 - 408240*x^3*y^2*z^2 + 288360*x^3*y^2*z - 54360*x^3*y^2 + 272160*x^3*y*z^3 - 288360*x^3*y*z^2 + 108720*x^3*y*z - 14988*x^3*y - 68040*x^3*z^4 + 96120*x^3*z^3 - 54360*x^3*z^2 + 14988*x^3*z - 1776*x^3 + 61236*x^2*y^5 - 306180*x^2*y^4*z + 107730*x^2*y^4 + 612360*x^2*y^3*z^2 - 430920*x^2*y^3*z + 81000*x^2*y^3 - 612360*x^2*y^2*z^3 + 646380*x^2*y^2*z^2 - 243000*x^2*y^2*z + 33426*x^2*y^2 + 306180*x^2*y*z^4 - 430920*x^2*y*z^3 + 243000*x^2*y*z^2 - 66852*x^2*y*z + 7896*x^2*y - 61236*x^2*z^5 + 107730*x^2*z^4 - 81000*x^2*z^3 + 33426*x^2*z^2 - 7884*x^2*z + 947*x^2 - 30618*x*y^6 + 183708*x*y^5*z - 64395*x*y^5 - 459270*x*y^4*z^2 + 321975*x*y^4*z - 60345*x*y^4 + 612360*x*y^3*z^3 - 643950*x*y^3*z^2 + 241380*x*y^3*z - 33129*x*y^3 - 459270*x*y^2*z^4 + 643950*x*y^2*z^3 - 362070*x*y^2*z^2 + 99387*x*y^2*z - 11700*x*y^2 + 183708*x*y*z^5 - 321975*x*y*z^4 + 241380*x*y*z^3 - 99387*x*y*z^2 + 23364*x*y*z - 2790*x*y - 30618*x*z^6 + 64395*x*z^5 - 60345*x*z^4 + 33129*x*z^3 - 11664*x*z^2 + 2775*x*z - 370*x + 6561*y^7 - 45927*y^6*z + 16038*y^6 + 137781*y^5*z^2 - 96228*y^5*z + 17982*y^5 - 229635*y^4*z^3 + 240570*y^4*z^2 - 89910*y^4*z + 12312*y^4 + 229635*y^3*z^4 - 320760*y^3*z^3 + 179820*y^3*z^2 - 49248*y^3*z + 5778*y^3 - 137781*y^2*z^5 + 240570*y^2*z^4 - 179820*y^2*z^3 + 73872*y^2*z^2 - 17307*y^2*z + 2055*y^2 + 45927*y*z^6 - 96228*y*z^5 + 89910*y*z^4 - 49248*y*z^3 + 17280*y*z^2 - 4089*y*z + 533*y - 6561*z^7 + 16038*z^6 - 17982*z^5 + 12312*z^4 - 5751*z^3 + 2034*z^2 - 527*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -2, 3, -3 ], [ -1, 1, -1 ], [ -1, 2, -1 ] ], "source_shift": [ 1, 2, 1 ], "target_matrix": [ [ -1, -1, -1 ], [ -1, -1, -2 ], [ 3, 2, 3 ] ], "target_shift": [ 29, 21, -86 ], "inverse_recipe": { "p": 2, "q": 5, "f": "x**2", "g": "x**5 + x**3 + x", "h": "2*y**2 + y*z + z", "polynomials": { "f": [ [ [ 2, 0, 0 ], 1 ] ], "g": [ [ [ 5, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ], [ [ 3, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 0, 1 ], 1 ], [ [ 0, 2, 0 ], 2 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 0.9780660789983813 ], "surface_features": [ 4.847953408069772, 5.810487910229379, 4.393259237826838, 0.0, 5.048179860306797, 5328 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B147", "pair": 147, "family": "bijection", "mapping": "F(x,y,z)=(66*x^7 - 440*x^6*y - 440*x^6*z + 859*x^6 + 1254*x^5*y^2 + 2508*x^5*y*z - 4890*x^5*y + 1254*x^5*z^2 - 4890*x^5*z + 4764*x^5 - 1980*x^4*y^3 - 5940*x^4*y^2*z + 11565*x^4*y^2 - 5940*x^4*y*z^2 + 23130*x^4*y*z - 22500*x^4*y - 1980*x^4*z^3 + 11565*x^4*z^2 - 22500*x^4*z + 14580*x^4 + 1870*x^3*y^4 + 7480*x^3*y^3*z - 14540*x^3*y^3 + 11220*x^3*y^2*z^2 - 43620*x^3*y^2*z + 42360*x^3*y^2 + 7480*x^3*y*z^3 - 43620*x^3*y*z^2 + 84720*x^3*y*z - 54800*x^3*y + 1870*x^3*z^4 - 14540*x^3*z^3 + 42360*x^3*z^2 - 54800*x^3*z + 26560*x^3 - 1056*x^2*y^5 - 5280*x^2*y^4*z + 10245*x^2*y^4 - 10560*x^2*y^3*z^2 + 40980*x^2*y^3*z - 39720*x^2*y^3 - 10560*x^2*y^2*z^3 + 61470*x^2*y^2*z^2 - 119160*x^2*y^2*z + 76920*x^2*y^2 - 5280*x^2*y*z^4 + 40980*x^2*y*z^3 - 119160*x^2*y*z^2 + 153840*x^2*y*z - 74400*x^2*y - 1056*x^2*z^5 + 10245*x^2*z^4 - 39720*x^2*z^3 + 76920*x^2*z^2 - 74400*x^2*z + 29016*x^2 + 330*x*y^6 + 1980*x*y^5*z - 3834*x*y^5 + 4950*x*y^4*z^2 - 19170*x*y^4*z + 18540*x*y^4 + 6600*x*y^3*z^3 - 38340*x*y^3*z^2 + 74160*x*y^3*z - 47760*x*y^3 + 4950*x*y^2*z^4 - 38340*x*y^2*z^3 + 111240*x*y^2*z^2 - 143280*x*y^2*z + 69120*x*y^2 + 1980*x*y*z^5 - 19170*x*y*z^4 + 74160*x*y*z^3 - 143280*x*y*z^2 + 138240*x*y*z - 53654*x*y + 330*x*z^6 - 3834*x*z^5 + 18540*x*z^4 - 47760*x*z^3 + 69120*x*z^2 - 53720*x*z + 17616*x - 44*y^7 - 308*y^6*z + 595*y^6 - 924*y^5*z^2 + 3570*y^5*z - 3444*y^5 - 1540*y^4*z^3 + 8925*y^4*z^2 - 17220*y^4*z + 11060*y^4 - 1540*y^3*z^4 + 11900*y^3*z^3 - 34440*y^3*z^2 + 44240*y^3*z - 21280*y^3 - 924*y^2*z^5 + 8925*y^2*z^4 - 34440*y^2*z^3 + 66360*y^2*z^2 - 63840*y^2*z + 24660*y^2 - 308*y*z^6 + 3570*y*z^5 - 17220*y*z^4 + 44240*y*z^3 - 63840*y*z^2 + 49364*y*z - 16055*y - 44*z^7 + 595*z^6 - 3444*z^5 + 11060*z^4 - 21280*z^3 + 24704*z^2 - 16122*z + 1, 33*x^7 - 220*x^6*y - 220*x^6*z + 431*x^6 + 627*x^5*y^2 + 1254*x^5*y*z - 2454*x^5*y + 627*x^5*z^2 - 2454*x^5*z + 2400*x^5 - 990*x^4*y^3 - 2970*x^4*y^2*z + 5805*x^4*y^2 - 2970*x^4*y*z^2 + 11610*x^4*y*z - 11340*x^4*y - 990*x^4*z^3 + 5805*x^4*z^2 - 11340*x^4*z + 7380*x^4 + 935*x^3*y^4 + 3740*x^3*y^3*z - 7300*x^3*y^3 + 5610*x^3*y^2*z^2 - 21900*x^3*y^2*z + 21360*x^3*y^2 + 3740*x^3*y*z^3 - 21900*x^3*y*z^2 + 42720*x^3*y*z - 27760*x^3*y + 935*x^3*z^4 - 7300*x^3*z^3 + 21360*x^3*z^2 - 27760*x^3*z + 13520*x^3 - 528*x^2*y^5 - 2640*x^2*y^4*z + 5145*x^2*y^4 - 5280*x^2*y^3*z^2 + 20580*x^2*y^3*z - 20040*x^2*y^3 - 5280*x^2*y^2*z^3 + 30870*x^2*y^2*z^2 - 60120*x^2*y^2*z + 39000*x^2*y^2 - 2640*x^2*y*z^4 + 20580*x^2*y*z^3 - 60120*x^2*y*z^2 + 78000*x^2*y*z - 37920*x^2*y - 528*x^2*z^5 + 5145*x^2*z^4 - 20040*x^2*z^3 + 39000*x^2*z^2 - 37920*x^2*z + 14868*x^2 + 165*x*y^6 + 990*x*y^5*z - 1926*x*y^5 + 2475*x*y^4*z^2 - 9630*x*y^4*z + 9360*x*y^4 + 3300*x*y^3*z^3 - 19260*x*y^3*z^2 + 37440*x*y^3*z - 24240*x*y^3 + 2475*x*y^2*z^4 - 19260*x*y^2*z^3 + 56160*x*y^2*z^2 - 72720*x*y^2*z + 35280*x*y^2 + 990*x*y*z^5 - 9630*x*y*z^4 + 37440*x*y*z^3 - 72720*x*y*z^2 + 70560*x*y*z - 27547*x*y + 165*x*z^6 - 1926*x*z^5 + 9360*x*z^4 - 24240*x*z^3 + 35280*x*z^2 - 27580*x*z + 9099*x - 22*y^7 - 154*y^6*z + 299*y^6 - 462*y^5*z^2 + 1794*y^5*z - 1740*y^5 - 770*y^4*z^3 + 4485*y^4*z^2 - 8700*y^4*z + 5620*y^4 - 770*y^3*z^4 + 5980*y^3*z^3 - 17400*y^3*z^2 + 22480*y^3*z - 10880*y^3 - 462*y^2*z^5 + 4485*y^2*z^4 - 17400*y^2*z^3 + 33720*y^2*z^2 - 32640*y^2*z + 12690*y^2 - 154*y*z^6 + 1794*y*z^5 - 8700*y*z^4 + 22480*y*z^3 - 32640*y*z^2 + 25402*y*z - 8318*y - 22*z^7 + 299*z^6 - 1740*z^5 + 5620*z^4 - 10880*z^3 + 12712*z^2 - 8353*z + 1, 33*x^7 - 220*x^6*y - 220*x^6*z + 430*x^6 + 627*x^5*y^2 + 1254*x^5*y*z - 2448*x^5*y + 627*x^5*z^2 - 2448*x^5*z + 2388*x^5 - 990*x^4*y^3 - 2970*x^4*y^2*z + 5790*x^4*y^2 - 2970*x^4*y*z^2 + 11580*x^4*y*z - 11280*x^4*y - 990*x^4*z^3 + 5790*x^4*z^2 - 11280*x^4*z + 7320*x^4 + 935*x^3*y^4 + 3740*x^3*y^3*z - 7280*x^3*y^3 + 5610*x^3*y^2*z^2 - 21840*x^3*y^2*z + 21240*x^3*y^2 + 3740*x^3*y*z^3 - 21840*x^3*y*z^2 + 42480*x^3*y*z - 27520*x^3*y + 935*x^3*z^4 - 7280*x^3*z^3 + 21240*x^3*z^2 - 27520*x^3*z + 13360*x^3 - 528*x^2*y^5 - 2640*x^2*y^4*z + 5130*x^2*y^4 - 5280*x^2*y^3*z^2 + 20520*x^2*y^3*z - 19920*x^2*y^3 - 5280*x^2*y^2*z^3 + 30780*x^2*y^2*z^2 - 59760*x^2*y^2*z + 38640*x^2*y^2 - 2640*x^2*y*z^4 + 20520*x^2*y*z^3 - 59760*x^2*y*z^2 + 77280*x^2*y*z - 37440*x^2*y - 528*x^2*z^5 + 5130*x^2*z^4 - 19920*x^2*z^3 + 38640*x^2*z^2 - 37440*x^2*z + 14628*x^2 + 165*x*y^6 + 990*x*y^5*z - 1920*x*y^5 + 2475*x*y^4*z^2 - 9600*x*y^4*z + 9300*x*y^4 + 3300*x*y^3*z^3 - 19200*x*y^3*z^2 + 37200*x*y^3*z - 24000*x*y^3 + 2475*x*y^2*z^4 - 19200*x*y^2*z^3 + 55800*x*y^2*z^2 - 72000*x*y^2*z + 34800*x*y^2 + 990*x*y*z^5 - 9600*x*y*z^4 + 37200*x*y*z^3 - 72000*x*y*z^2 + 69600*x*y*z - 27067*x*y + 165*x*z^6 - 1920*x*z^5 + 9300*x*z^4 - 24000*x*z^3 + 34800*x*z^2 - 27100*x*z + 8903*x - 22*y^7 - 154*y^6*z + 298*y^6 - 462*y^5*z^2 + 1788*y^5*z - 1728*y^5 - 770*y^4*z^3 + 4470*y^4*z^2 - 8640*y^4*z + 5560*y^4 - 770*y^3*z^4 + 5960*y^3*z^3 - 17280*y^3*z^2 + 22240*y^3*z - 10720*y^3 - 462*y^2*z^5 + 4470*y^2*z^4 - 17280*y^2*z^3 + 33360*y^2*z^2 - 32160*y^2*z + 12450*y^2 - 154*y*z^6 + 1788*y*z^5 - 8640*y*z^4 + 22240*y*z^3 - 32160*y*z^2 + 24922*y*z - 8123*y - 22*z^7 + 298*z^6 - 1728*z^5 + 5560*z^4 - 10720*z^3 + 12472*z^2 - 8157*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ 1, -1, -1 ], [ 3, -2, -3 ], [ 2, -1, -1 ] ], "source_shift": [ 2, 2, 1 ], "target_matrix": [ [ 2, 1, 1 ], [ 1, 2, 1 ], [ 1, 1, 1 ] ], "target_shift": [ -4553, -2375, -2307 ], "inverse_recipe": { "p": 6, "q": 1, "f": "x**6 + x", "g": "x", "h": "y*z", "polynomials": { "f": [ [ [ 6, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "g": [ [ [ 1, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ] ] }, "h_multiplier": 11 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 0.9815463070008263 ], "surface_features": [ 4.284736762871357, 5.187069271230423, 3.963764223743174, 0.0, 4.4072605946176395, 5141 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C243", "pair": 243, "family": "counterexample", "mapping": "F(x,y,z)=(16*x^7 - 88*x^6*y + 136*x^6*z - 220*x^6 + 204*x^5*y^2 - 624*x^5*y*z + 972*x^5*y + 468*x^5*z^2 - 1488*x^5*z + 1080*x^5 - 258*x^4*y^3 + 1170*x^4*y^2*z - 1767*x^4*y^2 - 1734*x^4*y*z^2 + 5334*x^4*y*z - 3888*x^4*y + 838*x^4*z^3 - 3939*x^4*z^2 + 5940*x^4*z - 2783*x^4 + 192*x^3*y^4 - 1146*x^3*y^3*z + 1692*x^3*y^3 + 2514*x^3*y^2*z^2 - 7557*x^3*y^2*z + 5520*x^3*y^2 - 2398*x^3*y*z^3 + 11010*x^3*y*z^2 - 16584*x^3*y*z + 7910*x^3*y + 838*x^3*z^4 - 5221*x^3*z^3 + 12168*x^3*z^2 - 12051*x^3*z + 4226*x^3 - 84*x^2*y^5 + 618*x^2*y^4*z - 900*x^2*y^4 - 1782*x^2*y^3*z^2 + 5289*x^2*y^3*z - 3867*x^2*y^3 + 2514*x^2*y^2*z^3 - 11412*x^2*y^2*z^2 + 17148*x^2*y^2*z - 8298*x^2*y^2 - 1734*x^2*y*z^4 + 10701*x^2*y*z^3 - 24759*x^2*y*z^2 + 24807*x^2*y*z - 8913*x^2*y + 468*x^2*z^5 - 3678*x^2*z^4 + 11610*x^2*z^3 - 18078*x^2*z^2 + 13512*x^2*z - 3851*x^2 + 20*x*y^6 - 174*x*y^5*z + 252*x*y^5 + 618*x*y^4*z^2 - 1827*x*y^4*z + 1338*x*y^4 - 1146*x*y^3*z^3 + 5190*x*y^3*z^2 - 7794*x*y^3*z + 3814*x*y^3 + 1170*x*y^2*z^4 - 7215*x*y^2*z^3 + 16644*x*y^2*z^2 - 16800*x*y^2*z + 6156*x*y^2 - 624*x*y*z^5 + 4908*x*y*z^4 - 15426*x*y*z^3 + 24054*x*y*z^2 - 18270*x*y*z + 5354*x*y + 136*x*z^6 - 1308*x*z^5 + 5238*x*z^4 - 11167*x*z^3 + 13182*x*z^2 - 8048*x*z + 1970*x - 2*y^7 + 20*y^6*z - 29*y^6 - 84*y^5*z^2 + 249*y^5*z - 183*y^5 + 192*y^4*z^3 - 873*y^4*z^2 + 1314*y^4*z - 649*y^4 - 258*y^3*z^4 + 1599*y^3*z^3 - 3693*y^3*z^2 + 3750*y^3*z - 1396*y^3 + 204*y^2*z^5 - 1614*y^2*z^4 + 5076*y^2*z^3 - 7932*y^2*z^2 + 6096*y^2*z - 1826*y^2 - 88*y*z^6 + 852*y*z^5 - 3414*y*z^4 + 7273*y*z^3 - 8637*y*z^2 + 5362*y*z - 1352*y + 16*z^7 - 184*z^6 + 900*z^5 - 2442*z^4 + 3964*z^3 - 3815*z^2 + 2006*z - 1, -8*x^7 + 44*x^6*y - 68*x^6*z + 128*x^6 - 102*x^5*y^2 + 312*x^5*y*z - 576*x^5*y - 234*x^5*z^2 + 888*x^5*z - 756*x^5 + 129*x^4*y^3 - 585*x^4*y^2*z + 1068*x^4*y^2 + 867*x^4*y*z^2 - 3252*x^4*y*z + 2772*x^4*y - 419*x^4*z^3 + 2424*x^4*z^2 - 4284*x^4*z + 2219*x^4 - 96*x^3*y^4 + 573*x^3*y^3*z - 1044*x^3*y^3 - 1257*x^3*y^2*z^2 + 4710*x^3*y^2*z - 4011*x^3*y^2 + 1199*x^3*y*z^3 - 6936*x^3*y*z^2 + 12198*x^3*y*z - 6428*x^3*y - 419*x^3*z^4 + 3326*x^3*z^3 - 9063*x^3*z^2 + 9937*x^3*z - 3692*x^3 + 42*x^2*y^5 - 309*x^2*y^4*z + 567*x^2*y^4 + 891*x^2*y^3*z^2 - 3369*x^2*y^3*z + 2865*x^2*y^3 - 1257*x^2*y^2*z^3 + 7353*x^2*y^2*z^2 - 12867*x^2*y^2*z + 6873*x^2*y^2 + 867*x^2*y*z^4 - 6975*x^2*y*z^3 + 18819*x^2*y*z^2 - 20853*x^2*y*z + 7944*x^2*y - 234*x^2*z^5 + 2424*x^2*z^4 - 8937*x^2*z^3 + 15429*x^2*z^2 - 12249*x^2*z + 3592*x^2 - 10*x*y^6 + 87*x*y^5*z - 162*x*y^5 - 309*x*y^4*z^2 + 1188*x*y^4*z - 1011*x*y^4 + 573*x*y^3*z^3 - 3414*x*y^3*z^2 + 5967*x*y^3*z - 3219*x*y^3 - 585*x*y^2*z^4 + 4800*x*y^2*z^3 - 12912*x*y^2*z^2 + 14388*x*y^2*z - 5595*x*y^2 + 312*x*y*z^5 - 3300*x*y*z^4 + 12123*x*y*z^3 - 20901*x*y*z^2 + 16887*x*y*z - 5104*x*y - 68*x*z^6 + 888*x*z^5 - 4167*x*z^4 + 9835*x*z^3 - 12396*x*z^2 + 7825*x*z - 1923*x + y^7 - 10*y^6*z + 19*y^6 + 42*y^5*z^2 - 165*y^5*z + 141*y^5 - 96*y^4*z^3 + 585*y^4*z^2 - 1026*y^4*z + 558*y^4 + 129*y^3*z^4 - 1083*y^3*z^3 + 2922*y^3*z^2 - 3271*y^3*z + 1293*y^3 - 102*y^2*z^5 + 1104*y^2*z^4 - 4068*y^2*z^3 + 7017*y^2*z^2 - 5739*y^2*z + 1777*y^2 + 44*y*z^6 - 588*y*z^5 + 2769*y*z^4 - 6520*y*z^3 + 8262*y*z^2 - 5318*y*z + 1353*y - 8*z^7 + 128*z^6 - 738*z^5 + 2216*z^4 - 3847*z^3 + 3856*z^2 - 2056*z + 1, 8*x^7 - 44*x^6*y + 68*x^6*z - 116*x^6 + 102*x^5*y^2 - 312*x^5*y*z + 516*x^5*y + 234*x^5*z^2 - 792*x^5*z + 612*x^5 - 129*x^4*y^3 + 585*x^4*y^2*z - 945*x^4*y^2 - 867*x^4*y*z^2 + 2862*x^4*y*z - 2220*x^4*y + 419*x^4*z^3 - 2121*x^4*z^2 + 3408*x^4*z - 1667*x^4 + 96*x^3*y^4 - 573*x^3*y^3*z + 912*x^3*y^3 + 1257*x^3*y^2*z^2 - 4089*x^3*y^2*z + 3177*x^3*y^2 - 1199*x^3*y*z^3 + 5982*x^3*y*z^2 - 9594*x^3*y*z + 4778*x^3*y + 419*x^3*z^4 - 2849*x^3*z^3 + 7077*x^3*z^2 - 7327*x^3*z + 2636*x^3 - 42*x^2*y^5 + 309*x^2*y^4*z - 489*x^2*y^4 - 891*x^2*y^3*z^2 + 2886*x^2*y^3*z - 2244*x^2*y^3 + 1257*x^2*y^2*z^3 - 6255*x^2*y^2*z^2 + 10005*x^2*y^2*z - 5055*x^2*y^2 - 867*x^2*y*z^4 + 5892*x^2*y*z^3 - 14526*x^2*y*z^2 + 15213*x^2*y*z - 5610*x^2*y + 234*x^2*z^5 - 2034*x^2*z^4 + 6849*x^2*z^3 - 11163*x^2*z^2 + 8571*x^2*z - 2473*x^2 + 10*x*y^6 - 87*x*y^5*z + 138*x*y^5 + 309*x*y^4*z^2 - 1005*x*y^4*z + 783*x*y^4 - 573*x*y^3*z^3 + 2868*x*y^3*z^2 - 4587*x*y^3*z + 2343*x*y^3 + 585*x*y^2*z^4 - 4005*x*y^2*z^3 + 9852*x*y^2*z^2 - 10389*x*y^2*z + 3909*x*y^2 - 312*x*y*z^5 + 2736*x*y*z^4 - 9183*x*y*z^3 + 14973*x*y*z^2 - 11691*x*y*z + 3472*x*y + 68*x*z^6 - 732*x*z^5 + 3135*x*z^4 - 6994*x*z^3 + 8502*x*z^2 - 5266*x*z + 1291*x - y^7 + 10*y^6*z - 16*y^6 - 42*y^5*z^2 + 138*y^5*z - 108*y^5 + 96*y^4*z^3 - 486*y^4*z^2 + 780*y^4*z - 402*y^4 - 129*y^3*z^4 + 894*y^3*z^3 - 2205*y^3*z^2 + 2338*y^3*z - 894*y^3 + 102*y^2*z^5 - 906*y^2*z^4 + 3048*y^2*z^3 - 4977*y^2*z^2 + 3933*y^2*z - 1195*y^2 - 44*y*z^6 + 480*y*z^5 - 2061*y*z^4 + 4591*y*z^3 - 5613*y*z^2 + 3539*y*z - 896*y + 8*z^7 - 104*z^6 + 546*z^5 - 1550*z^4 + 2593*z^3 - 2539*z^2 + 1344*z - 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -1, 1, -2 ], [ -2, 1, -1 ], [ -1, 1, -1 ] ], "source_shift": [ 1, 2, 1 ], "target_matrix": [ [ -2, -1, -1 ], [ 1, 2, 1 ], [ -1, -1, -1 ] ], "target_shift": [ 437, -439, 290 ], "collision_preimages": [ [ "3/4", "-3/4", "-1/4" ], [ "9", "20", "11/2" ], [ "6", "19", "15/2" ] ], "collision_image": [ "875/2", "-1757/4", "1161/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 0.962235678001889 ], "surface_features": [ 3.5662377480667797, 4.394574246649326, 3.2337573629655103, 0.0, 3.6764469920731777, 4883 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C153", "pair": 153, "family": "counterexample", "mapping": "F(x,y,z)=(-48*x^7 + 248*x^6*y - 264*x^6*z + 344*x^6 - 540*x^5*y^2 + 1152*x^5*y*z - 1500*x^5*y - 612*x^5*z^2 + 1596*x^5*z - 1068*x^5 + 642*x^4*y^3 - 2058*x^4*y^2*z + 2682*x^4*y^2 + 2190*x^4*y*z^2 - 5712*x^4*y*z + 3846*x^4*y - 774*x^4*z^3 + 3030*x^4*z^2 - 4086*x^4*z + 1882*x^4 - 450*x^3*y^4 + 1926*x^3*y^3*z - 2517*x^3*y^3 - 3078*x^3*y^2*z^2 + 8043*x^3*y^2*z - 5454*x^3*y^2 + 2178*x^3*y*z^3 - 8535*x^3*y*z^2 + 11586*x^3*y*z - 5395*x^3*y - 576*x^3*z^4 + 3009*x^3*z^3 - 6132*x^3*z^2 + 5723*x^3*z - 2056*x^3 + 186*x^2*y^5 - 996*x^2*y^4*z + 1308*x^2*y^4 + 2124*x^2*y^3*z^2 - 5571*x^2*y^3*z + 3807*x^2*y^3 - 2256*x^2*y^2*z^3 + 8865*x^2*y^2*z^2 - 12120*x^2*y^2*z + 5703*x^2*y^2 + 1194*x^2*y*z^4 - 6249*x^2*y*z^3 + 12819*x^2*y*z^2 - 12087*x^2*y*z + 4401*x^2*y - 252*x^2*z^5 + 1647*x^2*z^4 - 4506*x^2*z^3 + 6384*x^2*z^2 - 4668*x^2*z + 1409*x^2 - 42*x*y^6 + 270*x*y^5*z - 357*x*y^5 - 720*x*y^4*z^2 + 1899*x*y^4*z - 1308*x*y^4 + 1020*x*y^3*z^3 - 4026*x*y^3*z^2 + 5544*x*y^3*z - 2635*x*y^3 - 810*x*y^2*z^4 + 4254*x*y^2*z^3 - 8784*x*y^2*z^2 + 8361*x*y^2*z - 3084*x*y^2 + 342*x*y*z^5 - 2241*x*y*z^4 + 6168*x*y*z^3 - 8817*x*y*z^2 + 6528*x*y*z - 1999*x*y - 60*x*z^6 + 471*x*z^5 - 1620*x*z^4 + 3091*x*z^3 - 3444*x*z^2 + 2125*x*z - 568*x + 4*y^7 - 30*y^6*z + 40*y^6 + 96*y^5*z^2 - 255*y^5*z + 177*y^5 - 170*y^4*z^3 + 675*y^4*z^2 - 936*y^4*z + 449*y^4 + 180*y^3*z^4 - 950*y^3*z^3 + 1974*y^3*z^2 - 1895*y^3*z + 707*y^3 - 114*y^2*z^5 + 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2379*x*z^2 + 1496*x*z - 405*x + 2*y^7 - 15*y^6*z + 23*y^6 + 48*y^5*z^2 - 147*y^5*z + 111*y^5 - 85*y^4*z^3 + 390*y^4*z^2 - 588*y^4*z + 297*y^4 + 90*y^3*z^4 - 550*y^3*z^3 + 1242*y^3*z^2 - 1255*y^3*z + 483*y^3 - 57*y^2*z^5 + 435*y^2*z^4 - 1308*y^2*z^3 + 1983*y^2*z^2 - 1530*y^2*z + 484*y^2 + 20*y*z^6 - 183*y*z^5 + 687*y*z^4 - 1389*y*z^3 + 1611*y*z^2 - 1025*y*z + 281*y - 3*z^7 + 32*z^6 - 144*z^5 + 364*z^4 - 564*z^3 + 541*z^2 - 300*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -2, 1, -1 ], [ -1, 1, -1 ], [ -3, 2, -3 ] ], "source_shift": [ 1, 1, 2 ], "target_matrix": [ [ 2, 1, 1 ], [ 1, 2, 1 ], [ 1, 1, 1 ] ], "target_shift": [ -102, -118, -72 ], "collision_preimages": [ [ "0", "-3/4", "1/4" ], [ "-5/2", "-12", "-7" ], [ "5/2", "-3", "-6" ] ], "collision_image": [ "-205/2", "-473/4", "-289/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 0.9935778260005463 ], "surface_features": [ 3.3344537511509307, 4.107854147536859, 3.04766419460156, 0.0, 3.411449806373007, 4843 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C071", "pair": 71, "family": "counterexample", "mapping": "F(x,y,z)=(-16*x^7 - 104*x^6*y + 88*x^6*z + 284*x^6 - 276*x^5*y^2 + 480*x^5*y*z + 1416*x^5*y - 204*x^5*z^2 - 1260*x^5*z - 1800*x^5 - 390*x^4*y^3 + 1038*x^4*y^2*z + 2871*x^4*y^2 - 906*x^4*y*z^2 - 5166*x^4*y*z - 7080*x^4*y + 258*x^4*z^3 + 2295*x^4*z^2 + 6432*x^4*z + 5871*x^4 - 318*x^3*y^4 + 1146*x^3*y^3*z + 3027*x^3*y^3 - 1530*x^3*y^2*z^2 - 8250*x^3*y^2*z - 10944*x^3*y^2 + 894*x^3*y*z^3 + 7419*x^3*y*z^2 + 20004*x^3*y*z + 17813*x^3*y - 192*x^3*z^4 - 2196*x^3*z^3 - 9060*x^3*z^2 - 16350*x^3*z - 11010*x^3 - 150*x^2*y^5 + 684*x^2*y^4*z + 1749*x^2*y^4 - 1236*x^2*y^3*z^2 - 6411*x^2*y^3*z - 8295*x^2*y^3 + 1104*x^2*y^2*z^3 + 8739*x^2*y^2*z^2 + 22881*x^2*y^2*z + 19965*x^2*y^2 - 486*x^2*y*z^4 - 5241*x^2*y*z^3 - 20877*x^2*y*z^2 - 36795*x^2*y*z - 24357*x^2*y + 84*x^2*z^5 + 1164*x^2*z^4 + 6291*x^2*z^3 + 16830*x^2*z^2 + 22467*x^2*z + 12041*x^2 - 38*x*y^6 + 210*x*y^5*z + 525*x*y^5 - 480*x*y^4*z^2 - 2424*x*y^4*z - 3078*x*y^4 + 580*x*y^3*z^3 + 4446*x*y^3*z^2 + 11388*x*y^3*z + 9777*x*y^3 - 390*x*y^2*z^4 - 4044*x*y^2*z^3 - 15696*x*y^2*z^2 - 27150*x*y^2*z - 17718*x*y^2 + 138*x*y*z^5 + 1821*x*y*z^4 + 9540*x*y*z^3 + 24969*x*y*z^2 + 32790*x*y*z + 17345*x*y - 20*x*z^6 - 324*x*z^5 - 2154*x*z^4 - 7596*x*z^3 - 15072*x*z^2 - 16022*x*z - 7154*x - 4*y^7 + 26*y^6*z + 64*y^6 - 72*y^5*z^2 - 357*y^5*z - 447*y^5 + 110*y^4*z^3 + 825*y^4*z^2 + 2079*y^4*z + 1762*y^4 - 100*y^3*z^4 - 1010*y^3*z^3 - 3846*y^3*z^2 - 6555*y^3*z - 4227*y^3 + 54*y^2*z^5 + 690*y^2*z^4 + 3534*y^2*z^3 + 9093*y^2*z^2 + 11781*y^2*z + 6164*y^2 - 16*y*z^6 - 249*y*z^5 - 1611*y*z^4 - 5569*y*z^3 - 10881*y*z^2 - 11425*y*z - 5053*y + 2*z^7 + 37*z^6 + 291*z^5 + 1269*z^4 + 3327*z^3 + 5261*z^2 + 4661*z + 1, -8*x^7 - 52*x^6*y + 44*x^6*z + 148*x^6 - 138*x^5*y^2 + 240*x^5*y*z + 744*x^5*y - 102*x^5*z^2 - 660*x^5*z - 996*x^5 - 195*x^4*y^3 + 519*x^4*y^2*z + 1521*x^4*y^2 - 453*x^4*y*z^2 - 2730*x^4*y*z - 3960*x^4*y + 129*x^4*z^3 + 1209*x^4*z^2 + 3588*x^4*z + 3439*x^4 - 159*x^3*y^4 + 573*x^3*y^3*z + 1617*x^3*y^3 - 765*x^3*y^2*z^2 - 4398*x^3*y^2*z - 6189*x^3*y^2 + 447*x^3*y*z^3 + 3945*x^3*y*z^2 + 11286*x^3*y*z + 10557*x^3*y - 96*x^3*z^4 - 1164*x^3*z^3 - 5097*x^3*z^2 - 9670*x^3*z - 6776*x^3 - 75*x^2*y^5 + 342*x^2*y^4*z + 942*x^2*y^4 - 618*x^2*y^3*z^2 - 3447*x^2*y^3*z - 4743*x^2*y^3 + 552*x^2*y^2*z^3 + 4689*x^2*y^2*z^2 + 13056*x^2*y^2*z + 11973*x^2*y^2 - 243*x^2*y*z^4 - 2805*x^2*y*z^3 - 11883*x^2*y*z^2 - 22023*x^2*y*z - 15168*x^2*y + 42*x^2*z^5 + 621*x^2*z^4 + 3570*x^2*z^3 + 10050*x^2*z^2 + 13968*x^2*z + 7723*x^2 - 19*x*y^6 + 105*x*y^5*z + 285*x*y^5 - 240*x*y^4*z^2 - 1314*x*y^4*z - 1779*x*y^4 + 290*x*y^3*z^3 + 2406*x*y^3*z^2 + 6570*x*y^3*z + 5932*x*y^3 - 195*x*y^2*z^4 - 2184*x*y^2*z^3 - 9036*x*y^2*z^2 - 16443*x*y^2*z - 11163*x*y^2 + 69*x*y*z^5 + 981*x*y*z^4 + 5478*x*y*z^3 + 15090*x*y*z^2 + 20625*x*y*z + 11251*x*y - 10*x*z^6 - 174*x*z^5 - 1233*x*z^4 - 4579*x*z^3 - 9462*x*z^2 - 10378*x*z - 4745*x - 2*y^7 + 13*y^6*z + 35*y^6 - 36*y^5*z^2 - 195*y^5*z - 261*y^5 + 55*y^4*z^3 + 450*y^4*z^2 + 1212*y^4*z + 1081*y^4 - 50*y^3*z^4 - 550*y^3*z^3 - 2238*y^3*z^2 - 4015*y^3*z - 2693*y^3 + 27*y^2*z^5 + 375*y^2*z^4 + 2052*y^2*z^3 + 5559*y^2*z^2 + 7494*y^2*z + 4042*y^2 - 8*y*z^6 - 135*y*z^5 - 933*y*z^4 - 3397*y*z^3 - 6909*y*z^2 - 7481*y*z - 3387*y + z^7 + 20*z^6 + 168*z^5 + 772*z^4 + 2108*z^3 + 3439*z^2 + 3120*z + 1, -8*x^7 - 52*x^6*y + 44*x^6*z + 148*x^6 - 138*x^5*y^2 + 240*x^5*y*z + 744*x^5*y - 102*x^5*z^2 - 660*x^5*z - 996*x^5 - 195*x^4*y^3 + 519*x^4*y^2*z + 1521*x^4*y^2 - 453*x^4*y*z^2 - 2730*x^4*y*z - 3960*x^4*y + 129*x^4*z^3 + 1209*x^4*z^2 + 3588*x^4*z + 3438*x^4 - 159*x^3*y^4 + 573*x^3*y^3*z + 1617*x^3*y^3 - 765*x^3*y^2*z^2 - 4398*x^3*y^2*z - 6189*x^3*y^2 + 447*x^3*y*z^3 + 3945*x^3*y*z^2 + 11286*x^3*y*z + 10552*x^3*y - 96*x^3*z^4 - 1164*x^3*z^3 - 5097*x^3*z^2 - 9666*x^3*z - 6762*x^3 - 75*x^2*y^5 + 342*x^2*y^4*z + 942*x^2*y^4 - 618*x^2*y^3*z^2 - 3447*x^2*y^3*z - 4743*x^2*y^3 + 552*x^2*y^2*z^3 + 4689*x^2*y^2*z^2 + 13056*x^2*y^2*z + 11964*x^2*y^2 - 243*x^2*y*z^4 - 2805*x^2*y*z^3 - 11883*x^2*y*z^2 - 22008*x^2*y*z - 15123*x^2*y + 42*x^2*z^5 + 621*x^2*z^4 + 3570*x^2*z^3 + 10044*x^2*z^2 + 13929*x^2*z + 7669*x^2 - 19*x*y^6 + 105*x*y^5*z + 285*x*y^5 - 240*x*y^4*z^2 - 1314*x*y^4*z - 1779*x*y^4 + 290*x*y^3*z^3 + 2406*x*y^3*z^2 + 6570*x*y^3*z + 5925*x*y^3 - 195*x*y^2*z^4 - 2184*x*y^2*z^3 - 9036*x*y^2*z^2 - 16425*x*y^2*z - 11115*x*y^2 + 69*x*y*z^5 + 981*x*y*z^4 + 5478*x*y*z^3 + 15075*x*y*z^2 + 20541*x*y*z + 11143*x*y - 10*x*z^6 - 174*x*z^5 - 1233*x*z^4 - 4575*x*z^3 - 9426*x*z^2 - 10282*x*z - 4667*x - 2*y^7 + 13*y^6*z + 35*y^6 - 36*y^5*z^2 - 195*y^5*z - 261*y^5 + 55*y^4*z^3 + 450*y^4*z^2 + 1212*y^4*z + 1079*y^4 - 50*y^3*z^4 - 550*y^3*z^3 - 2238*y^3*z^2 - 4008*y^3*z - 2676*y^3 + 27*y^2*z^5 + 375*y^2*z^4 + 2052*y^2*z^3 + 5550*y^2*z^2 + 7449*y^2*z + 3988*y^2 - 8*y*z^6 - 135*y*z^5 - 933*y*z^4 - 3392*y*z^3 - 6870*y*z^2 - 7385*y*z - 3313*y + z^7 + 20*z^6 + 168*z^5 + 771*z^4 + 2097*z^3 + 3397*z^2 + 3054*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -1, -1, 1 ], [ -2, -1, 1 ], [ -1, -2, 1 ] ], "source_shift": [ 2, 2, 2 ], "target_matrix": [ [ 2, 1, 1 ], [ 1, 1, 1 ], [ 1, 1, 2 ] ], "target_shift": [ -1789, -1219, -1183 ], "collision_preimages": [ [ "0", "1/4", "-7/4" ], [ "5/2", "-11/2", "-4" ], [ "-5/2", "-15/2", "-13" ] ], "collision_image": [ "-3579/2", "-4877/4", "-4733/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0563407900008315 ], "surface_features": [ 3.6304891782070006, 4.565788807262067, 3.1834122119784256, 0.0, 3.772973966072845, 4876 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B024", "pair": 24, "family": "bijection", "mapping": "F(x,y,z)=(-2*x^7 - 14*x^6*y - 14*x^6*z - 36*x^6 - 42*x^5*y^2 - 84*x^5*y*z - 212*x^5*y - 42*x^5*z^2 - 212*x^5*z - 262*x^5 - 70*x^4*y^3 - 210*x^4*y^2*z - 520*x^4*y^2 - 210*x^4*y*z^2 - 1040*x^4*y*z - 1262*x^4*y - 70*x^4*z^3 - 520*x^4*z^2 - 1264*x^4*z - 1002*x^4 - 70*x^3*y^4 - 280*x^3*y^3*z - 680*x^3*y^3 - 420*x^3*y^2*z^2 - 2040*x^3*y^2*z - 2430*x^3*y^2 - 280*x^3*y*z^3 - 2040*x^3*y*z^2 - 4868*x^3*y*z - 3792*x^3*y - 70*x^3*z^4 - 680*x^3*z^3 - 2438*x^3*z^2 - 3816*x^3*z - 2184*x^3 - 42*x^2*y^5 - 210*x^2*y^4*z - 500*x^2*y^4 - 420*x^2*y^3*z^2 - 2000*x^2*y^3*z - 2338*x^2*y^3 - 420*x^2*y^2*z^3 - 3000*x^2*y^2*z^2 - 7026*x^2*y^2*z - 5376*x^2*y^2 - 210*x^2*y*z^4 - 2000*x^2*y*z^3 - 7038*x^2*y*z^2 - 10820*x^2*y*z - 6084*x^2*y - 42*x^2*z^5 - 500*x^2*z^4 - 2350*x^2*z^3 - 5444*x^2*z^2 - 6176*x^2*z - 2698*x^2 - 14*x*y^6 - 84*x*y^5*z - 196*x*y^5 - 210*x*y^4*z^2 - 980*x*y^4*z - 1124*x*y^4 - 280*x*y^3*z^3 - 1960*x*y^3*z^2 - 4504*x*y^3*z - 3384*x*y^3 - 210*x*y^2*z^4 - 1960*x*y^2*z^3 - 6768*x*y^2*z^2 - 10216*x*y^2*z - 5642*x*y^2 - 84*x*y*z^5 - 980*x*y*z^4 - 4520*x*y*z^3 - 10280*x*y*z^2 - 11444*x*y*z - 4920*x*y - 14*x*z^6 - 196*x*z^5 - 1132*x*z^4 - 3448*x*z^3 - 5802*x*z^2 - 5040*x*z - 1745*x - 2*y^7 - 14*y^6*z - 32*y^6 - 42*y^5*z^2 - 192*y^5*z - 216*y^5 - 70*y^4*z^3 - 480*y^4*z^2 - 1082*y^4*z - 798*y^4 - 70*y^3*z^4 - 640*y^3*z^3 - 2168*y^3*z^2 - 3212*y^3*z - 1742*y^3 - 42*y^2*z^5 - 480*y^2*z^4 - 2172*y^2*z^3 - 4848*y^2*z^2 - 5296*y^2*z - 2240*y^2 - 14*y*z^6 - 192*y*z^5 - 1088*y*z^4 - 3252*y*z^3 - 5366*y*z^2 - 4580*y*z - 1567*y - 2*z^7 - 32*z^6 - 218*z^5 - 818*z^4 - 1812*z^3 - 2340*z^2 - 1619*z - 1, 2*x^7 + 14*x^6*y + 14*x^6*z + 36*x^6 + 42*x^5*y^2 + 84*x^5*y*z + 212*x^5*y + 42*x^5*z^2 + 212*x^5*z + 262*x^5 + 70*x^4*y^3 + 210*x^4*y^2*z + 520*x^4*y^2 + 210*x^4*y*z^2 + 1040*x^4*y*z + 1262*x^4*y + 70*x^4*z^3 + 520*x^4*z^2 + 1264*x^4*z + 1002*x^4 + 70*x^3*y^4 + 280*x^3*y^3*z + 680*x^3*y^3 + 420*x^3*y^2*z^2 + 2040*x^3*y^2*z + 2430*x^3*y^2 + 280*x^3*y*z^3 + 2040*x^3*y*z^2 + 4868*x^3*y*z + 3792*x^3*y + 70*x^3*z^4 + 680*x^3*z^3 + 2438*x^3*z^2 + 3816*x^3*z + 2183*x^3 + 42*x^2*y^5 + 210*x^2*y^4*z + 500*x^2*y^4 + 420*x^2*y^3*z^2 + 2000*x^2*y^3*z + 2338*x^2*y^3 + 420*x^2*y^2*z^3 + 3000*x^2*y^2*z^2 + 7026*x^2*y^2*z + 5376*x^2*y^2 + 210*x^2*y*z^4 + 2000*x^2*y*z^3 + 7038*x^2*y*z^2 + 10820*x^2*y*z + 6081*x^2*y + 42*x^2*z^5 + 500*x^2*z^4 + 2350*x^2*z^3 + 5444*x^2*z^2 + 6173*x^2*z + 2690*x^2 + 14*x*y^6 + 84*x*y^5*z + 196*x*y^5 + 210*x*y^4*z^2 + 980*x*y^4*z + 1124*x*y^4 + 280*x*y^3*z^3 + 1960*x*y^3*z^2 + 4504*x*y^3*z + 3384*x*y^3 + 210*x*y^2*z^4 + 1960*x*y^2*z^3 + 6768*x*y^2*z^2 + 10216*x*y^2*z + 5639*x*y^2 + 84*x*y*z^5 + 980*x*y*z^4 + 4520*x*y*z^3 + 10280*x*y*z^2 + 11438*x*y*z + 4904*x*y + 14*x*z^6 + 196*x*z^5 + 1132*x*z^4 + 3448*x*z^3 + 5799*x*z^2 + 5024*x*z + 1720*x + 2*y^7 + 14*y^6*z + 32*y^6 + 42*y^5*z^2 + 192*y^5*z + 216*y^5 + 70*y^4*z^3 + 480*y^4*z^2 + 1082*y^4*z + 798*y^4 + 70*y^3*z^4 + 640*y^3*z^3 + 2168*y^3*z^2 + 3212*y^3*z + 1741*y^3 + 42*y^2*z^5 + 480*y^2*z^4 + 2172*y^2*z^3 + 4848*y^2*z^2 + 5293*y^2*z + 2232*y^2 + 14*y*z^6 + 192*y*z^5 + 1088*y*z^4 + 3252*y*z^3 + 5363*y*z^2 + 4564*y*z + 1544*y + 2*z^7 + 32*z^6 + 218*z^5 + 818*z^4 + 1811*z^3 + 2332*z^2 + 1595*z + 1, -2*x^7 - 14*x^6*y - 14*x^6*z - 36*x^6 - 42*x^5*y^2 - 84*x^5*y*z - 212*x^5*y - 42*x^5*z^2 - 212*x^5*z - 262*x^5 - 70*x^4*y^3 - 210*x^4*y^2*z - 520*x^4*y^2 - 210*x^4*y*z^2 - 1040*x^4*y*z - 1262*x^4*y - 70*x^4*z^3 - 520*x^4*z^2 - 1264*x^4*z - 1002*x^4 - 70*x^3*y^4 - 280*x^3*y^3*z - 680*x^3*y^3 - 420*x^3*y^2*z^2 - 2040*x^3*y^2*z - 2430*x^3*y^2 - 280*x^3*y*z^3 - 2040*x^3*y*z^2 - 4868*x^3*y*z - 3792*x^3*y - 70*x^3*z^4 - 680*x^3*z^3 - 2438*x^3*z^2 - 3816*x^3*z - 2183*x^3 - 42*x^2*y^5 - 210*x^2*y^4*z - 500*x^2*y^4 - 420*x^2*y^3*z^2 - 2000*x^2*y^3*z - 2338*x^2*y^3 - 420*x^2*y^2*z^3 - 3000*x^2*y^2*z^2 - 7026*x^2*y^2*z - 5376*x^2*y^2 - 210*x^2*y*z^4 - 2000*x^2*y*z^3 - 7038*x^2*y*z^2 - 10820*x^2*y*z - 6081*x^2*y - 42*x^2*z^5 - 500*x^2*z^4 - 2350*x^2*z^3 - 5444*x^2*z^2 - 6173*x^2*z - 2692*x^2 - 14*x*y^6 - 84*x*y^5*z - 196*x*y^5 - 210*x*y^4*z^2 - 980*x*y^4*z - 1124*x*y^4 - 280*x*y^3*z^3 - 1960*x*y^3*z^2 - 4504*x*y^3*z - 3384*x*y^3 - 210*x*y^2*z^4 - 1960*x*y^2*z^3 - 6768*x*y^2*z^2 - 10216*x*y^2*z - 5639*x*y^2 - 84*x*y*z^5 - 980*x*y*z^4 - 4520*x*y*z^3 - 10280*x*y*z^2 - 11438*x*y*z - 4908*x*y - 14*x*z^6 - 196*x*z^5 - 1132*x*z^4 - 3448*x*z^3 - 5799*x*z^2 - 5028*x*z - 1732*x - 2*y^7 - 14*y^6*z - 32*y^6 - 42*y^5*z^2 - 192*y^5*z - 216*y^5 - 70*y^4*z^3 - 480*y^4*z^2 - 1082*y^4*z - 798*y^4 - 70*y^3*z^4 - 640*y^3*z^3 - 2168*y^3*z^2 - 3212*y^3*z - 1741*y^3 - 42*y^2*z^5 - 480*y^2*z^4 - 2172*y^2*z^3 - 4848*y^2*z^2 - 5293*y^2*z - 2234*y^2 - 14*y*z^6 - 192*y*z^5 - 1088*y*z^4 - 3252*y*z^3 - 5363*y*z^2 - 4568*y*z - 1554*y - 2*z^7 - 32*z^6 - 218*z^5 - 818*z^4 - 1811*z^3 - 2334*z^2 - 1605*z - 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ 1, 1, 1 ], [ 1, 1, 2 ], [ 2, 1, 1 ] ], "source_shift": [ 2, 1, 1 ], "target_matrix": [ [ -1, -2, -1 ], [ 1, 1, 2 ], [ -1, -1, -1 ] ], "target_shift": [ 459, -440, 450 ], "inverse_recipe": { "p": 3, "q": 2, "f": "x**3", "g": "x**2", "h": "y*z**2", "polynomials": { "f": [ [ [ 3, 0, 0 ], 1 ] ], "g": [ [ [ 2, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 2 ], 1 ] ] }, "h_multiplier": 2 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0474472410023736 ], "surface_features": [ 3.2893820609863154, 4.058577849133225, 2.9537596917332287, 0.0, 3.399511732138733, 4778 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B039", "pair": 39, "family": "bijection", "mapping": "F(x,y,z)=(3328*x^7 - 11648*x^6*y + 11648*x^6*z - 13312*x^6 + 17472*x^5*y^2 - 34944*x^5*y*z + 39936*x^5*y + 17472*x^5*z^2 - 39936*x^5*z + 24544*x^5 - 14560*x^4*y^3 + 43680*x^4*y^2*z - 49920*x^4*y^2 - 43680*x^4*y*z^2 + 99840*x^4*y*z - 61568*x^4*y + 14560*x^4*z^3 - 49920*x^4*z^2 + 61984*x^4*z - 27888*x^4 + 7280*x^3*y^4 - 29120*x^3*y^3*z + 33280*x^3*y^3 + 43680*x^3*y^2*z^2 - 99840*x^3*y^2*z + 61776*x^3*y^2 - 29120*x^3*y*z^3 + 99840*x^3*y*z^2 - 124384*x^3*y*z + 56400*x^3*y + 7280*x^3*z^4 - 33280*x^3*z^3 + 62608*x^3*z^2 - 57440*x^3*z + 21336*x^3 - 2184*x^2*y^5 + 10920*x^2*y^4*z - 12480*x^2*y^4 - 21840*x^2*y^3*z^2 + 49920*x^2*y^3*z - 30992*x^2*y^3 + 21840*x^2*y^2*z^3 - 74880*x^2*y^2*z^2 + 93600*x^2*y^2*z - 42768*x^2*y^2 - 10920*x^2*y*z^4 + 49920*x^2*y*z^3 - 94224*x^2*y*z^2 + 87096*x^2*y*z - 32784*x^2*y + 2184*x^2*z^5 - 12480*x^2*z^4 + 31616*x^2*z^3 - 44328*x^2*z^2 + 33928*x^2*z - 11124*x^2 + 364*x*y^6 - 2184*x*y^5*z + 2496*x*y^5 + 5460*x*y^4*z^2 - 12480*x*y^4*z + 7774*x*y^4 - 7280*x*y^3*z^3 + 24960*x*y^3*z^2 - 31304*x*y^3*z + 14412*x*y^3 + 5460*x*y^2*z^4 - 24960*x*y^2*z^3 + 47268*x*y^2*z^2 - 44016*x*y^2*z + 16782*x*y^2 - 2184*x*y*z^5 + 12480*x*y*z^4 - 31720*x*y*z^3 + 44796*x*y*z^2 - 34708*x*y*z + 11670*x*y + 364*x*z^6 - 2496*x*z^5 + 7982*x*z^4 - 15192*x*z^3 + 17926*x*z^2 - 12424*x*z + 3764*x - 26*y^7 + 182*y^6*z - 208*y^6 - 546*y^5*z^2 + 1248*y^5*z - 780*y^5 + 910*y^4*z^3 - 3120*y^4*z^2 + 3926*y^4*z - 1821*y^4 - 910*y^3*z^4 + 4160*y^3*z^3 - 7904*y^3*z^2 + 7414*y^3*z - 2862*y^3 + 546*y^2*z^5 - 3120*y^2*z^4 + 7956*y^2*z^3 - 11316*y^2*z^2 + 8872*y^2*z - 3067*y^2 - 182*y*z^6 + 1248*y*z^5 - 4004*y*z^4 + 7674*y*z^3 - 9158*y*z^2 + 6550*y*z - 2077*y + 26*z^7 - 208*z^6 + 806*z^5 - 1951*z^4 + 3148*z^3 - 3509*z^2 + 2336*z - 1, -1664*x^7 + 5824*x^6*y - 5824*x^6*z + 6656*x^6 - 8736*x^5*y^2 + 17472*x^5*y*z - 19968*x^5*y - 8736*x^5*z^2 + 19968*x^5*z - 12272*x^5 + 7280*x^4*y^3 - 21840*x^4*y^2*z + 24960*x^4*y^2 + 21840*x^4*y*z^2 - 49920*x^4*y*z + 30784*x^4*y - 7280*x^4*z^3 + 24960*x^4*z^2 - 30992*x^4*z + 13952*x^4 - 3640*x^3*y^4 + 14560*x^3*y^3*z - 16640*x^3*y^3 - 21840*x^3*y^2*z^2 + 49920*x^3*y^2*z - 30888*x^3*y^2 + 14560*x^3*y*z^3 - 49920*x^3*y*z^2 + 62192*x^3*y*z - 28216*x^3*y - 3640*x^3*z^4 + 16640*x^3*z^3 - 31304*x^3*z^2 + 28736*x^3*z - 10676*x^3 + 1092*x^2*y^5 - 5460*x^2*y^4*z + 6240*x^2*y^4 + 10920*x^2*y^3*z^2 - 24960*x^2*y^3*z + 15496*x^2*y^3 - 10920*x^2*y^2*z^3 + 37440*x^2*y^2*z^2 - 46800*x^2*y^2*z + 21396*x^2*y^2 + 5460*x^2*y*z^4 - 24960*x^2*y*z^3 + 47112*x^2*y*z^2 - 43572*x^2*y*z + 16404*x^2*y - 1092*x^2*z^5 + 6240*x^2*z^4 - 15808*x^2*z^3 + 22176*x^2*z^2 - 16976*x^2*z + 5560*x^2 - 182*x*y^6 + 1092*x*y^5*z - 1248*x*y^5 - 2730*x*y^4*z^2 + 6240*x*y^4*z - 3887*x*y^4 + 3640*x*y^3*z^3 - 12480*x*y^3*z^2 + 15652*x*y^3*z - 7210*x*y^3 - 2730*x*y^2*z^4 + 12480*x*y^2*z^3 - 23634*x*y^2*z^2 + 22020*x*y^2*z - 8397*x*y^2 + 1092*x*y*z^5 - 6240*x*y*z^4 + 15860*x*y*z^3 - 22410*x*y*z^2 + 17366*x*y*z - 5833*x*y - 182*x*z^6 + 1248*x*z^5 - 3991*x*z^4 + 7600*x*z^3 - 8969*x*z^2 + 6210*x*z - 1877*x + 13*y^7 - 91*y^6*z + 104*y^6 + 273*y^5*z^2 - 624*y^5*z + 390*y^5 - 455*y^4*z^3 + 1560*y^4*z^2 - 1963*y^4*z + 911*y^4 + 455*y^3*z^4 - 2080*y^3*z^3 + 3952*y^3*z^2 - 3709*y^3*z + 1432*y^3 - 273*y^2*z^5 + 1560*y^2*z^4 - 3978*y^2*z^3 + 5661*y^2*z^2 - 4439*y^2*z + 1533*y^2 + 91*y*z^6 - 624*y*z^5 + 2002*y*z^4 - 3839*y*z^3 + 4582*y*z^2 - 3274*y*z + 1036*y - 13*z^7 + 104*z^6 - 403*z^5 + 976*z^4 - 1575*z^3 + 1754*z^2 - 1165*z + 1, 1664*x^7 - 5824*x^6*y + 5824*x^6*z - 6656*x^6 + 8736*x^5*y^2 - 17472*x^5*y*z + 19968*x^5*y + 8736*x^5*z^2 - 19968*x^5*z + 12272*x^5 - 7280*x^4*y^3 + 21840*x^4*y^2*z - 24960*x^4*y^2 - 21840*x^4*y*z^2 + 49920*x^4*y*z - 30784*x^4*y + 7280*x^4*z^3 - 24960*x^4*z^2 + 30992*x^4*z - 13952*x^4 + 3640*x^3*y^4 - 14560*x^3*y^3*z + 16640*x^3*y^3 + 21840*x^3*y^2*z^2 - 49920*x^3*y^2*z + 30888*x^3*y^2 - 14560*x^3*y*z^3 + 49920*x^3*y*z^2 - 62192*x^3*y*z + 28216*x^3*y + 3640*x^3*z^4 - 16640*x^3*z^3 + 31304*x^3*z^2 - 28736*x^3*z + 10660*x^3 - 1092*x^2*y^5 + 5460*x^2*y^4*z - 6240*x^2*y^4 - 10920*x^2*y^3*z^2 + 24960*x^2*y^3*z - 15496*x^2*y^3 + 10920*x^2*y^2*z^3 - 37440*x^2*y^2*z^2 + 46800*x^2*y^2*z - 21396*x^2*y^2 - 5460*x^2*y*z^4 + 24960*x^2*y*z^3 - 47112*x^2*y*z^2 + 43572*x^2*y*z - 16380*x^2*y + 1092*x^2*z^5 - 6240*x^2*z^4 + 15808*x^2*z^3 - 22176*x^2*z^2 + 16952*x^2*z - 5528*x^2 + 182*x*y^6 - 1092*x*y^5*z + 1248*x*y^5 + 2730*x*y^4*z^2 - 6240*x*y^4*z + 3887*x*y^4 - 3640*x*y^3*z^3 + 12480*x*y^3*z^2 - 15652*x*y^3*z + 7210*x*y^3 + 2730*x*y^2*z^4 - 12480*x*y^2*z^3 + 23634*x*y^2*z^2 - 22020*x*y^2*z + 8385*x*y^2 - 1092*x*y*z^5 + 6240*x*y*z^4 - 15860*x*y*z^3 + 22410*x*y*z^2 - 17342*x*y*z + 5801*x*y + 182*x*z^6 - 1248*x*z^5 + 3991*x*z^4 - 7600*x*z^3 + 8957*x*z^2 - 6178*x*z + 1851*x - 13*y^7 + 91*y^6*z - 104*y^6 - 273*y^5*z^2 + 624*y^5*z - 390*y^5 + 455*y^4*z^3 - 1560*y^4*z^2 + 1963*y^4*z - 911*y^4 - 455*y^3*z^4 + 2080*y^3*z^3 - 3952*y^3*z^2 + 3709*y^3*z - 1430*y^3 + 273*y^2*z^5 - 1560*y^2*z^4 + 3978*y^2*z^3 - 5661*y^2*z^2 + 4433*y^2*z - 1525*y^2 - 91*y*z^6 + 624*y*z^5 - 2002*y*z^4 + 3839*y*z^3 - 4576*y*z^2 + 3258*y*z - 1022*y + 13*z^7 - 104*z^6 + 403*z^5 - 976*z^4 + 1573*z^3 - 1746*z^2 + 1149*z - 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -2, 1, -1 ], [ -2, 2, -3 ], [ -1, 1, -2 ] ], "source_shift": [ 1, 2, 2 ], "target_matrix": [ [ -2, -1, -1 ], [ 1, 1, 1 ], [ -1, -1, -2 ] ], "target_shift": [ 646, -320, 310 ], "inverse_recipe": { "p": 4, "q": 3, "f": "x**4 + x**2 + x", "g": "x**3 + x**2 + x", "h": "y*z", "polynomials": { "f": [ [ [ 4, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "g": [ [ [ 3, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ] ] }, "h_multiplier": 13 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 0.9758984690015495 ], "surface_features": [ 4.188651671425824, 5.094764518951172, 3.8661100398443766, 0.0, 4.289798447235334, 5141 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C185", "pair": 185, "family": "counterexample", "mapping": "F(x,y,z)=(24*x^7 + 132*x^6*y - 156*x^6*z - 348*x^6 + 306*x^5*y^2 - 720*x^5*y*z - 1620*x^5*y + 414*x^5*z^2 + 1884*x^5*z + 2124*x^5 + 387*x^4*y^3 - 1359*x^4*y^2*z - 3087*x^4*y^2 + 1557*x^4*y*z^2 + 7134*x^4*y*z + 8136*x^4*y - 585*x^4*z^3 - 4047*x^4*z^2 - 9300*x^4*z - 7094*x^4 + 288*x^3*y^4 - 1341*x^3*y^3*z - 3078*x^3*y^3 + 2295*x^3*y^2*z^2 + 10596*x^3*y^2*z + 12231*x^3*y^2 - 1719*x^3*y*z^3 - 11958*x^3*y*z^2 - 27750*x^3*y*z - 21455*x^3*y + 477*x^3*z^4 + 4440*x^3*z^3 + 15519*x^3*z^2 + 24127*x^3*z + 14044*x^3 + 126*x^2*y^5 - 729*x^2*y^4*z - 1692*x^2*y^4 + 1656*x^2*y^3*z^2 + 7710*x^2*y^3*z + 9009*x^2*y^3 - 1854*x^2*y^2*z^3 - 12978*x^2*y^2*z^2 - 30420*x^2*y^2*z - 23835*x^2*y^2 + 1026*x^2*y*z^4 + 9594*x^2*y*z^3 + 33813*x^2*y*z^2 + 53178*x^2*y*z + 31413*x^2*y - 225*x^2*z^5 - 2634*x^2*z^4 - 12402*x^2*z^3 - 29343*x^2*z^2 - 34827*x^2*z - 16542*x^2 + 30*x*y^6 - 207*x*y^5*z - 486*x*y^5 + 585*x*y^4*z^2 + 2748*x*y^4*z + 3249*x*y^4 - 870*x*y^3*z^3 - 6132*x*y^3*z^2 - 14514*x*y^3*z - 11515*x*y^3 + 720*x*y^2*z^4 + 6768*x*y^2*z^3 + 24048*x*y^2*z^2 + 38229*x*y^2*z + 22896*x*y^2 - 315*x*y*z^5 - 3702*x*y*z^4 - 17550*x*y*z^3 - 41913*x*y*z^2 - 50355*x*y*z - 24294*x*y + 57*x*z^6 + 804*x*z^5 + 4767*x*z^4 + 15199*x*z^3 + 27459*x*z^2 + 26616*x*z + 10785*x + 3*y^7 - 24*y^6*z - 57*y^6 + 81*y^5*z^2 + 384*y^5*z + 459*y^5 - 150*y^4*z^3 - 1065*y^4*z^2 - 2544*y^4*z - 2041*y^4 + 165*y^3*z^4 + 1560*y^3*z^3 + 5586*y^3*z^2 + 8966*y^3*z + 5435*y^3 - 108*y^2*z^5 - 1275*y^2*z^4 - 6084*y^2*z^3 - 14652*y^2*z^2 - 17790*y^2*z - 8700*y^2 + 39*y*z^6 + 552*y*z^5 + 3291*y*z^4 + 10570*y*z^3 + 19275*y*z^2 + 18906*y*z + 7781*y - 6*z^7 - 99*z^6 - 708*z^5 - 2843*z^4 - 6920*z^3 - 10206*z^2 - 8440*z - 1, -8*x^7 - 44*x^6*y + 52*x^6*z + 148*x^6 - 102*x^5*y^2 + 240*x^5*y*z + 684*x^5*y - 138*x^5*z^2 - 804*x^5*z - 1044*x^5 - 129*x^4*y^3 + 453*x^4*y^2*z + 1293*x^4*y^2 - 519*x^4*y*z^2 - 3018*x^4*y*z - 3960*x^4*y + 195*x^4*z^3 + 1725*x^4*z^2 + 4572*x^4*z + 3834*x^4 - 96*x^3*y^4 + 447*x^3*y^3*z + 1278*x^3*y^3 - 765*x^3*y^2*z^2 - 4440*x^3*y^2*z - 5889*x^3*y^2 + 573*x^3*y*z^3 + 5046*x^3*y*z^2 + 13482*x^3*y*z + 11465*x^3*y - 159*x^3*z^4 - 1884*x^3*z^3 - 7593*x^3*z^2 - 12993*x^3*z - 8104*x^3 - 42*x^2*y^5 + 243*x^2*y^4*z + 696*x^2*y^4 - 552*x^2*y^3*z^2 - 3198*x^2*y^3*z - 4287*x^2*y^3 + 618*x^2*y^2*z^3 + 5418*x^2*y^2*z^2 + 14592*x^2*y^2*z + 12579*x^2*y^2 - 342*x^2*y*z^4 - 4026*x^2*y*z^3 - 16323*x^2*y*z^2 - 28254*x^2*y*z - 17895*x^2*y + 75*x^2*z^5 + 1110*x^2*z^4 + 6018*x^2*z^3 + 15675*x^2*z^2 + 19953*x^2*z + 9994*x^2 - 10*x*y^6 + 69*x*y^5*z + 198*x*y^5 - 195*x*y^4*z^2 - 1128*x*y^4*z - 1527*x*y^4 + 290*x*y^3*z^3 + 2532*x*y^3*z^2 + 6870*x*y^3*z + 5996*x*y^3 - 240*x*y^2*z^4 - 2808*x*y^2*z^3 - 11448*x*y^2*z^2 - 20022*x*y^2*z - 12861*x*y^2 + 105*x*y*z^5 + 1542*x*y*z^4 + 8394*x*y*z^3 + 22056*x*y*z^2 + 28419*x*y*z + 14462*x*y - 19*x*z^6 - 336*x*z^5 - 2289*x*z^4 - 8030*x*z^3 - 15558*x*z^2 - 15908*x*z - 6719*x - y^7 + 8*y^6*z + 23*y^6 - 27*y^5*z^2 - 156*y^5*z - 213*y^5 + 50*y^4*z^3 + 435*y^4*z^2 + 1188*y^4*z + 1048*y^4 - 55*y^3*z^4 - 640*y^3*z^3 - 2622*y^3*z^2 - 4627*y^3*z - 3008*y^3 + 36*y^2*z^5 + 525*y^2*z^4 + 2868*y^2*z^3 + 7593*y^2*z^2 + 9885*y^2*z + 5098*y^2 - 13*y*z^6 - 228*y*z^5 - 1557*y*z^4 - 5497*y*z^3 - 10746*y*z^2 - 11114*y*z - 4765*y + 2*z^7 + 41*z^6 + 336*z^5 + 1483*z^4 + 3869*z^3 + 6016*z^2 + 5182*z + 1, 8*x^7 + 44*x^6*y - 52*x^6*z - 124*x^6 + 102*x^5*y^2 - 240*x^5*y*z - 576*x^5*y + 138*x^5*z^2 + 672*x^5*z + 792*x^5 + 129*x^4*y^3 - 453*x^4*y^2*z - 1095*x^4*y^2 + 519*x^4*y*z^2 + 2538*x^4*y*z + 3024*x^4*y - 195*x^4*z^3 - 1443*x^4*z^2 - 3468*x^4*z - 2730*x^4 + 96*x^3*y^4 - 447*x^3*y^3*z - 1089*x^3*y^3 + 765*x^3*y^2*z^2 + 3759*x^3*y^2*z + 4530*x^3*y^2 - 573*x^3*y*z^3 - 4251*x^3*y*z^2 - 10308*x^3*y*z - 8225*x^3*y + 159*x^3*z^4 + 1581*x^3*z^3 + 5778*x^3*z^2 + 9273*x^3*z + 5524*x^3 + 42*x^2*y^5 - 243*x^2*y^4*z - 597*x^2*y^4 + 552*x^2*y^3*z^2 + 2727*x^2*y^3*z + 3324*x^2*y^3 - 618*x^2*y^2*z^3 - 4599*x^2*y^2*z^2 - 11253*x^2*y^2*z - 9099*x^2*y^2 + 342*x^2*y*z^4 + 3405*x^2*y*z^3 + 12534*x^2*y*z^2 + 20346*x^2*y*z + 12303*x^2*y - 75*x^2*z^5 - 936*x^2*z^4 - 4605*x^2*z^3 - 11247*x^2*z^2 - 13665*x^2*z - 6604*x^2 + 10*x*y^6 - 69*x*y^5*z - 171*x*y^5 + 195*x*y^4*z^2 + 969*x*y^4*z + 1194*x*y^4 - 290*x*y^3*z^3 - 2166*x*y^3*z^2 - 5346*x*y^3*z - 4376*x*y^3 + 240*x*y^2*z^4 + 2394*x*y^2*z^3 + 8874*x*y^2*z^2 + 14556*x*y^2*z + 8925*x*y^2 - 105*x*y*z^5 - 1311*x*y*z^4 - 6486*x*y*z^3 - 15984*x*y*z^2 - 19659*x*y*z - 9653*x*y + 19*x*z^6 + 285*x*z^5 + 1764*x*z^4 + 5804*x*z^3 + 10734*x*z^2 + 10589*x*z + 4348*x + y^7 - 8*y^6*z - 20*y^6 + 27*y^5*z^2 + 135*y^5*z + 168*y^5 - 50*y^4*z^3 - 375*y^4*z^2 - 933*y^4*z - 772*y^4 + 55*y^3*z^4 + 550*y^3*z^3 + 2052*y^3*z^2 + 3397*y^3*z + 2108*y^3 - 36*y^2*z^5 - 450*y^2*z^4 - 2238*y^2*z^3 - 5559*y^2*z^2 - 6909*y^2*z - 3439*y^2 + 13*y*z^6 + 195*y*z^5 + 1212*y*z^4 + 4015*y*z^3 + 7494*y*z^2 + 7481*y*z + 3120*y - 2*z^7 - 35*z^6 - 261*z^5 - 1081*z^4 - 2693*z^3 - 4042*z^2 - 3387*z - 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -2, -1, 1 ], [ -1, -1, 1 ], [ -1, -1, 2 ] ], "source_shift": [ 2, 2, 2 ], "target_matrix": [ [ -3, -2, -2 ], [ 1, 2, 1 ], [ -1, -1, -1 ] ], "target_shift": [ 3009, -1905, 1219 ], "collision_preimages": [ [ "0", "7/4", "-1/4" ], [ "-5/2", "14", "8" ], [ "5/2", "3", "5" ] ], "collision_image": [ "12039/4", "-7621/4", "4877/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0587659390002955 ], "surface_features": [ 3.7392681175159836, 4.725731999687966, 3.271609301378832, 0.0, 3.912503597097584, 4901 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B123", "pair": 123, "family": "bijection", "mapping": "F(x,y,z)=(6*x^7 - 42*x^6*y + 42*x^6*z - 96*x^6 + 126*x^5*y^2 - 252*x^5*y*z + 564*x^5*y + 126*x^5*z^2 - 564*x^5*z + 564*x^5 - 210*x^4*y^3 + 630*x^4*y^2*z - 1380*x^4*y^2 - 630*x^4*y*z^2 + 2760*x^4*y*z - 2712*x^4*y + 210*x^4*z^3 - 1380*x^4*z^2 + 2706*x^4*z - 1812*x^4 + 210*x^3*y^4 - 840*x^3*y^3*z + 1800*x^3*y^3 + 1260*x^3*y^2*z^2 - 5400*x^3*y^2*z + 5214*x^3*y^2 - 840*x^3*y*z^3 + 5400*x^3*y*z^2 - 10404*x^3*y*z + 6876*x^3*y + 210*x^3*z^4 - 1800*x^3*z^3 + 5190*x^3*z^2 - 6792*x^3*z + 3532*x^3 - 126*x^2*y^5 + 630*x^2*y^4*z - 1320*x^2*y^4 - 1260*x^2*y^3*z^2 + 5280*x^2*y^3*z - 5010*x^2*y^3 + 1260*x^2*y^2*z^3 - 7920*x^2*y^2*z^2 + 14994*x^2*y^2*z - 9780*x^2*y^2 - 630*x^2*y*z^4 + 5280*x^2*y*z^3 - 14958*x^2*y*z^2 + 19320*x^2*y*z - 9864*x^2*y + 126*x^2*z^5 - 1320*x^2*z^4 + 4974*x^2*z^3 - 9540*x^2*z^2 + 9594*x^2*z - 4063*x^2 + 42*x*y^6 - 252*x*y^5*z + 516*x*y^5 + 630*x*y^4*z^2 - 2580*x*y^4*z + 2406*x*y^4 - 840*x*y^3*z^3 + 5160*x*y^3*z^2 - 9600*x*y^3*z + 6180*x*y^3 + 630*x*y^2*z^4 - 5160*x*y^2*z^3 + 14364*x*y^2*z^2 - 18312*x*y^2*z + 9186*x*y^2 - 252*x*y*z^5 + 2580*x*y*z^4 - 9552*x*y*z^3 + 18084*x*y*z^2 - 17892*x*y*z + 7430*x*y + 42*x*z^6 - 516*x*z^5 + 2382*x*z^4 - 5952*x*z^3 + 8706*x*z^2 - 7094*x*z + 2495*x - 6*y^7 + 42*y^6*z - 84*y^6 - 126*y^5*z^2 + 504*y^5*z - 462*y^5 + 210*y^4*z^3 - 1260*y^4*z^2 + 2304*y^4*z - 1464*y^4 - 210*y^3*z^4 + 1680*y^3*z^3 - 4596*y^3*z^2 + 5784*y^3*z - 2854*y^3 + 126*y^2*z^5 - 1260*y^2*z^4 + 4584*y^2*z^3 - 8568*y^2*z^2 + 8346*y^2*z - 3415*y^2 - 42*y*z^6 + 504*y*z^5 - 2286*y*z^4 + 5640*y*z^3 - 8130*y*z^2 + 6518*y*z - 2252*y + 6*z^7 - 84*z^6 + 456*z^5 - 1392*z^4 + 2638*z^3 - 3115*z^2 + 2110*z - 1, 3*x^7 - 21*x^6*y + 21*x^6*z - 48*x^6 + 63*x^5*y^2 - 126*x^5*y*z + 282*x^5*y + 63*x^5*z^2 - 282*x^5*z + 282*x^5 - 105*x^4*y^3 + 315*x^4*y^2*z - 690*x^4*y^2 - 315*x^4*y*z^2 + 1380*x^4*y*z - 1356*x^4*y + 105*x^4*z^3 - 690*x^4*z^2 + 1353*x^4*z - 906*x^4 + 105*x^3*y^4 - 420*x^3*y^3*z + 900*x^3*y^3 + 630*x^3*y^2*z^2 - 2700*x^3*y^2*z + 2607*x^3*y^2 - 420*x^3*y*z^3 + 2700*x^3*y*z^2 - 5202*x^3*y*z + 3438*x^3*y + 105*x^3*z^4 - 900*x^3*z^3 + 2595*x^3*z^2 - 3396*x^3*z + 1765*x^3 - 63*x^2*y^5 + 315*x^2*y^4*z - 660*x^2*y^4 - 630*x^2*y^3*z^2 + 2640*x^2*y^3*z - 2505*x^2*y^3 + 630*x^2*y^2*z^3 - 3960*x^2*y^2*z^2 + 7497*x^2*y^2*z - 4890*x^2*y^2 - 315*x^2*y*z^4 + 2640*x^2*y*z^3 - 7479*x^2*y*z^2 + 9660*x^2*y*z - 4929*x^2*y + 63*x^2*z^5 - 660*x^2*z^4 + 2487*x^2*z^3 - 4770*x^2*z^2 + 4794*x^2*z - 2030*x^2 + 21*x*y^6 - 126*x*y^5*z + 258*x*y^5 + 315*x*y^4*z^2 - 1290*x*y^4*z + 1203*x*y^4 - 420*x*y^3*z^3 + 2580*x*y^3*z^2 - 4800*x*y^3*z + 3090*x*y^3 + 315*x*y^2*z^4 - 2580*x*y^2*z^3 + 7182*x*y^2*z^2 - 9156*x*y^2*z + 4590*x*y^2 - 126*x*y*z^5 + 1290*x*y*z^4 - 4776*x*y*z^3 + 9042*x*y*z^2 - 8940*x*y*z + 3712*x*y + 21*x*z^6 - 258*x*z^5 + 1191*x*z^4 - 2976*x*z^3 + 4350*x*z^2 - 3544*x*z + 1251*x - 3*y^7 + 21*y^6*z - 42*y^6 - 63*y^5*z^2 + 252*y^5*z - 231*y^5 + 105*y^4*z^3 - 630*y^4*z^2 + 1152*y^4*z - 732*y^4 - 105*y^3*z^4 + 840*y^3*z^3 - 2298*y^3*z^2 + 2892*y^3*z - 1426*y^3 + 63*y^2*z^5 - 630*y^2*z^4 + 2292*y^2*z^3 - 4284*y^2*z^2 + 4170*y^2*z - 1706*y^2 - 21*y*z^6 + 252*y*z^5 - 1143*y*z^4 + 2820*y*z^3 - 4062*y*z^2 + 3256*y*z - 1127*y + 3*z^7 - 42*z^6 + 228*z^5 - 696*z^4 + 1318*z^3 - 1556*z^2 + 1057*z - 1, -3*x^7 + 21*x^6*y - 21*x^6*z + 48*x^6 - 63*x^5*y^2 + 126*x^5*y*z - 282*x^5*y - 63*x^5*z^2 + 282*x^5*z - 282*x^5 + 105*x^4*y^3 - 315*x^4*y^2*z + 690*x^4*y^2 + 315*x^4*y*z^2 - 1380*x^4*y*z + 1356*x^4*y - 105*x^4*z^3 + 690*x^4*z^2 - 1353*x^4*z + 906*x^4 - 105*x^3*y^4 + 420*x^3*y^3*z - 900*x^3*y^3 - 630*x^3*y^2*z^2 + 2700*x^3*y^2*z - 2607*x^3*y^2 + 420*x^3*y*z^3 - 2700*x^3*y*z^2 + 5202*x^3*y*z - 3438*x^3*y - 105*x^3*z^4 + 900*x^3*z^3 - 2595*x^3*z^2 + 3396*x^3*z - 1765*x^3 + 63*x^2*y^5 - 315*x^2*y^4*z + 660*x^2*y^4 + 630*x^2*y^3*z^2 - 2640*x^2*y^3*z + 2505*x^2*y^3 - 630*x^2*y^2*z^3 + 3960*x^2*y^2*z^2 - 7497*x^2*y^2*z + 4890*x^2*y^2 + 315*x^2*y*z^4 - 2640*x^2*y*z^3 + 7479*x^2*y*z^2 - 9660*x^2*y*z + 4929*x^2*y - 63*x^2*z^5 + 660*x^2*z^4 - 2487*x^2*z^3 + 4770*x^2*z^2 - 4794*x^2*z + 2029*x^2 - 21*x*y^6 + 126*x*y^5*z - 258*x*y^5 - 315*x*y^4*z^2 + 1290*x*y^4*z - 1203*x*y^4 + 420*x*y^3*z^3 - 2580*x*y^3*z^2 + 4800*x*y^3*z - 3090*x*y^3 - 315*x*y^2*z^4 + 2580*x*y^2*z^3 - 7182*x*y^2*z^2 + 9156*x*y^2*z - 4590*x*y^2 + 126*x*y*z^5 - 1290*x*y*z^4 + 4776*x*y*z^3 - 9042*x*y*z^2 + 8940*x*y*z - 3710*x*y - 21*x*z^6 + 258*x*z^5 - 1191*x*z^4 + 2976*x*z^3 - 4350*x*z^2 + 3542*x*z - 1246*x + 3*y^7 - 21*y^6*z + 42*y^6 + 63*y^5*z^2 - 252*y^5*z + 231*y^5 - 105*y^4*z^3 + 630*y^4*z^2 - 1152*y^4*z + 732*y^4 + 105*y^3*z^4 - 840*y^3*z^3 + 2298*y^3*z^2 - 2892*y^3*z + 1426*y^3 - 63*y^2*z^5 + 630*y^2*z^4 - 2292*y^2*z^3 + 4284*y^2*z^2 - 4170*y^2*z + 1705*y^2 + 21*y*z^6 - 252*y*z^5 + 1143*y*z^4 - 2820*y*z^3 + 4062*y*z^2 - 3254*y*z + 1123*y - 3*z^7 + 42*z^6 - 228*z^5 + 696*z^4 - 1318*z^3 + 1555*z^2 - 1053*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -1, 1, -1 ], [ -2, 1, -1 ], [ -1, 2, -1 ] ], "source_shift": [ 1, 2, 1 ], "target_matrix": [ [ -2, -1, -1 ], [ -1, -2, -1 ], [ 1, 1, 1 ] ], "target_shift": [ 625, 316, -312 ], "inverse_recipe": { "p": 2, "q": 3, "f": "x**2 + x", "g": "x**3", "h": "y**3 + y**2*z + 2*z**2", "polynomials": { "f": [ [ [ 2, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "g": [ [ [ 3, 0, 0 ], 1 ] ], "h": [ [ [ 0, 2, 1 ], 1 ], [ [ 0, 3, 0 ], 1 ], [ [ 0, 0, 2 ], 2 ] ] }, "h_multiplier": 3 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 0.9701176119997399 ], "surface_features": [ 3.389833754108406, 4.286007122079474, 3.0879587894607328, 0.0, 3.512165741666634, 4824 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B079", "pair": 79, "family": "bijection", "mapping": "F(x,y,z)=(x^7 + 14*x^6*y - 7*x^6*z - 13*x^6 + 84*x^5*y^2 - 84*x^5*y*z - 154*x^5*y + 21*x^5*z^2 + 78*x^5*z + 74*x^5 + 280*x^4*y^3 - 420*x^4*y^2*z - 760*x^4*y^2 + 210*x^4*y*z^2 + 770*x^4*y*z + 720*x^4*y - 35*x^4*z^3 - 195*x^4*z^2 - 371*x^4*z - 239*x^4 + 560*x^3*y^4 - 1120*x^3*y^3*z - 2000*x^3*y^3 + 840*x^3*y^2*z^2 + 3040*x^3*y^2*z + 2801*x^3*y^2 - 280*x^3*y*z^3 - 1540*x^3*y*z^2 - 2888*x^3*y*z - 1828*x^3*y + 35*x^3*z^4 + 260*x^3*z^3 + 744*x^3*z^2 + 964*x^3*z + 473*x^3 + 672*x^2*y^5 - 1680*x^2*y^4*z - 2960*x^2*y^4 + 1680*x^2*y^3*z^2 + 6000*x^2*y^3*z + 5446*x^2*y^3 - 840*x^2*y^2*z^3 - 4560*x^2*y^2*z^2 - 8427*x^2*y^2*z - 5241*x^2*y^2 + 210*x^2*y*z^4 + 1540*x^2*y*z^3 + 4344*x^2*y*z^2 + 5530*x^2*y*z + 2656*x^2*y - 21*x^2*z^5 - 195*x^2*z^4 - 746*x^2*z^3 - 1458*x^2*z^2 - 1443*x^2*z - 570*x^2 + 448*x*y^6 - 1344*x*y^5*z - 2336*x*y^5 + 1680*x*y^4*z^2 + 5920*x*y^4*z + 5292*x*y^4 - 1120*x*y^3*z^3 - 6000*x*y^3*z^2 - 10924*x*y^3*z - 6676*x*y^3 + 420*x*y^2*z^4 + 3040*x*y^2*z^3 + 8451*x*y^2*z^2 + 10570*x*y^2*z + 4974*x*y^2 - 84*x*y*z^5 - 770*x*y*z^4 - 2904*x*y*z^3 - 5576*x*y*z^2 - 5400*x*y*z - 2080*x*y + 7*x*z^6 + 78*x*z^5 + 374*x*z^4 + 980*x*z^3 + 1467*x*z^2 + 1172*x*z + 383*x + 128*y^7 - 448*y^6*z - 768*y^6 + 672*y^5*z^2 + 2336*y^5*z + 2056*y^5 - 560*y^4*z^3 - 2960*y^4*z^2 - 5308*y^4*z - 3188*y^4 + 280*y^3*z^4 + 2000*y^3*z^3 + 5478*y^3*z^2 + 6732*y^3*z + 3106*y^3 - 84*y^2*z^5 - 760*y^2*z^4 - 2825*y^2*z^3 - 5329*y^2*z^2 - 5055*y^2*z - 1903*y^2 + 14*y*z^6 + 154*y*z^5 + 728*y*z^4 + 1874*y*z^3 + 2744*y*z^2 + 2136*y*z + 679*y - z^7 - 13*z^6 - 75*z^5 - 247*z^4 - 497*z^3 - 602*z^2 - 400*z - 1, 2*x^7 + 28*x^6*y - 14*x^6*z - 26*x^6 + 168*x^5*y^2 - 168*x^5*y*z - 308*x^5*y + 42*x^5*z^2 + 156*x^5*z + 148*x^5 + 560*x^4*y^3 - 840*x^4*y^2*z - 1520*x^4*y^2 + 420*x^4*y*z^2 + 1540*x^4*y*z + 1440*x^4*y - 70*x^4*z^3 - 390*x^4*z^2 - 742*x^4*z - 478*x^4 + 1120*x^3*y^4 - 2240*x^3*y^3*z - 4000*x^3*y^3 + 1680*x^3*y^2*z^2 + 6080*x^3*y^2*z + 5602*x^3*y^2 - 560*x^3*y*z^3 - 3080*x^3*y*z^2 - 5776*x^3*y*z - 3656*x^3*y + 70*x^3*z^4 + 520*x^3*z^3 + 1488*x^3*z^2 + 1928*x^3*z + 947*x^3 + 1344*x^2*y^5 - 3360*x^2*y^4*z - 5920*x^2*y^4 + 3360*x^2*y^3*z^2 + 12000*x^2*y^3*z + 10892*x^2*y^3 - 1680*x^2*y^2*z^3 - 9120*x^2*y^2*z^2 - 16854*x^2*y^2*z - 10482*x^2*y^2 + 420*x^2*y*z^4 + 3080*x^2*y*z^3 + 8688*x^2*y*z^2 + 11060*x^2*y*z + 5318*x^2*y - 42*x^2*z^5 - 390*x^2*z^4 - 1492*x^2*z^3 - 2916*x^2*z^2 - 2889*x^2*z - 1144*x^2 + 896*x*y^6 - 2688*x*y^5*z - 4672*x*y^5 + 3360*x*y^4*z^2 + 11840*x*y^4*z + 10584*x*y^4 - 2240*x*y^3*z^3 - 12000*x*y^3*z^2 - 21848*x*y^3*z - 13352*x*y^3 + 840*x*y^2*z^4 + 6080*x*y^2*z^3 + 16902*x*y^2*z^2 + 21140*x*y^2*z + 9960*x*y^2 - 168*x*y*z^5 - 1540*x*y*z^4 - 5808*x*y*z^3 - 11152*x*y*z^2 - 10812*x*y*z - 4176*x*y + 14*x*z^6 + 156*x*z^5 + 748*x*z^4 + 1960*x*z^3 + 2937*x*z^2 + 2352*x*z + 772*x 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3.0681858617461617, 0.0, 3.5355179947496063, 4822 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B240", "pair": 240, "family": "bijection", "mapping": "F(x,y,z)=(-128*x^7 + 448*x^6*y + 448*x^6*z - 640*x^6 - 672*x^5*y^2 - 1344*x^5*y*z + 1920*x^5*y - 672*x^5*z^2 + 1920*x^5*z - 1424*x^5 + 560*x^4*y^3 + 1680*x^4*y^2*z - 2400*x^4*y^2 + 1680*x^4*y*z^2 - 4800*x^4*y*z + 3568*x^4*y + 560*x^4*z^3 - 2400*x^4*z^2 + 3568*x^4*z - 1792*x^4 - 280*x^3*y^4 - 1120*x^3*y^3*z + 1600*x^3*y^3 - 1680*x^3*y^2*z^2 + 4800*x^3*y^2*z - 3576*x^3*y^2 - 1120*x^3*y*z^3 + 4800*x^3*y*z^2 - 7152*x^3*y*z + 3616*x^3*y - 280*x^3*z^4 + 1600*x^3*z^3 - 3576*x^3*z^2 + 3624*x^3*z - 1392*x^3 + 84*x^2*y^5 + 420*x^2*y^4*z - 600*x^2*y^4 + 840*x^2*y^3*z^2 - 2400*x^2*y^3*z + 1792*x^2*y^3 + 840*x^2*y^2*z^3 - 3600*x^2*y^2*z^2 + 5376*x^2*y^2*z - 2736*x^2*y^2 + 420*x^2*y*z^4 - 2400*x^2*y*z^3 + 5376*x^2*y*z^2 - 5484*x^2*y*z + 2132*x^2*y + 84*x^2*z^5 - 600*x^2*z^4 + 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distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ 1, 1, 1 ], [ 1, 2, 1 ], [ 2, 3, 3 ] ], "source_shift": [ 2, 1, 2 ], "target_matrix": [ [ 1, 1, 2 ], [ 1, 1, 1 ], [ 2, 3, 2 ] ], "target_shift": [ -204, -228, -626 ], "collision_preimages": [ [ "-15/4", "1", "3/4" ], [ "-15/2", "-3/2", "8" ], [ "-27/2", "7/2", "7" ] ], "collision_image": [ "-817/4", "-913/4", "-1253/2" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.039532845003123 ], "surface_features": [ 3.6352762121535687, 4.646060468555641, 3.236789099409293, 0.0, 3.7938558580862067, 4908 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B090", "pair": 90, "family": "bijection", "mapping": "F(x,y,z)=(128*x^7 + 448*x^6*y + 448*x^6*z + 992*x^6 + 672*x^5*y^2 + 1344*x^5*y*z + 3024*x^5*y + 672*x^5*z^2 + 2992*x^5*z + 3248*x^5 + 560*x^4*y^3 + 1680*x^4*y^2*z + 3840*x^4*y^2 + 1680*x^4*y*z^2 + 7600*x^4*y*z + 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183 ], "inverse_recipe": { "p": 2, "q": 5, "f": "x**2", "g": "x**5 + x**4 + x**3 + x", "h": "y*z", "polynomials": { "f": [ [ [ 2, 0, 0 ], 1 ] ], "g": [ [ [ 5, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ], [ [ 3, 0, 0 ], 1 ], [ [ 4, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0028853039984824 ], "surface_features": [ 3.4726817045908915, 4.270072962969794, 3.0492180226701815, 0.0, 3.5800648956531003, 4819 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B162", "pair": 162, "family": "bijection", "mapping": "F(x,y,z)=(3*x^7 + 21*x^6*y + 42*x^6*z + 24*x^6 + 63*x^5*y^2 + 252*x^5*y*z + 144*x^5*y + 252*x^5*z^2 + 285*x^5*z + 85*x^5 + 105*x^4*y^3 + 630*x^4*y^2*z + 360*x^4*y^2 + 1260*x^4*y*z^2 + 1425*x^4*y*z + 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3340*x*y^3*z + 716*x*y^3 + 5040*x*y^2*z^4 + 11160*x*y^2*z^3 + 9840*x*y^2*z^2 + 4197*x*y^2*z + 774*x*y^2 + 4032*x*y*z^5 + 11040*x*y*z^4 + 12880*x*y*z^3 + 8196*x*y*z^2 + 2994*x*y*z + 537*x*y + 1344*x*z^6 + 4368*x*z^5 + 6320*x*z^4 + 5332*x*z^3 + 2892*x*z^2 + 1008*x*z + 185*x + 3*y^7 + 42*y^6*z + 24*y^6 + 252*y^5*z^2 + 285*y^5*z + 85*y^5 + 840*y^4*z^3 + 1410*y^4*z^2 + 835*y^4*z + 179*y^4 + 1680*y^3*z^4 + 3720*y^3*z^3 + 3280*y^3*z^2 + 1399*y^3*z + 257*y^3 + 2016*y^2*z^5 + 5520*y^2*z^4 + 6440*y^2*z^3 + 4098*y^2*z^2 + 1491*y^2*z + 264*y^2 + 1344*y*z^6 + 4368*y*z^5 + 6320*y*z^4 + 5332*y*z^3 + 2880*y*z^2 + 993*y*z + 178*y + 384*z^7 + 1440*z^6 + 2480*z^5 + 2600*z^4 + 1852*z^3 + 936*z^2 + 318*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ 1, 1, 2 ], [ 2, 1, 1 ], [ 1, 1, 1 ] ], "source_shift": [ 1, 2, 1 ], "target_matrix": [ [ 1, 1, 2 ], [ 2, 1, 1 ], [ 1, 1, 1 ] ], "target_shift": [ -52, -111, -56 ], "inverse_recipe": { "p": 5, "q": 2, "f": "x**5 + x**3 + x**2 + x", "g": "x**2", "h": "y*z + z**2 + z", "polynomials": { "f": [ [ [ 5, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ], [ [ 3, 0, 0 ], 1 ] ], "g": [ [ [ 2, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 0, 2 ], 1 ], [ [ 0, 0, 1 ], 1 ] ] }, "h_multiplier": 3 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 0.9622900750000554 ], "surface_features": [ 3.5094246922028187, 4.40823996531185, 3.176958980586908, 0.0, 3.615271963774327, 4885 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C123", "pair": 123, "family": "counterexample", "mapping": "F(x,y,z)=(16*x^7 - 104*x^6*y + 88*x^6*z - 220*x^6 + 276*x^5*y^2 - 480*x^5*y*z + 1080*x^5*y + 204*x^5*z^2 - 972*x^5*z + 1080*x^5 - 390*x^4*y^3 + 1038*x^4*y^2*z - 2175*x^4*y^2 - 906*x^4*y*z^2 + 3942*x^4*y*z - 4212*x^4*y + 258*x^4*z^3 - 1767*x^4*z^2 + 3888*x^4*z - 2783*x^4 + 318*x^3*y^4 - 1146*x^3*y^3*z + 2295*x^3*y^3 + 1530*x^3*y^2*z^2 - 6282*x^3*y^2*z + 6504*x^3*y^2 - 894*x^3*y*z^3 + 5679*x^3*y*z^2 - 12024*x^3*y*z + 8457*x^3*y + 192*x^3*z^4 - 1692*x^3*z^3 + 5520*x^3*z^2 - 7910*x^3*z + 4226*x^3 - 150*x^2*y^5 + 684*x^2*y^4*z - 1335*x^2*y^4 - 1236*x^2*y^3*z^2 + 4905*x^2*y^3*z - 4962*x^2*y^3 + 1104*x^2*y^2*z^3 - 6705*x^2*y^2*z^2 + 13791*x^2*y^2*z - 9549*x^2*y^2 - 486*x^2*y*z^4 + 4035*x^2*y*z^3 - 12696*x^2*y*z^2 + 17847*x^2*y*z - 9462*x^2*y + 84*x^2*z^5 - 900*x^2*z^4 + 3867*x^2*z^3 - 8298*x^2*z^2 + 8913*x^2*z - 3851*x^2 + 38*x*y^6 - 210*x*y^5*z + 405*x*y^5 + 480*x*y^4*z^2 - 1872*x*y^4*z + 1866*x*y^4 - 580*x*y^3*z^3 + 3438*x*y^3*z^2 - 6936*x*y^3*z + 4744*x*y^3 + 390*x*y^2*z^4 - 3132*x*y^2*z^3 + 9612*x*y^2*z^2 - 13302*x*y^2*z + 6996*x*y^2 - 138*x*y*z^5 + 1413*x*y*z^4 - 5880*x*y*z^3 + 12372*x*y*z^2 - 13152*x*y*z + 5666*x*y + 20*x*z^6 - 252*x*z^5 + 1338*x*z^4 - 3814*x*z^3 + 6156*x*z^2 - 5354*x*z + 1970*x - 4*y^7 + 26*y^6*z - 50*y^6 - 72*y^5*z^2 + 279*y^5*z - 276*y^5 + 110*y^4*z^3 - 645*y^4*z^2 + 1287*y^4*z - 873*y^4 - 100*y^3*z^4 + 790*y^3*z^3 - 2388*y^3*z^2 + 3268*y^3*z - 1708*y^3 + 54*y^2*z^5 - 540*y^2*z^4 + 2202*y^2*z^3 - 4566*y^2*z^2 + 4812*y^2*z - 2066*y^2 - 16*y*z^6 + 195*y*z^5 - 1008*y*z^4 + 2820*y*z^3 - 4500*y*z^2 + 3892*y*z - 1432*y + 2*z^7 - 29*z^6 + 183*z^5 - 649*z^4 + 1396*z^3 - 1826*z^2 + 1352*z - 1, 8*x^7 - 52*x^6*y + 44*x^6*z - 128*x^6 + 138*x^5*y^2 - 240*x^5*y*z + 648*x^5*y + 102*x^5*z^2 - 576*x^5*z + 756*x^5 - 195*x^4*y^3 + 519*x^4*y^2*z - 1344*x^4*y^2 - 453*x^4*y*z^2 + 2412*x^4*y*z - 3024*x^4*y + 129*x^4*z^3 - 1068*x^4*z^2 + 2772*x^4*z - 2219*x^4 + 159*x^3*y^4 - 573*x^3*y^3*z + 1458*x^3*y^3 + 765*x^3*y^2*z^2 - 3960*x^3*y^2*z + 4791*x^3*y^2 - 447*x^3*y*z^3 + 3546*x^3*y*z^2 - 8802*x^3*y*z + 6899*x^3*y + 96*x^3*z^4 - 1044*x^3*z^3 + 4011*x^3*z^2 - 6428*x^3*z + 3692*x^3 - 75*x^2*y^5 + 342*x^2*y^4*z - 870*x^2*y^4 - 618*x^2*y^3*z^2 + 3177*x^2*y^3*z - 3750*x^2*y^3 + 552*x^2*y^2*z^3 - 4311*x^2*y^2*z^2 + 10365*x^2*y^2*z - 7974*x^2*y^2 - 243*x^2*y*z^4 + 2571*x^2*y*z^3 - 9480*x^2*y*z^2 + 14847*x^2*y*z - 8451*x^2*y + 42*x^2*z^5 - 567*x^2*z^4 + 2865*x^2*z^3 - 6873*x^2*z^2 + 7944*x^2*z - 3592*x^2 + 19*x*y^6 - 105*x*y^5*z + 270*x*y^5 + 240*x*y^4*z^2 - 1242*x*y^4*z + 1446*x*y^4 - 290*x*y^3*z^3 + 2268*x*y^3*z^2 - 5349*x*y^3*z + 4056*x*y^3 + 195*x*y^2*z^4 - 2052*x*y^2*z^3 + 7371*x*y^2*z^2 - 11331*x*y^2*z + 6390*x*y^2 - 69*x*y*z^5 + 918*x*y*z^4 - 4479*x*y*z^3 + 10494*x*y*z^2 - 11985*x*y*z + 5407*x*y + 10*x*z^6 - 162*x*z^5 + 1011*x*z^4 - 3219*x*z^3 + 5595*x*z^2 - 5104*x*z + 1923*x - 2*y^7 + 13*y^6*z - 34*y^6 - 36*y^5*z^2 + 189*y^5*z - 219*y^5 + 55*y^4*z^3 - 435*y^4*z^2 + 1017*y^4*z - 764*y^4 - 50*y^3*z^4 + 530*y^3*z^3 - 1878*y^3*z^2 + 2850*y^3*z - 1596*y^3 + 27*y^2*z^5 - 360*y^2*z^4 + 1722*y^2*z^3 - 3966*y^2*z^2 + 4485*y^2*z - 2017*y^2 - 8*y*z^6 + 129*y*z^5 - 783*y*z^4 + 2438*y*z^3 - 4182*y*z^2 + 3794*y*z - 1433*y + z^7 - 19*z^6 + 141*z^5 - 558*z^4 + 1293*z^3 - 1777*z^2 + 1353*z - 1, -8*x^7 + 52*x^6*y - 44*x^6*z + 116*x^6 - 138*x^5*y^2 + 240*x^5*y*z - 576*x^5*y - 102*x^5*z^2 + 516*x^5*z - 612*x^5 + 195*x^4*y^3 - 519*x^4*y^2*z + 1173*x^4*y^2 + 453*x^4*y*z^2 - 2118*x^4*y*z + 2412*x^4*y - 129*x^4*z^3 + 945*x^4*z^2 - 2220*x^4*z + 1667*x^4 - 159*x^3*y^4 + 573*x^3*y^3*z - 1251*x^3*y^3 - 765*x^3*y^2*z^2 + 3414*x^3*y^2*z - 3765*x^3*y^2 + 447*x^3*y*z^3 - 3075*x^3*y*z^2 + 6942*x^3*y*z - 5117*x^3*y - 96*x^3*z^4 + 912*x^3*z^3 - 3177*x^3*z^2 + 4778*x^3*z - 2636*x^3 + 75*x^2*y^5 - 342*x^2*y^4*z + 735*x^2*y^4 + 618*x^2*y^3*z^2 - 2694*x^2*y^3*z + 2904*x^2*y^3 - 552*x^2*y^2*z^3 + 3672*x^2*y^2*z^2 - 8052*x^2*y^2*z + 5838*x^2*y^2 + 243*x^2*y*z^4 - 2202*x^2*y*z^3 + 7392*x^2*y*z^2 - 10893*x^2*y*z + 5961*x^2*y - 42*x^2*z^5 + 489*x^2*z^4 - 2244*x^2*z^3 + 5055*x^2*z^2 - 5610*x^2*z + 2473*x^2 - 19*x*y^6 + 105*x*y^5*z - 225*x*y^5 - 240*x*y^4*z^2 + 1038*x*y^4*z - 1104*x*y^4 + 290*x*y^3*z^3 - 1902*x*y^3*z^2 + 4095*x*y^3*z - 2931*x*y^3 - 195*x*y^2*z^4 + 1728*x*y^2*z^3 - 5661*x*y^2*z^2 + 8205*x*y^2*z - 4452*x*y^2 + 69*x*y*z^5 - 777*x*y*z^4 + 3453*x*y*z^3 - 7617*x*y*z^2 + 8361*x*y*z - 3676*x*y - 10*x*z^6 + 138*x*z^5 - 783*x*z^4 + 2343*x*z^3 - 3909*x*z^2 + 3472*x*z - 1291*x + 2*y^7 - 13*y^6*z + 28*y^6 + 36*y^5*z^2 - 156*y^5*z + 165*y^5 - 55*y^4*z^3 + 360*y^4*z^2 - 768*y^4*z + 545*y^4 + 50*y^3*z^4 - 440*y^3*z^3 + 1422*y^3*z^2 - 2037*y^3*z + 1098*y^3 - 27*y^2*z^5 + 300*y^2*z^4 - 1308*y^2*z^3 + 2841*y^2*z^2 - 3090*y^2*z + 1354*y^2 + 8*y*z^6 - 108*y*z^5 + 597*y*z^4 - 1751*y*z^3 + 2886*y*z^2 - 2549*y*z + 949*y - z^7 + 16*z^6 - 108*z^5 + 402*z^4 - 894*z^3 + 1195*z^2 - 896*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -1, 1, -1 ], [ -2, 1, -1 ], [ -1, 2, -1 ] ], "source_shift": [ 1, 2, 1 ], "target_matrix": [ [ -2, -1, -1 ], [ -1, -2, -1 ], [ 1, 1, 1 ] ], "target_shift": [ 437, 439, -290 ], "collision_preimages": [ [ "1", "-1/4", "-1/4" ], [ "7/2", "11/2", "2" ], [ "-3/2", "15/2", "11" ] ], "collision_image": [ "875/2", "1757/4", "-1161/4" ] }, "probe_target": "No", "jev_answers": [ "Maybe" ], "seconds": [ 1.0191547829999763 ], "surface_features": [ 3.3804554636766864, 4.2515652236446275, 3.037027879755775, 0.0, 3.4870310149574992, 4818 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B086", "pair": 86, "family": "bijection", "mapping": "F(x,y,z)=(8*x^7 - 56*x^6*y - 56*x^6*z + 144*x^6 + 168*x^5*y^2 + 336*x^5*y*z - 880*x^5*y + 168*x^5*z^2 - 880*x^5*z + 1064*x^5 - 280*x^4*y^3 - 840*x^4*y^2*z + 2240*x^4*y^2 - 840*x^4*y*z^2 + 4480*x^4*y*z - 5520*x^4*y - 280*x^4*z^3 + 2240*x^4*z^2 - 5512*x^4*z + 4224*x^4 + 280*x^3*y^4 + 1120*x^3*y^3*z - 3040*x^3*y^3 + 1680*x^3*y^2*z^2 - 9120*x^3*y^2*z + 11448*x^3*y^2 + 1120*x^3*y*z^3 - 9120*x^3*y*z^2 + 22864*x^3*y*z - 17880*x^3*y + 280*x^3*z^4 - 3040*x^3*z^3 + 11416*x^3*z^2 - 17784*x^3*z + 9827*x^3 - 168*x^2*y^5 - 840*x^2*y^4*z + 2320*x^2*y^4 - 1680*x^2*y^3*z^2 + 9280*x^2*y^3*z - 11864*x^2*y^3 - 1680*x^2*y^2*z^3 + 13920*x^2*y^2*z^2 - 35544*x^2*y^2*z + 28360*x^2*y^2 - 840*x^2*y*z^4 + 9280*x^2*y*z^3 - 35496*x^2*y*z^2 + 56416*x^2*y*z - 31921*x^2*y - 168*x^2*z^5 + 2320*x^2*z^4 - 11816*x^2*z^3 + 28056*x^2*z^2 - 31545*x^2*z + 13358*x^2 + 56*x*y^6 + 336*x*y^5*z - 944*x*y^5 + 840*x*y^4*z^2 - 4720*x*y^4*z + 6144*x*y^4 + 1120*x*y^3*z^3 - 9440*x*y^3*z^2 + 24544*x*y^3*z - 19976*x*y^3 + 840*x*y^2*z^4 - 9440*x*y^2*z^3 + 36768*x*y^2*z^2 - 59608*x*y^2*z + 34569*x*y^2 + 336*x*y*z^5 - 4720*x*y*z^4 + 24480*x*y*z^3 - 59288*x*y*z^2 + 68290*x*y*z - 29628*x*y + 56*x*z^6 - 944*x*z^5 + 6112*x*z^4 - 19656*x*z^3 + 33721*x*z^2 - 29100*x*z + 9716*x - 8*y^7 - 56*y^6*z + 160*y^6 - 168*y^5*z^2 + 960*y^5*z - 1272*y^5 - 280*y^4*z^3 + 2400*y^4*z^2 - 6352*y^4*z + 5272*y^4 - 280*y^3*z^4 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17748*x^2*y*z^2 + 28208*x^2*y*z - 15962*x^2*y - 84*x^2*z^5 + 1160*x^2*z^4 - 5908*x^2*z^3 + 14028*x^2*z^2 - 15774*x^2*z + 6682*x^2 + 28*x*y^6 + 168*x*y^5*z - 472*x*y^5 + 420*x*y^4*z^2 - 2360*x*y^4*z + 3072*x*y^4 + 560*x*y^3*z^3 - 4720*x*y^3*z^2 + 12272*x*y^3*z - 9988*x*y^3 + 420*x*y^2*z^4 - 4720*x*y^2*z^3 + 18384*x*y^2*z^2 - 29804*x*y^2*z + 17286*x*y^2 + 168*x*y*z^5 - 2360*x*y*z^4 + 12240*x*y*z^3 - 29644*x*y*z^2 + 34148*x*y*z - 14820*x*y + 28*x*z^6 - 472*x*z^5 + 3056*x*z^4 - 9828*x*z^3 + 16862*x*z^2 - 14556*x*z + 4865*x - 4*y^7 - 28*y^6*z + 80*y^6 - 84*y^5*z^2 + 480*y^5*z - 636*y^5 - 140*y^4*z^3 + 1200*y^4*z^2 - 3176*y^4*z + 2636*y^4 - 140*y^3*z^4 + 1600*y^3*z^3 - 6344*y^3*z^2 + 10488*y^3*z - 6242*y^3 - 84*y^2*z^5 + 1200*y^2*z^4 - 6336*y^2*z^3 + 15648*y^2*z^2 - 18486*y^2*z + 8230*y^2 - 28*y*z^6 + 480*y*z^5 - 3164*y*z^4 + 10376*y*z^3 - 18246*y*z^2 + 16152*y*z - 5513*y - 4*z^7 + 80*z^6 - 632*z^5 + 2580*z^4 - 6002*z^3 + 7922*z^2 - 5391*z + 1, 4*x^7 - 28*x^6*y - 28*x^6*z + 72*x^6 + 84*x^5*y^2 + 168*x^5*y*z - 440*x^5*y + 84*x^5*z^2 - 440*x^5*z + 532*x^5 - 140*x^4*y^3 - 420*x^4*y^2*z + 1120*x^4*y^2 - 420*x^4*y*z^2 + 2240*x^4*y*z - 2760*x^4*y - 140*x^4*z^3 + 1120*x^4*z^2 - 2756*x^4*z + 2112*x^4 + 140*x^3*y^4 + 560*x^3*y^3*z - 1520*x^3*y^3 + 840*x^3*y^2*z^2 - 4560*x^3*y^2*z + 5724*x^3*y^2 + 560*x^3*y*z^3 - 4560*x^3*y*z^2 + 11432*x^3*y*z - 8940*x^3*y + 140*x^3*z^4 - 1520*x^3*z^3 + 5708*x^3*z^2 - 8892*x^3*z + 4913*x^3 - 84*x^2*y^5 - 420*x^2*y^4*z + 1160*x^2*y^4 - 840*x^2*y^3*z^2 + 4640*x^2*y^3*z - 5932*x^2*y^3 - 840*x^2*y^2*z^3 + 6960*x^2*y^2*z^2 - 17772*x^2*y^2*z + 14180*x^2*y^2 - 420*x^2*y*z^4 + 4640*x^2*y*z^3 - 17748*x^2*y*z^2 + 28208*x^2*y*z - 15959*x^2*y - 84*x^2*z^5 + 1160*x^2*z^4 - 5908*x^2*z^3 + 14028*x^2*z^2 - 15771*x^2*z + 6674*x^2 + 28*x*y^6 + 168*x*y^5*z - 472*x*y^5 + 420*x*y^4*z^2 - 2360*x*y^4*z + 3072*x*y^4 + 560*x*y^3*z^3 - 4720*x*y^3*z^2 + 12272*x*y^3*z - 9988*x*y^3 + 420*x*y^2*z^4 - 4720*x*y^2*z^3 + 18384*x*y^2*z^2 - 29804*x*y^2*z + 17283*x*y^2 + 168*x*y*z^5 - 2360*x*y*z^4 + 12240*x*y*z^3 - 29644*x*y*z^2 + 34142*x*y*z - 14804*x*y + 28*x*z^6 - 472*x*z^5 + 3056*x*z^4 - 9828*x*z^3 + 16859*x*z^2 - 14540*x*z + 4839*x - 4*y^7 - 28*y^6*z + 80*y^6 - 84*y^5*z^2 + 480*y^5*z - 636*y^5 - 140*y^4*z^3 + 1200*y^4*z^2 - 3176*y^4*z + 2636*y^4 - 140*y^3*z^4 + 1600*y^3*z^3 - 6344*y^3*z^2 + 10488*y^3*z - 6241*y^3 - 84*y^2*z^5 + 1200*y^2*z^4 - 6336*y^2*z^3 + 15648*y^2*z^2 - 18483*y^2*z + 8222*y^2 - 28*y*z^6 + 480*y*z^5 - 3164*y*z^4 + 10376*y*z^3 - 18243*y*z^2 + 16136*y*z - 5484*y - 4*z^7 + 80*z^6 - 632*z^5 + 2580*z^4 - 6001*z^3 + 7914*z^2 - 5363*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ 1, -1, -1 ], [ 1, -2, -1 ], [ 2, -3, -3 ] ], "source_shift": [ 2, 2, 1 ], "target_matrix": [ [ 2, 3, 2 ], [ 1, 2, 1 ], [ 1, 1, 2 ] ], "target_shift": [ -2899, -1455, -1433 ], "inverse_recipe": { "p": 3, "q": 2, "f": "x**3 + x", "g": "x**2", "h": "y*z**2 + y*z", "polynomials": { "f": [ [ [ 3, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "g": [ [ [ 2, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 2 ], 1 ], [ [ 0, 1, 1 ], 1 ] ] }, "h_multiplier": 4 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.004961900998751 ], "surface_features": [ 3.856015050356149, 4.834357112718405, 3.41161970596323, 0.0, 4.019625446224501, 4938 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B021", "pair": 21, "family": "bijection", "mapping": "F(x,y,z)=(384*x^7 - 4032*x^6*y - 2688*x^6*z + 2784*x^6 + 18144*x^5*y^2 + 24192*x^5*y*z - 25008*x^5*y + 8064*x^5*z^2 - 16704*x^5*z + 8800*x^5 - 45360*x^4*y^3 - 90720*x^4*y^2*z + 93600*x^4*y^2 - 60480*x^4*y*z^2 + 125040*x^4*y*z - 65760*x^4*y - 13440*x^4*z^3 + 41760*x^4*z^2 - 44000*x^4*z + 15728*x^4 + 68040*x^3*y^4 + 181440*x^3*y^3*z - 186840*x^3*y^3 + 181440*x^3*y^2*z^2 - 374400*x^3*y^2*z + 196560*x^3*y^2 + 80640*x^3*y*z^3 - 250080*x^3*y*z^2 + 263040*x^3*y*z - 93888*x^3*y + 13440*x^3*z^4 - 55680*x^3*z^3 + 88000*x^3*z^2 - 62912*x^3*z + 17188*x^3 - 61236*x^2*y^5 - 204120*x^2*y^4*z + 209790*x^2*y^4 - 272160*x^2*y^3*z^2 + 560520*x^2*y^3*z - 293760*x^2*y^3 - 181440*x^2*y^2*z^3 + 561600*x^2*y^2*z^2 - 589680*x^2*y^2*z + 210168*x^2*y^2 - 60480*x^2*y*z^4 + 250080*x^2*y*z^3 - 394560*x^2*y*z^2 + 281664*x^2*y*z - 76848*x^2*y - 8064*x^2*z^5 + 41760*x^2*z^4 - 88000*x^2*z^3 + 94368*x^2*z^2 - 51576*x^2*z + 11514*x^2 + 30618*x*y^6 + 122472*x*y^5*z - 125631*x*y^5 + 204120*x*y^4*z^2 - 419580*x*y^4*z + 219510*x*y^4 + 181440*x*y^3*z^3 - 560520*x*y^3*z^2 + 587520*x*y^3*z - 209088*x*y^3 + 90720*x*y^2*z^4 - 374400*x*y^2*z^3 + 589680*x*y^2*z^2 - 420336*x*y^2*z + 114525*x*y^2 + 24192*x*y*z^5 - 125040*x*y*z^4 + 263040*x*y*z^3 - 281664*x*y*z^2 + 153732*x*y*z - 34260*x*y + 2688*x*z^6 - 16704*x*z^5 + 44000*x*z^4 - 62912*x*z^3 + 51588*x*z^2 - 23055*x*z + 4405*x - 6561*y^7 - 30618*y^6*z + 31347*y^6 - 61236*y^5*z^2 + 125631*y^5*z - 65610*y^5 - 68040*y^4*z^3 + 209790*y^4*z^2 - 219510*y^4*z + 78003*y^4 - 45360*y^3*z^4 + 186840*y^3*z^3 - 293760*y^3*z^2 + 209088*y^3*z - 56889*y^3 - 18144*y^2*z^5 + 93600*y^2*z^4 - 196560*y^2*z^3 + 210168*y^2*z^2 - 114552*y^2*z + 25485*y^2 - 4032*y*z^6 + 25008*y*z^5 - 65760*y*z^4 + 93888*y*z^3 - 76884*y*z^2 + 34299*y*z - 6532*y - 384*z^7 + 2784*z^6 - 8800*z^5 + 15728*z^4 - 17200*z^3 + 11541*z^2 - 4425*z + 1, 256*x^7 - 2688*x^6*y - 1792*x^6*z + 1856*x^6 + 12096*x^5*y^2 + 16128*x^5*y*z - 16672*x^5*y + 5376*x^5*z^2 - 11136*x^5*z + 5856*x^5 - 30240*x^4*y^3 - 60480*x^4*y^2*z + 62400*x^4*y^2 - 40320*x^4*y*z^2 + 83360*x^4*y*z - 43760*x^4*y - 8960*x^4*z^3 + 27840*x^4*z^2 - 29280*x^4*z + 10432*x^4 + 45360*x^3*y^4 + 120960*x^3*y^3*z - 124560*x^3*y^3 + 120960*x^3*y^2*z^2 - 249600*x^3*y^2*z + 130800*x^3*y^2 + 53760*x^3*y*z^3 - 166720*x^3*y*z^2 + 175040*x^3*y*z - 62272*x^3*y + 8960*x^3*z^4 - 37120*x^3*z^3 + 58560*x^3*z^2 - 41728*x^3*z + 11352*x^3 - 40824*x^2*y^5 - 136080*x^2*y^4*z + 139860*x^2*y^4 - 181440*x^2*y^3*z^2 + 373680*x^2*y^3*z - 195480*x^2*y^3 - 120960*x^2*y^2*z^3 + 374400*x^2*y^2*z^2 - 392400*x^2*y^2*z + 139392*x^2*y^2 - 40320*x^2*y*z^4 + 166720*x^2*y*z^3 - 262560*x^2*y*z^2 + 186816*x^2*y*z - 50752*x^2*y - 5376*x^2*z^5 + 27840*x^2*z^4 - 58560*x^2*z^3 + 62592*x^2*z^2 - 34064*x^2*z + 7572*x^2 + 20412*x*y^6 + 81648*x*y^5*z - 83754*x*y^5 + 136080*x*y^4*z^2 - 279720*x*y^4*z + 146070*x*y^4 + 120960*x*y^3*z^3 - 373680*x*y^3*z^2 + 390960*x*y^3*z - 138672*x*y^3 + 60480*x*y^2*z^4 - 249600*x*y^2*z^3 + 392400*x*y^2*z^2 - 278784*x*y^2*z + 75630*x*y^2 + 16128*x*y*z^5 - 83360*x*y*z^4 + 175040*x*y*z^3 - 186816*x*y*z^2 + 101528*x*y*z - 22528*x*y + 1792*x*z^6 - 11136*x*z^5 + 29280*x*z^4 - 41728*x*z^3 + 34072*x*z^2 - 15162*x*z + 2889*x - 4374*y^7 - 20412*y^6*z + 20898*y^6 - 40824*y^5*z^2 + 83754*y^5*z - 43659*y^5 - 45360*y^4*z^3 + 139860*y^4*z^2 - 146070*y^4*z + 51732*y^4 - 30240*y^3*z^4 + 124560*y^3*z^3 - 195480*y^3*z^2 + 138672*y^3*z - 37566*y^3 - 12096*y^2*z^5 + 62400*y^2*z^4 - 130800*y^2*z^3 + 139392*y^2*z^2 - 75648*y^2*z + 16756*y^2 - 2688*y*z^6 + 16672*y*z^5 - 43760*y*z^4 + 62272*y*z^3 - 50776*y*z^2 + 22554*y*z - 4283*y - 256*z^7 + 1856*z^6 - 5856*z^5 + 10432*z^4 - 11360*z^3 + 7590*z^2 - 2902*z + 1, 128*x^7 - 1344*x^6*y - 896*x^6*z + 928*x^6 + 6048*x^5*y^2 + 8064*x^5*y*z - 8336*x^5*y + 2688*x^5*z^2 - 5568*x^5*z + 2976*x^5 - 15120*x^4*y^3 - 30240*x^4*y^2*z + 31200*x^4*y^2 - 20160*x^4*y*z^2 + 41680*x^4*y*z - 22240*x^4*y - 4480*x^4*z^3 + 13920*x^4*z^2 - 14880*x^4*z + 5456*x^4 + 22680*x^3*y^4 + 60480*x^3*y^3*z - 62280*x^3*y^3 + 60480*x^3*y^2*z^2 - 124800*x^3*y^2*z + 66480*x^3*y^2 + 26880*x^3*y*z^3 - 83360*x^3*y*z^2 + 88960*x^3*y*z - 32576*x^3*y + 4480*x^3*z^4 - 18560*x^3*z^3 + 29760*x^3*z^2 - 21824*x^3*z + 6156*x^3 - 20412*x^2*y^5 - 68040*x^2*y^4*z + 69930*x^2*y^4 - 90720*x^2*y^3*z^2 + 186840*x^2*y^3*z - 99360*x^2*y^3 - 60480*x^2*y^2*z^3 + 187200*x^2*y^2*z^2 - 199440*x^2*y^2*z + 72936*x^2*y^2 - 20160*x^2*y*z^4 + 83360*x^2*y*z^3 - 133440*x^2*y*z^2 + 97728*x^2*y*z - 27536*x^2*y - 2688*x^2*z^5 + 13920*x^2*z^4 - 29760*x^2*z^3 + 32736*x^2*z^2 - 18472*x^2*z + 4262*x^2 + 10206*x*y^6 + 40824*x*y^5*z - 41877*x*y^5 + 68040*x*y^4*z^2 - 139860*x*y^4*z + 74250*x*y^4 + 60480*x*y^3*z^3 - 186840*x*y^3*z^2 + 198720*x*y^3*z - 72576*x*y^3 + 30240*x*y^2*z^4 - 124800*x*y^2*z^3 + 199440*x*y^2*z^2 - 145872*x*y^2*z + 41055*x*y^2 + 8064*x*y*z^5 - 41680*x*y*z^4 + 88960*x*y*z^3 - 97728*x*y*z^2 + 55084*x*y*z - 12692*x*y + 896*x*z^6 - 5568*x*z^5 + 14880*x*z^4 - 21824*x*z^3 + 18476*x*z^2 - 8533*x*z + 1677*x - 2187*y^7 - 10206*y^6*z + 10449*y^6 - 20412*y^5*z^2 + 41877*y^5*z - 22194*y^5 - 22680*y^4*z^3 + 69930*y^4*z^2 - 74250*y^4*z + 27081*y^4 - 15120*y^3*z^4 + 62280*y^3*z^3 - 99360*y^3*z^2 + 72576*y^3*z - 20403*y^3 - 6048*y^2*z^5 + 31200*y^2*z^4 - 66480*y^2*z^3 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1800*y^4*z^2 + 1976*y^4*z + 741*y^4 + 1120*y^3*z^4 + 4800*y^3*z^3 + 7888*y^3*z^2 + 5894*y^3*z + 1694*y^3 + 1344*y^2*z^5 + 7200*y^2*z^4 + 15744*y^2*z^3 + 17580*y^2*z^2 + 10050*y^2*z + 2365*y^2 + 896*y*z^6 + 5760*y*z^5 + 15712*y*z^4 + 23304*y*z^3 + 19872*y*z^2 + 9272*y*z + 1855*y + 256*z^7 + 1920*z^6 + 6272*z^5 + 11584*z^4 + 13096*z^3 + 9088*z^2 + 3591*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ 1, 1, 2 ], [ 2, 1, 1 ], [ 3, 2, 2 ] ], "source_shift": [ 2, 2, 1 ], "target_matrix": [ [ 2, 1, 1 ], [ 1, 1, 1 ], [ 1, 1, 2 ] ], "target_shift": [ -1301, -649, -623 ], "inverse_recipe": { "p": 4, "q": 3, "f": "x**4 + x**2 + x", "g": "x**3 + x**2", "h": "y*z + z", "polynomials": { "f": [ [ [ 4, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "g": [ [ [ 3, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 0, 1 ], 1 ] ] }, "h_multiplier": 2 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0706217080005445 ], "surface_features": [ 3.8903904314185933, 4.849443984218218, 3.526339277389844, 0.0, 4.031297402062405, 4988 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C124", "pair": 124, "family": "counterexample", "mapping": "F(x,y,z)=(2*x^7 + 16*x^6*y - 20*x^6*z - 35*x^6 + 54*x^5*y^2 - 138*x^5*y*z - 237*x^5*y + 84*x^5*z^2 + 297*x^5*z + 246*x^5 + 100*x^4*y^3 - 390*x^4*y^2*z - 660*x^4*y^2 + 486*x^4*y*z^2 + 1689*x^4*y*z + 1374*x^4*y - 192*x^4*z^3 - 1029*x^4*z^2 - 1713*x^4*z - 906*x^4 + 110*x^3*y^4 - 580*x^3*y^3*z - 970*x^3*y^3 + 1104*x^3*y^2*z^2 + 3786*x^3*y^2*z + 3036*x^3*y^2 - 894*x^3*y*z^3 - 4719*x^3*y*z^2 - 7716*x^3*y*z - 4027*x^3*y + 258*x^3*z^4 + 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971*x + 2*y^7 - 19*y^6*z - 34*y^6 + 75*y^5*z^2 + 276*y^5*z + 225*y^5 - 159*y^4*z^3 - 903*y^4*z^2 - 1494*y^4*z - 779*y^4 + 195*y^3*z^4 + 1518*y^3*z^3 + 3822*y^3*z^2 + 4024*y^3*z + 1552*y^3 - 138*y^2*z^5 - 1377*y^2*z^4 - 4689*y^2*z^3 - 7470*y^2*z^2 - 5793*y^2*z - 1789*y^2 + 52*y*z^6 + 636*y*z^5 + 2748*y*z^4 + 5885*y*z^3 + 6864*y*z^2 + 4235*y*z + 1098*y - 8*z^7 - 116*z^6 - 612*z^5 - 1654*z^4 - 2578*z^3 - 2374*z^2 - 1217*z - 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -1, -1, 2 ], [ -1, -1, 1 ], [ -1, -2, 1 ] ], "source_shift": [ 2, 1, 1 ], "target_matrix": [ [ -2, -3, -2 ], [ 2, 1, 1 ], [ -1, -2, -1 ] ], "target_shift": [ 426, -212, 270 ], "collision_preimages": [ [ "-1/4", "1/4", "-1" ], [ "12", "-8", "3/2" ], [ "1", "-5", "-7/2" ] ], "collision_image": [ "853/2", "-425/2", "1081/4" ] }, "probe_target": "No", "jev_answers": [ "Maybe" ], "seconds": [ 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input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0539204889973917 ], "surface_features": [ 3.5817107818990346, 4.392204356370863, 3.1539672216454786, 0.0, 3.694310429640517, 4874 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B128", "pair": 128, "family": "bijection", "mapping": "F(x,y,z)=(-x^7 - 7*x^6*y + 14*x^6*z + 15*x^6 - 21*x^5*y^2 + 84*x^5*y*z + 90*x^5*y - 84*x^5*z^2 - 180*x^5*z - 98*x^5 - 35*x^4*y^3 + 210*x^4*y^2*z + 225*x^4*y^2 - 420*x^4*y*z^2 - 900*x^4*y*z - 490*x^4*y + 280*x^4*z^3 + 900*x^4*z^2 + 979*x^4*z + 364*x^4 - 35*x^3*y^4 + 280*x^3*y^3*z + 300*x^3*y^3 - 840*x^3*y^2*z^2 - 1800*x^3*y^2*z - 980*x^3*y^2 + 1120*x^3*y*z^3 + 3600*x^3*y*z^2 + 3916*x^3*y*z + 1455*x^3*y - 560*x^3*z^4 - 2400*x^3*z^3 - 3912*x^3*z^2 - 2900*x^3*z - 826*x^3 - 21*x^2*y^5 + 210*x^2*y^4*z + 225*x^2*y^4 - 840*x^2*y^3*z^2 - 1800*x^2*y^3*z - 980*x^2*y^3 + 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1800*x^3*y^2*z - 980*x^3*y^2 + 1120*x^3*y*z^3 + 3600*x^3*y*z^2 + 3916*x^3*y*z + 1455*x^3*y - 560*x^3*z^4 - 2400*x^3*z^3 - 3912*x^3*z^2 - 2900*x^3*z - 828*x^3 - 21*x^2*y^5 + 210*x^2*y^4*z + 225*x^2*y^4 - 840*x^2*y^3*z^2 - 1800*x^2*y^3*z - 980*x^2*y^3 + 1680*x^2*y^2*z^3 + 5400*x^2*y^2*z^2 + 5874*x^2*y^2*z + 2181*x^2*y^2 - 1680*x^2*y*z^4 - 7200*x^2*y*z^3 - 11736*x^2*y*z^2 - 8694*x^2*y*z - 2478*x^2*y + 672*x^2*z^5 + 3600*x^2*z^4 + 7816*x^2*z^3 + 8664*x^2*z^2 + 4920*x^2*z + 1148*x^2 - 7*x*y^6 + 84*x*y^5*z + 90*x*y^5 - 420*x*y^4*z^2 - 900*x*y^4*z - 490*x*y^4 + 1120*x*y^3*z^3 + 3600*x*y^3*z^2 + 3916*x*y^3*z + 1453*x*y^3 - 1680*x*y^2*z^4 - 7200*x*y^2*z^3 - 11736*x*y^2*z^2 - 8688*x*y^2*z - 2472*x*y^2 + 1344*x*y*z^5 + 7200*x*y*z^4 + 15632*x*y*z^3 + 17316*x*y*z^2 + 9816*x*y*z + 2282*x*y - 448*x*z^6 - 2880*x*z^5 - 7808*x*z^4 - 11504*x*z^3 - 9744*x*z^2 - 4504*x*z - 889*x - y^7 + 14*y^6*z + 15*y^6 - 84*y^5*z^2 - 180*y^5*z - 98*y^5 + 280*y^4*z^3 + 900*y^4*z^2 + 979*y^4*z + 363*y^4 - 560*y^3*z^4 - 2400*y^3*z^3 - 3912*y^3*z^2 - 2894*y^3*z - 822*y^3 + 672*y^2*z^5 + 3600*y^2*z^4 + 7816*y^2*z^3 + 8652*y^2*z^2 + 4896*y^2*z + 1134*y^2 - 448*y*z^6 - 2880*y*z^5 - 7808*y*z^4 - 11496*y*z^3 - 9720*y*z^2 - 4477*y*z - 877*y + 128*z^7 + 960*z^6 + 3120*z^5 + 5728*z^4 + 6432*z^3 + 4419*z^2 + 1718*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -1, -1, 2 ], [ -2, -1, 1 ], [ -1, -1, 1 ] ], "source_shift": [ 2, 1, 1 ], "target_matrix": [ [ 1, 1, 2 ], [ 2, 1, 1 ], [ 1, 1, 1 ] ], "target_shift": [ -268, -578, -290 ], "inverse_recipe": { "p": 4, "q": 3, "f": "x**4 + x**3", "g": "x**3 + x", "h": "y*z + z", "polynomials": { "f": [ [ [ 4, 0, 0 ], 1 ], [ [ 3, 0, 0 ], 1 ] ], "g": [ [ [ 3, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 0, 1 ], 1 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0590419950021897 ], "surface_features": [ 3.581653571074283, 4.538875222525595, 3.225309281725863, 0.0, 3.7226586633009147, 4851 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B081", "pair": 81, "family": "bijection", "mapping": "F(x,y,z)=(-2*x^7 + 14*x^6*y - 28*x^6*z + 30*x^6 - 42*x^5*y^2 + 168*x^5*y*z - 180*x^5*y - 168*x^5*z^2 + 360*x^5*z - 200*x^5 + 70*x^4*y^3 - 420*x^4*y^2*z + 450*x^4*y^2 + 840*x^4*y*z^2 - 1800*x^4*y*z + 998*x^4*y - 560*x^4*z^3 + 1800*x^4*z^2 - 1994*x^4*z + 762*x^4 - 70*x^3*y^4 + 560*x^3*y^3*z - 600*x^3*y^3 - 1680*x^3*y^2*z^2 + 3600*x^3*y^2*z - 1992*x^3*y^2 + 2240*x^3*y*z^3 - 7200*x^3*y*z^2 + 7960*x^3*y*z - 3030*x^3*y - 1120*x^3*z^4 + 4800*x^3*z^3 - 7952*x^3*z^2 + 6040*x^3*z - 1777*x^3 + 42*x^2*y^5 - 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896*y*z^6 - 5760*y*z^5 + 15776*y*z^4 - 23568*y*z^3 + 20284*y*z^2 - 9560*y*z + 1939*y - 256*z^7 + 1920*z^6 - 6304*z^5 + 11744*z^4 - 13424*z^3 + 9434*z^2 - 3791*z + 1, 2*x^7 - 14*x^6*y + 28*x^6*z - 30*x^6 + 42*x^5*y^2 - 168*x^5*y*z + 180*x^5*y + 168*x^5*z^2 - 360*x^5*z + 200*x^5 - 70*x^4*y^3 + 420*x^4*y^2*z - 450*x^4*y^2 - 840*x^4*y*z^2 + 1800*x^4*y*z - 998*x^4*y + 560*x^4*z^3 - 1800*x^4*z^2 + 1994*x^4*z - 764*x^4 + 70*x^3*y^4 - 560*x^3*y^3*z + 600*x^3*y^3 + 1680*x^3*y^2*z^2 - 3600*x^3*y^2*z + 1992*x^3*y^2 - 2240*x^3*y*z^3 + 7200*x^3*y*z^2 - 7960*x^3*y*z + 3038*x^3*y + 1120*x^3*z^4 - 4800*x^3*z^3 + 7952*x^3*z^2 - 6056*x^3*z + 1795*x^3 - 42*x^2*y^5 + 420*x^2*y^4*z - 450*x^2*y^4 - 1680*x^2*y^3*z^2 + 3600*x^2*y^3*z - 1988*x^2*y^3 + 3360*x^2*y^2*z^3 - 10800*x^2*y^2*z^2 + 11916*x^2*y^2*z - 4530*x^2*y^2 - 3360*x^2*y*z^4 + 14400*x^2*y*z^3 - 23808*x^2*y*z^2 + 18060*x^2*y*z - 5323*x^2*y + 1344*x^2*z^5 - 7200*x^2*z^4 + 15856*x^2*z^3 - 18000*x^2*z^2 + 10572*x^2*z - 2584*x^2 + 14*x*y^6 - 168*x*y^5*z 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20760*x*z^2 - 10030*x*z + 2121*x - 2*y^7 + 28*y^6*z - 30*y^6 - 168*y^5*z^2 + 360*y^5*z - 198*y^5 + 560*y^4*z^3 - 1800*y^4*z^2 + 1978*y^4*z - 746*y^4 - 1120*y^3*z^4 + 4800*y^3*z^3 - 7904*y^3*z^2 + 5948*y^3*z - 1734*y^3 + 1344*y^2*z^5 - 7200*y^2*z^4 + 15792*y^2*z^3 - 17784*y^2*z^2 + 10330*y^2*z - 2490*y^2 - 896*y*z^6 + 5760*y*z^5 - 15776*y*z^4 + 23632*y*z^3 - 20512*y*z^2 + 9832*y*z - 2053*y + 256*z^7 - 1920*z^6 + 6304*z^5 - 11776*z^4 + 13576*z^3 - 9706*z^2 + 4016*z - 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -1, 1, -2 ], [ -2, 1, -1 ], [ -1, 1, -1 ] ], "source_shift": [ 2, 2, 2 ], "target_matrix": [ [ 1, 1, 2 ], [ -1, -1, -1 ], [ -1, -2, -1 ] ], "target_shift": [ -677, 741, 753 ], "inverse_recipe": { "p": 3, "q": 4, "f": "x**3 + x", "g": "x**4 + x**3 + x**2 + x", "h": "y*z + y", "polynomials": { "f": [ [ [ 3, 0, 0 ], 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573*x^3*y^3*z - 1410*x^3*y^3 + 765*x^3*y^2*z^2 - 3852*x^3*y^2*z + 4572*x^3*y^2 + 447*x^3*y*z^3 - 3474*x^3*y*z^2 + 8373*x^3*y*z - 6463*x^3*y + 96*x^3*z^4 - 1032*x^3*z^3 + 3801*x^3*z^2 - 5928*x^3*z + 3363*x^3 - 75*x^2*y^5 - 342*x^2*y^4*z + 807*x^2*y^4 - 618*x^2*y^3*z^2 + 2964*x^2*y^3*z - 3408*x^2*y^3 - 552*x^2*y^2*z^3 + 4050*x^2*y^2*z^2 - 9420*x^2*y^2*z + 7074*x^2*y^2 - 243*x^2*y*z^4 + 2436*x^2*y*z^3 - 8616*x^2*y*z^2 + 13041*x^2*y*z - 7215*x^2*y - 42*x^2*z^5 + 543*x^2*z^4 - 2604*x^2*z^3 + 5967*x^2*z^2 - 6630*x^2*z + 2887*x^2 + 19*x*y^6 + 105*x*y^5*z - 240*x*y^5 + 240*x*y^4*z^2 - 1110*x*y^4*z + 1242*x*y^4 + 290*x*y^3*z^3 - 2040*x*y^3*z^2 + 4605*x*y^3*z - 3372*x*y^3 + 195*x*y^2*z^4 - 1860*x*y^2*z^3 + 6363*x*y^2*z^2 - 9375*x*y^2*z + 5067*x*y^2 + 69*x*y*z^5 - 840*x*y*z^4 + 3879*x*y*z^3 - 8634*x*y*z^2 + 9363*x*y*z - 3991*x*y + 10*x*z^6 - 150*x*z^5 + 879*x*z^4 - 2631*x*z^3 + 4296*x*z^2 - 3661*x*z + 1276*x - 2*y^7 - 13*y^6*z + 29*y^6 - 36*y^5*z^2 + 162*y^5*z - 177*y^5 - 55*y^4*z^3 + 375*y^4*z^2 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- 981*x*y*z^4 + 5121*x*y*z^3 - 12375*x*y*z^2 + 14223*x*y*z - 6346*x*y + 10*x*z^6 - 174*x*z^5 + 1155*x*z^4 - 3759*x*z^3 + 6510*x*z^2 - 5809*x*z + 2106*x - 2*y^7 - 13*y^6*z + 35*y^6 - 36*y^5*z^2 + 195*y^5*z - 240*y^5 - 55*y^4*z^3 + 450*y^4*z^2 - 1116*y^4*z + 863*y^4 - 50*y^3*z^4 + 550*y^3*z^3 - 2064*y^3*z^2 + 3210*y^3*z - 1794*y^3 - 27*y^2*z^5 + 375*y^2*z^4 - 1896*y^2*z^3 + 4452*y^2*z^2 - 4992*y^2*z + 2179*y^2 - 8*y*z^6 + 135*y*z^5 - 864*y*z^4 + 2726*y*z^3 - 4602*y*z^2 + 4019*y*z - 1437*y - z^7 + 20*z^6 - 156*z^5 + 621*z^4 - 1404*z^3 + 1840*z^2 - 1310*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ 1, -1, -1 ], [ 2, -1, -1 ], [ 1, -2, -1 ] ], "source_shift": [ 2, 1, 2 ], "target_matrix": [ [ 1, 1, 1 ], [ 2, 1, 1 ], [ 1, 2, 1 ] ], "target_shift": [ -228, -312, -397 ], "collision_preimages": [ [ "1", "1/4", "11/4" ], [ "-3/2", "-11/2", "5" ], [ "7/2", "-15/2", "14" ] ], "collision_image": [ "-913/4", "-625/2", "-1589/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 0.9642146969999885 ], "surface_features": [ 3.43705524099582, 4.316222037426815, 3.0637085593914173, 0.0, 3.5553011065698934, 4830 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B070", "pair": 70, "family": "bijection", "mapping": "F(x,y,z)=(-2*x^7 - 14*x^6*y + 28*x^6*z + 24*x^6 - 42*x^5*y^2 + 168*x^5*y*z + 144*x^5*y - 168*x^5*z^2 - 284*x^5*z - 120*x^5 - 70*x^4*y^3 + 420*x^4*y^2*z + 360*x^4*y^2 - 840*x^4*y*z^2 - 1420*x^4*y*z - 598*x^4*y + 560*x^4*z^3 + 1400*x^4*z^2 + 1162*x^4*z + 328*x^4 - 70*x^3*y^4 + 560*x^3*y^3*z + 480*x^3*y^3 - 1680*x^3*y^2*z^2 - 2840*x^3*y^2*z - 1192*x^3*y^2 + 2240*x^3*y*z^3 + 5600*x^3*y*z^2 + 4632*x^3*y*z + 1296*x^3*y - 1120*x^3*z^4 - 3680*x^3*z^3 - 4498*x^3*z^2 - 2472*x^3*z - 522*x^3 - 42*x^2*y^5 + 420*x^2*y^4*z + 360*x^2*y^4 - 1680*x^2*y^3*z^2 - 2840*x^2*y^3*z - 1188*x^2*y^3 + 3360*x^2*y^2*z^3 + 8400*x^2*y^2*z^2 + 6924*x^2*y^2*z + 1920*x^2*y^2 - 3360*x^2*y*z^4 - 11040*x^2*y*z^3 - 13446*x^2*y*z^2 - 7324*x^2*y*z - 1526*x^2*y + 1344*x^2*z^5 + 5440*x^2*z^4 + 8700*x^2*z^3 + 6980*x^2*z^2 + 2836*x^2*z + 454*x^2 - 14*x*y^6 + 168*x*y^5*z + 144*x*y^5 - 840*x*y^4*z^2 - 1420*x*y^4*z - 592*x*y^4 + 2240*x*y^3*z^3 + 5600*x*y^3*z^2 + 4600*x*y^3*z + 1264*x*y^3 - 3360*x*y^2*z^4 - 11040*x*y^2*z^3 - 13398*x*y^2*z^2 - 7232*x*y^2*z - 1486*x*y^2 + 2688*x*y*z^5 + 10880*x*y*z^4 + 17336*x*y*z^3 + 13784*x*y*z^2 + 5528*x*y*z + 876*x*y - 896*x*z^6 - 4288*x*z^5 - 8408*x*z^4 - 8752*x*z^3 - 5144*x*z^2 - 1592*x*z - 201*x - 2*y^7 + 28*y^6*z + 24*y^6 - 168*y^5*z^2 - 284*y^5*z - 118*y^5 + 560*y^4*z^3 + 1400*y^4*z^2 + 1146*y^4*z + 312*y^4 - 1120*y^3*z^4 - 3680*y^3*z^3 - 4450*y^3*z^2 - 2380*y^3*z - 482*y^3 + 1344*y^2*z^5 + 5440*y^2*z^4 + 8636*y^2*z^3 + 6804*y^2*z^2 + 2692*y^2*z + 422*y^2 - 896*y*z^6 - 4288*y*z^5 - 8376*y*z^4 - 8640*y*z^3 - 5014*y*z^2 - 1536*y*z - 191*y + 256*z^7 + 1408*z^6 + 3248*z^5 + 4112*z^4 + 3114*z^3 + 1400*z^2 + 342*z + 1, -2*x^7 - 14*x^6*y + 28*x^6*z + 24*x^6 - 42*x^5*y^2 + 168*x^5*y*z + 144*x^5*y - 168*x^5*z^2 - 284*x^5*z - 120*x^5 - 70*x^4*y^3 + 420*x^4*y^2*z + 360*x^4*y^2 - 840*x^4*y*z^2 - 1420*x^4*y*z - 598*x^4*y + 560*x^4*z^3 + 1400*x^4*z^2 + 1162*x^4*z + 328*x^4 - 70*x^3*y^4 + 560*x^3*y^3*z + 480*x^3*y^3 - 1680*x^3*y^2*z^2 - 2840*x^3*y^2*z - 1192*x^3*y^2 + 2240*x^3*y*z^3 + 5600*x^3*y*z^2 + 4632*x^3*y*z + 1296*x^3*y - 1120*x^3*z^4 - 3680*x^3*z^3 - 4498*x^3*z^2 - 2472*x^3*z - 521*x^3 - 42*x^2*y^5 + 420*x^2*y^4*z + 360*x^2*y^4 - 1680*x^2*y^3*z^2 - 2840*x^2*y^3*z - 1188*x^2*y^3 + 3360*x^2*y^2*z^3 + 8400*x^2*y^2*z^2 + 6924*x^2*y^2*z + 1920*x^2*y^2 - 3360*x^2*y*z^4 - 11040*x^2*y*z^3 - 13446*x^2*y*z^2 - 7324*x^2*y*z - 1523*x^2*y + 1344*x^2*z^5 + 5440*x^2*z^4 + 8700*x^2*z^3 + 6980*x^2*z^2 + 2830*x^2*z + 448*x^2 - 14*x*y^6 + 168*x*y^5*z + 144*x*y^5 - 840*x*y^4*z^2 - 1420*x*y^4*z - 592*x*y^4 + 2240*x*y^3*z^3 + 5600*x*y^3*z^2 + 4600*x*y^3*z + 1264*x*y^3 - 3360*x*y^2*z^4 - 11040*x*y^2*z^3 - 13398*x*y^2*z^2 - 7232*x*y^2*z - 1483*x*y^2 + 2688*x*y*z^5 + 10880*x*y*z^4 + 17336*x*y*z^3 + 13784*x*y*z^2 + 5516*x*y*z + 864*x*y - 896*x*z^6 - 4288*x*z^5 - 8408*x*z^4 - 8752*x*z^3 - 5132*x*z^2 - 1568*x*z - 185*x - 2*y^7 + 28*y^6*z + 24*y^6 - 168*y^5*z^2 - 284*y^5*z - 118*y^5 + 560*y^4*z^3 + 1400*y^4*z^2 + 1146*y^4*z + 312*y^4 - 1120*y^3*z^4 - 3680*y^3*z^3 - 4450*y^3*z^2 - 2380*y^3*z - 481*y^3 + 1344*y^2*z^5 + 5440*y^2*z^4 + 8636*y^2*z^3 + 6804*y^2*z^2 + 2686*y^2*z + 416*y^2 - 896*y*z^6 - 4288*y*z^5 - 8376*y*z^4 - 8640*y*z^3 - 5002*y*z^2 - 1512*y*z - 176*y + 256*z^7 + 1408*z^6 + 3248*z^5 + 4112*z^4 + 3106*z^3 + 1376*z^2 + 315*z + 1, -2*x^7 - 14*x^6*y + 28*x^6*z + 24*x^6 - 42*x^5*y^2 + 168*x^5*y*z + 144*x^5*y - 168*x^5*z^2 - 284*x^5*z - 120*x^5 - 70*x^4*y^3 + 420*x^4*y^2*z + 360*x^4*y^2 - 840*x^4*y*z^2 - 1420*x^4*y*z - 598*x^4*y + 560*x^4*z^3 + 1400*x^4*z^2 + 1162*x^4*z + 328*x^4 - 70*x^3*y^4 + 560*x^3*y^3*z + 480*x^3*y^3 - 1680*x^3*y^2*z^2 - 2840*x^3*y^2*z - 1192*x^3*y^2 + 2240*x^3*y*z^3 + 5600*x^3*y*z^2 + 4632*x^3*y*z + 1296*x^3*y - 1120*x^3*z^4 - 3680*x^3*z^3 - 4498*x^3*z^2 - 2472*x^3*z - 521*x^3 - 42*x^2*y^5 + 420*x^2*y^4*z + 360*x^2*y^4 - 1680*x^2*y^3*z^2 - 2840*x^2*y^3*z - 1188*x^2*y^3 + 3360*x^2*y^2*z^3 + 8400*x^2*y^2*z^2 + 6924*x^2*y^2*z + 1920*x^2*y^2 - 3360*x^2*y*z^4 - 11040*x^2*y*z^3 - 13446*x^2*y*z^2 - 7324*x^2*y*z - 1523*x^2*y + 1344*x^2*z^5 + 5440*x^2*z^4 + 8700*x^2*z^3 + 6980*x^2*z^2 + 2830*x^2*z + 450*x^2 - 14*x*y^6 + 168*x*y^5*z + 144*x*y^5 - 840*x*y^4*z^2 - 1420*x*y^4*z - 592*x*y^4 + 2240*x*y^3*z^3 + 5600*x*y^3*z^2 + 4600*x*y^3*z + 1264*x*y^3 - 3360*x*y^2*z^4 - 11040*x*y^2*z^3 - 13398*x*y^2*z^2 - 7232*x*y^2*z - 1483*x*y^2 + 2688*x*y*z^5 + 10880*x*y*z^4 + 17336*x*y*z^3 + 13784*x*y*z^2 + 5516*x*y*z + 868*x*y - 896*x*z^6 - 4288*x*z^5 - 8408*x*z^4 - 8752*x*z^3 - 5132*x*z^2 - 1576*x*z - 193*x - 2*y^7 + 28*y^6*z + 24*y^6 - 168*y^5*z^2 - 284*y^5*z - 118*y^5 + 560*y^4*z^3 + 1400*y^4*z^2 + 1146*y^4*z + 312*y^4 - 1120*y^3*z^4 - 3680*y^3*z^3 - 4450*y^3*z^2 - 2380*y^3*z - 481*y^3 + 1344*y^2*z^5 + 5440*y^2*z^4 + 8636*y^2*z^3 + 6804*y^2*z^2 + 2686*y^2*z + 418*y^2 - 896*y*z^6 - 4288*y*z^5 - 8376*y*z^4 - 8640*y*z^3 - 5002*y*z^2 - 1520*y*z - 184*y + 256*z^7 + 1408*z^6 + 3248*z^5 + 4112*z^4 + 3106*z^3 + 1384*z^2 + 329*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -1, -1, 2 ], [ -2, -1, 1 ], [ -2, -2, 3 ] ], "source_shift": [ 1, 1, 1 ], "target_matrix": [ [ 1, 2, 1 ], [ 1, 1, 2 ], [ 1, 1, 1 ] ], "target_shift": [ -36, -28, -32 ], "inverse_recipe": { "p": 3, "q": 2, "f": "x**3 + x**2 + x", "g": "x**2", "h": "y*z**2", "polynomials": { "f": [ [ [ 3, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "g": [ [ [ 2, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 2 ], 1 ] ] }, "h_multiplier": 2 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0286542909998388 ], "surface_features": [ 3.463119143019495, 4.238948898311762, 3.146128035678238, 0.0, 3.5481939340842956, 4878 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C106", "pair": 106, "family": "counterexample", "mapping": "F(x,y,z)=(27*x^7 + 243*x^6*y - 162*x^6*z - 351*x^6 + 900*x^5*y^2 - 1233*x^5*y*z - 2619*x^5*y + 414*x^5*z^2 + 1782*x^5*z + 1827*x^5 + 1772*x^4*y^3 - 3750*x^4*y^2*z - 7830*x^4*y^2 + 2589*x^4*y*z^2 + 10926*x^4*y*z + 11109*x^4*y - 584*x^4*z^3 - 3744*x^4*z^2 - 7644*x^4*z - 5072*x^4 + 2000*x^3*y^4 - 5816*x^3*y^3*z - 12008*x^3*y^3 + 6204*x^3*y^2*z^2 + 25752*x^3*y^2*z + 26088*x^3*y^2 - 2879*x^3*y*z^3 - 18099*x^3*y*z^2 - 36654*x^3*y*z - 24364*x^3*y + 491*x^3*z^4 + 4166*x^3*z^3 + 12699*x^3*z^2 + 16813*x^3*z + 8275*x^3 + 1296*x^2*y^5 - 4848*x^2*y^4*z - 9984*x^2*y^4 + 7104*x^2*y^3*z^2 + 29184*x^2*y^3*z + 29592*x^2*y^3 - 5094*x^2*y^2*z^3 - 31518*x^2*y^2*z^2 - 63588*x^2*y^2*z - 42336*x^2*y^2 + 1788*x^2*y*z^4 + 14880*x^2*y*z^3 + 44997*x^2*y*z^2 + 59604*x^2*y*z + 29373*x^2*y - 246*x^2*z^5 - 2589*x^2*z^4 - 10470*x^2*z^3 - 20730*x^2*z^2 - 20346*x^2*z - 7954*x^2 + 448*x*y^6 - 2064*x*y^5*z - 4272*x*y^5 + 3888*x*y^4*z^2 + 15936*x*y^4*z + 16176*x*y^4 - 3828*x*y^3*z^3 - 23460*x*y^3*z^2 - 47304*x*y^3*z - 31408*x*y^3 + 2076*x*y^2*z^4 + 17016*x*y^2*z^3 + 51240*x*y^2*z^2 + 67812*x*y^2*z + 33276*x*y^2 - 588*x*y*z^5 - 6072*x*y*z^4 - 24360*x*y*z^3 - 48180*x*y*z^2 - 47217*x*y*z - 18391*x*y + 68*x*z^6 + 852*x*z^5 + 4284*x*z^4 + 11267*x*z^3 + 16521*x*z^2 + 12860*x*z + 4185*x + 64*y^7 - 352*y^6*z - 736*y^6 + 816*y^5*z^2 + 3360*y^5*z + 3408*y^5 - 1032*y^4*z^3 - 6312*y^4*z^2 - 12720*y^4*z - 8384*y^4 + 768*y^3*z^4 + 6240*y^3*z^3 + 18756*y^3*z^2 + 24704*y^3*z + 11996*y^3 - 336*y^2*z^5 - 3420*y^2*z^4 - 13656*y^2*z^3 - 26928*y^2*z^2 - 26214*y^2*z - 10090*y^2 + 80*y*z^6 + 984*y*z^5 + 4908*y*z^4 + 12874*y*z^3 + 18804*y*z^2 + 14536*y*z + 4667*y - 8*z^7 - 116*z^6 - 696*z^5 - 2278*z^4 - 4432*z^3 - 5146*z^2 - 3324*z - 1, -54*x^7 - 486*x^6*y + 324*x^6*z + 702*x^6 - 1800*x^5*y^2 + 2466*x^5*y*z + 5238*x^5*y - 828*x^5*z^2 - 3564*x^5*z - 3654*x^5 - 3544*x^4*y^3 + 7500*x^4*y^2*z + 15660*x^4*y^2 - 5178*x^4*y*z^2 - 21852*x^4*y*z - 22218*x^4*y + 1168*x^4*z^3 + 7488*x^4*z^2 + 15288*x^4*z + 10171*x^4 - 4000*x^3*y^4 + 11632*x^3*y^3*z + 24016*x^3*y^3 - 12408*x^3*y^2*z^2 - 51504*x^3*y^2*z - 52176*x^3*y^2 + 5758*x^3*y*z^3 + 36198*x^3*y*z^2 + 73308*x^3*y*z + 48809*x^3*y - 982*x^3*z^4 - 8332*x^3*z^3 - 25398*x^3*z^2 - 33707*x^3*z - 16658*x^3 - 2592*x^2*y^5 + 9696*x^2*y^4*z + 19968*x^2*y^4 - 14208*x^2*y^3*z^2 - 58368*x^2*y^3*z - 59184*x^2*y^3 + 10188*x^2*y^2*z^3 + 63036*x^2*y^2*z^2 + 127176*x^2*y^2*z + 84762*x^2*y^2 - 3576*x^2*y*z^4 - 29760*x^2*y*z^3 - 89994*x^2*y*z^2 - 119388*x^2*y*z - 59016*x^2*y + 492*x^2*z^5 + 5178*x^2*z^4 + 20940*x^2*z^3 + 41550*x^2*z^2 + 40935*x^2*z + 16088*x^2 - 896*x*y^6 + 4128*x*y^5*z + 8544*x*y^5 - 7776*x*y^4*z^2 - 31872*x*y^4*z - 32352*x*y^4 + 7656*x*y^3*z^3 + 46920*x*y^3*z^2 + 94608*x*y^3*z + 62860*x*y^3 - 4152*x*y^2*z^4 - 34032*x*y^2*z^3 - 102480*x*y^2*z^2 - 135756*x*y^2*z - 66768*x*y^2 + 1176*x*y*z^5 + 12144*x*y*z^4 + 48720*x*y*z^3 + 96492*x*y*z^2 + 94830*x*y*z + 37082*x*y - 136*x*z^6 - 1704*x*z^5 - 8568*x*z^4 - 22578*x*z^3 - 33222*x*z^2 - 25984*x*z - 8492*x - 128*y^7 + 704*y^6*z + 1472*y^6 - 1632*y^5*z^2 - 6720*y^5*z - 6816*y^5 + 2064*y^4*z^3 + 12624*y^4*z^2 + 25440*y^4*z + 16776*y^4 - 1536*y^3*z^4 - 12480*y^3*z^3 - 37512*y^3*z^2 - 49440*y^3*z - 24048*y^3 + 672*y^2*z^5 + 6840*y^2*z^4 + 27312*y^2*z^3 + 53904*y^2*z^2 + 52584*y^2*z + 20300*y^2 - 160*y*z^6 - 1968*y*z^5 - 9816*y*z^4 - 25780*y*z^3 - 37752*y*z^2 - 29288*y*z - 9434*y + 16*z^7 + 232*z^6 + 1392*z^5 + 4564*z^4 + 8908*z^3 + 10388*z^2 + 6736*z + 1, 27*x^7 + 243*x^6*y - 162*x^6*z - 432*x^6 + 900*x^5*y^2 - 1233*x^5*y*z - 3186*x^5*y + 414*x^5*z^2 + 2187*x^5*z + 2475*x^5 + 1772*x^4*y^3 - 3750*x^4*y^2*z - 9396*x^4*y^2 + 2589*x^4*y*z^2 + 13248*x^4*y*z + 14835*x^4*y - 584*x^4*z^3 - 4581*x^4*z^2 - 10317*x^4*z - 7313*x^4 + 2000*x^3*y^4 - 5816*x^3*y^3*z - 14192*x^3*y^3 + 6204*x^3*y^2*z^2 + 30792*x^3*y^2*z + 34296*x^3*y^2 - 2879*x^3*y*z^3 - 21870*x^3*y*z^2 - 48750*x^3*y*z - 34543*x^3*y + 491*x^3*z^4 + 5081*x^3*z^3 + 17073*x^3*z^2 + 24130*x^3*z + 12493*x^3 + 1296*x^2*y^5 - 4848*x^2*y^4*z - 11616*x^2*y^4 + 7104*x^2*y^3*z^2 + 34368*x^2*y^3*z + 38280*x^2*y^3 - 5094*x^2*y^2*z^3 - 37548*x^2*y^2*z^2 - 83244*x^2*y^2*z - 59046*x^2*y^2 + 1788*x^2*y*z^4 + 17916*x^2*y*z^3 + 59595*x^2*y*z^2 + 84114*x^2*y*z + 43515*x^2*y - 246*x^2*z^5 - 3147*x^2*z^4 - 14019*x^2*z^3 - 29610*x^2*z^2 - 30543*x^2*z - 12451*x^2 + 448*x*y^6 - 2064*x*y^5*z - 4896*x*y^5 + 3888*x*y^4*z^2 + 18480*x*y^4*z + 20592*x*y^4 - 3828*x*y^3*z^3 - 27528*x*y^3*z^2 - 60936*x*y^3*z - 43092*x*y^3 + 2076*x*y^2*z^4 + 20196*x*y^2*z^3 + 66792*x*y^2*z^2 + 94128*x*y^2*z + 48396*x*y^2 - 588*x*y*z^5 - 7284*x*y*z^4 - 32124*x*y*z^3 - 67668*x*y*z^2 - 69549*x*y*z - 28138*x*y + 68*x*z^6 + 1032*x*z^5 + 5712*x*z^4 + 16015*x*z^3 + 24657*x*z^2 + 19979*x*z + 6699*x + 64*y^7 - 352*y^6*z - 832*y^6 + 816*y^5*z^2 + 3840*y^5*z + 4272*y^5 - 1032*y^4*z^3 - 7296*y^4*z^2 - 16128*y^4*z - 11320*y^4 + 768*y^3*z^4 + 7296*y^3*z^3 + 24060*y^3*z^2 + 33736*y^3*z + 17140*y^3 - 336*y^2*z^5 - 4044*y^2*z^4 - 17724*y^2*z^3 - 37200*y^2*z^2 - 37914*y^2*z - 15112*y^2 + 80*y*z^6 + 1176*y*z^5 + 6444*y*z^4 + 17994*y*z^3 + 27540*y*z^2 + 22084*y*z + 7277*y - 8*z^7 - 140*z^6 - 924*z^5 - 3222*z^4 - 6576*z^3 - 7936*z^2 - 5273*z - 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -3, -2, 2 ], [ -1, -2, 1 ], [ -1, -1, 1 ] ], "source_shift": [ 2, 2, 1 ], "target_matrix": [ [ -1, -1, -2 ], [ 2, 2, 3 ], [ -1, -2, -1 ] ], "target_shift": [ 924, -1877, 1488 ], "collision_preimages": [ [ "-1/2", "3/4", "-1" ], [ "12", "9", "53/2" ], [ "14", "6", "51/2" ] ], "collision_image": [ "3697/4", "-3755/2", "5953/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0831870720030565 ], "surface_features": [ 4.24896619587205, 5.132759033172907, 3.900093901543398, 0.0, 4.361672310073926, 5134 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B029", "pair": 29, "family": "bijection", "mapping": "F(x,y,z)=(-3*x^7 - 41*x^6*y + 21*x^6*z + 22*x^6 - 240*x^5*y^2 + 246*x^5*y*z + 258*x^5*y - 63*x^5*z^2 - 132*x^5*z - 90*x^5 - 780*x^4*y^3 + 1200*x^4*y^2*z + 1260*x^4*y^2 - 615*x^4*y*z^2 - 1290*x^4*y*z - 869*x^4*y + 105*x^4*z^3 + 330*x^4*z^2 + 456*x^4*z + 232*x^4 - 1520*x^3*y^4 + 3120*x^3*y^3*z + 3280*x^3*y^3 - 2400*x^3*y^2*z^2 - 5040*x^3*y^2*z - 3355*x^3*y^2 + 820*x^3*y*z^3 + 2580*x^3*y*z^2 + 3522*x^3*y*z + 1769*x^3*y - 105*x^3*z^4 - 440*x^3*z^3 - 924*x^3*z^2 - 954*x^3*z - 394*x^3 - 1776*x^2*y^5 + 4560*x^2*y^4*z + 4800*x^2*y^4 - 4680*x^2*y^3*z^2 - 9840*x^2*y^3*z - 6474*x^2*y^3 + 2400*x^2*y^2*z^3 + 7560*x^2*y^2*z^2 + 10197*x^2*y^2*z + 5055*x^2*y^2 - 615*x^2*y*z^4 - 2580*x^2*y*z^3 - 5352*x^2*y*z^2 - 5457*x^2*y*z - 2207*x^2*y + 63*x^2*z^5 + 330*x^2*z^4 + 936*x^2*z^3 + 1470*x^2*z^2 + 1245*x^2*z + 433*x^2 - 1152*x*y^6 + 3552*x*y^5*z + 3744*x*y^5 - 4560*x*y^4*z^2 - 9600*x*y^4*z - 6244*x*y^4 + 3120*x*y^3*z^3 + 9840*x*y^3*z^2 + 13116*x*y^3*z + 6416*x*y^3 - 1200*x*y^2*z^4 - 5040*x*y^2*z^3 - 10329*x*y^2*z^2 - 10398*x*y^2*z - 4124*x*y^2 + 246*x*y*z^5 + 1290*x*y*z^4 + 3614*x*y*z^3 + 5607*x*y*z^2 + 4644*x*y*z + 1570*x*y - 21*x*z^6 - 132*x*z^5 - 474*x*z^4 - 1006*x*z^3 - 1311*x*z^2 - 945*x*z - 264*x - 320*y^7 + 1152*y^6*z + 1216*y^6 - 1776*y^5*z^2 - 3744*y^5*z - 2408*y^5 + 1520*y^4*z^3 + 4800*y^4*z^2 + 6324*y^4*z + 3052*y^4 - 780*y^3*z^4 - 3280*y^3*z^3 - 6642*y^3*z^2 - 6600*y^3*z - 2570*y^3 + 240*y^2*z^5 + 1260*y^2*z^4 + 3487*y^2*z^3 + 5343*y^2*z^2 + 4335*y^2*z + 1428*y^2 - 41*y*z^6 - 258*y*z^5 - 915*y*z^4 - 1919*y*z^3 - 2442*y*z^2 - 1707*y*z - 465*y + 3*z^7 + 22*z^6 + 96*z^5 + 258*z^4 + 460*z^3 + 516*z^2 + 301*z + 1, -6*x^7 - 82*x^6*y + 42*x^6*z + 44*x^6 - 480*x^5*y^2 + 492*x^5*y*z + 516*x^5*y - 126*x^5*z^2 - 264*x^5*z - 180*x^5 - 1560*x^4*y^3 + 2400*x^4*y^2*z + 2520*x^4*y^2 - 1230*x^4*y*z^2 - 2580*x^4*y*z - 1738*x^4*y + 210*x^4*z^3 + 660*x^4*z^2 + 912*x^4*z + 464*x^4 - 3040*x^3*y^4 + 6240*x^3*y^3*z + 6560*x^3*y^3 - 4800*x^3*y^2*z^2 - 10080*x^3*y^2*z - 6710*x^3*y^2 + 1640*x^3*y*z^3 + 5160*x^3*y*z^2 + 7044*x^3*y*z + 3538*x^3*y - 210*x^3*z^4 - 880*x^3*z^3 - 1848*x^3*z^2 - 1908*x^3*z - 787*x^3 - 3552*x^2*y^5 + 9120*x^2*y^4*z + 9600*x^2*y^4 - 9360*x^2*y^3*z^2 - 19680*x^2*y^3*z - 12948*x^2*y^3 + 4800*x^2*y^2*z^3 + 15120*x^2*y^2*z^2 + 20394*x^2*y^2*z + 10110*x^2*y^2 - 1230*x^2*y*z^4 - 5160*x^2*y*z^3 - 10704*x^2*y*z^2 - 10914*x^2*y*z - 4408*x^2*y + 126*x^2*z^5 + 660*x^2*z^4 + 1872*x^2*z^3 + 2940*x^2*z^2 + 2487*x^2*z + 863*x^2 - 2304*x*y^6 + 7104*x*y^5*z + 7488*x*y^5 - 9120*x*y^4*z^2 - 19200*x*y^4*z - 12488*x*y^4 + 6240*x*y^3*z^3 + 19680*x*y^3*z^2 + 26232*x*y^3*z + 12832*x*y^3 - 2400*x*y^2*z^4 - 10080*x*y^2*z^3 - 20658*x*y^2*z^2 - 20796*x*y^2*z - 8236*x*y^2 + 492*x*y*z^5 + 2580*x*y*z^4 + 7228*x*y*z^3 + 11214*x*y*z^2 + 9276*x*y*z + 3128*x*y - 42*x*z^6 - 264*x*z^5 - 948*x*z^4 - 2012*x*z^3 - 2619*x*z^2 - 1884*x*z - 541*x - 640*y^7 + 2304*y^6*z + 2432*y^6 - 3552*y^5*z^2 - 7488*y^5*z - 4816*y^5 + 3040*y^4*z^3 + 9600*y^4*z^2 + 12648*y^4*z + 6104*y^4 - 1560*y^3*z^4 - 6560*y^3*z^3 - 13284*y^3*z^2 - 13200*y^3*z - 5132*y^3 + 480*y^2*z^5 + 2520*y^2*z^4 + 6974*y^2*z^3 + 10686*y^2*z^2 + 8658*y^2*z + 2844*y^2 - 82*y*z^6 - 516*y*z^5 - 1830*y*z^4 - 3838*y*z^3 - 4878*y*z^2 - 3402*y*z - 951*y + 6*z^7 + 44*z^6 + 192*z^5 + 516*z^4 + 919*z^3 + 1029*z^2 + 614*z + 1, -3*x^7 - 41*x^6*y + 21*x^6*z + 22*x^6 - 240*x^5*y^2 + 246*x^5*y*z + 258*x^5*y - 63*x^5*z^2 - 132*x^5*z - 90*x^5 - 780*x^4*y^3 + 1200*x^4*y^2*z + 1260*x^4*y^2 - 615*x^4*y*z^2 - 1290*x^4*y*z - 869*x^4*y + 105*x^4*z^3 + 330*x^4*z^2 + 456*x^4*z + 232*x^4 - 1520*x^3*y^4 + 3120*x^3*y^3*z + 3280*x^3*y^3 - 2400*x^3*y^2*z^2 - 5040*x^3*y^2*z - 3355*x^3*y^2 + 820*x^3*y*z^3 + 2580*x^3*y*z^2 + 3522*x^3*y*z + 1769*x^3*y - 105*x^3*z^4 - 440*x^3*z^3 - 924*x^3*z^2 - 954*x^3*z - 394*x^3 - 1776*x^2*y^5 + 4560*x^2*y^4*z + 4800*x^2*y^4 - 4680*x^2*y^3*z^2 - 9840*x^2*y^3*z - 6474*x^2*y^3 + 2400*x^2*y^2*z^3 + 7560*x^2*y^2*z^2 + 10197*x^2*y^2*z + 5055*x^2*y^2 - 615*x^2*y*z^4 - 2580*x^2*y*z^3 - 5352*x^2*y*z^2 - 5457*x^2*y*z - 2207*x^2*y + 63*x^2*z^5 + 330*x^2*z^4 + 936*x^2*z^3 + 1470*x^2*z^2 + 1245*x^2*z + 433*x^2 - 1152*x*y^6 + 3552*x*y^5*z + 3744*x*y^5 - 4560*x*y^4*z^2 - 9600*x*y^4*z - 6244*x*y^4 + 3120*x*y^3*z^3 + 9840*x*y^3*z^2 + 13116*x*y^3*z + 6416*x*y^3 - 1200*x*y^2*z^4 - 5040*x*y^2*z^3 - 10329*x*y^2*z^2 - 10398*x*y^2*z - 4124*x*y^2 + 246*x*y*z^5 + 1290*x*y*z^4 + 3614*x*y*z^3 + 5607*x*y*z^2 + 4644*x*y*z + 1570*x*y - 21*x*z^6 - 132*x*z^5 - 474*x*z^4 - 1006*x*z^3 - 1311*x*z^2 - 945*x*z - 270*x - 320*y^7 + 1152*y^6*z + 1216*y^6 - 1776*y^5*z^2 - 3744*y^5*z - 2408*y^5 + 1520*y^4*z^3 + 4800*y^4*z^2 + 6324*y^4*z + 3052*y^4 - 780*y^3*z^4 - 3280*y^3*z^3 - 6642*y^3*z^2 - 6600*y^3*z - 2570*y^3 + 240*y^2*z^5 + 1260*y^2*z^4 + 3487*y^2*z^3 + 5343*y^2*z^2 + 4335*y^2*z + 1428*y^2 - 41*y*z^6 - 258*y*z^5 - 915*y*z^4 - 1919*y*z^3 - 2442*y*z^2 - 1707*y*z - 475*y + 3*z^7 + 22*z^6 + 96*z^5 + 258*z^4 + 460*z^3 + 516*z^2 + 307*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -1, -2, 1 ], [ -1, -1, 2 ], [ -2, -3, 2 ] ], "source_shift": [ 1, 1, 2 ], "target_matrix": [ [ 1, 1, 2 ], [ 2, 1, 1 ], [ 1, 1, 1 ] ], "target_shift": [ -63, -142, -69 ], "inverse_recipe": { "p": 3, "q": 1, "f": "x**3 + x", "g": "x", "h": "y**2*z + y**2 + y*z**2 + y*z", "polynomials": { "f": [ [ [ 3, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "g": [ [ [ 1, 0, 0 ], 1 ] ], "h": [ [ [ 0, 2, 1 ], 1 ], [ [ 0, 2, 0 ], 1 ], [ [ 0, 1, 2 ], 1 ], [ [ 0, 1, 1 ], 1 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 0.9924933340007556 ], "surface_features": [ 3.52538282017277, 4.418831403661309, 3.248586243855104, 0.0, 3.6140823886450857, 4899 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C166", "pair": 166, "family": "counterexample", "mapping": "F(x,y,z)=(-54*x^7 - 351*x^6*y + 297*x^6*z + 486*x^6 - 972*x^5*y^2 + 1647*x^5*y*z + 2700*x^5*y - 693*x^5*z^2 - 2259*x^5*z - 1701*x^5 - 1485*x^4*y^3 + 3780*x^4*y^2*z + 6210*x^4*y^2 - 3186*x^4*y*z^2 - 10413*x^4*y*z - 7875*x^4*y + 889*x^4*z^3 + 4335*x^4*z^2 + 6561*x^4*z + 3190*x^4 - 1350*x^3*y^4 + 4590*x^3*y^3*z + 7560*x^3*y^3 - 5814*x^3*y^2*z^2 - 19062*x^3*y^2*z - 14490*x^3*y^2 + 3251*x^3*y*z^3 + 15912*x^3*y*z^2 + 24213*x^3*y*z + 11825*x^3*y - 677*x^3*z^4 - 4397*x^3*z^3 - 10044*x^3*z^2 - 9853*x^3*z - 3556*x^3 - 729*x^2*y^5 + 3105*x^2*y^4*z + 5130*x^2*y^4 - 5256*x^2*y^3*z^2 - 17298*x^2*y^3*z - 13230*x^2*y^3 + 4419*x^2*y^2*z^3 + 21726*x^2*y^2*z^2 + 33273*x^2*y^2*z + 16335*x^2*y^2 - 1845*x^2*y*z^4 - 12045*x^2*y*z^3 - 27699*x^2*y*z^2 - 27315*x^2*y*z - 9888*x^2*y + 306*x^2*z^5 + 2487*x^2*z^4 + 7632*x^2*z^3 + 11331*x^2*z^2 + 8256*x^2*z + 2398*x^2 - 216*x*y^6 + 1107*x*y^5*z + 1836*x*y^5 - 2349*x*y^4*z^2 - 7767*x*y^4*z - 5985*x*y^4 + 2641*x*y^3*z^3 + 13056*x*y^3*z^2 + 20151*x*y^3*z + 9955*x*y^3 - 1659*x*y^2*z^4 - 10899*x*y^2*z^3 - 25266*x*y^2*z^2 - 25071*x*y^2*z - 9108*x*y^2 + 552*x*y*z^5 + 4518*x*y*z^4 + 13980*x*y*z^3 + 20886*x*y*z^2 + 15273*x*y*z + 4436*x*y - 76*x*z^6 - 744*x*z^5 - 2880*x*z^4 - 5755*x*z^3 - 6348*x*z^2 - 3716*x*z - 920*x - 27*y^7 + 162*y^6*z + 270*y^6 - 414*y^5*z^2 - 1377*y^5*z - 1071*y^5 + 584*y^4*z^3 + 2907*y^4*z^2 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23760*x^2*y^2 + 3690*x^2*y*z^4 + 20541*x^2*y*z^3 + 42942*x^2*y*z^2 + 39924*x^2*y*z + 13950*x^2*y - 612*x^2*z^5 - 4260*x^2*z^4 - 11895*x^2*z^3 - 16647*x^2*z^2 - 11700*x^2*z - 3326*x^2 + 432*x*y^6 - 2214*x*y^5*z - 3105*x*y^5 + 4698*x*y^4*z^2 + 13185*x*y^4*z + 9216*x*y^4 - 5282*x*y^3*z^3 - 22251*x*y^3*z^2 - 31176*x*y^3*z - 14510*x*y^3 + 3318*x*y^2*z^4 + 18651*x*y^2*z^3 + 39282*x*y^2*z^2 + 36714*x*y^2*z + 12864*x*y^2 - 1104*x*y*z^5 - 7764*x*y*z^4 - 21846*x*y*z^3 - 30738*x*y*z^2 - 21666*x*y*z - 6157*x*y + 152*x*z^6 + 1284*x*z^5 + 4524*x*z^4 + 8514*x*z^3 + 9048*x*z^2 + 5179*x*z + 1264*x + 54*y^7 - 324*y^6*z - 459*y^6 + 828*y^5*z^2 + 2349*y^5*z + 1656*y^5 - 1168*y^4*z^3 - 4977*y^4*z^2 - 7035*y^4*z - 3295*y^4 + 982*y^3*z^4 + 5587*y^3*z^3 + 11874*y^3*z^2 + 11168*y^3*z + 3926*y^3 - 492*y^2*z^5 - 3504*y^2*z^4 - 9951*y^2*z^3 - 14091*y^2*z^2 - 9966*y^2*z - 2831*y^2 + 136*y*z^6 + 1164*y*z^5 + 4140*y*z^4 + 7842*y*z^3 + 8364*y*z^2 + 4789*y*z + 1164*y - 16*z^7 - 160*z^6 - 684*z^5 - 1624*z^4 - 2320*z^3 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39168*x*y^3*z^2 - 60453*x*y^3*z - 30000*x*y^3 + 4977*x*y^2*z^4 + 32697*x*y^2*z^3 + 75798*x*y^2*z^2 + 75510*x*y^2*z + 27702*x*y^2 - 1656*x*y*z^5 - 13554*x*y*z^4 - 41940*x*y*z^3 - 62874*x*y*z^2 - 46377*x*y*z - 13650*x*y + 228*x*z^6 + 2232*x*z^5 + 8640*x*z^4 + 17317*x*z^3 + 19248*x*z^2 + 11400*x*z + 2854*x + 81*y^7 - 486*y^6*z - 810*y^6 + 1242*y^5*z^2 + 4131*y^5*z + 3213*y^5 - 1752*y^4*z^3 - 8721*y^4*z^2 - 13590*y^4*z - 6792*y^4 + 1473*y^3*z^4 + 9753*y^3*z^3 + 22833*y^3*z^2 + 22908*y^3*z + 8436*y^3 - 738*y^2*z^5 - 6093*y^2*z^4 - 19044*y^2*z^3 - 28755*y^2*z^2 - 21294*y^2*z - 6267*y^2 + 204*y*z^6 + 2016*y*z^5 + 7884*y*z^4 + 15917*y*z^3 + 17766*y*z^2 + 10530*y*z + 2627*y - 24*z^7 - 276*z^6 - 1296*z^5 - 3278*z^4 - 4898*z^3 - 4380*z^2 - 2200*z - 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -3, -3, 2 ], [ -1, -1, 1 ], [ -2, -1, 1 ] ], "source_shift": [ 2, 1, 1 ], "target_matrix": [ [ 1, 1, 1 ], [ -2, -1, -1 ], [ -3, -3, -2 ] ], "target_shift": [ -155, 212, 483 ], "collision_preimages": [ [ "1/4", "-1/4", "-1" ], [ "-8", "4", "-13/2" ], [ "-5", "9", "9/2" ] ], "collision_image": [ "-621/4", "425/2", "1935/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 0.9787567739986116 ], "surface_features": [ 4.0552608639798065, 4.9992132147320945, 3.762265825409539, 0.0, 4.181251068482627, 5091 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C010", "pair": 10, "family": "counterexample", "mapping": "F(x,y,z)=(-x^7 + 10*x^6*y - 8*x^6*z + 13*x^6 - 42*x^5*y^2 + 69*x^5*y*z - 114*x^5*y - 27*x^5*z^2 + 87*x^5*z - 69*x^5 + 96*x^4*y^3 - 243*x^4*y^2*z + 411*x^4*y^2 + 195*x^4*y*z^2 - 636*x^4*y*z + 513*x^4*y - 50*x^4*z^3 + 240*x^4*z^2 - 378*x^4*z + 196*x^4 - 129*x^3*y^4 + 447*x^3*y^3*z - 780*x^3*y^3 - 552*x^3*y^2*z^2 + 1833*x^3*y^2*z - 1506*x^3*y^2 + 290*x^3*y*z^3 - 1404*x^3*y*z^2 + 2244*x^3*y*z - 1183*x^3*y - 55*x^3*z^4 + 350*x^3*z^3 - 822*x^3*z^2 + 850*x^3*z - 328*x^3 + 102*x^2*y^5 - 453*x^2*y^4*z + 822*x^2*y^4 + 765*x^2*y^3*z^2 - 2604*x^2*y^3*z + 2181*x^2*y^3 - 618*x^2*y^2*z^3 + 3033*x^2*y^2*z^2 - 4926*x^2*y^2*z + 2640*x^2*y^2 + 240*x^2*y*z^4 - 1536*x^2*y*z^3 + 3654*x^2*y*z^2 - 3837*x^2*y*z + 1509*x^2*y - 36*x^2*z^5 + 285*x^2*z^4 - 888*x^2*z^3 + 1374*x^2*z^2 - 1062*x^2*z + 331*x^2 - 44*x*y^6 + 240*x*y^5*z - 456*x*y^5 - 519*x*y^4*z^2 + 1824*x*y^4*z - 1557*x*y^4 + 573*x*y^3*z^3 - 2868*x*y^3*z^2 + 4737*x*y^3*z - 2580*x*y^3 - 342*x*y^2*z^4 + 2211*x*y^2*z^3 - 5334*x*y^2*z^2 + 5685*x*y^2*z - 2277*x*y^2 + 105*x*y*z^5 - 834*x*y*z^4 + 2628*x*y*z^3 - 4125*x*y*z^2 + 3246*x*y*z - 1033*x*y - 13*x*z^6 + 123*x*z^5 - 477*x*z^4 + 982*x*z^3 - 1140*x*z^2 + 716*x*z - 194*x + 8*y^7 - 52*y^6*z + 104*y^6 + 138*y^5*z^2 - 504*y^5*z + 438*y^5 - 195*y^4*z^3 + 1002*y^4*z^2 - 1683*y^4*z + 930*y^4 + 159*y^3*z^4 - 1044*y^3*z^3 + 2556*y^3*z^2 - 2764*y^3*z + 1125*y^3 - 75*y^2*z^5 + 600*y^2*z^4 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1501*z^4 + 1559*z^3 + 970*z^2 + 349*z + 1, -64*x^7 - 352*x^6*y - 448*x^6*z - 544*x^6 - 816*x^5*y^2 - 2064*x^5*y*z - 2448*x^5*y - 1296*x^5*z^2 - 3120*x^5*z - 1680*x^5 - 1032*x^4*y^3 - 3888*x^4*y^2*z - 4536*x^4*y^2 - 4848*x^4*y*z^2 - 11472*x^4*y*z - 6192*x^4*y - 2000*x^4*z^3 - 7200*x^4*z^2 - 7920*x^4*z - 2608*x^4 - 768*x^3*y^4 - 3828*x^3*y^3*z - 4428*x^3*y^3 - 7104*x^3*y^2*z^2 - 16668*x^3*y^2*z - 9000*x^3*y^2 - 5816*x^3*y*z^3 - 20760*x^3*y*z^2 - 22800*x^3*y*z - 7600*x^3*y - 1772*x^3*z^4 - 8552*x^3*z^3 - 14328*x^3*z^2 - 9720*x^3*z - 2312*x^3 - 336*x^2*y^5 - 2076*x^2*y^4*z - 2400*x^2*y^4 - 5094*x^2*y^3*z^2 - 11952*x^2*y^3*z - 6450*x^2*y^3 - 6204*x^2*y^2*z^3 - 22158*x^2*y^2*z^2 - 24276*x^2*y^2*z - 8166*x^2*y^2 - 3750*x^2*y*z^4 - 18120*x^2*y*z^3 - 30216*x^2*y*z^2 - 20652*x^2*y*z - 4980*x^2*y - 900*x^2*z^5 - 5514*x^2*z^4 - 12432*x^2*z^3 - 12948*x^2*z^2 - 6336*x^2*z - 1216*x^2 - 80*x*y^6 - 588*x*y^5*z - 684*x*y^5 - 1788*x*y^4*z^2 - 4224*x*y^4*z - 2280*x*y^4 - 2879*x*y^3*z^3 - 10359*x*y^3*z^2 - 11337*x*y^3*z - 3835*x*y^3 - 2589*x*y^2*z^4 - 12609*x*y^2*z^3 - 20976*x*y^2*z^2 - 14385*x*y^2*z - 3501*x*y^2 - 1233*x*y*z^5 - 7617*x*y*z^4 - 17112*x*y*z^3 - 17832*x*y*z^2 - 8784*x*y*z - 1714*x*y - 243*x*z^6 - 1827*x*z^5 - 5193*x*z^4 - 7302*x*z^3 - 5454*x*z^2 - 2158*x*z - 370*x - 8*y^7 - 68*y^6*z - 80*y^6 - 246*y^5*z^2 - 588*y^5*z - 318*y^5 - 491*y^4*z^3 - 1788*y^4*z^2 - 1959*y^4*z - 664*y^4 - 584*y^3*z^4 - 2879*y^3*z^3 - 4791*y^3*z^2 - 3284*y^3*z - 804*y^3 - 414*y^2*z^5 - 2589*y^2*z^4 - 5814*y^2*z^3 - 6039*y^2*z^2 - 2985*y^2*z - 589*y^2 - 162*y*z^6 - 1233*y*z^5 - 3501*y*z^4 - 4893*y*z^3 - 3657*y*z^2 - 1457*y*z - 259*y - 27*z^7 - 243*z^6 - 837*z^5 - 1474*z^4 - 1478*z^3 - 889*z^2 - 324*z - 1, -128*x^7 - 704*x^6*y - 896*x^6*z - 992*x^6 - 1632*x^5*y^2 - 4128*x^5*y*z - 4416*x^5*y - 2592*x^5*z^2 - 5616*x^5*z - 2736*x^5 - 2064*x^4*y^3 - 7776*x^4*y^2*z - 8088*x^4*y^2 - 9696*x^4*y*z^2 - 20400*x^4*y*z - 9984*x^4*y - 4000*x^4*z^3 - 12768*x^4*z^2 - 12720*x^4*z - 3992*x^4 - 1536*x^3*y^4 - 7656*x^3*y^3*z - 7800*x^3*y^3 - 14208*x^3*y^2*z^2 - 29268*x^3*y^2*z - 14364*x^3*y^2 - 11632*x^3*y*z^3 - 36336*x^3*y*z^2 - 36240*x^3*y*z - 11528*x^3*y - 3544*x^3*z^4 - 14920*x^3*z^3 - 22680*x^3*z^2 - 14676*x^3*z - 3460*x^3 - 672*x^2*y^5 - 4152*x^2*y^4*z - 4176*x^2*y^4 - 10188*x^2*y^3*z^2 - 20724*x^2*y^3*z - 10188*x^2*y^3 - 12408*x^2*y^2*z^3 - 38286*x^2*y^2*z^2 - 38184*x^2*y^2*z - 12282*x^2*y^2 - 7500*x^2*y*z^4 - 31200*x^2*y*z^3 - 47328*x^2*y*z^2 - 30924*x^2*y*z - 7416*x^2*y - 1800*x^2*z^5 - 9462*x^2*z^4 - 19392*x^2*z^3 - 19302*x^2*z^2 - 9408*x^2*z - 1826*x^2 - 160*x*y^6 - 1176*x*y^5*z - 1176*x*y^5 - 3576*x*y^4*z^2 - 7236*x*y^4*z - 3564*x*y^4 - 5758*x*y^3*z^3 - 17682*x*y^3*z^2 - 17646*x*y^3*z - 5726*x*y^3 - 5178*x*y^2*z^4 - 21447*x*y^2*z^3 - 32511*x*y^2*z^2 - 21393*x*y^2*z - 5199*x*y^2 - 2466*x*y*z^5 - 12912*x*y*z^4 - 26412*x*y*z^3 - 26418*x*y*z^2 - 13020*x*y*z - 2576*x*y - 486*x*z^6 - 3087*x*z^5 - 7983*x*z^4 - 10779*x*z^3 - 8073*x*z^2 - 3245*x*z - 555*x - 16*y^7 - 136*y^6*z - 136*y^6 - 492*y^5*z^2 - 996*y^5*z - 492*y^5 - 982*y^4*z^3 - 3018*y^4*z^2 - 3018*y^4*z - 986*y^4 - 1168*y^3*z^4 - 4843*y^3*z^3 - 7350*y^3*z^2 - 4861*y^3*z - 1194*y^3 - 828*y^2*z^5 - 4341*y^2*z^4 - 8883*y^2*z^3 - 8913*y^2*z^2 - 4431*y^2*z - 890*y^2 - 324*y*z^6 - 2061*y*z^5 - 5328*y*z^4 - 7203*y*z^3 - 5430*y*z^2 - 2209*y*z - 388*y - 54*z^7 - 405*z^6 - 1269*z^5 - 2165*z^4 - 2197*z^3 - 1355*z^2 - 485*z - 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ 2, 2, 3 ], [ 2, 1, 1 ], [ 1, 1, 1 ] ], "source_shift": [ 1, 1, 1 ], "target_matrix": [ [ 1, 1, 1 ], [ -1, -1, -2 ], [ -2, -1, -1 ] ], "target_shift": [ -53, 51, 75 ], "collision_preimages": [ [ "1/4", "-3", "3/2" ], [ "-8", "49/2", "-11" ], [ "-5", "47/2", "-13" ] ], "collision_image": [ "-213/4", "205/4", "151/2" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0799609249988862 ], "surface_features": [ 3.8066193240429165, 4.675118152316799, 3.550717423469283, 0.0, 3.8927358585024563, 5011 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B233", "pair": 233, "family": "bijection", "mapping": "F(x,y,z)=(32*x^7 - 256*x^6*y - 224*x^6*z + 256*x^6 + 872*x^5*y^2 + 1536*x^5*y*z - 1744*x^5*y + 672*x^5*z^2 - 1536*x^5*z + 1130*x^5 - 1640*x^4*y^3 - 4360*x^4*y^2*z + 4920*x^4*y^2 - 3840*x^4*y*z^2 + 8720*x^4*y*z - 6850*x^4*y - 1120*x^4*z^3 + 3840*x^4*z^2 - 5650*x^4*z + 3602*x^4 + 1840*x^3*y^4 + 6560*x^3*y^3*z - 7360*x^3*y^3 + 8720*x^3*y^2*z^2 - 19680*x^3*y^2*z + 16820*x^3*y^2 + 5120*x^3*y*z^3 - 17440*x^3*y*z^2 + 27400*x^3*y*z - 19080*x^3*y + 1120*x^3*z^4 - 5120*x^3*z^3 + 11300*x^3*z^2 - 14408*x^3*z + 7972*x^3 - 1232*x^2*y^5 - 5520*x^2*y^4*z + 6160*x^2*y^4 - 9840*x^2*y^3*z^2 + 22080*x^2*y^3*z - 20980*x^2*y^3 - 8720*x^2*y^2*z^3 + 29520*x^2*y^2*z^2 - 50460*x^2*y^2*z + 38596*x^2*y^2 - 3840*x^2*y*z^4 + 17440*x^2*y*z^3 - 41100*x^2*y*z^2 + 57240*x^2*y*z - 33532*x^2*y - 672*x^2*z^5 + 3840*x^2*z^4 - 11300*x^2*z^3 + 21612*x^2*z^2 - 23884*x^2*z + 10926*x^2 + 456*x*y^6 + 2464*x*y^5*z - 2736*x*y^5 + 5520*x*y^4*z^2 - 12320*x*y^4*z + 13330*x*y^4 + 6560*x*y^3*z^3 - 22080*x*y^3*z^2 + 41960*x*y^3*z - 35320*x*y^3 + 4360*x*y^2*z^4 - 19680*x*y^2*z^3 + 50460*x*y^2*z^2 - 77192*x*y^2*z + 47604*x*y^2 + 1536*x*y*z^5 - 8720*x*y*z^4 + 27400*x*y*z^3 - 57240*x*y*z^2 + 66968*x*y*z - 31436*x*y + 224*x*z^6 - 1536*x*z^5 + 5650*x*z^4 - 14408*x*z^3 + 23852*x*z^2 - 21756*x*z + 8116*x - 72*y^7 - 456*y^6*z + 504*y^6 - 1232*y^5*z^2 + 2736*y^5*z - 3458*y^5 - 1840*y^4*z^3 + 6160*y^4*z^2 - 13330*y^4*z + 12322*y^4 - 1640*y^3*z^4 + 7360*y^3*z^3 - 20980*y^3*z^2 + 35320*y^3*z - 22772*y^3 - 872*y^2*z^5 + 4920*y^2*z^4 - 16820*y^2*z^3 + 38596*y^2*z^2 - 47532*y^2*z + 22774*y^2 - 256*y*z^6 + 1744*y*z^5 - 6850*y*z^4 + 19080*y*z^3 - 33436*y*z^2 + 31292*y*z - 11766*y - 32*z^7 + 256*z^6 - 1130*z^5 + 3602*z^4 - 7940*z^3 + 10830*z^2 - 8042*z + 1, 32*x^7 - 256*x^6*y - 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19680*x*y^2*z^3 + 50430*x*y^2*z^2 - 77132*x*y^2*z + 47574*x*y^2 + 1536*x*y*z^5 - 8720*x*y*z^4 + 27380*x*y*z^3 - 57180*x*y*z^2 + 66908*x*y*z - 31414*x*y + 224*x*z^6 - 1536*x*z^5 + 5645*x*z^4 - 14388*x*z^3 + 23822*x*z^2 - 21734*x*z + 8113*x - 72*y^7 - 456*y^6*z + 504*y^6 - 1232*y^5*z^2 + 2736*y^5*z - 3457*y^5 - 1840*y^4*z^3 + 6160*y^4*z^2 - 13325*y^4*z + 12317*y^4 - 1640*y^3*z^4 + 7360*y^3*z^3 - 20970*y^3*z^2 + 35300*y^3*z - 22762*y^3 - 872*y^2*z^5 + 4920*y^2*z^4 - 16810*y^2*z^3 + 38566*y^2*z^2 - 47502*y^2*z + 22763*y^2 - 256*y*z^6 + 1744*y*z^5 - 6845*y*z^4 + 19060*y*z^3 - 33406*y*z^2 + 31270*y*z - 11767*y - 32*z^7 + 256*z^6 - 1129*z^5 + 3597*z^4 - 7930*z^3 + 10819*z^2 - 8040*z + 1, 16*x^7 - 128*x^6*y - 112*x^6*z + 128*x^6 + 436*x^5*y^2 + 768*x^5*y*z - 872*x^5*y + 336*x^5*z^2 - 768*x^5*z + 565*x^5 - 820*x^4*y^3 - 2180*x^4*y^2*z + 2460*x^4*y^2 - 1920*x^4*y*z^2 + 4360*x^4*y*z - 3425*x^4*y - 560*x^4*z^3 + 1920*x^4*z^2 - 2825*x^4*z + 1801*x^4 + 920*x^3*y^4 + 3280*x^3*y^3*z - 3680*x^3*y^3 + 4360*x^3*y^2*z^2 - 9840*x^3*y^2*z + 8410*x^3*y^2 + 2560*x^3*y*z^3 - 8720*x^3*y*z^2 + 13700*x^3*y*z - 9540*x^3*y + 560*x^3*z^4 - 2560*x^3*z^3 + 5650*x^3*z^2 - 7204*x^3*z + 3986*x^3 - 616*x^2*y^5 - 2760*x^2*y^4*z + 3080*x^2*y^4 - 4920*x^2*y^3*z^2 + 11040*x^2*y^3*z - 10490*x^2*y^3 - 4360*x^2*y^2*z^3 + 14760*x^2*y^2*z^2 - 25230*x^2*y^2*z + 19298*x^2*y^2 - 1920*x^2*y*z^4 + 8720*x^2*y*z^3 - 20550*x^2*y*z^2 + 28620*x^2*y*z - 16766*x^2*y - 336*x^2*z^5 + 1920*x^2*z^4 - 5650*x^2*z^3 + 10806*x^2*z^2 - 11942*x^2*z + 5463*x^2 + 228*x*y^6 + 1232*x*y^5*z - 1368*x*y^5 + 2760*x*y^4*z^2 - 6160*x*y^4*z + 6665*x*y^4 + 3280*x*y^3*z^3 - 11040*x*y^3*z^2 + 20980*x*y^3*z - 17660*x*y^3 + 2180*x*y^2*z^4 - 9840*x*y^2*z^3 + 25230*x*y^2*z^2 - 38596*x*y^2*z + 23802*x*y^2 + 768*x*y*z^5 - 4360*x*y*z^4 + 13700*x*y*z^3 - 28620*x*y*z^2 + 33484*x*y*z - 15718*x*y + 112*x*z^6 - 768*x*z^5 + 2825*x*z^4 - 7204*x*z^3 + 11926*x*z^2 - 10878*x*z + 4060*x - 36*y^7 - 228*y^6*z + 252*y^6 - 616*y^5*z^2 + 1368*y^5*z - 1729*y^5 - 920*y^4*z^3 + 3080*y^4*z^2 - 6665*y^4*z + 6161*y^4 - 820*y^3*z^4 + 3680*y^3*z^3 - 10490*y^3*z^2 + 17660*y^3*z - 11386*y^3 - 436*y^2*z^5 + 2460*y^2*z^4 - 8410*y^2*z^3 + 19298*y^2*z^2 - 23766*y^2*z + 11387*y^2 - 128*y*z^6 + 872*y*z^5 - 3425*y*z^4 + 9540*y*z^3 - 16718*y*z^2 + 15646*y*z - 5886*y - 16*z^7 + 128*z^6 - 565*z^5 + 1801*z^4 - 3970*z^3 + 5415*z^2 - 4023*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ 1, -1, -1 ], [ 3, -3, -2 ], [ 1, -2, -1 ] ], "source_shift": [ 1, 1, 2 ], "target_matrix": [ [ 2, 2, 3 ], [ 2, 1, 1 ], [ 1, 1, 1 ] ], "target_shift": [ -2463, -2471, -1234 ], "inverse_recipe": { "p": 5, "q": 1, "f": "x**5 + x**2 + x", "g": "x", "h": "y*z**2 + z**5 + z**3 + z", "polynomials": { "f": [ [ [ 5, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "g": [ [ [ 1, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 2 ], 1 ], [ [ 0, 0, 3 ], 1 ], [ [ 0, 0, 1 ], 1 ], [ [ 0, 0, 5 ], 1 ] ] }, "h_multiplier": 4 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0470912879973184 ], "surface_features": [ 4.059033426183395, 4.887572293394206, 3.7610252517113727, 0.0, 4.149769372976295, 5095 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C183", "pair": 183, "family": "counterexample", "mapping": "F(x,y,z)=(8*x^7 + 44*x^6*y - 68*x^6*z - 128*x^6 + 102*x^5*y^2 - 312*x^5*y*z - 576*x^5*y + 234*x^5*z^2 + 888*x^5*z + 756*x^5 + 129*x^4*y^3 - 585*x^4*y^2*z - 1068*x^4*y^2 + 867*x^4*y*z^2 + 3252*x^4*y*z + 2772*x^4*y - 419*x^4*z^3 - 2424*x^4*z^2 - 4284*x^4*z - 2219*x^4 + 96*x^3*y^4 - 573*x^3*y^3*z - 1044*x^3*y^3 + 1257*x^3*y^2*z^2 + 4710*x^3*y^2*z + 4011*x^3*y^2 - 1199*x^3*y*z^3 - 6936*x^3*y*z^2 - 12198*x^3*y*z - 6428*x^3*y + 419*x^3*z^4 + 3326*x^3*z^3 + 9063*x^3*z^2 + 9937*x^3*z + 3692*x^3 + 42*x^2*y^5 - 309*x^2*y^4*z - 567*x^2*y^4 + 891*x^2*y^3*z^2 + 3369*x^2*y^3*z + 2865*x^2*y^3 - 1257*x^2*y^2*z^3 - 7353*x^2*y^2*z^2 - 12867*x^2*y^2*z - 6873*x^2*y^2 + 867*x^2*y*z^4 + 6975*x^2*y*z^3 + 18819*x^2*y*z^2 + 20853*x^2*y*z + 7944*x^2*y - 234*x^2*z^5 - 2424*x^2*z^4 - 8937*x^2*z^3 - 15429*x^2*z^2 - 12249*x^2*z - 3592*x^2 + 10*x*y^6 - 87*x*y^5*z - 162*x*y^5 + 309*x*y^4*z^2 + 1188*x*y^4*z + 1011*x*y^4 - 573*x*y^3*z^3 - 3414*x*y^3*z^2 - 5967*x*y^3*z - 3219*x*y^3 + 585*x*y^2*z^4 + 4800*x*y^2*z^3 + 12912*x*y^2*z^2 + 14388*x*y^2*z + 5595*x*y^2 - 312*x*y*z^5 - 3300*x*y*z^4 - 12123*x*y*z^3 - 20901*x*y*z^2 - 16887*x*y*z - 5104*x*y + 68*x*z^6 + 888*x*z^5 + 4167*x*z^4 + 9835*x*z^3 + 12396*x*z^2 + 7825*x*z + 1923*x + y^7 - 10*y^6*z - 19*y^6 + 42*y^5*z^2 + 165*y^5*z + 141*y^5 - 96*y^4*z^3 - 585*y^4*z^2 - 1026*y^4*z - 558*y^4 + 129*y^3*z^4 + 1083*y^3*z^3 + 2922*y^3*z^2 + 3271*y^3*z + 1293*y^3 - 102*y^2*z^5 - 1104*y^2*z^4 - 4068*y^2*z^3 - 7017*y^2*z^2 - 5739*y^2*z - 1777*y^2 + 44*y*z^6 + 588*y*z^5 + 2769*y*z^4 + 6520*y*z^3 + 8262*y*z^2 + 5318*y*z + 1353*y - 8*z^7 - 128*z^6 - 738*z^5 - 2216*z^4 - 3847*z^3 - 3856*z^2 - 2056*z - 1, 8*x^7 + 44*x^6*y - 68*x^6*z - 116*x^6 + 102*x^5*y^2 - 312*x^5*y*z - 516*x^5*y + 234*x^5*z^2 + 792*x^5*z + 612*x^5 + 129*x^4*y^3 - 585*x^4*y^2*z - 945*x^4*y^2 + 867*x^4*y*z^2 + 2862*x^4*y*z + 2220*x^4*y - 419*x^4*z^3 - 2121*x^4*z^2 - 3408*x^4*z - 1666*x^4 + 96*x^3*y^4 - 573*x^3*y^3*z - 912*x^3*y^3 + 1257*x^3*y^2*z^2 + 4089*x^3*y^2*z + 3177*x^3*y^2 - 1199*x^3*y*z^3 - 5982*x^3*y*z^2 - 9594*x^3*y*z - 4774*x^3*y + 419*x^3*z^4 + 2849*x^3*z^3 + 7077*x^3*z^2 + 7320*x^3*z + 2626*x^3 + 42*x^2*y^5 - 309*x^2*y^4*z - 489*x^2*y^4 + 891*x^2*y^3*z^2 + 2886*x^2*y^3*z + 2244*x^2*y^3 - 1257*x^2*y^2*z^3 - 6255*x^2*y^2*z^2 - 10005*x^2*y^2*z - 5049*x^2*y^2 + 867*x^2*y*z^4 + 5892*x^2*y*z^3 + 14526*x^2*y*z^2 + 15192*x^2*y*z + 5583*x^2*y - 234*x^2*z^5 - 2034*x^2*z^4 - 6849*x^2*z^3 - 11145*x^2*z^2 - 8523*x^2*z - 2449*x^2 + 10*x*y^6 - 87*x*y^5*z - 138*x*y^5 + 309*x*y^4*z^2 + 1005*x*y^4*z + 783*x*y^4 - 573*x*y^3*z^3 - 2868*x*y^3*z^2 - 4587*x*y^3*z - 2339*x*y^3 + 585*x*y^2*z^4 + 4005*x*y^2*z^3 + 9852*x*y^2*z^2 + 10368*x*y^2*z + 3885*x*y^2 - 312*x*y*z^5 - 2736*x*y*z^4 - 9183*x*y*z^3 - 14937*x*y*z^2 - 11607*x*y*z - 3430*x*y + 68*x*z^6 + 732*x*z^5 + 3135*x*z^4 + 6974*x*z^3 + 8430*x*z^2 + 5191*x*z + 1271*x + y^7 - 10*y^6*z - 16*y^6 + 42*y^5*z^2 + 138*y^5*z + 108*y^5 - 96*y^4*z^3 - 486*y^4*z^2 - 780*y^4*z - 401*y^4 + 129*y^3*z^4 + 894*y^3*z^3 + 2205*y^3*z^2 + 2331*y^3*z + 887*y^3 - 102*y^2*z^5 - 906*y^2*z^4 - 3048*y^2*z^3 - 4959*y^2*z^2 - 3897*y^2*z - 1177*y^2 + 44*y*z^6 + 480*y*z^5 + 2061*y*z^4 + 4571*y*z^3 + 5553*y*z^2 + 3476*y*z + 879*y - 8*z^7 - 104*z^6 - 546*z^5 - 1542*z^4 - 2561*z^3 - 2485*z^2 - 1314*z - 1, -8*x^7 - 44*x^6*y + 68*x^6*z + 116*x^6 - 102*x^5*y^2 + 312*x^5*y*z + 516*x^5*y - 234*x^5*z^2 - 792*x^5*z - 612*x^5 - 129*x^4*y^3 + 585*x^4*y^2*z + 945*x^4*y^2 - 867*x^4*y*z^2 - 2862*x^4*y*z - 2220*x^4*y + 419*x^4*z^3 + 2121*x^4*z^2 + 3408*x^4*z + 1667*x^4 - 96*x^3*y^4 + 573*x^3*y^3*z + 912*x^3*y^3 - 1257*x^3*y^2*z^2 - 4089*x^3*y^2*z - 3177*x^3*y^2 + 1199*x^3*y*z^3 + 5982*x^3*y*z^2 + 9594*x^3*y*z + 4778*x^3*y - 419*x^3*z^4 - 2849*x^3*z^3 - 7077*x^3*z^2 - 7327*x^3*z - 2636*x^3 - 42*x^2*y^5 + 309*x^2*y^4*z + 489*x^2*y^4 - 891*x^2*y^3*z^2 - 2886*x^2*y^3*z - 2244*x^2*y^3 + 1257*x^2*y^2*z^3 + 6255*x^2*y^2*z^2 + 10005*x^2*y^2*z + 5055*x^2*y^2 - 867*x^2*y*z^4 - 5892*x^2*y*z^3 - 14526*x^2*y*z^2 - 15213*x^2*y*z - 5610*x^2*y + 234*x^2*z^5 + 2034*x^2*z^4 + 6849*x^2*z^3 + 11163*x^2*z^2 + 8571*x^2*z + 2473*x^2 - 10*x*y^6 + 87*x*y^5*z + 138*x*y^5 - 309*x*y^4*z^2 - 1005*x*y^4*z - 783*x*y^4 + 573*x*y^3*z^3 + 2868*x*y^3*z^2 + 4587*x*y^3*z + 2343*x*y^3 - 585*x*y^2*z^4 - 4005*x*y^2*z^3 - 9852*x*y^2*z^2 - 10389*x*y^2*z - 3909*x*y^2 + 312*x*y*z^5 + 2736*x*y*z^4 + 9183*x*y*z^3 + 14973*x*y*z^2 + 11691*x*y*z + 3472*x*y - 68*x*z^6 - 732*x*z^5 - 3135*x*z^4 - 6994*x*z^3 - 8502*x*z^2 - 5266*x*z - 1291*x - y^7 + 10*y^6*z + 16*y^6 - 42*y^5*z^2 - 138*y^5*z - 108*y^5 + 96*y^4*z^3 + 486*y^4*z^2 + 780*y^4*z + 402*y^4 - 129*y^3*z^4 - 894*y^3*z^3 - 2205*y^3*z^2 - 2338*y^3*z - 894*y^3 + 102*y^2*z^5 + 906*y^2*z^4 + 3048*y^2*z^3 + 4977*y^2*z^2 + 3933*y^2*z + 1195*y^2 - 44*y*z^6 - 480*y*z^5 - 2061*y*z^4 - 4591*y*z^3 - 5613*y*z^2 - 3539*y*z - 896*y + 8*z^7 + 104*z^6 + 546*z^5 + 1550*z^4 + 2593*z^3 + 2539*z^2 + 1344*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -1, -1, 2 ], [ -2, -1, 1 ], [ -1, -1, 1 ] ], "source_shift": [ 1, 2, 1 ], "target_matrix": [ [ -1, -2, -1 ], [ -1, -1, -2 ], [ 1, 1, 1 ] ], "target_shift": [ 439, 285, -290 ], "collision_preimages": [ [ "3/4", "3/4", "1/4" ], [ "9", "-20", "-11/2" ], [ "6", "-19", "-15/2" ] ], "collision_image": [ "1757/4", "1141/4", "-1161/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0009028169988596 ], "surface_features": [ 3.4855376719006035, 4.320167065254232, 3.0993352776859577, 0.0, 3.594037692001176, 4844 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C238", "pair": 238, "family": "counterexample", "mapping": "F(x,y,z)=(4*x^7 - 26*x^6*y - 38*x^6*z + 52*x^6 + 72*x^5*y^2 + 210*x^5*y*z - 291*x^5*y + 150*x^5*z^2 - 423*x^5*z + 297*x^5 - 110*x^4*y^3 - 480*x^4*y^2*z + 675*x^4*y^2 - 684*x^4*y*z^2 + 1962*x^4*y*z - 1392*x^4*y - 318*x^4*z^3 + 1401*x^4*z^2 - 2019*x^4*z + 966*x^4 + 100*x^3*y^4 + 580*x^3*y^3*z - 830*x^3*y^3 + 1236*x^3*y^2*z^2 - 3618*x^3*y^2*z + 2598*x^3*y^2 + 1146*x^3*y*z^3 - 5169*x^3*y*z^2 + 7548*x^3*y*z - 3640*x^3*y + 390*x^3*z^4 - 2421*x^3*z^3 + 5397*x^3*z^2 - 5272*x^3*z + 1932*x^3 - 54*x^2*y^5 - 390*x^2*y^4*z + 570*x^2*y^4 - 1104*x^2*y^3*z^2 + 3312*x^2*y^3*z - 2412*x^2*y^3 - 1530*x^2*y^2*z^3 + 7101*x^2*y^2*z^2 - 10530*x^2*y^2*z + 5124*x^2*y^2 - 1038*x^2*y*z^4 + 6660*x^2*y*z^3 - 15096*x^2*y*z^2 + 14886*x^2*y*z - 5484*x^2*y - 276*x^2*z^5 + 2307*x^2*z^4 - 7107*x^2*z^3 + 10644*x^2*z^2 - 7926*x^2*z + 2378*x^2 + 16*x*y^6 + 138*x*y^5*z - 207*x*y^5 + 486*x*y^4*z^2 - 1503*x*y^4*z + 1113*x*y^4 + 894*x*y^3*z^3 - 4299*x*y^3*z^2 + 6492*x*y^3*z - 3192*x*y^3 + 906*x*y^2*z^4 - 6057*x*y^2*z^3 + 14001*x*y^2*z^2 - 13956*x*y^2*z + 5172*x*y^2 + 480*x*y*z^5 - 4206*x*y*z^4 + 13230*x*y*z^3 - 20037*x*y*z^2 + 15012*x*y*z - 4516*x*y + 104*x*z^6 - 1152*x*z^5 + 4620*x*z^4 - 9441*x*z^3 + 10713*x*z^2 - 6506*x*z + 1672*x - 2*y^7 - 20*y^6*z + 31*y^6 - 84*y^5*z^2 + 270*y^5*z - 204*y^5 - 192*y^4*z^3 + 966*y^4*z^2 - 1491*y^4*z + 742*y^4 - 258*y^3*z^4 + 1818*y^3*z^3 - 4302*y^3*z^2 + 4342*y^3*z - 1620*y^3 - 204*y^2*z^5 + 1899*y^2*z^4 - 6123*y^2*z^3 + 9393*y^2*z^2 - 7086*y^2*z + 2138*y^2 - 88*y*z^6 + 1044*y*z^5 - 4296*y*z^4 + 8894*y*z^3 - 10164*y*z^2 + 6194*y*z - 1592*y - 16*z^7 + 236*z^6 - 1188*z^5 + 3107*z^4 - 4773*z^3 + 4400*z^2 - 2282*z + 1, 2*x^7 - 13*x^6*y - 19*x^6*z + 29*x^6 + 36*x^5*y^2 + 105*x^5*y*z - 162*x^5*y + 75*x^5*z^2 - 234*x^5*z + 177*x^5 - 55*x^4*y^3 - 240*x^4*y^2*z + 375*x^4*y^2 - 342*x^4*y*z^2 + 1083*x^4*y*z - 828*x^4*y - 159*x^4*z^3 + 768*x^4*z^2 - 1191*x^4*z + 602*x^4 + 50*x^3*y^4 + 290*x^3*y^3*z - 460*x^3*y^3 + 618*x^3*y^2*z^2 - 1992*x^3*y^2*z + 1542*x^3*y^2 + 573*x^3*y*z^3 - 2826*x^3*y*z^2 + 4443*x^3*y*z - 2265*x^3*y + 195*x^3*z^4 - 1314*x^3*z^3 + 3150*x^3*z^2 - 3252*x^3*z + 1241*x^3 - 27*x^2*y^5 - 195*x^2*y^4*z + 315*x^2*y^4 - 552*x^2*y^3*z^2 + 1818*x^2*y^3*z - 1428*x^2*y^3 - 765*x^2*y^2*z^3 + 3870*x^2*y^2*z^2 - 6183*x^2*y^2*z + 3183*x^2*y^2 - 519*x^2*y*z^4 + 3603*x^2*y*z^3 - 8790*x^2*y*z^2 + 9168*x^2*y*z - 3519*x^2*y - 138*x^2*z^5 + 1239*x^2*z^4 - 4104*x^2*z^3 + 6498*x^2*z^2 - 5040*x^2*z + 1558*x^2 + 8*x*y^6 + 69*x*y^5*z - 114*x*y^5 + 243*x*y^4*z^2 - 822*x*y^4*z + 657*x*y^4 + 447*x*y^3*z^3 - 2334*x*y^3*z^2 + 3801*x*y^3*z - 1979*x*y^3 + 453*x*y^2*z^4 - 3264*x*y^2*z^3 + 8130*x*y^2*z^2 - 8580*x*y^2*z + 3315*x*y^2 + 240*x*y*z^5 - 2250*x*y*z^4 + 7620*x*y*z^3 - 12213*x*y*z^2 + 9537*x*y*z - 2957*x*y + 52*x*z^6 - 612*x*z^5 + 2640*x*z^4 - 5705*x*z^3 + 6744*x*z^2 - 4219*x*z + 1108*x - y^7 - 10*y^6*z + 17*y^6 - 42*y^5*z^2 + 147*y^5*z - 120*y^5 - 96*y^4*z^3 + 522*y^4*z^2 - 870*y^4*z + 459*y^4 - 129*y^3*z^4 + 975*y^3*z^3 - 2490*y^3*z^2 + 2664*y^3*z - 1037*y^3 - 102*y^2*z^5 + 1011*y^2*z^4 - 3516*y^2*z^3 + 5715*y^2*z^2 - 4497*y^2*z + 1399*y^2 - 44*y*z^6 + 552*y*z^5 - 2448*y*z^4 + 5366*y*z^3 - 6393*y*z^2 + 4015*y*z - 1055*y - 8*z^7 + 124*z^6 - 672*z^5 + 1859*z^4 - 2975*z^3 + 2824*z^2 - 1495*z + 1, 2*x^7 - 13*x^6*y - 19*x^6*z + 29*x^6 + 36*x^5*y^2 + 105*x^5*y*z - 162*x^5*y + 75*x^5*z^2 - 234*x^5*z + 177*x^5 - 55*x^4*y^3 - 240*x^4*y^2*z + 375*x^4*y^2 - 342*x^4*y*z^2 + 1083*x^4*y*z - 828*x^4*y - 159*x^4*z^3 + 768*x^4*z^2 - 1191*x^4*z + 600*x^4 + 50*x^3*y^4 + 290*x^3*y^3*z - 460*x^3*y^3 + 618*x^3*y^2*z^2 - 1992*x^3*y^2*z + 1542*x^3*y^2 + 573*x^3*y*z^3 - 2826*x^3*y*z^2 + 4443*x^3*y*z - 2258*x^3*y + 195*x^3*z^4 - 1314*x^3*z^3 + 3150*x^3*z^2 - 3245*x^3*z + 1230*x^3 - 27*x^2*y^5 - 195*x^2*y^4*z + 315*x^2*y^4 - 552*x^2*y^3*z^2 + 1818*x^2*y^3*z - 1428*x^2*y^3 - 765*x^2*y^2*z^3 + 3870*x^2*y^2*z^2 - 6183*x^2*y^2*z + 3174*x^2*y^2 - 519*x^2*y*z^4 + 3603*x^2*y*z^3 - 8790*x^2*y*z^2 + 9150*x^2*y*z - 3489*x^2*y - 138*x^2*z^5 + 1239*x^2*z^4 - 4104*x^2*z^3 + 6489*x^2*z^2 - 5007*x^2*z + 1534*x^2 + 8*x*y^6 + 69*x*y^5*z - 114*x*y^5 + 243*x*y^4*z^2 - 822*x*y^4*z + 657*x*y^4 + 447*x*y^3*z^3 - 2334*x*y^3*z^2 + 3801*x*y^3*z - 1974*x*y^3 + 453*x*y^2*z^4 - 3264*x*y^2*z^3 + 8130*x*y^2*z^2 - 8565*x*y^2*z + 3288*x*y^2 + 240*x*y*z^5 - 2250*x*y*z^4 + 7620*x*y*z^3 - 12198*x*y*z^2 + 9477*x*y*z - 2912*x*y + 52*x*z^6 - 612*x*z^5 + 2640*x*z^4 - 5700*x*z^3 + 6711*x*z^2 - 4168*x*z + 1087*x - y^7 - 10*y^6*z + 17*y^6 - 42*y^5*z^2 + 147*y^5*z - 120*y^5 - 96*y^4*z^3 + 522*y^4*z^2 - 870*y^4*z + 458*y^4 - 129*y^3*z^4 + 975*y^3*z^3 - 2490*y^3*z^2 + 2660*y^3*z - 1029*y^3 - 102*y^2*z^5 + 1011*y^2*z^4 - 3516*y^2*z^3 + 5709*y^2*z^2 - 4470*y^2*z + 1378*y^2 - 44*y*z^6 + 552*y*z^5 - 2448*y*z^4 + 5362*y*z^3 - 6363*y*z^2 + 3967*y*z - 1035*y - 8*z^7 + 124*z^6 - 672*z^5 + 1858*z^4 - 2964*z^3 + 2797*z^2 - 1472*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ 1, -1, -1 ], [ 1, -1, -2 ], [ 2, -1, -1 ] ], "source_shift": [ 1, 2, 2 ], "target_matrix": [ [ 2, 1, 1 ], [ 1, 1, 1 ], [ 1, 1, 2 ] ], "target_shift": [ -517, -343, -337 ], "collision_preimages": [ [ "-5/4", "-5/4", "1" ], [ "9/2", "1", "7/2" ], [ "13/2", "10", "-3/2" ] ], "collision_image": [ "-1035/2", "-1373/4", "-1349/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.030764373997954 ], "surface_features": [ 3.3873210971609193, 4.301832698184397, 3.039612381896724, 0.0, 3.502264889348775, 4819 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C205", "pair": 205, "family": "counterexample", "mapping": "F(x,y,z)=(x^7 - 8*x^6*y - 10*x^6*z + 20*x^6 + 27*x^5*y^2 + 69*x^5*y*z - 135*x^5*y + 42*x^5*z^2 - 171*x^5*z + 168*x^5 - 50*x^4*y^3 - 195*x^4*y^2*z + 375*x^4*y^2 - 243*x^4*y*z^2 + 969*x^4*y*z - 933*x^4*y - 96*x^4*z^3 + 597*x^4*z^2 - 1194*x^4*z + 772*x^4 + 55*x^3*y^4 + 290*x^3*y^3*z - 550*x^3*y^3 + 552*x^3*y^2*z^2 - 2166*x^3*y^2*z + 2052*x^3*y^2 + 447*x^3*y*z^3 - 2727*x^3*y*z^2 + 5346*x^3*y*z - 3397*x^3*y + 129*x^3*z^4 - 1089*x^3*z^3 + 3324*x^3*z^2 - 4376*x^3*z + 2108*x^3 - 36*x^2*y^5 - 240*x^2*y^4*z + 450*x^2*y^4 - 618*x^2*y^3*z^2 + 2394*x^2*y^3*z - 2238*x^2*y^3 - 765*x^2*y^2*z^3 + 4599*x^2*y^2*z^2 - 8874*x^2*y^2*z + 5559*x^2*y^2 - 453*x^2*y*z^4 + 3759*x^2*y*z^3 - 11253*x^2*y*z^2 + 14556*x^2*y*z - 6909*x^2*y - 102*x^2*z^5 + 1095*x^2*z^4 - 4530*x^2*z^3 + 9099*x^2*z^2 - 8925*x^2*z + 3439*x^2 + 13*x*y^6 + 105*x*y^5*z - 195*x*y^5 + 342*x*y^4*z^2 - 1311*x*y^4*z + 1212*x*y^4 + 573*x*y^3*z^3 - 3405*x*y^3*z^2 + 6486*x*y^3*z - 4015*x*y^3 + 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3966*x^3*y^2*z - 3534*x^3*y^2 - 894*x^3*y*z^3 + 4983*x^3*y*z^2 - 9168*x^3*y*z + 5569*x^3*y - 258*x^3*z^4 + 1989*x^3*z^3 - 5685*x^3*z^2 + 7139*x^3*z - 3327*x^3 + 72*x^2*y^5 + 480*x^2*y^4*z - 825*x^2*y^4 + 1236*x^2*y^3*z^2 - 4374*x^2*y^3*z + 3846*x^2*y^3 + 1530*x^2*y^2*z^3 - 8379*x^2*y^2*z^2 + 15174*x^2*y^2*z - 9093*x^2*y^2 + 906*x^2*y*z^4 - 6837*x^2*y*z^3 + 19167*x^2*y*z^2 - 23673*x^2*y*z + 10881*x^2*y + 204*x^2*z^5 - 1992*x^2*z^4 + 7701*x^2*z^3 - 14736*x^2*z^2 + 13971*x^2*z - 5261*x^2 - 26*x*y^6 - 210*x*y^5*z + 357*x*y^5 - 684*x*y^4*z^2 + 2391*x*y^4*z - 2079*x*y^4 - 1146*x*y^3*z^3 + 6189*x*y^3*z^2 - 11064*x*y^3*z + 6555*x*y^3 - 1038*x*y^2*z^4 + 7707*x*y^2*z^3 - 21279*x*y^2*z^2 + 25929*x*y^2*z - 11781*x*y^2 - 480*x*y*z^5 + 4596*x*y*z^4 - 17442*x*y*z^3 + 32832*x*y*z^2 - 30696*x*y*z + 11425*x*y - 88*x*z^6 + 1044*x*z^5 - 5112*x*z^4 + 13230*x*z^3 - 19110*x*z^2 + 14641*x*z - 4661*x + 4*y^7 + 38*y^6*z - 64*y^6 + 150*y^5*z^2 - 519*y^5*z + 447*y^5 + 318*y^4*z^3 - 1698*y^4*z^2 + 3003*y^4*z - 1762*y^4 + 390*y^3*z^4 - 2859*y^3*z^3 + 7797*y^3*z^2 - 9395*y^3*z + 4227*y^3 + 276*y^2*z^5 - 2604*y^2*z^4 + 9741*y^2*z^3 - 18096*y^2*z^2 + 16725*y^2*z - 6164*y^2 + 104*y*z^6 - 1212*y*z^5 + 5832*y*z^4 - 14854*y*z^3 + 21162*y*z^2 - 16027*y*z + 5053*y + 16*z^7 - 224*z^6 + 1332*z^5 - 4364*z^4 + 8520*z^3 - 9938*z^2 + 6437*z - 1, -x^7 + 8*x^6*y + 10*x^6*z - 23*x^6 - 27*x^5*y^2 - 69*x^5*y*z + 156*x^5*y - 42*x^5*z^2 + 198*x^5*z - 213*x^5 + 50*x^4*y^3 + 195*x^4*y^2*z - 435*x^4*y^2 + 243*x^4*y*z^2 - 1128*x^4*y*z + 1188*x^4*y + 96*x^4*z^3 - 696*x^4*z^2 + 1527*x^4*z - 1048*x^4 - 55*x^3*y^4 - 290*x^3*y^3*z + 640*x^3*y^3 - 552*x^3*y^2*z^2 + 2532*x^3*y^2*z - 2622*x^3*y^2 - 447*x^3*y*z^3 + 3198*x^3*y*z^2 - 6870*x^3*y*z + 4627*x^3*y - 129*x^3*z^4 + 1278*x^3*z^3 - 4287*x^3*z^2 + 5996*x^3*z - 3008*x^3 + 36*x^2*y^5 + 240*x^2*y^4*z - 525*x^2*y^4 + 618*x^2*y^3*z^2 - 2808*x^2*y^3*z + 2868*x^2*y^3 + 765*x^2*y^2*z^3 - 5418*x^2*y^2*z^2 + 11448*x^2*y^2*z - 7593*x^2*y^2 + 453*x^2*y*z^4 - 4440*x^2*y*z^3 + 14592*x^2*y*z^2 - 20022*x^2*y*z + 9885*x^2*y + 102*x^2*z^5 - 1293*x^2*z^4 + 5889*x^2*z^3 - 12579*x^2*z^2 + 12861*x^2*z - 5098*x^2 - 13*x*y^6 - 105*x*y^5*z + 228*x*y^5 - 342*x*y^4*z^2 + 1542*x*y^4*z - 1557*x*y^4 - 573*x*y^3*z^3 + 4026*x*y^3*z^2 - 8394*x*y^3*z + 5497*x*y^3 - 519*x*y^2*z^4 + 5046*x*y^2*z^3 - 16323*x*y^2*z^2 + 22056*x*y^2*z - 10746*x*y^2 - 240*x*y*z^5 + 3018*x*y*z^4 - 13482*x*y*z^3 + 28254*x*y*z^2 - 28419*x*y*z + 11114*x*y - 44*x*z^6 + 684*x*z^5 - 3960*x*z^4 + 11465*x*z^3 - 17895*x*z^2 + 14462*x*z - 4765*x + 2*y^7 + 19*y^6*z - 41*y^6 + 75*y^5*z^2 - 336*y^5*z + 336*y^5 + 159*y^4*z^3 - 1110*y^4*z^2 + 2289*y^4*z - 1483*y^4 + 195*y^3*z^4 - 1884*y^3*z^3 + 6018*y^3*z^2 - 8030*y^3*z + 3869*y^3 + 138*y^2*z^5 - 1725*y^2*z^4 + 7593*y^2*z^3 - 15675*y^2*z^2 + 15558*y^2*z - 6016*y^2 + 52*y*z^6 - 804*y*z^5 + 4572*y*z^4 - 12993*y*z^3 + 19953*y*z^2 - 15908*y*z + 5182*y + 8*z^7 - 148*z^6 + 1044*z^5 - 3834*z^4 + 8104*z^3 - 9994*z^2 + 6719*z - 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ 1, -1, -2 ], [ 1, -1, -1 ], [ 1, -2, -1 ] ], "source_shift": [ 2, 2, 2 ], "target_matrix": [ [ 1, 1, 1 ], [ -2, -1, -1 ], [ -1, -2, -1 ] ], "target_shift": [ -1219, 1789, 1905 ], "collision_preimages": [ [ "-7/4", "1/4", "0" ], [ "-14", "-8", "-5/2" ], [ "-3", "-5", "5/2" ] ], "collision_image": [ "-4877/4", "3579/2", "7621/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0320747229998233 ], "surface_features": [ 3.642456550958693, 4.516297339095703, 3.1901915805753016, 0.0, 3.7768217786151426, 4879 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C077", "pair": 77, "family": "counterexample", "mapping": "F(x,y,z)=(-54*x^7 - 459*x^6*y + 297*x^6*z + 756*x^6 - 1584*x^5*y^2 + 2115*x^5*y*z + 5211*x^5*y - 693*x^5*z^2 - 3510*x^5*z - 3924*x^5 - 2860*x^4*y^3 + 5934*x^4*y^2*z 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"-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -1, -2, 1 ], [ -3, -2, 2 ], [ -2, -1, 1 ] ], "source_shift": [ 2, 2, 1 ], "target_matrix": [ [ 1, 1, 1 ], [ 1, 2, 1 ], [ 1, 1, 2 ] ], "target_shift": [ -952, -1488, -924 ], "collision_preimages": [ [ "1/2", "5/4", "1" ], [ "-29/2", "-8", "-63/2" ], [ "-23/2", "-3", "-41/2" ] ], "collision_image": [ "-3809/4", "-5953/4", "-3697/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 0.9162315030007449 ], "surface_features": [ 4.2736338516470935, 5.086320671210148, 3.9139462390580855, 0.0, 4.371449704457196, 5129 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C093", "pair": 93, "family": "counterexample", "mapping": "F(x,y,z)=(64*x^7 + 352*x^6*y - 448*x^6*z - 800*x^6 + 816*x^5*y^2 - 2064*x^5*y*z - 3648*x^5*y + 1296*x^5*z^2 + 4656*x^5*z + 3888*x^5 + 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7500*x^2*y*z^4 - 50832*x^2*y*z^3 - 125232*x^2*y*z^2 - 132252*x^2*y*z - 50376*x^2*y + 1800*x^2*z^5 + 15462*x^2*z^4 + 51576*x^2*z^3 + 83106*x^2*z^2 + 64428*x^2*z + 19238*x^2 - 160*x*y^6 + 1176*x*y^5*z + 1920*x*y^5 - 3576*x*y^4*z^2 - 11844*x*y^4*z - 9504*x*y^4 + 5758*x*y^3*z^3 + 29016*x*y^3*z^2 + 47268*x*y^3*z + 24836*x*y^3 - 5178*x*y^2*z^4 - 35283*x*y^2*z^3 - 87510*x*y^2*z^2 - 93432*x*y^2*z - 36168*x*y^2 + 2466*x*y*z^5 + 21294*x*y*z^4 + 71460*x*y*z^3 + 116274*x*y*z^2 + 91542*x*y*z + 27896*x*y - 486*x*z^6 - 5103*x*z^5 - 21714*x*z^4 - 47849*x*z^3 - 57474*x*z^2 - 35651*x*z - 8930*x - 16*y^7 + 136*y^6*z + 224*y^6 - 492*y^5*z^2 - 1644*y^5*z - 1332*y^5 + 982*y^4*z^3 + 4992*y^4*z^2 + 8208*y^4*z + 4364*y^4 - 1168*y^3*z^4 - 8027*y^3*z^3 - 20085*y^3*z^2 - 21682*y^3*z - 8520*y^3 + 828*y^2*z^5 + 7209*y^2*z^4 + 24393*y^2*z^3 + 40092*y^2*z^2 + 32010*y^2*z + 9938*y^2 - 324*y*z^6 - 3429*y*z^5 - 14703*y*z^4 - 32693*y*z^3 - 39771*y*z^2 - 25111*y*z - 6437*y + 54*z^7 + 675*z^6 + 3519*z^5 + 9919*z^4 + 16335*z^3 + 15731*z^2 + 8213*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -2, -2, 3 ], [ -2, -1, 1 ], [ -1, -1, 1 ] ], "source_shift": [ 2, 2, 2 ], "target_matrix": [ [ -1, -1, -1 ], [ -1, -1, -2 ], [ 2, 1, 1 ] ], "target_shift": [ 1219, 1183, -1789 ], "collision_preimages": [ [ "-1/4", "5", "5/2" ], [ "8", "-45/2", "-10" ], [ "5", "-43/2", "-12" ] ], "collision_image": [ "4877/4", "4733/4", "-3579/2" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 0.9812830520022544 ], "surface_features": [ 4.244887645780925, 5.121402248438007, 3.9143696054299086, 0.0, 4.352504823144055, 5133 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B032", "pair": 32, "family": "bijection", "mapping": "F(x,y,z)=(-384*x^7 + 1344*x^6*y + 1344*x^6*z - 1536*x^6 - 2016*x^5*y^2 - 4032*x^5*y*z + 4608*x^5*y - 2016*x^5*z^2 + 4608*x^5*z - 2640*x^5 + 1680*x^4*y^3 + 5040*x^4*y^2*z - 5760*x^4*y^2 + 5040*x^4*y*z^2 - 11520*x^4*y*z + 6672*x^4*y + 1680*x^4*z^3 - 5760*x^4*z^2 + 6624*x^4*z - 2752*x^4 - 840*x^3*y^4 - 3360*x^3*y^3*z + 3840*x^3*y^3 - 5040*x^3*y^2*z^2 + 11520*x^3*y^2*z - 6744*x^3*y^2 - 3360*x^3*y*z^3 + 11520*x^3*y*z^2 - 13392*x^3*y*z + 5696*x^3*y - 840*x^3*z^4 + 3840*x^3*z^3 - 6648*x^3*z^2 + 5576*x^3*z - 2024*x^3 + 252*x^2*y^5 + 1260*x^2*y^4*z - 1440*x^2*y^4 + 2520*x^2*y^3*z^2 - 5760*x^2*y^3*z + 3408*x^2*y^3 + 2520*x^2*y^2*z^3 - 8640*x^2*y^2*z^2 + 10152*x^2*y^2*z - 4416*x^2*y^2 + 1260*x^2*y*z^4 - 5760*x^2*y*z^3 + 10080*x^2*y*z^2 - 8652*x^2*y*z + 3216*x^2*y + 252*x^2*z^5 - 1440*x^2*z^4 + 3336*x^2*z^3 - 4236*x^2*z^2 + 3108*x^2*z - 1023*x^2 - 42*x*y^6 - 252*x*y^5*z + 288*x*y^5 - 630*x*y^4*z^2 + 1440*x*y^4*z - 861*x*y^4 - 840*x*y^3*z^3 + 2880*x*y^3*z^2 - 3420*x*y^3*z + 1520*x*y^3 - 630*x*y^2*z^4 + 2880*x*y^2*z^3 - 5094*x*y^2*z^2 + 4470*x*y^2*z - 1698*x*y^2 - 252*x*y*z^5 + 1440*x*y*z^4 - 3372*x*y*z^3 + 4380*x*y*z^2 - 3288*x*y*z + 1101*x*y - 42*x*z^6 + 288*x*z^5 - 837*x*z^4 + 1430*x*z^3 - 1590*x*z^2 + 1056*x*z - 319*x + 3*y^7 + 21*y^6*z - 24*y^6 + 63*y^5*z^2 - 144*y^5*z + 87*y^5 + 105*y^4*z^3 - 360*y^4*z^2 + 432*y^4*z - 196*y^4 + 105*y^3*z^4 - 480*y^3*z^3 + 858*y^3*z^2 - 769*y^3*z + 298*y^3 + 63*y^2*z^5 - 360*y^2*z^4 + 852*y^2*z^3 - 1131*y^2*z^2 + 867*y^2*z - 297*y^2 + 21*y*z^6 - 144*y*z^5 + 423*y*z^4 - 739*y*z^3 + 840*y*z^2 - 570*y*z + 184*y + 3*z^7 - 24*z^6 + 84*z^5 - 181*z^4 + 271*z^3 - 273*z^2 + 173*z - 1, 384*x^7 - 1344*x^6*y - 1344*x^6*z + 1536*x^6 + 2016*x^5*y^2 + 4032*x^5*y*z - 4608*x^5*y + 2016*x^5*z^2 - 4608*x^5*z + 2640*x^5 - 1680*x^4*y^3 - 5040*x^4*y^2*z + 5760*x^4*y^2 - 5040*x^4*y*z^2 + 11520*x^4*y*z - 6672*x^4*y - 1680*x^4*z^3 + 5760*x^4*z^2 - 6624*x^4*z + 2752*x^4 + 840*x^3*y^4 + 3360*x^3*y^3*z - 3840*x^3*y^3 + 5040*x^3*y^2*z^2 - 11520*x^3*y^2*z + 6744*x^3*y^2 + 3360*x^3*y*z^3 - 11520*x^3*y*z^2 + 13392*x^3*y*z - 5696*x^3*y + 840*x^3*z^4 - 3840*x^3*z^3 + 6648*x^3*z^2 - 5576*x^3*z + 2016*x^3 - 252*x^2*y^5 - 1260*x^2*y^4*z + 1440*x^2*y^4 - 2520*x^2*y^3*z^2 + 5760*x^2*y^3*z - 3408*x^2*y^3 - 2520*x^2*y^2*z^3 + 8640*x^2*y^2*z^2 - 10152*x^2*y^2*z + 4416*x^2*y^2 - 1260*x^2*y*z^4 + 5760*x^2*y*z^3 - 10080*x^2*y*z^2 + 8652*x^2*y*z - 3204*x^2*y - 252*x^2*z^5 + 1440*x^2*z^4 - 3336*x^2*z^3 + 4236*x^2*z^2 - 3096*x^2*z + 1011*x^2 + 42*x*y^6 + 252*x*y^5*z - 288*x*y^5 + 630*x*y^4*z^2 - 1440*x*y^4*z + 861*x*y^4 + 840*x*y^3*z^3 - 2880*x*y^3*z^2 + 3420*x*y^3*z - 1520*x*y^3 + 630*x*y^2*z^4 - 2880*x*y^2*z^3 + 5094*x*y^2*z^2 - 4470*x*y^2*z + 1692*x*y^2 + 252*x*y*z^5 - 1440*x*y*z^4 + 3372*x*y*z^3 - 4380*x*y*z^2 + 3276*x*y*z - 1089*x*y + 42*x*z^6 - 288*x*z^5 + 837*x*z^4 - 1430*x*z^3 + 1584*x*z^2 - 1044*x*z + 312*x - 3*y^7 - 21*y^6*z + 24*y^6 - 63*y^5*z^2 + 144*y^5*z - 87*y^5 - 105*y^4*z^3 + 360*y^4*z^2 - 432*y^4*z + 196*y^4 - 105*y^3*z^4 + 480*y^3*z^3 - 858*y^3*z^2 + 769*y^3*z - 297*y^3 - 63*y^2*z^5 + 360*y^2*z^4 - 852*y^2*z^3 + 1131*y^2*z^2 - 864*y^2*z + 294*y^2 - 21*y*z^6 + 144*y*z^5 - 423*y*z^4 + 739*y*z^3 - 837*y*z^2 + 564*y*z - 179*y - 3*z^7 + 24*z^6 - 84*z^5 + 181*z^4 - 270*z^3 + 270*z^2 - 169*z + 1, -768*x^7 + 2688*x^6*y + 2688*x^6*z - 3072*x^6 - 4032*x^5*y^2 - 8064*x^5*y*z + 9216*x^5*y - 4032*x^5*z^2 + 9216*x^5*z - 5280*x^5 + 3360*x^4*y^3 + 10080*x^4*y^2*z - 11520*x^4*y^2 + 10080*x^4*y*z^2 - 23040*x^4*y*z + 13344*x^4*y + 3360*x^4*z^3 - 11520*x^4*z^2 + 13248*x^4*z - 5536*x^4 - 1680*x^3*y^4 - 6720*x^3*y^3*z + 7680*x^3*y^3 - 10080*x^3*y^2*z^2 + 23040*x^3*y^2*z - 13488*x^3*y^2 - 6720*x^3*y*z^3 + 23040*x^3*y*z^2 - 26784*x^3*y*z + 11456*x^3*y - 1680*x^3*z^4 + 7680*x^3*z^3 - 13296*x^3*z^2 + 11216*x^3*z - 4104*x^3 + 504*x^2*y^5 + 2520*x^2*y^4*z - 2880*x^2*y^4 + 5040*x^2*y^3*z^2 - 11520*x^2*y^3*z + 6816*x^2*y^3 + 5040*x^2*y^2*z^3 - 17280*x^2*y^2*z^2 + 20304*x^2*y^2*z - 8880*x^2*y^2 + 2520*x^2*y*z^4 - 11520*x^2*y*z^3 + 20160*x^2*y*z^2 - 17400*x^2*y*z + 6516*x^2*y + 504*x^2*z^5 - 2880*x^2*z^4 + 6672*x^2*z^3 - 8520*x^2*z^2 + 6300*x^2*z - 2082*x^2 - 84*x*y^6 - 504*x*y^5*z + 576*x*y^5 - 1260*x*y^4*z^2 + 2880*x*y^4*z - 1722*x*y^4 - 1680*x*y^3*z^3 + 5760*x*y^3*z^2 - 6840*x*y^3*z + 3056*x*y^3 - 1260*x*y^2*z^4 + 5760*x*y^2*z^3 - 10188*x*y^2*z^2 + 8988*x*y^2*z - 3438*x*y^2 - 504*x*y*z^5 + 2880*x*y*z^4 - 6744*x*y*z^3 + 8808*x*y*z^2 - 6660*x*y*z + 2238*x*y - 84*x*z^6 + 576*x*z^5 - 1674*x*z^4 + 2876*x*z^3 - 3222*x*z^2 + 2148*x*z - 647*x + 6*y^7 + 42*y^6*z - 48*y^6 + 126*y^5*z^2 - 288*y^5*z + 174*y^5 + 210*y^4*z^3 - 720*y^4*z^2 + 864*y^4*z - 394*y^4 + 210*y^3*z^4 - 960*y^3*z^3 + 1716*y^3*z^2 - 1546*y^3*z + 603*y^3 + 126*y^2*z^5 - 720*y^2*z^4 + 1704*y^2*z^3 - 2274*y^2*z^2 + 1755*y^2*z - 603*y^2 + 42*y*z^6 - 288*y*z^5 + 846*y*z^4 - 1486*y*z^3 + 1701*y*z^2 - 1158*y*z + 369*y + 6*z^7 - 48*z^6 + 168*z^5 - 364*z^4 + 549*z^3 - 555*z^2 + 350*z - 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ 2, -1, -1 ], [ 1, -2, -1 ], [ 1, -1, -1 ] ], "source_shift": [ 1, 2, 1 ], "target_matrix": [ [ -1, -2, -1 ], [ 1, 1, 1 ], [ -2, -1, -1 ] ], "target_shift": [ 54, -51, 106 ], "inverse_recipe": { "p": 3, "q": 4, "f": "x**3", "g": "x**4 + x**3", "h": "y*z + 3*z", "polynomials": { "f": [ [ [ 3, 0, 0 ], 1 ] ], "g": [ [ [ 4, 0, 0 ], 1 ], [ [ 3, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 0, 1 ], 3 ] ] }, "h_multiplier": 3 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 0.8311229620012455 ], "surface_features": [ 3.525851068063375, 4.427875436313166, 3.2245330626060857, 0.0, 3.632340476658415, 4896 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C109", "pair": 109, "family": "counterexample", "mapping": "F(x,y,z)=(48*x^7 + 248*x^6*y + 392*x^6*z + 580*x^6 + 540*x^5*y^2 + 1680*x^5*y*z + 2496*x^5*y + 1284*x^5*z^2 + 3852*x^5*z + 2940*x^5 + 642*x^4*y^3 + 2946*x^4*y^2*z + 4401*x^4*y^2 + 4422*x^4*y*z^2 + 13350*x^4*y*z + 10272*x^4*y + 2166*x^4*z^3 + 9921*x^4*z^2 + 15468*x^4*z + 8149*x^4 + 450*x^3*y^4 + 2706*x^3*y^3*z + 4071*x^3*y^3 + 5982*x^3*y^2*z^2 + 18204*x^3*y^2*z + 14121*x^3*y^2 + 5750*x^3*y*z^3 + 26571*x^3*y*z^2 + 41754*x^3*y*z + 22199*x^3*y + 2024*x^3*z^4 + 12634*x^3*z^3 + 30165*x^3*z^2 + 32538*x^3*z + 13327*x^3 + 186*x^2*y^5 + 1374*x^2*y^4*z + 2085*x^2*y^4 + 3978*x^2*y^3*z^2 + 12222*x^2*y^3*z + 9558*x^2*y^3 + 5634*x^2*y^2*z^3 + 26307*x^2*y^2*z^2 + 41655*x^2*y^2*z + 22323*x^2*y^2 + 3900*x^2*y*z^4 + 24612*x^2*y*z^3 + 59148*x^2*y*z^2 + 64206*x^2*y*z + 26514*x^2*y + 1056*x^2*z^5 + 8442*x^2*z^4 + 27327*x^2*z^3 + 44994*x^2*z^2 + 37671*x^2*z + 12797*x^2 + 42*x*y^6 + 366*x*y^5*z + 561*x*y^5 + 1302*x*y^4*z^2 + 4044*x*y^4*z + 3189*x*y^4 + 2418*x*y^3*z^3 + 11421*x*y^3*z^2 + 18228*x*y^3*z + 9839*x*y^3 + 2472*x*y^2*z^4 + 15786*x*y^2*z^3 + 38217*x*y^2*z^2 + 41724*x*y^2*z + 17337*x*y^2 + 1320*x*y*z^5 + 10680*x*y*z^4 + 34818*x*y*z^3 + 57558*x*y*z^2 + 48348*x*y*z + 16505*x*y + 288*x*z^6 + 2832*x*z^5 + 11640*x*z^4 + 25826*x*z^3 + 32772*x*z^2 + 22598*x*z + 6615*x + 4*y^7 + 40*y^6*z + 62*y^6 + 168*y^5*z^2 + 528*y^5*z + 420*y^5 + 384*y^4*z^3 + 1836*y^4*z^2 + 2955*y^4*z + 1606*y^4 + 516*y^3*z^4 + 3336*y^3*z^3 + 8142*y^3*z^2 + 8940*y^3*z + 3734*y^3 + 408*y^2*z^5 + 3342*y^2*z^4 + 10983*y^2*z^3 + 18240*y^2*z^2 + 15369*y^2*z + 5264*y^2 + 176*y*z^6 + 1752*y*z^5 + 7260*y*z^4 + 16174*y*z^3 + 20556*y*z^2 + 14176*y*z + 4150*y + 32*z^7 + 376*z^6 + 1884*z^5 + 5268*z^4 + 8948*z^3 + 9272*z^2 + 5443*z + 1, 24*x^7 + 124*x^6*y + 196*x^6*z + 308*x^6 + 270*x^5*y^2 + 840*x^5*y*z + 1332*x^5*y + 642*x^5*z^2 + 2064*x^5*z + 1656*x^5 + 321*x^4*y^3 + 1473*x^4*y^2*z + 2361*x^4*y^2 + 2211*x^4*y*z^2 + 7194*x^4*y*z + 5832*x^4*y + 1083*x^4*z^3 + 5373*x^4*z^2 + 8844*x^4*z + 4841*x^4 + 225*x^3*y^4 + 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1140*z^5 + 3318*z^4 + 5770*z^3 + 6058*z^2 + 3583*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ 1, 1, 2 ], [ 2, 1, 1 ], [ 3, 2, 2 ] ], "source_shift": [ 2, 2, 1 ], "target_matrix": [ [ 2, 1, 1 ], [ 1, 1, 1 ], [ 1, 1, 2 ] ], "target_shift": [ -1397, -952, -924 ], "collision_preimages": [ [ "-11/4", "25/4", "-11/4" ], [ "-25/2", "63/2", "-10" ], [ "-13/2", "43/2", "-9" ] ], "collision_image": [ "-2795/2", "-3809/4", "-3697/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 0.7956720709989895 ], "surface_features": [ 3.922848960400242, 4.80757561444166, 3.5953859808091417, 0.0, 4.037770704069717, 5025 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B027", "pair": 27, "family": "bijection", "mapping": "F(x,y,z)=(-640*x^7 + 2240*x^6*y + 2240*x^6*z - 2560*x^6 - 3360*x^5*y^2 - 6720*x^5*y*z + 8000*x^5*y - 3360*x^5*z^2 + 7840*x^5*z - 4704*x^5 + 2800*x^4*y^3 + 8400*x^4*y^2*z - 10400*x^4*y^2 + 8400*x^4*y*z^2 - 20400*x^4*y*z + 12560*x^4*y + 2800*x^4*z^3 - 10000*x^4*z^2 + 12160*x^4*z - 5120*x^4 - 1400*x^3*y^4 - 5600*x^3*y^3*z + 7200*x^3*y^3 - 8400*x^3*y^2*z^2 + 21200*x^3*y^2*z - 13360*x^3*y^2 - 5600*x^3*y*z^3 + 20800*x^3*y*z^2 - 25920*x^3*y*z + 11120*x^3*y - 1400*x^3*z^4 + 6800*x^3*z^3 - 12560*x^3*z^2 + 10680*x^3*z - 3676*x^3 + 420*x^2*y^5 + 2100*x^2*y^4*z - 2800*x^2*y^4 + 4200*x^2*y^3*z^2 - 11000*x^2*y^3*z + 7080*x^2*y^3 + 4200*x^2*y^2*z^3 - 16200*x^2*y^2*z^2 + 20640*x^2*y^2*z - 9000*x^2*y^2 + 2100*x^2*y*z^4 - 10600*x^2*y*z^3 + 20040*x^2*y*z^2 - 17340*x^2*y*z + 6144*x^2*y + 420*x^2*z^5 - 2600*x^2*z^4 + 6480*x^2*z^3 - 8340*x^2*z^2 + 5824*x^2*z - 1846*x^2 - 70*x*y^6 - 420*x*y^5*z + 580*x*y^5 - 1050*x*y^4*z^2 + 2850*x*y^4*z - 1870*x*y^4 - 1400*x*y^3*z^3 + 5600*x*y^3*z^2 - 7280*x*y^3*z + 3220*x*y^3 - 1050*x*y^2*z^4 + 5500*x*y^2*z^3 - 10620*x*y^2*z^2 + 9330*x*y^2*z - 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10600*x^2*y*z^3 - 19920*x^2*y*z^2 + 17100*x^2*y*z - 6012*x^2*y - 420*x^2*z^5 + 2600*x^2*z^4 - 6440*x^2*z^3 + 8220*x^2*z^2 - 5692*x^2*z + 1794*x^2 + 70*x*y^6 + 420*x*y^5*z - 580*x*y^5 + 1050*x*y^4*z^2 - 2850*x*y^4*z + 1860*x*y^4 + 1400*x*y^3*z^3 - 5600*x*y^3*z^2 + 7240*x*y^3*z - 3180*x*y^3 + 1050*x*y^2*z^4 - 5500*x*y^2*z^3 + 10560*x*y^2*z^2 - 9210*x*y^2*z + 3321*x*y^2 + 420*x*y*z^5 - 2700*x*y*z^4 + 6840*x*y*z^3 - 8880*x*y*z^2 + 6322*x*y*z - 2149*x*y + 70*x*z^6 - 530*x*z^5 + 1660*x*z^4 - 2850*x*z^3 + 3001*x*z^2 - 1964*x*z + 624*x - 5*y^7 - 35*y^6*z + 50*y^6 - 105*y^5*z^2 + 295*y^5*z - 196*y^5 - 175*y^4*z^3 + 725*y^4*z^2 - 955*y^4*z + 425*y^4 - 175*y^3*z^4 + 950*y^3*z^3 - 1860*y^3*z^2 + 1645*y^3*z - 606*y^3 - 105*y^2*z^5 + 700*y^2*z^4 - 1810*y^2*z^3 + 2385*y^2*z^2 - 1738*y^2*z + 661*y^2 - 35*y*z^6 + 275*y*z^5 - 880*y*z^4 + 1535*y*z^3 - 1658*y*z^2 + 1192*y*z - 442*y - 5*z^7 + 45*z^6 - 171*z^5 + 370*z^4 - 526*z^3 + 541*z^2 - 373*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ 2, -1, -1 ], [ 1, -2, -1 ], [ 2, -3, -2 ] ], "source_shift": [ 1, 2, 2 ], "target_matrix": [ [ -1, -2, -1 ], [ -2, -2, -3 ], [ 1, 1, 1 ] ], "target_shift": [ 124, 233, -119 ], "inverse_recipe": { "p": 5, "q": 2, "f": "x**5 + x**3 + x", "g": "x**2", "h": "y*z + z**2", "polynomials": { "f": [ [ [ 5, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ], [ [ 3, 0, 0 ], 1 ] ], "g": [ [ [ 2, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 0, 2 ], 1 ] ] }, "h_multiplier": 5 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 0.8627210109989392 ], "surface_features": [ 3.797057577236828, 4.711975854351755, 3.469159027072012, 0.0, 3.906827611451223, 4997 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B148", "pair": 148, "family": "bijection", "mapping": "F(x,y,z)=(-7*x^7 + 49*x^6*y - 98*x^6*z + 77*x^6 - 147*x^5*y^2 + 588*x^5*y*z - 462*x^5*y - 588*x^5*z^2 + 917*x^5*z - 372*x^5 + 245*x^4*y^3 - 1470*x^4*y^2*z + 1155*x^4*y^2 + 2940*x^4*y*z^2 - 4585*x^4*y*z + 1860*x^4*y - 1960*x^4*z^3 + 4550*x^4*z^2 - 3657*x^4*z + 1035*x^4 - 245*x^3*y^4 + 1960*x^3*y^3*z - 1540*x^3*y^3 - 5880*x^3*y^2*z^2 + 9170*x^3*y^2*z - 3720*x^3*y^2 + 7840*x^3*y*z^3 - 18200*x^3*y*z^2 + 14628*x^3*y*z - 4140*x^3*y - 3920*x^3*z^4 + 12040*x^3*z^3 - 14376*x^3*z^2 + 8014*x^3*z - 1799*x^3 + 147*x^2*y^5 - 1470*x^2*y^4*z + 1155*x^2*y^4 + 5880*x^2*y^3*z^2 - 9170*x^2*y^3*z + 3720*x^2*y^3 - 11760*x^2*y^2*z^3 + 27300*x^2*y^2*z^2 - 21942*x^2*y^2*z + 6210*x^2*y^2 + 11760*x^2*y*z^4 - 36120*x^2*y*z^3 + 43128*x^2*y*z^2 - 24042*x^2*y*z + 5390*x^2*y - 4704*x^2*z^5 + 17920*x^2*z^4 - 28248*x^2*z^3 + 23265*x^2*z^2 - 10192*x^2*z + 1953*x^2 - 49*x*y^6 + 588*x*y^5*z - 462*x*y^5 - 2940*x*y^4*z^2 + 4585*x*y^4*z - 1860*x*y^4 + 7840*x*y^3*z^3 - 18200*x*y^3*z^2 + 14628*x*y^3*z - 4140*x*y^3 - 11760*x*y^2*z^4 + 36120*x*y^2*z^3 - 43128*x*y^2*z^2 + 24042*x*y^2*z - 5383*x*y^2 + 9408*x*y*z^5 - 35840*x*y*z^4 + 56496*x*y*z^3 - 46530*x*y*z^2 + 20356*x*y*z - 3878*x*y - 3136*x*z^6 + 14224*x*z^5 - 27744*x*z^4 + 30012*x*z^3 - 19264*x*z^2 + 7105*x*z - 1203*x + 7*y^7 - 98*y^6*z + 77*y^6 + 588*y^5*z^2 - 917*y^5*z + 372*y^5 - 1960*y^4*z^3 + 4550*y^4*z^2 - 3657*y^4*z + 1035*y^4 + 3920*y^3*z^4 - 12040*y^3*z^3 + 14376*y^3*z^2 - 8014*y^3*z + 1792*y^3 - 4704*y^2*z^5 + 17920*y^2*z^4 - 28248*y^2*z^3 + 23265*y^2*z^2 - 10164*y^2*z + 1925*y^2 + 3136*y*z^6 - 14224*y*z^5 + 27744*y*z^4 - 30012*y*z^3 + 19236*y*z^2 - 7056*y*z + 1181*y - 896*z^7 + 4704*z^6 - 10896*z^5 + 14516*z^4 - 12145*z^3 + 6489*z^2 - 2089*z + 1, -14*x^7 + 98*x^6*y - 196*x^6*z + 154*x^6 - 294*x^5*y^2 + 1176*x^5*y*z - 924*x^5*y - 1176*x^5*z^2 + 1834*x^5*z - 743*x^5 + 490*x^4*y^3 - 2940*x^4*y^2*z + 2310*x^4*y^2 + 5880*x^4*y*z^2 - 9170*x^4*y*z + 3715*x^4*y - 3920*x^4*z^3 + 9100*x^4*z^2 - 7304*x^4*z + 2064*x^4 - 490*x^3*y^4 + 3920*x^3*y^3*z - 3080*x^3*y^3 - 11760*x^3*y^2*z^2 + 18340*x^3*y^2*z - 7430*x^3*y^2 + 15680*x^3*y*z^3 - 36400*x^3*y*z^2 + 29216*x^3*y*z - 8256*x^3*y - 7840*x^3*z^4 + 24080*x^3*z^3 - 28712*x^3*z^2 + 15980*x^3*z - 3584*x^3 + 294*x^2*y^5 - 2940*x^2*y^4*z + 2310*x^2*y^4 + 11760*x^2*y^3*z^2 - 18340*x^2*y^3*z + 7430*x^2*y^3 - 23520*x^2*y^2*z^3 + 54600*x^2*y^2*z^2 - 43824*x^2*y^2*z + 12384*x^2*y^2 + 23520*x^2*y*z^4 - 72240*x^2*y*z^3 + 86136*x^2*y*z^2 - 47940*x^2*y*z + 10738*x^2*y - 9408*x^2*z^5 + 35840*x^2*z^4 - 56416*x^2*z^3 + 46386*x^2*z^2 - 20300*x^2*z + 3892*x^2 - 98*x*y^6 + 1176*x*y^5*z - 924*x*y^5 - 5880*x*y^4*z^2 + 9170*x*y^4*z - 3715*x*y^4 + 15680*x*y^3*z^3 - 36400*x*y^3*z^2 + 29216*x*y^3*z - 8256*x*y^3 - 23520*x*y^2*z^4 + 72240*x*y^2*z^3 - 86136*x*y^2*z^2 + 47940*x*y^2*z - 10724*x*y^2 + 18816*x*y*z^5 - 71680*x*y*z^4 + 112832*x*y*z^3 - 92772*x*y*z^2 + 40544*x*y*z - 7728*x*y - 6272*x*z^6 + 28448*x*z^5 - 55408*x*z^4 + 59832*x*z^3 - 38360*x*z^2 + 14154*x*z - 2401*x + 14*y^7 - 196*y^6*z + 154*y^6 + 1176*y^5*z^2 - 1834*y^5*z + 743*y^5 - 3920*y^4*z^3 + 9100*y^4*z^2 - 7304*y^4*z + 2064*y^4 + 7840*y^3*z^4 - 24080*y^3*z^3 + 28712*y^3*z^2 - 15980*y^3*z + 3570*y^3 - 9408*y^2*z^5 + 35840*y^2*z^4 - 56416*y^2*z^3 + 46386*y^2*z^2 - 20244*y^2*z + 3836*y^2 + 6272*y*z^6 - 28448*y*z^5 + 55408*y*z^4 - 59832*y*z^3 + 38304*y*z^2 - 14056*y*z + 2358*y - 1792*z^7 + 9408*z^6 - 21760*z^5 + 28936*z^4 - 24178*z^3 + 12922*z^2 - 4170*z + 1, -21*x^7 + 147*x^6*y - 294*x^6*z + 231*x^6 - 441*x^5*y^2 + 1764*x^5*y*z - 1386*x^5*y - 1764*x^5*z^2 + 2751*x^5*z - 1116*x^5 + 735*x^4*y^3 - 4410*x^4*y^2*z + 3465*x^4*y^2 + 8820*x^4*y*z^2 - 13755*x^4*y*z + 5580*x^4*y - 5880*x^4*z^3 + 13650*x^4*z^2 - 10971*x^4*z + 3105*x^4 - 735*x^3*y^4 + 5880*x^3*y^3*z - 4620*x^3*y^3 - 17640*x^3*y^2*z^2 + 27510*x^3*y^2*z - 11160*x^3*y^2 + 23520*x^3*y*z^3 - 54600*x^3*y*z^2 + 43884*x^3*y*z - 12420*x^3*y - 11760*x^3*z^4 + 36120*x^3*z^3 - 43128*x^3*z^2 + 24042*x^3*z - 5397*x^3 + 441*x^2*y^5 - 4410*x^2*y^4*z + 3465*x^2*y^4 + 17640*x^2*y^3*z^2 - 27510*x^2*y^3*z + 11160*x^2*y^3 - 35280*x^2*y^2*z^3 + 81900*x^2*y^2*z^2 - 65826*x^2*y^2*z + 18630*x^2*y^2 + 35280*x^2*y*z^4 - 108360*x^2*y*z^3 + 129384*x^2*y*z^2 - 72126*x^2*y*z + 16170*x^2*y - 14112*x^2*z^5 + 53760*x^2*z^4 - 84744*x^2*z^3 + 69795*x^2*z^2 - 30576*x^2*z + 5861*x^2 - 147*x*y^6 + 1764*x*y^5*z - 1386*x*y^5 - 8820*x*y^4*z^2 + 13755*x*y^4*z - 5580*x*y^4 + 23520*x*y^3*z^3 - 54600*x*y^3*z^2 + 43884*x*y^3*z - 12420*x*y^3 - 35280*x*y^2*z^4 + 108360*x*y^2*z^3 - 129384*x*y^2*z^2 + 72126*x*y^2*z - 16149*x*y^2 + 28224*x*y*z^5 - 107520*x*y*z^4 + 169488*x*y*z^3 - 139590*x*y*z^2 + 61068*x*y*z - 11638*x*y - 9408*x*z^6 + 42672*x*z^5 - 83232*x*z^4 + 90036*x*z^3 - 57792*x*z^2 + 21323*x*z - 3617*x + 21*y^7 - 294*y^6*z + 231*y^6 + 1764*y^5*z^2 - 2751*y^5*z + 1116*y^5 - 5880*y^4*z^3 + 13650*y^4*z^2 - 10971*y^4*z + 3105*y^4 + 11760*y^3*z^4 - 36120*y^3*z^3 + 43128*y^3*z^2 - 24042*y^3*z + 5376*y^3 - 14112*y^2*z^5 + 53760*y^2*z^4 - 84744*y^2*z^3 + 69795*y^2*z^2 - 30492*y^2*z + 5777*y^2 + 9408*y*z^6 - 42672*y*z^5 + 83232*y*z^4 - 90036*y*z^3 + 57708*y*z^2 - 21176*y*z + 3551*y - 2688*z^7 + 14112*z^6 - 32688*z^5 + 43548*z^4 - 36435*z^3 + 19475*z^2 - 6281*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -1, 1, -2 ], [ -3, 2, -2 ], [ -1, 1, -1 ] ], "source_shift": [ 1, 2, 1 ], "target_matrix": [ [ 1, 1, 1 ], [ 2, 1, 1 ], [ 3, 3, 2 ] ], "target_shift": [ -314, -630, -950 ], "inverse_recipe": { "p": 5, "q": 2, "f": "x**5 + x**4 + x", "g": "x**2 + x", "h": "y*z + z**3 + z", "polynomials": { "f": [ [ [ 5, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ], [ [ 4, 0, 0 ], 1 ] ], "g": [ [ [ 2, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 0, 3 ], 1 ], [ [ 0, 0, 1 ], 1 ] ] }, "h_multiplier": 7 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 0.8547349190012028 ], "surface_features": [ 4.291508954676907, 5.229138954938684, 3.962369335670021, 0.0, 4.42003717063912, 5153 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C084", "pair": 84, "family": "counterexample", "mapping": "F(x,y,z)=(-4*x^7 - 38*x^6*y - 26*x^6*z - 38*x^6 - 150*x^5*y^2 - 210*x^5*y*z - 309*x^5*y - 72*x^5*z^2 - 213*x^5*z - 162*x^5 - 318*x^4*y^3 - 684*x^4*y^2*z - 1014*x^4*y^2 - 480*x^4*y*z^2 - 1431*x^4*y*z - 1092*x^4*y - 110*x^4*z^3 - 495*x^4*z^2 - 759*x^4*z - 398*x^4 - 390*x^3*y^4 - 1146*x^3*y^3*z - 1713*x^3*y^3 - 1236*x^3*y^2*z^2 - 3717*x^3*y^2*z - 2844*x^3*y^2 - 580*x^3*y*z^3 - 2634*x^3*y*z^2 - 4056*x^3*y*z - 2125*x^3*y - 100*x^3*z^4 - 610*x^3*z^3 - 1416*x^3*z^2 - 1493*x^3*z - 608*x^3 - 276*x^2*y^5 - 1038*x^2*y^4*z - 1566*x^2*y^4 - 1530*x^2*y^3*z^2 - 4647*x^2*y^3*z - 3564*x^2*y^3 - 1104*x^2*y^2*z^3 - 5067*x^2*y^2*z^2 - 7833*x^2*y^2*z - 4092*x^2*y^2 - 390*x^2*y*z^4 - 2406*x^2*y*z^3 - 5616*x^2*y*z^2 - 5919*x^2*y*z - 2394*x^2*y - 54*x^2*z^5 - 420*x^2*z^4 - 1314*x^2*z^3 - 2091*x^2*z^2 - 1707*x^2*z - 578*x^2 - 104*x*y^6 - 480*x*y^5*z - 732*x*y^5 - 906*x*y^4*z^2 - 2784*x*y^4*z - 2142*x*y^4 - 894*x*y^3*z^3 - 4155*x*y^3*z^2 - 6450*x*y^3*z - 3355*x*y^3 - 486*x*y^2*z^4 - 3039*x*y^2*z^3 - 7134*x*y^2*z^2 - 7503*x*y^2*z - 3000*x*y^2 - 138*x*y*z^5 - 1089*x*y*z^4 - 3432*x*y*z^3 - 5463*x*y*z^2 - 4428*x*y*z - 1477*x*y - 16*x*z^6 - 153*x*z^5 - 606*x*z^4 - 1295*x*z^3 - 1590*x*z^2 - 1075*x*z - 318*x - 16*y^7 - 88*y^6*z - 136*y^6 - 204*y^5*z^2 - 636*y^5*z - 492*y^5 - 258*y^4*z^3 - 1218*y^4*z^2 - 1902*y^4*z - 986*y^4 - 192*y^3*z^4 - 1221*y^3*z^3 - 2886*y^3*z^2 - 3027*y^3*z - 1194*y^3 - 84*y^2*z^5 - 675*y^2*z^4 - 2145*y^2*z^3 - 3411*y^2*z^2 - 2733*y^2*z - 890*y^2 - 20*y*z^6 - 195*y*z^5 - 780*y*z^4 - 1669*y*z^3 - 2034*y*z^2 - 1351*y*z - 388*y - 2*z^7 - 23*z^6 - 111*z^5 - 299*z^4 - 491*z^3 - 497*z^2 - 291*z - 1, -2*x^7 - 19*x^6*y - 13*x^6*z - 28*x^6 - 75*x^5*y^2 - 105*x^5*y*z - 231*x^5*y - 36*x^5*z^2 - 156*x^5*z - 144*x^5 - 159*x^4*y^3 - 342*x^4*y^2*z - 768*x^4*y^2 - 240*x^4*y*z^2 - 1062*x^4*y*z - 987*x^4*y - 55*x^4*z^3 - 360*x^4*z^2 - 672*x^4*z - 396*x^4 - 195*x^3*y^4 - 573*x^3*y^3*z - 1311*x^3*y^3 - 618*x^3*y^2*z^2 - 2790*x^3*y^2*z - 2610*x^3*y^2 - 290*x^3*y*z^3 - 1938*x^3*y*z^2 - 3648*x^3*y*z - 2154*x^3*y - 50*x^3*z^4 - 440*x^3*z^3 - 1248*x^3*z^2 - 1481*x^3*z - 650*x^3 - 138*x^2*y^5 - 519*x^2*y^4*z - 1206*x^2*y^4 - 765*x^2*y^3*z^2 - 3516*x^2*y^3*z - 3312*x^2*y^3 - 552*x^2*y^2*z^3 - 3762*x^2*y^2*z^2 - 7143*x^2*y^2*z - 4218*x^2*y^2 - 195*x^2*y*z^4 - 1752*x^2*y*z^3 - 5022*x^2*y*z^2 - 5976*x^2*y*z - 2613*x^2*y - 27*x^2*z^5 - 300*x^2*z^4 - 1152*x^2*z^3 - 2067*x^2*z^2 - 1821*x^2*z - 649*x^2 - 52*x*y^6 - 240*x*y^5*z - 564*x*y^5 - 453*x*y^4*z^2 - 2112*x*y^4*z - 2007*x*y^4 - 447*x*y^3*z^3 - 3099*x*y^3*z^2 - 5943*x*y^3*z - 3504*x*y^3 - 243*x*y^2*z^4 - 2226*x*y^2*z^3 - 6456*x*y^2*z^2 - 7692*x*y^2*z - 3336*x*y^2 - 69*x*y*z^5 - 783*x*y*z^4 - 3048*x*y*z^3 - 5490*x*y*z^2 - 4818*x*y*z - 1700*x*y - 8*x*z^6 - 108*x*z^5 - 528*x*z^4 - 1275*x*z^3 - 1692*x*z^2 - 1205*x*z - 366*x - 8*y^7 - 44*y^6*z - 104*y^6 - 102*y^5*z^2 - 480*y^5*z - 462*y^5 - 129*y^4*z^3 - 906*y^4*z^2 - 1761*y^4*z - 1038*y^4 - 96*y^3*z^4 - 894*y^3*z^3 - 2631*y^3*z^2 - 3137*y^3*z - 1346*y^3 - 42*y^2*z^5 - 486*y^2*z^4 - 1923*y^2*z^3 - 3474*y^2*z^2 - 3024*y^2*z - 1045*y^2 - 10*y*z^6 - 138*y*z^5 - 687*y*z^4 - 1668*y*z^3 - 2205*y*z^2 - 1550*y*z - 461*y - z^7 - 16*z^6 - 96*z^5 - 293*z^4 - 521*z^3 - 556*z^2 - 335*z - 1, 2*x^7 + 19*x^6*y + 13*x^6*z + 22*x^6 + 75*x^5*y^2 + 105*x^5*y*z + 180*x^5*y + 36*x^5*z^2 + 123*x^5*z + 102*x^5 + 159*x^4*y^3 + 342*x^4*y^2*z + 594*x^4*y^2 + 240*x^4*y*z^2 + 831*x^4*y*z + 693*x^4*y + 55*x^4*z^3 + 285*x^4*z^2 + 477*x^4*z + 264*x^4 + 195*x^3*y^4 + 573*x^3*y^3*z + 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96*y^3*z^4 + 705*y^3*z^3 + 1839*y^3*z^2 + 2048*y^3*z + 836*y^3 + 42*y^2*z^5 + 387*y^2*z^4 + 1356*y^2*z^3 + 2289*y^2*z^2 + 1899*y^2*z + 631*y^2 + 10*y*z^6 + 111*y*z^5 + 489*y*z^4 + 1110*y*z^3 + 1401*y*z^2 + 950*y*z + 277*y + z^7 + 13*z^6 + 69*z^5 + 197*z^4 + 335*z^3 + 346*z^2 + 205*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ 1, 2, 1 ], [ 1, 1, 1 ], [ 2, 1, 1 ] ], "source_shift": [ 1, 1, 1 ], "target_matrix": [ [ -2, -1, -1 ], [ -1, -2, -1 ], [ 1, 1, 1 ] ], "target_shift": [ 75, 87, -53 ], "collision_preimages": [ [ "-1/4", "0", "-3/4" ], [ "8", "5/2", "-13" ], [ "5", "-5/2", "-2" ] ], "collision_image": [ "151/2", "349/4", "-213/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 0.9333106119993317 ], "surface_features": [ 3.1083695547501957, 3.893928126542607, 2.807873132003332, 0.0, 3.191738207620116, 4760 ], "component_degrees": [ 7, 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7600*x^3*z - 3438*x^3 + 210*x^2*y^5 + 1050*x^2*y^4*z - 2000*x^2*y^4 + 2100*x^2*y^3*z^2 - 7800*x^2*y^3*z + 7350*x^2*y^3 + 2100*x^2*y^2*z^3 - 11400*x^2*y^2*z^2 + 21030*x^2*y^2*z - 13200*x^2*y^2 + 1050*x^2*y*z^4 - 7400*x^2*y*z^3 + 20040*x^2*y*z^2 - 24680*x^2*y*z + 11514*x^2*y + 210*x^2*z^5 - 1800*x^2*z^4 + 6360*x^2*z^3 - 11520*x^2*z^2 + 10554*x^2*z - 3833*x^2 - 70*x*y^6 - 420*x*y^5*z + 820*x*y^5 - 1050*x*y^4*z^2 + 4000*x*y^4*z - 3860*x*y^4 - 1400*x*y^3*z^3 + 7800*x*y^3*z^2 - 14740*x*y^3*z + 9500*x*y^3 - 1050*x*y^2*z^4 + 7600*x*y^2*z^3 - 21090*x*y^2*z^2 + 26680*x*y^2*z - 12864*x*y^2 - 420*x*y*z^5 + 3700*x*y*z^4 - 13400*x*y*z^3 + 24940*x*y*z^2 - 23588*x*y*z + 8866*x*y - 70*x*z^6 + 720*x*z^5 - 3190*x*z^4 + 7760*x*z^3 - 10794*x*z^2 + 7986*x*z - 2392*x + 10*y^7 + 70*y^6*z - 140*y^6 + 210*y^5*z^2 - 820*y^5*z + 810*y^5 + 350*y^4*z^3 - 2000*y^4*z^2 + 3870*y^4*z - 2560*y^4 + 350*y^3*z^4 - 2600*y^3*z^3 + 7390*y^3*z^2 - 9600*y^3*z + 4798*y^3 + 210*y^2*z^5 - 1900*y^2*z^4 + 7050*y^2*z^3 - 13480*y^2*z^2 + 13194*y^2*z - 5153*y^2 + 70*y*z^6 - 740*y*z^5 + 3360*y*z^4 - 8400*y*z^3 + 12074*y*z^2 - 9266*y*z + 2871*y + 10*z^7 - 120*z^6 + 640*z^5 - 1960*z^4 + 3678*z^3 - 4153*z^2 + 2550*z - 1, 5*x^7 - 35*x^6*y - 35*x^6*z + 60*x^6 + 105*x^5*y^2 + 210*x^5*y*z - 370*x^5*y + 105*x^5*z^2 - 360*x^5*z + 315*x^5 - 175*x^4*y^3 - 525*x^4*y^2*z + 950*x^4*y^2 - 525*x^4*y*z^2 + 1850*x^4*y*z - 1660*x^4*y - 175*x^4*z^3 + 900*x^4*z^2 - 1580*x^4*z + 940*x^4 + 175*x^3*y^4 + 700*x^3*y^3*z - 1300*x^3*y^3 + 1050*x^3*y^2*z^2 - 3800*x^3*y^2*z + 3495*x^3*y^2 + 700*x^3*y*z^3 - 3700*x^3*y*z^2 + 6660*x^3*y*z - 4070*x^3*y + 175*x^3*z^4 - 1200*x^3*z^3 + 3170*x^3*z^2 - 3800*x^3*z + 1718*x^3 - 105*x^2*y^5 - 525*x^2*y^4*z + 1000*x^2*y^4 - 1050*x^2*y^3*z^2 + 3900*x^2*y^3*z - 3675*x^2*y^3 - 1050*x^2*y^2*z^3 + 5700*x^2*y^2*z^2 - 10515*x^2*y^2*z + 6600*x^2*y^2 - 525*x^2*y*z^4 + 3700*x^2*y*z^3 - 10020*x^2*y*z^2 + 12340*x^2*y*z - 5754*x^2*y - 105*x^2*z^5 + 900*x^2*z^4 - 3180*x^2*z^3 + 5760*x^2*z^2 - 5274*x^2*z + 1914*x^2 + 35*x*y^6 + 210*x*y^5*z - 410*x*y^5 + 525*x*y^4*z^2 - 2000*x*y^4*z + 1930*x*y^4 + 700*x*y^3*z^3 - 3900*x*y^3*z^2 + 7370*x*y^3*z - 4750*x*y^3 + 525*x*y^2*z^4 - 3800*x*y^2*z^3 + 10545*x*y^2*z^2 - 13340*x*y^2*z + 6429*x*y^2 + 210*x*y*z^5 - 1850*x*y*z^4 + 6700*x*y*z^3 - 12470*x*y*z^2 + 11788*x*y*z - 4428*x*y + 35*x*z^6 - 360*x*z^5 + 1595*x*z^4 - 3880*x*z^3 + 5394*x*z^2 - 3988*x*z + 1195*x - 5*y^7 - 35*y^6*z + 70*y^6 - 105*y^5*z^2 + 410*y^5*z - 405*y^5 - 175*y^4*z^3 + 1000*y^4*z^2 - 1935*y^4*z + 1280*y^4 - 175*y^3*z^4 + 1300*y^3*z^3 - 3695*y^3*z^2 + 4800*y^3*z - 2398*y^3 - 105*y^2*z^5 + 950*y^2*z^4 - 3525*y^2*z^3 + 6740*y^2*z^2 - 6594*y^2*z + 2574*y^2 - 35*y*z^6 + 370*y*z^5 - 1680*y*z^4 + 4200*y*z^3 - 6034*y*z^2 + 4628*y*z - 1435*y - 5*z^7 + 60*z^6 - 320*z^5 + 980*z^4 - 1838*z^3 + 2074*z^2 - 1273*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ 1, -1, -1 ], [ 1, -2, -1 ], [ 2, -3, -3 ] ], "source_shift": [ 1, 2, 2 ], "target_matrix": [ [ -1, -1, -1 ], [ -2, -1, -1 ], [ 1, 2, 1 ] ], "target_shift": [ 316, 637, -320 ], "inverse_recipe": { "p": 2, "q": 3, "f": "x**2 + x", "g": "x**3 + x**2", "h": "y**2*z", "polynomials": { "f": [ [ [ 2, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "g": [ [ [ 3, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "h": [ [ [ 0, 2, 1 ], 1 ] ] }, "h_multiplier": 5 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 0.9765963999998348 ], "surface_features": [ 3.528312021992966, 4.426185825244511, 3.2227164711475833, 0.0, 3.647050317277614, 4896 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C114", "pair": 114, "family": "counterexample", "mapping": "F(x,y,z)=(-24*x^7 + 196*x^6*y - 124*x^6*z + 344*x^6 - 642*x^5*y^2 + 840*x^5*y*z - 2220*x^5*y - 270*x^5*z^2 + 1464*x^5*z - 1638*x^5 + 1083*x^4*y^3 - 2211*x^4*y^2*z + 5538*x^4*y^2 + 1473*x^4*y*z^2 - 7596*x^4*y*z + 8271*x^4*y - 321*x^4*z^3 + 2550*x^4*z^2 - 5637*x^4*z + 3938*x^4 - 1012*x^3*y^4 + 2875*x^3*y^3*z - 6845*x^3*y^3 - 2991*x^3*y^2*z^2 + 14646*x^3*y^2*z - 15678*x^3*y^2 + 1353*x^3*y*z^3 - 10221*x^3*y*z^2 + 22089*x^3*y*z - 15376*x^3*y - 225*x^3*z^4 + 2328*x^3*z^3 - 7641*x^3*z^2 + 10663*x^3*z - 5545*x^3 + 528*x^2*y^5 - 1950*x^2*y^4*z + 4494*x^2*y^4 + 2817*x^2*y^3*z^2 - 13215*x^2*y^3*z + 14184*x^2*y^3 - 1989*x^2*y^2*z^3 + 14328*x^2*y^2*z^2 - 30681*x^2*y^2*z + 21549*x^2*y^2 + 687*x^2*y*z^4 - 6771*x^2*y*z^3 + 21849*x^2*y*z^2 - 30522*x^2*y*z + 15987*x^2*y - 93*x^2*z^5 + 1176*x^2*z^4 - 5106*x^2*z^3 + 10692*x^2*z^2 - 11157*x^2*z + 4690*x^2 - 144*x*y^6 + 660*x*y^5*z - 1500*x*y^5 - 1236*x*y^4*z^2 + 5652*x*y^4*z - 6120*x*y^4 + 1209*x*y^3*z^3 - 8403*x*y^3*z^2 + 18060*x*y^3*z - 12769*x*y^3 - 651*x*y^2*z^4 + 6150*x*y^2*z^3 - 19752*x*y^2*z^2 + 27765*x*y^2*z - 14634*x*y^2 + 183*x*y*z^5 - 2211*x*y*z^4 + 9483*x*y*z^3 - 19899*x*y*z^2 + 20871*x*y*z - 8848*x*y - 21*x*z^6 + 312*x*z^5 - 1683*x*z^4 + 4700*x*z^3 - 7368*x*z^2 + 6215*x*z - 2216*x + 16*y^7 - 88*y^6*z + 200*y^6 + 204*y^5*z^2 - 924*y^5*z + 1008*y^5 - 258*y^4*z^3 + 1758*y^4*z^2 - 3804*y^4*z + 2690*y^4 + 192*y^3*z^4 - 1761*y^3*z^3 + 5676*y^3*z^2 - 8000*y^3*z + 4199*y^3 - 84*y^2*z^5 + 978*y^2*z^4 - 4185*y^2*z^3 + 8811*y^2*z^2 - 9246*y^2*z + 3895*y^2 + 20*y*z^6 - 285*y*z^5 + 1524*y*z^4 - 4262*y*z^3 + 6693*y*z^2 - 5647*y*z + 2003*y - 2*z^7 + 34*z^6 - 219*z^5 + 764*z^4 - 1596*z^3 + 2017*z^2 - 1433*z + 1, 48*x^7 - 392*x^6*y + 248*x^6*z - 544*x^6 + 1284*x^5*y^2 - 1680*x^5*y*z + 3552*x^5*y + 540*x^5*z^2 - 2328*x^5*z + 2364*x^5 - 2166*x^4*y^3 + 4422*x^4*y^2*z - 9000*x^4*y^2 - 2946*x^4*y*z^2 + 12240*x^4*y*z - 12150*x^4*y + 642*x^4*z^3 - 4080*x^4*z^2 + 8202*x^4*z - 5392*x^4 + 2024*x^3*y^4 - 5750*x^3*y^3*z + 11344*x^3*y^3 + 5982*x^3*y^2*z^2 - 24006*x^3*y^2*z + 23520*x^3*y^2 - 2706*x^3*y*z^3 + 16596*x^3*y*z^2 - 32754*x^3*y*z + 21538*x^3*y + 450*x^3*z^4 - 3750*x^3*z^3 + 11214*x^3*z^2 - 14752*x^3*z + 7362*x^3 - 1056*x^2*y^5 + 3900*x^2*y^4*z - 7608*x^2*y^4 - 5634*x^2*y^3*z^2 + 22098*x^2*y^3*z - 21738*x^2*y^3 + 3978*x^2*y^2*z^3 - 23688*x^2*y^2*z^2 + 46458*x^2*y^2*z - 30834*x^2*y^2 - 1374*x^2*y*z^4 + 11082*x^2*y*z^3 - 32700*x^2*y*z^2 + 43164*x^2*y*z - 21720*x^2*y + 186*x^2*z^5 - 1908*x^2*z^4 + 7560*x^2*z^3 - 14940*x^2*z^2 + 14970*x^2*z - 6104*x^2 + 288*x*y^6 - 1320*x*y^5*z + 2592*x*y^5 + 2472*x*y^4*z^2 - 9648*x*y^4*z + 9576*x*y^4 - 2418*x*y^3*z^3 + 14172*x*y^3*z^2 - 27924*x*y^3*z + 18659*x*y^3 + 1302*x*y^2*z^4 - 10254*x*y^2*z^3 + 30180*x*y^2*z^2 - 40101*x*y^2*z + 20334*x*y^2 - 366*x*y*z^5 + 3648*x*y*z^4 - 14322*x*y*z^3 + 28407*x*y*z^2 - 28650*x*y*z + 11807*x*y + 42*x*z^6 - 510*x*z^5 + 2514*x*z^4 - 6631*x*z^3 + 9990*x*z^2 - 8185*x*z + 2868*x - 32*y^7 + 176*y^6*z - 352*y^6 - 408*y^5*z^2 + 1608*y^5*z - 1608*y^5 + 516*y^4*z^3 - 3024*y^4*z^2 + 6000*y^4*z - 4010*y^4 - 384*y^3*z^4 + 2994*y^3*z^3 - 8850*y^3*z^2 + 11792*y^3*z - 5961*y^3 + 168*y^2*z^5 - 1644*y^2*z^4 + 6450*y^2*z^3 - 12840*y^2*z^2 + 12972*y^2*z - 5327*y^2 - 40*y*z^6 + 474*y*z^5 - 2322*y*z^4 + 6140*y*z^3 - 9279*y*z^2 + 7622*y*z - 2667*y + 4*z^7 - 56*z^6 + 330*z^5 - 1088*z^4 + 2186*z^3 - 2687*z^2 + 1880*z - 1, 24*x^7 - 196*x^6*y + 124*x^6*z - 272*x^6 + 642*x^5*y^2 - 840*x^5*y*z + 1776*x^5*y + 270*x^5*z^2 - 1164*x^5*z + 1182*x^5 - 1083*x^4*y^3 + 2211*x^4*y^2*z - 4500*x^4*y^2 - 1473*x^4*y*z^2 + 6120*x^4*y*z - 6075*x^4*y + 321*x^4*z^3 - 2040*x^4*z^2 + 4101*x^4*z - 2708*x^4 + 1012*x^3*y^4 - 2875*x^3*y^3*z + 5672*x^3*y^3 + 2991*x^3*y^2*z^2 - 12003*x^3*y^2*z + 11760*x^3*y^2 - 1353*x^3*y*z^3 + 8298*x^3*y*z^2 - 16377*x^3*y*z + 10795*x^3*y + 225*x^3*z^4 - 1875*x^3*z^3 + 5607*x^3*z^2 - 7402*x^3*z + 3709*x^3 - 528*x^2*y^5 + 1950*x^2*y^4*z - 3804*x^2*y^4 - 2817*x^2*y^3*z^2 + 11049*x^2*y^3*z - 10869*x^2*y^3 + 1989*x^2*y^2*z^3 - 11844*x^2*y^2*z^2 + 23229*x^2*y^2*z - 15438*x^2*y^2 - 687*x^2*y*z^4 + 5541*x^2*y*z^3 - 16350*x^2*y*z^2 + 21624*x^2*y*z - 10914*x^2*y + 93*x^2*z^5 - 954*x^2*z^4 + 3780*x^2*z^3 - 7491*x^2*z^2 + 7533*x^2*z - 3085*x^2 + 144*x*y^6 - 660*x*y^5*z + 1296*x*y^5 + 1236*x*y^4*z^2 - 4824*x*y^4*z + 4788*x*y^4 - 1209*x*y^3*z^3 + 7086*x*y^3*z^2 - 13962*x*y^3*z + 9337*x*y^3 + 651*x*y^2*z^4 - 5127*x*y^2*z^3 + 15090*x*y^2*z^2 - 20073*x*y^2*z + 10200*x*y^2 - 183*x*y*z^5 + 1824*x*y*z^4 - 7161*x*y*z^3 + 14226*x*y*z^2 - 14385*x*y*z + 5947*x*y + 21*x*z^6 - 255*x*z^5 + 1257*x*z^4 - 3323*x*z^3 + 5022*x*z^2 - 4130*x*z + 1450*x - 16*y^7 + 88*y^6*z - 176*y^6 - 204*y^5*z^2 + 804*y^5*z - 804*y^5 + 258*y^4*z^3 - 1512*y^4*z^2 + 3000*y^4*z - 2006*y^4 - 192*y^3*z^4 + 1497*y^3*z^3 - 4425*y^3*z^2 + 5900*y^3*z - 2987*y^3 + 84*y^2*z^5 - 822*y^2*z^4 + 3225*y^2*z^3 - 6426*y^2*z^2 + 6504*y^2*z - 2677*y^2 - 20*y*z^6 + 237*y*z^5 - 1161*y*z^4 + 3074*y*z^3 - 4656*y*z^2 + 3835*y*z - 1344*y + 2*z^7 - 28*z^6 + 165*z^5 - 545*z^4 + 1098*z^3 - 1354*z^2 + 949*z - 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -2, 1, -1 ], [ -1, 2, -1 ], [ -3, 2, -2 ] ], "source_shift": [ 1, 2, 1 ], "target_matrix": [ [ 1, 2, 1 ], [ -2, -2, -3 ], [ -1, -1, -1 ] ], "target_shift": [ -439, 576, 290 ], "collision_preimages": [ [ "3/4", "-7/4", "-9/4" ], [ "11/2", "-9", "-20" ], [ "19/2", "-8", "-25" ] ], "collision_image": [ "-1757/4", "1153/2", "1161/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 0.9918873400019947 ], "surface_features": [ 3.785102937967266, 4.667060509647175, 3.4586378490256493, 0.0, 3.89229277296746, 4994 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B073", "pair": 73, "family": "bijection", "mapping": "F(x,y,z)=(-2*x^7 - 14*x^6*y - 28*x^6*z - 34*x^6 - 42*x^5*y^2 - 168*x^5*y*z - 204*x^5*y - 168*x^5*z^2 - 408*x^5*z - 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explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ 1, 2, 1 ], [ 1, 1, 1 ], [ 2, 1, 1 ] ], "source_shift": [ 1, 2, 1 ], "target_matrix": [ [ 1, 2, 1 ], [ 1, 1, 1 ], [ 2, 1, 1 ] ], "target_shift": [ -12, -8, -17 ], "inverse_recipe": { "p": 6, "q": 1, "f": "x**6 + x**5", "g": "x", "h": "y*z", "polynomials": { "f": [ [ [ 6, 0, 0 ], 1 ], [ [ 5, 0, 0 ], 1 ] ], "g": [ [ [ 1, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "Maybe" ], "seconds": [ 1.0332655490019533 ], "surface_features": [ 3.36022671896952, 4.249442961442583, 3.0242803760470798, 0.0, 3.4711740918258838, 4825 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C236", "pair": 236, "family": "counterexample", "mapping": "F(x,y,z)=(648*x^7 - 2916*x^6*y + 3564*x^6*z - 4860*x^6 + 5562*x^5*y^2 - 13608*x^5*y*z + 18036*x^5*y + 8262*x^5*z^2 - 22032*x^5*z + 14094*x^5 - 5835*x^4*y^3 + 21411*x^4*y^2*z - 27747*x^4*y^2 - 26001*x^4*y*z^2 + 67770*x^4*y*z - 42741*x^4*y + 10449*x^4*z^3 - 41067*x^4*z^2 + 52191*x^4*z - 21495*x^4 + 3639*x^3*y^4 - 17787*x^3*y^3*z + 22650*x^3*y^3 + 32373*x^3*y^2*z^2 - 82917*x^3*y^2*z + 51669*x^3*y^2 - 26001*x^3*y*z^3 + 100440*x^3*y*z^2 - 126027*x^3*y*z + 51486*x^3*y + 7776*x^3*z^4 - 40257*x^3*z^3 + 76248*x^3*z^2 - 62697*x^3*z + 19062*x^3 - 1350*x^2*y^5 + 8235*x^2*y^4*z - 10347*x^2*y^4 - 19953*x^2*y^3*z^2 + 50442*x^2*y^3*z - 31119*x^2*y^3 + 24003*x^2*y^2*z^3 - 91557*x^2*y^2*z^2 + 113679*x^2*y^2*z - 46113*x^2*y^2 - 14337*x^2*y*z^4 + 73332*x^2*y*z^3 - 137376*x^2*y*z^2 + 112086*x^2*y*z - 33906*x^2*y + 3402*x^2*z^5 - 21870*x^2*z^4 + 54918*x^2*z^3 - 67554*x^2*z^2 + 41055*x^2*z - 9981*x^2 + 276*x*y^6 - 2016*x*y^5*z + 2508*x*y^5 + 6093*x*y^4*z^2 - 15258*x*y^4*z + 9336*x*y^4 - 9753*x*y^3*z^3 + 36870*x*y^3*z^2 - 45390*x*y^3*z + 18299*x*y^3 + 8721*x*y^2*z^4 - 44235*x*y^2*z^3 + 82143*x*y^2*z^2 - 66573*x*y^2*z + 20046*x*y^2 - 4131*x*y*z^5 + 26352*x*y*z^4 - 65583*x*y*z^3 + 80091*x*y*z^2 - 48417*x*y*z + 11733*x*y + 810*x*z^6 - 6237*x*z^5 + 19494*x*z^4 - 31866*x*z^3 + 28977*x*z^2 - 14076*x*z + 2901*x - 24*y^7 + 204*y^6*z - 252*y^6 - 738*y^5*z^2 + 1836*y^5*z - 1116*y^5 + 1473*y^4*z^3 - 5535*y^4*z^2 + 6768*y^4*z - 2714*y^4 - 1752*y^3*z^4 + 8838*y^3*z^3 - 16299*y^3*z^2 + 13133*y^3*z - 3938*y^3 + 1242*y^2*z^5 - 7884*y^2*z^4 + 19485*y^2*z^3 - 23646*y^2*z^2 + 14226*y^2*z - 3438*y^2 - 486*y*z^6 + 3726*y*z^5 - 11565*y*z^4 + 18777*y*z^3 - 16983*y*z^2 + 8223*y*z - 1695*y + 81*z^7 - 729*z^6 + 2727*z^5 - 5550*z^4 + 6702*z^3 - 4869*z^2 + 2011*z - 1, 216*x^7 - 972*x^6*y + 1188*x^6*z - 2052*x^6 + 1854*x^5*y^2 - 4536*x^5*y*z + 7740*x^5*y + 2754*x^5*z^2 - 9504*x^5*z + 7290*x^5 - 1945*x^4*y^3 + 7137*x^4*y^2*z - 12093*x^4*y^2 - 8667*x^4*y*z^2 + 29718*x^4*y*z - 22383*x^4*y + 3483*x^4*z^3 - 18117*x^4*z^2 + 27549*x^4*z - 12592*x^4 + 1213*x^3*y^4 - 5929*x^3*y^3*z + 10018*x^3*y^3 + 10791*x^3*y^2*z^2 - 36927*x^3*y^2*z + 27387*x^3*y^2 - 8667*x^3*y*z^3 + 45036*x^3*y*z^2 - 67353*x^3*y*z + 30467*x^3*y + 2592*x^3*z^4 - 18171*x^3*z^3 + 41076*x^3*z^2 - 37423*x^3*z + 12137*x^3 - 450*x^2*y^5 + 2745*x^2*y^4*z - 4641*x^2*y^4 - 6651*x^2*y^3*z^2 + 22794*x^2*y^3*z - 16689*x^2*y^3 + 8001*x^2*y^2*z^3 - 41679*x^2*y^2*z^2 + 61485*x^2*y^2*z - 27561*x^2*y^2 - 4779*x^2*y*z^4 + 33624*x^2*y*z^3 - 74916*x^2*y*z^2 + 67581*x^2*y*z - 21789*x^2*y + 1134*x^2*z^5 - 10098*x^2*z^4 + 30186*x^2*z^3 - 41076*x^2*z^2 + 26631*x^2*z - 6760*x^2 + 92*x*y^6 - 672*x*y^5*z + 1140*x*y^5 + 2031*x*y^4*z^2 - 6990*x*y^4*z + 5064*x*y^4 - 3251*x*y^3*z^3 + 17022*x*y^3*z^2 - 24834*x*y^3*z + 11045*x*y^3 + 2907*x*y^2*z^4 - 20577*x*y^2*z^3 + 45321*x*y^2*z^2 - 40539*x*y^2*z + 13002*x*y^2 - 1377*x*y*z^5 + 12348*x*y*z^4 - 36477*x*y*z^3 + 49188*x*y*z^2 - 31701*x*y*z + 8023*x*y + 270*x*z^6 - 2943*x*z^5 + 10926*x*z^4 - 19730*x*z^3 + 19146*x*z^2 - 9730*x*z + 2061*x - 8*y^7 + 68*y^6*z - 116*y^6 - 246*y^5*z^2 + 852*y^5*z - 612*y^5 + 491*y^4*z^3 - 2589*y^4*z^2 + 3744*y^4*z - 1654*y^4 - 584*y^3*z^4 + 4166*y^3*z^3 - 9093*y^3*z^2 + 8075*y^3*z - 2578*y^3 + 414*y^2*z^5 - 3744*y^2*z^4 + 10959*y^2*z^3 - 14664*y^2*z^2 + 9402*y^2*z - 2374*y^2 - 162*y*z^6 + 1782*y*z^5 - 6555*y*z^4 + 11740*y*z^3 - 11328*y*z^2 + 5741*y*z - 1217*y + 27*z^7 - 351*z^6 + 1557*z^5 - 3497*z^4 + 4510*z^3 - 3436*z^2 + 1463*z - 1, -216*x^7 + 972*x^6*y - 1188*x^6*z + 1728*x^6 - 1854*x^5*y^2 + 4536*x^5*y*z - 6444*x^5*y - 2754*x^5*z^2 + 7884*x^5*z - 5346*x^5 + 1945*x^4*y^3 - 7137*x^4*y^2*z + 9960*x^4*y^2 + 8667*x^4*y*z^2 - 24372*x^4*y*z + 16281*x^4*y - 3483*x^4*z^3 + 14796*x^4*z^2 - 19935*x^4*z + 8515*x^4 - 1213*x^3*y^4 + 5929*x^3*y^3*z - 8167*x^3*y^3 - 10791*x^3*y^2*z^2 + 29961*x^3*y^2*z - 19764*x^3*y^2 + 8667*x^3*y*z^3 - 36369*x^3*y*z^2 + 48345*x^3*y*z - 20468*x^3*y - 2592*x^3*z^4 + 14607*x^3*z^3 - 29331*x^3*z^2 + 25003*x^3*z - 7766*x^3 + 450*x^2*y^5 - 2745*x^2*y^4*z + 3747*x^2*y^4 + 6651*x^2*y^3*z^2 - 18309*x^2*y^3*z + 11952*x^2*y^3 - 8001*x^2*y^2*z^3 + 33309*x^2*y^2*z^2 - 43791*x^2*y^2*z + 18396*x^2*y^2 + 4779*x^2*y*z^4 - 26739*x^2*y*z^3 + 53073*x^2*y*z^2 - 44856*x^2*y*z + 13854*x^2*y - 1134*x^2*z^5 + 7992*x^2*z^4 - 21276*x^2*z^3 + 27117*x^2*z^2 - 16827*x^2*z + 4138*x^2 - 92*x*y^6 + 672*x*y^5*z - 912*x*y^5 - 2031*x*y^4*z^2 + 5562*x*y^4*z - 3600*x*y^4 + 3251*x*y^3*z^3 - 13473*x*y^3*z^2 + 17556*x*y^3*z - 7325*x*y^3 - 2907*x*y^2*z^4 + 16203*x*y^2*z^3 - 31866*x*y^2*z^2 + 26733*x*y^2*z - 8214*x*y^2 + 1377*x*y*z^5 - 9675*x*y*z^4 + 25515*x*y*z^3 - 32259*x*y*z^2 + 19899*x*y*z - 4876*x*y - 270*x*z^6 + 2295*x*z^5 - 7605*x*z^4 + 12872*x*z^3 - 11943*x*z^2 + 5866*x*z - 1216*x + 8*y^7 - 68*y^6*z + 92*y^6 + 246*y^5*z^2 - 672*y^5*z + 432*y^5 - 491*y^4*z^3 + 2031*y^4*z^2 - 2628*y^4*z + 1090*y^4 + 584*y^3*z^4 - 3251*y^3*z^3 + 6348*y^3*z^2 - 5291*y^3*z + 1618*y^3 - 414*y^2*z^5 + 2907*y^2*z^4 - 7611*y^2*z^3 + 9555*y^2*z^2 - 5862*y^2*z + 1432*y^2 + 162*y*z^6 - 1377*y*z^5 + 4530*y*z^4 - 7609*y*z^3 + 7017*y*z^2 - 3434*y*z + 712*y - 27*z^7 + 270*z^6 - 1071*z^5 + 2255*z^4 - 2776*z^3 + 2038*z^2 - 847*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -3, 2, -3 ], [ -2, 1, -1 ], [ -1, 1, -1 ] ], "source_shift": [ 2, 1, 1 ], "target_matrix": [ [ -3, -2, -2 ], [ -1, -2, -1 ], [ 1, 1, 1 ] ], "target_shift": [ 368, 270, -155 ], "collision_preimages": [ [ "-1/4", "-7/4", "-1/4" ], [ "8", "35/2", "4" ], [ "5", "39/2", "9" ] ], "collision_image": [ "1475/4", "1081/4", "-621/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0266733399985242 ], "surface_features": [ 4.203379090069268, 5.137910866799345, 3.8937617620579434, 0.0, 4.332538298095683, 5132 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B190", "pair": 190, "family": "bijection", "mapping": "F(x,y,z)=(-3*x^7 - 21*x^6*y + 21*x^6*z + 51*x^6 - 63*x^5*y^2 + 126*x^5*y*z + 306*x^5*y - 63*x^5*z^2 - 309*x^5*z - 368*x^5 - 105*x^4*y^3 + 315*x^4*y^2*z + 765*x^4*y^2 - 315*x^4*y*z^2 - 1545*x^4*y*z - 1840*x^4*y + 105*x^4*z^3 + 780*x^4*z^2 + 1882*x^4*z + 1471*x^4 - 105*x^3*y^4 + 420*x^3*y^3*z + 1020*x^3*y^3 - 630*x^3*y^2*z^2 - 3090*x^3*y^2*z - 3680*x^3*y^2 + 420*x^3*y*z^3 + 3120*x^3*y*z^2 + 7528*x^3*y*z + 5884*x^3*y - 105*x^3*z^4 - 1050*x^3*z^3 - 3848*x^3*z^2 - 6118*x^3*z - 3549*x^3 - 63*x^2*y^5 + 315*x^2*y^4*z + 765*x^2*y^4 - 630*x^2*y^3*z^2 - 3090*x^2*y^3*z - 3680*x^2*y^3 + 630*x^2*y^2*z^3 + 4680*x^2*y^2*z^2 + 11292*x^2*y^2*z + 8826*x^2*y^2 - 315*x^2*y*z^4 - 3150*x^2*y*z^3 - 11544*x^2*y*z^2 - 18354*x^2*y*z - 10644*x^2*y + 63*x^2*z^5 + 795*x^2*z^4 + 3932*x^2*z^3 + 9537*x^2*z^2 + 11298*x^2*z + 5209*x^2 - 21*x*y^6 + 126*x*y^5*z + 306*x*y^5 - 315*x*y^4*z^2 - 1545*x*y^4*z - 1840*x*y^4 + 420*x*y^3*z^3 + 3120*x*y^3*z^2 + 7528*x*y^3*z + 5884*x*y^3 - 315*x*y^2*z^4 - 3150*x*y^2*z^3 - 11544*x*y^2*z^2 - 18354*x*y^2*z - 10641*x*y^2 + 126*x*y*z^5 + 1590*x*y*z^4 + 7864*x*y*z^3 + 19074*x*y*z^2 + 22590*x*y*z + 10403*x*y - 21*x*z^6 - 321*x*z^5 - 2008*x*z^4 - 6604*x*z^3 - 11994*x*z^2 - 11330*x*z - 4312*x - 3*y^7 + 21*y^6*z + 51*y^6 - 63*y^5*z^2 - 309*y^5*z - 368*y^5 + 105*y^4*z^3 + 780*y^4*z^2 + 1882*y^4*z + 1471*y^4 - 105*y^3*z^4 - 1050*y^3*z^3 - 3848*y^3*z^2 - 6118*y^3*z - 3546*y^3 + 63*y^2*z^5 + 795*y^2*z^4 + 3932*y^2*z^3 + 9537*y^2*z^2 + 11292*y^2*z + 5194*y^2 - 21*y*z^6 - 321*y*z^5 - 2008*y*z^4 - 6604*y*z^3 - 11991*y*z^2 - 11312*y*z - 4298*y + 3*z^7 + 54*z^6 + 410*z^5 + 1714*z^4 + 4248*z^3 + 6175*z^2 + 4827*z + 1, -3*x^7 - 21*x^6*y + 21*x^6*z + 51*x^6 - 63*x^5*y^2 + 126*x^5*y*z + 306*x^5*y - 63*x^5*z^2 - 309*x^5*z - 370*x^5 - 105*x^4*y^3 + 315*x^4*y^2*z + 765*x^4*y^2 - 315*x^4*y*z^2 - 1545*x^4*y*z - 1850*x^4*y + 105*x^4*z^3 + 780*x^4*z^2 + 1892*x^4*z + 1493*x^4 - 105*x^3*y^4 + 420*x^3*y^3*z + 1020*x^3*y^3 - 630*x^3*y^2*z^2 - 3090*x^3*y^2*z - 3700*x^3*y^2 + 420*x^3*y*z^3 + 3120*x^3*y*z^2 + 7568*x^3*y*z + 5972*x^3*y - 105*x^3*z^4 - 1050*x^3*z^3 - 3868*x^3*z^2 - 6206*x^3*z - 3645*x^3 - 63*x^2*y^5 + 315*x^2*y^4*z + 765*x^2*y^4 - 630*x^2*y^3*z^2 - 3090*x^2*y^3*z - 3700*x^2*y^3 + 630*x^2*y^2*z^3 + 4680*x^2*y^2*z^2 + 11352*x^2*y^2*z + 8958*x^2*y^2 - 315*x^2*y*z^4 - 3150*x^2*y*z^3 - 11604*x^2*y*z^2 - 18618*x^2*y*z - 10932*x^2*y + 63*x^2*z^5 + 795*x^2*z^4 + 3952*x^2*z^3 + 9669*x^2*z^2 + 11586*x^2*z + 5419*x^2 - 21*x*y^6 + 126*x*y^5*z + 306*x*y^5 - 315*x*y^4*z^2 - 1545*x*y^4*z - 1850*x*y^4 + 420*x*y^3*z^3 + 3120*x*y^3*z^2 + 7568*x*y^3*z + 5972*x*y^3 - 315*x*y^2*z^4 - 3150*x*y^2*z^3 - 11604*x*y^2*z^2 - 18618*x*y^2*z - 10929*x*y^2 + 126*x*y*z^5 + 1590*x*y*z^4 + 7904*x*y*z^3 + 19338*x*y*z^2 + 23166*x*y*z + 10823*x*y - 21*x*z^6 - 321*x*z^5 - 2018*x*z^4 - 6692*x*z^3 - 12282*x*z^2 - 11750*x*z - 4550*x - 3*y^7 + 21*y^6*z + 51*y^6 - 63*y^5*z^2 - 309*y^5*z - 370*y^5 + 105*y^4*z^3 + 780*y^4*z^2 + 1892*y^4*z + 1493*y^4 - 105*y^3*z^4 - 1050*y^3*z^3 - 3868*y^3*z^2 - 6206*y^3*z - 3642*y^3 + 63*y^2*z^5 + 795*y^2*z^4 + 3952*y^2*z^3 + 9669*y^2*z^2 + 11580*y^2*z + 5404*y^2 - 21*y*z^6 - 321*y*z^5 - 2018*y*z^4 - 6692*y*z^3 - 12279*y*z^2 - 11732*y*z - 4534*y + 3*z^7 + 54*z^6 + 412*z^5 + 1736*z^4 + 4344*z^3 + 6385*z^2 + 5063*z + 1, -3*x^7 - 21*x^6*y + 21*x^6*z + 51*x^6 - 63*x^5*y^2 + 126*x^5*y*z + 306*x^5*y - 63*x^5*z^2 - 309*x^5*z - 370*x^5 - 105*x^4*y^3 + 315*x^4*y^2*z + 765*x^4*y^2 - 315*x^4*y*z^2 - 1545*x^4*y*z - 1850*x^4*y + 105*x^4*z^3 + 780*x^4*z^2 + 1892*x^4*z + 1493*x^4 - 105*x^3*y^4 + 420*x^3*y^3*z + 1020*x^3*y^3 - 630*x^3*y^2*z^2 - 3090*x^3*y^2*z - 3700*x^3*y^2 + 420*x^3*y*z^3 + 3120*x^3*y*z^2 + 7568*x^3*y*z + 5972*x^3*y - 105*x^3*z^4 - 1050*x^3*z^3 - 3868*x^3*z^2 - 6206*x^3*z - 3645*x^3 - 63*x^2*y^5 + 315*x^2*y^4*z + 765*x^2*y^4 - 630*x^2*y^3*z^2 - 3090*x^2*y^3*z - 3700*x^2*y^3 + 630*x^2*y^2*z^3 + 4680*x^2*y^2*z^2 + 11352*x^2*y^2*z + 8958*x^2*y^2 - 315*x^2*y*z^4 - 3150*x^2*y*z^3 - 11604*x^2*y*z^2 - 18618*x^2*y*z - 10932*x^2*y + 63*x^2*z^5 + 795*x^2*z^4 + 3952*x^2*z^3 + 9669*x^2*z^2 + 11586*x^2*z + 5420*x^2 - 21*x*y^6 + 126*x*y^5*z + 306*x*y^5 - 315*x*y^4*z^2 - 1545*x*y^4*z - 1850*x*y^4 + 420*x*y^3*z^3 + 3120*x*y^3*z^2 + 7568*x*y^3*z + 5972*x*y^3 - 315*x*y^2*z^4 - 3150*x*y^2*z^3 - 11604*x*y^2*z^2 - 18618*x*y^2*z - 10929*x*y^2 + 126*x*y*z^5 + 1590*x*y*z^4 + 7904*x*y*z^3 + 19338*x*y*z^2 + 23166*x*y*z + 10825*x*y - 21*x*z^6 - 321*x*z^5 - 2018*x*z^4 - 6692*x*z^3 - 12282*x*z^2 - 11752*x*z - 4555*x - 3*y^7 + 21*y^6*z + 51*y^6 - 63*y^5*z^2 - 309*y^5*z - 370*y^5 + 105*y^4*z^3 + 780*y^4*z^2 + 1892*y^4*z + 1493*y^4 - 105*y^3*z^4 - 1050*y^3*z^3 - 3868*y^3*z^2 - 6206*y^3*z - 3642*y^3 + 63*y^2*z^5 + 795*y^2*z^4 + 3952*y^2*z^3 + 9669*y^2*z^2 + 11580*y^2*z + 5405*y^2 - 21*y*z^6 - 321*y*z^5 - 2018*y*z^4 - 6692*y*z^3 - 12279*y*z^2 - 11734*y*z - 4539*y + 3*z^7 + 54*z^6 + 412*z^5 + 1736*z^4 + 4344*z^3 + 6386*z^2 + 5069*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -1, -1, 1 ], [ -1, -1, 2 ], [ -2, -1, 1 ] ], "source_shift": [ 2, 2, 1 ], "target_matrix": [ [ 1, 1, 2 ], [ 1, 1, 1 ], [ 1, 2, 1 ] ], "target_shift": [ -1551, -1661, -1667 ], "inverse_recipe": { "p": 2, "q": 5, "f": "x**2", "g": "x**5 + x**4 + x**2 + x", "h": "y**3 + y**2 + y*z + y", "polynomials": { "f": [ [ [ 2, 0, 0 ], 1 ] ], "g": [ [ [ 5, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ], [ [ 4, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 3, 0 ], 1 ], [ [ 0, 1, 0 ], 1 ], [ [ 0, 2, 0 ], 1 ] ] }, "h_multiplier": 3 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0389847430014925 ], "surface_features": [ 3.5569862441417013, 4.364851052007693, 3.1815577738627865, 0.0, 3.6761908292322047, 4863 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C034", "pair": 34, "family": "counterexample", "mapping": "F(x,y,z)=(16*x^7 - 136*x^6*y + 88*x^6*z - 152*x^6 + 468*x^5*y^2 - 624*x^5*y*z + 1116*x^5*y + 204*x^5*z^2 - 708*x^5*z + 588*x^5 - 838*x^4*y^3 + 1734*x^4*y^2*z - 3246*x^4*y^2 - 1170*x^4*y*z^2 + 4212*x^4*y*z - 3570*x^4*y + 258*x^4*z^3 - 1350*x^4*z^2 + 2250*x^4*z - 1226*x^4 + 838*x^3*y^4 - 2398*x^3*y^3*z + 4757*x^3*y^3 + 2514*x^3*y^2*z^2 - 9489*x^3*y^2*z + 8208*x^3*y^2 - 1146*x^3*y*z^3 + 6231*x^3*y*z^2 - 10590*x^3*y*z + 5883*x^3*y + 192*x^3*z^4 - 1347*x^3*z^3 + 3378*x^3*z^2 - 3705*x^3*z + 1522*x^3 - 468*x^2*y^5 + 1734*x^2*y^4*z - 3681*x^2*y^4 - 2514*x^2*y^3*z^2 + 10056*x^2*y^3*z - 8853*x^2*y^3 + 1782*x^2*y^2*z^3 - 10170*x^2*y^2*z^2 + 17604*x^2*y^2*z - 9942*x^2*y^2 - 618*x^2*y*z^4 + 4512*x^2*y*z^3 - 11523*x^2*y*z^2 + 12864*x^2*y*z - 5409*x^2*y + 84*x^2*z^5 - 741*x^2*z^4 + 2484*x^2*z^3 - 4110*x^2*z^2 + 3414*x^2*z - 1157*x^2 + 136*x*y^6 - 624*x*y^5*z + 1416*x*y^5 + 1170*x*y^4*z^2 - 4986*x*y^4*z + 4440*x*y^4 - 1146*x*y^3*z^3 + 6921*x*y^3*z^2 - 12150*x*y^3*z + 6946*x*y^3 + 618*x*y^2*z^4 - 4737*x*y^2*z^3 + 12288*x*y^2*z^2 - 13908*x*y^2*z + 5946*x*y^2 - 174*x*y*z^5 + 1599*x*y*z^4 - 5448*x*y*z^3 + 9153*x*y*z^2 - 7752*x*y*z + 2693*x*y + 20*x*z^6 - 213*x*z^5 + 894*x*z^4 - 1981*x*z^3 + 2490*x*z^2 - 1711*x*z + 514*x - 16*y^7 + 88*y^6*z - 212*y^6 - 204*y^5*z^2 + 924*y^5*z - 828*y^5 + 258*y^4*z^3 - 1653*y^4*z^2 + 2928*y^4*z - 1680*y^4 - 192*y^3*z^4 + 1554*y^3*z^3 - 4077*y^3*z^2 + 4650*y^3*z - 2000*y^3 + 84*y^2*z^5 - 810*y^2*z^4 + 2796*y^2*z^3 - 4746*y^2*z^2 + 4062*y^2*z - 1433*y^2 - 20*y*z^6 + 222*y*z^5 - 945*y*z^4 + 2120*y*z^3 - 2703*y*z^2 + 1892*y*z - 575*y + 2*z^7 - 25*z^6 + 126*z^5 - 350*z^4 + 590*z^3 - 614*z^2 + 372*z - 1, -8*x^7 + 68*x^6*y - 44*x^6*z + 112*x^6 - 234*x^5*y^2 + 312*x^5*y*z - 792*x^5*y - 102*x^5*z^2 + 516*x^5*z - 546*x^5 + 419*x^4*y^3 - 867*x^4*y^2*z + 2208*x^4*y^2 + 585*x^4*y*z^2 - 2952*x^4*y*z + 3207*x^4*y - 129*x^4*z^3 + 972*x^4*z^2 - 2061*x^4*z + 1284*x^4 - 419*x^3*y^4 + 1199*x^3*y^3*z - 3094*x^3*y^3 - 1257*x^3*y^2*z^2 + 6369*x^3*y^2*z - 7101*x^3*y^2 + 573*x^3*y*z^3 - 4308*x^3*y*z^2 + 9363*x^3*y*z - 5951*x^3*y - 96*x^3*z^4 + 957*x^3*z^3 - 3048*x^3*z^2 + 3818*x^3*z - 1713*x^3 + 234*x^2*y^5 - 867*x^2*y^4*z + 2295*x^2*y^4 + 1257*x^2*y^3*z^2 - 6459*x^2*y^3*z + 7347*x^2*y^3 - 891*x^2*y^2*z^3 + 6732*x^2*y^2*z^2 - 14958*x^2*y^2*z + 9660*x^2*y^2 + 309*x^2*y*z^4 - 3075*x^2*y*z^3 + 10014*x^2*y*z^2 - 12768*x^2*y*z + 5853*x^2*y - 42*x^2*z^5 + 519*x^2*z^4 - 2205*x^2*z^3 + 4161*x^2*z^2 - 3768*x^2*z + 1357*x^2 - 68*x*y^6 + 312*x*y^5*z - 852*x*y^5 - 585*x*y^4*z^2 + 3078*x*y^4*z - 3525*x*y^4 + 573*x*y^3*z^3 - 4392*x*y^3*z^2 + 9891*x*y^3*z - 6453*x*y^3 - 309*x*y^2*z^4 + 3093*x*y^2*z^3 - 10248*x*y^2*z^2 + 13230*x*y^2*z - 6150*x*y^2 + 87*x*y*z^5 - 1074*x*y*z^4 + 4650*x*y*z^3 - 8904*x*y*z^2 + 8202*x*y*z - 3010*x*y - 10*x*z^6 + 147*x*z^5 - 780*x*z^4 + 1968*x*z^3 - 2691*x*z^2 + 1958*x*z - 611*x + 8*y^7 - 44*y^6*z + 124*y^6 + 102*y^5*z^2 - 552*y^5*z + 630*y^5 - 129*y^4*z^3 + 1011*y^4*z^2 - 2283*y^4*z + 1492*y^4 + 96*y^3*z^4 - 975*y^3*z^3 + 3258*y^3*z^2 - 4231*y^3*z + 1971*y^3 - 42*y^2*z^5 + 522*y^2*z^4 - 2289*y^2*z^3 + 4422*y^2*z^2 - 4104*y^2*z + 1519*y^2 + 10*y*z^6 - 147*y*z^5 + 792*y*z^4 - 2021*y*z^3 + 2796*y*z^2 - 2059*y*z + 642*y - z^7 + 17*z^6 - 108*z^5 + 341*z^4 - 624*z^3 + 685*z^2 - 428*z + 1, 8*x^7 - 68*x^6*y + 44*x^6*z - 88*x^6 + 234*x^5*y^2 - 312*x^5*y*z + 636*x^5*y + 102*x^5*z^2 - 408*x^5*z + 378*x^5 - 419*x^4*y^3 + 867*x^4*y^2*z - 1818*x^4*y^2 - 585*x^4*y*z^2 + 2388*x^4*y*z - 2259*x^4*y + 129*x^4*z^3 - 774*x^4*z^2 + 1437*x^4*z - 834*x^4 + 419*x^3*y^4 - 1199*x^3*y^3*z + 2617*x^3*y^3 + 1257*x^3*y^2*z^2 - 5286*x^3*y^2*z + 5103*x^3*y^2 - 573*x^3*y*z^3 + 3513*x^3*y*z^2 - 6651*x^3*y*z + 3938*x^3*y + 96*x^3*z^4 - 768*x^3*z^3 + 2142*x^3*z^2 - 2501*x^3*z + 1065*x^3 - 234*x^2*y^5 + 867*x^2*y^4*z - 1992*x^2*y^4 - 1257*x^2*y^3*z^2 + 5505*x^2*y^3*z - 5400*x^2*y^3 + 891*x^2*y^2*z^3 - 5634*x^2*y^2*z^2 + 10854*x^2*y^2*z - 6528*x^2*y^2 - 309*x^2*y*z^4 + 2529*x^2*y*z^3 - 7179*x^2*y*z^2 + 8532*x^2*y*z - 3726*x^2*y + 42*x^2*z^5 - 420*x^2*z^4 + 1563*x^2*z^3 - 2751*x^2*z^2 + 2370*x^2*z - 820*x^2 + 68*x*y^6 - 312*x*y^5*z + 756*x*y^5 + 585*x*y^4*z^2 - 2688*x*y^4*z + 2655*x*y^4 - 573*x*y^3*z^3 + 3771*x*y^3*z^2 - 7347*x*y^3*z + 4464*x*y^3 + 309*x*y^2*z^4 - 2610*x*y^2*z^3 + 7512*x*y^2*z^2 - 9039*x*y^2*z + 4014*x*y^2 - 87*x*y*z^5 + 891*x*y*z^4 - 3366*x*y*z^3 + 6012*x*y*z^2 - 5286*x*y*z + 1876*x*y + 10*x*z^6 - 120*x*z^5 + 558*x*z^4 - 1314*x*z^3 + 1713*x*z^2 - 1202*x*z + 367*x - 8*y^7 + 44*y^6*z - 112*y^6 - 102*y^5*z^2 + 492*y^5*z - 486*y^5 + 129*y^4*z^3 - 888*y^4*z^2 + 1737*y^4*z - 1057*y^4 - 96*y^3*z^4 + 843*y^3*z^3 - 2445*y^3*z^2 + 2959*y^3*z - 1320*y^3 + 42*y^2*z^5 - 444*y^2*z^4 + 1695*y^2*z^3 - 3054*y^2*z^2 + 2712*y^2*z - 976*y^2 - 10*y*z^6 + 123*y*z^5 - 579*y*z^4 + 1379*y*z^3 - 1824*y*z^2 + 1303*y*z - 400*y + z^7 - 14*z^6 + 78*z^5 - 230*z^4 + 402*z^3 - 427*z^2 + 262*z - 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -2, 1, -1 ], [ -1, 2, -1 ], [ -1, 1, -1 ] ], "source_shift": [ 1, 1, 2 ], "target_matrix": [ [ -2, -1, -1 ], [ 1, 2, 1 ], [ -1, -1, -1 ] ], "target_shift": [ 102, -118, 72 ], "collision_preimages": [ [ "-5/4", "5/4", "19/4" ], [ "9/2", "-7", "-16" ], [ "13/2", "-4", "-15" ] ], "collision_image": [ "205/2", "-473/4", "289/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 0.9418988729994453 ], "surface_features": [ 3.385694632713635, 4.2456113598909075, 3.073535065058784, 0.0, 3.4866486743655014, 4833 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C037", "pair": 37, "family": "counterexample", "mapping": "F(x,y,z)=(216*x^7 - 1188*x^6*y 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216*z^6 - 810*z^5 + 1658*z^4 - 2030*z^3 + 1521*z^2 - 665*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -3, 2, -2 ], [ -2, 1, -1 ], [ -1, 2, -1 ] ], "source_shift": [ 1, 1, 1 ], "target_matrix": [ [ -1, -1, -2 ], [ -2, -1, -1 ], [ 3, 2, 2 ] ], "target_shift": [ 51, 75, -129 ], "collision_preimages": [ [ "1", "-5/4", "-9/4" ], [ "5", "3", "-9/2" ], [ "-3", "8", "27/2" ] ], "collision_image": [ "205/4", "151/2", "-519/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 0.8738150999997742 ], "surface_features": [ 4.036646081410155, 4.918737848994417, 3.746128005464328, 0.0, 4.1513479576666334, 5086 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B035", "pair": 35, "family": "bijection", "mapping": "F(x,y,z)=(7*x^7 - 98*x^6*y + 49*x^6*z - 63*x^6 + 588*x^5*y^2 - 588*x^5*y*z + 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"determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -1, 2, -1 ], [ -2, 3, -2 ], [ -2, 1, -1 ] ], "source_shift": [ 1, 1, 1 ], "target_matrix": [ [ -1, -1, -1 ], [ -1, -1, -2 ], [ 2, 1, 1 ] ], "target_shift": [ 165, 159, -334 ], "inverse_recipe": { "p": 3, "q": 4, "f": "x**3 + x", "g": "x**4 + x**3", "h": "y**2 + y*z + 2*y", "polynomials": { "f": [ [ [ 3, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "g": [ [ [ 4, 0, 0 ], 1 ], [ [ 3, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 2, 0 ], 1 ], [ [ 0, 1, 0 ], 2 ] ] }, "h_multiplier": 7 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0041199410006811 ], "surface_features": [ 3.9817702992236272, 4.902916234446356, 3.6724673130680823, 0.0, 4.090313653124637, 5057 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B038", "pair": 38, "family": "bijection", "mapping": "F(x,y,z)=(-3*x^7 + 21*x^6*y - 21*x^6*z + 48*x^6 - 63*x^5*y^2 + 126*x^5*y*z - 291*x^5*y - 63*x^5*z^2 + 288*x^5*z - 325*x^5 + 105*x^4*y^3 - 315*x^4*y^2*z + 735*x^4*y^2 + 315*x^4*y*z^2 - 1455*x^4*y*z + 1658*x^4*y - 105*x^4*z^3 + 720*x^4*z^2 - 1625*x^4*z + 1211*x^4 - 105*x^3*y^4 + 420*x^3*y^3*z - 990*x^3*y^3 - 630*x^3*y^2*z^2 + 2940*x^3*y^2*z - 3382*x^3*y^2 + 420*x^3*y*z^3 - 2910*x^3*y*z^2 + 6632*x^3*y*z - 4988*x^3*y - 105*x^3*z^4 + 960*x^3*z^3 - 3250*x^3*z^2 + 4844*x^3*z - 2694*x^3 + 63*x^2*y^5 - 315*x^2*y^4*z + 750*x^2*y^4 + 630*x^2*y^3*z^2 - 2970*x^2*y^3*z + 3448*x^2*y^3 - 630*x^2*y^2*z^3 + 4410*x^2*y^2*z^2 - 10146*x^2*y^2*z + 7698*x^2*y^2 + 315*x^2*y*z^4 - 2910*x^2*y*z^3 + 9948*x^2*y*z^2 - 14964*x^2*y*z + 8397*x^2*y - 63*x^2*z^5 + 720*x^2*z^4 - 3250*x^2*z^3 + 7266*x^2*z^2 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5940*x^2*y^3*z - 6916*x^2*y^3 + 1260*x^2*y^2*z^3 - 8820*x^2*y^2*z^2 + 20352*x^2*y^2*z - 15528*x^2*y^2 - 630*x^2*y*z^4 + 5820*x^2*y*z^3 - 19956*x^2*y*z^2 + 30192*x^2*y*z - 17082*x^2*y + 126*x^2*z^5 - 1440*x^2*z^4 + 6520*x^2*z^3 - 14664*x^2*z^2 + 16458*x^2*z - 7399*x^2 + 42*x*y^6 - 252*x*y^5*z + 606*x*y^5 + 630*x*y^4*z^2 - 3000*x*y^4*z + 3524*x*y^4 - 840*x*y^3*z^3 + 5940*x*y^3*z^2 - 13832*x*y^3*z + 10640*x*y^3 + 630*x*y^2*z^4 - 5880*x*y^2*z^3 + 20352*x*y^2*z^2 - 31056*x*y^2*z + 17712*x*y^2 - 252*x*y*z^5 + 2910*x*y*z^4 - 13304*x*y*z^3 + 30192*x*y*z^2 - 34176*x*y*z + 15500*x*y + 42*x*z^6 - 576*x*z^5 + 3260*x*z^4 - 9776*x*z^3 + 16464*x*z^2 - 14828*x*z + 5575*x - 6*y^7 + 42*y^6*z - 102*y^6 - 126*y^5*z^2 + 606*y^5*z - 718*y^5 + 210*y^4*z^3 - 1500*y^4*z^2 + 3524*y^4*z - 2732*y^4 - 210*y^3*z^4 + 1980*y^3*z^3 - 6916*y^3*z^2 + 10640*y^3*z - 6114*y^3 + 126*y^2*z^5 - 1470*y^2*z^4 + 6784*y^2*z^3 - 15528*y^2*z^2 + 17718*y^2*z - 8101*y^2 - 42*y*z^6 + 582*y*z^5 - 3326*y*z^4 + 10064*y*z^3 - 17094*y*z^2 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7032*x^4*z^2 + 14280*x^4*z + 8528*x^4 - 2860*x^3*y^4 + 8360*x^3*y^3*z + 19144*x^3*y^3 - 9072*x^3*y^2*z^2 - 41352*x^3*y^2*z - 41016*x^3*y^2 + 4332*x^3*y*z^3 + 29628*x^3*y*z^2 + 59460*x^3*y*z + 35984*x^3*y - 768*x^3*z^4 - 7044*x^3*z^3 - 21504*x^3*z^2 - 26108*x^3*z - 10780*x^3 - 1584*x^2*y^5 + 5934*x^2*y^4*z + 14202*x^2*y^4 - 8796*x^2*y^3*z^2 - 41544*x^2*y^3*z - 41340*x^2*y^3 + 6450*x^2*y^2*z^3 + 45270*x^2*y^2*z^2 + 90582*x^2*y^2*z + 55356*x^2*y^2 - 2340*x^2*y*z^4 - 21792*x^2*y*z^3 - 65976*x^2*y*z^2 - 80928*x^2*y*z - 34392*x^2*y + 336*x^2*z^5 + 3912*x^2*z^4 + 15978*x^2*z^3 + 29514*x^2*z^2 + 24978*x^2*z + 7840*x^2 - 459*x*y^6 + 2115*x*y^5*z + 5319*x*y^5 - 4017*x*y^4*z^2 - 19863*x*y^4*z - 20184*x*y^4 + 4025*x*y^3*z^3 + 29409*x*y^3*z^2 + 59583*x*y^3*z + 36732*x*y^3 - 2244*x*y^2*z^4 - 21597*x*y^2*z^3 - 65694*x*y^2*z^2 - 81219*x*y^2*z - 35397*x*y^2 + 660*x*y*z^5 + 7872*x*y*z^4 + 32079*x*y*z^3 + 59709*x*y*z^2 + 51873*x*y*z + 16930*x*y - 80*x*z^6 - 1140*x*z^5 - 5856*x*z^4 - 14597*x*z^3 - 18957*x*z^2 - 12265*x*z - 3106*x - 54*y^7 + 297*y^6*z + 783*y^6 - 693*y^5*z^2 - 3591*y^5*z - 3762*y^5 + 889*y^4*z^3 + 6795*y^4*z^2 + 14118*y^4*z + 8864*y^4 - 677*y^3*z^4 - 6793*y^3*z^3 - 21051*y^3*z^2 - 26373*y^3*z - 11718*y^3 + 306*y^2*z^5 + 3786*y^2*z^4 + 15606*y^2*z^3 + 29334*y^2*z^2 + 26013*y^2*z + 8812*y^2 - 76*y*z^6 - 1116*y*z^5 - 5757*y*z^4 - 14460*y*z^3 - 19194*y*z^2 - 12896*y*z - 3433*y + 8*z^7 + 136*z^6 + 846*z^5 + 2666*z^4 + 4708*z^3 + 4705*z^2 + 2473*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -2, -3, 2 ], [ -2, -1, 1 ], [ -1, -2, 1 ] ], "source_shift": [ 1, 2, 2 ], "target_matrix": [ [ -2, -3, -2 ], [ -2, -1, -1 ], [ 1, 2, 1 ] ], "target_shift": [ 863, 517, -519 ], "collision_preimages": [ [ "13/4", "7/2", "8" ], [ "-1", "-9", "-29/2" ], [ "-6", "-11", "-47/2" ] ], "collision_image": [ "1727/2", "1035/2", "-2077/4" ] }, "probe_target": 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"pair": 80, "family": "counterexample", "mapping": "F(x,y,z)=(32*x^7 + 176*x^6*y + 288*x^6*z + 448*x^6 + 408*x^5*y^2 + 1320*x^5*y*z + 2040*x^5*y + 1056*x^5*z^2 + 3264*x^5*z + 2568*x^5 + 516*x^4*y^3 + 2472*x^4*y^2*z + 3816*x^4*y^2 + 3900*x^4*y*z^2 + 12072*x^4*y*z + 9480*x^4*y + 2024*x^4*z^3 + 9432*x^4*z^2 + 14856*x^4*z + 7906*x^4 + 384*x^3*y^4 + 2418*x^3*y^3*z + 3750*x^3*y^3 + 5634*x^3*y^2*z^2 + 17580*x^3*y^2*z + 13818*x^3*y^2 + 5750*x^3*y*z^3 + 27126*x^3*y*z^2 + 42804*x^3*y*z + 22768*x^3*y + 2166*x^3*z^4 + 13768*x^3*z^3 + 32706*x^3*z^2 + 34907*x^3*z + 14159*x^3 + 168*x^2*y^5 + 1302*x^2*y^4*z + 2040*x^2*y^4 + 3978*x^2*y^3*z^2 + 12588*x^2*y^3*z + 9936*x^2*y^3 + 5982*x^2*y^2*z^3 + 28746*x^2*y^2*z^2 + 45624*x^2*y^2*z + 24294*x^2*y^2 + 4422*x^2*y*z^4 + 28776*x^2*y*z^3 + 68904*x^2*y*z^2 + 73617*x^2*y*z + 29838*x^2*y + 1284*x^2*z^5 + 10650*x^2*z^4 + 34224*x^2*z^3 + 54960*x^2*z^2 + 44643*x^2*z + 14732*x^2 + 40*x*y^6 + 366*x*y^5*z + 582*x*y^5 + 1374*x*y^4*z^2 + 4428*x*y^4*z + 3522*x*y^4 + 2706*x*y^3*z^3 + 13290*x*y^3*z^2 + 21300*x*y^3*z + 11382*x*y^3 + 2946*x*y^2*z^4 + 19656*x*y^2*z^3 + 47658*x*y^2*z^2 + 51147*x*y^2*z + 20739*x*y^2 + 1680*x*y*z^5 + 14316*x*y*z^4 + 46752*x*y*z^3 + 75558*x*y*z^2 + 61368*x*y*z + 20198*x*y + 392*x*z^6 + 4104*x*z^5 + 16968*x*z^4 + 36722*x*z^3 + 44718*x*z^2 + 29396*x*z + 8204*x + 4*y^7 + 42*y^6*z + 68*y^6 + 186*y^5*z^2 + 612*y^5*z + 492*y^5 + 450*y^4*z^3 + 2262*y^4*z^2 + 3672*y^4*z + 1974*y^4 + 642*y^3*z^4 + 4392*y^3*z^3 + 10812*y^3*z^2 + 11689*y^3*z + 4752*y^3 + 540*y^2*z^5 + 4722*y^2*z^4 + 15696*y^2*z^3 + 25602*y^2*z^2 + 20865*y^2*z + 6866*y^2 + 248*y*z^6 + 2664*y*z^5 + 11232*y*z^4 + 24590*y*z^3 + 30114*y*z^2 + 19804*y*z + 5506*y + 48*z^7 + 616*z^6 + 3168*z^5 + 8740*z^4 + 14308*z^3 + 14084*z^2 + 7788*z + 1, -16*x^7 - 88*x^6*y - 144*x^6*z - 224*x^6 - 204*x^5*y^2 - 660*x^5*y*z - 1020*x^5*y - 528*x^5*z^2 - 1632*x^5*z - 1284*x^5 - 258*x^4*y^3 - 1236*x^4*y^2*z - 1908*x^4*y^2 - 1950*x^4*y*z^2 - 6036*x^4*y*z - 4740*x^4*y - 1012*x^4*z^3 - 4716*x^4*z^2 - 7428*x^4*z - 3954*x^4 - 192*x^3*y^4 - 1209*x^3*y^3*z - 1875*x^3*y^3 - 2817*x^3*y^2*z^2 - 8790*x^3*y^2*z - 6909*x^3*y^2 - 2875*x^3*y*z^3 - 13563*x^3*y*z^2 - 21402*x^3*y*z - 11388*x^3*y - 1083*x^3*z^4 - 6884*x^3*z^3 - 16353*x^3*z^2 - 17461*x^3*z - 7089*x^3 - 84*x^2*y^5 - 651*x^2*y^4*z - 1020*x^2*y^4 - 1989*x^2*y^3*z^2 - 6294*x^2*y^3*z - 4968*x^2*y^3 - 2991*x^2*y^2*z^3 - 14373*x^2*y^2*z^2 - 22812*x^2*y^2*z - 12153*x^2*y^2 - 2211*x^2*y*z^4 - 14388*x^2*y*z^3 - 34452*x^2*y*z^2 - 36831*x^2*y*z - 14946*x^2*y - 642*x^2*z^5 - 5325*x^2*z^4 - 17112*x^2*z^3 - 27501*x^2*z^2 - 22371*x^2*z - 7396*x^2 - 20*x*y^6 - 183*x*y^5*z - 291*x*y^5 - 687*x*y^4*z^2 - 2214*x*y^4*z - 1761*x*y^4 - 1353*x*y^3*z^3 - 6645*x*y^3*z^2 - 10650*x*y^3*z - 5695*x*y^3 - 1473*x*y^2*z^4 - 9828*x*y^2*z^3 - 23829*x*y^2*z^2 - 25596*x*y^2*z - 10395*x*y^2 - 840*x*y*z^5 - 7158*x*y*z^4 - 23376*x*y*z^3 - 37821*x*y*z^2 - 30777*x*y*z - 10153*x*y - 196*x*z^6 - 2052*x*z^5 - 8484*x*z^4 - 18387*x*z^3 - 22443*x*z^2 - 14794*x*z - 4139*x - 2*y^7 - 21*y^6*z - 34*y^6 - 93*y^5*z^2 - 306*y^5*z - 246*y^5 - 225*y^4*z^3 - 1131*y^4*z^2 - 1836*y^4*z - 988*y^4 - 321*y^3*z^4 - 2196*y^3*z^3 - 5406*y^3*z^2 - 5852*y^3*z - 2384*y^3 - 270*y^2*z^5 - 2361*y^2*z^4 - 7848*y^2*z^3 - 12822*y^2*z^2 - 10476*y^2*z - 3457*y^2 - 124*y*z^6 - 1332*y*z^5 - 5616*y*z^4 - 12321*y*z^3 - 15135*y*z^2 - 9986*y*z - 2784*y - 24*z^7 - 308*z^6 - 1584*z^5 - 4382*z^4 - 7200*z^3 - 7114*z^2 - 3946*z - 1, -16*x^7 - 88*x^6*y - 144*x^6*z - 248*x^6 - 204*x^5*y^2 - 660*x^5*y*z - 1140*x^5*y - 528*x^5*z^2 - 1836*x^5*z - 1560*x^5 - 258*x^4*y^3 - 1236*x^4*y^2*z - 2154*x^4*y^2 - 1950*x^4*y*z^2 - 6864*x^4*y*z - 5832*x^4*y - 1012*x^4*z^3 - 5406*x^4*z^2 - 9228*x^4*z - 5178*x^4 - 192*x^3*y^4 - 1209*x^3*y^3*z - 2139*x^3*y^3 - 2817*x^3*y^2*z^2 - 10107*x^3*y^2*z - 8610*x^3*y^2 - 2875*x^3*y*z^3 - 15729*x^3*y*z^2 - 26940*x^3*y*z - 15108*x^3*y - 1083*x^3*z^4 - 8057*x^3*z^3 - 20802*x^3*z^2 - 23413*x^3*z - 9849*x^3 - 84*x^2*y^5 - 651*x^2*y^4*z - 1176*x^2*y^4 - 1989*x^2*y^3*z^2 - 7317*x^2*y^3*z - 6270*x^2*y^3 - 2991*x^2*y^2*z^3 - 16857*x^2*y^2*z^2 - 29085*x^2*y^2*z - 16329*x^2*y^2 - 2211*x^2*y*z^4 - 17031*x^2*y*z^3 - 44388*x^2*y*z^2 - 50013*x^2*y*z - 21024*x^2*y - 642*x^2*z^5 - 6363*x^2*z^4 - 22281*x^2*z^3 - 37743*x^2*z^2 - 31809*x^2*z - 10774*x^2 - 20*x*y^6 - 183*x*y^5*z - 339*x*y^5 - 687*x*y^4*z^2 - 2601*x*y^4*z - 2250*x*y^4 - 1353*x*y^3*z^3 - 7875*x*y^3*z^2 - 13746*x*y^3*z - 7747*x*y^3 - 1473*x*y^2*z^4 - 11751*x*y^2*z^3 - 31074*x*y^2*z^2 - 35184*x*y^2*z - 14799*x*y^2 - 840*x*y*z^5 - 8634*x*y*z^4 - 30798*x*y*z^3 - 52533*x*y*z^2 - 44277*x*y*z - 14965*x*y - 196*x*z^6 - 2496*x*z^5 - 11292*x*z^4 - 25803*x*z^3 - 32619*x*z^2 - 22039*x*z - 6268*x - 2*y^7 - 21*y^6*z - 40*y^6 - 93*y^5*z^2 - 363*y^5*z - 318*y^5 - 225*y^4*z^3 - 1353*y^4*z^2 - 2397*y^4*z - 1360*y^4 - 321*y^3*z^4 - 2649*y^3*z^3 - 7128*y^3*z^2 - 8138*y^3*z - 3434*y^3 - 270*y^2*z^5 - 2871*y^2*z^4 - 10449*y^2*z^3 - 18012*y^2*z^2 - 15246*y^2*z - 5155*y^2 - 124*y*z^6 - 1632*y*z^5 - 7548*y*z^4 - 17481*y*z^3 - 22251*y*z^2 - 15047*y*z - 4264*y - 24*z^7 - 380*z^6 - 2148*z^5 - 6278*z^4 - 10692*z^3 - 10828*z^2 - 6107*z - 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ 1, 1, 2 ], [ 2, 1, 1 ], [ 2, 2, 3 ] ], "source_shift": [ 2, 2, 1 ], "target_matrix": [ [ 2, 2, 3 ], [ -1, -1, -1 ], [ -1, -2, -1 ] ], "target_shift": [ -1877, 952, 1488 ], "collision_preimages": [ [ "-11/4", "25/4", "-11/4" ], [ "-10", "24", "-15/2" ], [ "-9", "29", "-23/2" ] ], "collision_image": [ "-3755/2", "3809/4", "5953/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0125267570001597 ], "surface_features": [ 3.9939362507635896, 4.878280453743452, 3.644143050509919, 0.0, 4.119298783899469, 5046 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C202", "pair": 202, "family": "counterexample", "mapping": "F(x,y,z)=(216*x^7 - 1188*x^6*y - 972*x^6*z + 1836*x^6 + 2610*x^5*y^2 + 4464*x^5*y*z - 8100*x^5*y + 1854*x^5*z^2 - 6912*x^5*z + 6300*x^5 - 3043*x^4*y^3 - 8031*x^4*y^2*z + 14319*x^4*y^2 - 6933*x^4*y*z^2 + 25110*x^4*y*z - 21840*x^4*y - 1945*x^4*z^3 + 10791*x^4*z^2 - 19392*x^4*z + 11321*x^4 + 2060*x^3*y^4 + 7393*x^3*y^3*z - 13114*x^3*y^3 + 9819*x^3*y^2*z^2 - 35169*x^3*y^2*z + 29751*x^3*y^2 + 5699*x^3*y*z^3 - 30996*x^3*y*z^2 + 53586*x^3*y*z - 30134*x^3*y + 1213*x^3*z^4 - 8941*x^3*z^3 + 23835*x^3*z^2 - 27597*x^3*z + 11590*x^3 - 816*x^2*y^5 - 3714*x^2*y^4*z + 6603*x^2*y^4 - 6696*x^2*y^3*z^2 + 23955*x^2*y^3*z - 19959*x^2*y^3 - 5964*x^2*y^2*z^3 + 32247*x^2*y^2*z^2 - 54537*x^2*y^2*z + 29865*x^2*y^2 - 2616*x^2*y*z^4 + 19041*x^2*y*z^3 - 49197*x^2*y*z^2 + 55062*x^2*y*z - 22584*x^2*y - 450*x^2*z^5 + 4146*x^2*z^4 - 14619*x^2*z^3 + 25197*x^2*z^2 - 21141*x^2*z + 6841*x^2 + 176*x*y^6 + 972*x*y^5*z - 1740*x*y^5 + 2220*x*y^4*z^2 - 7980*x*y^4*z + 6606*x*y^4 + 2680*x*y^3*z^3 - 14520*x*y^3*z^2 + 24291*x*y^3*z - 13065*x*y^3 + 1800*x*y^2*z^4 - 13080*x*y^2*z^3 + 33237*x*y^2*z^2 - 36342*x*y^2*z + 14631*x*y^2 + 636*x*y*z^5 - 5820*x*y*z^4 + 20025*x*y*z^3 - 33489*x*y*z^2 + 27465*x*y*z - 8776*x*y + 92*x*z^6 - 1020*x*z^5 + 4473*x*z^4 - 10212*x*z^3 + 12834*x*z^2 - 8344*x*z + 2187*x - 16*y^7 - 104*y^6*z + 188*y^6 - 288*y^5*z^2 + 1044*y^5*z - 864*y^5 - 440*y^4*z^3 + 2400*y^4*z^2 - 4002*y^4*z + 2129*y^4 - 400*y^3*z^4 + 2920*y^3*z^3 - 7368*y^3*z^2 + 7937*y^3*z - 3151*y^3 - 216*y^2*z^5 + 1980*y^2*z^4 - 6732*y^2*z^3 + 11037*y^2*z^2 - 8895*y^2*z + 2812*y^2 - 64*y*z^6 + 708*y*z^5 - 3048*y*z^4 + 6779*y*z^3 - 8337*y*z^2 + 5351*y*z - 1397*y - 8*z^7 + 104*z^6 - 546*z^5 + 1550*z^4 - 2593*z^3 + 2539*z^2 - 1344*z + 1, 216*x^7 - 1188*x^6*y - 972*x^6*z + 1836*x^6 + 2610*x^5*y^2 + 4464*x^5*y*z - 8100*x^5*y + 1854*x^5*z^2 - 6912*x^5*z + 6300*x^5 - 3043*x^4*y^3 - 8031*x^4*y^2*z + 14319*x^4*y^2 - 6933*x^4*y*z^2 + 25110*x^4*y*z - 21840*x^4*y - 1945*x^4*z^3 + 10791*x^4*z^2 - 19392*x^4*z + 11294*x^4 + 2060*x^3*y^4 + 7393*x^3*y^3*z - 13114*x^3*y^3 + 9819*x^3*y^2*z^2 - 35169*x^3*y^2*z + 29751*x^3*y^2 + 5699*x^3*y*z^3 - 30996*x^3*y*z^2 + 53586*x^3*y*z - 30026*x^3*y + 1213*x^3*z^4 - 8941*x^3*z^3 + 23835*x^3*z^2 - 27516*x^3*z + 11482*x^3 - 816*x^2*y^5 - 3714*x^2*y^4*z + 6603*x^2*y^4 - 6696*x^2*y^3*z^2 + 23955*x^2*y^3*z - 19959*x^2*y^3 - 5964*x^2*y^2*z^3 + 32247*x^2*y^2*z^2 - 54537*x^2*y^2*z + 29721*x^2*y^2 - 2616*x^2*y*z^4 + 19041*x^2*y*z^3 - 49197*x^2*y*z^2 + 54828*x^2*y*z - 22341*x^2*y - 450*x^2*z^5 + 4146*x^2*z^4 - 14619*x^2*z^3 + 25107*x^2*z^2 - 20925*x^2*z + 6715*x^2 + 176*x*y^6 + 972*x*y^5*z - 1740*x*y^5 + 2220*x*y^4*z^2 - 7980*x*y^4*z + 6606*x*y^4 + 2680*x*y^3*z^3 - 14520*x*y^3*z^2 + 24291*x*y^3*z - 12985*x*y^3 + 1800*x*y^2*z^4 - 13080*x*y^2*z^3 + 33237*x*y^2*z^2 - 36138*x*y^2*z + 14451*x*y^2 + 636*x*y*z^5 - 5820*x*y*z^4 + 20025*x*y*z^3 - 33321*x*y*z^2 + 27141*x*y*z - 8602*x*y + 92*x*z^6 - 1020*x*z^5 + 4473*x*z^4 - 10168*x*z^3 + 12690*x*z^2 - 8179*x*z + 2141*x - 16*y^7 - 104*y^6*z + 188*y^6 - 288*y^5*z^2 + 1044*y^5*z - 864*y^5 - 440*y^4*z^3 + 2400*y^4*z^2 - 4002*y^4*z + 2113*y^4 - 400*y^3*z^4 + 2920*y^3*z^3 - 7368*y^3*z^2 + 7881*y^3*z - 3107*y^3 - 216*y^2*z^5 + 1980*y^2*z^4 - 6732*y^2*z^3 + 10965*y^2*z^2 - 8775*y^2*z + 2752*y^2 - 64*y*z^6 + 708*y*z^5 - 3048*y*z^4 + 6739*y*z^3 - 8229*y*z^2 + 5237*y*z - 1366*y - 8*z^7 + 104*z^6 - 546*z^5 + 1542*z^4 - 2561*z^3 + 2485*z^2 - 1314*z + 1, 648*x^7 - 3564*x^6*y - 2916*x^6*z + 5184*x^6 + 7830*x^5*y^2 + 13392*x^5*y*z - 22680*x^5*y + 5562*x^5*z^2 - 19440*x^5*z + 16956*x^5 - 9129*x^4*y^3 - 24093*x^4*y^2*z + 39852*x^4*y^2 - 20799*x^4*y*z^2 + 70092*x^4*y*z - 58392*x^4*y - 5835*x^4*z^3 + 30240*x^4*z^2 - 52020*x^4*z + 29454*x^4 + 6180*x^3*y^4 + 22179*x^3*y^3*z - 36330*x^3*y^3 + 29457*x^3*y^2*z^2 - 97632*x^3*y^2*z + 79101*x^3*y^2 + 17097*x^3*y*z^3 - 86274*x^3*y*z^2 + 142830*x^3*y*z - 78072*x^3*y + 3639*x^3*z^4 - 24972*x^3*z^3 + 63729*x^3*z^2 - 71622*x^3*z + 29502*x^3 - 2448*x^2*y^5 - 11142*x^2*y^4*z + 18225*x^2*y^4 - 20088*x^2*y^3*z^2 + 66219*x^2*y^3*z - 52812*x^2*y^3 - 17892*x^2*y^2*z^3 + 89307*x^2*y^2*z^2 - 144585*x^2*y^2*z + 77073*x^2*y^2 - 7848*x^2*y*z^4 + 52857*x^2*y*z^3 - 130734*x^2*y*z^2 + 142302*x^2*y*z - 57321*x^2*y - 1350*x^2*z^5 + 11544*x^2*z^4 - 38961*x^2*z^3 + 65229*x^2*z^2 - 53697*x^2*z + 17196*x^2 + 528*x*y^6 + 2916*x*y^5*z - 4788*x*y^5 + 6660*x*y^4*z^2 - 21984*x*y^4*z + 17406*x*y^4 + 8040*x*y^3*z^3 - 40056*x*y^3*z^2 + 64101*x*y^3*z - 33593*x*y^3 + 5400*x*y^2*z^4 - 36144*x*y^2*z^3 + 87867*x*y^2*z^2 - 93555*x*y^2*z + 37029*x*y^2 + 1908*x*y*z^5 - 16116*x*y*z^4 + 53055*x*y*z^3 - 86331*x*y*z^2 + 69555*x*y*z - 22008*x*y + 276*x*z^6 - 2832*x*z^5 + 11883*x*z^4 - 26369*x*z^3 + 32526*x*z^2 - 20928*x*z + 5465*x - 48*y^7 - 312*y^6*z + 516*y^6 - 864*y^5*z^2 + 2868*y^5*z - 2268*y^5 - 1320*y^4*z^3 + 6600*y^4*z^2 - 10518*y^4*z + 5455*y^4 - 1200*y^3*z^4 + 8040*y^3*z^3 - 19392*y^3*z^2 + 20357*y^3*z - 7952*y^3 - 648*y^2*z^5 + 5460*y^2*z^4 - 17748*y^2*z^3 + 28341*y^2*z^2 - 22461*y^2*z + 7035*y^2 - 192*y*z^6 + 1956*y*z^5 - 8052*y*z^4 + 17431*y*z^3 - 21066*y*z^2 + 13389*y*z - 3483*y - 24*z^7 + 288*z^6 - 1446*z^5 + 3992*z^4 - 6557*z^3 + 6354*z^2 - 3350*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ 3, -2, -2 ], [ 2, -1, -1 ], [ 1, -2, -1 ] ], "source_shift": [ 1, 2, 1 ], "target_matrix": [ [ 1, 1, 1 ], [ 1, 1, 2 ], [ 3, 2, 2 ] ], "target_shift": [ -290, -285, -728 ], "collision_preimages": [ [ "-3", "9/4", "-25/4" ], [ "-7", "-2", "-17/2" ], [ "1", "-7", "19/2" ] ], "collision_image": [ "-1161/4", "-1141/4", "-2915/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0669847919998574 ], "surface_features": [ 4.213871335034818, 5.160123239326862, 3.8673496171887924, 0.0, 4.364369169720787, 5126 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B115", "pair": 115, "family": "bijection", "mapping": "F(x,y,z)=(-2187*x^7 + 15309*x^6*y + 10206*x^6*z - 10206*x^6 - 45927*x^5*y^2 - 61236*x^5*y*z + 61236*x^5*y - 20412*x^5*z^2 + 40824*x^5*z - 20493*x^5 + 76545*x^4*y^3 + 153090*x^4*y^2*z - 153090*x^4*y^2 + 102060*x^4*y*z^2 - 204120*x^4*y*z + 102546*x^4*y + 22680*x^4*z^3 - 68040*x^4*z^2 + 68337*x^4*z - 23004*x^4 - 76545*x^3*y^4 - 204120*x^3*y^3*z + 204120*x^3*y^3 - 204120*x^3*y^2*z^2 + 408240*x^3*y^2*z - 205254*x^3*y^2 - 90720*x^3*y*z^3 + 272160*x^3*y*z^2 - 273564*x^3*y*z + 92232*x^3*y - 15120*x^3*z^4 + 60480*x^3*z^3 - 91152*x^3*z^2 + 61425*x^3*z - 15714*x^3 + 45927*x^2*y^5 + 153090*x^2*y^4*z - 153090*x^2*y^4 + 204120*x^2*y^3*z^2 - 408240*x^2*y^3*z + 205416*x^2*y^3 + 136080*x^2*y^2*z^3 - 408240*x^2*y^2*z^2 + 410670*x^2*y^2*z - 138672*x^2*y^2 + 45360*x^2*y*z^4 - 181440*x^2*y*z^3 + 273672*x^2*y*z^2 - 184707*x^2*y*z + 47358*x^2*y + 6048*x^2*z^5 - 30240*x^2*z^4 + 60792*x^2*z^3 - 61506*x^2*z^2 + 31518*x^2*z - 6616*x^2 - 15309*x*y^6 - 61236*x*y^5*z + 61236*x*y^5 - 102060*x*y^4*z^2 + 204120*x*y^4*z - 102789*x*y^4 - 90720*x*y^3*z^3 + 272160*x*y^3*z^2 - 273996*x*y^3*z + 92664*x*y^3 - 45360*x*y^2*z^4 + 181440*x*y^2*z^3 - 273888*x*y^2*z^2 + 185139*x*y^2*z - 47574*x*y^2 - 12096*x*y*z^5 + 60480*x*y*z^4 - 121680*x*y*z^3 + 123300*x*y*z^2 - 63324*x*y*z + 13332*x*y - 1344*x*z^6 + 8064*x*z^5 - 20272*x*z^4 + 27372*x*z^3 - 21072*x*z^2 + 8867*x*z - 1617*x + 2187*y^7 + 10206*y^6*z - 10206*y^6 + 20412*y^5*z^2 - 40824*y^5*z + 20574*y^5 + 22680*y^4*z^3 - 68040*y^4*z^2 + 68553*y^4*z - 23220*y^4 + 15120*y^3*z^4 - 60480*y^3*z^3 + 91368*y^3*z^2 - 61857*y^3*z + 15930*y^3 + 6048*y^2*z^5 - 30240*y^2*z^4 + 60888*y^2*z^3 - 61794*y^2*z^2 + 31806*y^2*z - 6716*y^2 + 1344*y*z^6 - 8064*y*z^5 + 20288*y*z^4 - 27436*y*z^3 + 21168*y*z^2 - 8934*y*z + 1639*y + 128*z^7 - 896*z^6 + 2704*z^5 - 4568*z^4 + 4696*z^3 - 2971*z^2 + 1088*z - 1, 2187*x^7 - 15309*x^6*y - 10206*x^6*z + 10206*x^6 + 45927*x^5*y^2 + 61236*x^5*y*z - 61236*x^5*y + 20412*x^5*z^2 - 40824*x^5*z + 20493*x^5 - 76545*x^4*y^3 - 153090*x^4*y^2*z + 153090*x^4*y^2 - 102060*x^4*y*z^2 + 204120*x^4*y*z - 102546*x^4*y - 22680*x^4*z^3 + 68040*x^4*z^2 - 68337*x^4*z + 22842*x^4 + 76545*x^3*y^4 + 204120*x^3*y^3*z - 204120*x^3*y^3 + 204120*x^3*y^2*z^2 - 408240*x^3*y^2*z + 205254*x^3*y^2 + 90720*x^3*y*z^3 - 272160*x^3*y*z^2 + 273564*x^3*y*z - 91584*x^3*y + 15120*x^3*z^4 - 60480*x^3*z^3 + 91152*x^3*z^2 - 60993*x^3*z + 15282*x^3 - 45927*x^2*y^5 - 153090*x^2*y^4*z + 153090*x^2*y^4 - 204120*x^2*y^3*z^2 + 408240*x^2*y^3*z - 205416*x^2*y^3 - 136080*x^2*y^2*z^3 + 408240*x^2*y^2*z^2 - 410670*x^2*y^2*z + 137700*x^2*y^2 - 45360*x^2*y*z^4 + 181440*x^2*y*z^3 - 273672*x^2*y*z^2 + 183411*x^2*y*z - 46062*x^2*y - 6048*x^2*z^5 + 30240*x^2*z^4 - 60792*x^2*z^3 + 61074*x^2*z^2 - 30654*x^2*z + 6184*x^2 + 15309*x*y^6 + 61236*x*y^5*z - 61236*x*y^5 + 102060*x*y^4*z^2 - 204120*x*y^4*z + 102789*x*y^4 + 90720*x*y^3*z^3 - 272160*x*y^3*z^2 + 273996*x*y^3*z - 92016*x*y^3 + 45360*x*y^2*z^4 - 181440*x*y^2*z^3 + 273888*x*y^2*z^2 - 183843*x*y^2*z + 46278*x*y^2 + 12096*x*y*z^5 - 60480*x*y*z^4 + 121680*x*y*z^3 - 122436*x*y*z^2 + 61596*x*y*z - 12468*x*y + 1344*x*z^6 - 8064*x*z^5 + 20272*x*z^4 - 27180*x*z^3 + 20496*x*z^2 - 8291*x*z + 1417*x - 2187*y^7 - 10206*y^6*z + 10206*y^6 - 20412*y^5*z^2 + 40824*y^5*z - 20574*y^5 - 22680*y^4*z^3 + 68040*y^4*z^2 - 68553*y^4*z + 23058*y^4 - 15120*y^3*z^4 + 60480*y^3*z^3 - 91368*y^3*z^2 + 61425*y^3*z - 15498*y^3 - 6048*y^2*z^5 + 30240*y^2*z^4 - 60888*y^2*z^3 + 61362*y^2*z^2 - 30942*y^2*z + 6284*y^2 - 1344*y*z^6 + 8064*y*z^5 - 20288*y*z^4 + 27244*y*z^3 - 20592*y*z^2 + 8358*y*z - 1439*y - 128*z^7 + 896*z^6 - 2704*z^5 + 4536*z^4 - 4568*z^3 + 2779*z^2 - 954*z + 1, -4374*x^7 + 30618*x^6*y + 20412*x^6*z - 20412*x^6 - 91854*x^5*y^2 - 122472*x^5*y*z + 122472*x^5*y - 40824*x^5*z^2 + 81648*x^5*z - 40986*x^5 + 153090*x^4*y^3 + 306180*x^4*y^2*z - 306180*x^4*y^2 + 204120*x^4*y*z^2 - 408240*x^4*y*z + 205092*x^4*y + 45360*x^4*z^3 - 136080*x^4*z^2 + 136674*x^4*z - 46170*x^4 - 153090*x^3*y^4 - 408240*x^3*y^3*z + 408240*x^3*y^3 - 408240*x^3*y^2*z^2 + 816480*x^3*y^2*z - 410508*x^3*y^2 - 181440*x^3*y*z^3 + 544320*x^3*y*z^2 - 547128*x^3*y*z + 185112*x^3*y - 30240*x^3*z^4 + 120960*x^3*z^3 - 182304*x^3*z^2 + 123282*x^3*z - 31833*x^3 + 91854*x^2*y^5 + 306180*x^2*y^4*z - 306180*x^2*y^4 + 408240*x^2*y^3*z^2 - 816480*x^2*y^3*z + 410832*x^2*y^3 + 272160*x^2*y^2*z^3 - 816480*x^2*y^2*z^2 + 821340*x^2*y^2*z - 278316*x^2*y^2 + 90720*x^2*y*z^4 - 362880*x^2*y*z^3 + 547344*x^2*y*z^2 - 370710*x^2*y*z + 95931*x^2*y + 12096*x^2*z^5 - 60480*x^2*z^4 + 121584*x^2*z^3 - 123444*x^2*z^2 + 63846*x^2*z - 13610*x^2 - 30618*x*y^6 - 122472*x*y^5*z + 122472*x*y^5 - 204120*x*y^4*z^2 + 408240*x*y^4*z - 205578*x*y^4 - 181440*x*y^3*z^3 + 544320*x*y^3*z^2 - 547992*x*y^3*z + 185976*x*y^3 - 90720*x*y^2*z^4 + 362880*x*y^2*z^3 - 547776*x*y^2*z^2 + 371574*x*y^2*z - 96363*x*y^2 - 24192*x*y*z^5 + 120960*x*y*z^4 - 243360*x*y*z^3 + 247464*x*y*z^2 - 128268*x*y*z + 27420*x*y - 2688*x*z^6 + 16128*x*z^5 - 40544*x*z^4 + 54936*x*z^3 - 42684*x*z^2 + 18238*x*z - 3397*x + 4374*y^7 + 20412*y^6*z - 20412*y^6 + 40824*y^5*z^2 - 81648*y^5*z + 41148*y^5 + 45360*y^4*z^3 - 136080*y^4*z^2 + 137106*y^4*z - 46602*y^4 + 30240*y^3*z^4 - 120960*y^3*z^3 + 182736*y^3*z^2 - 124146*y^3*z + 32265*y^3 + 12096*y^2*z^5 - 60480*y^2*z^4 + 121776*y^2*z^3 - 124020*y^2*z^2 + 64422*y^2*z - 13810*y^2 + 2688*y*z^6 - 16128*y*z^5 + 40576*y*z^4 - 55064*y*z^3 + 42876*y*z^2 - 18372*y*z + 3440*y + 256*z^7 - 1792*z^6 + 5408*z^5 - 9168*z^4 + 9512*z^3 - 6110*z^2 + 2285*z - 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ 3, -3, -2 ], [ 1, -2, -1 ], [ 1, -1, -1 ] ], "source_shift": [ 2, 2, 2 ], "target_matrix": [ [ -1, -1, -1 ], [ 1, 1, 2 ], [ -2, -1, -1 ] ], "target_shift": [ 181, -141, 393 ], "inverse_recipe": { "p": 3, "q": 4, "f": "x**3", "g": "x**4 + x", "h": "y*z + y", "polynomials": { "f": [ [ [ 3, 0, 0 ], 1 ] ], "g": [ [ [ 4, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 1, 0 ], 1 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "Maybe" ], "seconds": [ 1.0639454840020335 ], "surface_features": [ 5.04821090184323, 5.914522973875967, 4.784906013167753, 0.0, 5.151000589376546, 5463 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C239", "pair": 239, "family": "counterexample", "mapping": "F(x,y,z)=(-8*x^7 + 52*x^6*y + 44*x^6*z - 148*x^6 - 138*x^5*y^2 - 240*x^5*y*z + 744*x^5*y - 102*x^5*z^2 + 660*x^5*z - 996*x^5 + 195*x^4*y^3 + 519*x^4*y^2*z - 1521*x^4*y^2 + 453*x^4*y*z^2 - 2730*x^4*y*z + 3960*x^4*y + 129*x^4*z^3 - 1209*x^4*z^2 + 3588*x^4*z - 3439*x^4 - 159*x^3*y^4 - 573*x^3*y^3*z + 1617*x^3*y^3 - 765*x^3*y^2*z^2 + 4398*x^3*y^2*z - 6189*x^3*y^2 - 447*x^3*y*z^3 + 3945*x^3*y*z^2 - 11286*x^3*y*z + 10557*x^3*y - 96*x^3*z^4 + 1164*x^3*z^3 - 5097*x^3*z^2 + 9670*x^3*z - 6776*x^3 + 75*x^2*y^5 + 342*x^2*y^4*z - 942*x^2*y^4 + 618*x^2*y^3*z^2 - 3447*x^2*y^3*z + 4743*x^2*y^3 + 552*x^2*y^2*z^3 - 4689*x^2*y^2*z^2 + 13056*x^2*y^2*z - 11973*x^2*y^2 + 243*x^2*y*z^4 - 2805*x^2*y*z^3 + 11883*x^2*y*z^2 - 22023*x^2*y*z + 15168*x^2*y + 42*x^2*z^5 - 621*x^2*z^4 + 3570*x^2*z^3 - 10050*x^2*z^2 + 13968*x^2*z - 7723*x^2 - 19*x*y^6 - 105*x*y^5*z + 285*x*y^5 - 240*x*y^4*z^2 + 1314*x*y^4*z - 1779*x*y^4 - 290*x*y^3*z^3 + 2406*x*y^3*z^2 - 6570*x*y^3*z + 5932*x*y^3 - 195*x*y^2*z^4 + 2184*x*y^2*z^3 - 9036*x*y^2*z^2 + 16443*x*y^2*z - 11163*x*y^2 - 69*x*y*z^5 + 981*x*y*z^4 - 5478*x*y*z^3 + 15090*x*y*z^2 - 20625*x*y*z + 11251*x*y - 10*x*z^6 + 174*x*z^5 - 1233*x*z^4 + 4579*x*z^3 - 9462*x*z^2 + 10378*x*z - 4745*x + 2*y^7 + 13*y^6*z - 35*y^6 + 36*y^5*z^2 - 195*y^5*z + 261*y^5 + 55*y^4*z^3 - 450*y^4*z^2 + 1212*y^4*z - 1081*y^4 + 50*y^3*z^4 - 550*y^3*z^3 + 2238*y^3*z^2 - 4015*y^3*z + 2693*y^3 + 27*y^2*z^5 - 375*y^2*z^4 + 2052*y^2*z^3 - 5559*y^2*z^2 + 7494*y^2*z - 4042*y^2 + 8*y*z^6 - 135*y*z^5 + 933*y*z^4 - 3397*y*z^3 + 6909*y*z^2 - 7481*y*z + 3387*y + z^7 - 20*z^6 + 168*z^5 - 772*z^4 + 2108*z^3 - 3439*z^2 + 3120*z - 1, 8*x^7 - 52*x^6*y - 44*x^6*z + 160*x^6 + 138*x^5*y^2 + 240*x^5*y*z - 816*x^5*y + 102*x^5*z^2 - 720*x^5*z + 1188*x^5 - 195*x^4*y^3 - 519*x^4*y^2*z + 1692*x^4*y^2 - 453*x^4*y*z^2 + 3024*x^4*y*z - 4800*x^4*y - 129*x^4*z^3 + 1332*x^4*z^2 - 4332*x^4*z + 4447*x^4 + 159*x^3*y^4 + 573*x^3*y^3*z - 1824*x^3*y^3 + 765*x^3*y^2*z^2 - 4944*x^3*y^2*z + 7623*x^3*y^2 + 447*x^3*y*z^3 - 4416*x^3*y*z^2 + 13854*x^3*y*z - 13863*x^3*y + 96*x^3*z^4 - 1296*x^3*z^3 + 6231*x^3*z^2 - 12664*x^3*z + 9332*x^3 - 75*x^2*y^5 - 342*x^2*y^4*z + 1077*x^2*y^4 - 618*x^2*y^3*z^2 + 3930*x^2*y^3*z - 5934*x^2*y^3 - 552*x^2*y^2*z^3 + 5328*x^2*y^2*z^2 - 16287*x^2*y^2*z + 15963*x^2*y^2 - 243*x^2*y*z^4 + 3174*x^2*y*z^3 - 14772*x^2*y*z^2 + 29289*x^2*y*z - 21192*x^2*y - 42*x^2*z^5 + 699*x^2*z^4 - 4419*x^2*z^3 + 13326*x^2*z^2 - 19476*x^2*z + 11182*x^2 + 19*x*y^6 + 105*x*y^5*z - 330*x*y^5 + 240*x*y^4*z^2 - 1518*x*y^4*z + 2259*x*y^4 + 290*x*y^3*z^3 - 2772*x*y^3*z^2 + 8322*x*y^3*z - 8026*x*y^3 + 195*x*y^2*z^4 - 2508*x*y^2*z^3 + 11412*x*y^2*z^2 - 22197*x*y^2*z + 15819*x*y^2 + 69*x*y*z^5 - 1122*x*y*z^4 + 6894*x*y*z^3 - 20316*x*y*z^2 + 29169*x*y*z - 16516*x*y + 10*x*z^6 - 198*x*z^5 + 1545*x*z^4 - 6145*x*z^3 + 13350*x*z^2 - 15208*x*z + 7159*x - 2*y^7 - 13*y^6*z + 41*y^6 - 36*y^5*z^2 + 228*y^5*z - 336*y^5 - 55*y^4*z^3 + 525*y^4*z^2 - 1557*y^4*z + 1483*y^4 - 50*y^3*z^4 + 640*y^3*z^3 - 2868*y^3*z^2 + 5497*y^3*z - 3869*y^3 - 27*y^2*z^5 + 435*y^2*z^4 - 2622*y^2*z^3 + 7593*y^2*z^2 - 10746*y^2*z + 6016*y^2 - 8*y*z^6 + 156*y*z^5 - 1188*y*z^4 + 4627*y*z^3 - 9885*y*z^2 + 11114*y*z - 5182*y - z^7 + 23*z^6 - 213*z^5 + 1048*z^4 - 3008*z^3 + 5098*z^2 - 4765*z + 1, -16*x^7 + 104*x^6*y + 88*x^6*z - 308*x^6 - 276*x^5*y^2 - 480*x^5*y*z + 1560*x^5*y - 204*x^5*z^2 + 1380*x^5*z - 2184*x^5 + 390*x^4*y^3 + 1038*x^4*y^2*z - 3213*x^4*y^2 + 906*x^4*y*z^2 - 5754*x^4*y*z + 8760*x^4*y + 258*x^4*z^3 - 2541*x^4*z^2 + 7920*x^4*z - 7885*x^4 - 318*x^3*y^4 - 1146*x^3*y^3*z + 3441*x^3*y^3 - 1530*x^3*y^2*z^2 + 9342*x^3*y^2*z - 13812*x^3*y^2 - 894*x^3*y*z^3 + 8361*x^3*y*z^2 - 25140*x^3*y*z + 24415*x^3*y - 192*x^3*z^4 + 2460*x^3*z^3 - 11328*x^3*z^2 + 22330*x^3*z - 16094*x^3 + 150*x^2*y^5 + 684*x^2*y^4*z - 2019*x^2*y^4 + 1236*x^2*y^3*z^2 - 7377*x^2*y^3*z + 10677*x^2*y^3 + 1104*x^2*y^2*z^3 - 10017*x^2*y^2*z^2 + 29343*x^2*y^2*z - 27927*x^2*y^2 + 486*x^2*y*z^4 - 5979*x^2*y*z^3 + 26655*x^2*y*z^2 - 51297*x^2*y*z + 36315*x^2*y + 84*x^2*z^5 - 1320*x^2*z^4 + 7989*x^2*z^3 - 23370*x^2*z^2 + 33405*x^2*z - 18851*x^2 - 38*x*y^6 - 210*x*y^5*z + 615*x*y^5 - 480*x*y^4*z^2 + 2832*x*y^4*z - 4038*x*y^4 - 580*x*y^3*z^3 + 5178*x*y^3*z^2 - 14892*x*y^3*z + 13951*x*y^3 - 390*x*y^2*z^4 + 4692*x*y^2*z^3 - 20448*x*y^2*z^2 + 38622*x*y^2*z - 26934*x*y^2 - 138*x*y*z^5 + 2103*x*y*z^4 - 12372*x*y*z^3 + 35391*x*y*z^2 - 49710*x*y*z + 27659*x*y - 20*x*z^6 + 372*x*z^5 - 2778*x*z^4 + 10720*x*z^3 - 22776*x*z^2 + 25490*x*z - 11826*x + 4*y^7 + 26*y^6*z - 76*y^6 + 72*y^5*z^2 - 423*y^5*z + 597*y^5 + 110*y^4*z^3 - 975*y^4*z^2 + 2769*y^4*z - 2562*y^4 + 100*y^3*z^4 - 1190*y^3*z^3 + 5106*y^3*z^2 - 9505*y^3*z + 6545*y^3 + 54*y^2*z^5 - 810*y^2*z^4 + 4674*y^2*z^3 - 13143*y^2*z^2 + 18195*y^2*z - 10004*y^2 + 16*y*z^6 - 291*y*z^5 + 2121*y*z^4 - 8019*y*z^3 + 16755*y*z^2 - 18499*y*z + 8495*y + 2*z^7 - 43*z^6 + 381*z^5 - 1819*z^4 + 5105*z^3 - 8495*z^2 + 7819*z - 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ 1, -1, -1 ], [ 2, -1, -1 ], [ 1, -2, -1 ] ], "source_shift": [ 2, 2, 2 ], "target_matrix": [ [ -1, -1, -1 ], [ 1, 2, 1 ], [ -2, -3, -3 ] ], "target_shift": [ 1219, -1905, 3089 ], "collision_preimages": [ [ "0", "1/4", "7/4" ], [ "-5/2", "-11/2", "4" ], [ "5/2", "-15/2", "13" ] ], "collision_image": [ "4877/4", "-7621/4", "6179/2" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 0.9941302980005275 ], "surface_features": [ 3.7328343094546867, 4.710091967031052, 3.218666771495872, 0.0, 3.9058833835641003, 4895 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B191", "pair": 191, "family": "bijection", "mapping": "F(x,y,z)=(-192*x^7 - 704*x^6*y + 704*x^6*z + 1568*x^6 - 1104*x^5*y^2 + 2208*x^5*y*z + 4912*x^5*y - 1104*x^5*z^2 - 4912*x^5*z - 5488*x^5 - 960*x^4*y^3 + 2880*x^4*y^2*z + 6400*x^4*y^2 - 2880*x^4*y*z^2 - 12800*x^4*y*z - 14288*x^4*y + 960*x^4*z^3 + 6400*x^4*z^2 + 14288*x^4*z + 10672*x^4 - 500*x^3*y^4 + 2000*x^3*y^3*z + 4440*x^3*y^3 - 3000*x^3*y^2*z^2 - 13320*x^3*y^2*z - 14856*x^3*y^2 + 2000*x^3*y*z^3 + 13320*x^3*y*z^2 + 29712*x^3*y*z + 22176*x^3*y - 500*x^3*z^4 - 4440*x^3*z^3 - 14856*x^3*z^2 - 22176*x^3*z - 12448*x^3 - 156*x^2*y^5 + 780*x^2*y^4*z + 1730*x^2*y^4 - 1560*x^2*y^3*z^2 - 6920*x^2*y^3*z - 7712*x^2*y^3 + 1560*x^2*y^2*z^3 + 10380*x^2*y^2*z^2 + 23136*x^2*y^2*z + 17256*x^2*y^2 - 780*x^2*y*z^4 - 6920*x^2*y*z^3 - 23136*x^2*y*z^2 - 34512*x^2*y*z - 19360*x^2*y + 156*x^2*z^5 + 1730*x^2*z^4 + 7712*x^2*z^3 + 17256*x^2*z^2 + 19360*x^2*z + 8713*x^2 - 27*x*y^6 + 162*x*y^5*z + 359*x*y^5 - 405*x*y^4*z^2 - 1795*x*y^4*z - 1999*x*y^4 + 540*x*y^3*z^3 + 3590*x*y^3*z^2 + 7996*x*y^3*z + 5960*x*y^3 - 405*x*y^2*z^4 - 3590*x*y^2*z^3 - 11994*x*y^2*z^2 - 17880*x*y^2*z - 10024*x*y^2 + 162*x*y*z^5 + 1795*x*y*z^4 + 7996*x*y*z^3 + 17880*x*y*z^2 + 20048*x*y*z + 9023*x*y - 27*x*z^6 - 359*x*z^5 - 1999*x*z^4 - 5960*x*z^3 - 10024*x*z^2 - 9020*x*z - 3393*x - 2*y^7 + 14*y^6*z + 31*y^6 - 42*y^5*z^2 - 186*y^5*z - 207*y^5 + 70*y^4*z^3 + 465*y^4*z^2 + 1035*y^4*z + 771*y^4 - 70*y^3*z^4 - 620*y^3*z^3 - 2070*y^3*z^2 - 3084*y^3*z - 1728*y^3 + 42*y^2*z^5 + 465*y^2*z^4 + 2070*y^2*z^3 + 4626*y^2*z^2 + 5184*y^2*z + 2334*y^2 - 14*y*z^6 - 186*y*z^5 - 1035*y*z^4 - 3084*y*z^3 - 5184*y*z^2 - 4666*y*z - 1760*y + 2*z^7 + 31*z^6 + 207*z^5 + 771*z^4 + 1728*z^3 + 2332*z^2 + 1755*z + 1, -192*x^7 - 704*x^6*y + 704*x^6*z + 1568*x^6 - 1104*x^5*y^2 + 2208*x^5*y*z + 4912*x^5*y - 1104*x^5*z^2 - 4912*x^5*z - 5488*x^5 - 960*x^4*y^3 + 2880*x^4*y^2*z + 6400*x^4*y^2 - 2880*x^4*y*z^2 - 12800*x^4*y*z - 14288*x^4*y + 960*x^4*z^3 + 6400*x^4*z^2 + 14288*x^4*z + 10672*x^4 - 500*x^3*y^4 + 2000*x^3*y^3*z + 4440*x^3*y^3 - 3000*x^3*y^2*z^2 - 13320*x^3*y^2*z - 14856*x^3*y^2 + 2000*x^3*y*z^3 + 13320*x^3*y*z^2 + 29712*x^3*y*z + 22176*x^3*y - 500*x^3*z^4 - 4440*x^3*z^3 - 14856*x^3*z^2 - 22176*x^3*z - 12448*x^3 - 156*x^2*y^5 + 780*x^2*y^4*z + 1730*x^2*y^4 - 1560*x^2*y^3*z^2 - 6920*x^2*y^3*z - 7712*x^2*y^3 + 1560*x^2*y^2*z^3 + 10380*x^2*y^2*z^2 + 23136*x^2*y^2*z + 17256*x^2*y^2 - 780*x^2*y*z^4 - 6920*x^2*y*z^3 - 23136*x^2*y*z^2 - 34512*x^2*y*z - 19360*x^2*y + 156*x^2*z^5 + 1730*x^2*z^4 + 7712*x^2*z^3 + 17256*x^2*z^2 + 19360*x^2*z + 8713*x^2 - 27*x*y^6 + 162*x*y^5*z + 359*x*y^5 - 405*x*y^4*z^2 - 1795*x*y^4*z - 1999*x*y^4 + 540*x*y^3*z^3 + 3590*x*y^3*z^2 + 7996*x*y^3*z + 5960*x*y^3 - 405*x*y^2*z^4 - 3590*x*y^2*z^3 - 11994*x*y^2*z^2 - 17880*x*y^2*z - 10024*x*y^2 + 162*x*y*z^5 + 1795*x*y*z^4 + 7996*x*y*z^3 + 17880*x*y*z^2 + 20048*x*y*z + 9023*x*y - 27*x*z^6 - 359*x*z^5 - 1999*x*z^4 - 5960*x*z^3 - 10024*x*z^2 - 9020*x*z - 3399*x - 2*y^7 + 14*y^6*z + 31*y^6 - 42*y^5*z^2 - 186*y^5*z - 207*y^5 + 70*y^4*z^3 + 465*y^4*z^2 + 1035*y^4*z + 771*y^4 - 70*y^3*z^4 - 620*y^3*z^3 - 2070*y^3*z^2 - 3084*y^3*z - 1728*y^3 + 42*y^2*z^5 + 465*y^2*z^4 + 2070*y^2*z^3 + 4626*y^2*z^2 + 5184*y^2*z + 2334*y^2 - 14*y*z^6 - 186*y*z^5 - 1035*y*z^4 - 3084*y*z^3 - 5184*y*z^2 - 4666*y*z - 1764*y + 2*z^7 + 31*z^6 + 207*z^5 + 771*z^4 + 1728*z^3 + 2332*z^2 + 1759*z + 1, -192*x^7 - 704*x^6*y + 704*x^6*z + 1632*x^6 - 1104*x^5*y^2 + 2208*x^5*y*z + 5104*x^5*y - 1104*x^5*z^2 - 5104*x^5*z - 5904*x^5 - 960*x^4*y^3 + 2880*x^4*y^2*z + 6640*x^4*y^2 - 2880*x^4*y*z^2 - 13280*x^4*y*z - 15328*x^4*y + 960*x^4*z^3 + 6640*x^4*z^2 + 15328*x^4*z + 11808*x^4 - 500*x^3*y^4 + 2000*x^3*y^3*z + 4600*x^3*y^3 - 3000*x^3*y^2*z^2 - 13800*x^3*y^2*z - 15896*x^3*y^2 + 2000*x^3*y*z^3 + 13800*x^3*y*z^2 + 31792*x^3*y*z + 24448*x^3*y - 500*x^3*z^4 - 4600*x^3*z^3 - 15896*x^3*z^2 - 24448*x^3*z - 14112*x^3 - 156*x^2*y^5 + 780*x^2*y^4*z + 1790*x^2*y^4 - 1560*x^2*y^3*z^2 - 7160*x^2*y^3*z - 8232*x^2*y^3 + 1560*x^2*y^2*z^3 + 10740*x^2*y^2*z^2 + 24696*x^2*y^2*z + 18960*x^2*y^2 - 780*x^2*y*z^4 - 7160*x^2*y*z^3 - 24696*x^2*y*z^2 - 37920*x^2*y*z - 21856*x^2*y + 156*x^2*z^5 + 1790*x^2*z^4 + 8232*x^2*z^3 + 18960*x^2*z^2 + 21856*x^2*z + 10089*x^2 - 27*x*y^6 + 162*x*y^5*z + 371*x*y^5 - 405*x*y^4*z^2 - 1855*x*y^4*z - 2129*x*y^4 + 540*x*y^3*z^3 + 3710*x*y^3*z^2 + 8516*x*y^3*z + 6528*x*y^3 - 405*x*y^2*z^4 - 3710*x*y^2*z^3 - 12774*x*y^2*z^2 - 19584*x*y^2*z - 11272*x*y^2 + 162*x*y*z^5 + 1855*x*y*z^4 + 8516*x*y*z^3 + 19584*x*y*z^2 + 22544*x*y*z + 10399*x*y - 27*x*z^6 - 371*x*z^5 - 2129*x*z^4 - 6528*x*z^3 - 11272*x*z^2 - 10396*x*z - 4010*x - 2*y^7 + 14*y^6*z + 32*y^6 - 42*y^5*z^2 - 192*y^5*z - 220*y^5 + 70*y^4*z^3 + 480*y^4*z^2 + 1100*y^4*z + 842*y^4 - 70*y^3*z^4 - 640*y^3*z^3 - 2200*y^3*z^2 - 3368*y^3*z - 1936*y^3 + 42*y^2*z^5 + 480*y^2*z^4 + 2200*y^2*z^3 + 5052*y^2*z^2 + 5808*y^2*z + 2678*y^2 - 14*y*z^6 - 192*y*z^5 - 1100*y*z^4 - 3368*y*z^3 - 5808*y*z^2 - 5354*y*z - 2071*y + 2*z^7 + 32*z^6 + 220*z^5 + 842*z^4 + 1936*z^3 + 2676*z^2 + 2065*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -2, -1, 1 ], [ -1, -2, 1 ], [ -1, -1, 1 ] ], "source_shift": [ 2, 2, 2 ], "target_matrix": [ [ 1, 1, 2 ], [ 1, 1, 1 ], [ 1, 2, 1 ] ], "target_shift": [ -565, -573, -689 ], "inverse_recipe": { "p": 6, "q": 1, "f": "x**6 + x**5 + x**4 + x", "g": "x", "h": "y*z", "polynomials": { "f": [ [ [ 6, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ], [ [ 4, 0, 0 ], 1 ], [ [ 5, 0, 0 ], 1 ] ], "g": [ [ [ 1, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.060360938001395 ], "surface_features": [ 3.757666534964638, 4.5788683286660286, 3.3281756614383227, 0.0, 3.863291250674104, 4962 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B175", "pair": 175, "family": "bijection", "mapping": "F(x,y,z)=(384*x^7 - 1408*x^6*y + 1408*x^6*z - 2304*x^6 + 2208*x^5*y^2 - 4416*x^5*y*z + 7296*x^5*y + 2208*x^5*z^2 - 7296*x^5*z + 5856*x^5 - 1920*x^4*y^3 + 5760*x^4*y^2*z - 9600*x^4*y^2 - 5760*x^4*y*z^2 + 19200*x^4*y*z - 15616*x^4*y + 1920*x^4*z^3 - 9600*x^4*z^2 + 15616*x^4*z - 8274*x^4 + 1000*x^3*y^4 - 4000*x^3*y^3*z + 6720*x^3*y^3 + 6000*x^3*y^2*z^2 - 20160*x^3*y^2*z + 16592*x^3*y^2 - 4000*x^3*y*z^3 + 20160*x^3*y*z^2 - 33184*x^3*y*z + 18008*x^3*y + 1000*x^3*z^4 - 6720*x^3*z^3 + 16592*x^3*z^2 - 18008*x^3*z + 7128*x^3 - 312*x^2*y^5 + 1560*x^2*y^4*z - 2640*x^2*y^4 - 3120*x^2*y^3*z^2 + 10560*x^2*y^3*z - 8784*x^2*y^3 + 3120*x^2*y^2*z^3 - 15840*x^2*y^2*z^2 + 26352*x^2*y^2*z - 14628*x^2*y^2 - 1560*x^2*y*z^4 + 10560*x^2*y*z^3 - 26352*x^2*y*z^2 + 29256*x^2*y*z - 12096*x^2*y + 312*x^2*z^5 - 2640*x^2*z^4 + 8784*x^2*z^3 - 14628*x^2*z^2 + 12096*x^2*z - 3810*x^2 + 54*x*y^6 - 324*x*y^5*z + 552*x*y^5 + 810*x*y^4*z^2 - 2760*x*y^4*z + 2318*x*y^4 - 1080*x*y^3*z^3 + 5520*x*y^3*z^2 - 9272*x*y^3*z + 5262*x*y^3 + 810*x*y^2*z^4 - 5520*x*y^2*z^3 + 13908*x*y^2*z^2 - 15786*x*y^2*z + 6804*x*y^2 - 324*x*y*z^5 + 2760*x*y*z^4 - 9272*x*y*z^3 + 15786*x*y*z^2 - 13608*x*y*z + 4610*x*y + 54*x*z^6 - 552*x*z^5 + 2318*x*z^4 - 5262*x*z^3 + 6804*x*z^2 - 4616*x*z + 1207*x - 4*y^7 + 28*y^6*z - 48*y^6 - 84*y^5*z^2 + 288*y^5*z - 244*y^5 + 140*y^4*z^3 - 720*y^4*z^2 + 1220*y^4*z - 708*y^4 - 140*y^3*z^4 + 960*y^3*z^3 - 2440*y^3*z^2 + 2832*y^3*z - 1272*y^3 + 84*y^2*z^5 - 720*y^2*z^4 + 2440*y^2*z^3 - 4248*y^2*z^2 + 3816*y^2*z - 1380*y^2 - 28*y*z^6 + 288*y*z^5 - 1220*y*z^4 + 2832*y*z^3 - 3816*y*z^2 + 2764*y*z - 804*y + 4*z^7 - 48*z^6 + 244*z^5 - 708*z^4 + 1272*z^3 - 1384*z^2 + 804*z - 1, -192*x^7 + 704*x^6*y - 704*x^6*z + 1088*x^6 - 1104*x^5*y^2 + 2208*x^5*y*z - 3456*x^5*y - 1104*x^5*z^2 + 3456*x^5*z - 2544*x^5 + 960*x^4*y^3 - 2880*x^4*y^2*z + 4560*x^4*y^2 + 2880*x^4*y*z^2 - 9120*x^4*y*z + 6848*x^4*y - 960*x^4*z^3 + 4560*x^4*z^2 - 6848*x^4*z + 3161*x^4 - 500*x^3*y^4 + 2000*x^3*y^3*z - 3200*x^3*y^3 - 3000*x^3*y^2*z^2 + 9600*x^3*y^2*z - 7336*x^3*y^2 + 2000*x^3*y*z^3 - 9600*x^3*y*z^2 + 14672*x^3*y*z - 7052*x^3*y - 500*x^3*z^4 + 3200*x^3*z^3 - 7336*x^3*z^2 + 7052*x^3*z - 2212*x^3 + 156*x^2*y^5 - 780*x^2*y^4*z + 1260*x^2*y^4 + 1560*x^2*y^3*z^2 - 5040*x^2*y^3*z + 3912*x^2*y^3 - 1560*x^2*y^2*z^3 + 7560*x^2*y^2*z^2 - 11736*x^2*y^2*z + 5850*x^2*y^2 + 780*x^2*y*z^4 - 5040*x^2*y*z^3 + 11736*x^2*y*z^2 - 11700*x^2*y*z + 4020*x^2*y - 156*x^2*z^5 + 1260*x^2*z^4 - 3912*x^2*z^3 + 5850*x^2*z^2 - 4020*x^2*z + 825*x^2 - 27*x*y^6 + 162*x*y^5*z - 264*x*y^5 - 405*x*y^4*z^2 + 1320*x*y^4*z - 1039*x*y^4 + 540*x*y^3*z^3 - 2640*x*y^3*z^2 + 4156*x*y^3*z - 2143*x*y^3 - 405*x*y^2*z^4 + 2640*x*y^2*z^3 - 6234*x*y^2*z^2 + 6429*x*y^2*z - 2388*x*y^2 + 162*x*y*z^5 - 1320*x*y*z^4 + 4156*x*y*z^3 - 6429*x*y*z^2 + 4776*x*y*z - 1225*x*y - 27*x*z^6 + 264*x*z^5 - 1039*x*z^4 + 2143*x*z^3 - 2388*x*z^2 + 1228*x*z - 129*x + 2*y^7 - 14*y^6*z + 23*y^6 + 42*y^5*z^2 - 138*y^5*z + 110*y^5 - 70*y^4*z^3 + 345*y^4*z^2 - 550*y^4*z + 293*y^4 + 70*y^3*z^4 - 460*y^3*z^3 + 1100*y^3*z^2 - 1172*y^3*z + 467*y^3 - 42*y^2*z^5 + 345*y^2*z^4 - 1100*y^2*z^3 + 1758*y^2*z^2 - 1401*y^2*z + 420*y^2 + 14*y*z^6 - 138*y*z^5 + 550*y*z^4 - 1172*y*z^3 + 1401*y*z^2 - 842*y*z + 164*y - 2*z^7 + 23*z^6 - 110*z^5 + 293*z^4 - 467*z^3 + 422*z^2 - 163*z + 1, 192*x^7 - 704*x^6*y + 704*x^6*z - 1216*x^6 + 1104*x^5*y^2 - 2208*x^5*y*z + 3840*x^5*y + 1104*x^5*z^2 - 3840*x^5*z + 3312*x^5 - 960*x^4*y^3 + 2880*x^4*y^2*z - 5040*x^4*y^2 - 2880*x^4*y*z^2 + 10080*x^4*y*z - 8768*x^4*y + 960*x^4*z^3 - 5040*x^4*z^2 + 8768*x^4*z - 5113*x^4 + 500*x^3*y^4 - 2000*x^3*y^3*z + 3520*x^3*y^3 + 3000*x^3*y^2*z^2 - 10560*x^3*y^2*z + 9256*x^3*y^2 - 2000*x^3*y*z^3 + 10560*x^3*y*z^2 - 18512*x^3*y*z + 10956*x^3*y + 500*x^3*z^4 - 3520*x^3*z^3 + 9256*x^3*z^2 - 10956*x^3*z + 4916*x^3 - 156*x^2*y^5 + 780*x^2*y^4*z - 1380*x^2*y^4 - 1560*x^2*y^3*z^2 + 5520*x^2*y^3*z - 4872*x^2*y^3 + 1560*x^2*y^2*z^3 - 8280*x^2*y^2*z^2 + 14616*x^2*y^2*z - 8778*x^2*y^2 - 780*x^2*y*z^4 + 5520*x^2*y*z^3 - 14616*x^2*y*z^2 + 17556*x^2*y*z - 8076*x^2*y + 156*x^2*z^5 - 1380*x^2*z^4 + 4872*x^2*z^3 - 8778*x^2*z^2 + 8076*x^2*z - 2985*x^2 + 27*x*y^6 - 162*x*y^5*z + 288*x*y^5 + 405*x*y^4*z^2 - 1440*x*y^4*z + 1279*x*y^4 - 540*x*y^3*z^3 + 2880*x*y^3*z^2 - 5116*x*y^3*z + 3119*x*y^3 + 405*x*y^2*z^4 - 2880*x*y^2*z^3 + 7674*x*y^2*z^2 - 9357*x*y^2*z + 4416*x*y^2 - 162*x*y*z^5 + 1440*x*y*z^4 - 5116*x*y*z^3 + 9357*x*y*z^2 - 8832*x*y*z + 3385*x*y + 27*x*z^6 - 288*x*z^5 + 1279*x*z^4 - 3119*x*z^3 + 4416*x*z^2 - 3388*x*z + 1075*x - 2*y^7 + 14*y^6*z - 25*y^6 - 42*y^5*z^2 + 150*y^5*z - 134*y^5 + 70*y^4*z^3 - 375*y^4*z^2 + 670*y^4*z - 415*y^4 - 70*y^3*z^4 + 500*y^3*z^3 - 1340*y^3*z^2 + 1660*y^3*z - 805*y^3 + 42*y^2*z^5 - 375*y^2*z^4 + 1340*y^2*z^3 - 2490*y^2*z^2 + 2415*y^2*z - 960*y^2 - 14*y*z^6 + 150*y*z^5 - 670*y*z^4 + 1660*y*z^3 - 2415*y*z^2 + 1922*y*z - 638*y + 2*z^7 - 25*z^6 + 134*z^5 - 415*z^4 + 805*z^3 - 962*z^2 + 639*z - 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -2, 1, -1 ], [ -1, 1, -1 ], [ -1, 1, -2 ] ], "source_shift": [ 2, 1, 2 ], "target_matrix": [ [ -2, -3, -3 ], [ 1, 1, 2 ], [ -1, -1, -1 ] ], "target_shift": [ 180, 2, 178 ], "inverse_recipe": { "p": 1, "q": 6, "f": "x", "g": "x**6 + x**4 + x**3", "h": "y**4 + y*z + y", "polynomials": { "f": [ [ [ 1, 0, 0 ], 1 ] ], "g": [ [ [ 6, 0, 0 ], 1 ], [ [ 3, 0, 0 ], 1 ], [ [ 4, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 4, 0 ], 1 ], [ [ 0, 1, 0 ], 1 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.016315155000484 ], "surface_features": [ 3.5862840427643077, 4.520928734708947, 3.220108088040055, 0.0, 3.7087303419349444, 4906 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B246", "pair": 246, "family": "bijection", "mapping": "F(x,y,z)=(x^7 + 7*x^6*y - 14*x^6*z - 15*x^6 + 21*x^5*y^2 - 84*x^5*y*z - 90*x^5*y + 84*x^5*z^2 + 179*x^5*z + 94*x^5 + 35*x^4*y^3 - 210*x^4*y^2*z - 225*x^4*y^2 + 420*x^4*y*z^2 + 895*x^4*y*z + 470*x^4*y - 280*x^4*z^3 - 890*x^4*z^2 - 930*x^4*z - 322*x^4 + 35*x^3*y^4 - 280*x^3*y^3*z - 300*x^3*y^3 + 840*x^3*y^2*z^2 + 1790*x^3*y^2*z + 940*x^3*y^2 - 1120*x^3*y*z^3 - 3560*x^3*y*z^2 - 3720*x^3*y*z - 1288*x^3*y + 560*x^3*z^4 + 2360*x^3*z^3 + 3680*x^3*z^2 + 2535*x^3*z + 656*x^3 + 21*x^2*y^5 - 210*x^2*y^4*z - 225*x^2*y^4 + 840*x^2*y^3*z^2 + 1790*x^2*y^3*z + 940*x^2*y^3 - 1680*x^2*y^2*z^3 - 5340*x^2*y^2*z^2 - 5580*x^2*y^2*z - 1932*x^2*y^2 + 1680*x^2*y*z^4 + 7080*x^2*y*z^3 + 11040*x^2*y*z^2 + 7605*x^2*y*z + 1969*x^2*y - 672*x^2*z^5 - 3520*x^2*z^4 - 7280*x^2*z^3 - 7482*x^2*z^2 - 3848*x^2*z - 800*x^2 + 7*x*y^6 - 84*x*y^5*z - 90*x*y^5 + 420*x*y^4*z^2 + 895*x*y^4*z + 470*x*y^4 - 1120*x*y^3*z^3 - 3560*x*y^3*z^2 - 3720*x*y^3*z - 1288*x*y^3 + 1680*x*y^2*z^4 + 7080*x*y^2*z^3 + 11040*x*y^2*z^2 + 7605*x*y^2*z + 1970*x*y^2 - 1344*x*y*z^5 - 7040*x*y*z^4 - 14560*x*y*z^3 - 14964*x*y*z^2 - 7700*x*y*z - 1605*x*y + 448*x*z^6 + 2800*x*z^5 + 7200*x*z^4 + 9812*x*z^3 + 7520*x*z^2 + 3097*x*z + 540*x + y^7 - 14*y^6*z - 15*y^6 + 84*y^5*z^2 + 179*y^5*z + 94*y^5 - 280*y^4*z^3 - 890*y^4*z^2 - 930*y^4*z - 322*y^4 + 560*y^3*z^4 + 2360*y^3*z^3 + 3680*y^3*z^2 + 2535*y^3*z + 657*y^3 - 672*y^2*z^5 - 3520*y^2*z^4 - 7280*y^2*z^3 - 7482*y^2*z^2 - 3852*y^2*z - 805*y^2 + 448*y*z^6 + 2800*y*z^5 + 7200*y*z^4 + 9812*y*z^3 + 7524*y*z^2 + 3106*y*z + 543*y - 128*z^7 - 928*z^6 - 2848*z^5 - 4824*z^4 - 4896*z^3 - 2995*z^2 - 1028*z - 1, -2*x^7 - 14*x^6*y + 28*x^6*z + 30*x^6 - 42*x^5*y^2 + 168*x^5*y*z + 180*x^5*y - 168*x^5*z^2 - 358*x^5*z - 190*x^5 - 70*x^4*y^3 + 420*x^4*y^2*z + 450*x^4*y^2 - 840*x^4*y*z^2 - 1790*x^4*y*z - 950*x^4*y + 560*x^4*z^3 + 1780*x^4*z^2 + 1880*x^4*z + 664*x^4 - 70*x^3*y^4 + 560*x^3*y^3*z + 600*x^3*y^3 - 1680*x^3*y^2*z^2 - 3580*x^3*y^2*z - 1900*x^3*y^2 + 2240*x^3*y*z^3 + 7120*x^3*y*z^2 + 7520*x^3*y*z + 2656*x^3*y - 1120*x^3*z^4 - 4720*x^3*z^3 - 7440*x^3*z^2 - 5230*x^3*z - 1394*x^3 - 42*x^2*y^5 + 420*x^2*y^4*z + 450*x^2*y^4 - 1680*x^2*y^3*z^2 - 3580*x^2*y^3*z - 1900*x^2*y^3 + 3360*x^2*y^2*z^3 + 10680*x^2*y^2*z^2 + 11280*x^2*y^2*z + 3984*x^2*y^2 - 3360*x^2*y*z^4 - 14160*x^2*y*z^3 - 22320*x^2*y*z^2 - 15690*x^2*y*z - 4184*x^2*y + 1344*x^2*z^5 + 7040*x^2*z^4 + 14720*x^2*z^3 + 15444*x^2*z^2 + 8188*x^2*z + 1773*x^2 - 14*x*y^6 + 168*x*y^5*z + 180*x*y^5 - 840*x*y^4*z^2 - 1790*x*y^4*z - 950*x*y^4 + 2240*x*y^3*z^3 + 7120*x*y^3*z^2 + 7520*x*y^3*z + 2656*x*y^3 - 3360*x*y^2*z^4 - 14160*x*y^2*z^3 - 22320*x*y^2*z^2 - 15690*x*y^2*z - 4186*x*y^2 + 2688*x*y*z^5 + 14080*x*y*z^4 + 29440*x*y*z^3 + 30888*x*y*z^2 + 16384*x*y*z + 3556*x*y - 896*x*z^6 - 5600*x*z^5 - 14560*x*z^4 - 20264*x*z^3 - 16024*x*z^2 - 6886*x*z - 1271*x - 2*y^7 + 28*y^6*z + 30*y^6 - 168*y^5*z^2 - 358*y^5*z - 190*y^5 + 560*y^4*z^3 + 1780*y^4*z^2 + 1880*y^4*z + 664*y^4 - 1120*y^3*z^4 - 4720*y^3*z^3 - 7440*y^3*z^2 - 5230*y^3*z - 1396*y^3 + 1344*y^2*z^5 + 7040*y^2*z^4 + 14720*y^2*z^3 + 15444*y^2*z^2 + 8196*y^2*z + 1783*y^2 - 896*y*z^6 - 5600*y*z^5 - 14560*y*z^4 - 20264*y*z^3 - 16032*y*z^2 - 6904*y*z - 1279*y + 256*z^7 + 1856*z^6 + 5760*z^5 + 9968*z^4 + 10448*z^3 + 6682*z^2 + 2437*z + 1, x^7 + 7*x^6*y - 14*x^6*z - 15*x^6 + 21*x^5*y^2 - 84*x^5*y*z - 90*x^5*y + 84*x^5*z^2 + 179*x^5*z + 94*x^5 + 35*x^4*y^3 - 210*x^4*y^2*z - 225*x^4*y^2 + 420*x^4*y*z^2 + 895*x^4*y*z + 470*x^4*y - 280*x^4*z^3 - 890*x^4*z^2 - 930*x^4*z - 322*x^4 + 35*x^3*y^4 - 280*x^3*y^3*z - 300*x^3*y^3 + 840*x^3*y^2*z^2 + 1790*x^3*y^2*z + 940*x^3*y^2 - 1120*x^3*y*z^3 - 3560*x^3*y*z^2 - 3720*x^3*y*z - 1288*x^3*y + 560*x^3*z^4 + 2360*x^3*z^3 + 3680*x^3*z^2 + 2535*x^3*z + 656*x^3 + 21*x^2*y^5 - 210*x^2*y^4*z - 225*x^2*y^4 + 840*x^2*y^3*z^2 + 1790*x^2*y^3*z + 940*x^2*y^3 - 1680*x^2*y^2*z^3 - 5340*x^2*y^2*z^2 - 5580*x^2*y^2*z - 1932*x^2*y^2 + 1680*x^2*y*z^4 + 7080*x^2*y*z^3 + 11040*x^2*y*z^2 + 7605*x^2*y*z + 1969*x^2*y - 672*x^2*z^5 - 3520*x^2*z^4 - 7280*x^2*z^3 - 7482*x^2*z^2 - 3848*x^2*z - 801*x^2 + 7*x*y^6 - 84*x*y^5*z - 90*x*y^5 + 420*x*y^4*z^2 + 895*x*y^4*z + 470*x*y^4 - 1120*x*y^3*z^3 - 3560*x*y^3*z^2 - 3720*x*y^3*z - 1288*x*y^3 + 1680*x*y^2*z^4 + 7080*x*y^2*z^3 + 11040*x*y^2*z^2 + 7605*x*y^2*z + 1970*x*y^2 - 1344*x*y*z^5 - 7040*x*y*z^4 - 14560*x*y*z^3 - 14964*x*y*z^2 - 7700*x*y*z - 1607*x*y + 448*x*z^6 + 2800*x*z^5 + 7200*x*z^4 + 9812*x*z^3 + 7520*x*z^2 + 3101*x*z + 545*x + y^7 - 14*y^6*z - 15*y^6 + 84*y^5*z^2 + 179*y^5*z + 94*y^5 - 280*y^4*z^3 - 890*y^4*z^2 - 930*y^4*z - 322*y^4 + 560*y^3*z^4 + 2360*y^3*z^3 + 3680*y^3*z^2 + 2535*y^3*z + 657*y^3 - 672*y^2*z^5 - 3520*y^2*z^4 - 7280*y^2*z^3 - 7482*y^2*z^2 - 3852*y^2*z - 806*y^2 + 448*y*z^6 + 2800*y*z^5 + 7200*y*z^4 + 9812*y*z^3 + 7524*y*z^2 + 3110*y*z + 548*y - 128*z^7 - 928*z^6 - 2848*z^5 - 4824*z^4 - 4896*z^3 - 2999*z^2 - 1037*z - 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -1, -1, 2 ], [ -1, -1, 1 ], [ -1, -2, 1 ] ], "source_shift": [ 2, 1, 1 ], "target_matrix": [ [ -1, -1, -1 ], [ 2, 1, 1 ], [ -1, -2, -1 ] ], "target_shift": [ 152, -394, 157 ], "inverse_recipe": { "p": 2, "q": 5, "f": "x**2", "g": "x**5 + x**3 + x**2 + x", "h": "y*z + y", "polynomials": { "f": [ [ [ 2, 0, 0 ], 1 ] ], "g": [ [ [ 5, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ], [ [ 3, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 1, 0 ], 1 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 0.9836458669997228 ], "surface_features": [ 3.536294292694767, 4.489789788615993, 3.13640344813399, 0.0, 3.6778854152686544, 4840 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C105", "pair": 105, "family": "counterexample", "mapping": "F(x,y,z)=(24*x^7 - 124*x^6*y - 196*x^6*z + 244*x^6 + 270*x^5*y^2 + 840*x^5*y*z - 1068*x^5*y + 642*x^5*z^2 - 1680*x^5*z + 1056*x^5 - 321*x^4*y^3 - 1473*x^4*y^2*z + 1917*x^4*y^2 - 2211*x^4*y*z^2 + 5934*x^4*y*z - 3816*x^4*y - 1083*x^4*z^3 + 4509*x^4*z^2 - 5976*x^4*z + 2537*x^4 + 225*x^3*y^4 + 1353*x^3*y^3*z - 1806*x^3*y^3 + 2991*x^3*y^2*z^2 - 8247*x^3*y^2*z + 5424*x^3*y^2 + 2875*x^3*y*z^3 - 12312*x^3*y*z^2 + 16692*x^3*y*z - 7271*x^3*y + 1012*x^3*z^4 - 5995*x^3*z^3 + 12588*x^3*z^2 - 11326*x^3*z + 3677*x^3 - 93*x^2*y^5 - 687*x^2*y^4*z + 942*x^2*y^4 - 1989*x^2*y^3*z^2 + 5640*x^2*y^3*z - 3792*x^2*y^3 - 2817*x^2*y^2*z^3 + 12411*x^2*y^2*z^2 - 17196*x^2*y^2*z + 7674*x^2*y^2 - 1950*x^2*y*z^4 + 11874*x^2*y*z^3 - 25452*x^2*y*z^2 + 23448*x^2*y*z - 7830*x^2*y - 528*x^2*z^5 + 4161*x^2*z^4 - 12264*x^2*z^3 + 17517*x^2*z^2 - 12108*x^2*z + 3232*x^2 + 21*x*y^6 + 183*x*y^5*z - 258*x*y^5 + 651*x*y^4*z^2 - 1899*x*y^4*z + 1305*x*y^4 + 1209*x*y^3*z^3 - 5478*x*y^3*z^2 + 7755*x*y^3*z - 3538*x*y^3 + 1236*x*y^2*z^4 - 7731*x*y^2*z^3 + 16917*x*y^2*z^2 - 15906*x*y^2*z + 5448*x*y^2 + 660*x*y*z^5 - 5334*x*y*z^4 + 16029*x*y*z^3 - 23289*x*y*z^2 + 16485*x*y*z - 4537*x*y + 144*x*z^6 - 1440*x*z^5 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687*x^2*y^4*z - 942*x^2*y^4 + 1989*x^2*y^3*z^2 - 5640*x^2*y^3*z + 3792*x^2*y^3 + 2817*x^2*y^2*z^3 - 12411*x^2*y^2*z^2 + 17196*x^2*y^2*z - 7659*x^2*y^2 + 1950*x^2*y*z^4 - 11874*x^2*y*z^3 + 25452*x^2*y*z^2 - 23394*x^2*y*z + 7785*x^2*y + 528*x^2*z^5 - 4161*x^2*z^4 + 12264*x^2*z^3 - 17469*x^2*z^2 + 12027*x^2*z - 3199*x^2 - 21*x*y^6 - 183*x*y^5*z + 258*x*y^5 - 651*x*y^4*z^2 + 1899*x*y^4*z - 1305*x*y^4 - 1209*x*y^3*z^3 + 5478*x*y^3*z^2 - 7755*x*y^3*z + 3529*x*y^3 - 1236*x*y^2*z^4 + 7731*x*y^2*z^3 - 16917*x*y^2*z^2 + 15858*x*y^2*z - 5409*x*y^2 - 660*x*y*z^5 + 5334*x*y*z^4 - 16029*x*y*z^3 + 23205*x*y*z^2 - 16347*x*y*z + 4480*x*y - 144*x*z^6 + 1440*x*z^5 - 5562*x*z^4 + 11026*x*z^3 - 12036*x*z^2 + 6841*x*z - 1576*x + 2*y^7 + 20*y^6*z - 29*y^6 + 84*y^5*z^2 - 252*y^5*z + 177*y^5 + 192*y^4*z^3 - 894*y^4*z^2 + 1293*y^4*z - 600*y^4 + 258*y^3*z^4 - 1656*y^3*z^3 + 3699*y^3*z^2 - 3529*y^3*z + 1230*y^3 + 204*y^2*z^5 - 1689*y^2*z^4 + 5175*y^2*z^3 - 7605*y^2*z^2 + 5460*y^2*z - 1534*y^2 + 88*y*z^6 - 900*y*z^5 + 3540*y*z^4 - 7107*y*z^3 + 7872*y*z^2 - 4568*y*z + 1087*y + 16*z^7 - 196*z^6 + 948*z^5 - 2431*z^4 + 3678*z^3 - 3298*z^2 + 1633*z - 1, -48*x^7 + 248*x^6*y + 392*x^6*z - 452*x^6 - 540*x^5*y^2 - 1680*x^5*y*z + 1968*x^5*y - 1284*x^5*z^2 + 3084*x^5*z - 1836*x^5 + 642*x^4*y^3 + 2946*x^4*y^2*z - 3513*x^4*y^2 + 4422*x^4*y*z^2 - 10830*x^4*y*z + 6588*x^4*y + 2166*x^4*z^3 - 8193*x^4*z^2 + 10260*x^4*z - 4201*x^4 - 450*x^3*y^4 - 2706*x^3*y^3*z + 3291*x^3*y^3 - 5982*x^3*y^2*z^2 + 14964*x^3*y^2*z - 9297*x^3*y^2 - 5750*x^3*y*z^3 + 22239*x^3*y*z^2 - 28446*x^3*y*z + 11943*x^3*y - 2024*x^3*z^4 + 10778*x^3*z^3 - 21321*x^3*z^2 + 18478*x^3*z - 5871*x^3 + 186*x^2*y^5 + 1374*x^2*y^4*z - 1707*x^2*y^4 + 3978*x^2*y^3*z^2 - 10176*x^2*y^3*z + 6453*x^2*y^3 + 5634*x^2*y^2*z^3 - 22293*x^2*y^2*z^2 + 29091*x^2*y^2*z - 12504*x^2*y^2 + 3900*x^2*y*z^4 - 21234*x^2*y*z^3 + 42795*x^2*y*z^2 - 37944*x^2*y*z + 12393*x^2*y + 1056*x^2*z^5 - 7410*x^2*z^4 + 20493*x^2*z^3 - 28143*x^2*z^2 + 19017*x^2*z - 5024*x^2 - 42*x*y^6 - 366*x*y^5*z + 465*x*y^5 - 1302*x*y^4*z^2 + 3408*x*y^4*z - 2205*x*y^4 - 2418*x*y^3*z^3 + 9789*x*y^3*z^2 - 13026*x*y^3*z + 5720*x*y^3 - 2472*x*y^2*z^4 + 13758*x*y^2*z^3 - 28245*x*y^2*z^2 + 25542*x*y^2*z - 8550*x*y^2 - 1320*x*y*z^5 + 9456*x*y*z^4 - 26604*x*y*z^3 + 37143*x*y*z^2 - 25674*x*y*z + 6986*x*y - 288*x*z^6 + 2544*x*z^5 - 9180*x*z^4 + 17546*x*z^3 - 18783*x*z^2 + 10598*x*z - 2442*x + 4*y^7 + 40*y^6*z - 52*y^6 + 168*y^5*z^2 - 450*y^5*z + 297*y^5 + 384*y^4*z^3 - 1590*y^4*z^2 + 2157*y^4*z - 966*y^4 + 516*y^3*z^4 - 2934*y^3*z^3 + 6135*y^3*z^2 - 5648*y^3*z + 1932*y^3 + 408*y^2*z^5 - 2982*y^2*z^4 + 8535*y^2*z^3 - 12102*y^2*z^2 + 8526*y^2*z - 2378*y^2 + 176*y*z^6 - 1584*y*z^5 + 5808*y*z^4 - 11250*y*z^3 + 12225*y*z^2 - 7042*y*z + 1672*y + 32*z^7 - 344*z^6 + 1548*z^5 - 3830*z^4 + 5685*z^3 - 5060*z^2 + 2494*z - 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ 1, -1, -2 ], [ 2, -1, -1 ], [ 3, -2, -2 ] ], "source_shift": [ 1, 2, 2 ], "target_matrix": [ [ 1, 1, 1 ], [ -1, -1, -2 ], [ -2, -1, -1 ] ], "target_shift": [ -343, 337, 517 ], "collision_preimages": [ [ "-7/4", "-9/4", "3/4" ], [ "-23/2", "-55/2", "8" ], [ "-11/2", "-35/2", "7" ] ], "collision_image": [ "-1373/4", "1349/4", "1035/2" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0695047990011517 ], "surface_features": [ 3.7503665210059647, 4.631393030712758, 3.4586378490256493, 0.0, 3.8510219261155116, 4988 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B041", "pair": 41, "family": "bijection", "mapping": "F(x,y,z)=(4*x^7 + 28*x^6*y + 28*x^6*z + 60*x^6 + 84*x^5*y^2 + 168*x^5*y*z + 360*x^5*y + 84*x^5*z^2 + 360*x^5*z + 392*x^5 + 140*x^4*y^3 + 420*x^4*y^2*z + 900*x^4*y^2 + 420*x^4*y*z^2 + 1800*x^4*y*z + 1964*x^4*y + 140*x^4*z^3 + 900*x^4*z^2 + 1960*x^4*z + 1448*x^4 + 140*x^3*y^4 + 560*x^3*y^3*z + 1200*x^3*y^3 + 840*x^3*y^2*z^2 + 3600*x^3*y^2*z + 3936*x^3*y^2 + 560*x^3*y*z^3 + 3600*x^3*y*z^2 + 7856*x^3*y*z + 5828*x^3*y + 140*x^3*z^4 + 1200*x^3*z^3 + 3920*x^3*z^2 + 5796*x^3*z + 3273*x^3 + 84*x^2*y^5 + 420*x^2*y^4*z + 900*x^2*y^4 + 840*x^2*y^3*z^2 + 3600*x^2*y^3*z + 3944*x^2*y^3 + 840*x^2*y^2*z^3 + 5400*x^2*y^2*z^2 + 11808*x^2*y^2*z + 8796*x^2*y^2 + 420*x^2*y*z^4 + 3600*x^2*y*z^3 + 11784*x^2*y*z^2 + 17496*x^2*y*z + 9947*x^2*y + 84*x^2*z^5 + 900*x^2*z^4 + 3920*x^2*z^3 + 8700*x^2*z^2 + 9847*x^2*z + 4531*x^2 + 28*x*y^6 + 168*x*y^5*z + 360*x*y^5 + 420*x*y^4*z^2 + 1800*x*y^4*z + 1976*x*y^4 + 560*x*y^3*z^3 + 3600*x*y^3*z^2 + 7888*x*y^3*z + 5900*x*y^3 + 420*x*y^2*z^4 + 3600*x*y^2*z^3 + 11808*x*y^2*z^2 + 17604*x*y^2*z + 10075*x*y^2 + 168*x*y*z^5 + 1800*x*y*z^4 + 7856*x*y*z^3 + 17508*x*y*z^2 + 19950*x*y*z + 9286*x*y + 28*x*z^6 + 360*x*z^5 + 1960*x*z^4 + 5804*x*z^3 + 9875*x*z^2 + 9130*x*z + 3522*x + 4*y^7 + 28*y^6*z + 60*y^6 + 84*y^5*z^2 + 360*y^5*z + 396*y^5 + 140*y^4*z^3 + 900*y^4*z^2 + 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336*x^5*y*z + 720*x^5*y + 168*x^5*z^2 + 720*x^5*z + 784*x^5 + 280*x^4*y^3 + 840*x^4*y^2*z + 1800*x^4*y^2 + 840*x^4*y*z^2 + 3600*x^4*y*z + 3928*x^4*y + 280*x^4*z^3 + 1800*x^4*z^2 + 3920*x^4*z + 2900*x^4 + 280*x^3*y^4 + 1120*x^3*y^3*z + 2400*x^3*y^3 + 1680*x^3*y^2*z^2 + 7200*x^3*y^2*z + 7872*x^3*y^2 + 1120*x^3*y*z^3 + 7200*x^3*y*z^2 + 15712*x^3*y*z + 11672*x^3*y + 280*x^3*z^4 + 2400*x^3*z^3 + 7840*x^3*z^2 + 11608*x^3*z + 6579*x^3 + 168*x^2*y^5 + 840*x^2*y^4*z + 1800*x^2*y^4 + 1680*x^2*y^3*z^2 + 7200*x^2*y^3*z + 7888*x^2*y^3 + 1680*x^2*y^2*z^3 + 10800*x^2*y^2*z^2 + 23616*x^2*y^2*z + 17616*x^2*y^2 + 840*x^2*y*z^4 + 7200*x^2*y*z^3 + 23568*x^2*y*z^2 + 35040*x^2*y*z + 19993*x^2*y + 168*x^2*z^5 + 1800*x^2*z^4 + 7840*x^2*z^3 + 17424*x^2*z^2 + 19793*x^2*z + 9169*x^2 + 56*x*y^6 + 336*x*y^5*z + 720*x*y^5 + 840*x*y^4*z^2 + 3600*x*y^4*z + 3952*x*y^4 + 1120*x*y^3*z^3 + 7200*x*y^3*z^2 + 15776*x*y^3*z + 11816*x*y^3 + 840*x*y^2*z^4 + 7200*x*y^2*z^3 + 23616*x*y^2*z^2 + 35256*x*y^2*z + 20249*x*y^2 + 336*x*y*z^5 + 3600*x*y*z^4 + 15712*x*y*z^3 + 35064*x*y*z^2 + 40098*x*y*z + 18786*x*y + 56*x*z^6 + 720*x*z^5 + 3920*x*z^4 + 11624*x*z^3 + 19849*x*z^2 + 18474*x*z + 7217*x + 8*y^7 + 56*y^6*z + 120*y^6 + 168*y^5*z^2 + 720*y^5*z + 792*y^5 + 280*y^4*z^3 + 1800*y^4*z^2 + 3952*y^4*z + 2972*y^4 + 280*y^3*z^4 + 2400*y^3*z^3 + 7888*y^3*z^2 + 11824*y^3*z + 6835*y^3 + 168*y^2*z^5 + 1800*y^2*z^4 + 7872*y^2*z^3 + 17640*y^2*z^2 + 20305*y^2*z + 9625*y^2 + 56*y*z^6 + 720*y*z^5 + 3928*y*z^4 + 11696*y*z^3 + 20105*y*z^2 + 18930*y*z + 7512*y + 8*z^7 + 120*z^6 + 784*z^5 + 2908*z^4 + 6635*z^3 + 9305*z^2 + 7317*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ 1, 1, 1 ], [ 1, 2, 1 ], [ 2, 3, 3 ] ], "source_shift": [ 2, 1, 2 ], "target_matrix": [ [ 1, 1, 2 ], [ 1, 1, 1 ], [ 2, 3, 2 ] ], "target_shift": [ -1166, -1214, -2442 ], "inverse_recipe": { "p": 3, "q": 4, "f": "x**3 + x**2", "g": "x**4 + x**2 + x", "h": "y*z", "polynomials": { "f": [ [ [ 3, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "g": [ [ [ 4, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ] ] }, "h_multiplier": 4 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.072733885001071 ], "surface_features": [ 3.6581500411994914, 4.6031227115073365, 3.255272505103306, 0.0, 3.7992936298145445, 4882 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C000", "pair": 0, "family": "counterexample", "mapping": "F(x,y,z)=(16*x^7 + 88*x^6*y - 104*x^6*z - 176*x^6 + 204*x^5*y^2 - 480*x^5*y*z - 816*x^5*y + 276*x^5*z^2 + 936*x^5*z + 756*x^5 + 258*x^4*y^3 - 906*x^4*y^2*z - 1548*x^4*y^2 + 1038*x^4*y*z^2 + 3528*x^4*y*z + 2874*x^4*y - 390*x^4*z^3 - 1980*x^4*z^2 - 3198*x^4*z - 1660*x^4 + 192*x^3*y^4 - 894*x^3*y^3*z - 1536*x^3*y^3 + 1530*x^3*y^2*z^2 + 5214*x^3*y^2*z + 4284*x^3*y^2 - 1146*x^3*y*z^3 - 5820*x^3*y*z^2 - 9474*x^3*y*z - 4982*x^3*y + 318*x^3*z^4 + 2142*x^3*z^3 + 5190*x^3*z^2 + 5432*x^3*z + 2090*x^3 + 84*x^2*y^5 - 486*x^2*y^4*z - 840*x^2*y^4 + 1104*x^2*y^3*z^2 + 3774*x^2*y^3*z + 3126*x^2*y^3 - 1236*x^2*y^2*z^3 - 6282*x^2*y^2*z^2 - 10302*x^2*y^2*z - 5484*x^2*y^2 + 684*x^2*y*z^4 + 4602*x^2*y*z^3 + 11226*x^2*y*z^2 + 11898*x^2*y*z + 4668*x^2*y - 150*x^2*z^5 - 1254*x^2*z^4 - 4050*x^2*z^3 - 6414*x^2*z^2 - 5034*x^2*z - 1586*x^2 + 20*x*y^6 - 138*x*y^5*z - 240*x*y^5 + 390*x*y^4*z^2 + 1338*x*y^4*z + 1116*x*y^4 - 580*x*y^3*z^3 - 2952*x*y^3*z^2 - 4872*x*y^3*z - 2621*x*y^3 + 480*x*y^2*z^4 + 3228*x*y^2*z^3 + 7920*x*y^2*z^2 + 8481*x*y^2*z + 3384*x*y^2 - 210*x*y*z^5 - 1752*x*y*z^4 - 5688*x*y*z^3 - 9099*x*y*z^2 - 7260*x*y*z - 2341*x*y + 38*x*z^6 + 378*x*z^5 + 1524*x*z^4 + 3239*x*z^3 + 3876*x*z^2 + 2515*x*z + 710*x + 2*y^7 - 16*y^6*z - 28*y^6 + 54*y^5*z^2 + 186*y^5*z + 156*y^5 - 100*y^4*z^3 - 510*y^4*z^2 - 846*y^4*z - 459*y^4 + 110*y^3*z^4 + 740*y^3*z^3 + 1824*y^3*z^2 + 1969*y^3*z + 796*y^3 - 72*y^2*z^5 - 600*y^2*z^4 - 1956*y^2*z^3 - 3153*y^2*z^2 - 2547*y^2*z - 836*y^2 + 26*y*z^6 + 258*y*z^5 + 1044*y*z^4 + 2235*y*z^3 + 2706*y*z^2 + 1783*y*z + 510*y - 4*z^7 - 46*z^6 - 222*z^5 - 592*z^4 - 955*z^3 - 947*z^2 - 547*z - 1, -8*x^7 - 44*x^6*y + 52*x^6*z + 88*x^6 - 102*x^5*y^2 + 240*x^5*y*z + 408*x^5*y - 138*x^5*z^2 - 468*x^5*z - 378*x^5 - 129*x^4*y^3 + 453*x^4*y^2*z + 774*x^4*y^2 - 519*x^4*y*z^2 - 1764*x^4*y*z - 1437*x^4*y + 195*x^4*z^3 + 990*x^4*z^2 + 1599*x^4*z + 834*x^4 - 96*x^3*y^4 + 447*x^3*y^3*z + 768*x^3*y^3 - 765*x^3*y^2*z^2 - 2607*x^3*y^2*z - 2142*x^3*y^2 + 573*x^3*y*z^3 + 2910*x^3*y*z^2 + 4737*x^3*y*z + 2501*x^3*y - 159*x^3*z^4 - 1071*x^3*z^3 - 2595*x^3*z^2 - 2730*x^3*z - 1065*x^3 - 42*x^2*y^5 + 243*x^2*y^4*z + 420*x^2*y^4 - 552*x^2*y^3*z^2 - 1887*x^2*y^3*z - 1563*x^2*y^3 + 618*x^2*y^2*z^3 + 3141*x^2*y^2*z^2 + 5151*x^2*y^2*z + 2751*x^2*y^2 - 342*x^2*y*z^4 - 2301*x^2*y*z^3 - 5613*x^2*y*z^2 - 5973*x^2*y*z - 2370*x^2*y + 75*x^2*z^5 + 627*x^2*z^4 + 2025*x^2*z^3 + 3222*x^2*z^2 + 2559*x^2*z + 820*x^2 - 10*x*y^6 + 69*x*y^5*z + 120*x*y^5 - 195*x*y^4*z^2 - 669*x*y^4*z - 558*x*y^4 + 290*x*y^3*z^3 + 1476*x*y^3*z^2 + 2436*x*y^3*z + 1314*x*y^3 - 240*x*y^2*z^4 - 1614*x*y^2*z^3 - 3960*x*y^2*z^2 - 4254*x*y^2*z - 1713*x*y^2 + 105*x*y*z^5 + 876*x*y*z^4 + 2844*x*y*z^3 + 4566*x*y*z^2 + 3678*x*y*z + 1202*x*y - 19*x*z^6 - 189*x*z^5 - 762*x*z^4 - 1626*x*z^3 - 1965*x*z^2 - 1292*x*z - 367*x - y^7 + 8*y^6*z + 14*y^6 - 27*y^5*z^2 - 93*y^5*z - 78*y^5 + 50*y^4*z^3 + 255*y^4*z^2 + 423*y^4*z + 230*y^4 - 55*y^3*z^4 - 370*y^3*z^3 - 912*y^3*z^2 - 987*y^3*z - 402*y^3 + 36*y^2*z^5 + 300*y^2*z^4 + 978*y^2*z^3 + 1581*y^2*z^2 + 1287*y^2*z + 427*y^2 - 13*y*z^6 - 129*y*z^5 - 522*y*z^4 - 1121*y*z^3 - 1368*y*z^2 - 911*y*z - 262*y + 2*z^7 + 23*z^6 + 111*z^5 + 297*z^4 + 483*z^3 + 484*z^2 + 281*z + 1, 8*x^7 + 44*x^6*y 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105*x*y*z^5 - 1107*x*y*z^4 - 4020*x*y*z^3 - 6876*x*y*z^2 - 5790*x*y*z - 1958*x*y + 19*x*z^6 + 240*x*z^5 + 1080*x*z^4 + 2454*x*z^3 + 3099*x*z^2 + 2108*x*z + 611*x + y^7 - 8*y^6*z - 17*y^6 + 27*y^5*z^2 + 114*y^5*z + 108*y^5 - 50*y^4*z^3 - 315*y^4*z^2 - 588*y^4*z - 341*y^4 + 55*y^3*z^4 + 460*y^3*z^3 + 1272*y^3*z^2 + 1467*y^3*z + 624*y^3 - 36*y^2*z^5 - 375*y^2*z^4 - 1368*y^2*z^3 - 2355*y^2*z^2 - 2001*y^2*z - 685*y^2 + 13*y*z^6 + 162*y*z^5 + 732*y*z^4 + 1673*y*z^3 + 2130*y*z^2 + 1463*y*z + 428*y - 2*z^7 - 29*z^6 - 156*z^5 - 444*z^4 - 753*z^3 - 778*z^2 - 459*z - 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -2, -1, 1 ], [ -1, -1, 1 ], [ -1, -1, 2 ] ], "source_shift": [ 1, 1, 2 ], "target_matrix": [ [ -2, -2, -3 ], [ 1, 1, 1 ], [ -1, -2, -1 ] ], "target_shift": [ 142, -72, 118 ], "collision_preimages": [ [ "0", "-1/4", "-5/4" ], [ "-5/2", "12", "7" ], [ "5/2", "1", "4" ] ], "collision_image": [ "285/2", "-289/4", "473/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0095935889985412 ], "surface_features": [ 3.2056388621843044, 4.075473964588946, 2.8870543780509568, 0.0, 3.3128048333114046, 4771 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B180", "pair": 180, "family": "bijection", "mapping": "F(x,y,z)=(7*x^7 - 98*x^6*y + 49*x^6*z - 63*x^6 + 588*x^5*y^2 - 588*x^5*y*z + 742*x^5*y + 147*x^5*z^2 - 378*x^5*z + 259*x^5 - 1960*x^4*y^3 + 2940*x^4*y^2*z - 3640*x^4*y^2 - 1470*x^4*y*z^2 + 3710*x^4*y*z - 2492*x^4*y + 245*x^4*z^3 - 945*x^4*z^2 + 1302*x^4*z - 630*x^4 + 3920*x^3*y^4 - 7840*x^3*y^3*z + 9520*x^3*y^3 + 5880*x^3*y^2*z^2 - 14560*x^3*y^2*z + 9583*x^3*y^2 - 1960*x^3*y*z^3 + 7420*x^3*y*z^2 - 10024*x^3*y*z + 4718*x^3*y + 245*x^3*z^4 - 1260*x^3*z^3 + 2618*x^3*z^2 - 2562*x^3*z + 953*x^3 - 4704*x^2*y^5 + 11760*x^2*y^4*z - 14000*x^2*y^4 - 11760*x^2*y^3*z^2 + 28560*x^2*y^3*z - 18410*x^2*y^3 + 5880*x^2*y^2*z^3 - 21840*x^2*y^2*z^2 + 28917*x^2*y^2*z - 13237*x^2*y^2 - 1470*x^2*y*z^4 + 7420*x^2*y*z^3 - 15120*x^2*y*z^2 + 14392*x^2*y*z - 5158*x^2*y + 147*x^2*z^5 - 945*x^2*z^4 + 2632*x^2*z^3 - 3906*x^2*z^2 + 2950*x^2*z - 860*x^2 + 3136*x*y^6 - 9408*x*y^5*z + 10976*x*y^5 + 11760*x*y^4*z^2 - 28000*x*y^4*z + 17668*x*y^4 - 7840*x*y^3*z^3 + 28560*x*y^3*z^2 - 37044*x*y^3*z + 16492*x*y^3 + 2940*x*y^2*z^4 - 14560*x*y^2*z^3 + 29085*x*y^2*z^2 - 26922*x*y^2*z + 9322*x*y^2 - 588*x*y*z^5 + 3710*x*y*z^4 - 10136*x*y*z^3 + 14630*x*y*z^2 - 10638*x*y*z + 2978*x*y + 49*x*z^6 - 378*x*z^5 + 1323*x*z^4 - 2646*x*z^3 + 3041*x*z^2 - 1804*x*z + 415*x - 896*y^7 + 3136*y^6*z - 3584*y^6 - 4704*y^5*z^2 + 10976*y^5*z - 6776*y^5 + 3920*y^4*z^3 - 14000*y^4*z^2 + 17780*y^4*z - 7700*y^4 - 1960*y^3*z^4 + 9520*y^3*z^3 - 18634*y^3*z^2 + 16772*y^3*z - 5622*y^3 + 588*y^2*z^5 - 3640*y^2*z^4 + 9751*y^2*z^3 - 13685*y^2*z^2 + 9609*y^2*z - 2593*y^2 - 98*y*z^6 + 742*y*z^5 - 2548*y*z^4 + 4956*y*z^3 - 5480*y*z^2 + 3118*y*z - 692*y + 7*z^7 - 63*z^6 + 266*z^5 - 672*z^4 + 1044*z^3 - 944*z^2 + 444*z - 1, 14*x^7 - 196*x^6*y + 98*x^6*z - 126*x^6 + 1176*x^5*y^2 - 1176*x^5*y*z + 1484*x^5*y + 294*x^5*z^2 - 756*x^5*z + 518*x^5 - 3920*x^4*y^3 + 5880*x^4*y^2*z - 7280*x^4*y^2 - 2940*x^4*y*z^2 + 7420*x^4*y*z - 4984*x^4*y + 490*x^4*z^3 - 1890*x^4*z^2 + 2604*x^4*z - 1260*x^4 + 7840*x^3*y^4 - 15680*x^3*y^3*z + 19040*x^3*y^3 + 11760*x^3*y^2*z^2 - 29120*x^3*y^2*z + 19166*x^3*y^2 - 3920*x^3*y*z^3 + 14840*x^3*y*z^2 - 20048*x^3*y*z + 9436*x^3*y + 490*x^3*z^4 - 2520*x^3*z^3 + 5236*x^3*z^2 - 5124*x^3*z + 1905*x^3 - 9408*x^2*y^5 + 23520*x^2*y^4*z - 28000*x^2*y^4 - 23520*x^2*y^3*z^2 + 57120*x^2*y^3*z - 36820*x^2*y^3 + 11760*x^2*y^2*z^3 - 43680*x^2*y^2*z^2 + 57834*x^2*y^2*z - 26474*x^2*y^2 - 2940*x^2*y*z^4 + 14840*x^2*y*z^3 - 30240*x^2*y*z^2 + 28784*x^2*y*z - 10310*x^2*y + 294*x^2*z^5 - 1890*x^2*z^4 + 5264*x^2*z^3 - 7812*x^2*z^2 + 5897*x^2*z - 1723*x^2 + 6272*x*y^6 - 18816*x*y^5*z + 21952*x*y^5 + 23520*x*y^4*z^2 - 56000*x*y^4*z + 35336*x*y^4 - 15680*x*y^3*z^3 + 57120*x*y^3*z^2 - 74088*x*y^3*z + 32984*x*y^3 + 5880*x*y^2*z^4 - 29120*x*y^2*z^3 + 58170*x*y^2*z^2 - 53844*x*y^2*z + 18632*x*y^2 - 1176*x*y*z^5 + 7420*x*y*z^4 - 20272*x*y*z^3 + 29260*x*y*z^2 - 21264*x*y*z + 5968*x*y + 98*x*z^6 - 756*x*z^5 + 2646*x*z^4 - 5292*x*z^3 + 6079*x*z^2 - 3614*x*z + 842*x - 1792*y^7 + 6272*y^6*z - 7168*y^6 - 9408*y^5*z^2 + 21952*y^5*z - 13552*y^5 + 7840*y^4*z^3 - 28000*y^4*z^2 + 35560*y^4*z - 15400*y^4 - 3920*y^3*z^4 + 19040*y^3*z^3 - 37268*y^3*z^2 + 33544*y^3*z - 11236*y^3 + 1176*y^2*z^5 - 7280*y^2*z^4 + 19502*y^2*z^3 - 27370*y^2*z^2 + 19206*y^2*z - 5198*y^2 - 196*y*z^6 + 1484*y*z^5 - 5096*y*z^4 + 9912*y*z^3 - 10954*y*z^2 + 6248*y*z - 1404*y + 14*z^7 - 126*z^6 + 532*z^5 - 1344*z^4 + 2087*z^3 - 1891*z^2 + 899*z - 1, -7*x^7 + 98*x^6*y - 49*x^6*z + 63*x^6 - 588*x^5*y^2 + 588*x^5*y*z - 742*x^5*y - 147*x^5*z^2 + 378*x^5*z - 259*x^5 + 1960*x^4*y^3 - 2940*x^4*y^2*z + 3640*x^4*y^2 + 1470*x^4*y*z^2 - 3710*x^4*y*z + 2492*x^4*y - 245*x^4*z^3 + 945*x^4*z^2 - 1302*x^4*z + 630*x^4 - 3920*x^3*y^4 + 7840*x^3*y^3*z - 9520*x^3*y^3 - 5880*x^3*y^2*z^2 + 14560*x^3*y^2*z - 9583*x^3*y^2 + 1960*x^3*y*z^3 - 7420*x^3*y*z^2 + 10024*x^3*y*z - 4718*x^3*y - 245*x^3*z^4 + 1260*x^3*z^3 - 2618*x^3*z^2 + 2562*x^3*z - 953*x^3 + 4704*x^2*y^5 - 11760*x^2*y^4*z + 14000*x^2*y^4 + 11760*x^2*y^3*z^2 - 28560*x^2*y^3*z + 18410*x^2*y^3 - 5880*x^2*y^2*z^3 + 21840*x^2*y^2*z^2 - 28917*x^2*y^2*z + 13237*x^2*y^2 + 1470*x^2*y*z^4 - 7420*x^2*y*z^3 + 15120*x^2*y*z^2 - 14392*x^2*y*z + 5158*x^2*y - 147*x^2*z^5 + 945*x^2*z^4 - 2632*x^2*z^3 + 3906*x^2*z^2 - 2950*x^2*z + 862*x^2 - 3136*x*y^6 + 9408*x*y^5*z - 10976*x*y^5 - 11760*x*y^4*z^2 + 28000*x*y^4*z - 17668*x*y^4 + 7840*x*y^3*z^3 - 28560*x*y^3*z^2 + 37044*x*y^3*z - 16492*x*y^3 - 2940*x*y^2*z^4 + 14560*x*y^2*z^3 - 29085*x*y^2*z^2 + 26922*x*y^2*z - 9322*x*y^2 + 588*x*y*z^5 - 3710*x*y*z^4 + 10136*x*y*z^3 - 14630*x*y*z^2 + 10638*x*y*z - 2986*x*y - 49*x*z^6 + 378*x*z^5 - 1323*x*z^4 + 2646*x*z^3 - 3041*x*z^2 + 1808*x*z - 421*x + 896*y^7 - 3136*y^6*z + 3584*y^6 + 4704*y^5*z^2 - 10976*y^5*z + 6776*y^5 - 3920*y^4*z^3 + 14000*y^4*z^2 - 17780*y^4*z + 7700*y^4 + 1960*y^3*z^4 - 9520*y^3*z^3 + 18634*y^3*z^2 - 16772*y^3*z + 5622*y^3 - 588*y^2*z^5 + 3640*y^2*z^4 - 9751*y^2*z^3 + 13685*y^2*z^2 - 9609*y^2*z + 2601*y^2 + 98*y*z^6 - 742*y*z^5 + 2548*y*z^4 - 4956*y*z^3 + 5480*y*z^2 - 3126*y*z + 702*y - 7*z^7 + 63*z^6 - 266*z^5 + 672*z^4 - 1044*z^3 + 946*z^2 - 450*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -1, 2, -1 ], [ -2, 2, -3 ], [ -1, 1, -1 ] ], "source_shift": [ 1, 1, 1 ], "target_matrix": [ [ -1, -1, -2 ], [ -2, -1, -1 ], [ 1, 1, 1 ] ], "target_shift": [ 79, 168, -83 ], "inverse_recipe": { "p": 3, "q": 2, "f": "x**3 + x", "g": "x**2", "h": "y*z**2", "polynomials": { "f": [ [ [ 3, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "g": [ [ [ 2, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 2 ], 1 ] ] }, "h_multiplier": 7 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0379104159983399 ], "surface_features": [ 3.9555135059369424, 4.8697478711937014, 3.6724673130680823, 0.0, 4.060925014832887, 5041 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C139", "pair": 139, "family": "counterexample", "mapping": "F(x,y,z)=(54*x^7 - 351*x^6*y - 297*x^6*z + 756*x^6 + 972*x^5*y^2 + 1647*x^5*y*z - 4185*x^5*y + 693*x^5*z^2 - 3456*x^5*z + 3411*x^5 - 1485*x^4*y^3 - 3780*x^4*y^2*z + 9585*x^4*y^2 - 3186*x^4*y*z^2 + 15867*x^4*y*z - 15723*x^4*y - 889*x^4*z^3 + 6516*x^4*z^2 - 12705*x^4*z + 7532*x^4 + 1350*x^3*y^4 + 4590*x^3*y^3*z - 11610*x^3*y^3 + 5814*x^3*y^2*z^2 - 28908*x^3*y^2*z + 28782*x^3*y^2 + 3251*x^3*y*z^3 - 23811*x^3*y*z^2 + 46701*x^3*y*z - 27976*x^3*y + 677*x^3*z^4 - 6488*x^3*z^3 + 18798*x^3*z^2 - 22254*x^3*z + 9566*x^3 - 729*x^2*y^5 - 3105*x^2*y^4*z + 7830*x^2*y^4 - 5256*x^2*y^3*z^2 + 26082*x^2*y^3*z - 26118*x^2*y^3 - 4419*x^2*y^2*z^3 + 32337*x^2*y^2*z^2 - 63873*x^2*y^2*z + 38736*x^2*y^2 - 1845*x^2*y*z^4 + 17685*x^2*y*z^3 - 51666*x^2*y*z^2 + 61977*x^2*y*z - 27045*x^2*y - 306*x^2*z^5 + 3600*x^2*z^4 - 13827*x^2*z^3 + 24606*x^2*z^2 - 21228*x^2*z + 7159*x^2 + 216*x*y^6 + 1107*x*y^5*z - 2781*x*y^5 + 2349*x*y^4*z^2 - 11628*x*y^4*z + 11727*x*y^4 + 2641*x*y^3*z^3 - 19305*x*y^3*z^2 + 38463*x*y^3*z - 23672*x*y^3 + 1659*x*y^2*z^4 - 15906*x*y^2*z^3 + 46938*x*y^2*z^2 - 57192*x*y^2*z + 25392*x*y^2 + 552*x*y*z^5 - 6504*x*y*z^4 + 25263*x*y*z^3 - 45687*x*y*z^2 + 40083*x*y*z - 13724*x*y + 76*x*z^6 - 1056*x*z^5 + 5061*x*z^4 - 12069*x*z^3 + 15669*x*z^2 - 10523*x*z + 2839*x - 27*y^7 - 162*y^6*z + 405*y^6 - 414*y^5*z^2 + 2043*y^5*z - 2079*y^5 - 584*y^4*z^3 + 4263*y^4*z^2 - 8586*y^4*z + 5380*y^4 - 491*y^3*z^4 + 4709*y^3*z^3 - 14070*y^3*z^2 + 17469*y^3*z - 7913*y^3 - 246*y^2*z^5 + 2904*y^2*z^4 - 11436*y^2*z^3 + 21081*y^2*z^2 - 18855*y^2*z + 6565*y^2 - 68*y*z^6 + 948*y*z^5 - 4611*y*z^4 + 11208*y*z^3 - 14820*y*z^2 + 10106*y*z - 2759*y - 8*z^7 + 128*z^6 - 738*z^5 + 2216*z^4 - 3847*z^3 + 3856*z^2 - 2056*z + 1, -54*x^7 + 351*x^6*y + 297*x^6*z - 594*x^6 - 972*x^5*y^2 - 1647*x^5*y*z + 3294*x^5*y - 693*x^5*z^2 + 2727*x^5*z - 2439*x^5 + 1485*x^4*y^3 + 3780*x^4*y^2*z - 7560*x^4*y^2 + 3186*x^4*y*z^2 - 12546*x^4*y*z + 11268*x^4*y + 889*x^4*z^3 - 5166*x^4*z^2 + 9141*x^4*z - 5129*x^4 - 1350*x^3*y^4 - 4590*x^3*y^3*z + 9180*x^3*y^3 - 5814*x^3*y^2*z^2 + 22914*x^3*y^2*z - 20682*x^3*y^2 - 3251*x^3*y*z^3 + 18924*x^3*y*z^2 - 33687*x^3*y*z + 19093*x^3*y - 677*x^3*z^4 + 5171*x^3*z^3 - 13614*x^3*z^2 + 15261*x^3*z - 6338*x^3 + 729*x^2*y^5 + 3105*x^2*y^4*z - 6210*x^2*y^4 + 5256*x^2*y^3*z^2 - 20736*x^2*y^3*z + 18828*x^2*y^3 + 4419*x^2*y^2*z^3 - 25776*x^2*y^2*z^2 + 46215*x^2*y^2*z - 26505*x^2*y^2 + 1845*x^2*y*z^4 - 14136*x^2*y*z^3 + 37527*x^2*y*z^2 - 42600*x^2*y*z + 17940*x^2*y + 306*x^2*z^5 - 2886*x^2*z^4 + 10083*x^2*z^3 - 16989*x^2*z^2 + 14148*x^2*z - 4660*x^2 - 216*x*y^6 - 1107*x*y^5*z + 2214*x*y^5 - 2349*x*y^4*z^2 + 9279*x*y^4*z - 8487*x*y^4 - 2641*x*y^3*z^3 + 15444*x*y^3*z^2 - 27933*x*y^3*z + 16247*x*y^3 - 1659*x*y^2*z^4 + 12759*x*y^2*z^3 - 34212*x*y^2*z^2 + 39417*x*y^2*z - 16866*x*y^2 - 552*x*y*z^5 + 5232*x*y*z^4 - 18483*x*y*z^3 + 31620*x*y*z^2 - 26745*x*y*z + 8933*x*y - 76*x*z^6 + 852*x*z^5 - 3717*x*z^4 + 8388*x*z^3 - 10503*x*z^2 + 6890*x*z - 1845*x + 27*y^7 + 162*y^6*z - 324*y^6 + 414*y^5*z^2 - 1638*y^5*z + 1512*y^5 + 584*y^4*z^3 - 3426*y^4*z^2 + 6264*y^4*z - 3706*y^4 + 491*y^3*z^4 - 3794*y^3*z^3 + 10299*y^3*z^2 - 12078*y^3*z + 5264*y^3 + 246*y^2*z^5 - 2346*y^2*z^4 + 8400*y^2*z^3 - 14631*y^2*z^2 + 12597*y^2*z - 4273*y^2 + 68*y*z^6 - 768*y*z^5 + 3399*y*z^4 - 7809*y*z^3 + 9945*y*z^2 - 6617*y*z + 1792*y + 8*z^7 - 104*z^6 + 546*z^5 - 1550*z^4 + 2593*z^3 - 2539*z^2 + 1344*z - 1, -108*x^7 + 702*x^6*y + 594*x^6*z - 1026*x^6 - 1944*x^5*y^2 - 3294*x^5*y*z + 5697*x^5*y - 1386*x^5*z^2 + 4725*x^5*z - 3906*x^5 + 2970*x^4*y^3 + 7560*x^4*y^2*z - 13095*x^4*y^2 + 6372*x^4*y*z^2 - 21771*x^4*y*z + 18081*x^4*y + 1778*x^4*z^3 - 8982*x^4*z^2 + 14718*x^4*z - 7909*x^4 - 2700*x^3*y^4 - 9180*x^3*y^3*z + 15930*x^3*y^3 - 11628*x^3*y^2*z^2 + 39834*x^3*y^2*z - 33264*x^3*y^2 - 6502*x^3*y*z^3 + 32961*x^3*y*z^2 - 54360*x^3*y*z + 29492*x^3*y - 1354*x^3*z^4 + 9025*x^3*z^3 - 22044*x^3*z^2 + 23664*x^3*z - 9556*x^3 + 1458*x^2*y^5 + 6210*x^2*y^4*z - 10800*x^2*y^4 + 10512*x^2*y^3*z^2 - 36126*x^2*y^3*z + 30366*x^2*y^3 + 8838*x^2*y^2*z^3 - 44991*x^2*y^2*z^2 + 74772*x^2*y^2*z - 41022*x^2*y^2 + 3690*x^2*y*z^4 - 24723*x^2*y*z^3 + 60915*x^2*y*z^2 - 66174*x^2*y*z + 27072*x^2*y + 612*x^2*z^5 - 5058*x^2*z^4 + 16422*x^2*z^3 - 26487*x^2*z^2 + 21432*x^2*z - 6938*x^2 - 432*x*y^6 - 2214*x*y^5*z + 3861*x*y^5 - 4698*x*y^4*z^2 + 16209*x*y^4*z - 13734*x*y^4 - 5282*x*y^3*z^3 + 27027*x*y^3*z^2 - 45336*x*y^3*z + 25204*x*y^3 - 3318*x*y^2*z^4 + 22371*x*y^2*z^3 - 55698*x*y^2*z^2 + 61356*x*y^2*z - 25476*x*y^2 - 1104*x*y*z^5 + 9192*x*y*z^4 - 30186*x*y*z^3 + 49389*x*y*z^2 - 40548*x*y*z + 13300*x*y - 152*x*z^6 + 1500*x*z^5 - 6090*x*z^4 + 13147*x*z^3 - 15984*x*z^2 + 10306*x*z - 2740*x + 54*y^7 + 324*y^6*z - 567*y^6 + 828*y^5*z^2 - 2871*y^5*z + 2457*y^5 + 1168*y^4*z^3 - 6015*y^4*z^2 + 10206*y^4*z - 5765*y^4 + 982*y^3*z^4 - 6673*y^3*z^3 + 16827*y^3*z^2 - 18846*y^3*z + 7960*y^3 + 492*y^2*z^5 - 4134*y^2*z^4 + 13764*y^2*z^3 - 22902*y^2*z^2 + 19116*y^2*z - 6362*y^2 + 136*y*z^6 - 1356*y*z^5 + 5586*y*z^4 - 12263*y*z^3 + 15147*y*z^2 - 9898*y*z + 2660*y + 16*z^7 - 184*z^6 + 900*z^5 - 2442*z^4 + 3964*z^3 - 3815*z^2 + 2006*z - 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ 3, -3, -2 ], [ 1, -1, -1 ], [ 2, -1, -1 ] ], "source_shift": [ 1, 2, 1 ], "target_matrix": [ [ 1, 2, 1 ], [ -1, -1, -1 ], [ -2, -1, -1 ] ], "target_shift": [ -439, 290, 437 ], "collision_preimages": [ [ "3/4", "-9/4", "5" ], [ "9", "2", "21/2" ], [ "6", "7", "-1/2" ] ], "collision_image": [ "-1757/4", "1161/4", "875/2" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0986666700009664 ], "surface_features": [ 4.0552496041340405, 4.873738997302387, 3.7626410582667362, 0.0, 4.142053059737298, 5092 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C048", "pair": 48, "family": "counterexample", "mapping": "F(x,y,z)=(-2*x^7 - 20*x^6*y - 20*x^6*z - 35*x^6 - 84*x^5*y^2 - 174*x^5*y*z - 297*x^5*y - 84*x^5*z^2 - 306*x^5*z - 246*x^5 - 192*x^4*y^3 - 618*x^4*y^2*z - 1029*x^4*y^2 - 618*x^4*y*z^2 - 2193*x^4*y*z - 1713*x^4*y - 192*x^4*z^3 - 1098*x^4*z^2 - 1833*x^4*z - 905*x^4 - 258*x^3*y^4 - 1146*x^3*y^3*z - 1863*x^3*y^3 - 1782*x^3*y^2*z^2 - 6156*x^3*y^2*z - 4668*x^3*y^2 - 1146*x^3*y*z^3 - 6381*x^3*y*z^2 - 10356*x^3*y*z - 4936*x^3*y - 258*x^3*z^4 - 2070*x^3*z^3 - 5388*x^3*z^2 - 5522*x^3*z - 1906*x^3 - 204*x^2*y^5 - 1170*x^2*y^4*z - 1860*x^2*y^4 - 2514*x^2*y^3*z^2 - 8457*x^2*y^3*z - 6225*x^2*y^3 - 2514*x^2*y^2*z^3 - 13608*x^2*y^2*z^2 - 21441*x^2*y^2*z - 9861*x^2*y^2 - 1170*x^2*y*z^4 - 9147*x^2*y*z^3 - 23169*x^2*y*z^2 - 22920*x^2*y*z - 7590*x^2*y - 204*x^2*z^5 - 2163*x^2*z^4 - 7809*x^2*z^3 - 12471*x^2*z^2 - 8934*x^2*z - 2315*x^2 - 88*x*y^6 - 624*x*y^5*z - 972*x*y^5 - 1734*x*y^4*z^2 - 5688*x*y^4*z - 4068*x*y^4 - 2398*x*y^3*z^3 - 12609*x*y^3*z^2 - 19278*x*y^3*z - 8566*x*y^3 - 1734*x*y^2*z^4 - 13176*x*y^2*z^3 - 32418*x*y^2*z^2 - 30912*x*y^2*z - 9828*x*y^2 - 624*x*y*z^5 - 6462*x*y*z^4 - 22734*x*y*z^3 - 35058*x*y*z^2 - 24066*x*y*z - 5941*x*y - 88*x*z^6 - 1188*x*z^5 - 5580*x*z^4 - 12350*x*z^3 - 13788*x*z^2 - 7418*x*z - 1506*x - 16*y^7 - 136*y^6*z - 208*y^6 - 468*y^5*z^2 - 1500*y^5*z - 1044*y^5 - 838*y^4*z^3 - 4284*y^4*z^2 - 6360*y^4*z - 2736*y^4 - 838*y^3*z^4 - 6175*y^3*z^3 - 14745*y^3*z^2 - 13570*y^3*z - 4152*y^3 - 468*y^2*z^5 - 4719*y^2*z^4 - 16137*y^2*z^3 - 23955*y^2*z^2 - 15768*y^2*z - 3722*y^2 - 136*y*z^6 - 1800*y*z^5 - 8256*y*z^4 - 17670*y*z^3 - 18864*y*z^2 - 9641*y*z - 1865*y - 16*z^7 - 268*z^6 - 1572*z^5 - 4520*z^4 - 7008*z^3 - 5879*z^2 - 2461*z - 1, x^7 + 10*x^6*y + 10*x^6*z + 16*x^6 + 42*x^5*y^2 + 87*x^5*y*z + 135*x^5*y + 42*x^5*z^2 + 141*x^5*z + 105*x^5 + 96*x^4*y^3 + 309*x^4*y^2*z + 465*x^4*y^2 + 309*x^4*y*z^2 + 1005*x^4*y*z + 726*x^4*y + 96*x^4*z^3 + 510*x^4*z^2 + 789*x^4*z + 368*x^4 + 129*x^3*y^4 + 573*x^3*y^3*z + 837*x^3*y^3 + 891*x^3*y^2*z^2 + 2805*x^3*y^2*z + 1965*x^3*y^2 + 573*x^3*y*z^3 + 2949*x^3*y*z^2 + 4425*x^3*y*z + 1992*x^3*y + 129*x^3*z^4 + 969*x^3*z^3 + 2340*x^3*z^2 + 2264*x^3*z + 744*x^3 + 102*x^2*y^5 + 585*x^2*y^4*z + 831*x^2*y^4 + 1257*x^2*y^3*z^2 + 3831*x^2*y^3*z + 2604*x^2*y^3 + 1257*x^2*y^2*z^3 + 6255*x^2*y^2*z^2 + 9096*x^2*y^2*z + 3951*x^2*y^2 + 585*x^2*y*z^4 + 4263*x^2*y*z^3 + 9987*x^2*y*z^2 + 9324*x^2*y*z + 2934*x^2*y + 102*x^2*z^5 + 1020*x^2*z^4 + 3423*x^2*z^3 + 5157*x^2*z^2 + 3522*x^2*z + 865*x^2 + 44*x*y^6 + 312*x*y^5*z + 432*x*y^5 + 867*x*y^4*z^2 + 2562*x*y^4*z + 1692*x*y^4 + 1199*x*y^3*z^3 + 5763*x*y^3*z^2 + 8124*x*y^3*z + 3409*x*y^3 + 867*x*y^2*z^4 + 6111*x*y^2*z^3 + 13869*x*y^2*z^2 + 12483*x*y^2*z + 3762*x*y^2 + 312*x*y*z^5 + 3036*x*y*z^4 + 9891*x*y*z^3 + 14379*x*y*z^2 + 9399*x*y*z + 2186*x*y + 44*x*z^6 + 564*x*z^5 + 2469*x*z^4 + 5153*x*z^3 + 5487*x*z^2 + 2812*x*z + 535*x + 8*y^7 + 68*y^6*z + 92*y^6 + 234*y^5*z^2 + 672*y^5*z + 432*y^5 + 419*y^4*z^3 + 1947*y^4*z^2 + 2664*y^4*z + 1082*y^4 + 419*y^3*z^4 + 2849*y^3*z^3 + 6264*y^3*z^2 + 5443*y^3*z + 1574*y^3 + 234*y^2*z^5 + 2208*y^2*z^4 + 6966*y^2*z^3 + 9750*y^2*z^2 + 6102*y^2*z + 1348*y^2 + 68*y*z^6 + 852*y*z^5 + 3627*y*z^4 + 7309*y*z^3 + 7437*y*z^2 + 3604*y*z + 648*y + 8*z^7 + 128*z^6 + 702*z^5 + 1904*z^4 + 2815*z^3 + 2257*z^2 + 892*z + 1, -x^7 - 10*x^6*y - 10*x^6*z - 16*x^6 - 42*x^5*y^2 - 87*x^5*y*z - 135*x^5*y - 42*x^5*z^2 - 141*x^5*z - 105*x^5 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564*x*z^5 - 2469*x*z^4 - 5157*x*z^3 - 5523*x*z^2 - 2890*x*z - 577*x - 8*y^7 - 68*y^6*z - 92*y^6 - 234*y^5*z^2 - 672*y^5*z - 432*y^5 - 419*y^4*z^3 - 1947*y^4*z^2 - 2664*y^4*z - 1090*y^4 - 419*y^3*z^4 - 2849*y^3*z^3 - 6264*y^3*z^2 - 5463*y^3*z - 1618*y^3 - 234*y^2*z^5 - 2208*y^2*z^4 - 6966*y^2*z^3 - 9768*y^2*z^2 - 6198*y^2*z - 1432*y^2 - 68*y*z^6 - 852*y*z^5 - 3627*y*z^4 - 7316*y*z^3 - 7500*y*z^2 - 3736*y*z - 712*y - 8*z^7 - 128*z^6 - 702*z^5 - 1905*z^4 - 2828*z^3 - 2302*z^2 - 946*z - 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ 1, 2, 1 ], [ 1, 1, 2 ], [ 1, 1, 1 ] ], "source_shift": [ 2, 1, 1 ], "target_matrix": [ [ -2, -3, -3 ], [ 1, 1, 2 ], [ -1, -1, -1 ] ], "target_shift": [ 410, -139, 155 ], "collision_preimages": [ [ "-3/4", "-3/4", "1/4" ], [ "20", "-13/2", "-8" ], [ "19", "-17/2", "-5" ] ], "collision_image": [ "821/2", "-557/4", "621/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0356389330008824 ], "surface_features": [ 3.5631134143615895, 4.544787136693911, 3.2187979981117376, 0.0, 3.7247165448083996, 4865 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B120", "pair": 120, "family": "bijection", "mapping": "F(x,y,z)=(-6*x^7 + 42*x^6*y + 42*x^6*z - 66*x^6 - 126*x^5*y^2 - 252*x^5*y*z + 396*x^5*y - 126*x^5*z^2 + 408*x^5*z - 324*x^5 + 210*x^4*y^3 + 630*x^4*y^2*z - 990*x^4*y^2 + 630*x^4*y*z^2 - 2040*x^4*y*z + 1626*x^4*y + 210*x^4*z^3 - 1050*x^4*z^2 + 1722*x^4*z - 918*x^4 - 210*x^3*y^4 - 840*x^3*y^3*z + 1320*x^3*y^3 - 1260*x^3*y^2*z^2 + 4080*x^3*y^2*z - 3264*x^3*y^2 - 840*x^3*y*z^3 + 4200*x^3*y*z^2 - 6912*x^3*y*z + 3708*x^3*y - 210*x^3*z^4 + 1440*x^3*z^3 - 3654*x^3*z^2 + 4032*x^3*z - 1623*x^3 + 126*x^2*y^5 + 630*x^2*y^4*z - 990*x^2*y^4 + 1260*x^2*y^3*z^2 - 4080*x^2*y^3*z + 3276*x^2*y^3 + 1260*x^2*y^2*z^3 - 6300*x^2*y^2*z^2 + 10404*x^2*y^2*z - 5616*x^2*y^2 + 630*x^2*y*z^4 - 4320*x^2*y*z^3 + 10998*x^2*y*z^2 - 12216*x^2*y*z + 4959*x^2*y + 126*x^2*z^5 - 1110*x^2*z^4 + 3870*x^2*z^3 - 6630*x^2*z^2 + 5535*x^2*z - 1790*x^2 - 42*x*y^6 - 252*x*y^5*z + 396*x*y^5 - 630*x*y^4*z^2 + 2040*x*y^4*z - 1644*x*y^4 - 840*x*y^3*z^3 + 4200*x*y^3*z^2 - 6960*x*y^3*z + 3780*x*y^3 - 630*x*y^2*z^4 + 4320*x*y^2*z^3 - 11034*x*y^2*z^2 + 12336*x*y^2*z - 5049*x*y^2 - 252*x*y*z^5 + 2220*x*y*z^4 - 7764*x*y*z^3 + 13392*x*y*z^2 - 11286*x*y*z + 3688*x*y - 42*x*z^6 + 456*x*z^5 - 2046*x*z^4 + 4836*x*z^3 - 6291*x*z^2 + 4228*x*z - 1142*x + 6*y^7 + 42*y^6*z - 66*y^6 + 126*y^5*z^2 - 408*y^5*z + 330*y^5 + 210*y^4*z^3 - 1050*y^4*z^2 + 1746*y^4*z - 954*y^4 + 210*y^3*z^4 - 1440*y^3*z^3 + 3690*y^3*z^2 - 4152*y^3*z + 1713*y^3 + 126*y^2*z^5 - 1110*y^2*z^4 + 3894*y^2*z^3 - 6762*y^2*z^2 + 5751*y^2*z - 1898*y^2 + 42*y*z^6 - 456*y*z^5 + 2052*y*z^4 - 4884*y*z^3 + 6423*y*z^2 - 4372*y*z + 1199*y + 6*z^7 - 78*z^6 + 432*z^5 - 1320*z^4 + 2385*z^3 - 2510*z^2 + 1408*z - 1, -6*x^7 + 42*x^6*y + 42*x^6*z - 66*x^6 - 126*x^5*y^2 - 252*x^5*y*z + 396*x^5*y - 126*x^5*z^2 + 408*x^5*z - 324*x^5 + 210*x^4*y^3 + 630*x^4*y^2*z - 990*x^4*y^2 + 630*x^4*y*z^2 - 2040*x^4*y*z + 1626*x^4*y + 210*x^4*z^3 - 1050*x^4*z^2 + 1722*x^4*z - 918*x^4 - 210*x^3*y^4 - 840*x^3*y^3*z + 1320*x^3*y^3 - 1260*x^3*y^2*z^2 + 4080*x^3*y^2*z - 3264*x^3*y^2 - 840*x^3*y*z^3 + 4200*x^3*y*z^2 - 6912*x^3*y*z + 3708*x^3*y - 210*x^3*z^4 + 1440*x^3*z^3 - 3654*x^3*z^2 + 4032*x^3*z - 1621*x^3 + 126*x^2*y^5 + 630*x^2*y^4*z - 990*x^2*y^4 + 1260*x^2*y^3*z^2 - 4080*x^2*y^3*z + 3276*x^2*y^3 + 1260*x^2*y^2*z^3 - 6300*x^2*y^2*z^2 + 10404*x^2*y^2*z - 5616*x^2*y^2 + 630*x^2*y*z^4 - 4320*x^2*y*z^3 + 10998*x^2*y*z^2 - 12216*x^2*y*z + 4953*x^2*y + 126*x^2*z^5 - 1110*x^2*z^4 + 3870*x^2*z^3 - 6630*x^2*z^2 + 5529*x^2*z - 1784*x^2 - 42*x*y^6 - 252*x*y^5*z + 396*x*y^5 - 630*x*y^4*z^2 + 2040*x*y^4*z - 1644*x*y^4 - 840*x*y^3*z^3 + 4200*x*y^3*z^2 - 6960*x*y^3*z + 3780*x*y^3 - 630*x*y^2*z^4 + 4320*x*y^2*z^3 - 11034*x*y^2*z^2 + 12336*x*y^2*z - 5043*x*y^2 - 252*x*y*z^5 + 2220*x*y*z^4 - 7764*x*y*z^3 + 13392*x*y*z^2 - 11274*x*y*z + 3676*x*y - 42*x*z^6 + 456*x*z^5 - 2046*x*z^4 + 4836*x*z^3 - 6285*x*z^2 + 4216*x*z - 1136*x + 6*y^7 + 42*y^6*z - 66*y^6 + 126*y^5*z^2 - 408*y^5*z + 330*y^5 + 210*y^4*z^3 - 1050*y^4*z^2 + 1746*y^4*z - 954*y^4 + 210*y^3*z^4 - 1440*y^3*z^3 + 3690*y^3*z^2 - 4152*y^3*z + 1711*y^3 + 126*y^2*z^5 - 1110*y^2*z^4 + 3894*y^2*z^3 - 6762*y^2*z^2 + 5745*y^2*z - 1892*y^2 + 42*y*z^6 - 456*y*z^5 + 2052*y*z^4 - 4884*y*z^3 + 6417*y*z^2 - 4360*y*z + 1191*y + 6*z^7 - 78*z^6 + 432*z^5 - 1320*z^4 + 2383*z^3 - 2504*z^2 + 1404*z - 1, 3*x^7 - 21*x^6*y - 21*x^6*z + 33*x^6 + 63*x^5*y^2 + 126*x^5*y*z - 198*x^5*y + 63*x^5*z^2 - 204*x^5*z + 162*x^5 - 105*x^4*y^3 - 315*x^4*y^2*z + 495*x^4*y^2 - 315*x^4*y*z^2 + 1020*x^4*y*z - 813*x^4*y - 105*x^4*z^3 + 525*x^4*z^2 - 861*x^4*z + 459*x^4 + 105*x^3*y^4 + 420*x^3*y^3*z - 660*x^3*y^3 + 630*x^3*y^2*z^2 - 2040*x^3*y^2*z + 1632*x^3*y^2 + 420*x^3*y*z^3 - 2100*x^3*y*z^2 + 3456*x^3*y*z - 1854*x^3*y + 105*x^3*z^4 - 720*x^3*z^3 + 1827*x^3*z^2 - 2016*x^3*z + 812*x^3 - 63*x^2*y^5 - 315*x^2*y^4*z + 495*x^2*y^4 - 630*x^2*y^3*z^2 + 2040*x^2*y^3*z - 1638*x^2*y^3 - 630*x^2*y^2*z^3 + 3150*x^2*y^2*z^2 - 5202*x^2*y^2*z + 2808*x^2*y^2 - 315*x^2*y*z^4 + 2160*x^2*y*z^3 - 5499*x^2*y*z^2 + 6108*x^2*y*z - 2481*x^2*y - 63*x^2*z^5 + 555*x^2*z^4 - 1935*x^2*z^3 + 3315*x^2*z^2 - 2769*x^2*z + 897*x^2 + 21*x*y^6 + 126*x*y^5*z - 198*x*y^5 + 315*x*y^4*z^2 - 1020*x*y^4*z + 822*x*y^4 + 420*x*y^3*z^3 - 2100*x*y^3*z^2 + 3480*x*y^3*z - 1890*x*y^3 + 315*x*y^2*z^4 - 2160*x*y^2*z^3 + 5517*x*y^2*z^2 - 6168*x*y^2*z + 2526*x*y^2 + 126*x*y*z^5 - 1110*x*y*z^4 + 3882*x*y*z^3 - 6696*x*y*z^2 + 5646*x*y*z - 1848*x*y + 21*x*z^6 - 228*x*z^5 + 1023*x*z^4 - 2418*x*z^3 + 3147*x*z^2 - 2118*x*z + 574*x - 3*y^7 - 21*y^6*z + 33*y^6 - 63*y^5*z^2 + 204*y^5*z - 165*y^5 - 105*y^4*z^3 + 525*y^4*z^2 - 873*y^4*z + 477*y^4 - 105*y^3*z^4 + 720*y^3*z^3 - 1845*y^3*z^2 + 2076*y^3*z - 857*y^3 - 63*y^2*z^5 + 555*y^2*z^4 - 1947*y^2*z^3 + 3381*y^2*z^2 - 2877*y^2*z + 951*y^2 - 21*y*z^6 + 228*y*z^5 - 1026*y*z^4 + 2442*y*z^3 - 3213*y*z^2 + 2190*y*z - 603*y - 3*z^7 + 39*z^6 - 216*z^5 + 660*z^4 - 1193*z^3 + 1257*z^2 - 707*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ 1, -1, -1 ], [ 1, -2, -1 ], [ 1, -1, -2 ] ], "source_shift": [ 1, 1, 2 ], "target_matrix": [ [ -2, -3, -2 ], [ -2, -1, -1 ], [ 1, 2, 1 ] ], "target_shift": [ 322, 322, -162 ], "inverse_recipe": { "p": 3, "q": 2, "f": "x**3 + x**2", "g": "x**2", "h": "y*z**2 + z**3", "polynomials": { "f": [ [ [ 3, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "g": [ [ [ 2, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 2 ], 1 ], [ [ 0, 0, 3 ], 1 ] ] }, "h_multiplier": 3 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0571544669983268 ], "surface_features": [ 3.3337651257560625, 4.126845440649185, 3.03342375548695, 0.0, 3.4302780603555885, 4833 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C133", "pair": 133, "family": "counterexample", "mapping": "F(x,y,z)=(24*x^7 + 204*x^6*y - 132*x^6*z - 348*x^6 + 702*x^5*y^2 - 936*x^5*y*z - 2484*x^5*y + 306*x^5*z^2 + 1620*x^5*z + 2088*x^5 + 1257*x^4*y^3 - 2601*x^4*y^2*z - 7011*x^4*y^2 + 1755*x^4*y*z^2 + 9378*x^4*y*z + 12312*x^4*y - 387*x^4*z^3 - 3087*x^4*z^2 - 8028*x^4*z - 6866*x^4 + 1257*x^3*y^4 - 3597*x^3*y^3*z - 9978*x^3*y^3 + 3771*x^3*y^2*z^2 + 20538*x^3*y^2*z + 27666*x^3*y^2 - 1719*x^3*y*z^3 - 13878*x^3*y*z^2 - 36828*x^3*y*z - 32237*x^3*y + 288*x^3*z^4 + 3078*x^3*z^3 + 12114*x^3*z^2 + 20951*x^3*z + 13472*x^3 + 702*x^2*y^5 - 2601*x^2*y^4*z - 7533*x^2*y^4 + 3771*x^2*y^3*z^2 + 21195*x^2*y^3*z + 29448*x^2*y^3 - 2673*x^2*y^2*z^3 - 22059*x^2*y^2*z^2 - 60075*x^2*y^2*z - 53973*x^2*y^2 + 927*x^2*y*z^4 + 10053*x^2*y*z^3 + 40410*x^2*y*z^2 + 71523*x^2*y*z + 47121*x^2*y - 126*x^2*z^5 - 1692*x^2*z^4 - 8955*x^2*z^3 - 23472*x^2*z^2 - 30549*x^2*z - 15822*x^2 + 204*x*y^6 - 936*x*y^5*z - 2844*x*y^5 + 1755*x*y^4*z^2 + 10278*x*y^4*z + 14688*x*y^4 - 1719*x*y^3*z^3 - 14652*x*y^3*z^2 - 41004*x*y^3*z - 37744*x*y^3 + 927*x*y^2*z^4 + 10296*x*y^2*z^3 + 42408*x*y^2*z^2 + 76836*x*y^2*z + 51867*x*y^2 - 261*x*y*z^5 - 3564*x*y*z^4 - 19260*x*y*z^3 - 51591*x*y*z^2 - 68703*x*y*z - 36480*x*y + 30*x*z^6 + 486*x*z^5 + 3240*x*z^4 + 11425*x*z^3 + 22542*x*z^2 + 23652*x*z + 10324*x + 24*y^7 - 132*y^6*z - 420*y^6 + 306*y^5*z^2 + 1872*y^5*z + 2736*y^5 - 387*y^4*z^3 - 3429*y^4*z^2 - 9828*y^4*z - 9202*y^4 + 288*y^3*z^4 + 3303*y^3*z^3 + 13932*y^3*z^2 + 25738*y^3*z + 17680*y^3 - 126*y^2*z^5 - 1764*y^2*z^4 - 9747*y^2*z^3 - 26637*y^2*z^2 - 36171*y^2*z - 19617*y^2 + 30*y*z^6 + 495*y*z^5 + 3366*y*z^4 + 12103*y*z^3 + 24360*y*z^2 + 26115*y*z + 11682*y - 3*z^7 - 57*z^6 - 459*z^5 - 2038*z^4 - 5407*z^3 - 8598*z^2 - 7605*z - 1, -8*x^7 - 68*x^6*y + 44*x^6*z + 116*x^6 - 234*x^5*y^2 + 312*x^5*y*z + 828*x^5*y - 102*x^5*z^2 - 540*x^5*z - 696*x^5 - 419*x^4*y^3 + 867*x^4*y^2*z + 2337*x^4*y^2 - 585*x^4*y*z^2 - 3126*x^4*y*z - 4104*x^4*y + 129*x^4*z^3 + 1029*x^4*z^2 + 2676*x^4*z + 2286*x^4 - 419*x^3*y^4 + 1199*x^3*y^3*z + 3326*x^3*y^3 - 1257*x^3*y^2*z^2 - 6846*x^3*y^2*z - 9222*x^3*y^2 + 573*x^3*y*z^3 + 4626*x^3*y*z^2 + 12276*x^3*y*z + 10739*x^3*y - 96*x^3*z^4 - 1026*x^3*z^3 - 4038*x^3*z^2 - 6977*x^3*z - 4476*x^3 - 234*x^2*y^5 + 867*x^2*y^4*z + 2511*x^2*y^4 - 1257*x^2*y^3*z^2 - 7065*x^2*y^3*z - 9816*x^2*y^3 + 891*x^2*y^2*z^3 + 7353*x^2*y^2*z^2 + 20025*x^2*y^2*z + 17985*x^2*y^2 - 309*x^2*y*z^4 - 3351*x^2*y*z^3 - 13470*x^2*y*z^2 - 23829*x^2*y*z - 15675*x^2*y + 42*x^2*z^5 + 564*x^2*z^4 + 2985*x^2*z^3 + 7818*x^2*z^2 + 10155*x^2*z + 5242*x^2 - 68*x*y^6 + 312*x*y^5*z + 948*x*y^5 - 585*x*y^4*z^2 - 3426*x*y^4*z - 4896*x*y^4 + 573*x*y^3*z^3 + 4884*x*y^3*z^2 + 13668*x*y^3*z + 12579*x*y^3 - 309*x*y^2*z^4 - 3432*x*y^2*z^3 - 14136*x*y^2*z^2 - 25605*x*y^2*z - 17268*x*y^2 + 87*x*y*z^5 + 1188*x*y*z^4 + 6420*x*y*z^3 + 17190*x*y*z^2 + 22863*x*y*z + 12112*x*y - 10*x*z^6 - 162*x*z^5 - 1080*x*z^4 - 3806*x*z^3 - 7497*x*z^2 - 7844*x*z - 3412*x - 8*y^7 + 44*y^6*z + 140*y^6 - 102*y^5*z^2 - 624*y^5*z - 912*y^5 + 129*y^4*z^3 + 1143*y^4*z^2 + 3276*y^4*z + 3067*y^4 - 96*y^3*z^4 - 1101*y^3*z^3 - 4644*y^3*z^2 - 8578*y^3*z - 5889*y^3 + 42*y^2*z^5 + 588*y^2*z^4 + 3249*y^2*z^3 + 8877*y^2*z^2 + 12045*y^2*z + 6523*y^2 - 10*y*z^6 - 165*y*z^5 - 1122*y*z^4 - 4033*y*z^3 - 8109*y*z^2 - 8677*y*z - 3872*y + z^7 + 19*z^6 + 153*z^5 + 679*z^4 + 1799*z^3 + 2854*z^2 + 2517*z + 1, 16*x^7 + 136*x^6*y - 88*x^6*z - 208*x^6 + 468*x^5*y^2 - 624*x^5*y*z - 1500*x^5*y + 204*x^5*z^2 + 972*x^5*z + 1164*x^5 + 838*x^4*y^3 - 1734*x^4*y^2*z - 4284*x^4*y^2 + 1170*x^4*y*z^2 + 5688*x^4*y*z + 6960*x^4*y - 258*x^4*z^3 - 1860*x^4*z^2 - 4500*x^4*z - 3644*x^4 + 838*x^3*y^4 - 2398*x^3*y^3*z - 6175*x^3*y^3 + 2514*x^3*y^2*z^2 + 12609*x^3*y^2*z + 15879*x^3*y^2 - 1146*x^3*y*z^3 - 8457*x^3*y*z^2 - 20946*x^3*y*z - 17370*x^3*y + 192*x^3*z^4 + 1863*x^3*z^3 + 6831*x^3*z^2 + 11190*x^3*z + 6896*x^3 + 468*x^2*y^5 - 1734*x^2*y^4*z - 4719*x^2*y^4 + 2514*x^2*y^3*z^2 + 13176*x^2*y^3*z + 17169*x^2*y^3 - 1782*x^2*y^2*z^3 - 13608*x^2*y^2*z^2 - 34704*x^2*y^2*z - 29562*x^2*y^2 + 618*x^2*y*z^4 + 6156*x^2*y*z^3 + 23133*x^2*y*z^2 + 38814*x^2*y*z + 24510*x^2*y - 84*x^2*z^5 - 1029*x^2*z^4 - 5082*x^2*z^3 - 12624*x^2*z^2 - 15750*x^2*z - 7886*x^2 + 136*x*y^6 - 624*x*y^5*z - 1800*x*y^5 + 1170*x*y^4*z^2 + 6462*x*y^4*z + 8700*x*y^4 - 1146*x*y^3*z^3 - 9147*x*y^3*z^2 - 24066*x*y^3*z - 21043*x*y^3 + 618*x*y^2*z^4 + 6381*x*y^2*z^3 + 24663*x*y^2*z^2 + 42423*x*y^2*z + 27465*x*y^2 - 174*x*y*z^5 - 2193*x*y*z^4 - 11100*x*y*z^3 - 28215*x*y*z^2 - 36036*x*y*z - 18503*x*y + 20*x*z^6 + 297*x*z^5 + 1851*x*z^4 + 6191*x*z^3 + 11715*x*z^2 + 11887*x*z + 5051*x + 16*y^7 - 88*y^6*z - 268*y^6 + 204*y^5*z^2 + 1188*y^5*z + 1644*y^5 - 258*y^4*z^3 - 2163*y^4*z^2 - 5856*y^4*z - 5223*y^4 + 192*y^3*z^4 + 2070*y^3*z^3 + 8229*y^3*z^2 + 14466*y^3*z + 9547*y^3 - 84*y^2*z^5 - 1098*y^2*z^4 - 5706*y^2*z^3 - 14826*y^2*z^2 - 19332*y^2*z - 10151*y^2 + 20*y*z^6 + 306*y*z^5 + 1953*y*z^4 + 6672*y*z^3 + 12891*y*z^2 + 13376*y*z + 5831*y - 2*z^7 - 35*z^6 - 264*z^5 - 1113*z^4 - 2834*z^3 - 4361*z^2 - 3758*z - 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -2, -1, 1 ], [ -1, -2, 1 ], [ -1, -1, 1 ] ], "source_shift": [ 2, 2, 1 ], "target_matrix": [ [ -3, -3, -2 ], [ 1, 1, 1 ], [ -2, -1, -1 ] ], "target_shift": [ 2886, -952, 1397 ], "collision_preimages": [ [ "3/4", "3/4", "1/4" ], [ "13/2", "9", "21" ], [ "17/2", "6", "20" ] ], "collision_image": [ "11547/4", "-3809/4", "2795/2" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.088646542000788 ], "surface_features": [ 3.9142977504303205, 4.8855647478717925, 3.502836638621003, 0.0, 4.083940069528707, 4986 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C155", "pair": 155, "family": "counterexample", "mapping": "F(x,y,z)=(32*x^7 - 208*x^6*y + 256*x^6*z - 400*x^6 + 576*x^5*y^2 - 1416*x^5*y*z + 2256*x^5*y + 864*x^5*z^2 - 2856*x^5*z + 1944*x^5 - 880*x^4*y^3 + 3240*x^4*y^2*z - 5280*x^4*y^2 - 3948*x^4*y*z^2 + 13368*x^4*y*z - 9180*x^4*y + 1592*x^4*z^3 - 8412*x^4*z^2 + 11880*x^4*z - 4878*x^4 + 800*x^3*y^4 - 3920*x^3*y^3*z + 6560*x^3*y^3 + 7152*x^3*y^2*z^2 - 24912*x^3*y^2*z + 17280*x^3*y^2 - 5758*x^3*y*z^3 + 31356*x^3*y*z^2 - 44742*x^3*y*z + 18443*x^3*y + 1726*x^3*z^4 - 13078*x^3*z^3 + 28818*x^3*z^2 - 24365*x^3*z + 6807*x^3 - 432*x^2*y^5 + 2640*x^2*y^4*z - 4560*x^2*y^4 - 6408*x^2*y^3*z^2 + 23088*x^2*y^3*z - 16200*x^2*y^3 + 7722*x^2*y^2*z^3 - 43596*x^2*y^2*z^2 + 62946*x^2*y^2*z - 26061*x^2*y^2 - 4620*x^2*y*z^4 + 36378*x^2*y*z^3 - 81138*x^2*y*z^2 + 68892*x^2*y*z - 19257*x^2*y + 1098*x^2*z^5 - 11316*x^2*z^4 + 34686*x^2*z^3 - 45339*x^2*z^2 + 25974*x^2*z - 5207*x^2 + 128*x*y^6 - 936*x*y^5*z + 1680*x*y^5 + 2832*x*y^4*z^2 - 10632*x*y^4*z + 7560*x*y^4 - 4538*x*y^3*z^3 + 26772*x*y^3*z^2 - 39186*x*y^3*z + 16305*x*y^3 + 4062*x*y^2*z^4 - 33522*x*y^2*z^3 + 75822*x*y^2*z^2 - 64689*x*y^2*z + 18093*x*y^2 - 1926*x*y*z^5 + 20868*x*y*z^4 - 64890*x*y*z^3 + 85215*x*y*z^2 - 48822*x*y*z + 9766*x*y + 378*x*z^6 - 5166*x*z^5 + 20718*x*z^4 - 37261*x*z^3 + 32835*x*z^2 - 13434*x*z + 1976*x - 16*y^7 + 136*y^6*z - 256*y^6 - 492*y^5*z^2 + 1944*y^5*z - 1404*y^5 + 982*y^4*z^3 - 6120*y^4*z^2 + 9102*y^4*z - 3809*y^4 - 1168*y^3*z^4 + 10222*y^3*z^3 - 23502*y^3*z^2 + 20162*y^3*z - 5643*y^3 + 828*y^2*z^5 - 9552*y^2*z^4 + 30204*y^2*z^3 - 39876*y^2*z^2 + 22848*y^2*z - 4559*y^2 - 324*y*z^6 + 4734*y*z^5 - 19314*y*z^4 + 34916*y*z^3 - 30753*y*z^2 + 12534*y*z - 1838*y + 54*z^7 - 972*z^6 + 4914*z^5 - 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244*y^6 + 492*y^5*z^2 - 1860*y^5*z + 1236*y^5 - 982*y^4*z^3 + 5877*y^4*z^2 - 8064*y^4*z + 3100*y^4 + 1168*y^3*z^4 - 9850*y^3*z^3 + 20955*y^3*z^2 - 16522*y^3*z + 4304*y^3 - 828*y^2*z^5 + 9234*y^2*z^4 - 27102*y^2*z^3 + 32913*y^2*z^2 - 17520*y^2*z + 3329*y^2 + 324*y*z^6 - 4590*y*z^5 + 17439*y*z^4 - 29038*y*z^3 + 23718*y*z^2 - 9174*y*z + 1323*y - 54*z^7 + 945*z^6 - 4464*z^5 + 9571*z^4 - 10678*z^3 + 6309*z^2 - 1844*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -1, 1, -1 ], [ -2, 2, -3 ], [ -2, 1, -1 ] ], "source_shift": [ 2, 1, 1 ], "target_matrix": [ [ -2, -2, -3 ], [ -1, -1, -2 ], [ 2, 1, 1 ] ], "target_shift": [ 295, 139, -212 ], "collision_preimages": [ [ "-3/4", "-23/4", "-3" ], [ "-13/2", "-7", "1/2" ], [ "-17/2", "-18", "-13/2" ] ], "collision_image": [ "591/2", "557/4", "-425/2" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 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276*x^2*y^5 + 1038*x^2*y^4*z - 1698*x^2*y^4 + 1530*x^2*y^3*z^2 - 5025*x^2*y^3*z + 4056*x^2*y^3 + 1104*x^2*y^2*z^3 - 5463*x^2*y^2*z^2 + 8850*x^2*y^2*z - 4791*x^2*y^2 + 390*x^2*y*z^4 - 2586*x^2*y*z^3 + 6300*x^2*y*z^2 - 6855*x^2*y*z + 2850*x^2*y + 54*x^2*z^5 - 450*x^2*z^4 + 1464*x^2*z^3 - 2397*x^2*z^2 + 2004*x^2*z - 695*x^2 - 104*x*y^6 - 480*x*y^5*z + 804*x*y^5 - 906*x*y^4*z^2 + 3048*x*y^4*z - 2490*x*y^4 - 894*x*y^3*z^3 + 4533*x*y^3*z^2 - 7434*x*y^3*z + 4033*x*y^3 - 486*x*y^2*z^4 + 3303*x*y^2*z^3 - 8151*x*y^2*z^2 + 8901*x*y^2*z - 3681*x*y^2 - 138*x*y*z^5 + 1179*x*y*z^4 - 3888*x*y*z^3 + 6399*x*y*z^2 - 5340*x*y*z + 1837*x*y - 16*x*z^6 + 165*x*z^5 - 681*x*z^4 + 1499*x*z^3 - 1887*x*z^2 + 1309*x*z - 399*x + 16*y^7 + 88*y^6*z - 152*y^6 + 204*y^5*z^2 - 708*y^5*z + 588*y^5 + 258*y^4*z^3 - 1350*y^4*z^2 + 2250*y^4*z - 1226*y^4 + 192*y^3*z^4 - 1347*y^3*z^3 + 3378*y^3*z^2 - 3705*y^3*z + 1522*y^3 + 84*y^2*z^5 - 741*y^2*z^4 + 2484*y^2*z^3 - 4110*y^2*z^2 + 3414*y^2*z - 1157*y^2 + 20*y*z^6 - 213*y*z^5 + 894*y*z^4 - 1981*y*z^3 + 2490*y*z^2 - 1711*y*z + 514*y + 2*z^7 - 25*z^6 + 126*z^5 - 350*z^4 + 590*z^3 - 614*z^2 + 372*z - 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ 1, -2, -1 ], [ 1, -1, -1 ], [ 2, -1, -1 ] ], "source_shift": [ 1, 1, 2 ], "target_matrix": [ [ -3, -2, -2 ], [ 1, 1, 2 ], [ -2, -1, -1 ] ], "target_shift": [ 175, -69, 102 ], "collision_preimages": [ [ "-5/4", "0", "-1/4" ], [ "7", "-5/2", "12" ], [ "4", "5/2", "1" ] ], "collision_image": [ "703/4", "-277/4", "205/2" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.1813998389989138 ], "surface_features": [ 3.2743438989452316, 4.172427777151465, 2.972665592266111, 0.0, 3.3910624859120646, 4809 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C110", "pair": 110, "family": "counterexample", "mapping": "F(x,y,z)=(-2*x^7 + 21*x^6*y - 20*x^6*z + 28*x^6 - 93*x^5*y^2 + 183*x^5*y*z - 249*x^5*y - 84*x^5*z^2 + 243*x^5*z - 165*x^5 + 225*x^4*y^3 - 687*x^4*y^2*z + 909*x^4*y^2 + 651*x^4*y*z^2 - 1827*x^4*y*z + 1203*x^4*y - 192*x^4*z^3 + 864*x^4*z^2 - 1212*x^4*z + 530*x^4 - 321*x^3*y^4 + 1353*x^3*y^3*z - 1743*x^3*y^3 - 1989*x^3*y^2*z^2 + 5415*x^3*y^2*z - 3456*x^3*y^2 + 1209*x^3*y*z^3 - 5271*x^3*y*z^2 + 7167*x^3*y*z - 3029*x^3*y - 258*x^3*z^4 + 1611*x^3*z^3 - 3507*x^3*z^2 + 3170*x^3*z - 1000*x^3 + 270*x^2*y^5 - 1473*x^2*y^4*z + 1851*x^2*y^4 + 2991*x^2*y^3*z^2 - 7905*x^2*y^3*z + 4890*x^2*y^3 - 2817*x^2*y^2*z^3 + 11889*x^2*y^2*z^2 - 15657*x^2*y^2*z + 6393*x^2*y^2 + 1236*x^2*y*z^4 - 7473*x^2*y*z^3 + 15765*x^2*y*z^2 - 13767*x^2*y*z + 4182*x^2*y - 204*x^2*z^5 + 1662*x^2*z^4 - 4998*x^2*z^3 + 7011*x^2*z^2 - 4572*x^2*z + 1102*x^2 - 124*x*y^6 + 840*x*y^5*z - 1032*x*y^5 - 2211*x*y^4*z^2 + 5682*x*y^4*z - 3408*x*y^4 + 2875*x*y^3*z^3 - 11745*x*y^3*z^2 + 14976*x*y^3*z - 5909*x*y^3 - 1950*x*y^2*z^4 + 11397*x*y^2*z^3 - 23274*x*y^2*z^2 + 19623*x*y^2*z - 5742*x*y^2 + 660*x*y*z^5 - 5208*x*y*z^4 + 15177*x*y*z^3 - 20562*x*y*z^2 + 12903*x*y*z - 2984*x*y - 88*x*z^6 + 900*x*z^5 - 3510*x*z^4 + 6799*x*z^3 - 6879*x*z^2 + 3424*x*z - 650*x + 24*y^7 - 196*y^6*z + 236*y^6 + 642*y^5*z^2 - 1608*y^5*z + 936*y^5 - 1083*y^4*z^3 + 4287*y^4*z^2 - 5292*y^4*z + 2018*y^4 + 1012*y^3*z^4 - 5711*y^3*z^3 + 11286*y^3*z^2 - 9191*y^3*z + 2590*y^3 - 528*y^2*z^5 + 4026*y^2*z^4 - 11349*y^2*z^3 + 14832*y^2*z^2 - 8970*y^2*z + 1996*y^2 + 144*y*z^6 - 1428*y*z^5 + 5400*y*z^4 - 10102*y*z^3 + 9813*y*z^2 - 4678*y*z + 858*y - 16*z^7 + 200*z^6 - 972*z^5 + 2440*z^4 - 3413*z^3 + 2632*z^2 - 1019*z + 1, 4*x^7 - 42*x^6*y + 40*x^6*z - 50*x^6 + 186*x^5*y^2 - 366*x^5*y*z + 441*x^5*y + 168*x^5*z^2 - 438*x^5*z + 270*x^5 - 450*x^4*y^3 + 1374*x^4*y^2*z - 1596*x^4*y^2 - 1302*x^4*y*z^2 + 3267*x^4*y*z - 1947*x^4*y + 384*x^4*z^3 - 1572*x^4*z^2 + 2007*x^4*z - 813*x^4 + 642*x^3*y^4 - 2706*x^3*y^3*z + 3033*x^3*y^3 + 3978*x^3*y^2*z^2 - 9600*x^3*y^2*z + 5532*x^3*y^2 - 2418*x^3*y*z^3 + 9519*x^3*y*z^2 - 11736*x^3*y*z + 4594*x^3*y + 516*x^3*z^4 - 2958*x^3*z^3 + 5880*x^3*z^2 - 4920*x^3*z + 1464*x^3 - 540*x^2*y^5 + 2946*x^2*y^4*z - 3192*x^2*y^4 - 5982*x^2*y^3*z^2 + 13887*x^2*y^3*z - 7743*x^2*y^3 + 5634*x^2*y^2*z^3 - 21294*x^2*y^2*z^2 + 25347*x^2*y^2*z - 9591*x^2*y^2 - 2472*x^2*y*z^4 + 13629*x^2*y*z^3 - 26139*x^2*y*z^2 + 21114*x^2*y*z - 6060*x^2*y + 408*x^2*z^5 - 3078*x^2*z^4 + 8487*x^2*z^3 - 11019*x^2*z^2 + 6762*x^2*z - 1562*x^2 + 248*x*y^6 - 1680*x*y^5*z + 1764*x*y^5 + 4422*x*y^4*z^2 - 9888*x*y^4*z + 5340*x*y^4 - 5750*x*y^3*z^3 + 20847*x*y^3*z^2 - 23970*x*y^3*z + 8774*x*y^3 + 3900*x*y^2*z^4 - 20628*x*y^2*z^3 + 38142*x*y^2*z^2 - 29748*x*y^2*z + 8244*x*y^2 - 1320*x*y*z^5 + 9588*x*y*z^4 - 25494*x*y*z^3 + 31920*x*y*z^2 - 18870*x*y*z + 4195*x*y + 176*x*z^6 - 1680*x*z^5 + 6036*x*z^4 - 10828*x*z^3 + 10290*x*z^2 - 4892*x*z + 902*x - 48*y^7 + 392*y^6*z - 400*y^6 - 1284*y^5*z^2 + 2772*y^5*z - 1452*y^5 + 2166*y^4*z^3 - 7536*y^4*z^2 + 8376*y^4*z - 2968*y^4 - 2024*y^3*z^4 + 10249*y^3*z^3 - 18279*y^3*z^2 + 13778*y^3*z - 3688*y^3 + 1056*y^2*z^5 - 7362*y^2*z^4 + 18843*y^2*z^3 - 22749*y^2*z^2 + 12984*y^2*z - 2786*y^2 - 288*y*z^6 + 2652*y*z^5 - 9192*y*z^4 + 15884*y*z^3 - 14502*y*z^2 + 6623*y*z - 1183*y + 32*z^7 - 376*z^6 + 1692*z^5 - 3940*z^4 + 5168*z^3 - 3794*z^2 + 1423*z - 1, 6*x^7 - 63*x^6*y + 60*x^6*z - 84*x^6 + 279*x^5*y^2 - 549*x^5*y*z + 747*x^5*y + 252*x^5*z^2 - 729*x^5*z + 495*x^5 - 675*x^4*y^3 + 2061*x^4*y^2*z - 2727*x^4*y^2 - 1953*x^4*y*z^2 + 5481*x^4*y*z - 3609*x^4*y + 576*x^4*z^3 - 2592*x^4*z^2 + 3636*x^4*z - 1592*x^4 + 963*x^3*y^4 - 4059*x^3*y^3*z + 5229*x^3*y^3 + 5967*x^3*y^2*z^2 - 16245*x^3*y^2*z + 10368*x^3*y^2 - 3627*x^3*y*z^3 + 15813*x^3*y*z^2 - 21501*x^3*y*z + 9102*x^3*y + 774*x^3*z^4 - 4833*x^3*z^3 + 10521*x^3*z^2 - 9518*x^3*z + 3016*x^3 - 810*x^2*y^5 + 4419*x^2*y^4*z - 5553*x^2*y^4 - 8973*x^2*y^3*z^2 + 23715*x^2*y^3*z - 14670*x^2*y^3 + 8451*x^2*y^2*z^3 - 35667*x^2*y^2*z^2 + 46971*x^2*y^2*z - 19221*x^2*y^2 - 3708*x^2*y*z^4 + 22419*x^2*y*z^3 - 47295*x^2*y*z^2 + 41346*x^2*y*z - 12633*x^2*y + 612*x^2*z^5 - 4986*x^2*z^4 + 14994*x^2*z^3 - 21045*x^2*z^2 + 13767*x^2*z - 3351*x^2 + 372*x*y^6 - 2520*x*y^5*z + 3096*x*y^5 + 6633*x*y^4*z^2 - 17046*x*y^4*z + 10224*x*y^4 - 8625*x*y^3*z^3 + 35235*x*y^3*z^2 - 44928*x*y^3*z + 17779*x*y^3 + 5850*x*y^2*z^4 - 34191*x*y^2*z^3 + 69822*x*y^2*z^2 - 58953*x*y^2*z + 17382*x*y^2 - 1980*x*y*z^5 + 15624*x*y*z^4 - 45531*x*y*z^3 + 61731*x*y*z^2 - 38895*x*y*z + 9108*x*y + 264*x*z^6 - 2700*x*z^5 + 10530*x*z^4 - 20405*x*z^3 + 20691*x*z^2 - 10374*x*z + 2000*x - 72*y^7 + 588*y^6*z - 708*y^6 - 1926*y^5*z^2 + 4824*y^5*z - 2808*y^5 + 3249*y^4*z^3 - 12861*y^4*z^2 + 15876*y^4*z - 6078*y^4 - 3036*y^3*z^4 + 17133*y^3*z^3 - 33858*y^3*z^2 + 27625*y^3*z - 7862*y^3 + 1584*y^2*z^5 - 12078*y^2*z^4 + 34047*y^2*z^3 - 44538*y^2*z^2 + 27078*y^2*z - 6120*y^2 - 432*y*z^6 + 4284*y*z^5 - 16200*y*z^4 + 30321*y*z^3 - 29538*y*z^2 + 14214*y*z - 2654*y + 48*z^7 - 600*z^6 + 2916*z^5 - 7322*z^4 + 10258*z^3 - 7953*z^2 + 3119*z - 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -1, 2, -1 ], [ -1, 1, -2 ], [ -2, 3, -2 ] ], "source_shift": [ 2, 1, 1 ], "target_matrix": [ [ 1, 1, 1 ], [ -2, -1, -1 ], [ -3, -3, -2 ] ], "target_shift": [ -155, 212, 483 ], "collision_preimages": [ [ "-21/4", "-11/4", "7/4" ], [ "-23", "-15/2", "9" ], [ "-28", "-23/2", "8" ] ], "collision_image": [ "-621/4", "425/2", "1935/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.1054056819994003 ], "surface_features": [ 3.8847071182738766, 4.843992284703362, 3.53655844257153, 0.0, 4.025683979104693, 5016 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C230", "pair": 230, "family": "counterexample", "mapping": "F(x,y,z)=(16*x^7 + 104*x^6*y - 176*x^6*z - 212*x^6 + 288*x^5*y^2 - 972*x^5*y*z - 1176*x^5*y + 816*x^5*z^2 + 1968*x^5*z + 1140*x^5 + 440*x^4*y^3 - 2220*x^4*y^2*z - 2700*x^4*y^2 + 3714*x^4*y*z^2 + 9012*x^4*y*z + 5256*x^4*y - 2060*x^4*z^3 - 7485*x^4*z^2 - 8736*x^4*z - 3334*x^4 + 400*x^3*y^4 - 2680*x^3*y^3*z - 3280*x^3*y^3 + 6696*x^3*y^2*z^2 + 16368*x^3*y^2*z + 9624*x^3*y^2 - 7393*x^3*y*z^3 - 27096*x^3*y*z^2 - 31920*x^3*y*z - 12288*x^3*y + 3043*x^3*z^4 + 14881*x^3*z^3 + 26367*x^3*z^2 + 20365*x^3*z + 5838*x^3 + 216*x^2*y^5 - 1800*x^2*y^4*z - 2220*x^2*y^4 + 5964*x^2*y^3*z^2 + 14712*x^2*y^3*z + 8736*x^2*y^3 - 9819*x^2*y^2*z^3 - 36378*x^2*y^2*z^2 - 43344*x^2*y^2*z - 16860*x^2*y^2 + 8031*x^2*y*z^4 + 39780*x^2*y*z^3 + 71412*x^2*y*z^2 + 55803*x^2*y*z + 16152*x^2*y - 2610*x^2*z^5 - 16233*x^2*z^4 - 39078*x^2*z^3 - 46041*x^2*z^2 - 26754*x^2*z - 6178*x^2 + 64*x*y^6 - 636*x*y^5*z - 792*x*y^5 + 2616*x*y^4*z^2 + 6528*x*y^4*z + 3924*x*y^4 - 5699*x*y^3*z^3 - 21408*x*y^3*z^2 - 25872*x*y^3*z - 10192*x*y^3 + 6933*x*y^2*z^4 + 34917*x*y^2*z^3 + 63723*x*y^2*z^2 + 50511*x*y^2*z + 14790*x*y^2 - 4464*x*y*z^5 - 28326*x*y*z^4 - 69504*x*y*z^3 - 83214*x*y*z^2 - 48975*x*y*z - 11420*x*y + 1188*x*z^6 + 9144*x*z^5 + 28332*x*z^4 + 45582*x*z^3 + 40458*x*z^2 + 18931*x*z + 3679*x + 8*y^7 - 92*y^6*z - 116*y^6 + 450*y^5*z^2 + 1140*y^5*z + 696*y^5 - 1213*y^4*z^3 - 4641*y^4*z^2 - 5712*y^4*z - 2286*y^4 + 1945*y^3*z^4 + 10018*y^3*z^3 + 18678*y^3*z^2 + 15073*y^3*z + 4476*y^3 - 1854*y^2*z^5 - 12093*y^2*z^4 - 30426*y^2*z^3 - 37173*y^2*z^2 - 22221*y^2*z - 5242*y^2 + 972*y*z^6 + 7740*y*z^5 + 24696*y*z^4 + 40649*y*z^3 + 36702*y*z^2 + 17392*y*z + 3412*y - 216*z^7 - 2052*z^6 - 7992*z^5 - 16633*z^4 - 20171*z^3 - 14401*z^2 - 5662*z - 1, 48*x^7 + 312*x^6*y - 528*x^6*z - 588*x^6 + 864*x^5*y^2 - 2916*x^5*y*z - 3264*x^5*y + 2448*x^5*z^2 + 5472*x^5*z + 3024*x^5 + 1320*x^4*y^3 - 6660*x^4*y^2*z - 7500*x^4*y^2 + 11142*x^4*y*z^2 + 25080*x^4*y*z + 13956*x^4*y - 6180*x^4*z^3 - 20871*x^4*z^2 - 23256*x^4*z - 8586*x^4 + 1200*x^3*y^4 - 8040*x^3*y^3*z - 9120*x^3*y^3 + 20088*x^3*y^2*z^2 + 45600*x^3*y^2*z + 25584*x^3*y^2 - 22179*x^3*y*z^3 - 75642*x^3*y*z^2 - 85080*x^3*y*z - 31680*x^3*y + 9129*x^3*z^4 + 41631*x^3*z^3 + 70470*x^3*z^2 + 52659*x^3*z + 14718*x^3 + 648*x^2*y^5 - 5400*x^2*y^4*z - 6180*x^2*y^4 + 17892*x^2*y^3*z^2 + 41040*x^2*y^3*z + 23256*x^2*y^3 - 29457*x^2*y^2*z^3 - 101700*x^2*y^2*z^2 - 115704*x^2*y^2*z - 43524*x^2*y^2 + 24093*x^2*y*z^4 + 111465*x^2*y*z^3 + 191169*x^2*y*z^2 + 144489*x^2*y*z + 40764*x^2*y - 7830*x^2*z^5 - 45594*x^2*z^4 - 104913*x^2*z^3 - 119571*x^2*z^2 - 67734*x^2*z - 15330*x^2 + 192*x*y^6 - 1908*x*y^5*z - 2208*x*y^5 + 7848*x*y^4*z^2 + 18240*x*y^4*z + 10464*x*y^4 - 17097*x*y^3*z^3 - 59958*x*y^3*z^2 - 69192*x*y^3*z - 26352*x*y^3 + 20799*x*y^2*z^4 + 98037*x*y^2*z^3 + 170928*x*y^2*z^2 + 131001*x*y^2*z + 37374*x*y^2 - 13392*x*y*z^5 - 79740*x*y*z^4 - 187002*x*y*z^3 - 216480*x*y*z^2 - 124149*x*y*z - 28362*x*y + 3564*x*z^6 + 25812*x*z^5 + 76464*x*z^4 + 118944*x*z^3 + 102876*x*z^2 + 47169*x*z + 9026*x + 24*y^7 - 276*y^6*z - 324*y^6 + 1350*y^5*z^2 + 3192*y^5*z + 1860*y^5 - 3639*y^4*z^3 - 13029*y^4*z^2 - 15312*y^4*z - 5922*y^4 + 5835*y^3*z^4 + 28203*y^3*z^3 + 50229*y^3*z^2 + 39171*y^3*z + 11328*y^3 - 5562*y^2*z^5 - 34146*y^2*z^4 - 82089*y^2*z^3 - 96909*y^2*z^2 - 56415*y^2*z - 13032*y^2 + 2916*y*z^6 + 21924*y*z^5 + 66852*y*z^4 + 106305*y*z^3 + 93468*y*z^2 + 43377*y*z + 8375*y - 648*z^7 - 5832*z^6 - 21708*z^5 - 43635*z^4 - 51525*z^3 - 36030*z^2 - 13945*z - 1, -32*x^7 - 208*x^6*y + 352*x^6*z + 376*x^6 - 576*x^5*y^2 + 1944*x^5*y*z + 2088*x^5*y - 1632*x^5*z^2 - 3504*x^5*z - 1884*x^5 - 880*x^4*y^3 + 4440*x^4*y^2*z + 4800*x^4*y^2 - 7428*x^4*y*z^2 - 16068*x^4*y*z - 8700*x^4*y + 4120*x^4*z^3 + 13386*x^4*z^2 + 14520*x^4*z + 5268*x^4 - 800*x^3*y^4 + 5360*x^3*y^3*z + 5840*x^3*y^3 - 13392*x^3*y^2*z^2 - 29232*x^3*y^2*z - 15960*x^3*y^2 + 14786*x^3*y*z^3 + 48546*x^3*y*z^2 + 53160*x^3*y*z + 19448*x^3*y - 6086*x^3*z^4 - 26750*x^3*z^3 - 44103*x^3*z^2 - 32374*x^3*z - 8948*x^3 - 432*x^2*y^5 + 3600*x^2*y^4*z + 3960*x^2*y^4 - 11928*x^2*y^3*z^2 - 26328*x^2*y^3*z - 14520*x^2*y^3 + 19638*x^2*y^2*z^3 + 65322*x^2*y^2*z^2 + 72360*x^2*y^2*z + 26736*x^2*y^2 - 16062*x^2*y*z^4 - 71685*x^2*y*z^3 - 119757*x^2*y*z^2 - 88890*x^2*y*z - 24792*x^2*y + 5220*x^2*z^5 + 29361*x^2*z^4 + 65835*x^2*z^3 + 73674*x^2*z^2 + 41244*x^2*z + 9272*x^2 - 128*x*y^6 + 1272*x*y^5*z + 1416*x*y^5 - 5232*x*y^4*z^2 - 11712*x*y^4*z - 6540*x*y^4 + 11398*x*y^3*z^3 + 38550*x*y^3*z^2 + 43320*x*y^3*z + 16200*x*y^3 - 13866*x*y^2*z^4 - 63120*x*y^2*z^3 - 107205*x*y^2*z^2 - 80658*x*y^2*z - 22740*x*y^2 + 8928*x*y*z^5 + 51414*x*y*z^4 + 117498*x*y*z^3 + 133500*x*y*z^2 + 75630*x*y*z + 17158*x*y - 2376*x*z^6 - 16668*x*z^5 - 48132*x*z^4 - 73470*x*z^3 - 62751*x*z^2 - 28562*x*z - 5443*x - 16*y^7 + 184*y^6*z + 208*y^6 - 900*y^5*z^2 - 2052*y^5*z - 1164*y^5 + 2426*y^4*z^3 + 8388*y^4*z^2 + 9600*y^4*z + 3644*y^4 - 3890*y^3*z^4 - 18185*y^3*z^3 - 31551*y^3*z^2 - 24142*y^3*z - 6896*y^3 + 3708*y^2*z^5 + 22053*y^2*z^4 + 51663*y^2*z^3 + 59826*y^2*z^2 + 34386*y^2*z + 7886*y^2 - 1944*y*z^6 - 14184*y*z^5 - 42156*y*z^4 - 65737*y*z^3 - 57045*y*z^2 - 26273*y*z - 5051*y + 432*z^7 + 3780*z^6 + 13716*z^5 + 27029*z^4 + 31489*z^3 + 21845*z^2 + 8417*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -2, -2, 3 ], [ -1, -1, 2 ], [ -2, -1, 1 ] ], "source_shift": [ 2, 2, 1 ], "target_matrix": [ [ -1, -1, -1 ], [ -3, -2, -3 ], [ 2, 1, 1 ] ], "target_shift": [ 952, 2322, -1397 ], "collision_preimages": [ [ "5/4", "-13/4", "-2" ], [ "-3", "-11/2", "-6" ], [ "-8", "25/2", "2" ] ], "collision_image": [ "3809/4", "9291/4", "-2795/2" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0636272779993305 ], "surface_features": [ 4.408059254429644, 5.3354177792535475, 4.086021518808512, 0.0, 4.533939764185732, 5193 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B195", "pair": 195, "family": "bijection", "mapping": "F(x,y,z)=(-2187*x^7 - 10206*x^6*y + 10206*x^6*z + 10206*x^6 - 20412*x^5*y^2 + 40824*x^5*y*z + 40824*x^5*y - 20412*x^5*z^2 - 40824*x^5*z - 20979*x^5 - 22680*x^4*y^3 + 68040*x^4*y^2*z + 68040*x^4*y^2 - 68040*x^4*y*z^2 - 136080*x^4*y*z - 70038*x^4*y + 22680*x^4*z^3 + 68040*x^4*z^2 + 69957*x^4*z + 24543*x^4 - 15120*x^3*y^4 + 60480*x^3*y^3*z + 60480*x^3*y^3 - 90720*x^3*y^2*z^2 - 181440*x^3*y^2*z - 93528*x^3*y^2 + 60480*x^3*y*z^3 + 181440*x^3*y*z^2 + 186840*x^3*y*z + 65745*x^3*y - 15120*x^3*z^4 - 60480*x^3*z^3 - 93312*x^3*z^2 - 65529*x^3*z - 17640*x^3 - 6048*x^2*y^5 + 30240*x^2*y^4*z + 30240*x^2*y^4 - 60480*x^2*y^3*z^2 - 120960*x^2*y^3*z - 62448*x^2*y^3 + 60480*x^2*y^2*z^3 + 181440*x^2*y^2*z^2 + 187128*x^2*y^2*z + 66042*x^2*y^2 - 30240*x^2*y*z^4 - 120960*x^2*y*z^3 - 186912*x^2*y*z^2 - 131652*x^2*y*z - 35598*x^2*y + 6048*x^2*z^5 + 30240*x^2*z^4 + 62232*x^2*z^3 + 65610*x^2*z^2 + 35373*x^2*z + 7801*x^2 - 1344*x*y^6 + 8064*x*y^5*z + 8064*x*y^5 - 20160*x*y^4*z^2 - 40320*x*y^4*z - 20848*x*y^4 + 26880*x*y^3*z^3 + 80640*x*y^3*z^2 + 83296*x*y^3*z + 29484*x*y^3 - 20160*x*y^2*z^4 - 80640*x*y^2*z^3 - 124800*x*y^2*z^2 - 88164*x*y^2*z - 23944*x*y^2 + 8064*x*y*z^5 + 40320*x*y*z^4 + 83104*x*y*z^3 + 87876*x*y*z^2 + 47588*x*y*z + 10564*x*y - 1344*x*z^6 - 8064*x*z^5 - 20752*x*z^4 - 29196*x*z^3 - 23644*x*z^2 - 10452*x*z - 1971*x - 128*y^7 + 896*y^6*z + 896*y^6 - 2688*y^5*z^2 - 5376*y^5*z - 2784*y^5 + 4480*y^4*z^3 + 13440*y^4*z^2 + 13904*y^4*z + 4936*y^4 - 4480*y^3*z^4 - 17920*y^3*z^3 - 27776*y^3*z^2 - 19680*y^3*z - 5368*y^3 + 2688*y^2*z^5 + 13440*y^2*z^4 + 27744*y^2*z^3 + 29424*y^2*z^2 + 16004*y^2*z + 3576*y^2 - 896*y*z^6 - 5376*y*z^5 - 13856*y*z^4 - 19552*y*z^3 - 15904*y*z^2 - 7077*y*z - 1349*y + 128*z^7 + 896*z^6 + 2768*z^5 + 4872*z^4 + 5268*z^3 + 3501*z^2 + 1325*z + 1, 6561*x^7 + 30618*x^6*y - 30618*x^6*z - 30618*x^6 + 61236*x^5*y^2 - 122472*x^5*y*z - 122472*x^5*y + 61236*x^5*z^2 + 122472*x^5*z + 62937*x^5 + 68040*x^4*y^3 - 204120*x^4*y^2*z - 204120*x^4*y^2 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120960*x*y*z^4 - 249312*x*y*z^3 - 265932*x*y*z^2 - 147300*x*y*z - 34020*x*y + 4032*x*z^6 + 24192*x*z^5 + 62256*x*z^4 + 88356*x*z^3 + 73200*x*z^2 + 33684*x*z + 6769*x + 384*y^7 - 2688*y^6*z - 2688*y^6 + 8064*y^5*z^2 + 16128*y^5*z + 8352*y^5 - 13440*y^4*z^3 - 40320*y^4*z^2 - 41712*y^4*z - 14936*y^4 + 13440*y^3*z^4 + 53760*y^3*z^3 + 83328*y^3*z^2 + 59552*y^3*z + 16608*y^3 - 8064*y^2*z^5 - 40320*y^2*z^4 - 83232*y^2*z^3 - 89040*y^2*z^2 - 49524*y^2*z - 11504*y^2 + 2688*y*z^6 + 16128*y*z^5 + 41568*y*z^4 + 59168*y*z^3 + 49224*y*z^2 + 22783*y*z + 4619*y - 384*z^7 - 2688*z^6 - 8304*z^5 - 14744*z^4 - 16308*z^3 - 11279*z^2 - 4548*z - 1, 2187*x^7 + 10206*x^6*y - 10206*x^6*z - 10206*x^6 + 20412*x^5*y^2 - 40824*x^5*y*z - 40824*x^5*y + 20412*x^5*z^2 + 40824*x^5*z + 20979*x^5 + 22680*x^4*y^3 - 68040*x^4*y^2*z - 68040*x^4*y^2 + 68040*x^4*y*z^2 + 136080*x^4*y*z + 70038*x^4*y - 22680*x^4*z^3 - 68040*x^4*z^2 - 69957*x^4*z - 24705*x^4 + 15120*x^3*y^4 - 60480*x^3*y^3*z - 60480*x^3*y^3 + 90720*x^3*y^2*z^2 + 181440*x^3*y^2*z + 93528*x^3*y^2 - 60480*x^3*y*z^3 - 181440*x^3*y*z^2 - 186840*x^3*y*z - 66177*x^3*y + 15120*x^3*z^4 + 60480*x^3*z^3 + 93312*x^3*z^2 + 65961*x^3*z + 18072*x^3 + 6048*x^2*y^5 - 30240*x^2*y^4*z - 30240*x^2*y^4 + 60480*x^2*y^3*z^2 + 120960*x^2*y^3*z + 62448*x^2*y^3 - 60480*x^2*y^2*z^3 - 181440*x^2*y^2*z^2 - 187128*x^2*y^2*z - 66474*x^2*y^2 + 30240*x^2*y*z^4 + 120960*x^2*y*z^3 + 186912*x^2*y*z^2 + 132516*x^2*y*z + 36462*x^2*y - 6048*x^2*z^5 - 30240*x^2*z^4 - 62232*x^2*z^3 - 66042*x^2*z^2 - 36237*x^2*z - 8251*x^2 + 1344*x*y^6 - 8064*x*y^5*z - 8064*x*y^5 + 20160*x*y^4*z^2 + 40320*x*y^4*z + 20848*x*y^4 - 26880*x*y^3*z^3 - 80640*x*y^3*z^2 - 83296*x*y^3*z - 29676*x*y^3 + 20160*x*y^2*z^4 + 80640*x*y^2*z^3 + 124800*x*y^2*z^2 + 88740*x*y^2*z + 24520*x*y^2 - 8064*x*y*z^5 - 40320*x*y*z^4 - 83104*x*y*z^3 - 88452*x*y*z^2 - 48740*x*y*z - 11164*x*y + 1344*x*z^6 + 8064*x*z^5 + 20752*x*z^4 + 29388*x*z^3 + 24220*x*z^2 + 11052*x*z + 2195*x + 128*y^7 - 896*y^6*z - 896*y^6 + 2688*y^5*z^2 + 5376*y^5*z + 2784*y^5 - 4480*y^4*z^3 - 13440*y^4*z^2 - 13904*y^4*z - 4968*y^4 + 4480*y^3*z^4 + 17920*y^3*z^3 + 27776*y^3*z^2 + 19808*y^3*z + 5496*y^3 - 2688*y^2*z^5 - 13440*y^2*z^4 - 27744*y^2*z^3 - 29616*y^2*z^2 - 16388*y^2*z - 3776*y^2 + 896*y*z^6 + 5376*y*z^5 + 13856*y*z^4 + 19680*y*z^3 + 16288*y*z^2 + 7477*y*z + 1499*y - 128*z^7 - 896*z^6 - 2768*z^5 - 4904*z^4 - 5396*z^3 - 3701*z^2 - 1475*z - 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -3, -2, 2 ], [ -1, -2, 1 ], [ -1, -1, 1 ] ], "source_shift": [ 2, 1, 1 ], "target_matrix": [ [ 1, 1, 2 ], [ -3, -2, -2 ], [ -1, -1, -1 ] ], "target_shift": [ -219, 832, 265 ], "inverse_recipe": { "p": 3, "q": 4, "f": "x**3 + x", "g": "x**4 + x**2 + x", "h": "y*z + 2*z", "polynomials": { "f": [ [ [ 3, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "g": [ [ [ 4, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 0, 1 ], 2 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0432262729991635 ], "surface_features": [ 4.836834256191539, 5.749260030651944, 4.480581786829169, 0.0, 4.985548884210118, 5365 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C116", "pair": 116, "family": "counterexample", "mapping": "F(x,y,z)=(-8*x^7 + 68*x^6*y + 44*x^6*z - 140*x^6 - 234*x^5*y^2 - 312*x^5*y*z + 984*x^5*y - 102*x^5*z^2 + 648*x^5*z - 924*x^5 + 419*x^4*y^3 + 867*x^4*y^2*z - 2727*x^4*y^2 + 585*x^4*y*z^2 - 3690*x^4*y*z + 5352*x^4*y + 129*x^4*z^3 - 1227*x^4*z^2 + 3528*x^4*z - 3222*x^4 - 419*x^3*y^4 - 1199*x^3*y^3*z + 3803*x^3*y^3 - 1257*x^3*y^2*z^2 + 7929*x^3*y^2*z - 11787*x^3*y^2 - 573*x^3*y*z^3 + 5421*x^3*y*z^2 - 15882*x^3*y*z + 14867*x^3*y - 96*x^3*z^4 + 1215*x^3*z^3 - 5283*x^3*z^2 + 9761*x^3*z - 6576*x^3 + 234*x^2*y^5 + 867*x^2*y^4*z - 2814*x^2*y^4 + 1257*x^2*y^3*z^2 - 8019*x^2*y^3*z + 12279*x^2*y^3 + 891*x^2*y^2*z^3 - 8451*x^2*y^2*z^2 + 25371*x^2*y^2*z - 24411*x^2*y^2 + 309*x^2*y*z^4 - 3897*x^2*y*z^3 + 17277*x^2*y*z^2 - 32709*x^2*y*z + 22611*x^2*y + 42*x^2*z^5 - 663*x^2*z^4 + 3873*x^2*z^3 - 10848*x^2*z^2 + 14799*x^2*z - 7936*x^2 - 68*x*y^6 - 312*x*y^5*z + 1044*x*y^5 - 585*x*y^4*z^2 + 3816*x*y^4*z - 5988*x*y^4 - 573*x*y^3*z^3 + 5505*x*y^3*z^2 - 16938*x*y^3*z + 16701*x*y^3 - 309*x*y^2*z^4 + 3915*x*y^2*z^3 - 17745*x*y^2*z^2 + 34413*x*y^2*z - 24402*x*y^2 - 87*x*y*z^5 + 1371*x*y*z^4 - 8160*x*y*z^3 + 23376*x*y*z^2 - 32667*x*y*z + 17977*x*y - 10*x*z^6 + 189*x*z^5 - 1389*x*z^4 + 5234*x*z^3 - 10827*x*z^2 + 11765*x*z - 5273*x + 8*y^7 + 44*y^6*z - 152*y^6 + 102*y^5*z^2 - 684*y^5*z + 1092*y^5 + 129*y^4*z^3 - 1266*y^4*z^2 + 3972*y^4*z - 3979*y^4 + 96*y^3*z^4 - 1233*y^3*z^3 + 5703*y^3*z^2 - 11272*y^3*z + 8133*y^3 + 42*y^2*z^5 - 666*y^2*z^4 + 4041*y^2*z^3 - 11811*y^2*z^2 + 16839*y^2*z - 9466*y^2 + 10*y*z^6 - 189*y*z^5 + 1413*y*z^4 - 5431*y*z^3 + 11469*y*z^2 - 12739*y*z + 5851*y + z^7 - 22*z^6 + 195*z^5 - 925*z^4 + 2573*z^3 - 4237*z^2 + 3847*z - 1, -8*x^7 + 68*x^6*y + 44*x^6*z - 116*x^6 - 234*x^5*y^2 - 312*x^5*y*z + 828*x^5*y - 102*x^5*z^2 + 540*x^5*z - 696*x^5 + 419*x^4*y^3 + 867*x^4*y^2*z - 2337*x^4*y^2 + 585*x^4*y*z^2 - 3126*x^4*y*z + 4104*x^4*y + 129*x^4*z^3 - 1029*x^4*z^2 + 2676*x^4*z - 2278*x^4 - 419*x^3*y^4 - 1199*x^3*y^3*z + 3326*x^3*y^3 - 1257*x^3*y^2*z^2 + 6846*x^3*y^2*z - 9222*x^3*y^2 - 573*x^3*y*z^3 + 4626*x^3*y*z^2 - 12276*x^3*y*z + 10719*x^3*y - 96*x^3*z^4 + 1026*x^3*z^3 - 4038*x^3*z^2 + 6957*x^3*z - 4432*x^3 + 234*x^2*y^5 + 867*x^2*y^4*z - 2511*x^2*y^4 + 1257*x^2*y^3*z^2 - 7065*x^2*y^3*z + 9816*x^2*y^3 + 891*x^2*y^2*z^3 - 7353*x^2*y^2*z^2 + 20025*x^2*y^2*z - 17967*x^2*y^2 + 309*x^2*y*z^4 - 3351*x^2*y*z^3 + 13470*x^2*y*z^2 - 23793*x^2*y*z + 15579*x^2*y + 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696*x^5 - 419*x^4*y^3 - 867*x^4*y^2*z + 2337*x^4*y^2 - 585*x^4*y*z^2 + 3126*x^4*y*z - 4104*x^4*y - 129*x^4*z^3 + 1029*x^4*z^2 - 2676*x^4*z + 2286*x^4 + 419*x^3*y^4 + 1199*x^3*y^3*z - 3326*x^3*y^3 + 1257*x^3*y^2*z^2 - 6846*x^3*y^2*z + 9222*x^3*y^2 + 573*x^3*y*z^3 - 4626*x^3*y*z^2 + 12276*x^3*y*z - 10739*x^3*y + 96*x^3*z^4 - 1026*x^3*z^3 + 4038*x^3*z^2 - 6977*x^3*z + 4476*x^3 - 234*x^2*y^5 - 867*x^2*y^4*z + 2511*x^2*y^4 - 1257*x^2*y^3*z^2 + 7065*x^2*y^3*z - 9816*x^2*y^3 - 891*x^2*y^2*z^3 + 7353*x^2*y^2*z^2 - 20025*x^2*y^2*z + 17985*x^2*y^2 - 309*x^2*y*z^4 + 3351*x^2*y*z^3 - 13470*x^2*y*z^2 + 23829*x^2*y*z - 15675*x^2*y - 42*x^2*z^5 + 564*x^2*z^4 - 2985*x^2*z^3 + 7818*x^2*z^2 - 10155*x^2*z + 5242*x^2 + 68*x*y^6 + 312*x*y^5*z - 948*x*y^5 + 585*x*y^4*z^2 - 3426*x*y^4*z + 4896*x*y^4 + 573*x*y^3*z^3 - 4884*x*y^3*z^2 + 13668*x*y^3*z - 12579*x*y^3 + 309*x*y^2*z^4 - 3432*x*y^2*z^3 + 14136*x*y^2*z^2 - 25605*x*y^2*z + 17268*x*y^2 + 87*x*y*z^5 - 1188*x*y*z^4 + 6420*x*y*z^3 - 17190*x*y*z^2 + 22863*x*y*z - 12112*x*y + 10*x*z^6 - 162*x*z^5 + 1080*x*z^4 - 3806*x*z^3 + 7497*x*z^2 - 7844*x*z + 3412*x - 8*y^7 - 44*y^6*z + 140*y^6 - 102*y^5*z^2 + 624*y^5*z - 912*y^5 - 129*y^4*z^3 + 1143*y^4*z^2 - 3276*y^4*z + 3067*y^4 - 96*y^3*z^4 + 1101*y^3*z^3 - 4644*y^3*z^2 + 8578*y^3*z - 5889*y^3 - 42*y^2*z^5 + 588*y^2*z^4 - 3249*y^2*z^3 + 8877*y^2*z^2 - 12045*y^2*z + 6523*y^2 - 10*y*z^6 + 165*y*z^5 - 1122*y*z^4 + 4033*y*z^3 - 8109*y*z^2 + 8677*y*z - 3872*y - z^7 + 19*z^6 - 153*z^5 + 679*z^4 - 1799*z^3 + 2854*z^2 - 2517*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ 2, -1, -1 ], [ 1, -2, -1 ], [ 1, -1, -1 ] ], "source_shift": [ 2, 2, 1 ], "target_matrix": [ [ -1, -2, -1 ], [ -1, -1, -2 ], [ 1, 1, 1 ] ], "target_shift": [ 1488, 924, -952 ], "collision_preimages": [ [ "-3/4", "3/4", "-1/4" ], [ "-13/2", "9", "-21" ], [ "-17/2", "6", "-20" ] ], "collision_image": [ "5953/4", "3697/4", "-3809/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.055824365997978 ], "surface_features": [ 3.681900221375677, 4.536722534482417, 3.358315640082196, 0.0, 3.807745922562069, 4897 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B241", "pair": 241, "family": "bijection", "mapping": "F(x,y,z)=(12*x^7 + 168*x^6*y - 84*x^6*z - 216*x^6 + 1008*x^5*y^2 - 1008*x^5*y*z - 2520*x^5*y + 252*x^5*z^2 + 1272*x^5*z + 1584*x^5 + 3360*x^4*y^3 - 5040*x^4*y^2*z - 12240*x^4*y^2 + 2520*x^4*y*z^2 + 12360*x^4*y*z + 14964*x^4*y - 420*x^4*z^3 - 3120*x^4*z^2 - 7632*x^4*z - 6168*x^4 + 6720*x^3*y^4 - 13440*x^3*y^3*z - 31680*x^3*y^3 + 10080*x^3*y^2*z^2 + 48000*x^3*y^2*z + 56460*x^3*y^2 - 3360*x^3*y*z^3 - 24240*x^3*y*z^2 - 57624*x^3*y*z - 45240*x^3*y + 420*x^3*z^4 + 4080*x^3*z^3 + 14700*x^3*z^2 + 23352*x^3*z + 13791*x^3 + 8064*x^2*y^5 - 20160*x^2*y^4*z - 46080*x^2*y^4 + 20160*x^2*y^3*z^2 + 93120*x^2*y^3*z + 106344*x^2*y^3 - 10080*x^2*y^2*z^3 - 70560*x^2*y^2*z^2 - 162900*x^2*y^2*z - 124128*x^2*y^2 + 2520*x^2*y*z^4 + 23760*x^2*y*z^3 + 83160*x^2*y*z^2 + 128256*x^2*y*z + 73266*x^2*y - 252*x^2*z^5 - 3000*x^2*z^4 - 14148*x^2*z^3 - 33120*x^2*z^2 - 38397*x^2*z - 17414*x^2 + 5376*x*y^6 - 16128*x*y^5*z - 35712*x*y^5 + 20160*x*y^4*z^2 + 90240*x*y^4*z + 99984*x*y^4 - 13440*x*y^3*z^3 - 91200*x*y^3*z^2 - 204336*x*y^3*z - 151008*x*y^3 + 5040*x*y^2*z^4 + 46080*x*y^2*z^3 + 156564*x*y^2*z^2 + 234240*x*y^2*z + 129516*x*y^2 - 1008*x*y*z^5 - 11640*x*y*z^4 - 53304*x*y*z^3 - 121080*x*y*z^2 - 135780*x*y*z - 59576*x*y + 84*x*z^6 + 1176*x*z^5 + 6804*x*z^4 + 20856*x*z^3 + 35589*x*z^2 + 31708*x*z + 11420*x + 1536*y^7 - 5376*y^6*z - 11520*y^6 + 8064*y^5*z^2 + 34944*y^5*z + 37536*y^5 - 6720*y^4*z^3 - 44160*y^4*z^2 - 95952*y^4*z - 68736*y^4 + 3360*y^3*z^4 + 29760*y^3*z^3 + 98088*y^3*z^2 + 142272*y^3*z + 76188*y^3 - 1008*y^2*z^5 - 11280*y^2*z^4 - 50124*y^2*z^3 - 110400*y^2*z^2 - 119844*y^2*z - 50936*y^2 + 168*y*z^6 + 2280*y*z^5 + 12804*y*z^4 + 38064*y*z^3 + 62838*y*z^2 + 54176*y*z + 18949*y - 12*z^7 - 192*z^6 - 1308*z^5 - 4920*z^4 - 10983*z^3 - 14414*z^2 - 10224*z - 1, -6*x^7 - 84*x^6*y + 42*x^6*z + 108*x^6 - 504*x^5*y^2 + 504*x^5*y*z + 1260*x^5*y - 126*x^5*z^2 - 636*x^5*z - 792*x^5 - 1680*x^4*y^3 + 2520*x^4*y^2*z + 6120*x^4*y^2 - 1260*x^4*y*z^2 - 6180*x^4*y*z - 7482*x^4*y + 210*x^4*z^3 + 1560*x^4*z^2 + 3816*x^4*z + 3084*x^4 - 3360*x^3*y^4 + 6720*x^3*y^3*z + 15840*x^3*y^3 - 5040*x^3*y^2*z^2 - 24000*x^3*y^2*z - 28230*x^3*y^2 + 1680*x^3*y*z^3 + 12120*x^3*y*z^2 + 28812*x^3*y*z + 22620*x^3*y - 210*x^3*z^4 - 2040*x^3*z^3 - 7350*x^3*z^2 - 11676*x^3*z - 6895*x^3 - 4032*x^2*y^5 + 10080*x^2*y^4*z + 23040*x^2*y^4 - 10080*x^2*y^3*z^2 - 46560*x^2*y^3*z - 53172*x^2*y^3 + 5040*x^2*y^2*z^3 + 35280*x^2*y^2*z^2 + 81450*x^2*y^2*z + 62064*x^2*y^2 - 1260*x^2*y*z^4 - 11880*x^2*y*z^3 - 41580*x^2*y*z^2 - 64128*x^2*y*z - 36630*x^2*y + 126*x^2*z^5 + 1500*x^2*z^4 + 7074*x^2*z^3 + 16560*x^2*z^2 + 19197*x^2*z + 8702*x^2 - 2688*x*y^6 + 8064*x*y^5*z + 17856*x*y^5 - 10080*x*y^4*z^2 - 45120*x*y^4*z - 49992*x*y^4 + 6720*x*y^3*z^3 + 45600*x*y^3*z^2 + 102168*x*y^3*z + 75504*x*y^3 - 2520*x*y^2*z^4 - 23040*x*y^2*z^3 - 78282*x*y^2*z^2 - 117120*x*y^2*z - 64752*x*y^2 + 504*x*y*z^5 + 5820*x*y*z^4 + 26652*x*y*z^3 + 60540*x*y*z^2 + 67884*x*y*z + 29768*x*y - 42*x*z^6 - 588*x*z^5 - 3402*x*z^4 - 10428*x*z^3 - 17793*x*z^2 - 15844*x*z - 5691*x - 768*y^7 + 2688*y^6*z + 5760*y^6 - 4032*y^5*z^2 - 17472*y^5*z - 18768*y^5 + 3360*y^4*z^3 + 22080*y^4*z^2 + 47976*y^4*z + 34368*y^4 - 1680*y^3*z^4 - 14880*y^3*z^3 - 49044*y^3*z^2 - 71136*y^3*z - 38090*y^3 + 504*y^2*z^5 + 5640*y^2*z^4 + 25062*y^2*z^3 + 55200*y^2*z^2 + 59916*y^2*z + 25448*y^2 - 84*y*z^6 - 1140*y*z^5 - 6402*y*z^4 - 19032*y*z^3 - 31416*y*z^2 - 27068*y*z - 9443*y + 6*z^7 + 96*z^6 + 654*z^5 + 2460*z^4 + 5491*z^3 + 7202*z^2 + 5095*z + 1, 6*x^7 + 84*x^6*y - 42*x^6*z - 108*x^6 + 504*x^5*y^2 - 504*x^5*y*z - 1260*x^5*y + 126*x^5*z^2 + 636*x^5*z + 792*x^5 + 1680*x^4*y^3 - 2520*x^4*y^2*z - 6120*x^4*y^2 + 1260*x^4*y*z^2 + 6180*x^4*y*z + 7482*x^4*y - 210*x^4*z^3 - 1560*x^4*z^2 - 3816*x^4*z - 3084*x^4 + 3360*x^3*y^4 - 6720*x^3*y^3*z - 15840*x^3*y^3 + 5040*x^3*y^2*z^2 + 24000*x^3*y^2*z + 28230*x^3*y^2 - 1680*x^3*y*z^3 - 12120*x^3*y*z^2 - 28812*x^3*y*z - 22620*x^3*y + 210*x^3*z^4 + 2040*x^3*z^3 + 7350*x^3*z^2 + 11676*x^3*z + 6895*x^3 + 4032*x^2*y^5 - 10080*x^2*y^4*z - 23040*x^2*y^4 + 10080*x^2*y^3*z^2 + 46560*x^2*y^3*z + 53172*x^2*y^3 - 5040*x^2*y^2*z^3 - 35280*x^2*y^2*z^2 - 81450*x^2*y^2*z - 62064*x^2*y^2 + 1260*x^2*y*z^4 + 11880*x^2*y*z^3 + 41580*x^2*y*z^2 + 64128*x^2*y*z + 36630*x^2*y - 126*x^2*z^5 - 1500*x^2*z^4 - 7074*x^2*z^3 - 16560*x^2*z^2 - 19197*x^2*z - 8704*x^2 + 2688*x*y^6 - 8064*x*y^5*z - 17856*x*y^5 + 10080*x*y^4*z^2 + 45120*x*y^4*z + 49992*x*y^4 - 6720*x*y^3*z^3 - 45600*x*y^3*z^2 - 102168*x*y^3*z - 75504*x*y^3 + 2520*x*y^2*z^4 + 23040*x*y^2*z^3 + 78282*x*y^2*z^2 + 117120*x*y^2*z + 64752*x*y^2 - 504*x*y*z^5 - 5820*x*y*z^4 - 26652*x*y*z^3 - 60540*x*y*z^2 - 67884*x*y*z - 29776*x*y + 42*x*z^6 + 588*x*z^5 + 3402*x*z^4 + 10428*x*z^3 + 17793*x*z^2 + 15848*x*z + 5703*x + 768*y^7 - 2688*y^6*z - 5760*y^6 + 4032*y^5*z^2 + 17472*y^5*z + 18768*y^5 - 3360*y^4*z^3 - 22080*y^4*z^2 - 47976*y^4*z - 34368*y^4 + 1680*y^3*z^4 + 14880*y^3*z^3 + 49044*y^3*z^2 + 71136*y^3*z + 38090*y^3 - 504*y^2*z^5 - 5640*y^2*z^4 - 25062*y^2*z^3 - 55200*y^2*z^2 - 59916*y^2*z - 25456*y^2 + 84*y*z^6 + 1140*y*z^5 + 6402*y*z^4 + 19032*y*z^3 + 31416*y*z^2 + 27076*y*z + 9461*y - 6*z^7 - 96*z^6 - 654*z^5 - 2460*z^4 - 5491*z^3 - 7204*z^2 - 5105*z - 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -1, -2, 1 ], [ -2, -3, 2 ], [ -2, -1, 1 ] ], "source_shift": [ 2, 2, 1 ], "target_matrix": [ [ -2, -3, -2 ], [ 1, 1, 2 ], [ -1, -1, -1 ] ], "target_shift": [ 3013, -1491, 1501 ], "inverse_recipe": { "p": 3, "q": 2, "f": "x**3", "g": "x**2", "h": "y*z**2", "polynomials": { "f": [ [ [ 3, 0, 0 ], 1 ] ], "g": [ [ [ 2, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 2 ], 1 ] ] }, "h_multiplier": 6 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0647822940009064 ], "surface_features": [ 4.4014069555777064, 5.369661059378298, 4.003460532109506, 0.0, 4.544707118366591, 5166 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B188", "pair": 188, "family": "bijection", "mapping": "F(x,y,z)=(-3*x^7 + 42*x^6*y - 21*x^6*z + 24*x^6 - 252*x^5*y^2 + 252*x^5*y*z - 288*x^5*y - 63*x^5*z^2 + 144*x^5*z - 84*x^5 + 840*x^4*y^3 - 1260*x^4*y^2*z + 1440*x^4*y^2 + 630*x^4*y*z^2 - 1440*x^4*y*z + 837*x^4*y - 105*x^4*z^3 + 360*x^4*z^2 - 423*x^4*z + 172*x^4 - 1680*x^3*y^4 + 3360*x^3*y^3*z - 3840*x^3*y^3 - 2520*x^3*y^2*z^2 + 5760*x^3*y^2*z - 3336*x^3*y^2 + 840*x^3*y*z^3 - 2880*x^3*y*z^2 + 3372*x^3*y*z - 1355*x^3*y - 105*x^3*z^4 + 480*x^3*z^3 - 852*x^3*z^2 + 706*x^3*z - 222*x^3 + 2016*x^2*y^5 - 5040*x^2*y^4*z + 5760*x^2*y^4 + 5040*x^2*y^3*z^2 - 11520*x^2*y^3*z + 6648*x^2*y^3 - 2520*x^2*y^2*z^3 + 8640*x^2*y^2*z^2 - 10080*x^2*y^2*z + 4002*x^2*y^2 + 630*x^2*y*z^4 - 2880*x^2*y*z^3 + 5094*x^2*y*z^2 - 4173*x^2*y*z + 1287*x^2*y - 63*x^2*z^5 + 360*x^2*z^4 - 858*x^2*z^3 + 1086*x^2*z^2 - 702*x^2*z + 183*x^2 - 1344*x*y^6 + 4032*x*y^5*z - 4608*x*y^5 - 5040*x*y^4*z^2 + 11520*x*y^4*z - 6624*x*y^4 + 3360*x*y^3*z^3 - 11520*x*y^3*z^2 + 13392*x*y^3*z - 5252*x*y^3 - 1260*x*y^2*z^4 + 5760*x*y^2*z^3 - 10152*x*y^2*z^2 + 8220*x*y^2*z - 2484*x*y^2 + 252*x*y*z^5 - 1440*x*y*z^4 + 3420*x*y*z^3 - 4281*x*y*z^2 + 2718*x*y*z - 681*x*y - 21*x*z^6 + 144*x*z^5 - 432*x*z^4 + 742*x*z^3 - 738*x*z^2 + 405*x*z - 89*x + 384*y^7 - 1344*y^6*z + 1536*y^6 + 2016*y^5*z^2 - 4608*y^5*z + 2640*y^5 - 1680*y^4*z^3 + 5760*y^4*z^2 - 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] ], "target_shift": [ -18, -54, -43 ], "inverse_recipe": { "p": 3, "q": 4, "f": "x**3", "g": "x**4 + x**3", "h": "y*z", "polynomials": { "f": [ [ [ 3, 0, 0 ], 1 ] ], "g": [ [ [ 4, 0, 0 ], 1 ], [ [ 3, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ] ] }, "h_multiplier": 3 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "Maybe" ], "seconds": [ 1.1403530829993542 ], "surface_features": [ 3.6914020087486246, 4.603966695368848, 3.400624321661767, 0.0, 3.8191332072770154, 4949 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C192", "pair": 192, "family": "counterexample", "mapping": "F(x,y,z)=(8*x^7 + 52*x^6*y + 44*x^6*z + 116*x^6 + 138*x^5*y^2 + 240*x^5*y*z + 576*x^5*y + 102*x^5*z^2 + 516*x^5*z + 570*x^5 + 195*x^4*y^3 + 519*x^4*y^2*z + 1173*x^4*y^2 + 453*x^4*y*z^2 + 2118*x^4*y*z + 2229*x^4*y + 129*x^4*z^3 + 945*x^4*z^2 + 2055*x^4*z + 1318*x^4 + 159*x^3*y^4 + 573*x^3*y^3*z + 1251*x^3*y^3 + 765*x^3*y^2*z^2 + 3414*x^3*y^2*z + 3444*x^3*y^2 + 447*x^3*y*z^3 + 3075*x^3*y*z^2 + 6363*x^3*y*z + 3978*x^3*y + 96*x^3*z^4 + 912*x^3*z^3 + 2919*x^3*z^2 + 3706*x^3*z + 1699*x^3 + 75*x^2*y^5 + 342*x^2*y^4*z + 735*x^2*y^4 + 618*x^2*y^3*z^2 + 2694*x^2*y^3*z + 2622*x^2*y^3 + 552*x^2*y^2*z^3 + 3672*x^2*y^2*z^2 + 7287*x^2*y^2*z + 4452*x^2*y^2 + 243*x^2*y*z^4 + 2202*x^2*y*z^3 + 6708*x^2*y*z^2 + 8295*x^2*y*z + 3744*x^2*y + 42*x^2*z^5 + 489*x^2*z^4 + 2043*x^2*z^3 + 3843*x^2*z^2 + 3495*x^2*z + 1270*x^2 + 19*x*y^6 + 105*x*y^5*z + 225*x*y^5 + 240*x*y^4*z^2 + 1038*x*y^4*z + 981*x*y^4 + 290*x*y^3*z^3 + 1902*x*y^3*z^2 + 3648*x*y^3*z + 2186*x*y^3 + 195*x*y^2*z^4 + 1728*x*y^2*z^3 + 5058*x*y^2*z^2 + 6117*x*y^2*z + 2721*x*y^2 + 69*x*y*z^5 + 777*x*y*z^4 + 3096*x*y*z^3 + 5676*x*y*z^2 + 5076*x*y*z + 1822*x*y + 10*x*z^6 + 138*x*z^5 + 705*x*z^4 + 1745*x*z^3 + 2355*x*z^2 + 1693*x*z + 513*x + 2*y^7 + 13*y^6*z + 28*y^6 + 36*y^5*z^2 + 156*y^5*z + 144*y^5 + 55*y^4*z^3 + 360*y^4*z^2 + 672*y^4*z + 396*y^4 + 50*y^3*z^4 + 440*y^3*z^3 + 1248*y^3*z^2 + 1481*y^3*z + 650*y^3 + 27*y^2*z^5 + 300*y^2*z^4 + 1152*y^2*z^3 + 2067*y^2*z^2 + 1821*y^2*z + 649*y^2 + 8*y*z^6 + 108*y*z^5 + 528*y*z^4 + 1275*y*z^3 + 1692*y*z^2 + 1205*y*z + 366*y + z^7 + 16*z^6 + 96*z^5 + 293*z^4 + 521*z^3 + 556*z^2 + 335*z + 1, -8*x^7 - 52*x^6*y - 44*x^6*z - 104*x^6 - 138*x^5*y^2 - 240*x^5*y*z - 504*x^5*y - 102*x^5*z^2 - 456*x^5*z - 438*x^5 - 195*x^4*y^3 - 519*x^4*y^2*z - 1002*x^4*y^2 - 453*x^4*y*z^2 - 1824*x^4*y*z - 1683*x^4*y - 129*x^4*z^3 - 822*x^4*z^2 - 1557*x^4*z - 930*x^4 - 159*x^3*y^4 - 573*x^3*y^3*z - 1044*x^3*y^3 - 765*x^3*y^2*z^2 - 2868*x^3*y^2*z - 2556*x^3*y^2 - 447*x^3*y*z^3 - 2604*x^3*y*z^2 - 4737*x^3*y*z - 2764*x^3*y - 96*x^3*z^4 - 780*x^3*z^3 - 2181*x^3*z^2 - 2580*x^3*z - 1125*x^3 - 75*x^2*y^5 - 342*x^2*y^4*z - 600*x^2*y^4 - 618*x^2*y^3*z^2 - 2211*x^2*y^3*z - 1914*x^2*y^3 - 552*x^2*y^2*z^3 - 3033*x^2*y^2*z^2 - 5334*x^2*y^2*z - 3045*x^2*y^2 - 243*x^2*y*z^4 - 1833*x^2*y*z^3 - 4926*x^2*y*z^2 - 5685*x^2*y*z - 2436*x^2*y - 42*x^2*z^5 - 411*x^2*z^4 - 1506*x^2*z^3 - 2640*x^2*z^2 - 2277*x^2*z - 793*x^2 - 19*x*y^6 - 105*x*y^5*z - 180*x*y^5 - 240*x*y^4*z^2 - 834*x*y^4*z - 705*x*y^4 - 290*x*y^3*z^3 - 1536*x*y^3*z^2 - 2628*x*y^3*z - 1471*x*y^3 - 195*x*y^2*z^4 - 1404*x*y^2*z^3 - 3654*x*y^2*z^2 - 4125*x*y^2*z - 1737*x*y^2 - 69*x*y*z^5 - 636*x*y*z^4 - 2244*x*y*z^3 - 3837*x*y*z^2 - 3246*x*y*z - 1111*x*y - 10*x*z^6 - 114*x*z^5 - 513*x*z^4 - 1183*x*z^3 - 1509*x*z^2 - 1033*x*z - 305*x - 2*y^7 - 13*y^6*z - 22*y^6 - 36*y^5*z^2 - 123*y^5*z - 102*y^5 - 55*y^4*z^3 - 285*y^4*z^2 - 477*y^4*z - 262*y^4 - 50*y^3*z^4 - 350*y^3*z^3 - 888*y^3*z^2 - 982*y^3*z - 406*y^3 - 27*y^2*z^5 - 240*y^2*z^4 - 822*y^2*z^3 - 1374*y^2*z^2 - 1140*y^2*z - 385*y^2 - 8*y*z^6 - 87*y*z^5 - 378*y*z^4 - 850*y*z^3 - 1062*y*z^2 - 716*y*z - 212*y - z^7 - 13*z^6 - 69*z^5 - 196*z^4 - 328*z^3 - 331*z^2 - 194*z - 1, -8*x^7 - 52*x^6*y - 44*x^6*z - 104*x^6 - 138*x^5*y^2 - 240*x^5*y*z - 504*x^5*y - 102*x^5*z^2 - 456*x^5*z - 438*x^5 - 195*x^4*y^3 - 519*x^4*y^2*z - 1002*x^4*y^2 - 453*x^4*y*z^2 - 1824*x^4*y*z - 1683*x^4*y - 129*x^4*z^3 - 822*x^4*z^2 - 1557*x^4*z - 931*x^4 - 159*x^3*y^4 - 573*x^3*y^3*z - 1044*x^3*y^3 - 765*x^3*y^2*z^2 - 2868*x^3*y^2*z - 2556*x^3*y^2 - 447*x^3*y*z^3 - 2604*x^3*y*z^2 - 4737*x^3*y*z - 2769*x^3*y - 96*x^3*z^4 - 780*x^3*z^3 - 2181*x^3*z^2 - 2584*x^3*z - 1135*x^3 - 75*x^2*y^5 - 342*x^2*y^4*z - 600*x^2*y^4 - 618*x^2*y^3*z^2 - 2211*x^2*y^3*z - 1914*x^2*y^3 - 552*x^2*y^2*z^3 - 3033*x^2*y^2*z^2 - 5334*x^2*y^2*z - 3054*x^2*y^2 - 243*x^2*y*z^4 - 1833*x^2*y*z^3 - 4926*x^2*y*z^2 - 5700*x^2*y*z - 2466*x^2*y - 42*x^2*z^5 - 411*x^2*z^4 - 1506*x^2*z^3 - 2646*x^2*z^2 - 2304*x^2*z - 814*x^2 - 19*x*y^6 - 105*x*y^5*z - 180*x*y^5 - 240*x*y^4*z^2 - 834*x*y^4*z - 705*x*y^4 - 290*x*y^3*z^3 - 1536*x*y^3*z^2 - 2628*x*y^3*z - 1478*x*y^3 - 195*x*y^2*z^4 - 1404*x*y^2*z^3 - 3654*x*y^2*z^2 - 4143*x*y^2*z - 1767*x*y^2 - 69*x*y*z^5 - 636*x*y*z^4 - 2244*x*y*z^3 - 3852*x*y*z^2 - 3300*x*y*z - 1150*x*y - 10*x*z^6 - 114*x*z^5 - 513*x*z^4 - 1187*x*z^3 - 1533*x*z^2 - 1069*x*z - 319*x - 2*y^7 - 13*y^6*z - 22*y^6 - 36*y^5*z^2 - 123*y^5*z - 102*y^5 - 55*y^4*z^3 - 285*y^4*z^2 - 477*y^4*z - 264*y^4 - 50*y^3*z^4 - 350*y^3*z^3 - 888*y^3*z^2 - 989*y^3*z - 416*y^3 - 27*y^2*z^5 - 240*y^2*z^4 - 822*y^2*z^3 - 1383*y^2*z^2 - 1167*y^2*z - 403*y^2 - 8*y*z^6 - 87*y*z^5 - 378*y*z^4 - 855*y*z^3 - 1086*y*z^2 - 749*y*z - 224*y - z^7 - 13*z^6 - 69*z^5 - 197*z^4 - 335*z^3 - 346*z^2 - 205*z - 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ 1, 1, 1 ], [ 2, 1, 1 ], [ 1, 2, 1 ] ], "source_shift": [ 1, 1, 1 ], "target_matrix": [ [ 1, 2, 1 ], [ -1, -1, -2 ], [ -1, -1, -1 ] ], "target_shift": [ -87, 51, 53 ], "collision_preimages": [ [ "0", "-1/4", "-3/4" ], [ "-5/2", "11/2", "-3" ], [ "5/2", "15/2", "-12" ] ], "collision_image": [ "-349/4", "205/4", "213/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0392098629999964 ], "surface_features": [ 3.0669219720939966, 3.9188163903603797, 2.7505083948513462, 0.0, 3.157824340434864, 4736 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B202", "pair": 202, "family": "bijection", "mapping": "F(x,y,z)=(2187*x^7 - 10206*x^6*y - 10206*x^6*z + 5346*x^6 + 20412*x^5*y^2 + 40824*x^5*y*z - 21708*x^5*y + 20412*x^5*z^2 - 21465*x^5*z + 5994*x^5 - 22680*x^4*y^3 - 68040*x^4*y^2*z + 36720*x^4*y^2 - 68040*x^4*y*z^2 + 72630*x^4*y*z - 20520*x^4*y - 22680*x^4*z^3 + 35910*x^4*z^2 - 20115*x^4*z + 3996*x^4 + 15120*x^3*y^4 + 60480*x^3*y^3*z - 33120*x^3*y^3 + 90720*x^3*y^2*z^2 - 98280*x^3*y^2*z + 28080*x^3*y^2 + 60480*x^3*y*z^3 - 97200*x^3*y*z^2 + 55080*x^3*y*z - 11016*x^3*y + 15120*x^3*z^4 - 32040*x^3*z^3 + 27000*x^3*z^2 - 10746*x^3*z + 1710*x^3 - 6048*x^2*y^5 - 30240*x^2*y^4*z + 16800*x^2*y^4 - 60480*x^2*y^3*z^2 + 66480*x^2*y^3*z - 19200*x^2*y^3 - 60480*x^2*y^2*z^3 + 98640*x^2*y^2*z^2 - 56520*x^2*y^2*z + 11376*x^2*y^2 - 30240*x^2*y*z^4 + 65040*x^2*y*z^3 - 55440*x^2*y*z^2 + 22212*x^2*y*z - 3549*x^2*y - 6048*x^2*z^5 + 16080*x^2*z^4 - 18120*x^2*z^3 + 10836*x^2*z^2 - 3450*x^2*z + 485*x^2 + 1344*x*y^6 + 8064*x*y^5*z - 4544*x*y^5 + 20160*x*y^4*z^2 - 22480*x*y^4*z + 6560*x*y^4 + 26880*x*y^3*z^3 - 44480*x*y^3*z^2 + 25760*x*y^3*z - 5216*x*y^3 + 20160*x*y^2*z^4 - 44000*x*y^2*z^3 + 37920*x*y^2*z^2 - 15288*x*y^2*z + 2452*x*y^2 + 8064*x*y*z^5 - 21760*x*y*z^4 + 24800*x*y*z^3 - 14928*x*y*z^2 + 4772*x*y*z - 675*x*y + 1344*x*z^6 - 4304*x*z^5 + 6080*x*z^4 - 4856*x*z^3 + 2320*x*z^2 - 652*x*z + 93*x - 128*y^7 - 896*y^6*z + 512*y^6 - 2688*y^5*z^2 + 3040*y^5*z - 896*y^5 - 4480*y^4*z^3 + 7520*y^4*z^2 - 4400*y^4*z + 896*y^4 - 4480*y^3*z^4 + 9920*y^3*z^3 - 8640*y^3*z^2 + 3504*y^3*z - 564*y^3 - 2688*y^2*z^5 + 7360*y^2*z^4 - 8480*y^2*z^3 + 5136*y^2*z^2 - 1648*y^2*z + 234*y^2 - 896*y*z^6 + 2912*y*z^5 - 4160*y*z^4 + 3344*y*z^3 - 1604*y*z^2 + 453*y*z - 65*y - 128*z^7 + 480*z^6 - 816*z^5 + 816*z^4 - 520*z^3 + 219*z^2 - 62*z + 1, 2187*x^7 - 10206*x^6*y - 10206*x^6*z + 5346*x^6 + 20412*x^5*y^2 + 40824*x^5*y*z - 21708*x^5*y + 20412*x^5*z^2 - 21465*x^5*z + 5994*x^5 - 22680*x^4*y^3 - 68040*x^4*y^2*z + 36720*x^4*y^2 - 68040*x^4*y*z^2 + 72630*x^4*y*z - 20520*x^4*y - 22680*x^4*z^3 + 35910*x^4*z^2 - 20115*x^4*z + 3996*x^4 + 15120*x^3*y^4 + 60480*x^3*y^3*z - 33120*x^3*y^3 + 90720*x^3*y^2*z^2 - 98280*x^3*y^2*z + 28080*x^3*y^2 + 60480*x^3*y*z^3 - 97200*x^3*y*z^2 + 55080*x^3*y*z - 11016*x^3*y + 15120*x^3*z^4 - 32040*x^3*z^3 + 27000*x^3*z^2 - 10746*x^3*z + 1710*x^3 - 6048*x^2*y^5 - 30240*x^2*y^4*z + 16800*x^2*y^4 - 60480*x^2*y^3*z^2 + 66480*x^2*y^3*z - 19200*x^2*y^3 - 60480*x^2*y^2*z^3 + 98640*x^2*y^2*z^2 - 56520*x^2*y^2*z + 11376*x^2*y^2 - 30240*x^2*y*z^4 + 65040*x^2*y*z^3 - 55440*x^2*y*z^2 + 22212*x^2*y*z - 3549*x^2*y - 6048*x^2*z^5 + 16080*x^2*z^4 - 18120*x^2*z^3 + 10836*x^2*z^2 - 3450*x^2*z + 467*x^2 + 1344*x*y^6 + 8064*x*y^5*z - 4544*x*y^5 + 20160*x*y^4*z^2 - 22480*x*y^4*z + 6560*x*y^4 + 26880*x*y^3*z^3 - 44480*x*y^3*z^2 + 25760*x*y^3*z - 5216*x*y^3 + 20160*x*y^2*z^4 - 44000*x*y^2*z^3 + 37920*x*y^2*z^2 - 15288*x*y^2*z + 2452*x*y^2 + 8064*x*y*z^5 - 21760*x*y*z^4 + 24800*x*y*z^3 - 14928*x*y*z^2 + 4772*x*y*z - 651*x*y + 1344*x*z^6 - 4304*x*z^5 + 6080*x*z^4 - 4856*x*z^3 + 2320*x*z^2 - 628*x*z + 79*x - 128*y^7 - 896*y^6*z + 512*y^6 - 2688*y^5*z^2 + 3040*y^5*z - 896*y^5 - 4480*y^4*z^3 + 7520*y^4*z^2 - 4400*y^4*z + 896*y^4 - 4480*y^3*z^4 + 9920*y^3*z^3 - 8640*y^3*z^2 + 3504*y^3*z - 564*y^3 - 2688*y^2*z^5 + 7360*y^2*z^4 - 8480*y^2*z^3 + 5136*y^2*z^2 - 1648*y^2*z + 226*y^2 - 896*y*z^6 + 2912*y*z^5 - 4160*y*z^4 + 3344*y*z^3 - 1604*y*z^2 + 437*y*z - 53*y - 128*z^7 + 480*z^6 - 816*z^5 + 816*z^4 - 520*z^3 + 211*z^2 - 52*z + 1, 6561*x^7 - 30618*x^6*y - 30618*x^6*z + 16038*x^6 + 61236*x^5*y^2 + 122472*x^5*y*z - 65124*x^5*y + 61236*x^5*z^2 - 64395*x^5*z + 17739*x^5 - 68040*x^4*y^3 - 204120*x^4*y^2*z + 110160*x^4*y^2 - 204120*x^4*y*z^2 + 217890*x^4*y*z - 60750*x^4*y - 68040*x^4*z^3 + 107730*x^4*z^2 - 59535*x^4*z + 11583*x^4 + 45360*x^3*y^4 + 181440*x^3*y^3*z - 99360*x^3*y^3 + 272160*x^3*y^2*z^2 - 294840*x^3*y^2*z + 83160*x^3*y^2 + 181440*x^3*y*z^3 - 291600*x^3*y*z^2 + 163080*x^3*y*z - 31968*x^3*y + 45360*x^3*z^4 - 96120*x^3*z^3 + 79920*x^3*z^2 - 31158*x^3*z + 4860*x^3 - 18144*x^2*y^5 - 90720*x^2*y^4*z + 50400*x^2*y^4 - 181440*x^2*y^3*z^2 + 199440*x^2*y^3*z - 56880*x^2*y^3 - 181440*x^2*y^2*z^3 + 295920*x^2*y^2*z^2 - 167400*x^2*y^2*z + 33048*x^2*y^2 - 90720*x^2*y*z^4 + 195120*x^2*y*z^3 - 164160*x^2*y*z^2 + 64476*x^2*y*z - 10107*x^2*y - 18144*x^2*z^5 + 48240*x^2*z^4 - 53640*x^2*z^3 + 31428*x^2*z^2 - 9810*x^2*z + 1374*x^2 + 4032*x*y^6 + 24192*x*y^5*z - 13632*x*y^5 + 60480*x*y^4*z^2 - 67440*x*y^4*z + 19440*x*y^4 + 80640*x*y^3*z^3 - 133440*x*y^3*z^2 + 76320*x*y^3*z - 15168*x*y^3 + 60480*x*y^2*z^4 - 132000*x*y^2*z^3 + 112320*x*y^2*z^2 - 44424*x*y^2*z + 6996*x*y^2 + 24192*x*y*z^5 - 65280*x*y*z^4 + 73440*x*y*z^3 - 43344*x*y*z^2 + 13596*x*y*z - 1917*x*y + 4032*x*z^6 - 12912*x*z^5 + 18000*x*z^4 - 14088*x*z^3 + 6600*x*z^2 - 1848*x*z + 270*x - 384*y^7 - 2688*y^6*z + 1536*y^6 - 8064*y^5*z^2 + 9120*y^5*z - 2656*y^5 - 13440*y^4*z^3 + 22560*y^4*z^2 - 13040*y^4*z + 2608*y^4 - 13440*y^3*z^4 + 29760*y^3*z^3 - 25600*y^3*z^2 + 10192*y^3*z - 1612*y^3 - 8064*y^2*z^5 + 22080*y^2*z^4 - 25120*y^2*z^3 + 14928*y^2*z^2 - 4704*y^2*z + 666*y^2 - 2688*y*z^6 + 8736*y*z^5 - 12320*y*z^4 + 9712*y*z^3 - 4572*y*z^2 + 1287*y*z - 192*y - 384*z^7 + 1440*z^6 - 2416*z^5 + 2368*z^4 - 1480*z^3 + 621*z^2 - 181*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ 3, -2, -2 ], [ 2, -1, -1 ], [ 1, -2, -1 ] ], "source_shift": [ 1, 2, 1 ], "target_matrix": [ [ 1, 1, 1 ], [ 1, 1, 2 ], [ 3, 2, 2 ] ], "target_shift": [ -10, -6, -32 ], "inverse_recipe": { "p": 5, "q": 2, "f": "x**5 + x**2", "g": "x**2", "h": "y*z + z", "polynomials": { "f": [ [ [ 5, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "g": [ [ [ 2, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 0, 1 ], 1 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0937748149990512 ], "surface_features": [ 4.502886247568569, 5.471174318307337, 4.1309124421074745, 0.0, 4.687308473923641, 5204 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C072", "pair": 72, "family": "counterexample", "mapping": "F(x,y,z)=(-2*x^7 - 19*x^6*y + 13*x^6*z + 28*x^6 - 75*x^5*y^2 + 105*x^5*y*z + 225*x^5*y - 36*x^5*z^2 - 156*x^5*z - 165*x^5 - 159*x^4*y^3 + 342*x^4*y^2*z + 729*x^4*y^2 - 240*x^4*y*z^2 - 1035*x^4*y*z - 1083*x^4*y + 55*x^4*z^3 + 360*x^4*z^2 + 765*x^4*z + 530*x^4 - 195*x^3*y^4 + 573*x^3*y^3*z + 1215*x^3*y^3 - 618*x^3*y^2*z^2 - 2652*x^3*y^2*z - 2742*x^3*y^2 + 290*x^3*y*z^3 + 1890*x^3*y*z^2 + 3975*x^3*y*z + 2707*x^3*y - 50*x^3*z^4 - 440*x^3*z^3 - 1410*x^3*z^2 - 1959*x^3*z - 1000*x^3 - 138*x^2*y^5 + 519*x^2*y^4*z + 1095*x^2*y^4 - 765*x^2*y^3*z^2 - 3267*x^2*y^3*z - 3336*x^2*y^3 + 552*x^2*y^2*z^3 + 3582*x^2*y^2*z^2 + 7449*x^2*y^2*z + 4977*x^2*y^2 - 195*x^2*y*z^4 - 1710*x^2*y*z^3 - 5427*x^2*y*z^2 - 7413*x^2*y*z - 3690*x^2*y + 27*x^2*z^5 + 300*x^2*z^4 + 1290*x^2*z^3 + 2697*x^2*z^2 + 2754*x^2*z + 1102*x^2 - 52*x*y^6 + 240*x*y^5*z + 504*x*y^5 - 453*x*y^4*z^2 - 1926*x*y^4*z - 1944*x*y^4 + 447*x*y^3*z^3 + 2889*x*y^3*z^2 + 5940*x*y^3*z + 3893*x*y^3 - 243*x*y^2*z^4 - 2124*x*y^2*z^3 - 6672*x*y^2*z^2 - 8946*x*y^2*z - 4332*x*y^2 + 69*x*y*z^5 + 765*x*y*z^4 + 3261*x*y*z^3 + 6705*x*y*z^2 + 6675*x*y*z + 2576*x*y - 8*x*z^6 - 108*x*z^5 - 585*x*z^4 - 1637*x*z^3 - 2508*x*z^2 - 2000*x*z - 650*x - 8*y^7 + 44*y^6*z + 92*y^6 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7449*x^2*y^2*z + 4947*x^2*y^2 - 195*x^2*y*z^4 - 1710*x^2*y*z^3 - 5427*x^2*y*z^2 - 7380*x^2*y*z - 3615*x^2*y + 27*x^2*z^5 + 300*x^2*z^4 + 1290*x^2*z^3 + 2688*x^2*z^2 + 2712*x^2*z + 1057*x^2 - 52*x*y^6 + 240*x*y^5*z + 504*x*y^5 - 453*x*y^4*z^2 - 1926*x*y^4*z - 1944*x*y^4 + 447*x*y^3*z^3 + 2889*x*y^3*z^2 + 5940*x*y^3*z + 3865*x*y^3 - 243*x*y^2*z^4 - 2124*x*y^2*z^3 - 6672*x*y^2*z^2 - 8898*x*y^2*z - 4224*x*y^2 + 69*x*y*z^5 + 765*x*y*z^4 + 3261*x*y*z^3 + 6678*x*y*z^2 + 6549*x*y*z + 2444*x*y - 8*x*z^6 - 108*x*z^5 - 585*x*z^4 - 1632*x*z^3 - 2472*x*z^2 - 1922*x*z - 600*x - 8*y^7 + 44*y^6*z + 92*y^6 - 102*y^5*z^2 - 432*y^5*z - 432*y^5 + 129*y^4*z^3 + 831*y^4*z^2 + 1692*y^4*z + 1082*y^4 - 96*y^3*z^4 - 837*y^3*z^3 - 2604*y^3*z^2 - 3409*y^3*z - 1574*y^3 + 42*y^2*z^5 + 465*y^2*z^4 + 1965*y^2*z^3 + 3951*y^2*z^2 + 3762*y^2*z + 1348*y^2 - 10*y*z^6 - 135*y*z^5 - 726*y*z^4 - 1992*y*z^3 - 2934*y*z^2 - 2186*y*z - 648*y + z^7 + 16*z^6 + 105*z^5 + 368*z^4 + 744*z^3 + 865*z^2 + 535*z + 1, -4*x^7 - 38*x^6*y + 26*x^6*z + 50*x^6 - 150*x^5*y^2 + 210*x^5*y*z + 399*x^5*y - 72*x^5*z^2 - 279*x^5*z - 270*x^5 - 318*x^4*y^3 + 684*x^4*y^2*z + 1284*x^4*y^2 - 480*x^4*y*z^2 - 1839*x^4*y*z - 1755*x^4*y + 110*x^4*z^3 + 645*x^4*z^2 + 1254*x^4*z + 813*x^4 - 390*x^3*y^4 + 1146*x^3*y^3*z + 2127*x^3*y^3 - 1236*x^3*y^2*z^2 - 4683*x^3*y^2*z - 4404*x^3*y^2 + 580*x^3*y*z^3 + 3366*x^3*y*z^2 + 6456*x^3*y*z + 4110*x^3*y - 100*x^3*z^4 - 790*x^3*z^3 - 2316*x^3*z^2 - 3010*x^3*z - 1464*x^3 - 276*x^2*y^5 + 1038*x^2*y^4*z + 1908*x^2*y^4 - 1530*x^2*y^3*z^2 - 5739*x^2*y^3*z - 5319*x^2*y^3 + 1104*x^2*y^2*z^3 + 6345*x^2*y^2*z^2 + 12000*x^2*y^2*z + 7491*x^2*y^2 - 390*x^2*y*z^4 - 3054*x^2*y*z^3 - 8838*x^2*y*z^2 - 11280*x^2*y*z - 5352*x^2*y + 54*x^2*z^5 + 540*x^2*z^4 + 2124*x^2*z^3 + 4152*x^2*z^2 + 4038*x^2*z + 1562*x^2 - 104*x*y^6 + 480*x*y^5*z + 876*x*y^5 - 906*x*y^4*z^2 - 3372*x*y^4*z - 3084*x*y^4 + 894*x*y^3*z^3 + 5097*x*y^3*z^2 + 9510*x*y^3*z + 5822*x*y^3 - 486*x*y^2*z^4 - 3777*x*y^2*z^3 - 10788*x*y^2*z^2 - 13506*x*y^2*z - 6240*x*y^2 + 138*x*y*z^5 + 1371*x*y*z^4 + 5328*x*y*z^3 + 10230*x*y*z^2 + 9702*x*y*z + 3625*x*y - 16*x*z^6 - 195*x*z^5 - 966*x*z^4 - 2526*x*z^3 - 3684*x*z^2 - 2839*x*z - 902*x - 16*y^7 + 88*y^6*z + 160*y^6 - 204*y^5*z^2 - 756*y^5*z - 684*y^5 + 258*y^4*z^3 + 1464*y^4*z^2 + 2700*y^4*z + 1624*y^4 - 192*y^3*z^4 - 1485*y^3*z^3 - 4191*y^3*z^2 - 5150*y^3*z - 2320*y^3 + 84*y^2*z^5 + 831*y^2*z^4 + 3192*y^2*z^3 + 6015*y^2*z^2 + 5556*y^2*z + 2006*y^2 - 20*y*z^6 - 243*y*z^5 - 1191*y*z^4 - 3060*y*z^3 - 4350*y*z^2 - 3235*y*z - 983*y + 2*z^7 + 29*z^6 + 174*z^5 + 571*z^4 + 1110*z^3 + 1277*z^2 + 802*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -1, -2, 1 ], [ -1, -1, 1 ], [ -2, -1, 1 ] ], "source_shift": [ 2, 1, 1 ], "target_matrix": [ [ 1, 1, 1 ], [ 1, 1, 2 ], [ 2, 1, 1 ] ], "target_shift": [ -155, -139, -212 ], "collision_preimages": [ [ "1/4", "1", "1/4" ], [ "-8", "-3/2", "-12" ], [ "-5", "7/2", "-1" ] ], "collision_image": [ "-621/4", "-557/4", "-425/2" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0533338880013616 ], "surface_features": [ 3.2645686091534305, 4.130526745384164, 2.929929560084588, 0.0, 3.3646695079030065, 4792 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B052", "pair": 52, "family": "bijection", "mapping": "F(x,y,z)=(-128*x^7 + 896*x^6*y - 1344*x^6*z + 512*x^6 - 2688*x^5*y^2 + 8064*x^5*y*z - 3072*x^5*y - 6048*x^5*z^2 + 4608*x^5*z - 912*x^5 + 4480*x^4*y^3 - 20160*x^4*y^2*z + 7680*x^4*y^2 + 30240*x^4*y*z^2 - 23040*x^4*y*z + 4560*x^4*y - 15120*x^4*z^3 + 17280*x^4*z^2 - 6848*x^4*z + 952*x^4 - 4480*x^3*y^4 + 26880*x^3*y^3*z - 10240*x^3*y^3 - 60480*x^3*y^2*z^2 + 46080*x^3*y^2*z - 9120*x^3*y^2 + 60480*x^3*y*z^3 - 69120*x^3*y*z^2 + 27392*x^3*y*z - 3800*x^3*y - 22680*x^3*z^4 + 34560*x^3*z^3 - 20568*x^3*z^2 + 5716*x^3*z - 640*x^3 + 2688*x^2*y^5 - 20160*x^2*y^4*z + 7680*x^2*y^4 + 60480*x^2*y^3*z^2 - 46080*x^2*y^3*z + 9120*x^2*y^3 - 90720*x^2*y^2*z^3 + 103680*x^2*y^2*z^2 - 41088*x^2*y^2*z + 5688*x^2*y^2 + 68040*x^2*y*z^4 - 103680*x^2*y*z^3 + 61704*x^2*y*z^2 - 17112*x^2*y*z + 1908*x^2*y - 20412*x^2*z^5 + 38880*x^2*z^4 - 30888*x^2*z^3 + 12870*x^2*z^2 - 2876*x^2*z + 278*x^2 - 896*x*y^6 + 8064*x*y^5*z - 3072*x*y^5 - 30240*x*y^4*z^2 + 23040*x*y^4*z - 4560*x*y^4 + 60480*x*y^3*z^3 - 69120*x*y^3*z^2 + 27392*x*y^3*z - 3784*x*y^3 - 68040*x*y^2*z^4 + 103680*x*y^2*z^3 - 61704*x*y^2*z^2 + 17076*x*y^2*z - 1896*x*y^2 + 40824*x*y*z^5 - 77760*x*y*z^4 + 61776*x*y*z^3 - 25686*x*y*z^2 + 5716*x*y*z - 549*x*y - 10206*x*z^6 + 23328*x*z^5 - 23193*x*z^4 + 12879*x*z^3 - 4308*x*z^2 + 831*x*z - 73*x + 128*y^7 - 1344*y^6*z + 512*y^6 + 6048*y^5*z^2 - 4608*y^5*z + 912*y^5 - 15120*y^4*z^3 + 17280*y^4*z^2 - 6848*y^4*z + 944*y^4 + 22680*y^3*z^4 - 34560*y^3*z^3 + 20568*y^3*z^2 - 5680*y^3*z + 628*y^3 - 20412*y^2*z^5 + 38880*y^2*z^4 - 30888*y^2*z^3 + 12816*y^2*z^2 - 2840*y^2*z + 271*y^2 + 10206*y*z^6 - 23328*y*z^5 + 23193*y*z^4 - 12852*y*z^3 + 4281*y*z^2 - 820*y*z + 72*y - 2187*z^7 + 5832*z^6 - 6966*z^5 + 4833*z^4 - 2151*z^3 + 620*z^2 - 111*z + 1, 128*x^7 - 896*x^6*y + 1344*x^6*z - 512*x^6 + 2688*x^5*y^2 - 8064*x^5*y*z + 3072*x^5*y + 6048*x^5*z^2 - 4608*x^5*z + 912*x^5 - 4480*x^4*y^3 + 20160*x^4*y^2*z - 7680*x^4*y^2 - 30240*x^4*y*z^2 + 23040*x^4*y*z - 4560*x^4*y + 15120*x^4*z^3 - 17280*x^4*z^2 + 6848*x^4*z - 920*x^4 + 4480*x^3*y^4 - 26880*x^3*y^3*z + 10240*x^3*y^3 + 60480*x^3*y^2*z^2 - 46080*x^3*y^2*z + 9120*x^3*y^2 - 60480*x^3*y*z^3 + 69120*x^3*y*z^2 - 27392*x^3*y*z + 3672*x^3*y + 22680*x^3*z^4 - 34560*x^3*z^3 + 20568*x^3*z^2 - 5524*x^3*z + 552*x^3 - 2688*x^2*y^5 + 20160*x^2*y^4*z - 7680*x^2*y^4 - 60480*x^2*y^3*z^2 + 46080*x^2*y^3*z - 9120*x^2*y^3 + 90720*x^2*y^2*z^3 - 103680*x^2*y^2*z^2 + 41088*x^2*y^2*z - 5496*x^2*y^2 - 68040*x^2*y*z^4 + 103680*x^2*y*z^3 - 61704*x^2*y*z^2 + 16536*x^2*y*z - 1644*x^2*y + 20412*x^2*z^5 - 38880*x^2*z^4 + 30888*x^2*z^3 - 12438*x^2*z^2 + 2480*x^2*z - 186*x^2 + 896*x*y^6 - 8064*x*y^5*z + 3072*x*y^5 + 30240*x*y^4*z^2 - 23040*x*y^4*z + 4560*x*y^4 - 60480*x*y^3*z^3 + 69120*x*y^3*z^2 - 27392*x*y^3*z + 3656*x*y^3 + 68040*x*y^2*z^4 - 103680*x*y^2*z^3 + 61704*x*y^2*z^2 - 16500*x*y^2*z + 1632*x*y^2 - 40824*x*y*z^5 + 77760*x*y*z^4 - 61776*x*y*z^3 + 24822*x*y*z^2 - 4924*x*y*z + 365*x*y + 10206*x*z^6 - 23328*x*z^5 + 23193*x*z^4 - 12447*x*z^3 + 3714*x*z^2 - 555*x*z + 22*x - 128*y^7 + 1344*y^6*z - 512*y^6 - 6048*y^5*z^2 + 4608*y^5*z - 912*y^5 + 15120*y^4*z^3 - 17280*y^4*z^2 + 6848*y^4*z - 912*y^4 - 22680*y^3*z^4 + 34560*y^3*z^3 - 20568*y^3*z^2 + 5488*y^3*z - 540*y^3 + 20412*y^2*z^5 - 38880*y^2*z^4 + 30888*y^2*z^3 - 12384*y^2*z^2 + 2444*y^2*z - 179*y^2 - 10206*y*z^6 + 23328*y*z^5 - 23193*y*z^4 + 12420*y*z^3 - 3687*y*z^2 + 544*y*z - 23*y + 2187*z^7 - 5832*z^6 + 6966*z^5 - 4671*z^4 + 1854*z^3 - 413*z^2 + 38*z - 1, 128*x^7 - 896*x^6*y + 1344*x^6*z - 512*x^6 + 2688*x^5*y^2 - 8064*x^5*y*z + 3072*x^5*y + 6048*x^5*z^2 - 4608*x^5*z + 912*x^5 - 4480*x^4*y^3 + 20160*x^4*y^2*z - 7680*x^4*y^2 - 30240*x^4*y*z^2 + 23040*x^4*y*z - 4560*x^4*y + 15120*x^4*z^3 - 17280*x^4*z^2 + 6848*x^4*z - 952*x^4 + 4480*x^3*y^4 - 26880*x^3*y^3*z + 10240*x^3*y^3 + 60480*x^3*y^2*z^2 - 46080*x^3*y^2*z + 9120*x^3*y^2 - 60480*x^3*y*z^3 + 69120*x^3*y*z^2 - 27392*x^3*y*z + 3800*x^3*y + 22680*x^3*z^4 - 34560*x^3*z^3 + 20568*x^3*z^2 - 5716*x^3*z + 632*x^3 - 2688*x^2*y^5 + 20160*x^2*y^4*z - 7680*x^2*y^4 - 60480*x^2*y^3*z^2 + 46080*x^2*y^3*z - 9120*x^2*y^3 + 90720*x^2*y^2*z^3 - 103680*x^2*y^2*z^2 + 41088*x^2*y^2*z - 5688*x^2*y^2 - 68040*x^2*y*z^4 + 103680*x^2*y*z^3 - 61704*x^2*y*z^2 + 17112*x^2*y*z - 1884*x^2*y + 20412*x^2*z^5 - 38880*x^2*z^4 + 30888*x^2*z^3 - 12870*x^2*z^2 + 2840*x^2*z - 266*x^2 + 896*x*y^6 - 8064*x*y^5*z + 3072*x*y^5 + 30240*x*y^4*z^2 - 23040*x*y^4*z + 4560*x*y^4 - 60480*x*y^3*z^3 + 69120*x*y^3*z^2 - 27392*x*y^3*z + 3784*x*y^3 + 68040*x*y^2*z^4 - 103680*x*y^2*z^3 + 61704*x*y^2*z^2 - 17076*x*y^2*z + 1872*x*y^2 - 40824*x*y*z^5 + 77760*x*y*z^4 - 61776*x*y*z^3 + 25686*x*y*z^2 - 5644*x*y*z + 525*x*y + 10206*x*z^6 - 23328*x*z^5 + 23193*x*z^4 - 12879*x*z^3 + 4254*x*z^2 - 795*x*z + 66*x - 128*y^7 + 1344*y^6*z - 512*y^6 - 6048*y^5*z^2 + 4608*y^5*z - 912*y^5 + 15120*y^4*z^3 - 17280*y^4*z^2 + 6848*y^4*z - 944*y^4 - 22680*y^3*z^4 + 34560*y^3*z^3 - 20568*y^3*z^2 + 5680*y^3*z - 620*y^3 + 20412*y^2*z^5 - 38880*y^2*z^4 + 30888*y^2*z^3 - 12816*y^2*z^2 + 2804*y^2*z - 259*y^2 - 10206*y*z^6 + 23328*y*z^5 - 23193*y*z^4 + 12852*y*z^3 - 4227*y*z^2 + 784*y*z - 65*y + 2187*z^7 - 5832*z^6 + 6966*z^5 - 4833*z^4 + 2124*z^3 - 593*z^2 + 100*z - 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -2, 2, -3 ], [ -1, 1, -2 ], [ -2, 1, -1 ] ], "source_shift": [ 1, 2, 1 ], "target_matrix": [ [ 1, 2, 1 ], [ -1, -1, -2 ], [ -1, -1, -1 ] ], "target_shift": [ -11, -2, 8 ], "inverse_recipe": { "p": 3, "q": 4, "f": "x**3", "g": "x**4 + x**3 + x**2 + x", "h": "y*z", "polynomials": { "f": [ [ [ 3, 0, 0 ], 1 ] ], "g": [ [ [ 4, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ], [ [ 3, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0696580889998586 ], "surface_features": [ 4.303741224025618, 5.015694988526518, 3.959994838328416, 0.0, 4.391560263504985, 5154 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B113", "pair": 113, "family": "bijection", "mapping": "F(x,y,z)=(4*x^7 + 27*x^6*y + 27*x^6*z + 50*x^6 + 78*x^5*y^2 + 156*x^5*y*z + 288*x^5*y + 78*x^5*z^2 + 300*x^5*z + 252*x^5 + 125*x^4*y^3 + 375*x^4*y^2*z + 690*x^4*y^2 + 375*x^4*y*z^2 + 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"p": 2, "q": 1, "f": "x**2 + x", "g": "x", "h": "y**3*z + y*z + z**3 + z", "polynomials": { "f": [ [ [ 2, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "g": [ [ [ 1, 0, 0 ], 1 ] ], "h": [ [ [ 0, 3, 1 ], 1 ], [ [ 0, 0, 1 ], 1 ], [ [ 0, 1, 1 ], 1 ], [ [ 0, 0, 3 ], 1 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.036407360999874 ], "surface_features": [ 3.198374489829876, 3.923036668670786, 2.870111155364401, 0.0, 3.283591436432973, 4773 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B142", "pair": 142, "family": "bijection", "mapping": "F(x,y,z)=(-7*x^7 - 49*x^6*y - 49*x^6*z - 56*x^6 - 147*x^5*y^2 - 294*x^5*y*z - 336*x^5*y - 147*x^5*z^2 - 336*x^5*z - 217*x^5 - 245*x^4*y^3 - 735*x^4*y^2*z - 840*x^4*y^2 - 735*x^4*y*z^2 - 1680*x^4*y*z - 1085*x^4*y - 245*x^4*z^3 - 840*x^4*z^2 - 1078*x^4*z - 512*x^4 - 245*x^3*y^4 - 980*x^3*y^3*z - 1120*x^3*y^3 - 1470*x^3*y^2*z^2 - 3360*x^3*y^2*z - 2170*x^3*y^2 - 980*x^3*y*z^3 - 3360*x^3*y*z^2 - 4312*x^3*y*z - 2041*x^3*y - 245*x^3*z^4 - 1120*x^3*z^3 - 2142*x^3*z^2 - 2013*x^3*z - 786*x^3 - 147*x^2*y^5 - 735*x^2*y^4*z - 840*x^2*y^4 - 1470*x^2*y^3*z^2 - 3360*x^2*y^3*z - 2170*x^2*y^3 - 1470*x^2*y^2*z^3 - 5040*x^2*y^2*z^2 - 6468*x^2*y^2*z - 3051*x^2*y^2 - 735*x^2*y*z^4 - 3360*x^2*y*z^3 - 6426*x^2*y*z^2 - 6018*x^2*y*z - 2330*x^2*y - 147*x^2*z^5 - 840*x^2*z^4 - 2128*x^2*z^3 - 2967*x^2*z^2 - 2281*x^2*z - 811*x^2 - 49*x*y^6 - 294*x*y^5*z - 336*x*y^5 - 735*x*y^4*z^2 - 1680*x*y^4*z - 1085*x*y^4 - 980*x*y^3*z^3 - 3360*x*y^3*z^2 - 4312*x*y^3*z - 2027*x*y^3 - 735*x*y^2*z^4 - 3360*x*y^2*z^3 - 6426*x*y^2*z^2 - 5997*x*y^2*z - 2302*x*y^2 - 294*x*y*z^5 - 1680*x*y*z^4 - 4256*x*y*z^3 - 5913*x*y*z^2 - 4506*x*y*z - 1566*x*y - 49*x*z^6 - 336*x*z^5 - 1057*x*z^4 - 1943*x*z^3 - 2204*x*z^2 - 1503*x*z - 470*x - 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"target_shift": [ 110, 223, -114 ], "inverse_recipe": { "p": 4, "q": 3, "f": "x**4 + x**2 + x", "g": "x**3 + x**2", "h": "y*z + y", "polynomials": { "f": [ [ [ 4, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "g": [ [ [ 3, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 1, 0 ], 1 ] ] }, "h_multiplier": 7 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.008210156000132 ], "surface_features": [ 3.2939018646802305, 4.111800006898345, 3.0354297381845483, 0.0, 3.367634369706867, 4841 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C127", "pair": 127, "family": "counterexample", "mapping": "F(x,y,z)=(-216*x^7 - 972*x^6*y - 1188*x^6*z - 2376*x^6 - 1854*x^5*y^2 - 4536*x^5*y*z - 8964*x^5*y - 2754*x^5*z^2 - 11124*x^5*z - 9756*x^5 - 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4633*z^4 - 6687*z^3 - 5347*z^2 - 2179*z - 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ 3, 2, 3 ], [ 2, 1, 1 ], [ 1, 1, 1 ] ], "source_shift": [ 1, 2, 2 ], "target_matrix": [ [ -1, -2, -1 ], [ 2, 2, 3 ], [ -1, -1, -1 ] ], "target_shift": [ 519, -681, 343 ], "collision_preimages": [ [ "1/4", "-23/4", "13/4" ], [ "-8", "27/2", "-1" ], [ "-5", "31/2", "-6" ] ], "collision_image": [ "2077/4", "-1363/2", "1373/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0442964670000947 ], "surface_features": [ 4.3166643005710466, 5.204857656429958, 3.984842231405276, 0.0, 4.434362723522294, 5162 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C085", "pair": 85, "family": "counterexample", "mapping": "F(x,y,z)=(-8*x^7 - 44*x^6*y + 68*x^6*z + 116*x^6 - 102*x^5*y^2 + 312*x^5*y*z + 516*x^5*y - 234*x^5*z^2 - 792*x^5*z - 612*x^5 - 129*x^4*y^3 + 585*x^4*y^2*z + 945*x^4*y^2 - 867*x^4*y*z^2 - 2862*x^4*y*z - 2220*x^4*y + 419*x^4*z^3 + 2121*x^4*z^2 + 3408*x^4*z + 1666*x^4 - 96*x^3*y^4 + 573*x^3*y^3*z + 912*x^3*y^3 - 1257*x^3*y^2*z^2 - 4089*x^3*y^2*z - 3177*x^3*y^2 + 1199*x^3*y*z^3 + 5982*x^3*y*z^2 + 9594*x^3*y*z + 4774*x^3*y - 419*x^3*z^4 - 2849*x^3*z^3 - 7077*x^3*z^2 - 7320*x^3*z - 2626*x^3 - 42*x^2*y^5 + 309*x^2*y^4*z + 489*x^2*y^4 - 891*x^2*y^3*z^2 - 2886*x^2*y^3*z - 2244*x^2*y^3 + 1257*x^2*y^2*z^3 + 6255*x^2*y^2*z^2 + 10005*x^2*y^2*z + 5049*x^2*y^2 - 867*x^2*y*z^4 - 5892*x^2*y*z^3 - 14526*x^2*y*z^2 - 15192*x^2*y*z - 5583*x^2*y + 234*x^2*z^5 + 2034*x^2*z^4 + 6849*x^2*z^3 + 11145*x^2*z^2 + 8523*x^2*z + 2449*x^2 - 10*x*y^6 + 87*x*y^5*z + 138*x*y^5 - 309*x*y^4*z^2 - 1005*x*y^4*z - 783*x*y^4 + 573*x*y^3*z^3 + 2868*x*y^3*z^2 + 4587*x*y^3*z + 2339*x*y^3 - 585*x*y^2*z^4 - 4005*x*y^2*z^3 - 9852*x*y^2*z^2 - 10368*x*y^2*z - 3885*x*y^2 + 312*x*y*z^5 + 2736*x*y*z^4 + 9183*x*y*z^3 + 14937*x*y*z^2 + 11607*x*y*z + 3430*x*y - 68*x*z^6 - 732*x*z^5 - 3135*x*z^4 - 6974*x*z^3 - 8430*x*z^2 - 5191*x*z - 1271*x - y^7 + 10*y^6*z + 16*y^6 - 42*y^5*z^2 - 138*y^5*z - 108*y^5 + 96*y^4*z^3 + 486*y^4*z^2 + 780*y^4*z + 401*y^4 - 129*y^3*z^4 - 894*y^3*z^3 - 2205*y^3*z^2 - 2331*y^3*z - 887*y^3 + 102*y^2*z^5 + 906*y^2*z^4 + 3048*y^2*z^3 + 4959*y^2*z^2 + 3897*y^2*z + 1177*y^2 - 44*y*z^6 - 480*y*z^5 - 2061*y*z^4 - 4571*y*z^3 - 5553*y*z^2 - 3476*y*z - 879*y + 8*z^7 + 104*z^6 + 546*z^5 + 1542*z^4 + 2561*z^3 + 2485*z^2 + 1314*z + 1, 24*x^7 + 132*x^6*y - 204*x^6*z - 336*x^6 + 306*x^5*y^2 - 936*x^5*y*z - 1488*x^5*y + 702*x^5*z^2 + 2280*x^5*z + 1692*x^5 + 387*x^4*y^3 - 1755*x^4*y^2*z - 2712*x^4*y^2 + 2601*x^4*y*z^2 + 8196*x^4*y*z + 6108*x^4*y - 1257*x^4*z^3 - 6060*x^4*z^2 - 9348*x^4*z - 4449*x^4 + 288*x^3*y^4 - 1719*x^3*y^3*z - 2604*x^3*y^3 + 3771*x^3*y^2*z^2 + 11646*x^3*y^2*z + 8697*x^3*y^2 - 3597*x^3*y*z^3 - 16992*x^3*y*z^2 - 26178*x^3*y*z - 12684*x^3*y + 1257*x^3*z^4 + 8070*x^3*z^3 + 19245*x^3*z^2 + 19371*x^3*z + 6852*x^3 + 126*x^2*y^5 - 927*x^2*y^4*z - 1389*x^2*y^4 + 2673*x^2*y^3*z^2 + 8175*x^2*y^3*z + 6111*x^2*y^3 - 3771*x^2*y^2*z^3 - 17667*x^2*y^2*z^2 - 27153*x^2*y^2*z - 13347*x^2*y^2 + 2601*x^2*y*z^4 + 16593*x^2*y*z^3 + 39285*x^2*y*z^2 + 39999*x^2*y*z + 14496*x^2*y - 702*x^2*z^5 - 5712*x^2*z^4 - 18459*x^2*z^3 - 29223*x^2*z^2 - 22035*x^2*z - 6300*x^2 + 30*x*y^6 - 261*x*y^5*z - 390*x*y^5 + 927*x*y^4*z^2 + 2832*x*y^4*z + 2121*x*y^4 - 1719*x*y^3*z^3 - 8058*x*y^3*z^2 - 12381*x*y^3*z - 6153*x*y^3 + 1755*x*y^2*z^4 + 11220*x*y^2*z^3 + 26496*x*y^2*z^2 + 27168*x*y^2*z + 10041*x*y^2 - 936*x*y*z^5 - 7644*x*y*z^4 - 24609*x*y*z^3 - 38991*x*y*z^2 - 29877*x*y*z - 8784*x*y + 204*x*z^6 + 2040*x*z^5 + 8373*x*z^4 + 18141*x*z^3 + 21612*x*z^2 + 13239*x*z + 3241*x + 3*y^7 - 30*y^6*z - 45*y^6 + 126*y^5*z^2 + 387*y^5*z + 291*y^5 - 288*y^4*z^3 - 1359*y^4*z^2 - 2094*y^4*z - 1050*y^4 + 387*y^3*z^4 + 2493*y^3*z^3 + 5898*y^3*z^2 + 6081*y^3*z + 2283*y^3 - 306*y^2*z^5 - 2520*y^2*z^4 - 8124*y^2*z^3 - 12891*y^2*z^2 - 9993*y^2*z - 3003*y^2 + 132*y*z^6 + 1332*y*z^5 + 5475*y*z^4 + 11844*y*z^3 + 14190*y*z^2 + 8838*y*z + 2231*y - 24*z^7 - 288*z^6 - 1446*z^5 - 3984*z^4 - 6525*z^3 - 6300*z^2 - 3320*z - 1, 8*x^7 + 44*x^6*y - 68*x^6*z - 116*x^6 + 102*x^5*y^2 - 312*x^5*y*z - 516*x^5*y + 234*x^5*z^2 + 792*x^5*z + 612*x^5 + 129*x^4*y^3 - 585*x^4*y^2*z - 945*x^4*y^2 + 867*x^4*y*z^2 + 2862*x^4*y*z + 2220*x^4*y - 419*x^4*z^3 - 2121*x^4*z^2 - 3408*x^4*z - 1667*x^4 + 96*x^3*y^4 - 573*x^3*y^3*z - 912*x^3*y^3 + 1257*x^3*y^2*z^2 + 4089*x^3*y^2*z + 3177*x^3*y^2 - 1199*x^3*y*z^3 - 5982*x^3*y*z^2 - 9594*x^3*y*z - 4778*x^3*y + 419*x^3*z^4 + 2849*x^3*z^3 + 7077*x^3*z^2 + 7327*x^3*z + 2636*x^3 + 42*x^2*y^5 - 309*x^2*y^4*z - 489*x^2*y^4 + 891*x^2*y^3*z^2 + 2886*x^2*y^3*z + 2244*x^2*y^3 - 1257*x^2*y^2*z^3 - 6255*x^2*y^2*z^2 - 10005*x^2*y^2*z - 5055*x^2*y^2 + 867*x^2*y*z^4 + 5892*x^2*y*z^3 + 14526*x^2*y*z^2 + 15213*x^2*y*z + 5610*x^2*y - 234*x^2*z^5 - 2034*x^2*z^4 - 6849*x^2*z^3 - 11163*x^2*z^2 - 8571*x^2*z - 2473*x^2 + 10*x*y^6 - 87*x*y^5*z - 138*x*y^5 + 309*x*y^4*z^2 + 1005*x*y^4*z + 783*x*y^4 - 573*x*y^3*z^3 - 2868*x*y^3*z^2 - 4587*x*y^3*z - 2343*x*y^3 + 585*x*y^2*z^4 + 4005*x*y^2*z^3 + 9852*x*y^2*z^2 + 10389*x*y^2*z + 3909*x*y^2 - 312*x*y*z^5 - 2736*x*y*z^4 - 9183*x*y*z^3 - 14973*x*y*z^2 - 11691*x*y*z - 3472*x*y + 68*x*z^6 + 732*x*z^5 + 3135*x*z^4 + 6994*x*z^3 + 8502*x*z^2 + 5266*x*z + 1291*x + y^7 - 10*y^6*z - 16*y^6 + 42*y^5*z^2 + 138*y^5*z + 108*y^5 - 96*y^4*z^3 - 486*y^4*z^2 - 780*y^4*z - 402*y^4 + 129*y^3*z^4 + 894*y^3*z^3 + 2205*y^3*z^2 + 2338*y^3*z + 894*y^3 - 102*y^2*z^5 - 906*y^2*z^4 - 3048*y^2*z^3 - 4977*y^2*z^2 - 3933*y^2*z - 1195*y^2 + 44*y*z^6 + 480*y*z^5 + 2061*y*z^4 + 4591*y*z^3 + 5613*y*z^2 + 3539*y*z + 896*y - 8*z^7 - 104*z^6 - 546*z^5 - 1550*z^4 - 2593*z^3 - 2539*z^2 - 1344*z - 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -1, -1, 2 ], [ -2, -1, 1 ], [ -1, -1, 1 ] ], "source_shift": [ 1, 2, 1 ], "target_matrix": [ [ 1, 1, 2 ], [ -3, -2, -3 ], [ -1, -1, -1 ] ], "target_shift": [ -285, 723, 290 ], "collision_preimages": [ [ "3/4", "3/4", "1/4" ], [ "9", "-20", "-11/2" ], [ "6", "-19", "-15/2" ] ], "collision_image": [ "-1141/4", "2895/4", "1161/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.023903858000267 ], "surface_features": [ 3.6332175696470084, 4.602049133830196, 3.3140779917792127, 0.0, 3.8016137575243527, 4893 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B068", "pair": 68, "family": "bijection", "mapping": "F(x,y,z)=(1280*x^7 + 4480*x^6*y - 4480*x^6*z - 5760*x^6 + 6720*x^5*y^2 - 13440*x^5*y*z - 17600*x^5*y + 6720*x^5*z^2 + 18240*x^5*z + 11280*x^5 + 5600*x^4*y^3 - 16800*x^4*y^2*z - 22400*x^4*y^2 + 16800*x^4*y*z^2 + 46400*x^4*y*z + 29240*x^4*y - 5600*x^4*z^3 - 24000*x^4*z^2 - 31160*x^4*z - 12920*x^4 + 2800*x^3*y^4 - 11200*x^3*y^3*z - 15200*x^3*y^3 + 16800*x^3*y^2*z^2 + 47200*x^3*y^2*z + 30300*x^3*y^2 - 11200*x^3*y*z^3 - 48800*x^3*y*z^2 - 64520*x^3*y*z - 27480*x^3*y + 2800*x^3*z^4 + 16800*x^3*z^3 + 34300*x^3*z^2 + 30200*x^3*z + 9616*x^3 + 840*x^2*y^5 - 4200*x^2*y^4*z - 5800*x^2*y^4 + 8400*x^2*y^3*z^2 + 24000*x^2*y^3*z + 15690*x^2*y^3 - 8400*x^2*y^2*z^3 - 37200*x^2*y^2*z^2 - 50070*x^2*y^2*z - 21910*x^2*y^2 + 4200*x^2*y*z^4 + 25600*x^2*y*z^3 + 53190*x^2*y*z^2 + 48140*x^2*y*z + 15834*x^2*y - 840*x^2*z^5 - 6600*x^2*z^4 - 18810*x^2*z^3 - 26430*x^2*z^2 - 18004*x^2*z - 4726*x^2 + 140*x*y^6 - 840*x*y^5*z - 1180*x*y^5 + 2100*x*y^4*z^2 + 6100*x*y^4*z + 4060*x*y^4 - 2800*x*y^3*z^3 - 12600*x*y^3*z^2 - 17260*x*y^3*z - 7760*x*y^3 + 2100*x*y^2*z^4 + 13000*x*y^2*z^3 + 27480*x*y^2*z^2 + 25560*x*y^2*z + 8692*x*y^2 - 840*x*y*z^5 - 6700*x*y*z^4 - 19420*x*y*z^3 - 28040*x*y*z^2 - 19784*x*y*z - 5396*x*y + 140*x*z^6 + 1380*x*z^5 + 5140*x*z^4 + 10240*x*z^3 + 11272*x*z^2 + 6376*x*z + 1458*x + 10*y^7 - 70*y^6*z - 100*y^6 + 210*y^5*z^2 + 620*y^5*z + 420*y^5 - 350*y^4*z^3 - 1600*y^4*z^2 - 2230*y^4*z - 1030*y^4 + 350*y^3*z^4 + 2200*y^3*z^3 + 4730*y^3*z^2 + 4520*y^3*z + 1592*y^3 - 210*y^2*z^5 - 1700*y^2*z^4 - 5010*y^2*z^3 - 7430*y^2*z^2 - 5446*y^2*z - 1554*y^2 + 70*y*z^6 + 700*y*z^5 + 2650*y*z^4 + 5420*y*z^3 + 6226*y*z^2 + 3708*y*z + 895*y - 10*z^7 - 120*z^6 - 560*z^5 - 1480*z^4 - 2382*z^3 - 2234*z^2 - 1125*z - 1, 1280*x^7 + 4480*x^6*y - 4480*x^6*z - 5760*x^6 + 6720*x^5*y^2 - 13440*x^5*y*z - 17600*x^5*y + 6720*x^5*z^2 + 18240*x^5*z + 11280*x^5 + 5600*x^4*y^3 - 16800*x^4*y^2*z - 22400*x^4*y^2 + 16800*x^4*y*z^2 + 46400*x^4*y*z + 29240*x^4*y - 5600*x^4*z^3 - 24000*x^4*z^2 - 31160*x^4*z - 12920*x^4 + 2800*x^3*y^4 - 11200*x^3*y^3*z - 15200*x^3*y^3 + 16800*x^3*y^2*z^2 + 47200*x^3*y^2*z + 30300*x^3*y^2 - 11200*x^3*y*z^3 - 48800*x^3*y*z^2 - 64520*x^3*y*z - 27480*x^3*y + 2800*x^3*z^4 + 16800*x^3*z^3 + 34300*x^3*z^2 + 30200*x^3*z + 9608*x^3 + 840*x^2*y^5 - 4200*x^2*y^4*z - 5800*x^2*y^4 + 8400*x^2*y^3*z^2 + 24000*x^2*y^3*z + 15690*x^2*y^3 - 8400*x^2*y^2*z^3 - 37200*x^2*y^2*z^2 - 50070*x^2*y^2*z - 21910*x^2*y^2 + 4200*x^2*y*z^4 + 25600*x^2*y*z^3 + 53190*x^2*y*z^2 + 48140*x^2*y*z + 15822*x^2*y - 840*x^2*z^5 - 6600*x^2*z^4 - 18810*x^2*z^3 - 26430*x^2*z^2 - 17992*x^2*z - 4714*x^2 + 140*x*y^6 - 840*x*y^5*z - 1180*x*y^5 + 2100*x*y^4*z^2 + 6100*x*y^4*z + 4060*x*y^4 - 2800*x*y^3*z^3 - 12600*x*y^3*z^2 - 17260*x*y^3*z - 7760*x*y^3 + 2100*x*y^2*z^4 + 13000*x*y^2*z^3 + 27480*x*y^2*z^2 + 25560*x*y^2*z + 8686*x*y^2 - 840*x*y*z^5 - 6700*x*y*z^4 - 19420*x*y*z^3 - 28040*x*y*z^2 - 19772*x*y*z - 5384*x*y + 140*x*z^6 + 1380*x*z^5 + 5140*x*z^4 + 10240*x*z^3 + 11266*x*z^2 + 6364*x*z + 1450*x + 10*y^7 - 70*y^6*z - 100*y^6 + 210*y^5*z^2 + 620*y^5*z + 420*y^5 - 350*y^4*z^3 - 1600*y^4*z^2 - 2230*y^4*z - 1030*y^4 + 350*y^3*z^4 + 2200*y^3*z^3 + 4730*y^3*z^2 + 4520*y^3*z + 1591*y^3 - 210*y^2*z^5 - 1700*y^2*z^4 - 5010*y^2*z^3 - 7430*y^2*z^2 - 5443*y^2*z - 1551*y^2 + 70*y*z^6 + 700*y*z^5 + 2650*y*z^4 + 5420*y*z^3 + 6223*y*z^2 + 3702*y*z + 890*y - 10*z^7 - 120*z^6 - 560*z^5 - 1480*z^4 - 2381*z^3 - 2231*z^2 - 1119*z - 1, -3840*x^7 - 13440*x^6*y + 13440*x^6*z + 17280*x^6 - 20160*x^5*y^2 + 40320*x^5*y*z + 52800*x^5*y - 20160*x^5*z^2 - 54720*x^5*z - 33840*x^5 - 16800*x^4*y^3 + 50400*x^4*y^2*z + 67200*x^4*y^2 - 50400*x^4*y*z^2 - 139200*x^4*y*z - 87720*x^4*y + 16800*x^4*z^3 + 72000*x^4*z^2 + 93480*x^4*z + 38760*x^4 - 8400*x^3*y^4 + 33600*x^3*y^3*z + 45600*x^3*y^3 - 50400*x^3*y^2*z^2 - 141600*x^3*y^2*z - 90900*x^3*y^2 + 33600*x^3*y*z^3 + 146400*x^3*y*z^2 + 193560*x^3*y*z + 82440*x^3*y - 8400*x^3*z^4 - 50400*x^3*z^3 - 102900*x^3*z^2 - 90600*x^3*z - 28824*x^3 - 2520*x^2*y^5 + 12600*x^2*y^4*z + 17400*x^2*y^4 - 25200*x^2*y^3*z^2 - 72000*x^2*y^3*z - 47070*x^2*y^3 + 25200*x^2*y^2*z^3 + 111600*x^2*y^2*z^2 + 150210*x^2*y^2*z + 65730*x^2*y^2 - 12600*x^2*y*z^4 - 76800*x^2*y*z^3 - 159570*x^2*y*z^2 - 144420*x^2*y*z - 47466*x^2*y + 2520*x^2*z^5 + 19800*x^2*z^4 + 56430*x^2*z^3 + 79290*x^2*z^2 + 53976*x^2*z + 14150*x^2 - 420*x*y^6 + 2520*x*y^5*z + 3540*x*y^5 - 6300*x*y^4*z^2 - 18300*x*y^4*z - 12180*x*y^4 + 8400*x*y^3*z^3 + 37800*x*y^3*z^2 + 51780*x*y^3*z + 23280*x*y^3 - 6300*x*y^2*z^4 - 39000*x*y^2*z^3 - 82440*x*y^2*z^2 - 76680*x*y^2*z - 26058*x*y^2 + 2520*x*y*z^5 + 20100*x*y*z^4 + 58260*x*y*z^3 + 84120*x*y*z^2 + 59316*x*y*z + 16160*x*y - 420*x*z^6 - 4140*x*z^5 - 15420*x*z^4 - 30720*x*z^3 - 33798*x*z^2 - 19100*x*z - 4360*x - 30*y^7 + 210*y^6*z + 300*y^6 - 630*y^5*z^2 - 1860*y^5*z - 1260*y^5 + 1050*y^4*z^3 + 4800*y^4*z^2 + 6690*y^4*z + 3090*y^4 - 1050*y^3*z^4 - 6600*y^3*z^3 - 14190*y^3*z^2 - 13560*y^3*z - 4773*y^3 + 630*y^2*z^5 + 5100*y^2*z^4 + 15030*y^2*z^3 + 22290*y^2*z^2 + 16329*y^2*z + 4655*y^2 - 210*y*z^6 - 2100*y*z^5 - 7950*y*z^4 - 16260*y*z^3 - 18669*y*z^2 - 11110*y*z - 2676*y + 30*z^7 + 360*z^6 + 1680*z^5 + 4440*z^4 + 7143*z^3 + 6695*z^2 + 3365*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -2, -1, 1 ], [ -2, -2, 3 ], [ -1, -1, 2 ] ], "source_shift": [ 1, 2, 1 ], "target_matrix": [ [ -1, -2, -1 ], [ -1, -1, -1 ], [ 3, 3, 2 ] ], "target_shift": [ 232, 229, -693 ], "inverse_recipe": { "p": 3, "q": 2, "f": "x**3", "g": "x**2", "h": "y**2 + y*z**2 + z", "polynomials": { "f": [ [ [ 3, 0, 0 ], 1 ] ], "g": [ [ [ 2, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 2 ], 1 ], [ [ 0, 2, 0 ], 1 ], [ [ 0, 0, 1 ], 1 ] ] }, "h_multiplier": 10 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0573014660003537 ], "surface_features": [ 4.284351450631277, 5.286815613436587, 3.912487761332324, 0.0, 4.4490754880073045, 5154 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B048", "pair": 48, "family": "bijection", "mapping": "F(x,y,z)=(-2*x^7 - 28*x^6*y - 14*x^6*z - 30*x^6 - 168*x^5*y^2 - 168*x^5*y*z - 360*x^5*y - 42*x^5*z^2 - 180*x^5*z - 196*x^5 - 560*x^4*y^3 - 840*x^4*y^2*z - 1800*x^4*y^2 - 420*x^4*y*z^2 - 1800*x^4*y*z - 1958*x^4*y - 70*x^4*z^3 - 450*x^4*z^2 - 982*x^4*z - 714*x^4 - 1120*x^3*y^4 - 2240*x^3*y^3*z - 4800*x^3*y^3 - 1680*x^3*y^2*z^2 - 7200*x^3*y^2*z - 7824*x^3*y^2 - 560*x^3*y*z^3 - 3600*x^3*y*z^2 - 7848*x^3*y*z - 5694*x^3*y - 70*x^3*z^4 - 600*x^3*z^3 - 1968*x^3*z^2 - 2872*x^3*z - 1559*x^3 - 1344*x^2*y^5 - 3360*x^2*y^4*z - 7200*x^2*y^4 - 3360*x^2*y^3*z^2 - 14400*x^2*y^3*z - 15632*x^2*y^3 - 1680*x^2*y^2*z^3 - 10800*x^2*y^2*z^2 - 23520*x^2*y^2*z - 17028*x^2*y^2 - 420*x^2*y*z^4 - 3600*x^2*y*z^3 - 11796*x^2*y*z^2 - 17178*x^2*y*z - 9292*x^2*y - 42*x^2*z^5 - 450*x^2*z^4 - 1972*x^2*z^3 - 4332*x^2*z^2 - 4725*x^2*z - 2051*x^2 - 896*x*y^6 - 2688*x*y^5*z - 5760*x*y^5 - 3360*x*y^4*z^2 - 14400*x*y^4*z - 15616*x*y^4 - 2240*x*y^3*z^3 - 14400*x*y^3*z^2 - 31328*x*y^3*z - 22632*x*y^3 - 840*x*y^2*z^4 - 7200*x*y^2*z^3 - 23568*x*y^2*z^2 - 34248*x*y^2*z - 18460*x*y^2 - 168*x*y*z^5 - 1800*x*y*z^4 - 7880*x*y*z^3 - 17274*x*y*z^2 - 18776*x*y*z - 8102*x*y - 14*x*z^6 - 180*x*z^5 - 988*x*z^4 - 2904*x*z^3 - 4773*x*z^2 - 4168*x*z - 1532*x - 256*y^7 - 896*y^6*z - 1920*y^6 - 1344*y^5*z^2 - 5760*y^5*z - 6240*y^5 - 1120*y^4*z^3 - 7200*y^4*z^2 - 15648*y^4*z - 11280*y^4 - 560*y^3*z^4 - 4800*y^3*z^3 - 15696*y^3*z^2 - 22760*y^3*z - 12224*y^3 - 168*y^2*z^5 - 1800*y^2*z^4 - 7872*y^2*z^3 - 17220*y^2*z^2 - 18652*y^2*z - 8002*y^2 - 28*y*z^6 - 360*y*z^5 - 1974*y*z^4 - 5790*y*z^3 - 9484*y*z^2 - 8232*y*z - 2999*y - 2*z^7 - 30*z^6 - 198*z^5 - 730*z^4 - 1607*z^3 - 2117*z^2 - 1573*z - 1, x^7 + 14*x^6*y + 7*x^6*z + 15*x^6 + 84*x^5*y^2 + 84*x^5*y*z + 180*x^5*y + 21*x^5*z^2 + 90*x^5*z + 98*x^5 + 280*x^4*y^3 + 420*x^4*y^2*z + 900*x^4*y^2 + 210*x^4*y*z^2 + 900*x^4*y*z + 979*x^4*y + 35*x^4*z^3 + 225*x^4*z^2 + 491*x^4*z + 356*x^4 + 560*x^3*y^4 + 1120*x^3*y^3*z + 2400*x^3*y^3 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5624*y^4 + 280*y^3*z^4 + 2400*y^3*z^3 + 7848*y^3*z^2 + 11348*y^3*z + 6044*y^3 + 84*y^2*z^5 + 900*y^2*z^4 + 3936*y^2*z^3 + 8586*y^2*z^2 + 9224*y^2*z + 3891*y^2 + 14*y*z^6 + 180*y*z^5 + 987*y*z^4 + 2887*y*z^3 + 4691*y*z^2 + 4006*y*z + 1417*y + z^7 + 15*z^6 + 99*z^5 + 364*z^4 + 795*z^3 + 1031*z^2 + 744*z + 1, -x^7 - 14*x^6*y - 7*x^6*z - 15*x^6 - 84*x^5*y^2 - 84*x^5*y*z - 180*x^5*y - 21*x^5*z^2 - 90*x^5*z - 98*x^5 - 280*x^4*y^3 - 420*x^4*y^2*z - 900*x^4*y^2 - 210*x^4*y*z^2 - 900*x^4*y*z - 979*x^4*y - 35*x^4*z^3 - 225*x^4*z^2 - 491*x^4*z - 358*x^4 - 560*x^3*y^4 - 1120*x^3*y^3*z - 2400*x^3*y^3 - 840*x^3*y^2*z^2 - 3600*x^3*y^2*z - 3912*x^3*y^2 - 280*x^3*y*z^3 - 1800*x^3*y*z^2 - 3924*x^3*y*z - 2855*x^3*y - 35*x^3*z^4 - 300*x^3*z^3 - 984*x^3*z^2 - 1440*x^3*z - 787*x^3 - 672*x^2*y^5 - 1680*x^2*y^4*z - 3600*x^2*y^4 - 1680*x^2*y^3*z^2 - 7200*x^2*y^3*z - 7816*x^2*y^3 - 840*x^2*y^2*z^3 - 5400*x^2*y^2*z^2 - 11760*x^2*y^2*z - 8538*x^2*y^2 - 210*x^2*y*z^4 - 1800*x^2*y*z^3 - 5898*x^2*y*z^2 - 8613*x^2*y*z - 4691*x^2*y - 21*x^2*z^5 - 225*x^2*z^4 - 986*x^2*z^3 - 2172*x^2*z^2 - 2385*x^2*z - 1046*x^2 - 448*x*y^6 - 1344*x*y^5*z - 2880*x*y^5 - 1680*x*y^4*z^2 - 7200*x*y^4*z - 7808*x*y^4 - 1120*x*y^3*z^3 - 7200*x*y^3*z^2 - 15664*x*y^3*z - 11348*x*y^3 - 420*x*y^2*z^4 - 3600*x*y^2*z^3 - 11784*x*y^2*z^2 - 17172*x*y^2*z - 9320*x*y^2 - 84*x*y*z^5 - 900*x*y*z^4 - 3940*x*y*z^3 - 8661*x*y*z^2 - 9478*x*y*z - 4133*x*y - 7*x*z^6 - 90*x*z^5 - 494*x*z^4 - 1456*x*z^3 - 2409*x*z^2 - 2125*x*z - 790*x - 128*y^7 - 448*y^6*z - 960*y^6 - 672*y^5*z^2 - 2880*y^5*z - 3120*y^5 - 560*y^4*z^3 - 3600*y^4*z^2 - 7824*y^4*z - 5656*y^4 - 280*y^3*z^4 - 2400*y^3*z^3 - 7848*y^3*z^2 - 11412*y^3*z - 6172*y^3 - 84*y^2*z^5 - 900*y^2*z^4 - 3936*y^2*z^3 - 8634*y^2*z^2 - 9416*y^2*z - 4083*y^2 - 14*y*z^6 - 180*y*z^5 - 987*y*z^4 - 2903*y*z^3 - 4787*y*z^2 - 4198*y*z - 1547*y - z^7 - 15*z^6 - 99*z^5 - 366*z^4 - 811*z^3 - 1079*z^2 - 810*z - 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ 1, 2, 1 ], [ 1, 1, 2 ], [ 1, 1, 1 ] ], "source_shift": [ 2, 1, 1 ], "target_matrix": [ [ -2, -3, -3 ], [ 1, 1, 2 ], [ -1, -1, -1 ] ], "target_shift": [ 516, -233, 267 ], "inverse_recipe": { "p": 3, "q": 4, "f": "x**3 + x**2 + x", "g": "x**4", "h": "y*z + 2*y", "polynomials": { "f": [ [ [ 3, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "g": [ [ [ 4, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 1, 0 ], 2 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0091061249986524 ], "surface_features": [ 3.5754531019193374, 4.5346352148262525, 3.1947917577219247, 0.0, 3.7181117745384107, 4849 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B058", "pair": 58, "family": "bijection", "mapping": "F(x,y,z)=(-128*x^7 + 448*x^6*y + 448*x^6*z - 704*x^6 - 672*x^5*y^2 - 1344*x^5*y*z + 2144*x^5*y - 672*x^5*z^2 + 2144*x^5*z - 1720*x^5 + 560*x^4*y^3 + 1680*x^4*y^2*z - 2720*x^4*y^2 + 1680*x^4*y*z^2 - 5440*x^4*y*z + 4444*x^4*y + 560*x^4*z^3 - 2720*x^4*z^2 + 4428*x^4*z - 2424*x^4 - 280*x^3*y^4 - 1120*x^3*y^3*z + 1840*x^3*y^3 - 1680*x^3*y^2*z^2 + 5520*x^3*y^2*z - 4590*x^3*y^2 - 1120*x^3*y*z^3 + 5520*x^3*y*z^2 - 9148*x^3*y*z + 5120*x^3*y - 280*x^3*z^4 + 1840*x^3*z^3 - 4558*x^3*z^2 + 5064*x^3*z - 2155*x^3 + 84*x^2*y^5 + 420*x^2*y^4*z - 700*x^2*y^4 + 840*x^2*y^3*z^2 - 2800*x^2*y^3*z + 2369*x^2*y^3 + 840*x^2*y^2*z^3 - 4200*x^2*y^2*z^2 + 7083*x^2*y^2*z - 4054*x^2*y^2 + 420*x^2*y*z^4 - 2800*x^2*y*z^3 + 7059*x^2*y*z^2 - 8020*x^2*y*z + 3511*x^2*y + 84*x^2*z^5 - 700*x^2*z^4 + 2345*x^2*z^3 - 3966*x^2*z^2 + 3430*x^2*z - 1212*x^2 - 14*x*y^6 - 84*x*y^5*z + 142*x*y^5 - 210*x*y^4*z^2 + 710*x*y^4*z - 611*x*y^4 - 280*x*y^3*z^3 + 1420*x*y^3*z^2 - 2436*x*y^3*z + 1426*x*y^3 - 210*x*y^2*z^4 + 1420*x*y^2*z^3 - 3642*x*y^2*z^2 + 4232*x*y^2*z - 1908*x*y^2 - 84*x*y*z^5 + 710*x*y*z^4 - 2420*x*y*z^3 + 4186*x*y*z^2 - 3728*x*y*z + 1368*x*y - 14*x*z^6 + 142*x*z^5 - 603*x*z^4 + 1380*x*z^3 - 1820*x*z^2 + 1312*x*z - 396*x + y^7 + 7*y^6*z - 12*y^6 + 21*y^5*z^2 - 72*y^5*z + 63*y^5 + 35*y^4*z^3 - 180*y^4*z^2 + 314*y^4*z - 188*y^4 + 35*y^3*z^4 - 240*y^3*z^3 + 626*y^3*z^2 - 744*y^3*z + 346*y^3 + 21*y^2*z^5 - 180*y^2*z^4 + 624*y^2*z^3 - 1104*y^2*z^2 + 1014*y^2*z - 388*y^2 + 7*y*z^6 - 72*y*z^5 + 311*y*z^4 - 728*y*z^3 + 990*y*z^2 - 744*y*z + 239*y + z^7 - 12*z^6 + 62*z^5 - 180*z^4 + 322*z^3 - 356*z^2 + 221*z - 1, -256*x^7 + 896*x^6*y + 896*x^6*z - 1408*x^6 - 1344*x^5*y^2 - 2688*x^5*y*z + 4288*x^5*y - 1344*x^5*z^2 + 4288*x^5*z - 3440*x^5 + 1120*x^4*y^3 + 3360*x^4*y^2*z - 5440*x^4*y^2 + 3360*x^4*y*z^2 - 10880*x^4*y*z + 8888*x^4*y + 1120*x^4*z^3 - 5440*x^4*z^2 + 8856*x^4*z - 4848*x^4 - 560*x^3*y^4 - 2240*x^3*y^3*z + 3680*x^3*y^3 - 3360*x^3*y^2*z^2 + 11040*x^3*y^2*z - 9180*x^3*y^2 - 2240*x^3*y*z^3 + 11040*x^3*y*z^2 - 18296*x^3*y*z + 10240*x^3*y - 560*x^3*z^4 + 3680*x^3*z^3 - 9116*x^3*z^2 + 10128*x^3*z - 4302*x^3 + 168*x^2*y^5 + 840*x^2*y^4*z - 1400*x^2*y^4 + 1680*x^2*y^3*z^2 - 5600*x^2*y^3*z + 4738*x^2*y^3 + 1680*x^2*y^2*z^3 - 8400*x^2*y^2*z^2 + 14166*x^2*y^2*z - 8108*x^2*y^2 + 840*x^2*y*z^4 - 5600*x^2*y*z^3 + 14118*x^2*y*z^2 - 16040*x^2*y*z + 7010*x^2*y + 168*x^2*z^5 - 1400*x^2*z^4 + 4690*x^2*z^3 - 7932*x^2*z^2 + 6848*x^2*z - 2432*x^2 - 28*x*y^6 - 168*x*y^5*z + 284*x*y^5 - 420*x*y^4*z^2 + 1420*x*y^4*z - 1222*x*y^4 - 560*x*y^3*z^3 + 2840*x*y^3*z^2 - 4872*x*y^3*z + 2852*x*y^3 - 420*x*y^2*z^4 + 2840*x*y^2*z^3 - 7284*x*y^2*z^2 + 8464*x*y^2*z - 3810*x*y^2 - 168*x*y*z^5 + 1420*x*y*z^4 - 4840*x*y*z^3 + 8372*x*y*z^2 - 7444*x*y*z + 2744*x*y - 28*x*z^6 + 284*x*z^5 - 1206*x*z^4 + 2760*x*z^3 - 3634*x*z^2 + 2632*x*z - 823*x + 2*y^7 + 14*y^6*z - 24*y^6 + 42*y^5*z^2 - 144*y^5*z + 126*y^5 + 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21249*x^2*y^2*z + 12162*x^2*y^2 - 1260*x^2*y*z^4 + 8400*x^2*y*z^3 - 21177*x^2*y*z^2 + 24060*x^2*y*z - 10521*x^2*y - 252*x^2*z^5 + 2100*x^2*z^4 - 7035*x^2*z^3 + 11898*x^2*z^2 - 10278*x^2*z + 3652*x^2 + 42*x*y^6 + 252*x*y^5*z - 426*x*y^5 + 630*x*y^4*z^2 - 2130*x*y^4*z + 1833*x*y^4 + 840*x*y^3*z^3 - 4260*x*y^3*z^2 + 7308*x*y^3*z - 4278*x*y^3 + 630*x*y^2*z^4 - 4260*x*y^2*z^3 + 10926*x*y^2*z^2 - 12696*x*y^2*z + 5718*x*y^2 + 252*x*y*z^5 - 2130*x*y*z^4 + 7260*x*y*z^3 - 12558*x*y*z^2 + 11172*x*y*z - 4120*x*y + 42*x*z^6 - 426*x*z^5 + 1809*x*z^4 - 4140*x*z^3 + 5454*x*z^2 - 3952*x*z + 1233*x - 3*y^7 - 21*y^6*z + 36*y^6 - 63*y^5*z^2 + 216*y^5*z - 189*y^5 - 105*y^4*z^3 + 540*y^4*z^2 - 942*y^4*z + 564*y^4 - 105*y^3*z^4 + 720*y^3*z^3 - 1878*y^3*z^2 + 2232*y^3*z - 1037*y^3 - 63*y^2*z^5 + 540*y^2*z^4 - 1872*y^2*z^3 + 3312*y^2*z^2 - 3039*y^2*z + 1168*y^2 - 21*y*z^6 + 216*y*z^5 - 933*y*z^4 + 2184*y*z^3 - 2967*y*z^2 + 2240*y*z - 742*y - 3*z^7 + 36*z^6 - 186*z^5 + 540*z^4 - 965*z^3 + 1072*z^2 - 689*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ 2, -1, -1 ], [ 1, -2, -1 ], [ 1, -1, -1 ] ], "source_shift": [ 1, 1, 2 ], "target_matrix": [ [ -1, -1, -2 ], [ -2, -1, -1 ], [ 3, 2, 2 ] ], "target_shift": [ 54, 130, -193 ], "inverse_recipe": { "p": 3, "q": 2, "f": "x**3 + x**2", "g": "x**2 + x", "h": "y*z**2 + y + z**2", "polynomials": { "f": [ [ [ 3, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "g": [ [ [ 2, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 2 ], 1 ], [ [ 0, 0, 2 ], 1 ], [ [ 0, 1, 0 ], 1 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0431000790013059 ], "surface_features": [ 3.5034324049812873, 4.438447410654374, 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42*y*z^6 - 321*y*z^5 + 1045*y*z^4 - 1810*y*z^3 + 1785*y*z^2 - 1004*y*z + 280*y + 3*z^7 - 27*z^6 + 106*z^5 - 230*z^4 + 304*z^3 - 266*z^2 + 169*z - 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ 1, -2, -1 ], [ 1, -1, -2 ], [ 1, -1, -1 ] ], "source_shift": [ 1, 2, 2 ], "target_matrix": [ [ 2, 1, 1 ], [ -1, -2, -1 ], [ -1, -1, -1 ] ], "target_shift": [ -117, 60, 56 ], "inverse_recipe": { "p": 5, "q": 2, "f": "x**5 + x", "g": "x**2 + x", "h": "y*z + y", "polynomials": { "f": [ [ [ 5, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "g": [ [ [ 2, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 1, 0 ], 1 ] ] }, "h_multiplier": 3 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0303504189978412 ], "surface_features": [ 3.5794256641579283, 4.5094713521025485, 3.251273113674373, 0.0, 3.6958797783674915, 4914 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B213", "pair": 213, "family": "bijection", "mapping": "F(x,y,z)=(4*x^7 + 56*x^6*y - 28*x^6*z - 28*x^6 + 336*x^5*y^2 - 336*x^5*y*z - 336*x^5*y + 84*x^5*z^2 + 168*x^5*z + 96*x^5 + 1120*x^4*y^3 - 1680*x^4*y^2*z - 1680*x^4*y^2 + 840*x^4*y*z^2 + 1680*x^4*y*z + 956*x^4*y - 140*x^4*z^3 - 420*x^4*z^2 - 480*x^4*z - 218*x^4 + 2240*x^3*y^4 - 4480*x^3*y^3*z - 4480*x^3*y^3 + 3360*x^3*y^2*z^2 + 6720*x^3*y^2*z + 3808*x^3*y^2 - 1120*x^3*y*z^3 - 3360*x^3*y*z^2 - 3824*x^3*y*z - 1716*x^3*y + 140*x^3*z^4 + 560*x^3*z^3 + 960*x^3*z^2 + 868*x^3*z + 333*x^3 + 2688*x^2*y^5 - 6720*x^2*y^4*z - 6720*x^2*y^4 + 6720*x^2*y^3*z^2 + 13440*x^2*y^3*z + 7584*x^2*y^3 - 3360*x^2*y^2*z^3 - 10080*x^2*y^2*z^2 - 11424*x^2*y^2*z - 5064*x^2*y^2 + 840*x^2*y*z^4 + 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23104*y^3*z + 13112*y^3 - 168*y^2*z^5 - 1800*y^2*z^4 - 7824*y^2*z^3 - 17400*y^2*z^2 - 19860*y^2*z - 9294*y^2 + 28*y*z^6 + 360*y*z^5 + 1958*y*z^4 + 5824*y*z^3 + 10026*y*z^2 + 9464*y*z + 3811*y - 2*z^7 - 30*z^6 - 196*z^5 - 731*z^4 - 1687*z^3 - 2410*z^2 - 1972*z - 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -1, -2, 1 ], [ -1, -1, 2 ], [ -1, -1, 1 ] ], "source_shift": [ 2, 1, 2 ], "target_matrix": [ [ 1, 1, 1 ], [ -3, -2, -3 ], [ -2, -1, -1 ] ], "target_shift": [ -351, 1030, 702 ], "inverse_recipe": { "p": 4, "q": 3, "f": "x**4 + x**3", "g": "x**3 + x", "h": "y*z + y + 2*z", "polynomials": { "f": [ [ [ 4, 0, 0 ], 1 ], [ [ 3, 0, 0 ], 1 ] ], "g": [ [ [ 3, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 0, 1 ], 2 ], [ [ 0, 1, 0 ], 1 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.042572332000418 ], "surface_features": [ 3.7617046645942214, 4.7175706537129605, 3.380211241711606, 0.0, 3.9134003247863047, 4925 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B183", "pair": 183, "family": "bijection", "mapping": "F(x,y,z)=(2*x^7 + 14*x^6*y - 28*x^6*z - 26*x^6 + 42*x^5*y^2 - 168*x^5*y*z - 152*x^5*y + 168*x^5*z^2 + 300*x^5*z + 138*x^5 + 70*x^4*y^3 - 420*x^4*y^2*z - 370*x^4*y^2 + 840*x^4*y*z^2 + 1460*x^4*y*z + 658*x^4*y - 560*x^4*z^3 - 1440*x^4*z^2 - 1282*x^4*z - 404*x^4 + 70*x^3*y^4 - 560*x^3*y^3*z - 480*x^3*y^3 + 1680*x^3*y^2*z^2 + 2840*x^3*y^2*z + 1254*x^3*y^2 - 2240*x^3*y*z^3 - 5600*x^3*y*z^2 - 4884*x^3*y*z - 1512*x^3*y + 1120*x^3*z^4 + 3680*x^3*z^3 + 4754*x^3*z^2 + 2900*x^3*z + 708*x^3 + 42*x^2*y^5 - 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9010*x*y^2 + 336*x*y*z^5 + 2840*x*y*z^4 + 10000*x*y*z^3 + 18340*x*y*z^2 + 17652*x*y*z + 7144*x*y + 56*x*z^6 + 568*x*z^5 + 2492*x*z^4 + 6052*x*z^3 + 8642*x*z^2 + 6892*x*z + 2407*x + 4*y^7 + 28*y^6*z + 48*y^6 + 84*y^5*z^2 + 288*y^5*z + 260*y^5 + 140*y^4*z^3 + 720*y^4*z^2 + 1296*y^4*z + 820*y^4 + 140*y^3*z^4 + 960*y^3*z^3 + 2584*y^3*z^2 + 3248*y^3*z + 1625*y^3 + 84*y^2*z^5 + 720*y^2*z^4 + 2576*y^2*z^3 + 4824*y^2*z^2 + 4775*y^2*z + 2006*y^2 + 28*y*z^6 + 288*y*z^5 + 1284*y*z^4 + 3184*y*z^3 + 4675*y*z^2 + 3868*y*z + 1426*y + 4*z^7 + 48*z^6 + 256*z^5 + 788*z^4 + 1525*z^3 + 1862*z^2 + 1337*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ 2, 1, 1 ], [ 1, 2, 1 ], [ 1, 1, 1 ] ], "source_shift": [ 1, 2, 2 ], "target_matrix": [ [ -2, -3, -2 ], [ -1, -1, -1 ], [ 2, 1, 1 ] ], "target_shift": [ 440, 217, -438 ], "inverse_recipe": { "p": 3, "q": 2, "f": "x**3 + x**2 + x", "g": "x**2 + x", "h": "y*z**2 + y*z + 2*y", "polynomials": { "f": [ [ [ 3, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "g": [ [ [ 2, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 2 ], 1 ], [ [ 0, 1, 1 ], 1 ], [ [ 0, 1, 0 ], 2 ] ] }, "h_multiplier": 2 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0833786100010911 ], "surface_features": [ 3.7514662226856235, 4.578318240585007, 3.4294292643817874, 0.0, 3.8525131663026277, 4966 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C215", "pair": 215, "family": "counterexample", "mapping": "F(x,y,z)=(8*x^7 - 68*x^6*y - 76*x^6*z + 116*x^6 + 246*x^5*y^2 + 552*x^5*y*z - 876*x^5*y + 306*x^5*z^2 - 936*x^5*z + 612*x^5 - 491*x^4*y^3 - 1659*x^4*y^2*z + 2745*x^4*y^2 - 1845*x^4*y*z^2 + 5862*x^4*y*z - 3900*x^4*y - 677*x^4*z^3 + 3117*x^4*z^2 - 4092*x^4*z + 1667*x^4 + 584*x^3*y^4 + 2641*x^3*y^3*z - 4568*x^3*y^3 + 4419*x^3*y^2*z^2 - 14619*x^3*y^2*z + 9897*x^3*y^2 + 3251*x^3*y*z^3 - 15534*x^3*y*z^2 + 20742*x^3*y*z - 8558*x^3*y + 889*x^3*z^4 - 5483*x^3*z^3 + 10845*x^3*z^2 - 8897*x^3*z + 2636*x^3 - 414*x^2*y^5 - 2349*x^2*y^4*z + 4257*x^2*y^4 - 5256*x^2*y^3*z^2 + 18144*x^2*y^3*z - 12498*x^2*y^3 - 5814*x^2*y^2*z^3 + 28890*x^2*y^2*z^2 - 39234*x^2*y^2*z + 16395*x^2*y^2 - 3186*x^2*y*z^4 + 20376*x^2*y*z^3 - 40974*x^2*y*z^2 + 34041*x^2*y*z - 10206*x^2*y - 693*x^2*z^5 + 5373*x^2*z^4 - 14238*x^2*z^3 + 17646*x^2*z^2 - 10557*x^2*z + 2473*x^2 + 162*x*y^6 + 1107*x*y^5*z - 2106*x*y^5 + 3105*x*y^4*z^2 - 11205*x*y^4*z + 7851*x*y^4 + 4590*x*y^3*z^3 - 23760*x*y^3*z^2 + 32811*x*y^3*z - 13885*x*y^3 + 3780*x*y^2*z^4 - 25110*x*y^2*z^3 + 51327*x*y^2*z^2 - 43179*x*y^2*z + 13101*x*y^2 + 1647*x*y*z^5 - 13230*x*y*z^4 + 35625*x*y*z^3 - 44703*x*y*z^2 + 27063*x*y*z - 6420*x*y + 297*x*z^6 - 2781*x*z^5 + 9258*x*z^4 - 15409*x*z^3 + 13962*x*z^2 - 6624*x*z + 1291*x - 27*y^7 - 216*y^6*z + 432*y^6 - 729*y^5*z^2 + 2754*y^5*z - 1962*y^5 - 1350*y^4*z^3 + 7290*y^4*z^2 - 10233*y^4*z + 4384*y^4 - 1485*y^3*z^4 + 10260*y^3*z^3 - 21312*y^3*z^2 + 18149*y^3*z - 5572*y^3 - 972*y^2*z^5 + 8100*y^2*z^4 - 22158*y^2*z^3 + 28143*y^2*z^2 - 17238*y^2*z + 4143*y^2 - 351*y*z^6 + 3402*y*z^5 - 11502*y*z^4 + 19375*y*z^3 - 17760*y*z^2 + 8535*y*z - 1686*y - 54*z^7 + 594*z^6 - 2385*z^5 + 4997*z^4 - 6094*z^3 + 4392*z^2 - 1739*z + 1, -8*x^7 + 68*x^6*y + 76*x^6*z - 116*x^6 - 246*x^5*y^2 - 552*x^5*y*z + 876*x^5*y - 306*x^5*z^2 + 936*x^5*z - 612*x^5 + 491*x^4*y^3 + 1659*x^4*y^2*z - 2745*x^4*y^2 + 1845*x^4*y*z^2 - 5862*x^4*y*z + 3900*x^4*y + 677*x^4*z^3 - 3117*x^4*z^2 + 4092*x^4*z - 1666*x^4 - 584*x^3*y^4 - 2641*x^3*y^3*z + 4568*x^3*y^3 - 4419*x^3*y^2*z^2 + 14619*x^3*y^2*z - 9897*x^3*y^2 - 3251*x^3*y*z^3 + 15534*x^3*y*z^2 - 20742*x^3*y*z + 8554*x^3*y - 889*x^3*z^4 + 5483*x^3*z^3 - 10845*x^3*z^2 + 8892*x^3*z - 2626*x^3 + 414*x^2*y^5 + 2349*x^2*y^4*z - 4257*x^2*y^4 + 5256*x^2*y^3*z^2 - 18144*x^2*y^3*z + 12498*x^2*y^3 + 5814*x^2*y^2*z^3 - 28890*x^2*y^2*z^2 + 39234*x^2*y^2*z - 16389*x^2*y^2 + 3186*x^2*y*z^4 - 20376*x^2*y*z^3 + 40974*x^2*y*z^2 - 34026*x^2*y*z + 10173*x^2*y + 693*x^2*z^5 - 5373*x^2*z^4 + 14238*x^2*z^3 - 17637*x^2*z^2 + 10521*x^2*z - 2449*x^2 - 162*x*y^6 - 1107*x*y^5*z + 2106*x*y^5 - 3105*x*y^4*z^2 + 11205*x*y^4*z - 7851*x*y^4 - 4590*x*y^3*z^3 + 23760*x*y^3*z^2 - 32811*x*y^3*z + 13881*x*y^3 - 3780*x*y^2*z^4 + 25110*x*y^2*z^3 - 51327*x*y^2*z^2 + 43164*x*y^2*z - 13065*x*y^2 - 1647*x*y*z^5 + 13230*x*y*z^4 - 35625*x*y*z^3 + 44685*x*y*z^2 - 26985*x*y*z + 6366*x*y - 297*x*z^6 + 2781*x*z^5 - 9258*x*z^4 + 15402*x*z^3 - 13920*x*z^2 + 6567*x*z - 1271*x + 27*y^7 + 216*y^6*z - 432*y^6 + 729*y^5*z^2 - 2754*y^5*z + 1962*y^5 + 1350*y^4*z^3 - 7290*y^4*z^2 + 10233*y^4*z - 4383*y^4 + 1485*y^3*z^4 - 10260*y^3*z^3 + 21312*y^3*z^2 - 18144*y^3*z + 5559*y^3 + 972*y^2*z^5 - 8100*y^2*z^4 + 22158*y^2*z^3 - 28134*y^2*z^2 + 17196*y^2*z - 4113*y^2 + 351*y*z^6 - 3402*y*z^5 + 11502*y*z^4 - 19368*y*z^3 + 17715*y*z^2 - 8472*y*z + 1663*y + 54*z^7 - 594*z^6 + 2385*z^5 - 4995*z^4 + 6078*z^3 - 4359*z^2 + 1715*z - 1, -24*x^7 + 204*x^6*y + 228*x^6*z - 336*x^6 - 738*x^5*y^2 - 1656*x^5*y*z + 2544*x^5*y - 918*x^5*z^2 + 2712*x^5*z - 1692*x^5 + 1473*x^4*y^3 + 4977*x^4*y^2*z - 7992*x^4*y^2 + 5535*x^4*y*z^2 - 17028*x^4*y*z + 10812*x^4*y + 2031*x^4*z^3 - 9036*x^4*z^2 + 11328*x^4*z - 4449*x^4 - 1752*x^3*y^4 - 7923*x^3*y^3*z + 13332*x^3*y^3 - 13257*x^3*y^2*z^2 + 42570*x^3*y^2*z - 27513*x^3*y^2 - 9753*x^3*y*z^3 + 45144*x^3*y*z^2 - 57582*x^3*y*z + 22908*x^3*y - 2667*x^3*z^4 + 15906*x^3*z^3 - 30069*x^3*z^2 + 23793*x^3*z - 6852*x^3 + 1242*x^2*y^5 + 7047*x^2*y^4*z - 12453*x^2*y^4 + 15768*x^2*y^3*z^2 - 52959*x^2*y^3*z + 34839*x^2*y^3 + 17442*x^2*y^2*z^3 - 84159*x^2*y^2*z^2 + 109224*x^2*y^2*z - 44019*x^2*y^2 + 9558*x^2*y*z^4 - 59253*x^2*y*z^3 + 113931*x^2*y*z^2 - 91317*x^2*y*z + 26616*x^2*y + 2079*x^2*z^5 - 15600*x^2*z^4 + 39546*x^2*z^3 - 47298*x^2*z^2 + 27513*x^2*z - 6300*x^2 - 486*x*y^6 - 3321*x*y^5*z + 6174*x*y^5 - 9315*x*y^4*z^2 + 32778*x*y^4*z - 21945*x*y^4 - 13770*x*y^3*z^3 + 69372*x*y^3*z^2 - 91599*x*y^3*z + 37395*x*y^3 - 11340*x*y^2*z^4 + 73188*x*y^2*z^3 - 143127*x*y^2*z^2 + 116196*x*y^2*z - 34281*x*y^2 - 4941*x*y*z^5 + 38502*x*y*z^4 - 99237*x*y*z^3 + 120207*x*y*z^2 - 70773*x*y*z + 16416*x*y - 891*x*z^6 + 8082*x*z^5 - 25764*x*z^4 + 41406*x*z^3 - 36492*x*z^2 + 16929*x*z - 3241*x + 81*y^7 + 648*y^6*z - 1269*y^6 + 2187*y^5*z^2 - 8073*y^5*z + 5499*y^5 + 4050*y^4*z^3 - 21330*y^4*z^2 + 28647*y^4*z - 11844*y^4 + 4455*y^3*z^4 - 29970*y^3*z^3 + 59598*y^3*z^2 - 48996*y^3*z + 14631*y^3 + 2916*y^2*z^5 - 23625*y^2*z^4 + 61902*y^2*z^3 - 75924*y^2*z^2 + 45240*y^2*z - 10635*y^2 + 1053*y*z^6 - 9909*y*z^5 + 32103*y*z^4 - 52236*y*z^3 + 46587*y*z^2 - 21900*y*z + 4251*y + 162*z^7 - 1728*z^6 + 6651*z^5 - 13464*z^4 + 15978*z^3 - 11265*z^2 + 4383*z - 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ 1, -1, -1 ], [ 2, -3, -3 ], [ 1, -1, -2 ] ], "source_shift": [ 1, 2, 1 ], "target_matrix": [ [ 1, 1, 1 ], [ -1, -1, -2 ], [ -3, -2, -3 ] ], "target_shift": [ -290, 285, 723 ], "collision_preimages": [ [ "-1", "-1/4", "1/4" ], [ "7/2", "9", "-11/2" ], [ "-11/2", "4", "-15/2" ] ], "collision_image": [ "-1161/4", "1141/4", "2895/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 0.9951828209996165 ], "surface_features": [ 4.164489584138846, 5.155721568385414, 3.801609488027319, 0.0, 4.324686492311525, 5115 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C165", "pair": 165, "family": "counterexample", "mapping": "F(x,y,z)=(6*x^7 + 57*x^6*y - 39*x^6*z - 69*x^6 + 225*x^5*y^2 - 315*x^5*y*z - 567*x^5*y + 108*x^5*z^2 + 387*x^5*z + 333*x^5 + 477*x^4*y^3 - 1026*x^4*y^2*z - 1881*x^4*y^2 + 720*x^4*y*z^2 + 2628*x^4*y*z + 2286*x^4*y - 165*x^4*z^3 - 900*x^4*z^2 - 1566*x^4*z - 893*x^4 + 585*x^3*y^4 - 1719*x^3*y^3*z - 3213*x^3*y^3 + 1854*x^3*y^2*z^2 + 6903*x^3*y^2*z + 6075*x^3*y^2 - 870*x^3*y*z^3 - 4842*x^3*y*z^2 - 8532*x^3*y*z - 4891*x^3*y + 150*x^3*z^4 + 1110*x^3*z^3 + 2934*x^3*z^2 + 3370*x^3*z + 1460*x^3 + 414*x^2*y^5 - 1557*x^2*y^4*z - 2970*x^2*y^4 + 2295*x^2*y^3*z^2 + 8730*x^2*y^3*z + 7785*x^2*y^3 - 1656*x^2*y^2*z^3 - 9423*x^2*y^2*z^2 - 16839*x^2*y^2*z - 9696*x^2*y^2 + 585*x^2*y*z^4 + 4428*x^2*y*z^3 + 11880*x^2*y*z^2 + 13731*x^2*y*z + 5949*x^2*y - 81*x^2*z^5 - 765*x^2*z^4 - 2736*x^2*z^3 - 4752*x^2*z^2 - 4134*x^2*z - 1473*x^2 + 156*x*y^6 - 720*x*y^5*z - 1404*x*y^5 + 1359*x*y^4*z^2 + 5292*x*y^4*z + 4797*x*y^4 - 1341*x*y^3*z^3 - 7821*x*y^3*z^2 - 14211*x*y^3*z - 8218*x*y^3 + 729*x*y^2*z^4 + 5661*x*y^2*z^3 + 15453*x*y^2*z^2 + 17967*x*y^2*z + 7761*x*y^2 - 207*x*y*z^5 - 2007*x*y*z^4 - 7308*x*y*z^3 - 12789*x*y*z^2 - 11130*x*y*z - 3945*x*y + 24*x*z^6 + 279*x*z^5 + 1269*x*z^4 + 2966*x*z^3 + 3888*x*z^2 + 2772*x*z + 858*x + 24*y^7 - 132*y^6*z - 264*y^6 + 306*y^5*z^2 + 1224*y^5*z + 1134*y^5 - 387*y^4*z^3 - 2322*y^4*z^2 - 4311*y^4*z - 2510*y^4 + 288*y^3*z^4 + 2304*y^3*z^3 + 6426*y^3*z^2 + 7523*y^3*z + 3235*y^3 - 126*y^2*z^5 - 1260*y^2*z^4 - 4689*y^2*z^3 - 8271*y^2*z^2 - 7182*y^2*z - 2514*y^2 + 30*y*z^6 + 360*y*z^5 + 1674*y*z^4 + 3949*y*z^3 + 5181*y*z^2 + 3669*y*z + 1125*y - 3*z^7 - 42*z^6 - 234*z^5 - 691*z^4 - 1214*z^3 - 1299*z^2 - 800*z - 1, 2*x^7 + 19*x^6*y - 13*x^6*z - 23*x^6 + 75*x^5*y^2 - 105*x^5*y*z - 189*x^5*y + 36*x^5*z^2 + 129*x^5*z + 111*x^5 + 159*x^4*y^3 - 342*x^4*y^2*z - 627*x^4*y^2 + 240*x^4*y*z^2 + 876*x^4*y*z + 762*x^4*y - 55*x^4*z^3 - 300*x^4*z^2 - 522*x^4*z - 297*x^4 + 195*x^3*y^4 - 573*x^3*y^3*z - 1071*x^3*y^3 + 618*x^3*y^2*z^2 + 2301*x^3*y^2*z + 2025*x^3*y^2 - 290*x^3*y*z^3 - 1614*x^3*y*z^2 - 2844*x^3*y*z - 1626*x^3*y + 50*x^3*z^4 + 370*x^3*z^3 + 978*x^3*z^2 + 1121*x^3*z + 483*x^3 + 138*x^2*y^5 - 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1713*y*z^2 + 1202*y*z + 367*y - z^7 - 14*z^6 - 78*z^5 - 230*z^4 - 402*z^3 - 427*z^2 - 262*z - 1, -4*x^7 - 38*x^6*y + 26*x^6*z + 40*x^6 - 150*x^5*y^2 + 210*x^5*y*z + 327*x^5*y - 72*x^5*z^2 - 225*x^5*z - 177*x^5 - 318*x^4*y^3 + 684*x^4*y^2*z + 1080*x^4*y^2 - 480*x^4*y*z^2 - 1521*x^4*y*z - 1206*x^4*y + 110*x^4*z^3 + 525*x^4*z^2 + 834*x^4*z + 449*x^4 - 390*x^3*y^4 + 1146*x^3*y^3*z + 1839*x^3*y^3 - 1236*x^3*y^2*z^2 - 3981*x^3*y^2*z - 3183*x^3*y^2 + 580*x^3*y*z^3 + 2814*x^3*y*z^2 + 4512*x^3*y*z + 2437*x^3*y - 100*x^3*z^4 - 650*x^3*z^3 - 1566*x^3*z^2 - 1697*x^3*z - 707*x^3 - 276*x^2*y^5 + 1038*x^2*y^4*z + 1698*x^2*y^4 - 1530*x^2*y^3*z^2 - 5025*x^2*y^3*z - 4056*x^2*y^3 + 1104*x^2*y^2*z^3 + 5463*x^2*y^2*z^2 + 8850*x^2*y^2*z + 4791*x^2*y^2 - 390*x^2*y*z^4 - 2586*x^2*y*z^3 - 6300*x^2*y*z^2 - 6855*x^2*y*z - 2850*x^2*y + 54*x^2*z^5 + 450*x^2*z^4 + 1464*x^2*z^3 + 2397*x^2*z^2 + 2004*x^2*z + 695*x^2 - 104*x*y^6 + 480*x*y^5*z + 804*x*y^5 - 906*x*y^4*z^2 - 3048*x*y^4*z - 2490*x*y^4 + 894*x*y^3*z^3 + 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[ 2, 1, 1 ] ], "target_shift": [ 221, 72, -102 ], "collision_preimages": [ [ "5/4", "0", "1/4" ], [ "-7", "-5/2", "-12" ], [ "-4", "5/2", "-1" ] ], "collision_image": [ "887/4", "289/4", "-205/2" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0501122339992435 ], "surface_features": [ 3.308881452532317, 4.254475567803871, 2.9923325590474645, 0.0, 3.447128097903515, 4812 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C207", "pair": 207, "family": "counterexample", "mapping": "F(x,y,z)=(-48*x^7 - 264*x^6*y - 248*x^6*z - 364*x^6 - 612*x^5*y^2 - 1152*x^5*y*z - 1644*x^5*y - 540*x^5*z^2 - 1560*x^5*z - 1152*x^5 - 774*x^4*y^3 - 2190*x^4*y^2*z - 3051*x^4*y^2 - 2058*x^4*y*z^2 - 5790*x^4*y*z - 4224*x^4*y - 642*x^4*z^3 - 2739*x^4*z^2 - 4020*x^4*z - 2044*x^4 - 576*x^3*y^4 - 2178*x^3*y^3*z - 2976*x^3*y^3 - 3078*x^3*y^2*z^2 - 8475*x^3*y^2*z - 6114*x^3*y^2 - 1926*x^3*y*z^3 - 8022*x^3*y*z^2 - 11628*x^3*y*z - 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1029*x^4*y*z^2 - 3240*x^4*y*z - 2559*x^4*y - 321*x^4*z^3 - 1530*x^4*z^2 - 2433*x^4*z - 1298*x^4 - 288*x^3*y^4 - 1089*x^3*y^3*z - 1686*x^3*y^3 - 1539*x^3*y^2*z^2 - 4794*x^3*y^2*z - 3747*x^3*y^2 - 963*x^3*y*z^3 - 4530*x^3*y*z^2 - 7119*x^3*y*z - 3758*x^3*y - 225*x^3*z^4 - 1422*x^3*z^3 - 3372*x^3*z^2 - 3574*x^3*z - 1443*x^3 - 126*x^2*y^5 - 597*x^2*y^4*z - 921*x^2*y^4 - 1128*x^2*y^3*z^2 - 3495*x^2*y^3*z - 2706*x^2*y^3 - 1062*x^2*y^2*z^3 - 4959*x^2*y^2*z^2 - 7710*x^2*y^2*z - 4026*x^2*y^2 - 498*x^2*y*z^4 - 3117*x^2*y*z^3 - 7302*x^2*y*z^2 - 7647*x^2*y*z - 3051*x^2*y - 93*x^2*z^5 - 732*x^2*z^4 - 2298*x^2*z^3 - 3621*x^2*z^2 - 2892*x^2*z - 949*x^2 - 30*x*y^6 - 171*x*y^5*z - 264*x*y^5 - 405*x*y^4*z^2 - 1254*x*y^4*z - 963*x*y^4 - 510*x*y^3*z^3 - 2376*x*y^3*z^2 - 3660*x*y^3*z - 1891*x*y^3 - 360*x*y^2*z^4 - 2244*x*y^2*z^3 - 5202*x*y^2*z^2 - 5385*x*y^2*z - 2121*x*y^2 - 135*x*y*z^5 - 1056*x*y*z^4 - 3276*x*y*z^3 - 5097*x*y*z^2 - 4014*x*y*z - 1297*x*y - 21*x*z^6 - 198*x*z^5 - 771*x*z^4 - 1603*x*z^3 - 1893*x*z^2 - 1219*x*z - 341*x - 3*y^7 - 20*y^6*z - 31*y^6 - 57*y^5*z^2 - 177*y^5*z - 135*y^5 - 90*y^4*z^3 - 420*y^4*z^2 - 642*y^4*z - 328*y^4 - 85*y^3*z^4 - 530*y^3*z^3 - 1218*y^3*z^2 - 1246*y^3*z - 484*y^3 - 48*y^2*z^5 - 375*y^2*z^4 - 1152*y^2*z^3 - 1770*y^2*z^2 - 1374*y^2*z - 439*y^2 - 15*y*z^6 - 141*y*z^5 - 543*y*z^4 - 1114*y*z^3 - 1296*y*z^2 - 824*y*z - 230*y - 2*z^7 - 22*z^6 - 102*z^5 - 262*z^4 - 406*z^3 - 385*z^2 - 212*z - 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ 1, 1, 1 ], [ 2, 1, 1 ], [ 3, 3, 2 ] ], "source_shift": [ 1, 1, 1 ], "target_matrix": [ [ -2, -1, -1 ], [ 1, 1, 1 ], [ -1, -1, -2 ] ], "target_shift": [ 75, -53, 51 ], "collision_preimages": [ [ "0", "3/4", "-7/4" ], [ "-5/2", "8", "-11/2" ], [ "5/2", "7", "-23/2" ] ], "collision_image": [ "151/2", "-213/4", "205/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 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19254*x^2*y*z^2 + 19362*x^2*y*z - 7182*x^2*y - 204*x^2*z^5 + 1899*x^2*z^4 - 6123*x^2*z^3 + 9393*x^2*z^2 - 7086*x^2*z + 2138*x^2 + 88*x*y^6 + 624*x*y^5*z - 924*x*y^5 + 1734*x*y^4*z^2 - 5340*x*y^4*z + 3930*x*y^4 + 2398*x*y^3*z^3 - 11655*x*y^3*z^2 + 17682*x*y^3*z - 8647*x*y^3 + 1734*x*y^2*z^4 - 11964*x*y^2*z^3 + 28098*x*y^2*z^2 - 28122*x*y^2*z + 10356*x*y^2 + 624*x*y*z^5 - 5766*x*y*z^4 + 18570*x*y*z^3 - 28425*x*y*z^2 + 21348*x*y*z - 6394*x*y + 88*x*z^6 - 1044*x*z^5 + 4296*x*z^4 - 8894*x*z^3 + 10164*x*z^2 - 6194*x*z + 1592*x - 16*y^7 - 136*y^6*z + 200*y^6 - 468*y^5*z^2 + 1428*y^5*z - 1044*y^5 - 838*y^4*z^3 + 4026*y^4*z^2 - 6066*y^4*z + 2946*y^4 - 838*y^3*z^4 + 5711*y^3*z^3 - 13338*y^3*z^2 + 13273*y^3*z - 4848*y^3 - 468*y^2*z^5 + 4287*y^2*z^4 - 13779*y^2*z^3 + 21021*y^2*z^2 - 15687*y^2*z + 4652*y^2 - 136*y*z^6 + 1608*y*z^5 - 6624*y*z^4 + 13707*y*z^3 - 15624*y*z^2 + 9470*y*z - 2414*y - 16*z^7 + 236*z^6 - 1188*z^5 + 3107*z^4 - 4773*z^3 + 4400*z^2 - 2282*z + 1, -x^7 + 10*x^6*y + 10*x^6*z - 20*x^6 - 42*x^5*y^2 - 87*x^5*y*z + 174*x^5*y - 42*x^5*z^2 + 171*x^5*z - 156*x^5 + 96*x^4*y^3 + 309*x^4*y^2*z - 618*x^4*y^2 + 309*x^4*y*z^2 - 1257*x^4*y*z + 1143*x^4*y + 96*x^4*z^3 - 600*x^4*z^2 + 1119*x^4*z - 636*x^4 - 129*x^3*y^4 - 573*x^3*y^3*z + 1146*x^3*y^3 - 891*x^3*y^2*z^2 + 3621*x^3*y^2*z - 3279*x^3*y^2 - 573*x^3*y*z^3 + 3576*x^3*y*z^2 - 6657*x^3*y*z + 3772*x^3*y - 129*x^3*z^4 + 1107*x^3*z^3 - 3168*x^3*z^2 + 3654*x^3*z - 1499*x^3 + 102*x^2*y^5 + 585*x^2*y^4*z - 1170*x^2*y^4 + 1257*x^2*y^3*z^2 - 5103*x^2*y^3*z + 4599*x^2*y^3 + 1257*x^2*y^2*z^3 - 7830*x^2*y^2*z^2 + 14532*x^2*y^2*z - 8199*x^2*y^2 + 585*x^2*y*z^4 - 5013*x^2*y*z^3 + 14352*x^2*y*z^2 - 16533*x^2*y*z + 6756*x^2*y + 102*x^2*z^5 - 1134*x^2*z^4 + 4425*x^2*z^3 - 7758*x^2*z^2 + 6432*x^2*z - 2080*x^2 - 44*x*y^6 - 312*x*y^5*z + 624*x*y^5 - 867*x*y^4*z^2 + 3516*x*y^4*z - 3153*x*y^4 - 1199*x*y^3*z^3 + 7452*x*y^3*z^2 - 13773*x*y^3*z + 7729*x*y^3 - 867*x*y^2*z^4 + 7413*x*y^2*z^3 - 21204*x*y^2*z^2 + 24366*x*y^2*z - 9903*x*y^2 - 312*x*y*z^5 + 3468*x*y*z^4 - 13578*x*y*z^3 + 23823*x*y*z^2 - 19707*x*y*z + 6341*x*y - 44*x*z^6 + 612*x*z^5 - 3048*x*z^4 + 7208*x*z^3 - 9045*x*z^2 + 5899*x*z - 1593*x + 8*y^7 + 68*y^6*z - 136*y^6 + 234*y^5*z^2 - 948*y^5*z + 846*y^5 + 419*y^4*z^3 - 2598*y^4*z^2 + 4779*y^4*z - 2666*y^4 + 419*y^3*z^4 - 3571*y^3*z^3 + 10188*y^3*z^2 - 11662*y^3*z + 4708*y^3 + 234*y^2*z^5 - 2598*y^2*z^4 + 10197*y^2*z^3 - 17886*y^2*z^2 + 14736*y^2*z - 4705*y^2 + 68*y*z^6 - 948*y*z^5 + 4752*y*z^4 - 11269*y*z^3 + 14139*y*z^2 - 9196*y*z + 2473*y + 8*z^7 - 136*z^6 + 828*z^5 - 2471*z^4 + 4163*z^3 - 4099*z^2 + 2226*z - 1, -x^7 + 10*x^6*y + 10*x^6*z - 17*x^6 - 42*x^5*y^2 - 87*x^5*y*z + 147*x^5*y - 42*x^5*z^2 + 147*x^5*z - 120*x^5 + 96*x^4*y^3 + 309*x^4*y^2*z - 519*x^4*y^2 + 309*x^4*y*z^2 - 1074*x^4*y*z + 873*x^4*y + 96*x^4*z^3 - 522*x^4*z^2 + 870*x^4*z - 459*x^4 - 129*x^3*y^4 - 573*x^3*y^3*z + 957*x^3*y^3 - 891*x^3*y^2*z^2 + 3075*x^3*y^2*z - 2487*x^3*y^2 - 573*x^3*y*z^3 + 3093*x^3*y*z^2 - 5139*x^3*y*z + 2701*x^3*y - 129*x^3*z^4 + 975*x^3*z^3 - 2490*x^3*z^2 + 2664*x^3*z - 1037*x^3 + 102*x^2*y^5 + 585*x^2*y^4*z - 972*x^2*y^4 + 1257*x^2*y^3*z^2 - 4308*x^2*y^3*z + 3465*x^2*y^3 + 1257*x^2*y^2*z^3 - 6732*x^2*y^2*z^2 + 11139*x^2*y^2*z - 5826*x^2*y^2 + 585*x^2*y*z^4 - 4392*x^2*y*z^3 + 11202*x^2*y*z^2 - 11958*x^2*y*z + 4632*x^2*y + 102*x^2*z^5 - 1011*x^2*z^4 + 3516*x^2*z^3 - 5715*x^2*z^2 + 4497*x^2*z - 1399*x^2 - 44*x*y^6 - 312*x*y^5*z + 516*x*y^5 - 867*x*y^4*z^2 + 2952*x*y^4*z - 2361*x*y^4 - 1199*x*y^3*z^3 + 6369*x*y^3*z^2 - 10485*x*y^3*z + 5452*x*y^3 - 867*x*y^2*z^4 + 6459*x*y^2*z^3 - 16434*x*y^2*z^2 + 17484*x*y^2*z - 6729*x*y^2 - 312*x*y*z^5 + 3078*x*y*z^4 - 10716*x*y*z^3 + 17409*x*y*z^2 - 13653*x*y*z + 4220*x*y - 44*x*z^6 + 552*x*z^5 - 2448*x*z^4 + 5366*x*z^3 - 6393*x*z^2 + 4015*x*z - 1055*x + 8*y^7 + 68*y^6*z - 112*y^6 + 234*y^5*z^2 - 792*y^5*z + 630*y^5 + 419*y^4*z^3 - 2208*y^4*z^2 + 3615*y^4*z - 1868*y^4 + 419*y^3*z^4 - 3094*y^3*z^3 + 7842*y^3*z^2 - 8305*y^3*z + 3172*y^3 + 234*y^2*z^5 - 2295*y^2*z^4 + 7992*y^2*z^3 - 12963*y^2*z^2 + 10113*y^2*z - 3097*y^2 + 68*y*z^6 - 852*y*z^5 + 3792*y*z^4 - 8323*y*z^3 + 9903*y*z^2 - 6193*y*z + 1617*y + 8*z^7 - 124*z^6 + 672*z^5 - 1859*z^4 + 2975*z^3 - 2824*z^2 + 1495*z - 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ 1, -2, -1 ], [ 1, -1, -2 ], [ 1, -1, -1 ] ], "source_shift": [ 1, 2, 2 ], "target_matrix": [ [ 2, 1, 1 ], [ -1, -2, -1 ], [ -1, -1, -1 ] ], "target_shift": [ -517, 519, 343 ], "collision_preimages": [ [ "-15/4", "-5/4", "-1/4" ], [ "17", "9/2", "8" ], [ "16", "13/2", "5" ] ], "collision_image": [ "-1035/2", "2077/4", "1373/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.1140974449990608 ], "surface_features": [ 3.6204327735583113, 4.453700473359772, 3.255272505103306, 0.0, 3.7369102044530633, 4899 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B093", "pair": 93, "family": "bijection", "mapping": "F(x,y,z)=(128*x^7 + 896*x^6*y - 1344*x^6*z - 928*x^6 + 2688*x^5*y^2 - 8064*x^5*y*z - 5568*x^5*y + 6048*x^5*z^2 + 8336*x^5*z + 2976*x^5 + 4480*x^4*y^3 - 20160*x^4*y^2*z - 13920*x^4*y^2 + 30240*x^4*y*z^2 + 41680*x^4*y*z + 14880*x^4*y - 15120*x^4*z^3 - 31200*x^4*z^2 - 22240*x^4*z - 5480*x^4 + 4480*x^3*y^4 - 26880*x^3*y^3*z - 18560*x^3*y^3 + 60480*x^3*y^2*z^2 + 83360*x^3*y^2*z + 29760*x^3*y^2 - 60480*x^3*y*z^3 - 124800*x^3*y*z^2 - 88960*x^3*y*z - 21920*x^3*y + 22680*x^3*z^4 + 62280*x^3*z^3 + 66480*x^3*z^2 + 32716*x^3*z + 6284*x^3 + 2688*x^2*y^5 - 20160*x^2*y^4*z - 13920*x^2*y^4 + 60480*x^2*y^3*z^2 + 83360*x^2*y^3*z + 29760*x^2*y^3 - 90720*x^2*y^2*z^3 - 187200*x^2*y^2*z^2 - 133440*x^2*y^2*z - 32880*x^2*y^2 + 68040*x^2*y*z^4 + 186840*x^2*y*z^3 + 199440*x^2*y*z^2 + 98148*x^2*y*z + 18848*x^2*y - 20412*x^2*z^5 - 69930*x^2*z^4 - 99360*x^2*z^3 - 73242*x^2*z^2 - 28092*x^2*z - 4513*x^2 + 896*x*y^6 - 8064*x*y^5*z - 5568*x*y^5 + 30240*x*y^4*z^2 + 41680*x*y^4*z + 14880*x*y^4 - 60480*x*y^3*z^3 - 124800*x*y^3*z^2 - 88960*x*y^3*z - 21920*x*y^3 + 68040*x*y^2*z^4 + 186840*x*y^2*z^3 + 199440*x*y^2*z^2 + 98148*x*y^2*z + 18844*x*y^2 - 40824*x*y*z^5 - 139860*x*y*z^4 - 198720*x*y*z^3 - 146484*x*y*z^2 - 56172*x*y*z - 9017*x*y + 10206*x*z^6 + 41877*x*z^5 + 74250*x*z^4 + 72873*x*z^3 + 41859*x*z^2 + 13414*x*z + 1896*x + 128*y^7 - 1344*y^6*z - 928*y^6 + 6048*y^5*z^2 + 8336*y^5*z + 2976*y^5 - 15120*y^4*z^3 - 31200*y^4*z^2 - 22240*y^4*z - 5480*y^4 + 22680*y^3*z^4 + 62280*y^3*z^3 + 66480*y^3*z^2 + 32716*y^3*z + 6280*y^3 - 20412*y^2*z^5 - 69930*y^2*z^4 - 99360*y^2*z^3 - 73242*y^2*z^2 - 28080*y^2*z - 4504*y^2 + 10206*y*z^6 + 41877*y*z^5 + 74250*y*z^4 + 72873*y*z^3 + 41850*y*z^2 + 13401*y*z + 1889*y - 2187*z^7 - 10449*z^6 - 22194*z^5 - 27189*z^4 - 20790*z^3 - 9968*z^2 - 2801*z - 1, 128*x^7 + 896*x^6*y - 1344*x^6*z - 928*x^6 + 2688*x^5*y^2 - 8064*x^5*y*z - 5568*x^5*y + 6048*x^5*z^2 + 8336*x^5*z + 2976*x^5 + 4480*x^4*y^3 - 20160*x^4*y^2*z - 13920*x^4*y^2 + 30240*x^4*y*z^2 + 41680*x^4*y*z + 14880*x^4*y - 15120*x^4*z^3 - 31200*x^4*z^2 - 22240*x^4*z - 5480*x^4 + 4480*x^3*y^4 - 26880*x^3*y^3*z - 18560*x^3*y^3 + 60480*x^3*y^2*z^2 + 83360*x^3*y^2*z + 29760*x^3*y^2 - 60480*x^3*y*z^3 - 124800*x^3*y*z^2 - 88960*x^3*y*z - 21920*x^3*y + 22680*x^3*z^4 + 62280*x^3*z^3 + 66480*x^3*z^2 + 32716*x^3*z + 6284*x^3 + 2688*x^2*y^5 - 20160*x^2*y^4*z - 13920*x^2*y^4 + 60480*x^2*y^3*z^2 + 83360*x^2*y^3*z + 29760*x^2*y^3 - 90720*x^2*y^2*z^3 - 187200*x^2*y^2*z^2 - 133440*x^2*y^2*z - 32880*x^2*y^2 + 68040*x^2*y*z^4 + 186840*x^2*y*z^3 + 199440*x^2*y*z^2 + 98148*x^2*y*z + 18848*x^2*y - 20412*x^2*z^5 - 69930*x^2*z^4 - 99360*x^2*z^3 - 73242*x^2*z^2 - 28092*x^2*z - 4505*x^2 + 896*x*y^6 - 8064*x*y^5*z - 5568*x*y^5 + 30240*x*y^4*z^2 + 41680*x*y^4*z + 14880*x*y^4 - 60480*x*y^3*z^3 - 124800*x*y^3*z^2 - 88960*x*y^3*z - 21920*x*y^3 + 68040*x*y^2*z^4 + 186840*x*y^2*z^3 + 199440*x*y^2*z^2 + 98148*x*y^2*z + 18844*x*y^2 - 40824*x*y*z^5 - 139860*x*y*z^4 - 198720*x*y*z^3 - 146484*x*y*z^2 - 56172*x*y*z - 9001*x*y + 10206*x*z^6 + 41877*x*z^5 + 74250*x*z^4 + 72873*x*z^3 + 41859*x*z^2 + 13390*x*z + 1878*x + 128*y^7 - 1344*y^6*z - 928*y^6 + 6048*y^5*z^2 + 8336*y^5*z + 2976*y^5 - 15120*y^4*z^3 - 31200*y^4*z^2 - 22240*y^4*z - 5480*y^4 + 22680*y^3*z^4 + 62280*y^3*z^3 + 66480*y^3*z^2 + 32716*y^3*z + 6280*y^3 - 20412*y^2*z^5 - 69930*y^2*z^4 - 99360*y^2*z^3 - 73242*y^2*z^2 - 28080*y^2*z - 4496*y^2 + 10206*y*z^6 + 41877*y*z^5 + 74250*y*z^4 + 72873*y*z^3 + 41850*y*z^2 + 13377*y*z + 1871*y - 2187*z^7 - 10449*z^6 - 22194*z^5 - 27189*z^4 - 20790*z^3 - 9950*z^2 - 2775*z - 1, -256*x^7 - 1792*x^6*y + 2688*x^6*z + 1856*x^6 - 5376*x^5*y^2 + 16128*x^5*y*z + 11136*x^5*y - 12096*x^5*z^2 - 16672*x^5*z - 5920*x^5 - 8960*x^4*y^3 + 40320*x^4*y^2*z + 27840*x^4*y^2 - 60480*x^4*y*z^2 - 83360*x^4*y*z - 29600*x^4*y + 30240*x^4*z^3 + 62400*x^4*z^2 + 44240*x^4*z + 10800*x^4 - 8960*x^3*y^4 + 53760*x^3*y^3*z + 37120*x^3*y^3 - 120960*x^3*y^2*z^2 - 166720*x^3*y^2*z - 59200*x^3*y^2 + 120960*x^3*y*z^3 + 249600*x^3*y*z^2 + 176960*x^3*y*z + 43200*x^3*y - 45360*x^3*z^4 - 124560*x^3*z^3 - 132240*x^3*z^2 - 64472*x^3*z - 12240*x^3 - 5376*x^2*y^5 + 40320*x^2*y^4*z + 27840*x^2*y^4 - 120960*x^2*y^3*z^2 - 166720*x^2*y^3*z - 59200*x^2*y^3 + 181440*x^2*y^2*z^3 + 374400*x^2*y^2*z^2 + 265440*x^2*y^2*z + 64800*x^2*y^2 - 136080*x^2*y*z^4 - 373680*x^2*y*z^3 - 396720*x^2*y*z^2 - 193416*x^2*y*z - 36712*x^2*y + 40824*x^2*z^5 + 139860*x^2*z^4 + 197640*x^2*z^3 + 144324*x^2*z^2 + 54708*x^2*z + 8686*x^2 - 1792*x*y^6 + 16128*x*y^5*z + 11136*x*y^5 - 60480*x*y^4*z^2 - 83360*x*y^4*z - 29600*x*y^4 + 120960*x*y^3*z^3 + 249600*x*y^3*z^2 + 176960*x*y^3*z + 43200*x*y^3 - 136080*x*y^2*z^4 - 373680*x*y^2*z^3 - 396720*x*y^2*z^2 - 193416*x*y^2*z - 36704*x*y^2 + 81648*x*y*z^5 + 279720*x*y*z^4 + 395280*x*y*z^3 + 288648*x*y*z^2 + 109392*x*y*z + 17354*x*y - 20412*x*z^6 - 83754*x*z^5 - 147690*x*z^4 - 143586*x*z^3 - 81504*x*z^2 - 25808*x*z - 3614*x - 256*y^7 + 2688*y^6*z + 1856*y^6 - 12096*y^5*z^2 - 16672*y^5*z - 5920*y^5 + 30240*y^4*z^3 + 62400*y^4*z^2 + 44240*y^4*z + 10800*y^4 - 45360*y^3*z^4 - 124560*y^3*z^3 - 132240*y^3*z^2 - 64472*y^3*z - 12232*y^3 + 40824*y^2*z^5 + 139860*y^2*z^4 + 197640*y^2*z^3 + 144324*y^2*z^2 + 54684*y^2*z + 8668*y^2 - 20412*y*z^6 - 83754*y*z^5 - 147690*y*z^4 - 143586*y*z^3 - 81486*y*z^2 - 25782*y*z - 3601*y + 4374*z^7 + 20898*z^6 + 44145*z^5 + 53568*z^4 + 40473*z^3 + 19171*z^2 + 5336*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -2, -2, 3 ], [ -2, -1, 1 ], [ -1, -1, 1 ] ], "source_shift": [ 2, 2, 2 ], "target_matrix": [ [ -1, -1, -1 ], [ -1, -1, -2 ], [ 2, 1, 1 ] ], "target_shift": [ 361, 349, -687 ], "inverse_recipe": { "p": 5, "q": 2, "f": "x**5 + x**3 + x**2 + x", "g": "x**2", "h": "y*z + z**2", "polynomials": { "f": [ [ [ 5, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ], [ [ 3, 0, 0 ], 1 ] ], "g": [ [ [ 2, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 0, 2 ], 1 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.053687308998633 ], "surface_features": [ 4.76184044059425, 5.5984840952830535, 4.480581786829169, 0.0, 4.854209378112198, 5354 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C089", "pair": 89, "family": "counterexample", "mapping": "F(x,y,z)=(-8*x^7 - 44*x^6*y + 52*x^6*z + 80*x^6 - 102*x^5*y^2 + 240*x^5*y*z + 372*x^5*y - 138*x^5*z^2 - 432*x^5*z - 318*x^5 - 129*x^4*y^3 + 453*x^4*y^2*z + 708*x^4*y^2 - 519*x^4*y*z^2 - 1632*x^4*y*z - 1221*x^4*y + 195*x^4*z^3 + 924*x^4*z^2 + 1383*x^4*z + 664*x^4 - 96*x^3*y^4 + 447*x^3*y^3*z + 705*x^3*y^3 - 765*x^3*y^2*z^2 - 2418*x^3*y^2*z - 1839*x^3*y^2 + 573*x^3*y*z^3 + 2721*x^3*y*z^2 + 4131*x^3*y*z + 2028*x^3*y - 159*x^3*z^4 - 1008*x^3*z^3 - 2292*x^3*z^2 - 2249*x^3*z - 804*x^3 - 42*x^2*y^5 + 243*x^2*y^4*z + 387*x^2*y^4 - 552*x^2*y^3*z^2 - 1755*x^2*y^3*z - 1356*x^2*y^3 + 618*x^2*y^2*z^3 + 2943*x^2*y^2*z^2 + 4530*x^2*y^2*z + 2271*x^2*y^2 - 342*x^2*y*z^4 - 2169*x^2*y*z^3 - 4992*x^2*y*z^2 - 5001*x^2*y*z - 1839*x^2*y + 75*x^2*z^5 + 594*x^2*z^4 + 1818*x^2*z^3 + 2730*x^2*z^2 + 2016*x^2*z + 589*x^2 - 10*x*y^6 + 69*x*y^5*z + 111*x*y^5 - 195*x*y^4*z^2 - 624*x*y^4*z - 489*x*y^4 + 290*x*y^3*z^3 + 1386*x*y^3*z^2 + 2160*x*y^3*z + 1103*x*y^3 - 240*x*y^2*z^4 - 1524*x*y^2*z^3 - 3546*x*y^2*z^2 - 3615*x*y^2*z - 1365*x*y^2 + 105*x*y*z^5 + 831*x*y*z^4 + 2568*x*y*z^3 + 3921*x*y*z^2 + 2970*x*y*z + 899*x*y - 19*x*z^6 - 180*x*z^5 - 693*x*z^4 - 1409*x*z^3 - 1605*x*z^2 - 983*x*z - 259*x - y^7 + 8*y^6*z + 13*y^6 - 27*y^5*z^2 - 87*y^5*z - 69*y^5 + 50*y^4*z^3 + 240*y^4*z^2 + 378*y^4*z + 196*y^4 - 55*y^3*z^4 - 350*y^3*z^3 - 822*y^3*z^2 - 850*y^3*z - 328*y^3 + 36*y^2*z^5 + 285*y^2*z^4 + 888*y^2*z^3 + 1374*y^2*z^2 + 1062*y^2*z + 331*y^2 - 13*y*z^6 - 123*y*z^5 - 477*y*z^4 - 982*y*z^3 - 1140*y*z^2 - 716*y*z - 194*y + 2*z^7 + 22*z^6 + 102*z^5 + 262*z^4 + 406*z^3 + 385*z^2 + 212*z + 1, 8*x^7 + 44*x^6*y - 52*x^6*z - 80*x^6 + 102*x^5*y^2 - 240*x^5*y*z - 372*x^5*y + 138*x^5*z^2 + 432*x^5*z + 318*x^5 + 129*x^4*y^3 - 453*x^4*y^2*z - 708*x^4*y^2 + 519*x^4*y*z^2 + 1632*x^4*y*z + 1221*x^4*y - 195*x^4*z^3 - 924*x^4*z^2 - 1383*x^4*z - 672*x^4 + 96*x^3*y^4 - 447*x^3*y^3*z - 705*x^3*y^3 + 765*x^3*y^2*z^2 + 2418*x^3*y^2*z + 1839*x^3*y^2 - 573*x^3*y*z^3 - 2721*x^3*y*z^2 - 4131*x^3*y*z - 2048*x^3*y + 159*x^3*z^4 + 1008*x^3*z^3 + 2292*x^3*z^2 + 2277*x^3*z + 836*x^3 + 42*x^2*y^5 - 243*x^2*y^4*z - 387*x^2*y^4 + 552*x^2*y^3*z^2 + 1755*x^2*y^3*z + 1356*x^2*y^3 - 618*x^2*y^2*z^3 - 2943*x^2*y^2*z^2 - 4530*x^2*y^2*z - 2289*x^2*y^2 + 342*x^2*y*z^4 + 2169*x^2*y*z^3 + 4992*x^2*y*z^2 + 5049*x^2*y*z + 1899*x^2*y - 75*x^2*z^5 - 594*x^2*z^4 - 1818*x^2*z^3 - 2760*x^2*z^2 - 2088*x^2*z - 631*x^2 + 10*x*y^6 - 69*x*y^5*z - 111*x*y^5 + 195*x*y^4*z^2 + 624*x*y^4*z + 489*x*y^4 - 290*x*y^3*z^3 - 1386*x*y^3*z^2 - 2160*x*y^3*z - 1110*x*y^3 + 240*x*y^2*z^4 + 1524*x*y^2*z^3 + 3546*x*y^2*z^2 + 3642*x*y^2*z + 1401*x*y^2 - 105*x*y*z^5 - 831*x*y*z^4 - 2568*x*y*z^3 - 3954*x*y*z^2 - 3054*x*y*z - 950*x*y + 19*x*z^6 + 180*x*z^5 + 693*x*z^4 + 1422*x*z^3 + 1653*x*z^2 + 1040*x*z + 277*x + y^7 - 8*y^6*z - 13*y^6 + 27*y^5*z^2 + 87*y^5*z + 69*y^5 - 50*y^4*z^3 - 240*y^4*z^2 - 378*y^4*z - 197*y^4 + 55*y^3*z^4 + 350*y^3*z^3 + 822*y^3*z^2 + 855*y^3*z + 335*y^3 - 36*y^2*z^5 - 285*y^2*z^4 - 888*y^2*z^3 - 1383*y^2*z^2 - 1086*y^2*z - 346*y^2 + 13*y*z^6 + 123*y*z^5 + 477*y*z^4 + 989*y*z^3 + 1167*y*z^2 + 749*y*z + 205*y - 2*z^7 - 22*z^6 - 102*z^5 - 264*z^4 - 416*z^3 - 403*z^2 - 224*z - 1, 8*x^7 + 44*x^6*y - 52*x^6*z - 104*x^6 + 102*x^5*y^2 - 240*x^5*y*z - 480*x^5*y + 138*x^5*z^2 + 564*x^5*z + 462*x^5 + 129*x^4*y^3 - 453*x^4*y^2*z - 906*x^4*y^2 + 519*x^4*y*z^2 + 2112*x^4*y*z + 1761*x^4*y - 195*x^4*z^3 - 1206*x^4*z^2 - 2007*x^4*z - 1038*x^4 + 96*x^3*y^4 - 447*x^3*y^3*z - 894*x^3*y^3 + 765*x^3*y^2*z^2 + 3099*x^3*y^2*z + 2631*x^3*y^2 - 573*x^3*y*z^3 - 3516*x^3*y*z^2 - 5943*x^3*y*z - 3137*x^3*y + 159*x^3*z^4 + 1311*x^3*z^3 + 3312*x^3*z^2 + 3504*x^3*z + 1346*x^3 + 42*x^2*y^5 - 243*x^2*y^4*z - 486*x^2*y^4 + 552*x^2*y^3*z^2 + 2226*x^2*y^3*z + 1923*x^2*y^3 - 618*x^2*y^2*z^3 - 3762*x^2*y^2*z^2 - 6456*x^2*y^2*z - 3474*x^2*y^2 + 342*x^2*y*z^4 + 2790*x^2*y*z^3 + 7143*x^2*y*z^2 + 7692*x^2*y*z + 3024*x^2*y - 75*x^2*z^5 - 768*x^2*z^4 - 2610*x^2*z^3 - 4218*x^2*z^2 - 3336*x^2*z - 1045*x^2 + 10*x*y^6 - 69*x*y^5*z - 138*x*y^5 + 195*x*y^4*z^2 + 783*x*y^4*z + 687*x*y^4 - 290*x*y^3*z^3 - 1752*x*y^3*z^2 - 3048*x*y^3*z - 1668*x*y^3 + 240*x*y^2*z^4 + 1938*x*y^2*z^3 + 5022*x*y^2*z^2 + 5490*x*y^2*z + 2205*x*y^2 - 105*x*y*z^5 - 1062*x*y*z^4 - 3648*x*y*z^3 - 5976*x*y*z^2 - 4818*x*y*z - 1550*x*y + 19*x*z^6 + 231*x*z^5 + 987*x*z^4 + 2154*x*z^3 + 2613*x*z^2 + 1700*x*z + 461*x + y^7 - 8*y^6*z - 16*y^6 + 27*y^5*z^2 + 108*y^5*z + 96*y^5 - 50*y^4*z^3 - 300*y^4*z^2 - 528*y^4*z - 293*y^4 + 55*y^3*z^4 + 440*y^3*z^3 + 1152*y^3*z^2 + 1275*y^3*z + 521*y^3 - 36*y^2*z^5 - 360*y^2*z^4 - 1248*y^2*z^3 - 2067*y^2*z^2 - 1692*y^2*z - 556*y^2 + 13*y*z^6 + 156*y*z^5 + 672*y*z^4 + 1481*y*z^3 + 1821*y*z^2 + 1205*y*z + 335*y - 2*z^7 - 28*z^6 - 144*z^5 - 396*z^4 - 650*z^3 - 649*z^2 - 366*z - 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -2, -1, 1 ], [ -1, -1, 1 ], [ -1, -1, 2 ] ], "source_shift": [ 1, 1, 1 ], "target_matrix": [ [ 1, 1, 2 ], [ -1, -1, -1 ], [ -1, -2, -1 ] ], "target_shift": [ -51, 53, 87 ], "collision_preimages": [ [ "0", "3/4", "-1/4" ], [ "-5/2", "13", "8" ], [ "5/2", "2", "5" ] ], "collision_image": [ "-205/4", "213/4", "349/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 0.9993349470023531 ], "surface_features": [ 3.038456779631996, 3.8860392755664424, 2.74350976472843, 0.0, 3.128360373433823, 4726 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C246", "pair": 246, "family": "counterexample", "mapping": "F(x,y,z)=(x^7 + 8*x^6*y - 10*x^6*z - 16*x^6 + 27*x^5*y^2 - 69*x^5*y*z - 108*x^5*y + 42*x^5*z^2 + 135*x^5*z + 105*x^5 + 50*x^4*y^3 - 195*x^4*y^2*z - 300*x^4*y^2 + 243*x^4*y*z^2 + 765*x^4*y*z + 585*x^4*y - 96*x^4*z^3 - 465*x^4*z^2 - 726*x^4*z - 369*x^4 + 55*x^3*y^4 - 290*x^3*y^3*z - 440*x^3*y^3 + 552*x^3*y^2*z^2 + 1710*x^3*y^2*z + 1290*x^3*y^2 - 447*x^3*y*z^3 - 2124*x^3*y*z^2 - 3261*x^3*y*z - 1637*x^3*y + 129*x^3*z^4 + 837*x^3*z^3 + 1965*x^3*z^2 + 1999*x^3*z + 754*x^3 + 36*x^2*y^5 - 240*x^2*y^4*z - 360*x^2*y^4 + 618*x^2*y^3*z^2 + 1890*x^2*y^3*z + 1410*x^2*y^3 - 765*x^2*y^2*z^3 - 3582*x^2*y^2*z^2 - 5427*x^2*y^2*z - 2697*x^2*y^2 + 453*x^2*y*z^4 + 2889*x^2*y*z^3 + 6672*x^2*y*z^2 + 6705*x^2*y*z + 2508*x^2*y - 102*x^2*z^5 - 831*x^2*z^4 - 2604*x^2*z^3 - 3969*x^2*z^2 - 2985*x^2*z - 898*x^2 + 13*x*y^6 - 105*x*y^5*z - 156*x*y^5 + 342*x*y^4*z^2 + 1035*x*y^4*z + 765*x*y^4 - 573*x*y^3*z^3 - 2652*x*y^3*z^2 - 3975*x*y^3*z - 1959*x*y^3 + 519*x*y^2*z^4 + 3267*x*y^2*z^3 + 7449*x*y^2*z^2 + 7413*x*y^2*z + 2754*x*y^2 - 240*x*y*z^5 - 1926*x*y*z^4 - 5940*x*y*z^3 - 8946*x*y*z^2 - 6675*x*y*z - 2000*x*y + 44*x*z^6 + 432*x*z^5 + 1692*x*z^4 + 3429*x*z^3 + 3846*x*z^2 + 2294*x*z + 577*x + 2*y^7 - 19*y^6*z - 28*y^6 + 75*y^5*z^2 + 225*y^5*z + 165*y^5 - 159*y^4*z^3 - 729*y^4*z^2 - 1083*y^4*z - 530*y^4 + 195*y^3*z^4 + 1215*y^3*z^3 + 2742*y^3*z^2 + 2707*y^3*z + 1000*y^3 - 138*y^2*z^5 - 1095*y^2*z^4 - 3336*y^2*z^3 - 4977*y^2*z^2 - 3690*y^2*z - 1102*y^2 + 52*y*z^6 + 504*y*z^5 + 1944*y*z^4 + 3893*y*z^3 + 4332*y*z^2 + 2576*y*z + 650*y - 8*z^7 - 92*z^6 - 432*z^5 - 1090*z^4 - 1618*z^3 - 1432*z^2 - 712*z - 1, -2*x^7 - 16*x^6*y + 20*x^6*z + 29*x^6 - 54*x^5*y^2 + 138*x^5*y*z + 195*x^5*y - 84*x^5*z^2 - 243*x^5*z - 174*x^5 - 100*x^4*y^3 + 390*x^4*y^2*z + 540*x^4*y^2 - 486*x^4*y*z^2 - 1371*x^4*y*z - 966*x^4*y + 192*x^4*z^3 + 831*x^4*z^2 + 1191*x^4*z + 571*x^4 - 110*x^3*y^4 + 580*x^3*y^3*z + 790*x^3*y^3 - 1104*x^3*y^2*z^2 - 3054*x^3*y^2*z - 2124*x^3*y^2 + 894*x^3*y*z^3 + 3777*x^3*y*z^2 + 5328*x^3*y*z + 2526*x^3*y - 258*x^3*z^4 - 1485*x^3*z^3 - 3192*x^3*z^2 - 3060*x^3*z - 1110*x^3 - 72*x^2*y^5 + 480*x^2*y^4*z + 645*x^2*y^4 - 1236*x^2*y^3*z^2 - 3366*x^2*y^3*z - 2316*x^2*y^3 + 1530*x^2*y^2*z^3 + 6345*x^2*y^2*z^2 + 8838*x^2*y^2*z + 4152*x^2*y^2 - 906*x^2*y*z^4 - 5097*x^2*y*z^3 - 10788*x^2*y*z^2 - 10230*x^2*y*z - 3684*x^2*y + 204*x^2*z^5 + 1464*x^2*z^4 + 4191*x^2*z^3 + 6015*x^2*z^2 + 4350*x^2*z + 1277*x^2 - 26*x*y^6 + 210*x*y^5*z + 279*x*y^5 - 684*x*y^4*z^2 - 1839*x*y^4*z - 1254*x*y^4 + 1146*x*y^3*z^3 + 4683*x*y^3*z^2 + 6456*x*y^3*z + 3010*x*y^3 - 1038*x*y^2*z^4 - 5739*x*y^2*z^3 - 12000*x*y^2*z^2 - 11280*x*y^2*z - 4038*x*y^2 + 480*x*y*z^5 + 3372*x*y*z^4 + 9510*x*y*z^3 + 13506*x*y*z^2 + 9702*x*y*z + 2839*x*y - 88*x*z^6 - 756*x*z^5 - 2700*x*z^4 - 5150*x*z^3 - 5556*x*z^2 - 3235*x*z - 802*x - 4*y^7 + 38*y^6*z + 50*y^6 - 150*y^5*z^2 - 399*y^5*z - 270*y^5 + 318*y^4*z^3 + 1284*y^4*z^2 + 1755*y^4*z + 813*y^4 - 390*y^3*z^4 - 2127*y^3*z^3 - 4404*y^3*z^2 - 4110*y^3*z - 1464*y^3 + 276*y^2*z^5 + 1908*y^2*z^4 + 5319*y^2*z^3 + 7491*y^2*z^2 + 5352*y^2*z + 1562*y^2 - 104*y*z^6 - 876*y*z^5 - 3084*y*z^4 - 5822*y*z^3 - 6240*y*z^2 - 3625*y*z - 902*y + 16*z^7 + 160*z^6 + 684*z^5 + 1624*z^4 + 2320*z^3 + 2006*z^2 + 983*z + 1, x^7 + 8*x^6*y - 10*x^6*z - 19*x^6 + 27*x^5*y^2 - 69*x^5*y*z - 129*x^5*y + 42*x^5*z^2 + 162*x^5*z + 141*x^5 + 50*x^4*y^3 - 195*x^4*y^2*z - 360*x^4*y^2 + 243*x^4*y*z^2 + 924*x^4*y*z + 789*x^4*y - 96*x^4*z^3 - 564*x^4*z^2 - 987*x^4*z - 537*x^4 + 55*x^3*y^4 - 290*x^3*y^3*z - 530*x^3*y^3 + 552*x^3*y^2*z^2 + 2076*x^3*y^2*z + 1746*x^3*y^2 - 447*x^3*y*z^3 - 2595*x^3*y*z^2 - 4455*x^3*y*z - 2390*x^3*y + 129*x^3*z^4 + 1026*x^3*z^3 + 2703*x^3*z^2 + 2944*x^3*z + 1162*x^3 + 36*x^2*y^5 - 240*x^2*y^4*z - 435*x^2*y^4 + 618*x^2*y^3*z^2 + 2304*x^2*y^3*z + 1914*x^2*y^3 - 765*x^2*y^2*z^3 - 4401*x^2*y^2*z^2 - 7443*x^2*y^2*z - 3948*x^2*y^2 + 453*x^2*y*z^4 + 3570*x^2*y*z^3 + 9228*x^2*y*z^2 + 9912*x^2*y*z + 3876*x^2*y - 102*x^2*z^5 - 1029*x^2*z^4 - 3621*x^2*z^3 - 5910*x^2*z^2 - 4656*x^2*z - 1450*x^2 + 13*x*y^6 - 105*x*y^5*z - 189*x*y^5 + 342*x*y^4*z^2 + 1266*x*y^4*z + 1041*x*y^4 - 573*x*y^3*z^3 - 3273*x*y^3*z^2 - 5469*x*y^3*z - 2874*x*y^3 + 519*x*y^2*z^4 + 4062*x*y^2*z^3 + 10347*x*y^2*z^2 + 10992*x*y^2*z + 4266*x*y^2 - 240*x*y*z^5 - 2406*x*y*z^4 - 8310*x*y*z^3 - 13380*x*y*z^2 - 10449*x*y*z - 3239*x*y + 44*x*z^6 + 540*x*z^5 + 2376*x*z^4 + 5157*x*z^3 + 6066*x*z^2 + 3755*x*z + 971*x + 2*y^7 - 19*y^6*z - 34*y^6 + 75*y^5*z^2 + 276*y^5*z + 225*y^5 - 159*y^4*z^3 - 903*y^4*z^2 - 1494*y^4*z - 779*y^4 + 195*y^3*z^4 + 1518*y^3*z^3 + 3822*y^3*z^2 + 4024*y^3*z + 1552*y^3 - 138*y^2*z^5 - 1377*y^2*z^4 - 4689*y^2*z^3 - 7470*y^2*z^2 - 5793*y^2*z - 1789*y^2 + 52*y*z^6 + 636*y*z^5 + 2748*y*z^4 + 5885*y*z^3 + 6864*y*z^2 + 4235*y*z + 1098*y - 8*z^7 - 116*z^6 - 612*z^5 - 1654*z^4 - 2578*z^3 - 2374*z^2 - 1217*z - 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -1, -1, 2 ], [ -1, -1, 1 ], [ -1, -2, 1 ] ], "source_shift": [ 2, 1, 1 ], "target_matrix": [ [ -1, -1, -1 ], [ 2, 1, 1 ], [ -1, -2, -1 ] ], "target_shift": [ 155, -212, 270 ], "collision_preimages": [ [ "-1/4", "1/4", "-1" ], [ "12", "-8", "3/2" ], [ "1", "-5", "-7/2" ] ], "collision_image": [ "621/4", "-425/2", "1081/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0567979679981363 ], "surface_features": [ 3.3138268108311246, 4.130526745384164, 2.9934362304976116, 0.0, 3.413758112610727, 4803 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B034", "pair": 34, "family": "bijection", "mapping": "F(x,y,z)=(512*x^7 - 1792*x^6*y + 1792*x^6*z - 2304*x^6 + 2688*x^5*y^2 - 5376*x^5*y*z + 6912*x^5*y + 2688*x^5*z^2 - 6912*x^5*z + 4416*x^5 - 2240*x^4*y^3 + 6720*x^4*y^2*z - 8640*x^4*y^2 - 6720*x^4*y*z^2 + 17280*x^4*y*z - 11072*x^4*y + 2240*x^4*z^3 - 8640*x^4*z^2 + 11072*x^4*z - 4944*x^4 + 1120*x^3*y^4 - 4480*x^3*y^3*z + 5760*x^3*y^3 + 6720*x^3*y^2*z^2 - 17280*x^3*y^2*z + 11104*x^3*y^2 - 4480*x^3*y*z^3 + 17280*x^3*y*z^2 - 22208*x^3*y*z + 10048*x^3*y + 1120*x^3*z^4 - 5760*x^3*z^3 + 11104*x^3*z^2 - 10016*x^3*z + 3720*x^3 - 336*x^2*y^5 + 1680*x^2*y^4*z - 2160*x^2*y^4 - 3360*x^2*y^3*z^2 + 8640*x^2*y^3*z - 5568*x^2*y^3 + 3360*x^2*y^2*z^3 - 12960*x^2*y^2*z^2 + 16704*x^2*y^2*z - 7656*x^2*y^2 - 1680*x^2*y*z^4 + 8640*x^2*y*z^3 - 16704*x^2*y*z^2 + 15264*x^2*y*z - 5780*x^2*y + 336*x^2*z^5 - 2160*x^2*z^4 + 5568*x^2*z^3 - 7608*x^2*z^2 + 5732*x^2*z - 1912*x^2 + 56*x*y^6 - 336*x*y^5*z + 432*x*y^5 + 840*x*y^4*z^2 - 2160*x*y^4*z + 1396*x*y^4 - 1120*x*y^3*z^3 + 4320*x*y^3*z^2 - 5584*x*y^3*z + 2592*x*y^3 + 840*x*y^2*z^4 - 4320*x*y^2*z^3 + 8376*x*y^2*z^2 - 7752*x*y^2*z + 2990*x*y^2 - 336*x*y*z^5 + 2160*x*y*z^4 - 5584*x*y*z^3 + 7728*x*y*z^2 - 5932*x*y*z + 2020*x*y + 56*x*z^6 - 432*x*z^5 + 1396*x*z^4 - 2568*x*z^3 + 2942*x*z^2 - 1992*x*z + 621*x - 4*y^7 + 28*y^6*z - 36*y^6 - 84*y^5*z^2 + 216*y^5*z - 140*y^5 + 140*y^4*z^3 - 540*y^4*z^2 + 700*y^4*z - 329*y^4 - 140*y^3*z^4 + 720*y^3*z^3 - 1400*y^3*z^2 + 1312*y^3*z - 515*y^3 + 84*y^2*z^5 - 540*y^2*z^4 + 1400*y^2*z^3 - 1962*y^2*z^2 + 1533*y^2*z - 536*y^2 - 28*y*z^6 + 216*y*z^5 - 700*y*z^4 + 1304*y*z^3 - 1521*y*z^2 + 1056*y*z - 357*y + 4*z^7 - 36*z^6 + 140*z^5 - 325*z^4 + 503*z^3 - 520*z^2 + 340*z - 1, -256*x^7 + 896*x^6*y - 896*x^6*z + 1152*x^6 - 1344*x^5*y^2 + 2688*x^5*y*z - 3456*x^5*y - 1344*x^5*z^2 + 3456*x^5*z - 2208*x^5 + 1120*x^4*y^3 - 3360*x^4*y^2*z + 4320*x^4*y^2 + 3360*x^4*y*z^2 - 8640*x^4*y*z + 5536*x^4*y - 1120*x^4*z^3 + 4320*x^4*z^2 - 5536*x^4*z + 2496*x^4 - 560*x^3*y^4 + 2240*x^3*y^3*z - 2880*x^3*y^3 - 3360*x^3*y^2*z^2 + 8640*x^3*y^2*z - 5552*x^3*y^2 + 2240*x^3*y*z^3 - 8640*x^3*y*z^2 + 11104*x^3*y*z - 5072*x^3*y - 560*x^3*z^4 + 2880*x^3*z^3 - 5552*x^3*z^2 + 5056*x^3*z - 1912*x^3 + 168*x^2*y^5 - 840*x^2*y^4*z + 1080*x^2*y^4 + 1680*x^2*y^3*z^2 - 4320*x^2*y^3*z + 2784*x^2*y^3 - 1680*x^2*y^2*z^3 + 6480*x^2*y^2*z^2 - 8352*x^2*y^2*z + 3864*x^2*y^2 + 840*x^2*y*z^4 - 4320*x^2*y*z^3 + 8352*x^2*y*z^2 - 7704*x^2*y*z + 2968*x^2*y - 168*x^2*z^5 + 1080*x^2*z^4 - 2784*x^2*z^3 + 3840*x^2*z^2 - 2944*x^2*z + 1004*x^2 - 28*x*y^6 + 168*x*y^5*z - 216*x*y^5 - 420*x*y^4*z^2 + 1080*x*y^4*z - 698*x*y^4 + 560*x*y^3*z^3 - 2160*x*y^3*z^2 + 2792*x*y^3*z - 1308*x*y^3 - 420*x*y^2*z^4 + 2160*x*y^2*z^3 - 4188*x*y^2*z^2 + 3912*x*y^2*z - 1534*x*y^2 + 168*x*y*z^5 - 1080*x*y*z^4 + 2792*x*y*z^3 - 3900*x*y*z^2 + 3044*x*y*z - 1058*x*y - 28*x*z^6 + 216*x*z^5 - 698*x*z^4 + 1296*x*z^3 - 1510*x*z^2 + 1044*x*z - 332*x + 2*y^7 - 14*y^6*z + 18*y^6 + 42*y^5*z^2 - 108*y^5*z + 70*y^5 - 70*y^4*z^3 + 270*y^4*z^2 - 350*y^4*z + 166*y^4 + 70*y^3*z^4 - 360*y^3*z^3 + 700*y^3*z^2 - 662*y^3*z + 264*y^3 - 42*y^2*z^5 + 270*y^2*z^4 - 700*y^2*z^3 + 990*y^2*z^2 - 786*y^2*z + 280*y^2 + 14*y*z^6 - 108*y*z^5 + 350*y*z^4 - 658*y*z^3 + 780*y*z^2 - 552*y*z + 191*y - 2*z^7 + 18*z^6 - 70*z^5 + 164*z^4 - 258*z^3 + 272*z^2 - 181*z + 1, 256*x^7 - 896*x^6*y + 896*x^6*z - 1152*x^6 + 1344*x^5*y^2 - 2688*x^5*y*z + 3456*x^5*y + 1344*x^5*z^2 - 3456*x^5*z + 2208*x^5 - 1120*x^4*y^3 + 3360*x^4*y^2*z - 4320*x^4*y^2 - 3360*x^4*y*z^2 + 8640*x^4*y*z - 5536*x^4*y + 1120*x^4*z^3 - 4320*x^4*z^2 + 5536*x^4*z - 2480*x^4 + 560*x^3*y^4 - 2240*x^3*y^3*z + 2880*x^3*y^3 + 3360*x^3*y^2*z^2 - 8640*x^3*y^2*z + 5552*x^3*y^2 - 2240*x^3*y*z^3 + 8640*x^3*y*z^2 - 11104*x^3*y*z + 5040*x^3*y + 560*x^3*z^4 - 2880*x^3*z^3 + 5552*x^3*z^2 - 5024*x^3*z + 1872*x^3 - 168*x^2*y^5 + 840*x^2*y^4*z - 1080*x^2*y^4 - 1680*x^2*y^3*z^2 + 4320*x^2*y^3*z - 2784*x^2*y^3 + 1680*x^2*y^2*z^3 - 6480*x^2*y^2*z^2 + 8352*x^2*y^2*z - 3840*x^2*y^2 - 840*x^2*y*z^4 + 4320*x^2*y*z^3 - 8352*x^2*y*z^2 + 7656*x^2*y*z - 2908*x^2*y + 168*x^2*z^5 - 1080*x^2*z^4 + 2784*x^2*z^3 - 3816*x^2*z^2 + 2884*x^2*z - 964*x^2 + 28*x*y^6 - 168*x*y^5*z + 216*x*y^5 + 420*x*y^4*z^2 - 1080*x*y^4*z + 698*x*y^4 - 560*x*y^3*z^3 + 2160*x*y^3*z^2 - 2792*x*y^3*z + 1300*x*y^3 + 420*x*y^2*z^4 - 2160*x*y^2*z^3 + 4188*x*y^2*z^2 - 3888*x*y^2*z + 1504*x*y^2 - 168*x*y*z^5 + 1080*x*y*z^4 - 2792*x*y*z^3 + 3876*x*y*z^2 - 2984*x*y*z + 1018*x*y + 28*x*z^6 - 216*x*z^5 + 698*x*z^4 - 1288*x*z^3 + 1480*x*z^2 - 1004*x*z + 313*x - 2*y^7 + 14*y^6*z - 18*y^6 - 42*y^5*z^2 + 108*y^5*z - 70*y^5 + 70*y^4*z^3 - 270*y^4*z^2 + 350*y^4*z - 165*y^4 - 70*y^3*z^4 + 360*y^3*z^3 - 700*y^3*z^2 + 658*y^3*z - 259*y^3 + 42*y^2*z^5 - 270*y^2*z^4 + 700*y^2*z^3 - 984*y^2*z^2 + 771*y^2*z - 270*y^2 - 14*y*z^6 + 108*y*z^5 - 350*y*z^4 + 654*y*z^3 - 765*y*z^2 + 532*y*z - 180*y + 2*z^7 - 18*z^6 + 70*z^5 - 163*z^4 + 253*z^3 - 262*z^2 + 171*z - 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -2, 1, -1 ], [ -1, 2, -1 ], [ -1, 1, -1 ] ], "source_shift": [ 1, 1, 2 ], "target_matrix": [ [ -2, -1, -1 ], [ 1, 2, 1 ], [ -1, -1, -1 ] ], "target_shift": [ 111, -58, 54 ], "inverse_recipe": { "p": 4, "q": 3, "f": "x**4 + x**3 + x**2", "g": "x**3", "h": "y*z + y + z**2 + z", "polynomials": { "f": [ [ [ 4, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ], [ [ 3, 0, 0 ], 1 ] ], "g": [ [ [ 3, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 1, 0 ], 1 ], [ [ 0, 0, 1 ], 1 ], [ [ 0, 0, 2 ], 1 ] ] }, "h_multiplier": 2 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0761382540003979 ], "surface_features": [ 3.4240722497746083, 4.346509448774761, 3.1132746924643504, 0.0, 3.5338725180944413, 4858 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B237", "pair": 237, "family": "bijection", "mapping": "F(x,y,z)=(-3*x^7 - 21*x^6*y - 21*x^6*z - 45*x^6 - 63*x^5*y^2 - 126*x^5*y*z - 273*x^5*y - 63*x^5*z^2 - 270*x^5*z - 283*x^5 - 105*x^4*y^3 - 315*x^4*y^2*z - 690*x^4*y^2 - 315*x^4*y*z^2 - 1365*x^4*y*z - 1445*x^4*y - 105*x^4*z^3 - 675*x^4*z^2 - 1415*x^4*z - 976*x^4 - 105*x^3*y^4 - 420*x^3*y^3*z - 930*x^3*y^3 - 630*x^3*y^2*z^2 - 2760*x^3*y^2*z - 2950*x^3*y^2 - 420*x^3*y*z^3 - 2730*x^3*y*z^2 - 5780*x^3*y*z - 4024*x^3*y - 105*x^3*z^4 - 900*x^3*z^3 - 2830*x^3*z^2 - 3904*x^3*z - 2023*x^3 - 63*x^2*y^5 - 315*x^2*y^4*z - 705*x^2*y^4 - 630*x^2*y^3*z^2 - 2790*x^2*y^3*z - 3010*x^2*y^3 - 630*x^2*y^2*z^3 - 4140*x^2*y^2*z^2 - 8850*x^2*y^2*z - 6216*x^2*y^2 - 315*x^2*y*z^4 - 2730*x^2*y*z^3 - 8670*x^2*y*z^2 - 12072*x^2*y*z - 6318*x^2*y - 63*x^2*z^5 - 675*x^2*z^4 - 2830*x^2*z^3 - 5856*x^2*z^2 - 6072*x^2*z - 2563*x^2 - 21*x*y^6 - 126*x*y^5*z - 285*x*y^5 - 315*x*y^4*z^2 - 1410*x*y^4*z - 1535*x*y^4 - 420*x*y^3*z^3 - 2790*x*y^3*z^2 - 6020*x*y^3*z - 4264*x*y^3 - 315*x*y^2*z^4 - 2760*x*y^2*z^3 - 8850*x*y^2*z^2 - 12432*x*y^2*z - 6567*x*y^2 - 126*x*y*z^5 - 1365*x*y*z^4 - 5780*x*y*z^3 - 12072*x*y*z^2 - 12642*x*y*z - 5414*x*y - 21*x*z^6 - 270*x*z^5 - 1415*x*z^4 - 3904*x*z^3 - 6075*x*z^2 - 5141*x*z - 1844*x - 3*y^7 - 21*y^6*z - 48*y^6 - 63*y^5*z^2 - 285*y^5*z - 313*y^5 - 105*y^4*z^3 - 705*y^4*z^2 - 1535*y^4*z - 1096*y^4 - 105*y^3*z^4 - 930*y^3*z^3 - 3010*y^3*z^2 - 4264*y^3*z - 2272*y^3 - 63*y^2*z^5 - 690*y^2*z^4 - 2950*y^2*z^3 - 6216*y^2*z^2 - 6570*y^2*z - 2854*y^2 - 21*y*z^6 - 273*y*z^5 - 1445*y*z^4 - 4024*y*z^3 - 6324*y*z^2 - 5432*y*z - 1989*y - 3*z^7 - 45*z^6 - 283*z^5 - 976*z^4 - 2026*z^3 - 2578*z^2 - 1857*z - 1, 6*x^7 + 42*x^6*y + 42*x^6*z + 90*x^6 + 126*x^5*y^2 + 252*x^5*y*z + 546*x^5*y + 126*x^5*z^2 + 540*x^5*z + 568*x^5 + 210*x^4*y^3 + 630*x^4*y^2*z + 1380*x^4*y^2 + 630*x^4*y*z^2 + 2730*x^4*y*z + 2900*x^4*y + 210*x^4*z^3 + 1350*x^4*z^2 + 2840*x^4*z + 1972*x^4 + 210*x^3*y^4 + 840*x^3*y^3*z + 1860*x^3*y^3 + 1260*x^3*y^2*z^2 + 5520*x^3*y^2*z + 5920*x^3*y^2 + 840*x^3*y*z^3 + 5460*x^3*y*z^2 + 11600*x^3*y*z + 8128*x^3*y + 210*x^3*z^4 + 1800*x^3*z^3 + 5680*x^3*z^2 + 7888*x^3*z + 4126*x^3 + 126*x^2*y^5 + 630*x^2*y^4*z + 1410*x^2*y^4 + 1260*x^2*y^3*z^2 + 5580*x^2*y^3*z + 6040*x^2*y^3 + 1260*x^2*y^2*z^3 + 8280*x^2*y^2*z^2 + 17760*x^2*y^2*z + 12552*x^2*y^2 + 630*x^2*y*z^4 + 5460*x^2*y*z^3 + 17400*x^2*y*z^2 + 24384*x^2*y*z + 12876*x^2*y + 126*x^2*z^5 + 1350*x^2*z^4 + 5680*x^2*z^3 + 11832*x^2*z^2 + 12384*x^2*z + 5287*x^2 + 42*x*y^6 + 252*x*y^5*z + 570*x*y^5 + 630*x*y^4*z^2 + 2820*x*y^4*z + 3080*x*y^4 + 840*x*y^3*z^3 + 5580*x*y^3*z^2 + 12080*x*y^3*z + 8608*x*y^3 + 630*x*y^2*z^4 + 5520*x*y^2*z^3 + 17760*x*y^2*z^2 + 25104*x*y^2*z + 13374*x*y^2 + 252*x*y*z^5 + 2730*x*y*z^4 + 11600*x*y*z^3 + 24384*x*y*z^2 + 25764*x*y*z + 11150*x*y + 42*x*z^6 + 540*x*z^5 + 2840*x*z^4 + 7888*x*z^3 + 12390*x*z^2 + 10604*x*z + 3855*x + 6*y^7 + 42*y^6*z + 96*y^6 + 126*y^5*z^2 + 570*y^5*z + 628*y^5 + 210*y^4*z^3 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- 2730*x^2*y*z^3 - 8670*x^2*y*z^2 - 12072*x^2*y*z - 6318*x^2*y - 63*x^2*z^5 - 675*x^2*z^4 - 2830*x^2*z^3 - 5856*x^2*z^2 - 6072*x^2*z - 2564*x^2 - 21*x*y^6 - 126*x*y^5*z - 285*x*y^5 - 315*x*y^4*z^2 - 1410*x*y^4*z - 1535*x*y^4 - 420*x*y^3*z^3 - 2790*x*y^3*z^2 - 6020*x*y^3*z - 4264*x*y^3 - 315*x*y^2*z^4 - 2760*x*y^2*z^3 - 8850*x*y^2*z^2 - 12432*x*y^2*z - 6567*x*y^2 - 126*x*y*z^5 - 1365*x*y*z^4 - 5780*x*y*z^3 - 12072*x*y*z^2 - 12642*x*y*z - 5416*x*y - 21*x*z^6 - 270*x*z^5 - 1415*x*z^4 - 3904*x*z^3 - 6075*x*z^2 - 5143*x*z - 1849*x - 3*y^7 - 21*y^6*z - 48*y^6 - 63*y^5*z^2 - 285*y^5*z - 313*y^5 - 105*y^4*z^3 - 705*y^4*z^2 - 1535*y^4*z - 1096*y^4 - 105*y^3*z^4 - 930*y^3*z^3 - 3010*y^3*z^2 - 4264*y^3*z - 2272*y^3 - 63*y^2*z^5 - 690*y^2*z^4 - 2950*y^2*z^3 - 6216*y^2*z^2 - 6570*y^2*z - 2855*y^2 - 21*y*z^6 - 273*y*z^5 - 1445*y*z^4 - 4024*y*z^3 - 6324*y*z^2 - 5434*y*z - 1995*y - 3*z^7 - 45*z^6 - 283*z^5 - 976*z^4 - 2026*z^3 - 2579*z^2 - 1862*z - 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ 1, 1, 1 ], [ 1, 2, 1 ], [ 2, 2, 3 ] ], "source_shift": [ 2, 1, 1 ], "target_matrix": [ [ -1, -1, -1 ], [ 2, 1, 1 ], [ -1, -2, -1 ] ], "target_shift": [ 577, -1224, 582 ], "inverse_recipe": { "p": 2, "q": 5, "f": "x**2", "g": "x**5 + x**2", "h": "y**2 + y*z + y", "polynomials": { "f": [ [ [ 2, 0, 0 ], 1 ] ], "g": [ [ [ 5, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 1, 0 ], 1 ], [ [ 0, 2, 0 ], 1 ] ] }, "h_multiplier": 3 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0618771670015121 ], "surface_features": [ 3.4955632228561098, 4.411013290483874, 3.1492191126553797, 0.0, 3.632740640199953, 4849 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B164", "pair": 164, "family": "bijection", "mapping": "F(x,y,z)=(256*x^7 - 2688*x^6*y - 2688*x^6*z + 1088*x^6 + 12096*x^5*y^2 + 24192*x^5*y*z - 9760*x^5*y + 12096*x^5*z^2 - 9760*x^5*z + 1760*x^5 - 30240*x^4*y^3 - 90720*x^4*y^2*z + 36480*x^4*y^2 - 90720*x^4*y*z^2 + 72960*x^4*y*z - 13120*x^4*y - 30240*x^4*z^3 + 36480*x^4*z^2 - 13120*x^4*z + 1440*x^4 + 45360*x^3*y^4 + 181440*x^3*y^3*z - 72720*x^3*y^3 + 272160*x^3*y^2*z^2 - 218160*x^3*y^2*z + 39120*x^3*y^2 + 181440*x^3*y*z^3 - 218160*x^3*y*z^2 + 78240*x^3*y*z - 8560*x^3*y + 45360*x^3*z^4 - 72720*x^3*z^3 + 39120*x^3*z^2 - 8560*x^3*z + 664*x^3 - 40824*x^2*y^5 - 204120*x^2*y^4*z + 81540*x^2*y^4 - 408240*x^2*y^3*z^2 + 326160*x^2*y^3*z - 58320*x^2*y^3 - 408240*x^2*y^2*z^3 + 489240*x^2*y^2*z^2 - 174960*x^2*y^2*z + 19080*x^2*y^2 - 204120*x^2*y*z^4 + 326160*x^2*y*z^3 - 174960*x^2*y*z^2 + 38160*x^2*y*z - 2952*x^2*y - 40824*x^2*z^5 + 81540*x^2*z^4 - 58320*x^2*z^3 + 19080*x^2*z^2 - 2944*x^2*z + 222*x^2 + 20412*x*y^6 + 122472*x*y^5*z - 48762*x*y^5 + 306180*x*y^4*z^2 - 243810*x*y^4*z + 43470*x*y^4 + 408240*x*y^3*z^3 - 487620*x*y^3*z^2 + 173880*x*y^3*z - 18900*x*y^3 + 306180*x*y^2*z^4 - 487620*x*y^2*z^3 + 260820*x*y^2*z^2 - 56700*x*y^2*z + 4374*x*y^2 + 122472*x*y*z^5 - 243810*x*y*z^4 + 173880*x*y*z^3 - 56700*x*y*z^2 + 8724*x*y*z - 660*x*y + 20412*x*z^6 - 48762*x*z^5 + 43470*x*z^4 - 18900*x*z^3 + 4350*x*z^2 - 646*x*z + 74*x - 4374*y^7 - 30618*y^6*z + 12150*y^6 - 91854*y^5*z^2 + 72900*y^5*z - 12960*y^5 - 153090*y^4*z^3 + 182250*y^4*z^2 - 64800*y^4*z + 7020*y^4 - 153090*y^3*z^4 + 243000*y^3*z^3 - 129600*y^3*z^2 + 28080*y^3*z - 2160*y^3 - 91854*y^2*z^5 + 182250*y^2*z^4 - 129600*y^2*z^3 + 42120*y^2*z^2 - 6462*y^2*z + 490*y^2 - 30618*y*z^6 + 72900*y*z^5 - 64800*y*z^4 + 28080*y*z^3 - 6444*y*z^2 + 960*y*z - 106*y - 4374*z^7 + 12150*z^6 - 12960*z^5 + 7020*z^4 - 2142*z^3 + 470*z^2 - 106*z + 1, -128*x^7 + 1344*x^6*y + 1344*x^6*z - 544*x^6 - 6048*x^5*y^2 - 12096*x^5*y*z + 4880*x^5*y - 6048*x^5*z^2 + 4880*x^5*z - 848*x^5 + 15120*x^4*y^3 + 45360*x^4*y^2*z - 18240*x^4*y^2 + 45360*x^4*y*z^2 - 36480*x^4*y*z + 6320*x^4*y + 15120*x^4*z^3 - 18240*x^4*z^2 + 6320*x^4*z - 640*x^4 - 22680*x^3*y^4 - 90720*x^3*y^3*z + 36360*x^3*y^3 - 136080*x^3*y^2*z^2 + 109080*x^3*y^2*z - 18840*x^3*y^2 - 90720*x^3*y*z^3 + 109080*x^3*y*z^2 - 37680*x^3*y*z + 3800*x^3*y - 22680*x^3*z^4 + 36360*x^3*z^3 - 18840*x^3*z^2 + 3800*x^3*z - 252*x^3 + 20412*x^2*y^5 + 102060*x^2*y^4*z - 40770*x^2*y^4 + 204120*x^2*y^3*z^2 - 163080*x^2*y^3*z + 28080*x^2*y^3 + 204120*x^2*y^2*z^3 - 244620*x^2*y^2*z^2 + 84240*x^2*y^2*z - 8460*x^2*y^2 + 102060*x^2*y*z^4 - 163080*x^2*y*z^3 + 84240*x^2*y*z^2 - 16920*x^2*y*z + 1116*x^2*y + 20412*x^2*z^5 - 40770*x^2*z^4 + 28080*x^2*z^3 - 8460*x^2*z^2 + 1112*x^2*z - 71*x^2 - 10206*x*y^6 - 61236*x*y^5*z + 24381*x*y^5 - 153090*x*y^4*z^2 + 121905*x*y^4*z - 20925*x*y^4 - 204120*x*y^3*z^3 + 243810*x*y^3*z^2 - 83700*x*y^3*z + 8370*x*y^3 - 153090*x*y^2*z^4 + 243810*x*y^2*z^3 - 125550*x*y^2*z^2 + 25110*x*y^2*z - 1647*x*y^2 - 61236*x*y*z^5 + 121905*x*y*z^4 - 83700*x*y*z^3 + 25110*x*y*z^2 - 3282*x*y*z + 210*x*y - 10206*x*z^6 + 24381*x*z^5 - 20925*x*z^4 + 8370*x*z^3 - 1635*x*z^2 + 203*x*z - 24*x + 2187*y^7 + 15309*y^6*z - 6075*y^6 + 45927*y^5*z^2 - 36450*y^5*z + 6237*y^5 + 76545*y^4*z^3 - 91125*y^4*z^2 + 31185*y^4*z - 3105*y^4 + 76545*y^3*z^4 - 121500*y^3*z^3 + 62370*y^3*z^2 - 12420*y^3*z + 810*y^3 + 45927*y^2*z^5 - 91125*y^2*z^4 + 62370*y^2*z^3 - 18630*y^2*z^2 + 2421*y^2*z - 155*y^2 + 15309*y*z^6 - 36450*y*z^5 + 31185*y*z^4 - 12420*y*z^3 + 2412*y*z^2 - 300*y*z + 33*y + 2187*z^7 - 6075*z^6 + 6237*z^5 - 3105*z^4 + 801*z^3 - 145*z^2 + 34*z - 1, -256*x^7 + 2688*x^6*y + 2688*x^6*z - 1088*x^6 - 12096*x^5*y^2 - 24192*x^5*y*z + 9760*x^5*y - 12096*x^5*z^2 + 9760*x^5*z - 1888*x^5 + 30240*x^4*y^3 + 90720*x^4*y^2*z - 36480*x^4*y^2 + 90720*x^4*y*z^2 - 72960*x^4*y*z + 14080*x^4*y + 30240*x^4*z^3 - 36480*x^4*z^2 + 14080*x^4*z - 1760*x^4 - 45360*x^3*y^4 - 181440*x^3*y^3*z + 72720*x^3*y^3 - 272160*x^3*y^2*z^2 + 218160*x^3*y^2*z - 42000*x^3*y^2 - 181440*x^3*y*z^3 + 218160*x^3*y*z^2 - 84000*x^3*y*z + 10480*x^3*y - 45360*x^3*z^4 + 72720*x^3*z^3 - 42000*x^3*z^2 + 10480*x^3*z - 984*x^3 + 40824*x^2*y^5 + 204120*x^2*y^4*z - 81540*x^2*y^4 + 408240*x^2*y^3*z^2 - 326160*x^2*y^3*z + 62640*x^2*y^3 + 408240*x^2*y^2*z^3 - 489240*x^2*y^2*z^2 + 187920*x^2*y^2*z - 23400*x^2*y^2 + 204120*x^2*y*z^4 - 326160*x^2*y*z^3 + 187920*x^2*y*z^2 - 46800*x^2*y*z + 4392*x^2*y + 40824*x^2*z^5 - 81540*x^2*z^4 + 62640*x^2*z^3 - 23400*x^2*z^2 + 4384*x^2*z - 378*x^2 - 20412*x*y^6 - 122472*x*y^5*z + 48762*x*y^5 - 306180*x*y^4*z^2 + 243810*x*y^4*z - 46710*x*y^4 - 408240*x*y^3*z^3 + 487620*x*y^3*z^2 - 186840*x*y^3*z + 23220*x*y^3 - 306180*x*y^2*z^4 + 487620*x*y^2*z^3 - 280260*x*y^2*z^2 + 69660*x*y^2*z - 6534*x*y^2 - 122472*x*y*z^5 + 243810*x*y*z^4 - 186840*x*y*z^3 + 69660*x*y*z^2 - 13044*x*y*z + 1128*x*y - 20412*x*z^6 + 48762*x*z^5 - 46710*x*z^4 + 23220*x*z^3 - 6510*x*z^2 + 1114*x*z - 119*x + 4374*y^7 + 30618*y^6*z - 12150*y^6 + 91854*y^5*z^2 - 72900*y^5*z + 13932*y^5 + 153090*y^4*z^3 - 182250*y^4*z^2 + 69660*y^4*z - 8640*y^4 + 153090*y^3*z^4 - 243000*y^3*z^3 + 139320*y^3*z^2 - 34560*y^3*z + 3240*y^3 + 91854*y^2*z^5 - 182250*y^2*z^4 + 139320*y^2*z^3 - 51840*y^2*z^2 + 9702*y^2*z - 841*y^2 + 30618*y*z^6 - 72900*y*z^5 + 69660*y*z^4 - 34560*y*z^3 + 9684*y*z^2 - 1662*y*z + 176*y + 4374*z^7 - 12150*z^6 + 13932*z^5 - 8640*z^4 + 3222*z^3 - 821*z^2 + 172*z - 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ 2, -3, -3 ], [ 1, -1, -1 ], [ 1, -2, -1 ] ], "source_shift": [ 1, 1, 2 ], "target_matrix": [ [ 2, 2, 3 ], [ -1, -1, -2 ], [ -2, -1, -1 ] ], "target_shift": [ -7, -1, 20 ], "inverse_recipe": { "p": 2, "q": 5, "f": "x**2 + x", "g": "x**5 + x", "h": "y*z", "polynomials": { "f": [ [ [ 2, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "g": [ [ [ 5, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0479779639972548 ], "surface_features": [ 4.875892174120414, 5.689521957499782, 4.485976818715875, 0.0, 5.016796116077183, 5333 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B102", "pair": 102, "family": "bijection", "mapping": "F(x,y,z)=(-x^7 - 7*x^6*y + 14*x^6*z + 16*x^6 - 21*x^5*y^2 + 84*x^5*y*z + 96*x^5*y - 84*x^5*z^2 - 192*x^5*z - 111*x^5 - 35*x^4*y^3 + 210*x^4*y^2*z + 240*x^4*y^2 - 420*x^4*y*z^2 - 960*x^4*y*z - 556*x^4*y + 280*x^4*z^3 + 960*x^4*z^2 + 1109*x^4*z + 436*x^4 - 35*x^3*y^4 + 280*x^3*y^3*z + 320*x^3*y^3 - 840*x^3*y^2*z^2 - 1920*x^3*y^2*z - 1114*x^3*y^2 + 1120*x^3*y*z^3 + 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640*y^6 - 816*y^5*z^2 - 2880*y^5*z - 2400*y^5 + 1032*y^4*z^3 + 5328*y^4*z^2 + 8760*y^4*z + 4608*y^4 - 768*y^3*z^4 - 5184*y^3*z^3 - 12612*y^3*z^2 - 13128*y^3*z - 4992*y^3 + 336*y^2*z^5 + 2796*y^2*z^4 + 8952*y^2*z^3 + 13812*y^2*z^2 + 10374*y^2*z + 3076*y^2 - 80*y*z^6 - 792*y*z^5 - 3132*y*z^4 - 6362*y*z^3 - 7062*y*z^2 - 4114*y*z - 1005*y + 8*z^7 + 92*z^6 + 432*z^5 + 1082*z^4 + 1574*z^3 + 1348*z^2 + 648*z + 1, 81*x^7 + 729*x^6*y - 486*x^6*z - 729*x^6 + 2700*x^5*y^2 - 3699*x^5*y*z - 5589*x^5*y + 1242*x^5*z^2 + 3726*x^5*z + 2727*x^5 + 5316*x^4*y^3 - 11250*x^4*y^2*z - 17226*x^4*y^2 + 7767*x^4*y*z^2 + 23490*x^4*y*z + 17424*x^4*y - 1752*x^4*z^3 - 7884*x^4*z^2 - 11565*x^4*z - 5550*x^4 + 6000*x^3*y^4 - 17448*x^3*y^3*z - 27288*x^3*y^3 + 18612*x^3*y^2*z^2 + 57096*x^3*y^2*z + 43020*x^3*y^2 - 8637*x^3*y*z^3 - 39213*x^3*y*z^2 - 58293*x^3*y*z - 28443*x^3*y + 1473*x^3*z^4 + 8838*x^3*z^3 + 19485*x^3*z^2 + 18777*x^3*z + 6702*x^3 + 3888*x^2*y^5 - 14544*x^2*y^4*z - 23424*x^2*y^4 + 21312*x^2*y^3*z^2 + 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-2, 2 ], [ -1, -2, 1 ], [ -1, -1, 1 ] ], "source_shift": [ 2, 1, 1 ], "target_matrix": [ [ 1, 1, 2 ], [ -3, -2, -2 ], [ -1, -1, -1 ] ], "target_shift": [ -139, 368, 155 ], "collision_preimages": [ [ "-1/2", "-1/4", "-2" ], [ "12", "8", "51/2" ], [ "14", "5", "49/2" ] ], "collision_image": [ "-557/4", "1475/4", "621/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0502921100014646 ], "surface_features": [ 4.047293801085636, 5.02649240705284, 3.72607487021537, 0.0, 4.1921055516136905, 5084 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C173", "pair": 173, "family": "counterexample", "mapping": "F(x,y,z)=(-2*x^7 + 20*x^6*y + 20*x^6*z - 25*x^6 - 84*x^5*y^2 - 174*x^5*y*z + 213*x^5*y - 84*x^5*z^2 + 222*x^5*z - 126*x^5 + 192*x^4*y^3 + 618*x^4*y^2*z - 741*x^4*y^2 + 618*x^4*y*z^2 - 1599*x^4*y*z + 894*x^4*y + 192*x^4*z^3 - 810*x^4*z^2 + 945*x^4*z - 350*x^4 - 258*x^3*y^4 - 1146*x^3*y^3*z + 1347*x^3*y^3 - 1782*x^3*y^2*z^2 + 4512*x^3*y^2*z - 2484*x^3*y^2 - 1146*x^3*y*z^3 + 4737*x^3*y*z^2 - 5448*x^3*y*z + 1981*x^3*y - 258*x^3*z^4 + 1554*x^3*z^3 - 2796*x^3*z^2 + 2120*x^3*z - 590*x^3 + 204*x^2*y^5 + 1170*x^2*y^4*z - 1350*x^2*y^4 + 2514*x^2*y^3*z^2 - 6231*x^2*y^3*z + 3378*x^2*y^3 + 2514*x^2*y^2*z^3 - 10170*x^2*y^2*z^2 + 11523*x^2*y^2*z - 4110*x^2*y^2 + 1170*x^2*y*z^4 - 6921*x^2*y*z^3 + 12288*x^2*y*z^2 - 9153*x^2*y*z + 2490*x^2*y + 204*x^2*z^5 - 1653*x^2*z^4 + 4077*x^2*z^3 - 4746*x^2*z^2 + 2703*x^2*z - 614*x^2 - 88*x*y^6 - 624*x*y^5*z + 708*x*y^5 - 1734*x*y^4*z^2 + 4212*x*y^4*z - 2250*x*y^4 - 2398*x*y^3*z^3 + 9489*x*y^3*z^2 - 10590*x*y^3*z + 3705*x*y^3 - 1734*x*y^2*z^4 + 10056*x*y^2*z^3 - 17604*x*y^2*z^2 + 12864*x*y^2*z - 3414*x*y^2 - 624*x*y*z^5 + 4986*x*y*z^4 - 12150*x*y*z^3 + 13908*x*y*z^2 - 7752*x*y*z + 1711*x*y - 88*x*z^6 + 924*x*z^5 - 2928*x*z^4 + 4650*x*z^3 - 4062*x*z^2 + 1892*x*z - 372*x + 16*y^7 + 136*y^6*z - 152*y^6 + 468*y^5*z^2 - 1116*y^5*z + 588*y^5 + 838*y^4*z^3 - 3246*y^4*z^2 + 3570*y^4*z - 1226*y^4 + 838*y^3*z^4 - 4757*y^3*z^3 + 8208*y^3*z^2 - 5883*y^3*z + 1522*y^3 + 468*y^2*z^5 - 3681*y^2*z^4 + 8853*y^2*z^3 - 9942*y^2*z^2 + 5409*y^2*z - 1157*y^2 + 136*y*z^6 - 1416*y*z^5 + 4440*y*z^4 - 6946*y*z^3 + 5946*y*z^2 - 2693*y*z + 514*y + 16*z^7 - 212*z^6 + 828*z^5 - 1680*z^4 + 2000*z^3 - 1433*z^2 + 575*z - 1, -x^7 + 10*x^6*y + 10*x^6*z - 17*x^6 - 42*x^5*y^2 - 87*x^5*y*z + 147*x^5*y - 42*x^5*z^2 + 147*x^5*z - 108*x^5 + 96*x^4*y^3 + 309*x^4*y^2*z - 519*x^4*y^2 + 309*x^4*y*z^2 - 1074*x^4*y*z + 780*x^4*y + 96*x^4*z^3 - 522*x^4*z^2 + 792*x^4*z - 341*x^4 - 129*x^3*y^4 - 573*x^3*y^3*z + 957*x^3*y^3 - 891*x^3*y^2*z^2 + 3075*x^3*y^2*z - 2205*x^3*y^2 - 573*x^3*y*z^3 + 3093*x^3*y*z^2 - 4650*x^3*y*z + 1968*x^3*y - 129*x^3*z^4 + 975*x^3*z^3 - 2289*x^3*z^2 + 2021*x^3*z - 624*x^3 + 102*x^2*y^5 + 585*x^2*y^4*z - 972*x^2*y^4 + 1257*x^2*y^3*z^2 - 4308*x^2*y^3*z + 3048*x^2*y^3 + 1257*x^2*y^2*z^3 - 6732*x^2*y^2*z^2 + 10014*x^2*y^2*z - 4161*x^2*y^2 + 585*x^2*y*z^4 - 4392*x^2*y*z^3 + 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120*x^5*y + 42*x^5*z^2 - 123*x^5*z + 78*x^5 - 96*x^4*y^3 - 309*x^4*y^2*z + 420*x^4*y^2 - 309*x^4*y*z^2 + 891*x^4*y*z - 558*x^4*y - 96*x^4*z^3 + 444*x^4*z^2 - 579*x^4*z + 230*x^4 + 129*x^3*y^4 + 573*x^3*y^3*z - 768*x^3*y^3 + 891*x^3*y^2*z^2 - 2529*x^3*y^2*z + 1563*x^3*y^2 + 573*x^3*y*z^3 - 2610*x^3*y*z^2 + 3366*x^3*y*z - 1314*x^3*y + 129*x^3*z^4 - 843*x^3*z^3 + 1695*x^3*z^2 - 1379*x^3*z + 402*x^3 - 102*x^2*y^5 - 585*x^2*y^4*z + 774*x^2*y^4 - 1257*x^2*y^3*z^2 + 3513*x^2*y^3*z - 2142*x^2*y^3 - 1257*x^2*y^2*z^3 + 5634*x^2*y^2*z^2 - 7179*x^2*y^2*z + 2751*x^2*y^2 - 585*x^2*y*z^4 + 3771*x^2*y*z^3 - 7512*x^2*y*z^2 + 6012*x^2*y*z - 1713*x^2*y - 102*x^2*z^5 + 888*x^2*z^4 - 2445*x^2*z^3 + 3054*x^2*z^2 - 1824*x^2*z + 427*x^2 + 44*x*y^6 + 312*x*y^5*z - 408*x*y^5 + 867*x*y^4*z^2 - 2388*x*y^4*z + 1437*x*y^4 + 1199*x*y^3*z^3 - 5286*x*y^3*z^2 + 6651*x*y^3*z - 2501*x*y^3 + 867*x*y^2*z^4 - 5505*x*y^2*z^3 + 10854*x*y^2*z^2 - 8532*x*y^2*z + 2370*x*y^2 + 312*x*y*z^5 - 2688*x*y*z^4 + 7347*x*y*z^3 - 9039*x*y*z^2 + 5286*x*y*z - 1202*x*y + 44*x*z^6 - 492*x*z^5 + 1737*x*z^4 - 2959*x*z^3 + 2712*x*z^2 - 1303*x*z + 262*x - 8*y^7 - 68*y^6*z + 88*y^6 - 234*y^5*z^2 + 636*y^5*z - 378*y^5 - 419*y^4*z^3 + 1818*y^4*z^2 - 2259*y^4*z + 834*y^4 - 419*y^3*z^4 + 2617*y^3*z^3 - 5103*y^3*z^2 + 3938*y^3*z - 1065*y^3 - 234*y^2*z^5 + 1992*y^2*z^4 - 5400*y^2*z^3 + 6528*y^2*z^2 - 3726*y^2*z + 820*y^2 - 68*y*z^6 + 756*y*z^5 - 2655*y*z^4 + 4464*y*z^3 - 4014*y*z^2 + 1876*y*z - 367*y - 8*z^7 + 112*z^6 - 486*z^5 + 1057*z^4 - 1320*z^3 + 976*z^2 - 400*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ 1, -2, -1 ], [ 1, -1, -2 ], [ 1, -1, -1 ] ], "source_shift": [ 1, 1, 2 ], "target_matrix": [ [ -2, -1, -1 ], [ -1, -2, -1 ], [ 1, 1, 1 ] ], "target_shift": [ 102, 118, -72 ], "collision_preimages": [ [ "-19/4", "-5/4", "-5/4" ], [ "16", "9/2", "7" ], [ "15", "13/2", "4" ] ], "collision_image": [ "205/2", "473/4", "-289/4" ] }, "probe_target": "No", "jev_answers": [ "Maybe" ], "seconds": [ 1.0798455320000357 ], "surface_features": [ 3.385694632713635, 4.2456113598909075, 3.073535065058784, 0.0, 3.4866486743655014, 4834 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C242", "pair": 242, "family": "counterexample", "mapping": "F(x,y,z)=(8*x^7 + 44*x^6*y - 68*x^6*z - 136*x^6 + 102*x^5*y^2 - 312*x^5*y*z - 600*x^5*y + 234*x^5*z^2 + 912*x^5*z + 774*x^5 + 129*x^4*y^3 - 585*x^4*y^2*z - 1086*x^4*y^2 + 867*x^4*y*z^2 + 3252*x^4*y*z + 2733*x^4*y - 419*x^4*z^3 - 2382*x^4*z^2 - 4059*x^4*z - 2175*x^4 + 96*x^3*y^4 - 573*x^3*y^3*z - 1032*x^3*y^3 + 1257*x^3*y^2*z^2 + 4566*x^3*y^2*z + 3801*x^3*y^2 - 1199*x^3*y*z^3 - 6588*x^3*y*z^2 - 11103*x^3*y*z - 5928*x^3*y + 419*x^3*z^4 + 3094*x^3*z^3 + 7920*x^3*z^2 + 8549*x^3*z + 3363*x^3 + 42*x^2*y^5 - 309*x^2*y^4*z - 543*x^2*y^4 + 891*x^2*y^3*z^2 + 3156*x^2*y^3*z + 2604*x^2*y^3 - 1257*x^2*y^2*z^3 - 6732*x^2*y^2*z^2 - 11229*x^2*y^2*z - 5967*x^2*y^2 + 867*x^2*y*z^4 + 6240*x^2*y*z^3 + 15777*x^2*y*z^2 + 16911*x^2*y*z + 6630*x^2*y - 234*x^2*z^5 - 2121*x^2*z^4 - 7218*x^2*z^3 - 11685*x^2*z^2 - 9216*x^2*z - 2887*x^2 + 10*x*y^6 - 87*x*y^5*z - 150*x*y^5 + 309*x*y^4*z^2 + 1074*x*y^4*z + 879*x*y^4 - 573*x*y^3*z^3 - 3012*x*y^3*z^2 - 4977*x*y^3*z - 2631*x*y^3 + 585*x*y^2*z^4 + 4134*x*y^2*z^3 + 10338*x*y^2*z^2 + 11004*x*y^2*z + 4296*x*y^2 - 312*x*y*z^5 - 2778*x*y*z^4 - 9336*x*y*z^3 - 14985*x*y*z^2 - 11745*x*y*z - 3661*x*y + 68*x*z^6 + 732*x*z^5 + 3096*x*z^4 + 6651*x*z^3 + 7830*x*z^2 + 4879*x*z + 1276*x + y^7 - 10*y^6*z - 17*y^6 + 42*y^5*z^2 + 144*y^5*z + 117*y^5 - 96*y^4*z^3 - 498*y^4*z^2 - 816*y^4*z - 429*y^4 + 129*y^3*z^4 + 900*y^3*z^3 + 2229*y^3*z^2 + 2356*y^3*z + 915*y^3 - 102*y^2*z^5 - 897*y^2*z^4 - 2982*y^2*z^3 - 4746*y^2*z^2 - 3693*y^2*z - 1144*y^2 + 44*y*z^6 + 468*y*z^5 + 1956*y*z^4 + 4161*y*z^3 + 4854*y*z^2 + 2999*y*z + 781*y - 8*z^7 - 100*z^6 - 504*z^5 - 1342*z^4 - 2082*z^3 - 1918*z^2 - 994*z - 1, -16*x^7 - 88*x^6*y + 136*x^6*z + 260*x^6 - 204*x^5*y^2 + 624*x^5*y*z + 1140*x^5*y - 468*x^5*z^2 - 1728*x^5*z - 1368*x^5 - 258*x^4*y^3 + 1170*x^4*y^2*z + 2049*x^4*y^2 - 1734*x^4*y*z^2 - 6114*x^4*y*z - 4776*x^4*y + 838*x^4*z^3 + 4461*x^4*z^2 + 7032*x^4*z + 3556*x^4 - 192*x^3*y^4 + 1146*x^3*y^3*z + 1932*x^3*y^3 - 2514*x^3*y^2*z^2 - 8511*x^3*y^2*z - 6564*x^3*y^2 + 2398*x^3*y*z^3 + 12222*x^3*y*z^2 + 18996*x^3*y*z + 9562*x^3*y - 838*x^3*z^4 - 5711*x^3*z^3 - 13416*x^3*z^2 - 13622*x^3*z - 5144*x^3 - 84*x^2*y^5 + 618*x^2*y^4*z + 1008*x^2*y^4 - 1782*x^2*y^3*z^2 - 5829*x^2*y^3*z - 4443*x^2*y^3 + 2514*x^2*y^2*z^3 + 12366*x^2*y^2*z^2 + 18975*x^2*y^2*z + 9501*x^2*y^2 - 1734*x^2*y*z^4 - 11397*x^2*y*z^3 - 26397*x^2*y*z^2 - 26610*x^2*y*z - 10014*x^2*y + 468*x^2*z^5 + 3852*x^2*z^4 + 11961*x^2*z^3 + 18177*x^2*z^2 + 13746*x^2*z + 4193*x^2 - 20*x*y^6 + 174*x*y^5*z + 276*x*y^5 - 618*x*y^4*z^2 - 1965*x*y^4*z - 1482*x*y^4 + 1146*x*y^3*z^3 + 5478*x*y^3*z^2 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1164*x^3*y^3 + 1257*x^3*y^2*z^2 + 5187*x^3*y^2*z + 4839*x^3*y^2 - 1199*x^3*y*z^3 - 7542*x^3*y*z^2 - 14313*x^3*y*z - 8226*x^3*y + 419*x^3*z^4 + 3571*x^3*z^3 + 10344*x^3*z^2 + 12032*x^3*z + 4959*x^3 + 42*x^2*y^5 - 309*x^2*y^4*z - 621*x^2*y^4 + 891*x^2*y^3*z^2 + 3639*x^2*y^3*z + 3369*x^2*y^3 - 1257*x^2*y^2*z^3 - 7830*x^2*y^2*z^2 - 14712*x^2*y^2*z - 8406*x^2*y^2 + 867*x^2*y*z^4 + 7323*x^2*y*z^3 + 20934*x^2*y*z^2 + 24144*x^2*y*z + 9915*x^2*y - 234*x^2*z^5 - 2511*x^2*z^4 - 9693*x^2*z^3 - 16896*x^2*z^2 - 13971*x^2*z - 4519*x^2 + 10*x*y^6 - 87*x*y^5*z - 174*x*y^5 + 309*x*y^4*z^2 + 1257*x*y^4*z + 1155*x*y^4 - 573*x*y^3*z^3 - 3558*x*y^3*z^2 - 6621*x*y^3*z - 3759*x*y^3 + 585*x*y^2*z^4 + 4929*x*y^2*z^3 + 13920*x*y^2*z^2 + 15921*x*y^2*z + 6510*x*y^2 - 312*x*y*z^5 - 3342*x*y*z^4 - 12714*x*y*z^3 - 21939*x*y*z^2 - 18027*x*y*z - 5809*x*y + 68*x*z^6 + 888*x*z^5 + 4260*x*z^4 + 9843*x*z^3 + 12162*x*z^2 + 7852*x*z + 2106*x + y^7 - 10*y^6*z - 20*y^6 + 42*y^5*z^2 + 171*y^5*z + 156*y^5 - 96*y^4*z^3 - 597*y^4*z^2 - 1101*y^4*z - 621*y^4 + 129*y^3*z^4 + 1089*y^3*z^3 + 3042*y^3*z^2 + 3451*y^3*z + 1404*y^3 - 102*y^2*z^5 - 1095*y^2*z^4 - 4113*y^2*z^3 - 7029*y^2*z^2 - 5736*y^2*z - 1840*y^2 + 44*y*z^6 + 576*y*z^5 + 2724*y*z^4 + 6225*y*z^3 + 7626*y*z^2 + 4892*y*z + 1310*y - 8*z^7 - 124*z^6 - 708*z^5 - 2026*z^4 - 3306*z^3 - 3172*z^2 - 1697*z - 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -1, -1, 2 ], [ -2, -1, 1 ], [ -1, -1, 1 ] ], "source_shift": [ 2, 1, 2 ], "target_matrix": [ [ -1, -1, -1 ], [ 2, 1, 1 ], [ -1, -2, -1 ] ], "target_shift": [ 228, -312, 397 ], "collision_preimages": [ [ "-5/4", "15/4", "1/4" ], [ "7", "-17", "-11/2" ], [ "4", "-16", "-15/2" ] ], "collision_image": [ "913/4", "-625/2", "1589/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0750735500005248 ], "surface_features": [ 3.5887816644246855, 4.425044874551388, 3.2336305801644634, 0.0, 3.703030236595741, 4892 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C031", "pair": 31, "family": "counterexample", "mapping": "F(x,y,z)=(16*x^7 - 184*x^6*y - 128*x^6*z + 160*x^6 + 900*x^5*y^2 + 1272*x^5*y*z - 1596*x^5*y + 432*x^5*z^2 - 1080*x^5*z + 684*x^5 - 2426*x^4*y^3 - 5232*x^4*y^2*z + 6600*x^4*y^2 - 3600*x^4*y*z^2 + 9024*x^4*y*z - 5736*x^4*y - 800*x^4*z^3 + 3000*x^4*z^2 - 3804*x^4*z + 1624*x^4 + 3890*x^3*y^4 + 11398*x^3*y^3*z - 14483*x^3*y^3 + 11928*x^3*y^2*z^2 - 30018*x^3*y^2*z + 19167*x^3*y^2 + 5360*x^3*y*z^3 - 20136*x^3*y*z^2 + 25608*x^3*y*z - 10974*x^3*y + 880*x^3*z^4 - 4400*x^3*z^3 + 8376*x^3*z^2 - 7168*x^3*z + 2320*x^3 - 3708*x^2*y^5 - 13866*x^2*y^4*z + 17787*x^2*y^4 - 19638*x^2*y^3*z^2 + 49692*x^2*y^3*z - 31905*x^2*y^3 - 13392*x^2*y^2*z^3 + 50454*x^2*y^2*z^2 - 64401*x^2*y^2*z + 27717*x^2*y^2 - 4440*x^2*y*z^4 + 22224*x^2*y*z^3 - 42408*x^2*y*z^2 + 36414*x^2*y*z - 11832*x^2*y - 576*x^2*z^5 + 3600*x^2*z^4 - 9144*x^2*z^3 + 11760*x^2*z^2 - 7644*x^2*z + 2006*x^2 + 1944*x*y^6 + 8928*x*y^5*z - 11592*x*y^5 + 16062*x*y^4*z^2 - 40938*x*y^4*z + 26460*x*y^4 + 14786*x*y^3*z^3 - 55935*x*y^3*z^2 + 71718*x*y^3*z - 31018*x*y^3 + 7428*x*y^2*z^4 - 37254*x*y^2*z^3 + 71301*x*y^2*z^2 - 61452*x*y^2*z + 20052*x*y^2 + 1944*x*y*z^5 - 12156*x*y*z^4 + 30936*x*y*z^3 - 39906*x*y*z^2 + 26034*x*y*z - 6857*x*y + 208*x*z^6 - 1560*x*z^5 + 4956*x*z^4 - 8512*x*z^3 + 8328*x*z^2 - 4402*x*z + 983*x - 432*y^7 - 2376*y^6*z + 3132*y^6 - 5220*y^5*z^2 + 13428*y^5*z - 8748*y^5 - 6086*y^4*z^3 + 23151*y^4*z^2 - 29844*y^4*z + 12980*y^4 - 4120*y^3*z^4 + 20726*y^3*z^3 - 39813*y^3*z^2 + 34458*y^3*z - 11296*y^3 - 1632*y^2*z^5 + 10218*y^2*z^4 - 26067*y^2*z^3 + 33735*y^2*z^2 - 22092*y^2*z + 5843*y^2 - 352*y*z^6 + 2640*y*z^5 - 8400*y*z^4 + 14466*y*z^3 - 14202*y*z^2 + 7532*y*z - 1685*y - 32*z^7 + 280*z^6 - 1068*z^5 + 2296*z^4 - 3004*z^3 + 2396*z^2 - 1083*z + 1, 8*x^7 - 92*x^6*y - 64*x^6*z + 92*x^6 + 450*x^5*y^2 + 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13854*x*y^2 + 972*x*y*z^5 - 7056*x*y*z^4 + 19620*x*y*z^3 - 26775*x*y*z^2 + 18105*x*y*z - 4876*x*y + 104*x*z^6 - 912*x*z^5 + 3168*x*z^4 - 5752*x*z^3 + 5826*x*z^2 - 3146*x*z + 712*x - 216*y^7 - 1188*y^6*z + 1728*y^6 - 2610*y^5*z^2 + 7524*y^5*z - 5346*y^5 - 3043*y^4*z^3 + 13128*y^4*z^2 - 18459*y^4*z + 8515*y^4 - 2060*y^3*z^4 + 11869*y^3*z^3 - 24879*y^3*z^2 + 22809*y^3*z - 7766*y^3 - 816*y^2*z^5 + 5901*y^2*z^4 - 16437*y^2*z^3 + 22512*y^2*z^2 - 15291*y^2*z + 4138*y^2 - 176*y*z^6 + 1536*y*z^5 - 5340*y*z^4 + 9725*y*z^3 - 9891*y*z^2 + 5362*y*z - 1216*y - 16*z^7 + 164*z^6 - 684*z^5 + 1554*z^4 - 2104*z^3 + 1714*z^2 - 785*z + 1, 8*x^7 - 92*x^6*y - 64*x^6*z + 92*x^6 + 450*x^5*y^2 + 636*x^5*y*z - 912*x^5*y + 216*x^5*z^2 - 624*x^5*z + 432*x^5 - 1213*x^4*y^3 - 2616*x^4*y^2*z + 3747*x^4*y^2 - 1800*x^4*y*z^2 + 5184*x^4*y*z - 3600*x^4*y - 400*x^4*z^3 + 1740*x^4*z^2 - 2412*x^4*z + 1082*x^4 + 1945*x^3*y^4 + 5699*x^3*y^3*z - 8167*x^3*y^3 + 5964*x^3*y^2*z^2 - 17142*x^3*y^2*z + 11952*x^3*y^2 + 2680*x^3*y*z^3 - 11616*x^3*y*z^2 + 16140*x^3*y*z - 7281*x^3*y + 440*x^3*z^4 - 2560*x^3*z^3 + 5328*x^3*z^2 - 4784*x^3*z + 1574*x^3 - 1854*x^2*y^5 - 6933*x^2*y^4*z + 9960*x^2*y^4 - 9819*x^2*y^3*z^2 + 28203*x^2*y^3*z - 19764*x^2*y^3 - 6696*x^2*y^2*z^3 + 28944*x^2*y^2*z^2 - 40341*x^2*y^2*z + 18306*x^2*y^2 - 2220*x^2*y*z^4 + 12864*x^2*y*z^3 - 26820*x^2*y*z^2 + 24207*x^2*y*z - 8022*x^2*y - 288*x^2*z^5 + 2100*x^2*z^4 - 5832*x^2*z^3 + 7860*x^2*z^2 - 5184*x^2*z + 1348*x^2 + 972*x*y^6 + 4464*x*y^5*z - 6444*x*y^5 + 8031*x*y^4*z^2 - 23088*x*y^4*z + 16281*x*y^4 + 7393*x*y^3*z^3 - 31905*x*y^3*z^2 + 44643*x*y^3*z - 20387*x*y^3 + 3714*x*y^2*z^4 - 21450*x*y^2*z^3 + 44826*x*y^2*z^2 - 40674*x*y^2*z + 13575*x*y^2 + 972*x*y*z^5 - 7056*x*y*z^4 + 19620*x*y*z^3 - 26571*x*y*z^2 + 17649*x*y*z - 4624*x*y + 104*x*z^6 - 912*x*z^5 + 3168*x*z^4 - 5696*x*z^3 + 5646*x*z^2 - 2954*x*z + 648*x - 216*y^7 - 1188*y^6*z + 1728*y^6 - 2610*y^5*z^2 + 7524*y^5*z - 5346*y^5 - 3043*y^4*z^3 + 13128*y^4*z^2 - 18459*y^4*z + 8488*y^4 - 2060*y^3*z^4 + 11869*y^3*z^3 - 24879*y^3*z^2 + 22701*y^3*z - 7631*y^3 - 816*y^2*z^5 + 5901*y^2*z^4 - 16437*y^2*z^3 + 22368*y^2*z^2 - 14958*y^2*z + 3949*y^2 - 176*y*z^6 + 1536*y*z^5 - 5340*y*z^4 + 9645*y*z^3 - 9627*y*z^2 + 5074*y*z - 1118*y - 16*z^7 + 164*z^6 - 684*z^5 + 1538*z^4 - 2036*z^3 + 1606*z^2 - 713*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ 2, -3, -2 ], [ 1, -2, -1 ], [ 1, -1, -2 ] ], "source_shift": [ 2, 1, 1 ], "target_matrix": [ [ 2, 1, 1 ], [ 1, 1, 1 ], [ 1, 1, 2 ] ], "target_shift": [ -212, -155, -139 ], "collision_preimages": [ [ "-3/4", "0", "1/4" ], [ "3/2", "4", "-4" ], [ "-33/2", "-4", "-9" ] ], "collision_image": [ "-425/2", "-621/4", "-557/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.1176247570001578 ], "surface_features": [ 4.012452420222745, 4.855628169887986, 3.7292458072253067, 0.0, 4.1030310135498445, 5073 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C190", "pair": 190, "family": "counterexample", "mapping": "F(x,y,z)=(-2*x^7 - 13*x^6*y + 19*x^6*z + 34*x^6 - 36*x^5*y^2 + 105*x^5*y*z + 189*x^5*y - 75*x^5*z^2 - 276*x^5*z - 246*x^5 - 55*x^4*y^3 + 240*x^4*y^2*z + 435*x^4*y^2 - 342*x^4*y*z^2 - 1269*x^4*y*z - 1137*x^4*y + 159*x^4*z^3 + 909*x^4*z^2 + 1668*x^4*z + 986*x^4 - 50*x^3*y^4 + 290*x^3*y^3*z + 530*x^3*y^3 - 618*x^3*y^2*z^2 - 2316*x^3*y^2*z - 2088*x^3*y^2 + 573*x^3*y*z^3 + 3315*x^3*y*z^2 + 6126*x^3*y*z + 3636*x^3*y - 195*x^3*z^4 - 1554*x^3*z^3 - 4422*x^3*z^2 - 5379*x^3*z - 2368*x^3 - 27*x^2*y^5 + 195*x^2*y^4*z + 360*x^2*y^4 - 552*x^2*y^3*z^2 - 2094*x^2*y^3*z - 1902*x^2*y^3 + 765*x^2*y^2*z^3 + 4491*x^2*y^2*z^2 + 8370*x^2*y^2*z + 4992*x^2*y^2 - 519*x^2*y*z^4 - 4209*x^2*y*z^3 - 12093*x^2*y*z^2 - 14787*x^2*y*z - 6525*x^2*y + 138*x^2*z^5 + 1455*x^2*z^4 + 5736*x^2*z^3 + 10791*x^2*z^2 + 9765*x^2*z + 3409*x^2 - 8*x*y^6 + 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159*x^4*z^3 + 909*x^4*z^2 + 1668*x^4*z + 988*x^4 - 50*x^3*y^4 + 290*x^3*y^3*z + 530*x^3*y^3 - 618*x^3*y^2*z^2 - 2316*x^3*y^2*z - 2088*x^3*y^2 + 573*x^3*y*z^3 + 3315*x^3*y*z^2 + 6126*x^3*y*z + 3643*x^3*y - 195*x^3*z^4 - 1554*x^3*z^3 - 4422*x^3*z^2 - 5386*x^3*z - 2384*x^3 - 27*x^2*y^5 + 195*x^2*y^4*z + 360*x^2*y^4 - 552*x^2*y^3*z^2 - 2094*x^2*y^3*z - 1902*x^2*y^3 + 765*x^2*y^2*z^3 + 4491*x^2*y^2*z^2 + 8370*x^2*y^2*z + 5001*x^2*y^2 - 519*x^2*y*z^4 - 4209*x^2*y*z^3 - 12093*x^2*y*z^2 - 14805*x^2*y*z - 6567*x^2*y + 138*x^2*z^5 + 1455*x^2*z^4 + 5736*x^2*z^3 + 10800*x^2*z^2 + 9810*x^2*z + 3457*x^2 - 8*x*y^6 + 69*x*y^5*z + 129*x*y^5 - 243*x*y^4*z^2 - 936*x*y^4*z - 858*x*y^4 + 447*x*y^3*z^3 + 2673*x*y^3*z^2 + 5034*x*y^3*z + 3025*x*y^3 - 453*x*y^2*z^4 - 3756*x*y^2*z^3 - 10920*x*y^2*z^2 - 13452*x*y^2*z - 5982*x*y^2 + 240*x*y*z^5 + 2598*x*y*z^4 + 10380*x*y*z^3 + 19677*x*y*z^2 + 17919*x*y*z + 6311*x*y - 52*x*z^6 - 708*x*z^5 - 3648*x*z^4 - 9465*x*z^3 - 13245*x*z^2 - 9550*x*z - 2784*x - y^7 + 10*y^6*z 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8133*z^3 + 9466*z^2 + 5851*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -1, -1, 1 ], [ -1, -1, 2 ], [ -2, -1, 1 ] ], "source_shift": [ 2, 2, 1 ], "target_matrix": [ [ 1, 1, 2 ], [ 1, 1, 1 ], [ 1, 2, 1 ] ], "target_shift": [ -924, -952, -1488 ], "collision_preimages": [ [ "-3/4", "11/4", "0" ], [ "-13/2", "5", "-5/2" ], [ "-17/2", "14", "5/2" ] ], "collision_image": [ "-3697/4", "-3809/4", "-5953/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.047535744997731 ], "surface_features": [ 3.538977415987282, 4.419278183187153, 3.0580462303952816, 0.0, 3.6718358213552857, 4837 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C179", "pair": 179, "family": "counterexample", "mapping": "F(x,y,z)=(-6*x^7 - 39*x^6*y - 57*x^6*z - 81*x^6 - 108*x^5*y^2 - 315*x^5*y*z - 453*x^5*y - 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114*x*y^5 + 243*x*y^4*z^2 + 822*x*y^4*z + 657*x*y^4 + 447*x*y^3*z^3 + 2334*x*y^3*z^2 + 3801*x*y^3*z + 1974*x*y^3 + 453*x*y^2*z^4 + 3264*x*y^2*z^3 + 8130*x*y^2*z^2 + 8565*x*y^2*z + 3288*x*y^2 + 240*x*y*z^5 + 2250*x*y*z^4 + 7620*x*y*z^3 + 12198*x*y*z^2 + 9477*x*y*z + 2912*x*y + 52*x*z^6 + 612*x*z^5 + 2640*x*z^4 + 5700*x*z^3 + 6711*x*z^2 + 4168*x*z + 1087*x + y^7 + 10*y^6*z + 17*y^6 + 42*y^5*z^2 + 147*y^5*z + 120*y^5 + 96*y^4*z^3 + 522*y^4*z^2 + 870*y^4*z + 458*y^4 + 129*y^3*z^4 + 975*y^3*z^3 + 2490*y^3*z^2 + 2660*y^3*z + 1029*y^3 + 102*y^2*z^5 + 1011*y^2*z^4 + 3516*y^2*z^3 + 5709*y^2*z^2 + 4470*y^2*z + 1378*y^2 + 44*y*z^6 + 552*y*z^5 + 2448*y*z^4 + 5362*y*z^3 + 6363*y*z^2 + 3967*y*z + 1035*y + 8*z^7 + 124*z^6 + 672*z^5 + 1858*z^4 + 2964*z^3 + 2797*z^2 + 1472*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ 1, 1, 1 ], [ 1, 1, 2 ], [ 2, 1, 1 ] ], "source_shift": [ 1, 2, 2 ], "target_matrix": [ [ -3, -2, -3 ], [ -1, -1, -1 ], [ 1, 1, 2 ] ], "target_shift": [ 855, 343, -337 ], "collision_preimages": [ [ "-5/4", "5/4", "-1" ], [ "9/2", "-1", "-7/2" ], [ "13/2", "-10", "3/2" ] ], "collision_image": [ "3423/4", "1373/4", "-1349/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0413631080009509 ], "surface_features": [ 3.491588510654704, 4.508327674547124, 3.102262149494273, 0.0, 3.660975234670547, 4844 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C178", "pair": 178, "family": "counterexample", "mapping": "F(x,y,z)=(2*x^7 + 13*x^6*y - 19*x^6*z - 29*x^6 + 36*x^5*y^2 - 105*x^5*y*z - 162*x^5*y + 75*x^5*z^2 + 240*x^5*z + 177*x^5 + 55*x^4*y^3 - 240*x^4*y^2*z - 375*x^4*y^2 + 342*x^4*y*z^2 + 1110*x^4*y*z + 825*x^4*y - 159*x^4*z^3 - 807*x^4*z^2 - 1242*x^4*z - 588*x^4 + 50*x^3*y^4 - 290*x^3*y^3*z - 460*x^3*y^3 + 618*x^3*y^2*z^2 + 2040*x^3*y^2*z + 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680*x^3*z^3 + 2488*x^3*z^2 + 4044*x^3*z + 2444*x^3 + 42*x^2*y^5 + 210*x^2*y^4*z + 550*x^2*y^4 + 420*x^2*y^3*z^2 + 2160*x^2*y^3*z + 2774*x^2*y^3 + 420*x^2*y^2*z^3 + 3180*x^2*y^2*z^2 + 8034*x^2*y^2*z + 6752*x^2*y^2 + 210*x^2*y*z^4 + 2080*x^2*y*z^3 + 7752*x^2*y*z^2 + 12838*x^2*y*z + 7910*x^2*y + 42*x^2*z^5 + 510*x^2*z^4 + 2492*x^2*z^3 + 6096*x^2*z^2 + 7404*x^2*z + 3542*x^2 + 14*x*y^6 + 84*x*y^5*z + 224*x*y^5 + 210*x*y^4*z^2 + 1100*x*y^4*z + 1438*x*y^4 + 280*x*y^3*z^3 + 2160*x*y^3*z^2 + 5556*x*y^3*z + 4754*x*y^3 + 210*x*y^2*z^4 + 2120*x*y^2*z^3 + 8046*x*y^2*z^2 + 13572*x*y^2*z + 8524*x*y^2 + 84*x*y*z^5 + 1040*x*y*z^4 + 5176*x*y*z^3 + 12902*x*y*z^2 + 15984*x*y*z + 7800*x*y + 14*x*z^6 + 204*x*z^5 + 1248*x*z^4 + 4084*x*z^3 + 7476*x*z^2 + 7194*x*z + 2825*x + 2*y^7 + 14*y^6*z + 38*y^6 + 42*y^5*z^2 + 224*y^5*z + 298*y^5 + 70*y^4*z^3 + 550*y^4*z^2 + 1440*y^4*z + 1254*y^4 + 70*y^3*z^4 + 720*y^3*z^3 + 2782*y^3*z^2 + 4778*y^3*z + 3058*y^3 + 42*y^2*z^5 + 530*y^2*z^4 + 2686*y^2*z^3 + 6820*y^2*z^2 + 8618*y^2*z + 4290*y^2 + 14*y*z^6 + 208*y*z^5 + 1296*y*z^4 + 4322*y*z^3 + 8074*y*z^2 + 7932*y*z + 3181*y + 2*z^7 + 34*z^6 + 250*z^5 + 1026*z^4 + 2516*z^3 + 3652*z^2 + 2883*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ 1, 1, 1 ], [ 1, 2, 1 ], [ 1, 1, 2 ] ], "source_shift": [ 2, 1, 2 ], "target_matrix": [ [ 1, 1, 2 ], [ 1, 1, 1 ], [ 1, 2, 1 ] ], "target_shift": [ -912, -944, -949 ], "inverse_recipe": { "p": 2, "q": 3, "f": "x**2", "g": "x**3 + x**2 + x", "h": "y**2*z + y*z + y", "polynomials": { "f": [ [ [ 2, 0, 0 ], 1 ] ], "g": [ [ [ 3, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "h": [ [ [ 0, 2, 1 ], 1 ], [ [ 0, 1, 1 ], 1 ], [ [ 0, 1, 0 ], 1 ] ] }, "h_multiplier": 2 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0539761149993865 ], "surface_features": [ 3.3953273639926604, 4.203685470881907, 3.0141003215196207, 0.0, 3.517815072018337, 4788 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B031", "pair": 31, "family": "bijection", "mapping": "F(x,y,z)=(256*x^7 - 2688*x^6*y - 1792*x^6*z + 1792*x^6 + 12096*x^5*y^2 + 16128*x^5*y*z - 16128*x^5*y + 5376*x^5*z^2 - 10752*x^5*z + 5472*x^5 - 30240*x^4*y^3 - 60480*x^4*y^2*z + 60480*x^4*y^2 - 40320*x^4*y*z^2 + 80640*x^4*y*z - 41024*x^4*y - 8960*x^4*z^3 + 26880*x^4*z^2 - 27392*x^4*z + 9536*x^4 + 45360*x^3*y^4 + 120960*x^3*y^3*z - 120960*x^3*y^3 + 120960*x^3*y^2*z^2 - 241920*x^3*y^2*z + 123024*x^3*y^2 + 53760*x^3*y*z^3 - 161280*x^3*y*z^2 + 164288*x^3*y*z - 57160*x^3*y + 8960*x^3*z^4 - 35840*x^3*z^3 + 54848*x^3*z^2 - 38272*x^3*z + 10304*x^3 - 40824*x^2*y^5 - 136080*x^2*y^4*z + 136080*x^2*y^4 - 181440*x^2*y^3*z^2 + 362880*x^2*y^3*z - 184464*x^2*y^3 - 120960*x^2*y^2*z^3 + 362880*x^2*y^2*z^2 - 369504*x^2*y^2*z + 128484*x^2*y^2 - 40320*x^2*y*z^4 + 161280*x^2*y*z^3 - 246720*x^2*y*z^2 + 172056*x^2*y*z - 46296*x^2*y - 5376*x^2*z^5 + 26880*x^2*z^4 - 54912*x^2*z^3 + 57600*x^2*z^2 - 31104*x^2*z + 6930*x^2 + 20412*x*y^6 + 81648*x*y^5*z - 81648*x*y^5 + 136080*x*y^4*z^2 - 272160*x*y^4*z + 138294*x*y^4 + 120960*x*y^3*z^3 - 362880*x*y^3*z^2 + 369360*x*y^3*z - 128358*x*y^3 + 60480*x*y^2*z^4 - 241920*x*y^2*z^3 + 369936*x*y^2*z^2 - 257832*x*y^2*z + 69336*x*y^2 + 16128*x*y*z^5 - 80640*x*y*z^4 + 164672*x*y*z^3 - 172632*x*y*z^2 + 93168*x*y*z - 20750*x*y + 1792*x*z^6 - 10752*x*z^5 + 27488*x*z^4 - 38528*x*z^3 + 31296*x*z^2 - 13994*x*z + 2705*x - 4374*y^7 - 20412*y^6*z + 20412*y^6 - 40824*y^5*z^2 + 81648*y^5*z - 41472*y^5 - 45360*y^4*z^3 + 136080*y^4*z^2 - 138456*y^4*z + 48087*y^4 - 30240*y^3*z^4 + 120960*y^3*z^3 - 184896*y^3*z^2 + 128790*y^3*z - 34614*y^3 - 12096*y^2*z^5 + 60480*y^2*z^4 - 123456*y^2*z^3 + 129348*y^2*z^2 - 69768*y^2*z + 15532*y^2 - 2688*y*z^6 + 16128*y*z^5 - 41216*y*z^4 + 57736*y*z^3 - 46872*y*z^2 + 20952*y*z - 4051*y - 256*z^7 + 1792*z^6 - 5504*z^5 + 9664*z^4 - 10496*z^3 + 7064*z^2 - 2743*z + 1, 128*x^7 - 1344*x^6*y - 896*x^6*z + 896*x^6 + 6048*x^5*y^2 + 8064*x^5*y*z - 8064*x^5*y + 2688*x^5*z^2 - 5376*x^5*z + 2736*x^5 - 15120*x^4*y^3 - 30240*x^4*y^2*z + 30240*x^4*y^2 - 20160*x^4*y*z^2 + 40320*x^4*y*z - 20512*x^4*y - 4480*x^4*z^3 + 13440*x^4*z^2 - 13696*x^4*z + 4776*x^4 + 22680*x^3*y^4 + 60480*x^3*y^3*z - 60480*x^3*y^3 + 60480*x^3*y^2*z^2 - 120960*x^3*y^2*z + 61512*x^3*y^2 + 26880*x^3*y*z^3 - 80640*x^3*y*z^2 + 82144*x^3*y*z - 28628*x^3*y + 4480*x^3*z^4 - 17920*x^3*z^3 + 27424*x^3*z^2 - 19168*x^3*z + 5176*x^3 - 20412*x^2*y^5 - 68040*x^2*y^4*z + 68040*x^2*y^4 - 90720*x^2*y^3*z^2 + 181440*x^2*y^3*z - 92232*x^2*y^3 - 60480*x^2*y^2*z^3 + 181440*x^2*y^2*z^2 - 184752*x^2*y^2*z + 64350*x^2*y^2 - 20160*x^2*y*z^4 + 80640*x^2*y*z^3 - 123360*x^2*y*z^2 + 86172*x^2*y*z - 23256*x^2*y - 2688*x^2*z^5 + 13440*x^2*z^4 - 27456*x^2*z^3 + 28848*x^2*z^2 - 15624*x^2*z + 3489*x^2 + 10206*x*y^6 + 40824*x*y^5*z - 40824*x*y^5 + 68040*x*y^4*z^2 - 136080*x*y^4*z + 69147*x*y^4 + 60480*x*y^3*z^3 - 181440*x*y^3*z^2 + 184680*x*y^3*z - 64287*x*y^3 + 30240*x*y^2*z^4 - 120960*x*y^2*z^3 + 184968*x*y^2*z^2 - 129132*x*y^2*z + 34830*x*y^2 + 8064*x*y*z^5 - 40320*x*y*z^4 + 82336*x*y*z^3 - 86460*x*y*z^2 + 46800*x*y*z - 10447*x*y + 896*x*z^6 - 5376*x*z^5 + 13744*x*z^4 - 19296*x*z^3 + 15720*x*z^2 - 7045*x*z + 1359*x - 2187*y^7 - 10206*y^6*z + 10206*y^6 - 20412*y^5*z^2 + 40824*y^5*z - 20736*y^5 - 22680*y^4*z^3 + 68040*y^4*z^2 - 69228*y^4*z + 24084*y^4 - 15120*y^3*z^4 + 60480*y^3*z^3 - 92448*y^3*z^2 + 64503*y^3*z - 17388*y^3 - 6048*y^2*z^5 + 30240*y^2*z^4 - 61728*y^2*z^3 + 64782*y^2*z^2 - 35046*y^2*z + 7820*y^2 - 1344*y*z^6 + 8064*y*z^5 - 20608*y*z^4 + 28916*y*z^3 - 23544*y*z^2 + 10548*y*z - 2036*y - 128*z^7 + 896*z^6 - 2752*z^5 + 4840*z^4 - 5272*z^3 + 3556*z^2 - 1377*z + 1, 128*x^7 - 1344*x^6*y - 896*x^6*z + 896*x^6 + 6048*x^5*y^2 + 8064*x^5*y*z - 8064*x^5*y + 2688*x^5*z^2 - 5376*x^5*z + 2736*x^5 - 15120*x^4*y^3 - 30240*x^4*y^2*z + 30240*x^4*y^2 - 20160*x^4*y*z^2 + 40320*x^4*y*z - 20512*x^4*y - 4480*x^4*z^3 + 13440*x^4*z^2 - 13696*x^4*z + 4776*x^4 + 22680*x^3*y^4 + 60480*x^3*y^3*z - 60480*x^3*y^3 + 60480*x^3*y^2*z^2 - 120960*x^3*y^2*z + 61512*x^3*y^2 + 26880*x^3*y*z^3 - 80640*x^3*y*z^2 + 82144*x^3*y*z - 28628*x^3*y + 4480*x^3*z^4 - 17920*x^3*z^3 + 27424*x^3*z^2 - 19168*x^3*z + 5160*x^3 - 20412*x^2*y^5 - 68040*x^2*y^4*z + 68040*x^2*y^4 - 90720*x^2*y^3*z^2 + 181440*x^2*y^3*z - 92232*x^2*y^3 - 60480*x^2*y^2*z^3 + 181440*x^2*y^2*z^2 - 184752*x^2*y^2*z + 64350*x^2*y^2 - 20160*x^2*y*z^4 + 80640*x^2*y*z^3 - 123360*x^2*y*z^2 + 86172*x^2*y*z - 23184*x^2*y - 2688*x^2*z^5 + 13440*x^2*z^4 - 27456*x^2*z^3 + 28848*x^2*z^2 - 15576*x^2*z + 3441*x^2 + 10206*x*y^6 + 40824*x*y^5*z - 40824*x*y^5 + 68040*x*y^4*z^2 - 136080*x*y^4*z + 69147*x*y^4 + 60480*x*y^3*z^3 - 181440*x*y^3*z^2 + 184680*x*y^3*z - 64287*x*y^3 + 30240*x*y^2*z^4 - 120960*x*y^2*z^3 + 184968*x*y^2*z^2 - 129132*x*y^2*z + 34722*x*y^2 + 8064*x*y*z^5 - 40320*x*y*z^4 + 82336*x*y*z^3 - 86460*x*y*z^2 + 46656*x*y*z - 10303*x*y + 896*x*z^6 - 5376*x*z^5 + 13744*x*z^4 - 19296*x*z^3 + 15672*x*z^2 - 6949*x*z + 1305*x - 2187*y^7 - 10206*y^6*z + 10206*y^6 - 20412*y^5*z^2 + 40824*y^5*z - 20736*y^5 - 22680*y^4*z^3 + 68040*y^4*z^2 - 69228*y^4*z + 24084*y^4 - 15120*y^3*z^4 + 60480*y^3*z^3 - 92448*y^3*z^2 + 64503*y^3*z - 17334*y^3 - 6048*y^2*z^5 + 30240*y^2*z^4 - 61728*y^2*z^3 + 64782*y^2*z^2 - 34938*y^2*z + 7712*y^2 - 1344*y*z^6 + 8064*y*z^5 - 20608*y*z^4 + 28916*y*z^3 - 23472*y*z^2 + 10404*y*z - 1956*y - 128*z^7 + 896*z^6 - 2752*z^5 + 4840*z^4 - 5256*z^3 + 3508*z^2 - 1321*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ 2, -3, -2 ], [ 1, -2, -1 ], [ 1, -1, -2 ] ], "source_shift": [ 2, 1, 1 ], "target_matrix": [ [ 2, 1, 1 ], [ 1, 1, 1 ], [ 1, 1, 2 ] ], "target_shift": [ -478, -237, -215 ], "inverse_recipe": { "p": 4, "q": 3, "f": "x**4 + x", "g": "x**3 + x", "h": "y*z + y + z", "polynomials": { "f": [ [ [ 4, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "g": [ [ [ 3, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 1, 0 ], 1 ], [ [ 0, 0, 1 ], 1 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.011462559003121 ], "surface_features": [ 4.735578031595341, 5.568126596361596, 4.458456540444431, 0.0, 4.830863226036515, 5350 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B009", "pair": 9, "family": "bijection", "mapping": "F(x,y,z)=(-2187*x^7 + 10206*x^6*y + 10206*x^6*z - 5103*x^6 - 20412*x^5*y^2 - 40824*x^5*y*z + 20412*x^5*y - 20412*x^5*z^2 + 20412*x^5*z - 5427*x^5 + 22680*x^4*y^3 + 68040*x^4*y^2*z - 34020*x^4*y^2 + 68040*x^4*y*z^2 - 68040*x^4*y*z + 18117*x^4*y + 22680*x^4*z^3 - 34020*x^4*z^2 + 18117*x^4*z - 3699*x^4 - 15120*x^3*y^4 - 60480*x^3*y^3*z + 30240*x^3*y^3 - 90720*x^3*y^2*z^2 + 90720*x^3*y^2*z - 24192*x^3*y^2 - 60480*x^3*y*z^3 + 90720*x^3*y*z^2 - 48384*x^3*y*z + 9936*x^3*y - 15120*x^3*z^4 + 30240*x^3*z^3 - 24192*x^3*z^2 + 9909*x^3*z - 1683*x^3 + 6048*x^2*y^5 + 30240*x^2*y^4*z - 15120*x^2*y^4 + 60480*x^2*y^3*z^2 - 60480*x^2*y^3*z + 16152*x^2*y^3 + 60480*x^2*y^2*z^3 - 90720*x^2*y^2*z^2 + 48456*x^2*y^2*z - 10008*x^2*y^2 + 30240*x^2*y*z^4 - 60480*x^2*y*z^3 + 48456*x^2*y*z^2 - 19962*x^2*y*z + 3423*x^2*y + 6048*x^2*z^5 - 15120*x^2*z^4 + 16152*x^2*z^3 - 9954*x^2*z^2 + 3396*x^2*z - 499*x^2 - 1344*x*y^6 - 8064*x*y^5*z + 4032*x*y^5 - 20160*x*y^4*z^2 + 20160*x*y^4*z - 5392*x*y^4 - 26880*x*y^3*z^3 + 40320*x*y^3*z^2 - 21568*x*y^3*z + 4480*x*y^3 - 20160*x*y^2*z^4 + 40320*x*y^2*z^3 - 32352*x*y^2*z^2 + 13404*x*y^2*z - 2320*x*y^2 - 8064*x*y*z^5 + 20160*x*y*z^4 - 21568*x*y*z^3 + 13368*x*y*z^2 - 4604*x*y*z + 686*x*y - 1344*x*z^6 + 4032*x*z^5 - 5392*x*z^4 + 4444*x*z^3 - 2284*x*z^2 + 676*x*z - 101*x + 128*y^7 + 896*y^6*z - 448*y^6 + 2688*y^5*z^2 - 2688*y^5*z + 720*y^5 + 4480*y^4*z^3 - 6720*y^4*z^2 + 3600*y^4*z - 752*y^4 + 4480*y^3*z^4 - 8960*y^3*z^3 + 7200*y^3*z^2 - 3000*y^3*z + 524*y^3 + 2688*y^2*z^5 - 6720*y^2*z^4 + 7200*y^2*z^3 - 4488*y^2*z^2 + 1560*y^2*z - 236*y^2 + 896*y*z^6 - 2688*y*z^5 + 3600*y*z^4 - 2984*y*z^3 + 1548*y*z^2 - 465*y*z + 73*y + 128*z^7 - 448*z^6 + 720*z^5 - 744*z^4 + 512*z^3 - 229*z^2 + 69*z - 1, 2187*x^7 - 10206*x^6*y - 10206*x^6*z + 5103*x^6 + 20412*x^5*y^2 + 40824*x^5*y*z - 20412*x^5*y + 20412*x^5*z^2 - 20412*x^5*z + 5427*x^5 - 22680*x^4*y^3 - 68040*x^4*y^2*z + 34020*x^4*y^2 - 68040*x^4*y*z^2 + 68040*x^4*y*z - 18117*x^4*y - 22680*x^4*z^3 + 34020*x^4*z^2 - 18117*x^4*z + 3699*x^4 + 15120*x^3*y^4 + 60480*x^3*y^3*z - 30240*x^3*y^3 + 90720*x^3*y^2*z^2 - 90720*x^3*y^2*z + 24192*x^3*y^2 + 60480*x^3*y*z^3 - 90720*x^3*y*z^2 + 48384*x^3*y*z - 9936*x^3*y + 15120*x^3*z^4 - 30240*x^3*z^3 + 24192*x^3*z^2 - 9909*x^3*z + 1737*x^3 - 6048*x^2*y^5 - 30240*x^2*y^4*z + 15120*x^2*y^4 - 60480*x^2*y^3*z^2 + 60480*x^2*y^3*z - 16152*x^2*y^3 - 60480*x^2*y^2*z^3 + 90720*x^2*y^2*z^2 - 48456*x^2*y^2*z + 10008*x^2*y^2 - 30240*x^2*y*z^4 + 60480*x^2*y*z^3 - 48456*x^2*y*z^2 + 19962*x^2*y*z - 3531*x^2*y - 6048*x^2*z^5 + 15120*x^2*z^4 - 16152*x^2*z^3 + 9954*x^2*z^2 - 3504*x^2*z + 553*x^2 + 1344*x*y^6 + 8064*x*y^5*z - 4032*x*y^5 + 20160*x*y^4*z^2 - 20160*x*y^4*z + 5392*x*y^4 + 26880*x*y^3*z^3 - 40320*x*y^3*z^2 + 21568*x*y^3*z - 4480*x*y^3 + 20160*x*y^2*z^4 - 40320*x*y^2*z^3 + 32352*x*y^2*z^2 - 13404*x*y^2*z + 2392*x*y^2 + 8064*x*y*z^5 - 20160*x*y*z^4 + 21568*x*y*z^3 - 13368*x*y*z^2 + 4748*x*y*z - 758*x*y + 1344*x*z^6 - 4032*x*z^5 + 5392*x*z^4 - 4444*x*z^3 + 2356*x*z^2 - 748*x*z + 121*x - 128*y^7 - 896*y^6*z + 448*y^6 - 2688*y^5*z^2 + 2688*y^5*z - 720*y^5 - 4480*y^4*z^3 + 6720*y^4*z^2 - 3600*y^4*z + 752*y^4 - 4480*y^3*z^4 + 8960*y^3*z^3 - 7200*y^3*z^2 + 3000*y^3*z - 540*y^3 - 2688*y^2*z^5 + 6720*y^2*z^4 - 7200*y^2*z^3 + 4488*y^2*z^2 - 1608*y^2*z + 260*y^2 - 896*y*z^6 + 2688*y*z^5 - 3600*y*z^4 + 2984*y*z^3 - 1596*y*z^2 + 513*y*z - 87*y - 128*z^7 + 448*z^6 - 720*z^5 + 744*z^4 - 528*z^3 + 253*z^2 - 83*z + 1, -2187*x^7 + 10206*x^6*y + 10206*x^6*z - 5103*x^6 - 20412*x^5*y^2 - 40824*x^5*y*z + 20412*x^5*y - 20412*x^5*z^2 + 20412*x^5*z - 5427*x^5 + 22680*x^4*y^3 + 68040*x^4*y^2*z - 34020*x^4*y^2 + 68040*x^4*y*z^2 - 68040*x^4*y*z + 18117*x^4*y + 22680*x^4*z^3 - 34020*x^4*z^2 + 18117*x^4*z - 3780*x^4 - 15120*x^3*y^4 - 60480*x^3*y^3*z + 30240*x^3*y^3 - 90720*x^3*y^2*z^2 + 90720*x^3*y^2*z - 24192*x^3*y^2 - 60480*x^3*y*z^3 + 90720*x^3*y*z^2 - 48384*x^3*y*z + 10152*x^3*y - 15120*x^3*z^4 + 30240*x^3*z^3 - 24192*x^3*z^2 + 10125*x^3*z - 1845*x^3 + 6048*x^2*y^5 + 30240*x^2*y^4*z - 15120*x^2*y^4 + 60480*x^2*y^3*z^2 - 60480*x^2*y^3*z + 16152*x^2*y^3 + 60480*x^2*y^2*z^3 - 90720*x^2*y^2*z^2 + 48456*x^2*y^2*z - 10224*x^2*y^2 + 30240*x^2*y*z^4 - 60480*x^2*y*z^3 + 48456*x^2*y*z^2 - 20394*x^2*y*z + 3747*x^2*y + 6048*x^2*z^5 - 15120*x^2*z^4 + 16152*x^2*z^3 - 10170*x^2*z^2 + 3720*x^2*z - 616*x^2 - 1344*x*y^6 - 8064*x*y^5*z + 4032*x*y^5 - 20160*x*y^4*z^2 + 20160*x*y^4*z - 5392*x*y^4 - 26880*x*y^3*z^3 + 40320*x*y^3*z^2 - 21568*x*y^3*z + 4576*x*y^3 - 20160*x*y^2*z^4 + 40320*x*y^2*z^3 - 32352*x*y^2*z^2 + 13692*x*y^2*z - 2536*x*y^2 - 8064*x*y*z^5 + 20160*x*y*z^4 - 21568*x*y*z^3 + 13656*x*y*z^2 - 5036*x*y*z + 842*x*y - 1344*x*z^6 + 4032*x*z^5 - 5392*x*z^4 + 4540*x*z^3 - 2500*x*z^2 + 832*x*z - 143*x + 128*y^7 + 896*y^6*z - 448*y^6 + 2688*y^5*z^2 - 2688*y^5*z + 720*y^5 + 4480*y^4*z^3 - 6720*y^4*z^2 + 3600*y^4*z - 768*y^4 + 4480*y^3*z^4 - 8960*y^3*z^3 + 7200*y^3*z^2 - 3064*y^3*z + 572*y^3 + 2688*y^2*z^5 - 6720*y^2*z^4 + 7200*y^2*z^3 - 4584*y^2*z^2 + 1704*y^2*z - 288*y^2 + 896*y*z^6 - 2688*y*z^5 + 3600*y*z^4 - 3048*y*z^3 + 1692*y*z^2 - 569*y*z + 103*y + 128*z^7 - 448*z^6 + 720*z^5 - 760*z^4 + 560*z^3 - 281*z^2 + 98*z - 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ 3, -2, -2 ], [ 1, -2, -1 ], [ 1, -1, -1 ] ], "source_shift": [ 1, 1, 1 ], "target_matrix": [ [ -1, -1, -2 ], [ 1, 1, 1 ], [ -1, -2, -1 ] ], "target_shift": [ 10, -14, 18 ], "inverse_recipe": { "p": 4, "q": 3, "f": "x**4 + x**2 + x", "g": "x**3", "h": "y*z + y + z", "polynomials": { "f": [ [ [ 4, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "g": [ [ [ 3, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 1, 0 ], 1 ], [ [ 0, 0, 1 ], 1 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.083078530999046 ], "surface_features": [ 4.255702364472767, 4.9577030415488315, 3.90655051910145, 0.0, 4.350525987008448, 5139 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C083", "pair": 83, "family": "counterexample", "mapping": "F(x,y,z)=(-8*x^7 - 68*x^6*y + 44*x^6*z + 148*x^6 - 234*x^5*y^2 + 312*x^5*y*z + 1044*x^5*y - 102*x^5*z^2 - 684*x^5*z - 1044*x^5 - 419*x^4*y^3 + 867*x^4*y^2*z + 2901*x^4*y^2 - 585*x^4*y*z^2 - 3906*x^4*y*z - 6048*x^4*y + 129*x^4*z^3 + 1293*x^4*z^2 + 3960*x^4*z + 3834*x^4 - 419*x^3*y^4 + 1199*x^3*y^3*z + 4048*x^3*y^3 - 1257*x^3*y^2*z^2 - 8406*x^3*y^2*z - 13257*x^3*y^2 + 573*x^3*y*z^3 + 5724*x^3*y*z^2 + 17790*x^3*y*z + 17531*x^3*y - 96*x^3*z^4 - 1278*x^3*z^3 - 5889*x^3*z^2 - 11465*x^3*z - 8104*x^3 - 234*x^2*y^5 + 867*x^2*y^4*z + 2988*x^2*y^4 - 1257*x^2*y^3*z^2 - 8496*x^2*y^3*z - 13671*x^2*y^3 + 891*x^2*y^2*z^3 + 8928*x^2*y^2*z^2 + 28233*x^2*y^2*z + 28326*x^2*y^2 - 309*x^2*y*z^4 - 4104*x^2*y*z^3 - 19185*x^2*y*z^2 - 37983*x^2*y*z - 27315*x^2*y + 42*x^2*z^5 + 696*x^2*z^4 + 4287*x^2*z^3 + 12579*x^2*z^2 + 17895*x^2*z + 9994*x^2 - 68*x*y^6 + 312*x*y^5*z + 1104*x*y^5 - 585*x*y^4*z^2 - 4032*x*y^4*z - 6588*x*y^4 + 573*x*y^3*z^3 + 5808*x*y^3*z^2 + 18666*x*y^3*z + 19005*x*y^3 - 309*x*y^2*z^4 - 4122*x*y^2*z^3 - 19563*x*y^2*z^2 - 39327*x*y^2*z - 28716*x*y^2 + 87*x*y*z^5 + 1440*x*y*z^4 + 8988*x*y*z^3 + 26772*x*y*z^2 + 38679*x*y*z + 21952*x*y - 10*x*z^6 - 198*x*z^5 - 1527*x*z^4 - 5996*x*z^3 - 12861*x*z^2 - 14462*x*z - 6719*x - 8*y^7 + 44*y^6*z + 160*y^6 - 102*y^5*z^2 - 720*y^5*z - 1188*y^5 + 129*y^4*z^3 + 1332*y^4*z^2 + 4332*y^4*z + 4447*y^4 - 96*y^3*z^4 - 1296*y^3*z^3 - 6231*y^3*z^2 - 12664*y^3*z - 9332*y^3 + 42*y^2*z^5 + 699*y^2*z^4 + 4419*y^2*z^3 + 13326*y^2*z^2 + 19476*y^2*z + 11182*y^2 - 10*y*z^6 - 198*y*z^5 - 1545*y*z^4 - 6145*y*z^3 - 13350*y*z^2 - 15208*y*z - 7159*y + z^7 + 23*z^6 + 213*z^5 + 1048*z^4 + 3008*z^3 + 5098*z^2 + 4765*z + 1, -24*x^7 - 204*x^6*y + 132*x^6*z + 348*x^6 - 702*x^5*y^2 + 936*x^5*y*z + 2508*x^5*y - 306*x^5*z^2 - 1620*x^5*z - 2124*x^5 - 1257*x^4*y^3 + 2601*x^4*y^2*z + 7143*x^4*y^2 - 1755*x^4*y*z^2 - 9462*x^4*y*z - 12624*x^4*y + 387*x^4*z^3 + 3087*x^4*z^2 + 8136*x^4*z + 7094*x^4 - 1257*x^3*y^4 + 3597*x^3*y^3*z + 10236*x^3*y^3 - 3771*x^3*y^2*z^2 - 20886*x^3*y^2*z - 28479*x^3*y^2 + 1719*x^3*y*z^3 + 13992*x^3*y*z^2 + 37554*x^3*y*z + 33317*x^3*y - 288*x^3*z^4 - 3078*x^3*z^3 - 12231*x^3*z^2 - 21455*x^3*z - 14044*x^3 - 702*x^2*y^5 + 2601*x^2*y^4*z + 7752*x^2*y^4 - 3771*x^2*y^3*z^2 - 21672*x^2*y^3*z - 30285*x^2*y^3 + 2673*x^2*y^2*z^3 + 22392*x^2*y^2*z^2 + 61407*x^2*y^2*z + 55476*x^2*y^2 - 927*x^2*y*z^4 - 10128*x^2*y*z^3 - 40995*x^2*y*z^2 - 73137*x^2*y*z - 48693*x^2*y + 126*x^2*z^5 + 1692*x^2*z^4 + 9009*x^2*z^3 + 23835*x^2*z^2 + 31413*x^2*z + 16542*x^2 - 204*x*y^6 + 936*x*y^5*z + 2928*x*y^5 - 1755*x*y^4*z^2 - 10536*x*y^4*z - 15060*x*y^4 + 1719*x*y^3*z^3 + 14940*x*y^3*z^2 + 41886*x*y^3*z + 38470*x*y^3 - 927*x*y^2*z^4 - 10434*x*y^2*z^3 - 43101*x*y^2*z^2 - 78162*x*y^2*z - 52857*x*y^2 + 261*x*y*z^5 + 3588*x*y*z^4 + 19452*x*y*z^3 + 52281*x*y*z^2 + 69987*x*y*z + 37476*x*y - 30*x*z^6 - 486*x*z^5 - 3249*x*z^4 - 11515*x*z^3 - 22896*x*z^2 - 24294*x*z - 10785*x - 24*y^7 + 132*y^6*z + 432*y^6 - 306*y^5*z^2 - 1920*y^5*z - 2796*y^5 + 387*y^4*z^3 + 3504*y^4*z^2 + 10020*y^4*z + 9310*y^4 - 288*y^3*z^4 - 3360*y^3*z^3 - 14157*y^3*z^2 - 26020*y^3*z - 17786*y^3 + 126*y^2*z^5 + 1785*y^2*z^4 + 9861*y^2*z^3 + 26880*y^2*z^2 + 36435*y^2*z + 19764*y^2 - 30*y*z^6 - 498*y*z^5 - 3387*y*z^4 - 12175*y*z^3 - 24534*y*z^2 - 26400*y*z - 11899*y + 3*z^7 + 57*z^6 + 459*z^5 + 2041*z^4 + 5435*z^3 + 8700*z^2 + 7781*z + 1, -16*x^7 - 136*x^6*y + 88*x^6*z + 224*x^6 - 468*x^5*y^2 + 624*x^5*y*z + 1620*x^5*y - 204*x^5*z^2 - 1044*x^5*z - 1332*x^5 - 838*x^4*y^3 + 1734*x^4*y^2*z + 4632*x^4*y^2 - 1170*x^4*y*z^2 - 6120*x^4*y*z - 7956*x^4*y + 258*x^4*z^3 + 1992*x^4*z^2 + 5112*x^4*z + 4364*x^4 - 838*x^3*y^4 + 2398*x^3*y^3*z + 6665*x^3*y^3 - 2514*x^3*y^2*z^2 - 13563*x^3*y^2*z - 18045*x^3*y^2 + 1146*x^3*y*z^3 + 9063*x^3*y*z^2 + 23718*x^3*y*z + 20610*x^3*y - 192*x^3*z^4 - 1989*x^3*z^3 - 7701*x^3*z^2 - 13230*x^3*z - 8520*x^3 - 468*x^2*y^5 + 1734*x^2*y^4*z + 5067*x^2*y^4 - 2514*x^2*y^3*z^2 - 14130*x^2*y^3*z - 19296*x^2*y^3 + 1782*x^2*y^2*z^3 + 14562*x^2*y^2*z^2 + 38997*x^2*y^2*z + 34530*x^2*y^2 - 618*x^2*y*z^4 - 6570*x^2*y*z^3 - 25950*x^2*y*z^2 - 45366*x^2*y*z - 29718*x^2*y + 84*x^2*z^5 + 1095*x^2*z^4 + 5685*x^2*z^3 + 14736*x^2*z^2 + 19110*x^2*z + 9938*x^2 - 136*x*y^6 + 624*x*y^5*z + 1920*x*y^5 - 1170*x*y^4*z^2 - 6894*x*y^4*z - 9648*x*y^4 + 1146*x*y^3*z^3 + 9753*x*y^3*z^2 + 26748*x*y^3*z + 24103*x*y^3 - 618*x*y^2*z^4 - 6795*x*y^2*z^3 - 27435*x*y^2*z^2 - 48795*x*y^2*z - 32481*x*y^2 + 174*x*y*z^5 + 2331*x*y*z^4 + 12342*x*y*z^3 + 32523*x*y*z^2 + 42852*x*y*z + 22661*x*y - 20*x*z^6 - 315*x*z^5 - 2055*x*z^4 - 7139*x*z^3 - 13971*x*z^2 - 14641*x*z - 6437*x - 16*y^7 + 88*y^6*z + 284*y^6 - 204*y^5*z^2 - 1260*y^5*z - 1800*y^5 + 258*y^4*z^3 + 2295*y^4*z^2 + 6432*y^4*z + 5871*y^4 - 192*y^3*z^4 - 2196*y^3*z^3 - 9060*y^3*z^2 - 16350*y^3*z - 11010*y^3 + 84*y^2*z^5 + 1164*y^2*z^4 + 6291*y^2*z^3 + 16830*y^2*z^2 + 22467*y^2*z + 12041*y^2 - 20*y*z^6 - 324*y*z^5 - 2154*y*z^4 - 7596*y*z^3 - 15072*y*z^2 - 16022*y*z - 7154*y + 2*z^7 + 37*z^6 + 291*z^5 + 1269*z^4 + 3327*z^3 + 5261*z^2 + 4661*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -2, -1, 1 ], [ -1, -2, 1 ], [ -1, -1, 1 ] ], "source_shift": [ 2, 2, 2 ], "target_matrix": [ [ 1, 2, 1 ], [ 3, 2, 2 ], [ 2, 1, 1 ] ], "target_shift": [ -1905, -3009, -1789 ], "collision_preimages": [ [ "-1/4", "-1/4", "-11/4" ], [ "11/2", "8", "18" ], [ "15/2", "5", "17" ] ], "collision_image": [ "-7621/4", "-12039/4", "-3579/2" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0155956690032326 ], "surface_features": 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14052*x^2*y - 114*x^2*z^5 + 1230*x^2*z^4 - 5034*x^2*z^3 + 9927*x^2*z^2 - 9501*x^2*z + 3527*x^2 + 248*x*y^6 + 1152*x*y^5*z - 2568*x*y^5 + 2190*x*y^4*z^2 - 9642*x*y^4*z + 10380*x*y^4 + 2178*x*y^3*z^3 - 14253*x*y^3*z^2 + 30264*x*y^3*z - 20897*x*y^3 + 1194*x*y^2*z^4 - 10359*x*y^2*z^3 + 32583*x*y^2*z^2 - 44265*x*y^2*z + 21975*x*y^2 + 342*x*y*z^5 - 3699*x*y*z^4 + 15336*x*y*z^3 - 30783*x*y*z^2 + 30078*x*y*z - 11423*x*y + 40*x*z^6 - 519*x*z^5 + 2661*x*z^4 - 7021*x*z^3 + 10137*x*z^2 - 7597*x*z + 2291*x - 48*y^7 - 264*y^6*z + 596*y^6 - 612*y^5*z^2 + 2724*y^5*z - 2976*y^5 - 774*y^4*z^3 + 5109*y^4*z^2 - 11004*y^4*z + 7747*y^4 - 576*y^3*z^4 + 5028*y^3*z^3 - 16032*y^3*z^2 + 22198*y^3*z - 11286*y^3 - 252*y^2*z^5 + 2736*y^2*z^4 - 11493*y^2*z^3 + 23493*y^2*z^2 - 23457*y^2*z + 9140*y^2 - 60*y*z^6 + 780*y*z^5 - 4050*y*z^4 + 10874*y*z^3 - 16026*y*z^2 + 12290*y*z - 3782*y - 6*z^7 + 91*z^6 - 561*z^5 + 1856*z^4 - 3591*z^3 + 4070*z^2 - 2491*z + 1, -2*x^7 + 21*x^6*y + 15*x^6*z - 35*x^6 - 93*x^5*y^2 - 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90*x^3*z^4 + 700*x^3*z^3 - 2010*x^3*z^2 + 2512*x^3*z - 1144*x^3 + 270*x^2*y^5 + 1029*x^2*y^4*z - 2106*x^2*y^4 + 1539*x^2*y^3*z^2 - 6192*x^2*y^3*z + 6180*x^2*y^3 + 1128*x^2*y^2*z^3 - 6714*x^2*y^2*z^2 + 13206*x^2*y^2*z - 8541*x^2*y^2 + 405*x^2*y*z^4 - 3180*x^2*y*z^3 + 9261*x^2*y*z^2 - 11793*x^2*y*z + 5499*x^2*y + 57*x^2*z^5 - 555*x^2*z^4 + 2130*x^2*z^3 - 4008*x^2*z^2 + 3672*x^2*z - 1297*x^2 - 124*x*y^6 - 576*x*y^5*z + 1200*x*y^5 - 1095*x*y^4*z^2 + 4476*x*y^4*z - 4545*x*y^4 - 1089*x*y^3*z^3 + 6570*x*y^3*z^2 - 13131*x*y^3*z + 8666*x*y^3 - 597*x*y^2*z^4 + 4740*x*y^2*z^3 - 14010*x*y^2*z^2 + 18186*x*y^2*z - 8688*x*y^2 - 171*x*y*z^5 + 1680*x*y*z^4 - 6537*x*y*z^3 + 12531*x*y*z^2 - 11757*x*y*z + 4273*x*y - 20*x*z^6 + 234*x*z^5 - 1125*x*z^4 + 2832*x*z^3 - 3912*x*z^2 + 2786*x*z - 792*x + 24*y^7 + 132*y^6*z - 280*y^6 + 306*y^5*z^2 - 1272*y^5*z + 1314*y^5 + 387*y^4*z^3 - 2370*y^4*z^2 + 4815*y^4*z - 3242*y^4 + 288*y^3*z^4 - 2316*y^3*z^3 + 6951*y^3*z^2 - 9200*y^3*z + 4507*y^3 + 126*y^2*z^5 - 1251*y^2*z^4 + 4938*y^2*z^3 - 9645*y^2*z^2 + 9267*y^2*z - 3472*y^2 + 30*y*z^6 - 354*y*z^5 + 1725*y*z^4 - 4423*y*z^3 + 6258*y*z^2 - 4591*y*z + 1340*y + 3*z^7 - 41*z^6 + 237*z^5 - 748*z^4 + 1384*z^3 - 1489*z^2 + 857*z - 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ 1, -1, -1 ], [ 1, -2, -1 ], [ 2, -3, -3 ] ], "source_shift": [ 2, 1, 2 ], "target_matrix": [ [ 2, 3, 2 ], [ -1, -2, -1 ], [ -1, -1, -2 ] ], "target_shift": [ -626, 397, 204 ], "collision_preimages": [ [ "-15/4", "-1", "-3/4" ], [ "-15/2", "3/2", "-8" ], [ "-27/2", "-7/2", "-7" ] ], "collision_image": [ "-1253/2", "1589/4", "817/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 0.9762701330000709 ], "surface_features": [ 3.6703793585142286, 4.646060468555641, 3.3093107157881754, 0.0, 3.8138608946415733, 4922 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B159", "pair": 159, "family": "bijection", "mapping": "F(x,y,z)=(-5*x^7 + 35*x^6*y + 35*x^6*z - 80*x^6 - 105*x^5*y^2 - 210*x^5*y*z + 480*x^5*y - 105*x^5*z^2 + 470*x^5*z - 495*x^5 + 175*x^4*y^3 + 525*x^4*y^2*z - 1200*x^4*y^2 + 525*x^4*y*z^2 - 2350*x^4*y*z + 2470*x^4*y + 175*x^4*z^3 - 1150*x^4*z^2 + 2370*x^4*z - 1565*x^4 - 175*x^3*y^4 - 700*x^3*y^3*z + 1600*x^3*y^3 - 1050*x^3*y^2*z^2 + 4700*x^3*y^2*z - 4930*x^3*y^2 - 700*x^3*y*z^3 + 4600*x^3*y*z^2 - 9460*x^3*y*z + 6200*x^3*y - 175*x^3*z^4 + 1500*x^3*z^3 - 4535*x^3*z^2 + 5850*x^3*z - 2882*x^3 + 105*x^2*y^5 + 525*x^2*y^4*z - 1200*x^2*y^4 + 1050*x^2*y^3*z^2 - 4700*x^2*y^3*z + 4920*x^2*y^3 + 1050*x^2*y^2*z^3 - 6900*x^2*y^2*z^2 + 14160*x^2*y^2*z - 9210*x^2*y^2 + 525*x^2*y*z^4 - 4500*x^2*y*z^3 + 13575*x^2*y*z^2 - 17380*x^2*y*z + 8426*x^2*y + 105*x^2*z^5 - 1100*x^2*z^4 + 4335*x^2*z^3 - 8190*x^2*z^2 + 7796*x^2*z - 3106*x^2 - 35*x*y^6 - 210*x*y^5*z + 480*x*y^5 - 525*x*y^4*z^2 + 2350*x*y^4*z - 2455*x*y^4 - 700*x*y^3*z^3 + 4600*x*y^3*z^2 - 9420*x*y^3*z + 6080*x*y^3 - 525*x*y^2*z^4 + 4500*x*y^2*z^3 - 13545*x*y^2*z^2 + 17210*x*y^2*z - 8206*x*y^2 - 210*x*y*z^5 + 2200*x*y*z^4 - 8650*x*y*z^3 + 16220*x*y*z^2 - 15212*x*y*z + 5972*x*y - 35*x*z^6 + 430*x*z^5 - 2070*x*z^4 + 5090*x*z^3 - 7041*x*z^2 + 5392*x*z - 1763*x + 5*y^7 + 35*y^6*z - 80*y^6 + 105*y^5*z^2 - 470*y^5*z + 490*y^5 + 175*y^4*z^3 - 1150*y^4*z^2 + 2350*y^4*z - 1505*y^4 + 175*y^3*z^4 - 1500*y^3*z^3 + 4505*y^3*z^2 - 5680*y^3*z + 2662*y^3 + 105*y^2*z^5 - 1100*y^2*z^4 + 4315*y^2*z^3 - 8030*y^2*z^2 + 7416*y^2*z - 2866*y^2 + 35*y*z^6 - 430*y*z^5 + 2065*y*z^4 - 5040*y*z^3 + 6876*y*z^2 - 5192*y*z + 1681*y + 5*z^7 - 70*z^6 + 395*z^5 - 1185*z^4 + 2122*z^3 - 2351*z^2 + 1483*z - 1, 10*x^7 - 70*x^6*y - 70*x^6*z + 160*x^6 + 210*x^5*y^2 + 420*x^5*y*z - 960*x^5*y + 210*x^5*z^2 - 940*x^5*z + 990*x^5 - 350*x^4*y^3 - 1050*x^4*y^2*z + 2400*x^4*y^2 - 1050*x^4*y*z^2 + 4700*x^4*y*z - 4940*x^4*y - 350*x^4*z^3 + 2300*x^4*z^2 - 4740*x^4*z + 3130*x^4 + 350*x^3*y^4 + 1400*x^3*y^3*z - 3200*x^3*y^3 + 2100*x^3*y^2*z^2 - 9400*x^3*y^2*z + 9860*x^3*y^2 + 1400*x^3*y*z^3 - 9200*x^3*y*z^2 + 18920*x^3*y*z - 12400*x^3*y + 350*x^3*z^4 - 3000*x^3*z^3 + 9070*x^3*z^2 - 11700*x^3*z + 5763*x^3 - 210*x^2*y^5 - 1050*x^2*y^4*z + 2400*x^2*y^4 - 2100*x^2*y^3*z^2 + 9400*x^2*y^3*z - 9840*x^2*y^3 - 2100*x^2*y^2*z^3 + 13800*x^2*y^2*z^2 - 28320*x^2*y^2*z + 18420*x^2*y^2 - 1050*x^2*y*z^4 + 9000*x^2*y*z^3 - 27150*x^2*y*z^2 + 34760*x^2*y*z - 16849*x^2*y - 210*x^2*z^5 + 2200*x^2*z^4 - 8670*x^2*z^3 + 16380*x^2*z^2 - 15589*x^2*z + 6206*x^2 + 70*x*y^6 + 420*x*y^5*z - 960*x*y^5 + 1050*x*y^4*z^2 - 4700*x*y^4*z + 4910*x*y^4 + 1400*x*y^3*z^3 - 9200*x*y^3*z^2 + 18840*x*y^3*z - 12160*x*y^3 + 1050*x*y^2*z^4 - 9000*x*y^2*z^3 + 27090*x*y^2*z^2 - 34420*x*y^2*z + 16409*x*y^2 + 420*x*y*z^5 - 4400*x*y*z^4 + 17300*x*y*z^3 - 32440*x*y*z^2 + 30418*x*y*z - 11932*x*y + 70*x*z^6 - 860*x*z^5 + 4140*x*z^4 - 10180*x*z^3 + 14079*x*z^2 - 10772*x*z + 3507*x - 10*y^7 - 70*y^6*z + 160*y^6 - 210*y^5*z^2 + 940*y^5*z - 980*y^5 - 350*y^4*z^3 + 2300*y^4*z^2 - 4700*y^4*z + 3010*y^4 - 350*y^3*z^4 + 3000*y^3*z^3 - 9010*y^3*z^2 + 11360*y^3*z - 5323*y^3 - 210*y^2*z^5 + 2200*y^2*z^4 - 8630*y^2*z^3 + 16060*y^2*z^2 - 14829*y^2*z + 5726*y^2 - 70*y*z^6 + 860*y*z^5 - 4130*y*z^4 + 10080*y*z^3 - 13749*y*z^2 + 10372*y*z - 3344*y - 10*z^7 + 140*z^6 - 790*z^5 + 2370*z^4 - 4243*z^3 + 4696*z^2 - 2950*z + 1, -5*x^7 + 35*x^6*y + 35*x^6*z - 80*x^6 - 105*x^5*y^2 - 210*x^5*y*z + 480*x^5*y - 105*x^5*z^2 + 470*x^5*z - 495*x^5 + 175*x^4*y^3 + 525*x^4*y^2*z - 1200*x^4*y^2 + 525*x^4*y*z^2 - 2350*x^4*y*z + 2470*x^4*y + 175*x^4*z^3 - 1150*x^4*z^2 + 2370*x^4*z - 1565*x^4 - 175*x^3*y^4 - 700*x^3*y^3*z + 1600*x^3*y^3 - 1050*x^3*y^2*z^2 + 4700*x^3*y^2*z - 4930*x^3*y^2 - 700*x^3*y*z^3 + 4600*x^3*y*z^2 - 9460*x^3*y*z + 6200*x^3*y - 175*x^3*z^4 + 1500*x^3*z^3 - 4535*x^3*z^2 + 5850*x^3*z - 2881*x^3 + 105*x^2*y^5 + 525*x^2*y^4*z - 1200*x^2*y^4 + 1050*x^2*y^3*z^2 - 4700*x^2*y^3*z + 4920*x^2*y^3 + 1050*x^2*y^2*z^3 - 6900*x^2*y^2*z^2 + 14160*x^2*y^2*z - 9210*x^2*y^2 + 525*x^2*y*z^4 - 4500*x^2*y*z^3 + 13575*x^2*y*z^2 - 17380*x^2*y*z + 8423*x^2*y + 105*x^2*z^5 - 1100*x^2*z^4 + 4335*x^2*z^3 - 8190*x^2*z^2 + 7793*x^2*z - 3102*x^2 - 35*x*y^6 - 210*x*y^5*z + 480*x*y^5 - 525*x*y^4*z^2 + 2350*x*y^4*z - 2455*x*y^4 - 700*x*y^3*z^3 + 4600*x*y^3*z^2 - 9420*x*y^3*z + 6080*x*y^3 - 525*x*y^2*z^4 + 4500*x*y^2*z^3 - 13545*x*y^2*z^2 + 17210*x*y^2*z - 8203*x*y^2 - 210*x*y*z^5 + 2200*x*y*z^4 - 8650*x*y*z^3 + 16220*x*y*z^2 - 15206*x*y*z + 5964*x*y - 35*x*z^6 + 430*x*z^5 - 2070*x*z^4 + 5090*x*z^3 - 7038*x*z^2 + 5384*x*z - 1756*x + 5*y^7 + 35*y^6*z - 80*y^6 + 105*y^5*z^2 - 470*y^5*z + 490*y^5 + 175*y^4*z^3 - 1150*y^4*z^2 + 2350*y^4*z - 1505*y^4 + 175*y^3*z^4 - 1500*y^3*z^3 + 4505*y^3*z^2 - 5680*y^3*z + 2661*y^3 + 105*y^2*z^5 - 1100*y^2*z^4 + 4315*y^2*z^3 - 8030*y^2*z^2 + 7413*y^2*z - 2862*y^2 + 35*y*z^6 - 430*y*z^5 + 2065*y*z^4 - 5040*y*z^3 + 6873*y*z^2 - 5184*y*z + 1675*y + 5*z^7 - 70*z^6 + 395*z^5 - 1185*z^4 + 2121*z^3 - 2347*z^2 + 1477*z - 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ 1, -1, -1 ], [ 2, -1, -1 ], [ 3, -3, -2 ] ], "source_shift": [ 1, 2, 2 ], "target_matrix": [ [ -1, -2, -1 ], [ 2, 3, 3 ], [ -1, -1, -1 ] ], "target_shift": [ 400, -789, 396 ], "inverse_recipe": { "p": 3, "q": 2, "f": "x**3 + x**2", "g": "x**2 + x", "h": "y*z**2 + z**2", "polynomials": { "f": [ [ [ 3, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "g": [ [ [ 2, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 2 ], 1 ], [ [ 0, 0, 2 ], 1 ] ] }, "h_multiplier": 5 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 0.9971495890022197 ], "surface_features": [ 3.6381138709909058, 4.541079767776629, 3.322219294733919, 0.0, 3.7682229700946337, 4917 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C184", "pair": 184, "family": "counterexample", "mapping": "F(x,y,z)=(216*x^7 - 1188*x^6*y - 972*x^6*z + 2484*x^6 + 2754*x^5*y^2 + 4536*x^5*y*z - 11988*x^5*y + 1854*x^5*z^2 - 9324*x^5*z + 8640*x^5 - 3483*x^4*y^3 - 8667*x^4*y^2*z + 23733*x^4*y^2 - 7137*x^4*y*z^2 + 37170*x^4*y*z - 35316*x^4*y - 1945*x^4*z^3 + 14493*x^4*z^2 - 26928*x^4*z + 14586*x^4 + 2592*x^3*y^4 + 8667*x^3*y^3*z - 24624*x^3*y^3 + 10791*x^3*y^2*z^2 - 58302*x^3*y^2*z + 56916*x^3*y^2 + 5929*x^3*y*z^3 - 45804*x^3*y*z^2 + 87372*x^3*y*z - 48579*x^3*y + 1213*x^3*z^4 - 11942*x^3*z^3 + 33408*x^3*z^2 - 36489*x^3*z + 13602*x^3 - 1134*x^2*y^5 - 4779*x^2*y^4*z + 14094*x^2*y^4 - 8001*x^2*y^3*z^2 + 44892*x^2*y^3*z - 45117*x^2*y^3 - 6651*x^2*y^2*z^3 + 53352*x^2*y^2*z^2 - 104688*x^2*y^2*z + 59868*x^2*y^2 - 2745*x^2*y*z^4 + 28044*x^2*y*z^3 - 80637*x^2*y*z^2 + 90507*x^2*y*z - 34704*x^2*y - 450*x^2*z^5 + 5502*x^2*z^4 - 20622*x^2*z^3 + 34089*x^2*z^2 - 25704*x^2*z + 7158*x^2 + 270*x*y^6 + 1377*x*y^5*z - 4212*x*y^5 + 2907*x*y^4*z^2 - 16938*x*y^4*z + 17550*x*y^4 + 3251*x*y^3*z^3 - 27096*x*y^3*z^2 + 54783*x*y^3*z - 32286*x*y^3 + 2031*x*y^2*z^4 - 21558*x*y^2*z^3 + 63825*x*y^2*z^2 - 73758*x*y^2*z + 29169*x*y^2 + 672*x*y*z^5 - 8532*x*y*z^4 + 32901*x*y*z^3 - 55947*x*y*z^2 + 43455*x*y*z - 12465*x*y + 92*x*z^6 - 1344*x*z^5 + 6333*x*z^4 - 14093*x*z^3 + 16134*x*z^2 - 9105*x*z + 1984*x - 27*y^7 - 162*y^6*z + 513*y^6 - 414*y^5*z^2 + 2502*y^5*z - 2673*y^5 - 584*y^4*z^3 + 5055*y^4*z^2 - 10539*y^4*z + 6411*y^4 - 491*y^3*z^4 + 5416*y^3*z^3 - 16530*y^3*z^2 + 19701*y^3*z - 8061*y^3 - 246*y^2*z^5 + 3246*y^2*z^4 - 12897*y^2*z^3 + 22599*y^2*z^2 - 18132*y^2*z + 5370*y^2 - 68*y*z^6 + 1032*y*z^5 - 5007*y*z^4 + 11472*y*z^3 - 13548*y*z^2 + 7887*y*z - 1771*y - 8*z^7 + 136*z^6 - 774*z^5 + 2175*z^4 - 3363*z^3 + 2887*z^2 - 1276*z + 1, -432*x^7 + 2376*x^6*y + 1944*x^6*z - 5184*x^6 - 5508*x^5*y^2 - 9072*x^5*y*z + 24948*x^5*y - 3708*x^5*z^2 + 19476*x^5*z - 19476*x^5 + 6966*x^4*y^3 + 17334*x^4*y^2*z - 49248*x^4*y^2 + 14274*x^4*y*z^2 - 77400*x^4*y*z + 79128*x^4*y + 3890*x^4*z^3 - 30288*x^4*z^2 + 60612*x^4*z - 34840*x^4 - 5184*x^3*y^4 - 17334*x^3*y^3*z + 50949*x^3*y^3 - 21582*x^3*y^2*z^2 + 121023*x^3*y^2*z - 126693*x^3*y^2 - 11858*x^3*y*z^3 + 95403*x^3*y*z^2 - 195414*x^3*y*z + 115306*x^3*y - 2426*x^3*z^4 + 24961*x^3*z^3 - 75069*x^3*z^2 + 86974*x^3*z - 33824*x^3 + 2268*x^2*y^5 + 9558*x^2*y^4*z - 29079*x^2*y^4 + 16002*x^2*y^3*z^2 - 92898*x^2*y^3*z + 99729*x^2*y^3 + 13302*x^2*y^2*z^3 - 110754*x^2*y^2*z^2 + 232542*x^2*y^2*z - 141093*x^2*y^2 + 5490*x^2*y*z^4 - 58410*x^2*y*z^3 + 179985*x^2*y*z^2 - 214284*x^2*y*z + 85836*x^2*y + 900*x^2*z^5 - 11499*x^2*z^4 + 46248*x^2*z^3 - 81063*x^2*z^2 + 63792*x^2*z - 18294*x^2 - 540*x*y^6 - 2754*x*y^5*z + 8667*x*y^5 - 5814*x*y^4*z^2 + 34947*x*y^4*z - 38511*x*y^4 - 6502*x*y^3*z^3 + 56067*x*y^3*z^2 - 120810*x*y^3*z + 75476*x*y^3 - 4062*x*y^2*z^4 + 44745*x*y^2*z^3 - 141447*x*y^2*z^2 + 173304*x*y^2*z - 71700*x*y^2 - 1344*x*y*z^5 + 17766*x*y*z^4 - 73272*x*y*z^3 + 132090*x*y*z^2 - 107226*x*y*z + 31731*x*y - 184*x*z^6 + 2808*x*z^5 - 14172*x*z^4 + 33426*x*z^3 - 39954*x*z^2 + 23235*x*z - 5161*x + 54*y^7 + 324*y^6*z - 1053*y^6 + 828*y^5*z^2 - 5148*y^5*z + 5823*y^5 + 1168*y^4*z^3 - 10428*y^4*z^2 + 23070*y^4*z - 14849*y^4 + 982*y^3*z^4 - 11204*y^3*z^3 + 36363*y^3*z^2 - 45890*y^3*z + 19672*y^3 + 492*y^2*z^5 - 6735*y^2*z^4 + 28512*y^2*z^3 - 52923*y^2*z^2 + 44442*y^2*z - 13605*y^2 + 136*y*z^6 - 2148*y*z^5 + 11124*y*z^4 - 27002*y*z^3 + 33342*y*z^2 - 20040*y*z + 4595*y + 16*z^7 - 284*z^6 + 1728*z^5 - 5144*z^4 + 8308*z^3 - 7355*z^2 + 3316*z - 1, -216*x^7 + 1188*x^6*y + 972*x^6*z - 2484*x^6 - 2754*x^5*y^2 - 4536*x^5*y*z + 11988*x^5*y - 1854*x^5*z^2 + 9324*x^5*z - 8640*x^5 + 3483*x^4*y^3 + 8667*x^4*y^2*z - 23733*x^4*y^2 + 7137*x^4*y*z^2 - 37170*x^4*y*z + 35316*x^4*y + 1945*x^4*z^3 - 14493*x^4*z^2 + 26928*x^4*z - 14578*x^4 - 2592*x^3*y^4 - 8667*x^3*y^3*z + 24624*x^3*y^3 - 10791*x^3*y^2*z^2 + 58302*x^3*y^2*z - 56916*x^3*y^2 - 5929*x^3*y*z^3 + 45804*x^3*y*z^2 - 87372*x^3*y*z + 48559*x^3*y - 1213*x^3*z^4 + 11942*x^3*z^3 - 33408*x^3*z^2 + 36469*x^3*z - 13526*x^3 + 1134*x^2*y^5 + 4779*x^2*y^4*z - 14094*x^2*y^4 + 8001*x^2*y^3*z^2 - 44892*x^2*y^3*z + 45117*x^2*y^3 + 6651*x^2*y^2*z^3 - 53352*x^2*y^2*z^2 + 104688*x^2*y^2*z - 59850*x^2*y^2 + 2745*x^2*y*z^4 - 28044*x^2*y*z^3 + 80637*x^2*y*z^2 - 90471*x^2*y*z + 34560*x^2*y + 450*x^2*z^5 - 5502*x^2*z^4 + 20622*x^2*z^3 - 34071*x^2*z^2 + 25572*x^2*z - 7002*x^2 - 270*x*y^6 - 1377*x*y^5*z + 4212*x*y^5 - 2907*x*y^4*z^2 + 16938*x*y^4*z - 17550*x*y^4 - 3251*x*y^3*z^3 + 27096*x*y^3*z^2 - 54783*x*y^3*z + 32279*x*y^3 - 2031*x*y^2*z^4 + 21558*x*y^2*z^3 - 63825*x*y^2*z^2 + 73737*x*y^2*z - 29082*x*y^2 - 672*x*y*z^5 + 8532*x*y*z^4 - 32901*x*y*z^3 + 55926*x*y*z^2 - 43293*x*y*z + 12261*x*y - 92*x*z^6 + 1344*x*z^5 - 6333*x*z^4 + 14086*x*z^3 - 16059*x*z^2 + 8925*x*z - 1872*x + 27*y^7 + 162*y^6*z - 513*y^6 + 414*y^5*z^2 - 2502*y^5*z + 2673*y^5 + 584*y^4*z^3 - 5055*y^4*z^2 + 10539*y^4*z - 6410*y^4 + 491*y^3*z^4 - 5416*y^3*z^3 + 16530*y^3*z^2 - 19697*y^3*z + 8044*y^3 + 246*y^2*z^5 - 3246*y^2*z^4 + 12897*y^2*z^3 - 22593*y^2*z^2 + 18084*y^2*z - 5307*y^2 + 68*y*z^6 - 1032*y*z^5 + 5007*y*z^4 - 11468*y*z^3 + 13503*y*z^2 - 7773*y*z + 1693*y + 8*z^7 - 136*z^6 + 774*z^5 - 2174*z^4 + 3349*z^3 - 2836*z^2 + 1210*z - 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ 2, -1, -1 ], [ 3, -3, -2 ], [ 1, -1, -1 ] ], "source_shift": [ 2, 1, 2 ], "target_matrix": [ [ 1, 1, 1 ], [ -2, -3, -3 ], [ -1, -1, -2 ] ], "target_shift": [ -228, 602, 204 ], "collision_preimages": [ [ "1/4", "-13/4", "23/4" ], [ "-11/2", "6", "-16" ], [ "-15/2", "1", "-13" ] ], "collision_image": [ "-913/4", "1205/2", "817/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.015242120000039 ], "surface_features": [ 4.39033547876703, 5.366501403337462, 4.022799404511688, 0.0, 4.546817983180565, 5179 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B144", "pair": 144, "family": "bijection", "mapping": "F(x,y,z)=(-128*x^7 - 896*x^6*y - 1344*x^6*z - 448*x^6 - 2688*x^5*y^2 - 8064*x^5*y*z - 2688*x^5*y - 6048*x^5*z^2 - 4032*x^5*z - 688*x^5 - 4480*x^4*y^3 - 20160*x^4*y^2*z - 6720*x^4*y^2 - 30240*x^4*y*z^2 - 20160*x^4*y*z - 3440*x^4*y - 15120*x^4*z^3 - 15120*x^4*z^2 - 5152*x^4*z - 656*x^4 - 4480*x^3*y^4 - 26880*x^3*y^3*z - 8960*x^3*y^3 - 60480*x^3*y^2*z^2 - 40320*x^3*y^2*z - 6880*x^3*y^2 - 60480*x^3*y*z^3 - 60480*x^3*y*z^2 - 20608*x^3*y*z - 2616*x^3*y - 22680*x^3*z^4 - 30240*x^3*z^3 - 15432*x^3*z^2 - 3904*x^3*z - 424*x^3 - 2688*x^2*y^5 - 20160*x^2*y^4*z - 6720*x^2*y^4 - 60480*x^2*y^3*z^2 - 40320*x^2*y^3*z - 6880*x^2*y^3 - 90720*x^2*y^2*z^3 - 90720*x^2*y^2*z^2 - 30912*x^2*y^2*z - 3912*x^2*y^2 - 68040*x^2*y*z^4 - 90720*x^2*y*z^3 - 46296*x^2*y*z^2 - 11676*x^2*y*z - 1260*x^2*y - 20412*x^2*z^5 - 34020*x^2*z^4 - 23112*x^2*z^3 - 8712*x^2*z^2 - 1872*x^2*z - 180*x^2 - 896*x*y^6 - 8064*x*y^5*z - 2688*x*y^5 - 30240*x*y^4*z^2 - 20160*x*y^4*z - 3440*x*y^4 - 60480*x*y^3*z^3 - 60480*x*y^3*z^2 - 20608*x*y^3*z - 2600*x*y^3 - 68040*x*y^2*z^4 - 90720*x*y^2*z^3 - 46296*x*y^2*z^2 - 11640*x*y^2*z - 1248*x*y^2 - 40824*x*y*z^5 - 68040*x*y*z^4 - 46224*x*y*z^3 - 17370*x*y*z^2 - 3708*x*y*z - 353*x*y - 10206*x*z^6 - 20412*x*z^5 - 17307*x*z^4 - 8640*x*z^3 - 2754*x*z^2 - 520*x*z - 52*x - 128*y^7 - 1344*y^6*z - 448*y^6 - 6048*y^5*z^2 - 4032*y^5*z - 688*y^5 - 15120*y^4*z^3 - 15120*y^4*z^2 - 5152*y^4*z - 648*y^4 - 22680*y^3*z^4 - 30240*y^3*z^3 - 15432*y^3*z^2 - 3868*y^3*z - 412*y^3 - 20412*y^2*z^5 - 34020*y^2*z^4 - 23112*y^2*z^3 - 8658*y^2*z^2 - 1836*y^2*z - 173*y^2 - 10206*y*z^6 - 20412*y*z^5 - 17307*y*z^4 - 8613*y*z^3 - 2727*y*z^2 - 510*y*z - 49*y - 2187*z^7 - 5103*z^6 - 5184*z^5 - 3213*z^4 - 1350*z^3 - 376*z^2 - 71*z - 1, -128*x^7 - 896*x^6*y - 1344*x^6*z - 448*x^6 - 2688*x^5*y^2 - 8064*x^5*y*z - 2688*x^5*y - 6048*x^5*z^2 - 4032*x^5*z - 688*x^5 - 4480*x^4*y^3 - 20160*x^4*y^2*z - 6720*x^4*y^2 - 30240*x^4*y*z^2 - 20160*x^4*y*z - 3440*x^4*y - 15120*x^4*z^3 - 15120*x^4*z^2 - 5152*x^4*z - 656*x^4 - 4480*x^3*y^4 - 26880*x^3*y^3*z - 8960*x^3*y^3 - 60480*x^3*y^2*z^2 - 40320*x^3*y^2*z - 6880*x^3*y^2 - 60480*x^3*y*z^3 - 60480*x^3*y*z^2 - 20608*x^3*y*z - 2616*x^3*y - 22680*x^3*z^4 - 30240*x^3*z^3 - 15432*x^3*z^2 - 3904*x^3*z - 408*x^3 - 2688*x^2*y^5 - 20160*x^2*y^4*z - 6720*x^2*y^4 - 60480*x^2*y^3*z^2 - 40320*x^2*y^3*z - 6880*x^2*y^3 - 90720*x^2*y^2*z^3 - 90720*x^2*y^2*z^2 - 30912*x^2*y^2*z - 3912*x^2*y^2 - 68040*x^2*y*z^4 - 90720*x^2*y*z^3 - 46296*x^2*y*z^2 - 11676*x^2*y*z - 1212*x^2*y - 20412*x^2*z^5 - 34020*x^2*z^4 - 23112*x^2*z^3 - 8712*x^2*z^2 - 1800*x^2*z - 156*x^2 - 896*x*y^6 - 8064*x*y^5*z - 2688*x*y^5 - 30240*x*y^4*z^2 - 20160*x*y^4*z - 3440*x*y^4 - 60480*x*y^3*z^3 - 60480*x*y^3*z^2 - 20608*x*y^3*z - 2600*x*y^3 - 68040*x*y^2*z^4 - 90720*x*y^2*z^3 - 46296*x*y^2*z^2 - 11640*x*y^2*z - 1200*x*y^2 - 40824*x*y*z^5 - 68040*x*y*z^4 - 46224*x*y*z^3 - 17370*x*y*z^2 - 3564*x*y*z - 305*x*y - 10206*x*z^6 - 20412*x*z^5 - 17307*x*z^4 - 8640*x*z^3 - 2646*x*z^2 - 448*x*z - 38*x - 128*y^7 - 1344*y^6*z - 448*y^6 - 6048*y^5*z^2 - 4032*y^5*z - 688*y^5 - 15120*y^4*z^3 - 15120*y^4*z^2 - 5152*y^4*z - 648*y^4 - 22680*y^3*z^4 - 30240*y^3*z^3 - 15432*y^3*z^2 - 3868*y^3*z - 396*y^3 - 20412*y^2*z^5 - 34020*y^2*z^4 - 23112*y^2*z^3 - 8658*y^2*z^2 - 1764*y^2*z - 149*y^2 - 10206*y*z^6 - 20412*y*z^5 - 17307*y*z^4 - 8613*y*z^3 - 2619*y*z^2 - 438*y*z - 35*y - 2187*z^7 - 5103*z^6 - 5184*z^5 - 3213*z^4 - 1296*z^3 - 322*z^2 - 51*z - 1, 256*x^7 + 1792*x^6*y + 2688*x^6*z + 896*x^6 + 5376*x^5*y^2 + 16128*x^5*y*z + 5376*x^5*y + 12096*x^5*z^2 + 8064*x^5*z + 1376*x^5 + 8960*x^4*y^3 + 40320*x^4*y^2*z + 13440*x^4*y^2 + 60480*x^4*y*z^2 + 40320*x^4*y*z + 6880*x^4*y + 30240*x^4*z^3 + 30240*x^4*z^2 + 10304*x^4*z + 1296*x^4 + 8960*x^3*y^4 + 53760*x^3*y^3*z + 17920*x^3*y^3 + 120960*x^3*y^2*z^2 + 80640*x^3*y^2*z + 13760*x^3*y^2 + 120960*x^3*y*z^3 + 120960*x^3*y*z^2 + 41216*x^3*y*z + 5168*x^3*y + 45360*x^3*z^4 + 60480*x^3*z^3 + 30864*x^3*z^2 + 7712*x^3*z + 832*x^3 + 5376*x^2*y^5 + 40320*x^2*y^4*z + 13440*x^2*y^4 + 120960*x^2*y^3*z^2 + 80640*x^2*y^3*z + 13760*x^2*y^3 + 181440*x^2*y^2*z^3 + 181440*x^2*y^2*z^2 + 61824*x^2*y^2*z + 7728*x^2*y^2 + 136080*x^2*y*z^4 + 181440*x^2*y*z^3 + 92592*x^2*y*z^2 + 23064*x^2*y*z + 2472*x^2*y + 40824*x^2*z^5 + 68040*x^2*z^4 + 46224*x^2*z^3 + 17208*x^2*z^2 + 3672*x^2*z + 360*x^2 + 1792*x*y^6 + 16128*x*y^5*z + 5376*x*y^5 + 60480*x*y^4*z^2 + 40320*x*y^4*z + 6880*x*y^4 + 120960*x*y^3*z^3 + 120960*x*y^3*z^2 + 41216*x*y^3*z + 5136*x*y^3 + 136080*x*y^2*z^4 + 181440*x*y^2*z^3 + 92592*x*y^2*z^2 + 22992*x*y^2*z + 2448*x*y^2 + 81648*x*y*z^5 + 136080*x*y*z^4 + 92448*x*y*z^3 + 34308*x*y*z^2 + 7272*x*y*z + 706*x*y + 20412*x*z^6 + 40824*x*z^5 + 34614*x*z^4 + 17064*x*z^3 + 5400*x*z^2 + 1040*x*z + 106*x + 256*y^7 + 2688*y^6*z + 896*y^6 + 12096*y^5*z^2 + 8064*y^5*z + 1376*y^5 + 30240*y^4*z^3 + 30240*y^4*z^2 + 10304*y^4*z + 1280*y^4 + 45360*y^3*z^4 + 60480*y^3*z^3 + 30864*y^3*z^2 + 7640*y^3*z + 808*y^3 + 40824*y^2*z^5 + 68040*y^2*z^4 + 46224*y^2*z^3 + 17100*y^2*z^2 + 3600*y^2*z + 346*y^2 + 20412*y*z^6 + 40824*y*z^5 + 34614*y*z^4 + 17010*y*z^3 + 5346*y*z^2 + 1020*y*z + 101*y + 4374*z^7 + 10206*z^6 + 10368*z^5 + 6345*z^4 + 2646*z^3 + 752*z^2 + 146*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ 2, 2, 3 ], [ 2, 1, 1 ], [ 1, 1, 1 ] ], "source_shift": [ 1, 2, 1 ], "target_matrix": [ [ -1, -1, -1 ], [ -1, -1, -2 ], [ 2, 1, 1 ] ], "target_shift": [ 8, 4, -17 ], "inverse_recipe": { "p": 4, "q": 3, "f": "x**4 + x", "g": "x**3", "h": "y*z", "polynomials": { "f": [ [ [ 4, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "g": [ [ [ 3, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0453038660016318 ], "surface_features": [ 4.372668871100826, 5.258733037212813, 3.9523080096621253, 0.0, 4.5156626563987246, 5170 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B204", "pair": 204, "family": "bijection", "mapping": "F(x,y,z)=(x^7 + 14*x^6*y - 7*x^6*z - 15*x^6 + 84*x^5*y^2 - 84*x^5*y*z - 180*x^5*y + 21*x^5*z^2 + 90*x^5*z + 97*x^5 + 280*x^4*y^3 - 420*x^4*y^2*z - 900*x^4*y^2 + 210*x^4*y*z^2 + 900*x^4*y*z + 969*x^4*y - 35*x^4*z^3 - 225*x^4*z^2 - 485*x^4*z - 353*x^4 + 560*x^3*y^4 - 1120*x^3*y^3*z - 2400*x^3*y^3 + 840*x^3*y^2*z^2 + 3600*x^3*y^2*z + 3872*x^3*y^2 - 280*x^3*y*z^3 - 1800*x^3*y*z^2 - 3876*x^3*y*z - 2814*x^3*y + 35*x^3*z^4 + 300*x^3*z^3 + 970*x^3*z^2 + 1413*x^3*z + 782*x^3 + 672*x^2*y^5 - 1680*x^2*y^4*z - 3600*x^2*y^4 + 1680*x^2*y^3*z^2 + 7200*x^2*y^3*z + 7736*x^2*y^3 - 840*x^2*y^2*z^3 - 5400*x^2*y^2*z^2 - 11616*x^2*y^2*z - 8412*x^2*y^2 + 210*x^2*y*z^4 + 1800*x^2*y*z^3 + 5814*x^2*y*z^2 + 8448*x^2*y*z + 4656*x^2*y - 21*x^2*z^5 - 225*x^2*z^4 - 970*x^2*z^3 - 2121*x^2*z^2 - 2352*x^2*z - 1053*x^2 + 448*x*y^6 - 1344*x*y^5*z - 2880*x*y^5 + 1680*x*y^4*z^2 + 7200*x*y^4*z + 7728*x*y^4 - 1120*x*y^3*z^3 - 7200*x*y^3*z^2 - 15472*x*y^3*z - 11176*x*y^3 + 420*x*y^2*z^4 + 3600*x*y^2*z^3 + 11616*x*y^2*z^2 + 16836*x*y^2*z + 9240*x*y^2 - 84*x*y*z^5 - 900*x*y*z^4 - 3876*x*y*z^3 - 8454*x*y*z^2 - 9336*x*y*z - 4154*x*y + 7*x*z^6 + 90*x*z^5 + 485*x*z^4 + 1415*x*z^3 + 2358*x*z^2 + 2119*x*z + 795*x + 128*y^7 - 448*y^6*z - 960*y^6 + 672*y^5*z^2 + 2880*y^5*z + 3088*y^5 - 560*y^4*z^3 - 3600*y^4*z^2 - 7728*y^4*z - 5568*y^4 + 280*y^3*z^4 + 2400*y^3*z^3 + 7736*y^3*z^2 + 11184*y^3*z + 6112*y^3 - 84*y^2*z^5 - 900*y^2*z^4 - 3872*y^2*z^3 - 8424*y^2*z^2 - 9264*y^2*z - 4097*y^2 + 14*y*z^6 + 180*y*z^5 + 969*y*z^4 + 2820*y*z^3 + 4680*y*z^2 + 4179*y*z + 1556*y - z^7 - 15*z^6 - 97*z^5 - 354*z^4 - 788*z^3 - 1066*z^2 - 806*z - 1, 2*x^7 + 28*x^6*y - 14*x^6*z - 30*x^6 + 168*x^5*y^2 - 168*x^5*y*z - 360*x^5*y + 42*x^5*z^2 + 180*x^5*z + 194*x^5 + 560*x^4*y^3 - 840*x^4*y^2*z - 1800*x^4*y^2 + 420*x^4*y*z^2 + 1800*x^4*y*z + 1938*x^4*y - 70*x^4*z^3 - 450*x^4*z^2 - 970*x^4*z - 705*x^4 + 1120*x^3*y^4 - 2240*x^3*y^3*z - 4800*x^3*y^3 + 1680*x^3*y^2*z^2 + 7200*x^3*y^2*z + 7744*x^3*y^2 - 560*x^3*y*z^3 - 3600*x^3*y*z^2 - 7752*x^3*y*z - 5620*x^3*y + 70*x^3*z^4 + 600*x^3*z^3 + 1940*x^3*z^2 + 2822*x^3*z + 1557*x^3 + 1344*x^2*y^5 - 3360*x^2*y^4*z - 7200*x^2*y^4 + 3360*x^2*y^3*z^2 + 14400*x^2*y^3*z + 15472*x^2*y^3 - 1680*x^2*y^2*z^3 - 10800*x^2*y^2*z^2 - 23232*x^2*y^2*z - 16800*x^2*y^2 + 420*x^2*y*z^4 + 3600*x^2*y*z^3 + 11628*x^2*y*z^2 + 16872*x^2*y*z + 9270*x^2*y - 42*x^2*z^5 - 450*x^2*z^4 - 1940*x^2*z^3 - 4236*x^2*z^2 - 4683*x^2*z - 2088*x^2 + 896*x*y^6 - 2688*x*y^5*z - 5760*x*y^5 + 3360*x*y^4*z^2 + 14400*x*y^4*z + 15456*x*y^4 - 2240*x*y^3*z^3 - 14400*x*y^3*z^2 - 30944*x*y^3*z - 22320*x*y^3 + 840*x*y^2*z^4 + 7200*x*y^2*z^3 + 23232*x*y^2*z^2 + 33624*x*y^2*z + 18396*x*y^2 - 168*x*y*z^5 - 1800*x*y*z^4 - 7752*x*y*z^3 - 16884*x*y*z^2 - 18588*x*y*z - 8236*x*y + 14*x*z^6 + 180*x*z^5 + 970*x*z^4 + 2826*x*z^3 + 4695*x*z^2 + 4202*x*z + 1571*x + 256*y^7 - 896*y^6*z - 1920*y^6 + 1344*y^5*z^2 + 5760*y^5*z + 6176*y^5 - 1120*y^4*z^3 - 7200*y^4*z^2 - 15456*y^4*z - 11120*y^4 + 560*y^3*z^4 + 4800*y^3*z^3 + 15472*y^3*z^2 + 22336*y^3*z + 12168*y^3 - 168*y^2*z^5 - 1800*y^2*z^4 - 7744*y^2*z^3 - 16824*y^2*z^2 - 18444*y^2*z - 8122*y^2 + 28*y*z^6 + 360*y*z^5 + 1938*y*z^4 + 5632*y*z^3 + 9318*y*z^2 + 8286*y*z + 3073*y - 2*z^7 - 30*z^6 - 194*z^5 - 707*z^4 - 1569*z^3 - 2114*z^2 - 1592*z - 1, -x^7 - 14*x^6*y + 7*x^6*z + 15*x^6 - 84*x^5*y^2 + 84*x^5*y*z + 180*x^5*y - 21*x^5*z^2 - 90*x^5*z - 97*x^5 - 280*x^4*y^3 + 420*x^4*y^2*z + 900*x^4*y^2 - 210*x^4*y*z^2 - 900*x^4*y*z - 969*x^4*y + 35*x^4*z^3 + 225*x^4*z^2 + 485*x^4*z + 354*x^4 - 560*x^3*y^4 + 1120*x^3*y^3*z + 2400*x^3*y^3 - 840*x^3*y^2*z^2 - 3600*x^3*y^2*z - 3872*x^3*y^2 + 280*x^3*y*z^3 + 1800*x^3*y*z^2 + 3876*x^3*y*z + 2822*x^3*y - 35*x^3*z^4 - 300*x^3*z^3 - 970*x^3*z^2 - 1417*x^3*z - 791*x^3 - 672*x^2*y^5 + 1680*x^2*y^4*z + 3600*x^2*y^4 - 1680*x^2*y^3*z^2 - 7200*x^2*y^3*z - 7736*x^2*y^3 + 840*x^2*y^2*z^3 + 5400*x^2*y^2*z^2 + 11616*x^2*y^2*z + 8436*x^2*y^2 - 210*x^2*y*z^4 - 1800*x^2*y*z^3 - 5814*x^2*y*z^2 - 8472*x^2*y*z - 4710*x^2*y + 21*x^2*z^5 + 225*x^2*z^4 + 970*x^2*z^3 + 2127*x^2*z^2 + 2379*x^2*z + 1083*x^2 - 448*x*y^6 + 1344*x*y^5*z + 2880*x*y^5 - 1680*x*y^4*z^2 - 7200*x*y^4*z - 7728*x*y^4 + 1120*x*y^3*z^3 + 7200*x*y^3*z^2 + 15472*x*y^3*z + 11208*x*y^3 - 420*x*y^2*z^4 - 3600*x*y^2*z^3 - 11616*x*y^2*z^2 - 16884*x*y^2*z - 9348*x*y^2 + 84*x*y*z^5 + 900*x*y*z^4 + 3876*x*y*z^3 + 8478*x*y*z^2 + 9444*x*y*z + 4274*x*y - 7*x*z^6 - 90*x*z^5 - 485*x*z^4 - 1419*x*z^3 - 2385*x*z^2 - 2179*x*z - 840*x - 128*y^7 + 448*y^6*z + 960*y^6 - 672*y^5*z^2 - 2880*y^5*z - 3088*y^5 + 560*y^4*z^3 + 3600*y^4*z^2 + 7728*y^4*z + 5584*y^4 - 280*y^3*z^4 - 2400*y^3*z^3 - 7736*y^3*z^2 - 11216*y^3*z - 6184*y^3 + 84*y^2*z^5 + 900*y^2*z^4 + 3872*y^2*z^3 + 8448*y^2*z^2 + 9372*y^2*z + 4217*y^2 - 14*y*z^6 - 180*y*z^5 - 969*y*z^4 - 2828*y*z^3 - 4734*y*z^2 - 4299*y*z - 1645*y + z^7 + 15*z^6 + 97*z^5 + 355*z^4 + 797*z^3 + 1096*z^2 + 852*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -1, -2, 1 ], [ -1, -1, 2 ], [ -1, -1, 1 ] ], "source_shift": [ 2, 1, 2 ], "target_matrix": [ [ -1, -1, -1 ], [ -2, -1, -1 ], [ 1, 2, 1 ] ], "target_shift": [ 256, 508, -281 ], "inverse_recipe": { "p": 4, "q": 3, "f": "x**4 + x**3", "g": "x**3", "h": "y*z", "polynomials": { "f": [ [ [ 4, 0, 0 ], 1 ], [ [ 3, 0, 0 ], 1 ] ], "g": [ [ [ 3, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0056131390010705 ], "surface_features": [ 3.572338729230921, 4.526649376997381, 3.1990691962517417, 0.0, 3.712622561083267, 4849 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C149", "pair": 149, "family": "counterexample", "mapping": "F(x,y,z)=(54*x^7 + 351*x^6*y - 297*x^6*z - 783*x^6 + 972*x^5*y^2 - 1647*x^5*y*z - 4347*x^5*y + 693*x^5*z^2 + 3591*x^5*z + 3762*x^5 + 1485*x^4*y^3 - 3780*x^4*y^2*z - 9990*x^4*y^2 + 3186*x^4*y*z^2 + 16542*x^4*y*z + 17478*x^4*y - 889*x^4*z^3 - 6795*x^4*z^2 - 14118*x^4*z - 8864*x^4 + 1350*x^3*y^4 - 4590*x^3*y^3*z - 12150*x^3*y^3 + 5814*x^3*y^2*z^2 + 30258*x^3*y^2*z + 32292*x^3*y^2 - 3251*x^3*y*z^3 - 24927*x^3*y*z^2 - 52353*x^3*y*z - 33304*x^3*y + 677*x^3*z^4 + 6793*x^3*z^3 + 21051*x^3*z^2 + 26373*x^3*z + 11718*x^3 + 729*x^2*y^5 - 3105*x^2*y^4*z - 8235*x^2*y^4 + 5256*x^2*y^3*z^2 + 27432*x^2*y^3*z + 29628*x^2*y^3 - 4419*x^2*y^2*z^3 - 34011*x^2*y^2*z^2 - 72351*x^2*y^2*z - 46728*x^2*y^2 + 1845*x^2*y*z^4 + 18600*x^2*y*z^3 + 58425*x^2*y*z^2 + 74334*x^2*y*z + 33501*x^2*y - 306*x^2*z^5 - 3786*x^2*z^4 - 15606*x^2*z^3 - 29334*x^2*z^2 - 26013*x^2*z - 8812*x^2 + 216*x*y^6 - 1107*x*y^5*z - 2943*x*y^5 + 2349*x*y^4*z^2 + 12303*x*y^4*z + 13482*x*y^4 - 2641*x*y^3*z^3 - 20421*x*y^3*z^2 - 44115*x*y^3*z - 29000*x*y^3 + 1659*x*y^2*z^4 + 16821*x*y^2*z^3 + 53697*x*y^2*z^2 + 69549*x*y^2*z + 31848*x*y^2 - 552*x*y*z^5 - 6876*x*y*z^4 - 28821*x*y*z^3 - 55143*x*y*z^2 - 49653*x*y*z - 17030*x*y + 76*x*z^6 + 1116*x*z^5 + 5757*x*z^4 + 14460*x*z^3 + 19194*x*z^2 + 12896*x*z + 3433*x + 27*y^7 - 162*y^6*z - 432*y^6 + 414*y^5*z^2 + 2178*y^5*z + 2430*y^5 - 584*y^4*z^3 - 4542*y^4*z^2 - 9999*y^4*z - 6712*y^4 + 491*y^3*z^4 + 5014*y^3*z^3 + 16323*y^3*z^2 + 21588*y^3*z + 10065*y^3 - 246*y^2*z^5 - 3090*y^2*z^4 - 13215*y^2*z^3 - 25809*y^2*z^2 - 23640*y^2*z - 8218*y^2 + 68*y*z^6 + 1008*y*z^5 + 5307*y*z^4 + 13599*y*z^3 + 18345*y*z^2 + 12479*y*z + 3353*y - 8*z^7 - 136*z^6 - 846*z^5 - 2666*z^4 - 4708*z^3 - 4705*z^2 - 2473*z - 1, 54*x^7 + 351*x^6*y - 297*x^6*z - 621*x^6 + 972*x^5*y^2 - 1647*x^5*y*z - 3456*x^5*y + 693*x^5*z^2 + 2862*x^5*z + 2709*x^5 + 1485*x^4*y^3 - 3780*x^4*y^2*z - 7965*x^4*y^2 + 3186*x^4*y*z^2 + 13221*x^4*y*z + 12618*x^4*y - 889*x^4*z^3 - 5445*x^4*z^2 - 10230*x^4*z - 6002*x^4 + 1350*x^3*y^4 - 4590*x^3*y^3*z - 9720*x^3*y^3 + 5814*x^3*y^2*z^2 + 24264*x^3*y^2*z + 23382*x^3*y^2 - 3251*x^3*y*z^3 - 20040*x^3*y*z^2 - 38043*x^3*y*z - 22612*x^3*y + 677*x^3*z^4 + 5476*x^3*z^3 + 15354*x^3*z^2 + 18003*x^3*z + 7626*x^3 + 729*x^2*y^5 - 3105*x^2*y^4*z - 6615*x^2*y^4 + 5256*x^2*y^3*z^2 + 22086*x^2*y^3*z + 21528*x^2*y^3 - 4419*x^2*y^2*z^3 - 27450*x^2*y^2*z^2 - 52749*x^2*y^2*z - 31824*x^2*y^2 + 1845*x^2*y*z^4 + 15051*x^2*y*z^3 + 42747*x^2*y*z^2 + 50880*x^2*y*z + 21831*x^2*y - 306*x^2*z^5 - 3072*x^2*z^4 - 11460*x^2*z^3 - 20178*x^2*z^2 - 17061*x^2*z - 5590*x^2 + 216*x*y^6 - 1107*x*y^5*z - 2376*x*y^5 + 2349*x*y^4*z^2 + 9954*x*y^4*z + 9837*x*y^4 - 2641*x*y^3*z^3 - 16560*x*y^3*z^2 - 32289*x*y^3*z - 19820*x*y^3 + 1659*x*y^2*z^4 + 13674*x*y^2*z^3 + 39432*x*y^2*z^2 + 47751*x*y^2*z + 20784*x*y^2 - 552*x*y*z^5 - 5604*x*y*z^4 - 21237*x*y*z^3 - 38034*x*y*z^2 - 32607*x*y*z - 10802*x*y + 76*x*z^6 + 912*x*z^5 + 4257*x*z^4 + 10019*x*z^3 + 12681*x*z^2 + 8246*x*z + 2179*x + 27*y^7 - 162*y^6*z - 351*y^6 + 414*y^5*z^2 + 1773*y^5*z + 1782*y^5 - 584*y^4*z^3 - 3705*y^4*z^2 - 7353*y^4*z - 4606*y^4 + 491*y^3*z^4 + 4099*y^3*z^3 + 12039*y^3*z^2 + 14874*y^3*z + 6579*y^3 - 246*y^2*z^5 - 2532*y^2*z^4 - 9777*y^2*z^3 - 17856*y^2*z^2 - 15546*y^2*z - 5212*y^2 + 68*y*z^6 + 828*y*z^5 + 3939*y*z^4 + 9448*y*z^3 + 12135*y*z^2 + 7979*y*z + 2127*y - 8*z^7 - 112*z^6 - 630*z^5 - 1860*z^4 - 3132*z^3 - 3031*z^2 - 1581*z - 1, -54*x^7 - 351*x^6*y + 297*x^6*z + 621*x^6 - 972*x^5*y^2 + 1647*x^5*y*z + 3456*x^5*y - 693*x^5*z^2 - 2862*x^5*z - 2709*x^5 - 1485*x^4*y^3 + 3780*x^4*y^2*z + 7965*x^4*y^2 - 3186*x^4*y*z^2 - 13221*x^4*y*z - 12618*x^4*y + 889*x^4*z^3 + 5445*x^4*z^2 + 10230*x^4*z + 6056*x^4 - 1350*x^3*y^4 + 4590*x^3*y^3*z + 9720*x^3*y^3 - 5814*x^3*y^2*z^2 - 24264*x^3*y^2*z - 23382*x^3*y^2 + 3251*x^3*y*z^3 + 20040*x^3*y*z^2 + 38043*x^3*y*z + 22801*x^3*y - 677*x^3*z^4 - 5476*x^3*z^3 - 15354*x^3*z^2 - 18138*x^3*z - 7761*x^3 - 729*x^2*y^5 + 3105*x^2*y^4*z + 6615*x^2*y^4 - 5256*x^2*y^3*z^2 - 22086*x^2*y^3*z - 21528*x^2*y^3 + 4419*x^2*y^2*z^3 + 27450*x^2*y^2*z^2 + 52749*x^2*y^2*z + 32067*x^2*y^2 - 1845*x^2*y*z^4 - 15051*x^2*y*z^3 - 42747*x^2*y*z^2 - 51231*x^2*y*z - 22209*x^2*y + 306*x^2*z^5 + 3072*x^2*z^4 + 11460*x^2*z^3 + 20304*x^2*z^2 + 17331*x^2*z + 5734*x^2 - 216*x*y^6 + 1107*x*y^5*z + 2376*x*y^5 - 2349*x*y^4*z^2 - 9954*x*y^4*z - 9837*x*y^4 + 2641*x*y^3*z^3 + 16560*x*y^3*z^2 + 32289*x*y^3*z + 19955*x*y^3 - 1659*x*y^2*z^4 - 13674*x*y^2*z^3 - 39432*x*y^2*z^2 - 48048*x*y^2*z - 21135*x*y^2 + 552*x*y*z^5 + 5604*x*y*z^4 + 21237*x*y*z^3 + 38250*x*y*z^2 + 33111*x*y*z + 11081*x*y - 76*x*z^6 - 912*x*z^5 - 4257*x*z^4 - 10071*x*z^3 - 12861*x*z^2 - 8441*x*z - 2232*x - 27*y^7 + 162*y^6*z + 351*y^6 - 414*y^5*z^2 - 1773*y^5*z - 1782*y^5 + 584*y^4*z^3 + 3705*y^4*z^2 + 7353*y^4*z + 4633*y^4 - 491*y^3*z^4 - 4099*y^3*z^3 - 12039*y^3*z^2 - 14955*y^3*z - 6687*y^3 + 246*y^2*z^5 + 2532*y^2*z^4 + 9777*y^2*z^3 + 17946*y^2*z^2 + 15780*y^2*z + 5347*y^2 - 68*y*z^6 - 828*y*z^5 - 3939*y*z^4 - 9492*y*z^3 - 12303*y*z^2 - 8168*y*z - 2179*y + 8*z^7 + 112*z^6 + 630*z^5 + 1868*z^4 + 3172*z^3 + 3097*z^2 + 1617*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -3, -3, 2 ], [ -1, -1, 1 ], [ -2, -1, 1 ] ], "source_shift": [ 1, 2, 2 ], "target_matrix": [ [ -1, -2, -1 ], [ -1, -1, -2 ], [ 1, 1, 1 ] ], "target_shift": [ 519, 337, -343 ], "collision_preimages": [ [ "1/4", "-13/4", "-5" ], [ "-8", "1", "-21/2" ], [ "-5", "6", "1/2" ] ], "collision_image": [ "2077/4", "1349/4", "-1373/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 0.9938711920003698 ], "surface_features": [ 4.040800390716639, 4.871187503328807, 3.7206553565517244, 0.0, 4.137021035407215, 5073 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B134", "pair": 134, "family": "bijection", "mapping": "F(x,y,z)=(4374*x^7 + 20412*x^6*y + 30618*x^6*z + 10692*x^6 + 40824*x^5*y^2 + 122472*x^5*y*z + 42930*x^5*y + 91854*x^5*z^2 + 64152*x^5*z + 10530*x^5 + 45360*x^4*y^3 + 204120*x^4*y^2*z + 71820*x^4*y^2 + 306180*x^4*y*z^2 + 214650*x^4*y*z + 35370*x^4*y + 153090*x^4*z^3 + 160380*x^4*z^2 + 52650*x^4*z + 5400*x^4 + 30240*x^3*y^4 + 181440*x^3*y^3*z + 64080*x^3*y^3 + 408240*x^3*y^2*z^2 + 287280*x^3*y^2*z + 47520*x^3*y^2 + 408240*x^3*y*z^3 + 429300*x^3*y*z^2 + 141480*x^3*y*z + 14580*x^3*y + 153090*x^3*z^4 + 213840*x^3*z^3 + 105300*x^3*z^2 + 21600*x^3*z + 1602*x^3 + 12096*x^2*y^5 + 90720*x^2*y^4*z + 32160*x^2*y^4 + 272160*x^2*y^3*z^2 + 192240*x^2*y^3*z + 31920*x^2*y^3 + 408240*x^2*y^2*z^3 + 430920*x^2*y^2*z^2 + 142560*x^2*y^2*z + 14760*x^2*y^2 + 306180*x^2*y*z^4 + 429300*x^2*y*z^3 + 212220*x^2*y*z^2 + 43740*x^2*y*z + 3270*x^2*y + 91854*x^2*z^5 + 160380*x^2*z^4 + 105300*x^2*z^3 + 32400*x^2*z^2 + 4824*x^2*z + 338*x^2 + 2688*x*y^6 + 24192*x*y^5*z + 8608*x*y^5 + 90720*x*y^4*z^2 + 64320*x*y^4*z + 10720*x*y^4 + 181440*x*y^3*z^3 + 192240*x*y^3*z^2 + 63840*x*y^3*z + 6640*x*y^3 + 204120*x*y^2*z^4 + 287280*x*y^2*z^3 + 142560*x*y^2*z^2 + 29520*x*y^2*z + 2224*x*y^2 + 122472*x*y*z^5 + 214650*x*y*z^4 + 141480*x*y*z^3 + 43740*x*y*z^2 + 6564*x*y*z + 468*x*y + 30618*x*z^6 + 64152*x*z^5 + 52650*x*z^4 + 21600*x*z^3 + 4842*x*z^2 + 690*x*z + 62*x + 256*y^7 + 2688*y^6*z + 960*y^6 + 12096*y^5*z^2 + 8608*y^5*z + 1440*y^5 + 30240*y^4*z^3 + 32160*y^4*z^2 + 10720*y^4*z + 1120*y^4 + 45360*y^3*z^4 + 64080*y^3*z^3 + 31920*y^3*z^2 + 6640*y^3*z + 504*y^3 + 40824*y^2*z^5 + 71820*y^2*z^4 + 47520*y^2*z^3 + 14760*y^2*z^2 + 2232*y^2*z + 162*y^2 + 20412*y*z^6 + 42930*y*z^5 + 35370*y*z^4 + 14580*y*z^3 + 3294*y*z^2 + 478*y*z + 44*y + 4374*z^7 + 10692*z^6 + 10530*z^5 + 5400*z^4 + 1620*z^3 + 352*z^2 + 62*z + 1, -2187*x^7 - 10206*x^6*y - 15309*x^6*z - 5346*x^6 - 20412*x^5*y^2 - 61236*x^5*y*z - 21465*x^5*y - 45927*x^5*z^2 - 32076*x^5*z - 5022*x^5 - 22680*x^4*y^3 - 102060*x^4*y^2*z - 35910*x^4*y^2 - 153090*x^4*y*z^2 - 107325*x^4*y*z - 16875*x^4*y - 76545*x^4*z^3 - 80190*x^4*z^2 - 25110*x^4*z - 2295*x^4 - 15120*x^3*y^4 - 90720*x^3*y^3*z - 32040*x^3*y^3 - 204120*x^3*y^2*z^2 - 143640*x^3*y^2*z - 22680*x^3*y^2 - 204120*x^3*y*z^3 - 214650*x^3*y*z^2 - 67500*x^3*y*z - 6210*x^3*y - 76545*x^3*z^4 - 106920*x^3*z^3 - 50220*x^3*z^2 - 9180*x^3*z - 531*x^3 - 6048*x^2*y^5 - 45360*x^2*y^4*z - 16080*x^2*y^4 - 136080*x^2*y^3*z^2 - 96120*x^2*y^3*z - 15240*x^2*y^3 - 204120*x^2*y^2*z^3 - 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- 2187*z^7 - 5346*z^6 - 5022*z^5 - 2295*z^4 - 540*z^3 - 86*z^2 - 11*z - 1, -4374*x^7 - 20412*x^6*y - 30618*x^6*z - 10692*x^6 - 40824*x^5*y^2 - 122472*x^5*y*z - 42930*x^5*y - 91854*x^5*z^2 - 64152*x^5*z - 11502*x^5 - 45360*x^4*y^3 - 204120*x^4*y^2*z - 71820*x^4*y^2 - 306180*x^4*y*z^2 - 214650*x^4*y*z - 38610*x^4*y - 153090*x^4*z^3 - 160380*x^4*z^2 - 57510*x^4*z - 7020*x^4 - 30240*x^3*y^4 - 181440*x^3*y^3*z - 64080*x^3*y^3 - 408240*x^3*y^2*z^2 - 287280*x^3*y^2*z - 51840*x^3*y^2 - 408240*x^3*y*z^3 - 429300*x^3*y*z^2 - 154440*x^3*y*z - 18900*x^3*y - 153090*x^3*z^4 - 213840*x^3*z^3 - 115020*x^3*z^2 - 28080*x^3*z - 2682*x^3 - 12096*x^2*y^5 - 90720*x^2*y^4*z - 32160*x^2*y^4 - 272160*x^2*y^3*z^2 - 192240*x^2*y^3*z - 34800*x^2*y^3 - 408240*x^2*y^2*z^3 - 430920*x^2*y^2*z^2 - 155520*x^2*y^2*z - 19080*x^2*y^2 - 306180*x^2*y*z^4 - 429300*x^2*y*z^3 - 231660*x^2*y*z^2 - 56700*x^2*y*z - 5430*x^2*y - 91854*x^2*z^5 - 160380*x^2*z^4 - 115020*x^2*z^3 - 42120*x^2*z^2 - 8064*x^2*z - 689*x^2 - 2688*x*y^6 - 24192*x*y^5*z - 8608*x*y^5 - 90720*x*y^4*z^2 - 64320*x*y^4*z - 11680*x*y^4 - 181440*x*y^3*z^3 - 192240*x*y^3*z^2 - 69600*x*y^3*z - 8560*x*y^3 - 204120*x*y^2*z^4 - 287280*x*y^2*z^3 - 155520*x*y^2*z^2 - 38160*x*y^2*z - 3664*x*y^2 - 122472*x*y*z^5 - 214650*x*y*z^4 - 154440*x*y*z^3 - 56700*x*y*z^2 - 10884*x*y*z - 936*x*y - 30618*x*z^6 - 64152*x*z^5 - 57510*x*z^4 - 28080*x*z^3 - 8082*x*z^2 - 1392*x*z - 131*x - 256*y^7 - 2688*y^6*z - 960*y^6 - 12096*y^5*z^2 - 8608*y^5*z - 1568*y^5 - 30240*y^4*z^3 - 32160*y^4*z^2 - 11680*y^4*z - 1440*y^4 - 45360*y^3*z^4 - 64080*y^3*z^3 - 34800*y^3*z^2 - 8560*y^3*z - 824*y^3 - 40824*y^2*z^5 - 71820*y^2*z^4 - 51840*y^2*z^3 - 19080*y^2*z^2 - 3672*y^2*z - 318*y^2 - 20412*y*z^6 - 42930*y*z^5 - 38610*y*z^4 - 18900*y*z^3 - 5454*y*z^2 - 946*y*z - 91*y - 4374*z^7 - 10692*z^6 - 11502*z^5 - 7020*z^4 - 2700*z^3 - 703*z^2 - 135*z - 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ 3, 2, 3 ], [ 1, 1, 1 ], [ 1, 1, 2 ] ], "source_shift": [ 1, 2, 1 ], "target_matrix": [ [ 2, 2, 3 ], [ -1, -1, -2 ], [ -2, -1, -1 ] ], "target_shift": [ -7, 0, 16 ], "inverse_recipe": { "p": 2, "q": 5, "f": "x**2", "g": "x**5 + x", "h": "y*z", "polynomials": { "f": [ [ [ 2, 0, 0 ], 1 ] ], "g": [ [ [ 5, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0408503210019262 ], "surface_features": [ 4.836195721700471, 5.634396651173698, 4.480581786829169, 0.0, 4.984611728216747, 5315 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C187", "pair": 187, "family": "counterexample", "mapping": "F(x,y,z)=(27*x^7 - 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3251*x^3*y^3*z + 5483*x^3*y^3 - 8001*x^3*y^2*z^2 + 27342*x^3*y^2*z - 18951*x^3*y^2 - 8667*x^3*y*z^3 + 45117*x^3*y*z^2 - 62832*x^3*y*z + 26679*x^3*y - 3483*x^3*z^4 + 24624*x^3*z^3 - 51723*x^3*z^2 + 43965*x^3*z - 13335*x^3 + 246*x^2*y^5 + 2031*x^2*y^4*z - 3303*x^2*y^4 + 6651*x^2*y^3*z^2 - 21870*x^2*y^3*z + 14916*x^2*y^3 + 10791*x^2*y^2*z^3 - 53919*x^2*y^2*z^2 + 73842*x^2*y^2*z - 31011*x^2*y^2 + 8667*x^2*y*z^4 - 58644*x^2*y*z^3 + 121050*x^2*y*z^2 - 101727*x^2*y*z + 30552*x^2*y + 2754*x^2*z^5 - 23733*x^2*z^4 + 65700*x^2*z^3 - 82890*x^2*z^2 + 49674*x^2*z - 11532*x^2 - 68*x*y^6 - 672*x*y^5*z + 1056*x*y^5 - 2745*x*y^4*z^2 + 8706*x*y^4*z - 5844*x*y^4 - 5929*x*y^3*z^3 + 28515*x*y^3*z^2 - 38412*x*y^3*z + 15954*x*y^3 - 7137*x*y^2*z^4 + 46368*x*y^2*z^3 - 94083*x*y^2*z^2 + 78168*x*y^2*z - 23247*x*y^2 - 4536*x*y*z^5 + 37422*x*y*z^4 - 101760*x*y*z^3 + 126891*x*y*z^2 - 75291*x*y*z + 17307*x*y - 1188*x*z^6 + 11988*x*z^5 - 41004*x*z^4 + 68247*x*z^3 - 60609*x*z^2 + 27726*x*z - 5149*x + 8*y^7 + 92*y^6*z - 140*y^6 + 450*y^5*z^2 - 1380*y^5*z + 912*y^5 + 1213*y^4*z^3 - 5631*y^4*z^2 + 7464*y^4*z - 3066*y^4 + 1945*y^3*z^4 - 12172*y^3*z^3 + 24288*y^3*z^2 - 19952*y^3*z + 5876*y^3 + 1854*y^2*z^5 - 14697*y^2*z^4 + 39276*y^2*z^3 - 48411*y^2*z^2 + 28443*y^2*z - 6475*y^2 + 972*y*z^6 - 9396*y*z^5 + 31560*y*z^4 - 51909*y*z^3 + 45645*y*z^2 - 20682*y*z + 3806*y + 216*z^7 - 2484*z^6 + 10080*z^5 - 20754*z^4 + 24288*z^3 - 16434*z^2 + 6010*z - 1, -81*x^7 + 486*x^6*y + 810*x^6*z - 1512*x^6 - 1242*x^5*y^2 - 4131*x^5*y*z + 7389*x^5*y - 3402*x^5*z^2 + 12393*x^5*z - 8577*x^5 + 1752*x^4*y^3 + 8721*x^4*y^2*z - 14964*x^4*y^2 + 14337*x^4*y*z^2 - 49959*x^4*y*z + 33987*x^4*y + 7776*x^4*z^3 - 41391*x^4*z^2 + 56547*x^4*z - 23481*x^4 - 1473*x^3*y^4 - 9753*x^3*y^3*z + 16077*x^3*y^3 - 24003*x^3*y^2*z^2 + 80151*x^3*y^2*z - 53595*x^3*y^2 - 26001*x^3*y*z^3 + 132237*x^3*y*z^2 - 177456*x^3*y*z + 72855*x^3*y - 10449*x^3*z^4 + 72171*x^3*z^3 - 145899*x^3*z^2 + 119808*x^3*z - 35331*x^3 + 738*x^2*y^5 + 6093*x^2*y^4*z - 9666*x^2*y^4 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130449*y^2*z^2 + 74526*y^2*z - 16626*y^2 + 2916*y*z^6 - 27360*y*z^5 + 88080*y*z^4 - 139683*y*z^3 + 119415*y*z^2 - 53037*y*z + 9637*y + 648*z^7 - 7236*z^6 + 28116*z^5 - 55782*z^4 + 63456*z^3 - 42096*z^2 + 15207*z - 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ 1, -1, -2 ], [ 3, -2, -3 ], [ 1, -1, -1 ] ], "source_shift": [ 2, 2, 1 ], "target_matrix": [ [ 1, 1, 1 ], [ -1, -1, -2 ], [ -3, -2, -3 ] ], "target_shift": [ -952, 924, 2322 ], "collision_preimages": [ [ "5/4", "7/4", "3/4" ], [ "-8", "-20", "13/2" ], [ "-3", "-17", "17/2" ] ], "collision_image": [ "-3809/4", "3697/4", "9291/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 0.9996932670001115 ], "surface_features": [ 4.541728905542369, 5.535239671548926, 4.167228694175993, 0.0, 4.719045965062429, 5221 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], 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5424*x*y^4 - 3360*x*y^3*z^3 - 10080*x*y^3*z^2 - 10896*x*y^3*z - 4268*x*y^3 - 1260*x*y^2*z^4 - 5040*x*y^2*z^3 - 8208*x*y^2*z^2 - 6528*x*y^2*z - 2052*x*y^2 - 252*x*y*z^5 - 1260*x*y*z^4 - 2748*x*y*z^3 - 3327*x*y*z^2 - 2142*x*y*z - 591*x*y - 21*x*z^6 - 126*x*z^5 - 345*x*z^4 - 565*x*z^3 - 558*x*z^2 - 327*x*z - 83*x - 384*y^7 - 1344*y^6*z - 1344*y^6 - 2016*y^5*z^2 - 4032*y^5*z - 2160*y^5 - 1680*y^4*z^3 - 5040*y^4*z^2 - 5424*y^4*z - 2104*y^4 - 840*y^3*z^4 - 3360*y^3*z^3 - 5448*y^3*z^2 - 4292*y^3*z - 1332*y^3 - 252*y^2*z^5 - 1260*y^2*z^4 - 2736*y^2*z^3 - 3282*y^2*z^2 - 2088*y^2*z - 561*y^2 - 42*y*z^6 - 252*y*z^5 - 687*y*z^4 - 1115*y*z^3 - 1089*y*z^2 - 618*y*z - 151*y - 3*z^7 - 21*z^6 - 69*z^5 - 142*z^4 - 189*z^3 - 171*z^2 - 90*z - 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ 1, 2, 1 ], [ 1, 1, 2 ], [ 2, 3, 2 ] ], "source_shift": [ 1, 1, 1 ], "target_matrix": [ [ -2, -1, -1 ], [ 1, 1, 1 ], [ -1, -1, -2 ] ], "target_shift": [ 48, -23, 19 ], "inverse_recipe": { "p": 4, "q": 3, "f": "x**4 + x", "g": "x**3", "h": "y*z + z", "polynomials": { "f": [ [ [ 4, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "g": [ [ [ 3, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 0, 1 ], 1 ] ] }, "h_multiplier": 3 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0315191770023375 ], "surface_features": [ 3.4505172511346007, 4.338297090232691, 3.1166077439882485, 0.0, 3.5596275009449414, 4877 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B158", "pair": 158, "family": "bijection", "mapping": "F(x,y,z)=(-256*x^7 - 896*x^6*y + 896*x^6*z + 1408*x^6 - 1344*x^5*y^2 + 2688*x^5*y*z + 4224*x^5*y - 1344*x^5*z^2 - 4224*x^5*z - 3232*x^5 - 1120*x^4*y^3 + 3360*x^4*y^2*z + 5280*x^4*y^2 - 3360*x^4*y*z^2 - 10560*x^4*y*z - 8096*x^4*y + 1120*x^4*z^3 + 5280*x^4*z^2 + 8096*x^4*z + 4128*x^4 - 560*x^3*y^4 + 2240*x^3*y^3*z + 3520*x^3*y^3 - 3360*x^3*y^2*z^2 - 10560*x^3*y^2*z - 8112*x^3*y^2 + 2240*x^3*y*z^3 + 10560*x^3*y*z^2 + 16224*x^3*y*z + 8352*x^3*y - 560*x^3*z^4 - 3520*x^3*z^3 - 8112*x^3*z^2 - 8336*x^3*z - 3240*x^3 - 168*x^2*y^5 + 840*x^2*y^4*z + 1320*x^2*y^4 - 1680*x^2*y^3*z^2 - 5280*x^2*y^3*z - 4064*x^2*y^3 + 1680*x^2*y^2*z^3 + 7920*x^2*y^2*z^2 + 12192*x^2*y^2*z + 6336*x^2*y^2 - 840*x^2*y*z^4 - 5280*x^2*y*z^3 - 12192*x^2*y*z^2 - 12648*x^2*y*z - 5008*x^2*y + 168*x^2*z^5 + 1320*x^2*z^4 + 4064*x^2*z^3 + 6312*x^2*z^2 + 4976*x^2*z + 1582*x^2 - 28*x*y^6 + 168*x*y^5*z + 264*x*y^5 - 420*x*y^4*z^2 - 1320*x*y^4*z - 1018*x*y^4 + 560*x*y^3*z^3 + 2640*x*y^3*z^2 + 4072*x*y^3*z + 2136*x*y^3 - 420*x*y^2*z^4 - 2640*x*y^2*z^3 - 6108*x*y^2*z^2 - 6396*x*y^2*z - 2578*x*y^2 + 168*x*y*z^5 + 1320*x*y*z^4 + 4072*x*y*z^3 + 6384*x*y*z^2 + 5124*x*y*z + 1674*x*y - 28*x*z^6 - 264*x*z^5 - 1018*x*z^4 - 2124*x*z^3 - 2546*x*z^2 - 1652*x*z - 452*x - 2*y^7 + 14*y^6*z + 22*y^6 - 42*y^5*z^2 - 132*y^5*z - 102*y^5 + 70*y^4*z^3 + 330*y^4*z^2 + 510*y^4*z + 270*y^4 - 70*y^3*z^4 - 440*y^3*z^3 - 1020*y^3*z^2 - 1078*y^3*z - 442*y^3 + 42*y^2*z^5 + 330*y^2*z^4 + 1020*y^2*z^3 + 1614*y^2*z^2 + 1318*y^2*z + 444*y^2 - 14*y*z^6 - 132*y*z^5 - 510*y*z^4 - 1074*y*z^3 - 1310*y*z^2 - 876*y*z - 253*y + 2*z^7 + 22*z^6 + 102*z^5 + 268*z^4 + 434*z^3 + 432*z^2 + 245*z + 1, -512*x^7 - 1792*x^6*y + 1792*x^6*z + 2816*x^6 - 2688*x^5*y^2 + 5376*x^5*y*z + 8448*x^5*y - 2688*x^5*z^2 - 8448*x^5*z - 6464*x^5 - 2240*x^4*y^3 + 6720*x^4*y^2*z + 10560*x^4*y^2 - 6720*x^4*y*z^2 - 21120*x^4*y*z - 16192*x^4*y + 2240*x^4*z^3 + 10560*x^4*z^2 + 16192*x^4*z + 8240*x^4 - 1120*x^3*y^4 + 4480*x^3*y^3*z + 7040*x^3*y^3 - 6720*x^3*y^2*z^2 - 21120*x^3*y^2*z - 16224*x^3*y^2 + 4480*x^3*y*z^3 + 21120*x^3*y*z^2 + 32448*x^3*y*z + 16672*x^3*y - 1120*x^3*z^4 - 7040*x^3*z^3 - 16224*x^3*z^2 - 16640*x^3*z - 6440*x^3 - 336*x^2*y^5 + 1680*x^2*y^4*z + 2640*x^2*y^4 - 3360*x^2*y^3*z^2 - 10560*x^2*y^3*z - 8128*x^2*y^3 + 3360*x^2*y^2*z^3 + 15840*x^2*y^2*z^2 + 24384*x^2*y^2*z + 12648*x^2*y^2 - 1680*x^2*y*z^4 - 10560*x^2*y*z^3 - 24384*x^2*y*z^2 - 25248*x^2*y*z - 9956*x^2*y + 336*x^2*z^5 + 2640*x^2*z^4 + 8128*x^2*z^3 + 12600*x^2*z^2 + 9892*x^2*z + 3128*x^2 - 56*x*y^6 + 336*x*y^5*z + 528*x*y^5 - 840*x*y^4*z^2 - 2640*x*y^4*z - 2036*x*y^4 + 1120*x*y^3*z^3 + 5280*x*y^3*z^2 + 8144*x*y^3*z + 4264*x*y^3 - 840*x*y^2*z^4 - 5280*x*y^2*z^3 - 12216*x*y^2*z^2 - 12768*x*y^2*z - 5126*x*y^2 + 336*x*y*z^5 + 2640*x*y*z^4 + 8144*x*y*z^3 + 12744*x*y*z^2 + 10188*x*y*z + 3312*x*y - 56*x*z^6 - 528*x*z^5 - 2036*x*z^4 - 4240*x*z^3 - 5062*x*z^2 - 3268*x*z - 889*x - 4*y^7 + 28*y^6*z + 44*y^6 - 84*y^5*z^2 - 264*y^5*z - 204*y^5 + 140*y^4*z^3 + 660*y^4*z^2 + 1020*y^4*z + 539*y^4 - 140*y^3*z^4 - 880*y^3*z^3 - 2040*y^3*z^2 - 2152*y^3*z - 879*y^3 + 84*y^2*z^5 + 660*y^2*z^4 + 2040*y^2*z^3 + 3222*y^2*z^2 + 2621*y^2*z + 879*y^2 - 28*y*z^6 - 264*y*z^5 - 1020*y*z^4 - 2144*y*z^3 - 2605*y*z^2 - 1734*y*z - 497*y + 4*z^7 + 44*z^6 + 204*z^5 + 535*z^4 + 863*z^3 + 855*z^2 + 482*z + 1, -512*x^7 - 1792*x^6*y + 1792*x^6*z + 2816*x^6 - 2688*x^5*y^2 + 5376*x^5*y*z + 8448*x^5*y - 2688*x^5*z^2 - 8448*x^5*z - 6464*x^5 - 2240*x^4*y^3 + 6720*x^4*y^2*z + 10560*x^4*y^2 - 6720*x^4*y*z^2 - 21120*x^4*y*z - 16192*x^4*y + 2240*x^4*z^3 + 10560*x^4*z^2 + 16192*x^4*z + 8208*x^4 - 1120*x^3*y^4 + 4480*x^3*y^3*z + 7040*x^3*y^3 - 6720*x^3*y^2*z^2 - 21120*x^3*y^2*z - 16224*x^3*y^2 + 4480*x^3*y*z^3 + 21120*x^3*y*z^2 + 32448*x^3*y*z + 16608*x^3*y - 1120*x^3*z^4 - 7040*x^3*z^3 - 16224*x^3*z^2 - 16576*x^3*z - 6376*x^3 - 336*x^2*y^5 + 1680*x^2*y^4*z + 2640*x^2*y^4 - 3360*x^2*y^3*z^2 - 10560*x^2*y^3*z - 8128*x^2*y^3 + 3360*x^2*y^2*z^3 + 15840*x^2*y^2*z^2 + 24384*x^2*y^2*z + 12600*x^2*y^2 - 1680*x^2*y*z^4 - 10560*x^2*y*z^3 - 24384*x^2*y*z^2 - 25152*x^2*y*z - 9860*x^2*y + 336*x^2*z^5 + 2640*x^2*z^4 + 8128*x^2*z^3 + 12552*x^2*z^2 + 9796*x^2*z + 3088*x^2 - 56*x*y^6 + 336*x*y^5*z + 528*x*y^5 - 840*x*y^4*z^2 - 2640*x*y^4*z - 2036*x*y^4 + 1120*x*y^3*z^3 + 5280*x*y^3*z^2 + 8144*x*y^3*z + 4248*x*y^3 - 840*x*y^2*z^4 - 5280*x*y^2*z^3 - 12216*x*y^2*z^2 - 12720*x*y^2*z - 5078*x*y^2 + 336*x*y*z^5 + 2640*x*y*z^4 + 8144*x*y*z^3 + 12696*x*y*z^2 + 10092*x*y*z + 3272*x*y - 56*x*z^6 - 528*x*z^5 - 2036*x*z^4 - 4224*x*z^3 - 5014*x*z^2 - 3228*x*z - 881*x - 4*y^7 + 28*y^6*z + 44*y^6 - 84*y^5*z^2 - 264*y^5*z - 204*y^5 + 140*y^4*z^3 + 660*y^4*z^2 + 1020*y^4*z + 537*y^4 - 140*y^3*z^4 - 880*y^3*z^3 - 2040*y^3*z^2 - 2144*y^3*z - 871*y^3 + 84*y^2*z^5 + 660*y^2*z^4 + 2040*y^2*z^3 + 3210*y^2*z^2 + 2597*y^2*z + 869*y^2 - 28*y*z^6 - 264*y*z^5 - 1020*y*z^4 - 2136*y*z^3 - 2581*y*z^2 - 1714*y*z - 491*y + 4*z^7 + 44*z^6 + 204*z^5 + 533*z^4 + 855*z^3 + 845*z^2 + 478*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -2, -1, 1 ], [ -1, -2, 1 ], [ -1, -1, 1 ] ], "source_shift": [ 1, 2, 1 ], "target_matrix": [ [ 1, 2, 1 ], [ 2, 3, 2 ], [ 2, 1, 1 ] ], "target_shift": [ -62, -121, -119 ], "inverse_recipe": { "p": 4, "q": 3, "f": "x**4 + x**3", "g": "x**3 + x**2", "h": "y*z + 2*z**2", "polynomials": { "f": [ [ [ 4, 0, 0 ], 1 ], [ [ 3, 0, 0 ], 1 ] ], "g": [ [ [ 3, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 0, 2 ], 2 ] ] }, "h_multiplier": 2 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 0.9825399740002467 ], "surface_features": [ 3.6589614034279463, 4.511187933317223, 3.332034277027518, 0.0, 3.767372287138234, 4936 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C181", "pair": 181, "family": "counterexample", "mapping": "F(x,y,z)=(-6*x^7 - 60*x^6*y - 45*x^6*z - 96*x^6 - 252*x^5*y^2 - 387*x^5*y*z - 825*x^5*y - 144*x^5*z^2 - 618*x^5*z - 666*x^5 - 576*x^4*y^3 - 1359*x^4*y^2*z - 2904*x^4*y^2 - 1035*x^4*y*z^2 - 4437*x^4*y*z - 4794*x^4*y - 255*x^4*z^3 - 1650*x^4*z^2 - 3579*x^4*z - 2594*x^4 - 774*x^3*y^4 - 2493*x^3*y^3*z - 5361*x^3*y^3 - 2916*x^3*y^2*z^2 - 12522*x^3*y^2*z - 13602*x^3*y^2 - 1470*x^3*y*z^3 - 9498*x^3*y*z^2 - 20646*x^3*y*z - 15041*x^3*y - 270*x^3*z^4 - 2340*x^3*z^3 - 7656*x^3*z^2 - 11178*x^3*z - 6118*x^3 - 612*x^2*y^5 - 2520*x^2*y^4*z - 5478*x^2*y^4 - 4023*x^2*y^3*z^2 - 17367*x^2*y^3*z - 19026*x^2*y^3 - 3114*x^2*y^2*z^3 - 20142*x^2*y^2*z^2 - 43983*x^2*y^2*z - 32295*x^2*y^2 - 1170*x^2*y*z^4 - 10122*x^2*y*z^3 - 33174*x^2*y*z^2 - 48657*x^2*y*z - 26832*x^2*y - 171*x^2*z^5 - 1860*x^2*z^4 - 8154*x^2*z^3 - 17970*x^2*z^2 - 19839*x^2*z - 8721*x^2 - 264*x*y^6 - 1332*x*y^5*z - 2940*x*y^5 - 2718*x*y^4*z^2 - 11844*x*y^4*z - 13128*x*y^4 - 2871*x*y^3*z^3 - 18651*x*y^3*z^2 - 41028*x*y^3*z - 30452*x*y^3 - 1656*x*y^2*z^4 - 14334*x*y^2*z^3 - 47160*x*y^2*z^2 - 69648*x*y^2*z - 38814*x*y^2 - 495*x*y*z^5 - 5373*x*y*z^4 - 23586*x*y*z^3 - 52191*x*y*z^2 - 58011*x*y*z - 25755*x*y - 60*x*z^6 - 786*x*z^5 - 4326*x*z^4 - 12782*x*z^3 - 21324*x*z^2 - 18948*x*z - 6934*x - 48*y^7 - 288*y^6*z - 648*y^6 - 720*y^5*z^2 - 3180*y^5*z - 3576*y^5 - 972*y^4*z^3 - 6366*y^4*z^2 - 14148*y^4*z - 10640*y^4 - 765*y^3*z^4 - 6645*y^3*z^3 - 22002*y^3*z^2 - 32802*y^3*z - 18526*y^3 - 351*y^2*z^5 - 3810*y^2*z^4 - 16779*y^2*z^3 - 37353*y^2*z^2 - 41904*y^2*z - 18870*y^2 - 87*y*z^6 - 1137*y*z^5 - 6264*y*z^4 - 18575*y*z^3 - 31179*y*z^2 - 27951*y*z - 10362*y - 9*z^7 - 138*z^6 - 915*z^5 - 3396*z^4 - 7603*z^3 - 10227*z^2 - 7593*z - 1, 2*x^7 + 20*x^6*y + 15*x^6*z + 34*x^6 + 84*x^5*y^2 + 129*x^5*y*z + 291*x^5*y + 48*x^5*z^2 + 219*x^5*z + 246*x^5 + 192*x^4*y^3 + 453*x^4*y^2*z + 1020*x^4*y^2 + 345*x^4*y*z^2 + 1566*x^4*y*z + 1761*x^4*y + 85*x^4*z^3 + 585*x^4*z^2 + 1323*x^4*z + 988*x^4 + 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[ 2, 1, 1 ] ], "target_shift": [ -439, -290, -437 ], "collision_preimages": [ [ "3/4", "1", "-15/4" ], [ "9", "7/2", "-16" ], [ "6", "-3/2", "-5" ] ], "collision_image": [ "-1757/4", "-1161/4", "-875/2" ] }, "probe_target": "No", "jev_answers": [ "Maybe" ], "seconds": [ 1.0524461099994369 ], "surface_features": [ 3.385260606854612, 4.245092976101281, 3.041787318971752, 0.0, 3.4933646079179623, 4815 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B036", "pair": 36, "family": "bijection", "mapping": "F(x,y,z)=(-2*x^7 - 28*x^6*y - 14*x^6*z - 16*x^6 - 168*x^5*y^2 - 168*x^5*y*z - 192*x^5*y - 42*x^5*z^2 - 96*x^5*z - 58*x^5 - 560*x^4*y^3 - 840*x^4*y^2*z - 960*x^4*y^2 - 420*x^4*y*z^2 - 960*x^4*y*z - 578*x^4*y - 70*x^4*z^3 - 240*x^4*z^2 - 290*x^4*z - 132*x^4 - 1120*x^3*y^4 - 2240*x^3*y^3*z - 2560*x^3*y^3 - 1680*x^3*y^2*z^2 - 3840*x^3*y^2*z - 2304*x^3*y^2 - 560*x^3*y*z^3 - 1920*x^3*y*z^2 - 2312*x^3*y*z - 1040*x^3*y - 70*x^3*z^4 - 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720*x^4*z^2 - 870*x^4*z - 398*x^4 - 3360*x^3*y^4 - 6720*x^3*y^3*z - 7680*x^3*y^3 - 5040*x^3*y^2*z^2 - 11520*x^3*y^2*z - 6912*x^3*y^2 - 1680*x^3*y*z^3 - 5760*x^3*y*z^2 - 6936*x^3*y*z - 3136*x^3*y - 210*x^3*z^4 - 960*x^3*z^3 - 1740*x^3*z^2 - 1586*x^3*z - 618*x^3 - 4032*x^2*y^5 - 10080*x^2*y^4*z - 11520*x^2*y^4 - 10080*x^2*y^3*z^2 - 23040*x^2*y^3*z - 13776*x^2*y^3 - 5040*x^2*y^2*z^3 - 17280*x^2*y^2*z^2 - 20736*x^2*y^2*z - 9264*x^2*y^2 - 1260*x^2*y*z^4 - 5760*x^2*y*z^3 - 10404*x^2*y*z^2 - 9372*x^2*y*z - 3600*x^2*y - 126*x^2*z^5 - 720*x^2*z^4 - 1740*x^2*z^3 - 2370*x^2*z^2 - 1836*x^2*z - 636*x^2 - 2688*x*y^6 - 8064*x*y^5*z - 9216*x*y^5 - 10080*x*y^4*z^2 - 23040*x*y^4*z - 13728*x*y^4 - 6720*x*y^3*z^3 - 23040*x*y^3*z^2 - 27552*x*y^3*z - 12160*x*y^3 - 2520*x*y^2*z^4 - 11520*x*y^2*z^3 - 20736*x*y^2*z^2 - 18456*x*y^2*z - 6984*x*y^2 - 504*x*y*z^5 - 2880*x*y*z^4 - 6936*x*y*z^3 - 9336*x*y*z^2 - 7128*x*y*z - 2400*x*y - 42*x*z^6 - 288*x*z^5 - 870*x*z^4 - 1574*x*z^3 - 1818*x*z^2 - 1242*x*z - 421*x - 768*y^7 - 2688*y^6*z - 3072*y^6 - 4032*y^5*z^2 - 9216*y^5*z - 5472*y^5 - 3360*y^4*z^3 - 11520*y^4*z^2 - 13728*y^4*z - 5984*y^4 - 1680*y^3*z^4 - 7680*y^3*z^3 - 13776*y^3*z^2 - 12112*y^3*z - 4512*y^3 - 504*y^2*z^5 - 2880*y^2*z^4 - 6912*y^2*z^3 - 9192*y^2*z^2 - 6912*y^2*z - 2274*y^2 - 84*y*z^6 - 576*y*z^5 - 1734*y*z^4 - 3100*y*z^3 - 3528*y*z^2 - 2346*y*z - 720*y - 6*z^7 - 48*z^6 - 174*z^5 - 392*z^4 - 600*z^3 - 606*z^2 - 389*z - 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ 1, 2, 1 ], [ 1, 1, 1 ], [ 2, 1, 1 ] ], "source_shift": [ 1, 2, 1 ], "target_matrix": [ [ -1, -1, -1 ], [ 1, 1, 2 ], [ -3, -2, -2 ] ], "target_shift": [ 42, -36, 130 ], "inverse_recipe": { "p": 3, "q": 4, "f": "x**3 + x", "g": "x**4 + x**3", "h": "y*z + y + 2*z", "polynomials": { "f": [ [ [ 3, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "g": [ [ [ 4, 0, 0 ], 1 ], [ [ 3, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 0, 1 ], 2 ], [ [ 0, 1, 0 ], 1 ] ] }, "h_multiplier": 2 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0030317069977173 ], "surface_features": [ 3.455904751073886, 4.44015312977356, 3.0835026198302673, 0.0, 3.615085981137108, 4852 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B192", "pair": 192, "family": "bijection", "mapping": "F(x,y,z)=(4*x^7 + 28*x^6*y + 28*x^6*z + 40*x^6 + 84*x^5*y^2 + 168*x^5*y*z + 248*x^5*y + 84*x^5*z^2 + 240*x^5*z + 176*x^5 + 140*x^4*y^3 + 420*x^4*y^2*z + 640*x^4*y^2 + 420*x^4*y*z^2 + 1240*x^4*y*z + 932*x^4*y + 140*x^4*z^3 + 600*x^4*z^2 + 876*x^4*z + 436*x^4 + 140*x^3*y^4 + 560*x^3*y^3*z + 880*x^3*y^3 + 840*x^3*y^2*z^2 + 2560*x^3*y^2*z + 1972*x^3*y^2 + 560*x^3*y*z^3 + 2480*x^3*y*z^2 + 3712*x^3*y*z + 1888*x^3*y + 140*x^3*z^4 + 800*x^3*z^3 + 1744*x^3*z^2 + 1720*x^3*z + 646*x^3 + 84*x^2*y^5 + 420*x^2*y^4*z + 680*x^2*y^4 + 840*x^2*y^3*z^2 + 2640*x^2*y^3*z + 2084*x^2*y^3 + 840*x^2*y^2*z^3 + 3840*x^2*y^2*z^2 + 5892*x^2*y^2*z + 3056*x^2*y^2 + 420*x^2*y*z^4 + 2480*x^2*y*z^3 + 5544*x^2*y*z^2 + 5584*x^2*y*z + 2142*x^2*y + 84*x^2*z^5 + 600*x^2*z^4 + 1736*x^2*z^3 + 2544*x^2*z^2 + 1886*x^2*z + 562*x^2 + 28*x*y^6 + 168*x*y^5*z + 280*x*y^5 + 420*x*y^4*z^2 + 1360*x*y^4*z + 1100*x*y^4 + 560*x*y^3*z^3 + 2640*x*y^3*z^2 + 4152*x*y^3*z + 2192*x*y^3 + 420*x*y^2*z^4 + 2560*x*y^2*z^3 + 5868*x*y^2*z^2 + 6024*x*y^2*z + 2350*x*y^2 + 168*x*y*z^5 + 1240*x*y*z^4 + 3680*x*y*z^3 + 5504*x*y*z^2 + 4156*x*y*z + 1260*x*y + 28*x*z^6 + 240*x*z^5 + 864*x*z^4 + 1672*x*z^3 + 1834*x*z^2 + 1076*x*z + 265*x + 4*y^7 + 28*y^6*z + 48*y^6 + 84*y^5*z^2 + 280*y^5*z + 232*y^5 + 140*y^4*z^3 + 680*y^4*z^2 + 1096*y^4*z + 588*y^4 + 140*y^3*z^4 + 880*y^3*z^3 + 2068*y^3*z^2 + 2160*y^3*z + 850*y^3 + 84*y^2*z^5 + 640*y^2*z^4 + 1948*y^2*z^3 + 2968*y^2*z^2 + 2270*y^2*z + 694*y^2 + 28*y*z^6 + 248*y*z^5 + 916*y*z^4 + 1808*y*z^3 + 2014*y*z^2 + 1196*y*z + 293*y + 4*z^7 + 40*z^6 + 172*z^5 + 412*z^4 + 594*z^3 + 514*z^2 + 247*z + 1, -4*x^7 - 28*x^6*y - 28*x^6*z - 40*x^6 - 84*x^5*y^2 - 168*x^5*y*z - 248*x^5*y - 84*x^5*z^2 - 240*x^5*z - 176*x^5 - 140*x^4*y^3 - 420*x^4*y^2*z - 640*x^4*y^2 - 420*x^4*y*z^2 - 1240*x^4*y*z - 932*x^4*y - 140*x^4*z^3 - 600*x^4*z^2 - 876*x^4*z - 436*x^4 - 140*x^3*y^4 - 560*x^3*y^3*z - 880*x^3*y^3 - 840*x^3*y^2*z^2 - 2560*x^3*y^2*z - 1972*x^3*y^2 - 560*x^3*y*z^3 - 2480*x^3*y*z^2 - 3712*x^3*y*z - 1888*x^3*y - 140*x^3*z^4 - 800*x^3*z^3 - 1744*x^3*z^2 - 1720*x^3*z - 645*x^3 - 84*x^2*y^5 - 420*x^2*y^4*z - 680*x^2*y^4 - 840*x^2*y^3*z^2 - 2640*x^2*y^3*z - 2084*x^2*y^3 - 840*x^2*y^2*z^3 - 3840*x^2*y^2*z^2 - 5892*x^2*y^2*z - 3056*x^2*y^2 - 420*x^2*y*z^4 - 2480*x^2*y*z^3 - 5544*x^2*y*z^2 - 5584*x^2*y*z - 2139*x^2*y - 84*x^2*z^5 - 600*x^2*z^4 - 1736*x^2*z^3 - 2544*x^2*z^2 - 1883*x^2*z - 556*x^2 - 28*x*y^6 - 168*x*y^5*z - 280*x*y^5 - 420*x*y^4*z^2 - 1360*x*y^4*z - 1100*x*y^4 - 560*x*y^3*z^3 - 2640*x*y^3*z^2 - 4152*x*y^3*z - 2192*x*y^3 - 420*x*y^2*z^4 - 2560*x*y^2*z^3 - 5868*x*y^2*z^2 - 6024*x*y^2*z - 2347*x*y^2 - 168*x*y*z^5 - 1240*x*y*z^4 - 3680*x*y*z^3 - 5504*x*y*z^2 - 4150*x*y*z - 1248*x*y - 28*x*z^6 - 240*x*z^5 - 864*x*z^4 - 1672*x*z^3 - 1831*x*z^2 - 1064*x*z - 252*x - 4*y^7 - 28*y^6*z - 48*y^6 - 84*y^5*z^2 - 280*y^5*z - 232*y^5 - 140*y^4*z^3 - 680*y^4*z^2 - 1096*y^4*z - 588*y^4 - 140*y^3*z^4 - 880*y^3*z^3 - 2068*y^3*z^2 - 2160*y^3*z - 849*y^3 - 84*y^2*z^5 - 640*y^2*z^4 - 1948*y^2*z^3 - 2968*y^2*z^2 - 2267*y^2*z - 688*y^2 - 28*y*z^6 - 248*y*z^5 - 916*y*z^4 - 1808*y*z^3 - 2011*y*z^2 - 1184*y*z - 279*y - 4*z^7 - 40*z^6 - 172*z^5 - 412*z^4 - 593*z^3 - 508*z^2 - 235*z - 1, -4*x^7 - 28*x^6*y - 28*x^6*z - 40*x^6 - 84*x^5*y^2 - 168*x^5*y*z - 248*x^5*y - 84*x^5*z^2 - 240*x^5*z - 176*x^5 - 140*x^4*y^3 - 420*x^4*y^2*z - 640*x^4*y^2 - 420*x^4*y*z^2 - 1240*x^4*y*z - 932*x^4*y - 140*x^4*z^3 - 600*x^4*z^2 - 876*x^4*z - 436*x^4 - 140*x^3*y^4 - 560*x^3*y^3*z - 880*x^3*y^3 - 840*x^3*y^2*z^2 - 2560*x^3*y^2*z - 1972*x^3*y^2 - 560*x^3*y*z^3 - 2480*x^3*y*z^2 - 3712*x^3*y*z - 1888*x^3*y - 140*x^3*z^4 - 800*x^3*z^3 - 1744*x^3*z^2 - 1720*x^3*z - 645*x^3 - 84*x^2*y^5 - 420*x^2*y^4*z - 680*x^2*y^4 - 840*x^2*y^3*z^2 - 2640*x^2*y^3*z - 2084*x^2*y^3 - 840*x^2*y^2*z^3 - 3840*x^2*y^2*z^2 - 5892*x^2*y^2*z - 3056*x^2*y^2 - 420*x^2*y*z^4 - 2480*x^2*y*z^3 - 5544*x^2*y*z^2 - 5584*x^2*y*z - 2139*x^2*y - 84*x^2*z^5 - 600*x^2*z^4 - 1736*x^2*z^3 - 2544*x^2*z^2 - 1883*x^2*z - 558*x^2 - 28*x*y^6 - 168*x*y^5*z - 280*x*y^5 - 420*x*y^4*z^2 - 1360*x*y^4*z - 1100*x*y^4 - 560*x*y^3*z^3 - 2640*x*y^3*z^2 - 4152*x*y^3*z - 2192*x*y^3 - 420*x*y^2*z^4 - 2560*x*y^2*z^3 - 5868*x*y^2*z^2 - 6024*x*y^2*z - 2347*x*y^2 - 168*x*y*z^5 - 1240*x*y*z^4 - 3680*x*y*z^3 - 5504*x*y*z^2 - 4150*x*y*z - 1252*x*y - 28*x*z^6 - 240*x*z^5 - 864*x*z^4 - 1672*x*z^3 - 1831*x*z^2 - 1068*x*z - 258*x - 4*y^7 - 28*y^6*z - 48*y^6 - 84*y^5*z^2 - 280*y^5*z - 232*y^5 - 140*y^4*z^3 - 680*y^4*z^2 - 1096*y^4*z - 588*y^4 - 140*y^3*z^4 - 880*y^3*z^3 - 2068*y^3*z^2 - 2160*y^3*z - 849*y^3 - 84*y^2*z^5 - 640*y^2*z^4 - 1948*y^2*z^3 - 2968*y^2*z^2 - 2267*y^2*z - 690*y^2 - 28*y*z^6 - 248*y*z^5 - 916*y*z^4 - 1808*y*z^3 - 2011*y*z^2 - 1188*y*z - 287*y - 4*z^7 - 40*z^6 - 172*z^5 - 412*z^4 - 593*z^3 - 510*z^2 - 241*z - 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ 1, 1, 1 ], [ 2, 1, 1 ], [ 1, 2, 1 ] ], "source_shift": [ 1, 1, 1 ], "target_matrix": [ [ 1, 2, 1 ], [ -1, -1, -2 ], [ -1, -1, -1 ] ], "target_shift": [ -50, 43, 47 ], "inverse_recipe": { "p": 3, "q": 2, "f": "x**3 + x**2", "g": "x**2", "h": "y*z**2", "polynomials": { "f": [ [ [ 3, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "g": [ [ [ 2, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 2 ], 1 ] ] }, "h_multiplier": 4 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0075651529987226 ], "surface_features": [ 3.096778739726786, 3.7798849631926443, 2.821513528404773, 0.0, 3.1588565023576187, 4757 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B089", "pair": 89, "family": "bijection", "mapping": "F(x,y,z)=(-128*x^7 - 448*x^6*y + 448*x^6*z + 576*x^6 - 672*x^5*y^2 + 1344*x^5*y*z + 1728*x^5*y - 672*x^5*z^2 - 1728*x^5*z - 1200*x^5 - 560*x^4*y^3 + 1680*x^4*y^2*z + 2160*x^4*y^2 - 1680*x^4*y*z^2 - 4320*x^4*y*z - 3008*x^4*y + 560*x^4*z^3 + 2160*x^4*z^2 + 3008*x^4*z + 1448*x^4 - 280*x^3*y^4 + 1120*x^3*y^3*z + 1440*x^3*y^3 - 1680*x^3*y^2*z^2 - 4320*x^3*y^2*z - 3016*x^3*y^2 + 1120*x^3*y*z^3 + 4320*x^3*y*z^2 + 6032*x^3*y*z + 2924*x^3*y - 280*x^3*z^4 - 1440*x^3*z^3 - 3016*x^3*z^2 - 2932*x^3*z - 1124*x^3 - 84*x^2*y^5 + 420*x^2*y^4*z + 540*x^2*y^4 - 840*x^2*y^3*z^2 - 2160*x^2*y^3*z - 1512*x^2*y^3 + 840*x^2*y^2*z^3 + 3240*x^2*y^2*z^2 + 4536*x^2*y^2*z + 2214*x^2*y^2 - 420*x^2*y*z^4 - 2160*x^2*y*z^3 - 4536*x^2*y*z^2 - 4440*x^2*y*z - 1724*x^2*y + 84*x^2*z^5 + 540*x^2*z^4 + 1512*x^2*z^3 + 2226*x^2*z^2 + 1740*x^2*z + 572*x^2 - 14*x*y^6 + 84*x*y^5*z + 108*x*y^5 - 210*x*y^4*z^2 - 540*x*y^4*z - 379*x*y^4 + 280*x*y^3*z^3 + 1080*x*y^3*z^2 + 1516*x*y^3*z + 745*x*y^3 - 210*x*y^2*z^4 - 1080*x*y^2*z^3 - 2274*x*y^2*z^2 - 2241*x*y^2*z - 881*x*y^2 + 84*x*y*z^5 + 540*x*y*z^4 + 1516*x*y*z^3 + 2247*x*y*z^2 + 1778*x*y*z + 599*x*y - 14*x*z^6 - 108*x*z^5 - 379*x*z^4 - 751*x*z^3 - 897*x*z^2 - 612*x*z - 171*x - y^7 + 7*y^6*z + 9*y^6 - 21*y^5*z^2 - 54*y^5*z - 38*y^5 + 35*y^4*z^3 + 135*y^4*z^2 + 190*y^4*z + 94*y^4 - 35*y^3*z^4 - 180*y^3*z^3 - 380*y^3*z^2 - 377*y^3*z - 150*y^3 + 21*y^2*z^5 + 135*y^2*z^4 + 380*y^2*z^3 + 567*y^2*z^2 + 454*y^2*z + 157*y^2 - 7*y*z^6 - 54*y*z^5 - 190*y*z^4 - 379*y*z^3 - 458*y*z^2 - 321*y*z - 92*y + z^7 + 9*z^6 + 38*z^5 + 95*z^4 + 154*z^3 + 164*z^2 + 92*z + 1, 128*x^7 + 448*x^6*y - 448*x^6*z - 576*x^6 + 672*x^5*y^2 - 1344*x^5*y*z - 1728*x^5*y + 672*x^5*z^2 + 1728*x^5*z + 1200*x^5 + 560*x^4*y^3 - 1680*x^4*y^2*z - 2160*x^4*y^2 + 1680*x^4*y*z^2 + 4320*x^4*y*z + 3008*x^4*y - 560*x^4*z^3 - 2160*x^4*z^2 - 3008*x^4*z - 1480*x^4 + 280*x^3*y^4 - 1120*x^3*y^3*z - 1440*x^3*y^3 + 1680*x^3*y^2*z^2 + 4320*x^3*y^2*z + 3016*x^3*y^2 - 1120*x^3*y*z^3 - 4320*x^3*y*z^2 - 6032*x^3*y*z - 2988*x^3*y + 280*x^3*z^4 + 1440*x^3*z^3 + 3016*x^3*z^2 + 2996*x^3*z + 1188*x^3 + 84*x^2*y^5 - 420*x^2*y^4*z - 540*x^2*y^4 + 840*x^2*y^3*z^2 + 2160*x^2*y^3*z + 1512*x^2*y^3 - 840*x^2*y^2*z^3 - 3240*x^2*y^2*z^2 - 4536*x^2*y^2*z - 2262*x^2*y^2 + 420*x^2*y*z^4 + 2160*x^2*y*z^3 + 4536*x^2*y*z^2 + 4536*x^2*y*z + 1820*x^2*y - 84*x^2*z^5 - 540*x^2*z^4 - 1512*x^2*z^3 - 2274*x^2*z^2 - 1836*x^2*z - 628*x^2 + 14*x*y^6 - 84*x*y^5*z - 108*x*y^5 + 210*x*y^4*z^2 + 540*x*y^4*z + 379*x*y^4 - 280*x*y^3*z^3 - 1080*x*y^3*z^2 - 1516*x*y^3*z - 761*x*y^3 + 210*x*y^2*z^4 + 1080*x*y^2*z^3 + 2274*x*y^2*z^2 + 2289*x*y^2*z + 929*x*y^2 - 84*x*y*z^5 - 540*x*y*z^4 - 1516*x*y*z^3 - 2295*x*y*z^2 - 1874*x*y*z - 655*x*y + 14*x*z^6 + 108*x*z^5 + 379*x*z^4 + 767*x*z^3 + 945*x*z^2 + 668*x*z + 201*x + y^7 - 7*y^6*z - 9*y^6 + 21*y^5*z^2 + 54*y^5*z + 38*y^5 - 35*y^4*z^3 - 135*y^4*z^2 - 190*y^4*z - 96*y^4 + 35*y^3*z^4 + 180*y^3*z^3 + 380*y^3*z^2 + 385*y^3*z + 158*y^3 - 21*y^2*z^5 - 135*y^2*z^4 - 380*y^2*z^3 - 579*y^2*z^2 - 478*y^2*z - 171*y^2 + 7*y*z^6 + 54*y*z^5 + 190*y*z^4 + 387*y*z^3 + 482*y*z^2 + 349*y*z + 108*y - z^7 - 9*z^6 - 38*z^5 - 97*z^4 - 162*z^3 - 178*z^2 - 110*z - 1, 128*x^7 + 448*x^6*y - 448*x^6*z - 576*x^6 + 672*x^5*y^2 - 1344*x^5*y*z - 1728*x^5*y + 672*x^5*z^2 + 1728*x^5*z + 1200*x^5 + 560*x^4*y^3 - 1680*x^4*y^2*z - 2160*x^4*y^2 + 1680*x^4*y*z^2 + 4320*x^4*y*z + 3008*x^4*y - 560*x^4*z^3 - 2160*x^4*z^2 - 3008*x^4*z - 1480*x^4 + 280*x^3*y^4 - 1120*x^3*y^3*z - 1440*x^3*y^3 + 1680*x^3*y^2*z^2 + 4320*x^3*y^2*z + 3016*x^3*y^2 - 1120*x^3*y*z^3 - 4320*x^3*y*z^2 - 6032*x^3*y*z - 2988*x^3*y + 280*x^3*z^4 + 1440*x^3*z^3 + 3016*x^3*z^2 + 2996*x^3*z + 1196*x^3 + 84*x^2*y^5 - 420*x^2*y^4*z - 540*x^2*y^4 + 840*x^2*y^3*z^2 + 2160*x^2*y^3*z + 1512*x^2*y^3 - 840*x^2*y^2*z^3 - 3240*x^2*y^2*z^2 - 4536*x^2*y^2*z - 2262*x^2*y^2 + 420*x^2*y*z^4 + 2160*x^2*y*z^3 + 4536*x^2*y*z^2 + 4536*x^2*y*z + 1832*x^2*y - 84*x^2*z^5 - 540*x^2*z^4 - 1512*x^2*z^3 - 2274*x^2*z^2 - 1848*x^2*z - 644*x^2 + 14*x*y^6 - 84*x*y^5*z - 108*x*y^5 + 210*x*y^4*z^2 + 540*x*y^4*z + 379*x*y^4 - 280*x*y^3*z^3 - 1080*x*y^3*z^2 - 1516*x*y^3*z - 761*x*y^3 + 210*x*y^2*z^4 + 1080*x*y^2*z^3 + 2274*x*y^2*z^2 + 2289*x*y^2*z + 935*x*y^2 - 84*x*y*z^5 - 540*x*y*z^4 - 1516*x*y*z^3 - 2295*x*y*z^2 - 1886*x*y*z - 671*x*y + 14*x*z^6 + 108*x*z^5 + 379*x*z^4 + 767*x*z^3 + 951*x*z^2 + 684*x*z + 214*x + y^7 - 7*y^6*z - 9*y^6 + 21*y^5*z^2 + 54*y^5*z + 38*y^5 - 35*y^4*z^3 - 135*y^4*z^2 - 190*y^4*z - 96*y^4 + 35*y^3*z^4 + 180*y^3*z^3 + 380*y^3*z^2 + 385*y^3*z + 159*y^3 - 21*y^2*z^5 - 135*y^2*z^4 - 380*y^2*z^3 - 579*y^2*z^2 - 481*y^2*z - 175*y^2 + 7*y*z^6 + 54*y*z^5 + 190*y*z^4 + 387*y*z^3 + 485*y*z^2 + 357*y*z + 115*y - z^7 - 9*z^6 - 38*z^5 - 97*z^4 - 163*z^3 - 182*z^2 - 117*z - 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -2, -1, 1 ], [ -1, -1, 1 ], [ -1, -1, 2 ] ], "source_shift": [ 1, 1, 1 ], "target_matrix": [ [ 1, 1, 2 ], [ -1, -1, -1 ], [ -1, -2, -1 ] ], "target_shift": [ -20, 28, 32 ], "inverse_recipe": { "p": 3, "q": 4, "f": "x**3 + x**2 + x", "g": "x**4 + x**2 + x", "h": "y**2 + y*z", "polynomials": { "f": [ [ [ 3, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "g": [ [ [ 4, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 2, 0 ], 1 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.053326871999161 ], "surface_features": [ 3.0302330455194055, 3.7804613328617176, 2.7403626894942437, 0.0, 3.0951607974863027, 4718 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C213", "pair": 213, "family": "counterexample", "mapping": "F(x,y,z)=(6*x^7 + 60*x^6*y - 40*x^6*z - 53*x^6 + 252*x^5*y^2 - 342*x^5*y*z - 453*x^5*y + 114*x^5*z^2 + 303*x^5*z + 213*x^5 + 576*x^4*y^3 - 1194*x^4*y^2*z - 1581*x^4*y^2 + 810*x^4*y*z^2 + 2151*x^4*y*z + 1506*x^4*y - 180*x^4*z^3 - 720*x^4*z^2 - 1014*x^4*z - 497*x^4 + 774*x^3*y^4 - 2178*x^3*y^3*z - 2883*x^3*y^3 + 2256*x^3*y^2*z^2 + 5985*x^3*y^2*z + 4167*x^3*y^2 - 1020*x^3*y*z^3 - 4074*x^3*y*z^2 - 5712*x^3*y*z - 2779*x^3*y + 170*x^3*z^4 + 910*x^3*z^3 + 1926*x^3*z^2 + 1889*x^3*z + 725*x^3 + 612*x^2*y^5 - 2190*x^2*y^4*z - 2898*x^2*y^4 + 3078*x^2*y^3*z^2 + 8157*x^2*y^3*z + 5640*x^2*y^3 - 2124*x^2*y^2*z^3 - 8469*x^2*y^2*z^2 - 11802*x^2*y^2*z - 5685*x^2*y^2 + 720*x^2*y*z^4 + 3846*x^2*y*z^3 + 8100*x^2*y*z^2 + 7881*x^2*y*z + 2988*x^2*y - 96*x^2*z^5 - 645*x^2*z^4 - 1824*x^2*z^3 - 2685*x^2*z^2 - 2058*x^2*z - 659*x^2 + 264*x*y^6 - 1152*x*y^5*z - 1524*x*y^5 + 2058*x*y^4*z^2 + 5448*x*y^4*z + 3738*x*y^4 - 1926*x*y^3*z^3 - 7665*x*y^3*z^2 - 10602*x*y^3*z - 5045*x*y^3 + 996*x*y^2*z^4 + 5307*x*y^2*z^3 + 11103*x*y^2*z^2 + 10689*x*y^2*z + 3987*x*y^2 - 270*x*y*z^5 - 1809*x*y*z^4 - 5088*x*y*z^3 - 7425*x*y*z^2 - 5616*x*y*z - 1765*x*y + 30*x*z^6 + 243*x*z^5 + 861*x*z^4 + 1691*x*z^3 + 1941*x*z^2 + 1237*x*z + 345*x + 48*y^7 - 248*y^6*z - 328*y^6 + 540*y^5*z^2 + 1428*y^5*z + 972*y^5 - 642*y^4*z^3 - 2550*y^4*z^2 - 3498*y^4*z - 1642*y^4 + 450*y^3*z^4 + 2391*y^3*z^3 + 4962*y^3*z^2 + 4717*y^3*z + 1728*y^3 - 186*y^2*z^5 - 1242*y^2*z^4 - 3468*y^2*z^3 - 5004*y^2*z^2 - 3720*y^2*z - 1142*y^2 + 42*y*z^6 + 339*y*z^5 + 1194*y*z^4 + 2323*y*z^3 + 2628*y*z^2 + 1639*y*z + 442*y - 4*z^7 - 38*z^6 - 162*z^5 - 398*z^4 - 608*z^3 - 578*z^2 - 318*z - 1, 3*x^7 + 30*x^6*y - 20*x^6*z - 40*x^6 + 126*x^5*y^2 - 171*x^5*y*z - 348*x^5*y + 57*x^5*z^2 + 228*x^5*z + 192*x^5 + 288*x^4*y^3 - 597*x^4*y^2*z - 1236*x^4*y^2 + 405*x^4*y*z^2 + 1647*x^4*y*z + 1383*x^4*y - 90*x^4*z^3 - 540*x^4*z^2 - 912*x^4*z - 499*x^4 + 387*x^3*y^4 - 1089*x^3*y^3*z - 2292*x^3*y^3 + 1128*x^3*y^2*z^2 + 4662*x^3*y^2*z + 3897*x^3*y^2 - 510*x^3*y*z^3 - 3108*x^3*y*z^2 - 5232*x^3*y*z - 2846*x^3*y + 85*x^3*z^4 + 680*x^3*z^3 + 1728*x^3*z^2 + 1893*x^3*z + 779*x^3 + 306*x^2*y^5 - 1095*x^2*y^4*z - 2340*x^2*y^4 + 1539*x^2*y^3*z^2 + 6459*x^2*y^3*z + 5367*x^2*y^3 - 1062*x^2*y^2*z^3 - 6570*x^2*y^2*z^2 - 11004*x^2*y^2*z - 5934*x^2*y^2 + 360*x^2*y*z^4 + 2922*x^2*y*z^3 + 7398*x^2*y*z^2 + 8052*x^2*y*z + 3279*x^2*y - 48*x^2*z^5 - 480*x^2*z^4 - 1632*x^2*z^3 - 2685*x^2*z^2 - 2208*x^2*z - 742*x^2 + 132*x*y^6 - 576*x*y^5*z - 1248*x*y^5 + 1029*x*y^4*z^2 + 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321*y^4*z^3 + 1530*y^4*z^2 + 2289*y^4*z + 1130*y^4 - 225*y^3*z^4 - 1422*y^3*z^3 - 3216*y^3*z^2 - 3219*y^3*z - 1214*y^3 + 93*y^2*z^5 + 732*y^2*z^4 + 2226*y^2*z^3 + 3384*y^2*z^2 + 2592*y^2*z + 811*y^2 - 21*y*z^6 - 198*y*z^5 - 759*y*z^4 - 1556*y*z^3 - 1815*y*z^2 - 1154*y*z - 315*y + 2*z^7 + 22*z^6 + 102*z^5 + 264*z^4 + 416*z^3 + 403*z^2 + 224*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -1, -2, 1 ], [ -1, -1, 1 ], [ -3, -3, 2 ] ], "source_shift": [ 1, 1, 1 ], "target_matrix": [ [ -2, -1, -1 ], [ -1, -2, -1 ], [ 1, 1, 1 ] ], "target_shift": [ 75, 87, -53 ], "collision_preimages": [ [ "-3/4", "0", "-7/4" ], [ "-8", "-5/2", "-13" ], [ "-7", "5/2", "-4" ] ], "collision_image": [ "151/2", "349/4", "-213/4" ] }, "probe_target": "No", "jev_answers": [ "Maybe" ], "seconds": [ 1.0276491759977944 ], "surface_features": [ 3.2977289072161646, 4.071955610302981, 3.0394141191761372, 0.0, 3.3723670528037317, 4836 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B179", "pair": 179, "family": "bijection", "mapping": "F(x,y,z)=(-12*x^7 - 84*x^6*y - 84*x^6*z - 168*x^6 - 252*x^5*y^2 - 504*x^5*y*z - 984*x^5*y - 252*x^5*z^2 - 984*x^5*z - 948*x^5 - 420*x^4*y^3 - 1260*x^4*y^2*z - 2400*x^4*y^2 - 1260*x^4*y*z^2 - 4800*x^4*y*z - 4524*x^4*y - 420*x^4*z^3 - 2400*x^4*z^2 - 4536*x^4*z - 2832*x^4 - 420*x^3*y^4 - 1680*x^3*y^3*z - 3120*x^3*y^3 - 2520*x^3*y^2*z^2 - 9360*x^3*y^2*z - 8628*x^3*y^2 - 1680*x^3*y*z^3 - 9360*x^3*y*z^2 - 17304*x^3*y*z - 10608*x^3*y - 420*x^3*z^4 - 3120*x^3*z^3 - 8676*x^3*z^2 - 10728*x^3*z - 4970*x^3 - 252*x^2*y^5 - 1260*x^2*y^4*z - 2280*x^2*y^4 - 2520*x^2*y^3*z^2 - 9120*x^2*y^3*z - 8220*x^2*y^3 - 2520*x^2*y^2*z^3 - 13680*x^2*y^2*z^2 - 24732*x^2*y^2*z - 14880*x^2*y^2 - 1260*x^2*y*z^4 - 9120*x^2*y*z^3 - 24804*x^2*y*z^2 - 30096*x^2*y*z - 13710*x^2*y - 252*x^2*z^5 - 2280*x^2*z^4 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140*x^4*y^3 - 420*x^4*y^2*z - 800*x^4*y^2 - 420*x^4*y*z^2 - 1600*x^4*y*z - 1508*x^4*y - 140*x^4*z^3 - 800*x^4*z^2 - 1512*x^4*z - 944*x^4 - 140*x^3*y^4 - 560*x^3*y^3*z - 1040*x^3*y^3 - 840*x^3*y^2*z^2 - 3120*x^3*y^2*z - 2876*x^3*y^2 - 560*x^3*y*z^3 - 3120*x^3*y*z^2 - 5768*x^3*y*z - 3536*x^3*y - 140*x^3*z^4 - 1040*x^3*z^3 - 2892*x^3*z^2 - 3576*x^3*z - 1657*x^3 - 84*x^2*y^5 - 420*x^2*y^4*z - 760*x^2*y^4 - 840*x^2*y^3*z^2 - 3040*x^2*y^3*z - 2740*x^2*y^3 - 840*x^2*y^2*z^3 - 4560*x^2*y^2*z^2 - 8244*x^2*y^2*z - 4960*x^2*y^2 - 420*x^2*y*z^4 - 3040*x^2*y*z^3 - 8268*x^2*y*z^2 - 10032*x^2*y*z - 4571*x^2*y - 84*x^2*z^5 - 760*x^2*z^4 - 2764*x^2*z^3 - 5072*x^2*z^2 - 4703*x^2*z - 1746*x^2 - 28*x*y^6 - 168*x*y^5*z - 296*x*y^5 - 420*x*y^4*z^2 - 1480*x*y^4*z - 1304*x*y^4 - 560*x*y^3*z^3 - 2960*x*y^3*z^2 - 5232*x*y^3*z - 3088*x*y^3 - 420*x*y^2*z^4 - 2960*x*y^2*z^3 - 7872*x*y^2*z^2 - 9368*x*y^2*z - 4195*x*y^2 - 168*x*y*z^5 - 1480*x*y*z^4 - 5264*x*y*z^3 - 9472*x*y*z^2 - 8614*x*y*z - 3140*x*y - 28*x*z^6 - 296*x*z^5 - 1320*x*z^4 - 3192*x*z^3 - 4419*x*z^2 - 3300*x*z - 1021*x - 4*y^7 - 28*y^6*z - 48*y^6 - 84*y^5*z^2 - 288*y^5*z - 248*y^5 - 140*y^4*z^3 - 720*y^4*z^2 - 1244*y^4*z - 720*y^4 - 140*y^3*z^4 - 960*y^3*z^3 - 2496*y^3*z^2 - 2912*y^3*z - 1281*y^3 - 84*y^2*z^5 - 720*y^2*z^4 - 2504*y^2*z^3 - 4416*y^2*z^2 - 3939*y^2*z - 1410*y^2 - 28*y*z^6 - 288*y*z^5 - 1256*y*z^4 - 2976*y*z^3 - 4035*y*z^2 - 2948*y*z - 895*y - 4*z^7 - 48*z^6 - 252*z^5 - 752*z^4 - 1377*z^3 - 1538*z^2 - 960*z - 1, 4*x^7 + 28*x^6*y + 28*x^6*z + 56*x^6 + 84*x^5*y^2 + 168*x^5*y*z + 328*x^5*y + 84*x^5*z^2 + 328*x^5*z + 316*x^5 + 140*x^4*y^3 + 420*x^4*y^2*z + 800*x^4*y^2 + 420*x^4*y*z^2 + 1600*x^4*y*z + 1508*x^4*y + 140*x^4*z^3 + 800*x^4*z^2 + 1512*x^4*z + 944*x^4 + 140*x^3*y^4 + 560*x^3*y^3*z + 1040*x^3*y^3 + 840*x^3*y^2*z^2 + 3120*x^3*y^2*z + 2876*x^3*y^2 + 560*x^3*y*z^3 + 3120*x^3*y*z^2 + 5768*x^3*y*z + 3536*x^3*y + 140*x^3*z^4 + 1040*x^3*z^3 + 2892*x^3*z^2 + 3576*x^3*z + 1657*x^3 + 84*x^2*y^5 + 420*x^2*y^4*z + 760*x^2*y^4 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887*y + 4*z^7 + 48*z^6 + 252*z^5 + 752*z^4 + 1377*z^3 + 1536*z^2 + 952*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ 1, 1, 1 ], [ 1, 1, 2 ], [ 2, 1, 1 ] ], "source_shift": [ 1, 2, 2 ], "target_matrix": [ [ -3, -2, -3 ], [ -1, -1, -1 ], [ 1, 1, 2 ] ], "target_shift": [ 754, 252, -244 ], "inverse_recipe": { "p": 3, "q": 2, "f": "x**3 + x**2", "g": "x**2 + x", "h": "y*z**2", "polynomials": { "f": [ [ [ 3, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "g": [ [ [ 2, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 2 ], 1 ] ] }, "h_multiplier": 4 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.014002438998432 ], "surface_features": [ 3.5087249392735016, 4.4785087782063036, 3.12057393120585, 0.0, 3.6773810293149496, 4856 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C163", "pair": 163, "family": "counterexample", "mapping": "F(x,y,z)=(8*x^7 + 44*x^6*y + 68*x^6*z + 136*x^6 + 102*x^5*y^2 + 312*x^5*y*z + 600*x^5*y + 234*x^5*z^2 + 912*x^5*z + 774*x^5 + 129*x^4*y^3 + 585*x^4*y^2*z + 1086*x^4*y^2 + 867*x^4*y*z^2 + 3252*x^4*y*z + 2733*x^4*y + 419*x^4*z^3 + 2382*x^4*z^2 + 4059*x^4*z + 2175*x^4 + 96*x^3*y^4 + 573*x^3*y^3*z + 1032*x^3*y^3 + 1257*x^3*y^2*z^2 + 4566*x^3*y^2*z + 3801*x^3*y^2 + 1199*x^3*y*z^3 + 6588*x^3*y*z^2 + 11103*x^3*y*z + 5928*x^3*y + 419*x^3*z^4 + 3094*x^3*z^3 + 7920*x^3*z^2 + 8549*x^3*z + 3363*x^3 + 42*x^2*y^5 + 309*x^2*y^4*z + 543*x^2*y^4 + 891*x^2*y^3*z^2 + 3156*x^2*y^3*z + 2604*x^2*y^3 + 1257*x^2*y^2*z^3 + 6732*x^2*y^2*z^2 + 11229*x^2*y^2*z + 5967*x^2*y^2 + 867*x^2*y*z^4 + 6240*x^2*y*z^3 + 15777*x^2*y*z^2 + 16911*x^2*y*z + 6630*x^2*y + 234*x^2*z^5 + 2121*x^2*z^4 + 7218*x^2*z^3 + 11685*x^2*z^2 + 9216*x^2*z + 2887*x^2 + 10*x*y^6 + 87*x*y^5*z + 150*x*y^5 + 309*x*y^4*z^2 + 1074*x*y^4*z + 879*x*y^4 + 573*x*y^3*z^3 + 3012*x*y^3*z^2 + 4977*x*y^3*z + 2631*x*y^3 + 585*x*y^2*z^4 + 4134*x*y^2*z^3 + 10338*x*y^2*z^2 + 11004*x*y^2*z + 4296*x*y^2 + 312*x*y*z^5 + 2778*x*y*z^4 + 9336*x*y*z^3 + 14985*x*y*z^2 + 11745*x*y*z + 3661*x*y + 68*x*z^6 + 732*x*z^5 + 3096*x*z^4 + 6651*x*z^3 + 7830*x*z^2 + 4879*x*z + 1276*x + y^7 + 10*y^6*z + 17*y^6 + 42*y^5*z^2 + 144*y^5*z + 117*y^5 + 96*y^4*z^3 + 498*y^4*z^2 + 816*y^4*z + 429*y^4 + 129*y^3*z^4 + 900*y^3*z^3 + 2229*y^3*z^2 + 2356*y^3*z + 915*y^3 + 102*y^2*z^5 + 897*y^2*z^4 + 2982*y^2*z^3 + 4746*y^2*z^2 + 3693*y^2*z + 1144*y^2 + 44*y*z^6 + 468*y*z^5 + 1956*y*z^4 + 4161*y*z^3 + 4854*y*z^2 + 2999*y*z + 781*y + 8*z^7 + 100*z^6 + 504*z^5 + 1342*z^4 + 2082*z^3 + 1918*z^2 + 994*z + 1, -8*x^7 - 44*x^6*y - 68*x^6*z - 136*x^6 - 102*x^5*y^2 - 312*x^5*y*z - 600*x^5*y - 234*x^5*z^2 - 912*x^5*z - 774*x^5 - 129*x^4*y^3 - 585*x^4*y^2*z - 1086*x^4*y^2 - 867*x^4*y*z^2 - 3252*x^4*y*z - 2733*x^4*y - 419*x^4*z^3 - 2382*x^4*z^2 - 4059*x^4*z - 2174*x^4 - 96*x^3*y^4 - 573*x^3*y^3*z - 1032*x^3*y^3 - 1257*x^3*y^2*z^2 - 4566*x^3*y^2*z - 3801*x^3*y^2 - 1199*x^3*y*z^3 - 6588*x^3*y*z^2 - 11103*x^3*y*z - 5924*x^3*y - 419*x^3*z^4 - 3094*x^3*z^3 - 7920*x^3*z^2 - 8542*x^3*z - 3349*x^3 - 42*x^2*y^5 - 309*x^2*y^4*z - 543*x^2*y^4 - 891*x^2*y^3*z^2 - 3156*x^2*y^3*z - 2604*x^2*y^3 - 1257*x^2*y^2*z^3 - 6732*x^2*y^2*z^2 - 11229*x^2*y^2*z - 5961*x^2*y^2 - 867*x^2*y*z^4 - 6240*x^2*y*z^3 - 15777*x^2*y*z^2 - 16890*x^2*y*z - 6591*x^2*y - 234*x^2*z^5 - 2121*x^2*z^4 - 7218*x^2*z^3 - 11667*x^2*z^2 - 9147*x^2*z - 2836*x^2 - 10*x*y^6 - 87*x*y^5*z - 150*x*y^5 - 309*x*y^4*z^2 - 1074*x*y^4*z - 879*x*y^4 - 573*x*y^3*z^3 - 3012*x*y^3*z^2 - 4977*x*y^3*z - 2627*x*y^3 - 585*x*y^2*z^4 - 4134*x*y^2*z^3 - 10338*x*y^2*z^2 - 10983*x*y^2*z - 4260*x*y^2 - 312*x*y*z^5 - 2778*x*y*z^4 - 9336*x*y*z^3 - 14949*x*y*z^2 - 11619*x*y*z - 3571*x*y - 68*x*z^6 - 732*x*z^5 - 3096*x*z^4 - 6631*x*z^3 - 7722*x*z^2 - 4723*x*z - 1210*x - y^7 - 10*y^6*z - 17*y^6 - 42*y^5*z^2 - 144*y^5*z - 117*y^5 - 96*y^4*z^3 - 498*y^4*z^2 - 816*y^4*z - 428*y^4 - 129*y^3*z^4 - 900*y^3*z^3 - 2229*y^3*z^2 - 2349*y^3*z - 904*y^3 - 102*y^2*z^5 - 897*y^2*z^4 - 2982*y^2*z^3 - 4728*y^2*z^2 - 3636*y^2*z - 1105*y^2 - 44*y*z^6 - 468*y*z^5 - 1956*y*z^4 - 4141*y*z^3 - 4758*y*z^2 - 2867*y*z - 727*y - 8*z^7 - 100*z^6 - 504*z^5 - 1334*z^4 - 2030*z^3 - 1810*z^2 - 906*z - 1, -16*x^7 - 88*x^6*y - 136*x^6*z - 260*x^6 - 204*x^5*y^2 - 624*x^5*y*z - 1140*x^5*y - 468*x^5*z^2 - 1728*x^5*z - 1368*x^5 - 258*x^4*y^3 - 1170*x^4*y^2*z - 2049*x^4*y^2 - 1734*x^4*y*z^2 - 6114*x^4*y*z - 4776*x^4*y - 838*x^4*z^3 - 4461*x^4*z^2 - 7032*x^4*z - 3556*x^4 - 192*x^3*y^4 - 1146*x^3*y^3*z - 1932*x^3*y^3 - 2514*x^3*y^2*z^2 - 8511*x^3*y^2*z - 6564*x^3*y^2 - 2398*x^3*y*z^3 - 12222*x^3*y*z^2 - 18996*x^3*y*z - 9562*x^3*y - 838*x^3*z^4 - 5711*x^3*z^3 - 13416*x^3*z^2 - 13622*x^3*z - 5144*x^3 - 84*x^2*y^5 - 618*x^2*y^4*z - 1008*x^2*y^4 - 1782*x^2*y^3*z^2 - 5829*x^2*y^3*z - 4443*x^2*y^3 - 2514*x^2*y^2*z^3 - 12366*x^2*y^2*z^2 - 18975*x^2*y^2*z - 9501*x^2*y^2 - 1734*x^2*y*z^4 - 11397*x^2*y*z^3 - 26397*x^2*y*z^2 - 26610*x^2*y*z - 10014*x^2*y - 468*x^2*z^5 - 3852*x^2*z^4 - 11961*x^2*z^3 - 18177*x^2*z^2 - 13746*x^2*z - 4193*x^2 - 20*x*y^6 - 174*x*y^5*z - 276*x*y^5 - 618*x*y^4*z^2 - 1965*x*y^4*z - 1482*x*y^4 - 1146*x*y^3*z^3 - 5478*x*y^3*z^2 - 8310*x*y^3*z - 4138*x*y^3 - 1170*x*y^2*z^4 - 7473*x*y^2*z^3 - 17094*x*y^2*z^2 - 17112*x*y^2*z - 6414*x*y^2 - 624*x*y*z^5 - 4992*x*y*z^4 - 15294*x*y*z^3 - 23052*x*y*z^2 - 17334*x*y*z - 5264*x*y - 136*x*z^6 - 1308*x*z^5 - 5028*x*z^4 - 10130*x*z^3 - 11436*x*z^2 - 6941*x*z - 1788*x - 2*y^7 - 20*y^6*z - 31*y^6 - 84*y^5*z^2 - 261*y^5*z - 195*y^5 - 192*y^4*z^3 - 897*y^4*z^2 - 1347*y^4*z - 667*y^4 - 258*y^3*z^4 - 1611*y^3*z^3 - 3645*y^3*z^2 - 3624*y^3*z - 1352*y^3 - 204*y^2*z^5 - 1596*y^2*z^4 - 4833*y^2*z^3 - 7227*y^2*z^2 - 5400*y^2*z - 1631*y^2 - 88*y*z^6 - 828*y*z^5 - 3144*y*z^4 - 6278*y*z^3 - 7032*y*z^2 - 4237*y*z - 1087*y - 16*z^7 - 176*z^6 - 804*z^5 - 2008*z^4 - 2992*z^3 - 2690*z^2 - 1373*z - 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ 1, 1, 2 ], [ 2, 1, 1 ], [ 1, 1, 1 ] ], "source_shift": [ 2, 1, 2 ], "target_matrix": [ [ 1, 1, 1 ], [ -1, -1, -2 ], [ -2, -1, -1 ] ], "target_shift": [ -228, 204, 312 ], "collision_preimages": [ [ "5/4", "-15/4", "1/4" ], [ "-7", "17", "-11/2" ], [ "-4", "16", "-15/2" ] ], "collision_image": [ "-913/4", "817/4", "625/2" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0038059830003476 ], "surface_features": [ 3.5462783461247356, 4.425044874551388, 3.187238619831479, 0.0, 3.6629230589171926, 4880 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B245", "pair": 245, "family": "bijection", "mapping": "F(x,y,z)=(-2*x^7 - 14*x^6*y + 14*x^6*z + 32*x^6 - 42*x^5*y^2 + 84*x^5*y*z + 192*x^5*y - 42*x^5*z^2 - 196*x^5*z - 218*x^5 - 70*x^4*y^3 + 210*x^4*y^2*z + 480*x^4*y^2 - 210*x^4*y*z^2 - 980*x^4*y*z - 1088*x^4*y + 70*x^4*z^3 + 500*x^4*z^2 + 1132*x^4*z + 818*x^4 - 70*x^3*y^4 + 280*x^3*y^3*z + 640*x^3*y^3 - 420*x^3*y^2*z^2 - 1960*x^3*y^2*z - 2172*x^3*y^2 + 280*x^3*y*z^3 + 2000*x^3*y*z^2 + 4520*x^3*y*z + 3252*x^3*y - 70*x^3*z^4 - 680*x^3*z^3 - 2350*x^3*z^2 - 3448*x^3*z - 1808*x^3 - 42*x^2*y^5 + 210*x^2*y^4*z + 480*x^2*y^4 - 420*x^2*y^3*z^2 - 1960*x^2*y^3*z - 2168*x^2*y^3 + 420*x^2*y^2*z^3 + 3000*x^2*y^2*z^2 + 6768*x^2*y^2*z + 4848*x^2*y^2 - 210*x^2*y*z^4 - 2040*x^2*y*z^3 - 7038*x^2*y*z^2 - 10280*x^2*y*z - 5354*x^2*y + 42*x^2*z^5 + 520*x^2*z^4 + 2438*x^2*z^3 + 5444*x^2*z^2 + 5790*x^2*z + 2320*x^2 - 14*x*y^6 + 84*x*y^5*z + 192*x*y^5 - 210*x*y^4*z^2 - 980*x*y^4*z - 1082*x*y^4 + 280*x*y^3*z^3 + 2000*x*y^3*z^2 + 4504*x*y^3*z + 3212*x*y^3 - 210*x*y^2*z^4 - 2040*x*y^2*z^3 - 7026*x*y^2*z^2 - 10216*x*y^2*z - 5284*x*y^2 + 84*x*y*z^5 + 1040*x*y*z^4 + 4868*x*y*z^3 + 10820*x*y*z^2 + 11420*x*y*z + 4540*x*y - 14*x*z^6 - 212*x*z^5 - 1264*x*z^4 - 3816*x*z^3 - 6164*x*z^2 - 5000*x*z - 1583*x - 2*y^7 + 14*y^6*z + 32*y^6 - 42*y^5*z^2 - 196*y^5*z - 216*y^5 + 70*y^4*z^3 + 500*y^4*z^2 + 1124*y^4*z + 798*y^4 - 70*y^3*z^4 - 680*y^3*z^3 - 2338*y^3*z^2 - 3384*y^3*z - 1738*y^3 + 42*y^2*z^5 + 520*y^2*z^4 + 2430*y^2*z^3 + 5376*y^2*z^2 + 5630*y^2*z + 2220*y^2 - 14*y*z^6 - 212*y*z^5 - 1262*y*z^4 - 3792*y*z^3 - 6072*y*z^2 - 4880*y*z - 1535*y + 2*z^7 + 36*z^6 + 262*z^5 + 1002*z^4 + 2180*z^3 + 2678*z^2 + 1717*z + 1, -2*x^7 - 14*x^6*y + 14*x^6*z + 32*x^6 - 42*x^5*y^2 + 84*x^5*y*z + 192*x^5*y - 42*x^5*z^2 - 196*x^5*z - 218*x^5 - 70*x^4*y^3 + 210*x^4*y^2*z + 480*x^4*y^2 - 210*x^4*y*z^2 - 980*x^4*y*z - 1088*x^4*y + 70*x^4*z^3 + 500*x^4*z^2 + 1132*x^4*z + 818*x^4 - 70*x^3*y^4 + 280*x^3*y^3*z + 640*x^3*y^3 - 420*x^3*y^2*z^2 - 1960*x^3*y^2*z - 2172*x^3*y^2 + 280*x^3*y*z^3 + 2000*x^3*y*z^2 + 4520*x^3*y*z + 3252*x^3*y - 70*x^3*z^4 - 680*x^3*z^3 - 2350*x^3*z^2 - 3448*x^3*z - 1808*x^3 - 42*x^2*y^5 + 210*x^2*y^4*z + 480*x^2*y^4 - 420*x^2*y^3*z^2 - 1960*x^2*y^3*z - 2168*x^2*y^3 + 420*x^2*y^2*z^3 + 3000*x^2*y^2*z^2 + 6768*x^2*y^2*z + 4848*x^2*y^2 - 210*x^2*y*z^4 - 2040*x^2*y*z^3 - 7038*x^2*y*z^2 - 10280*x^2*y*z - 5354*x^2*y + 42*x^2*z^5 + 520*x^2*z^4 + 2438*x^2*z^3 + 5444*x^2*z^2 + 5790*x^2*z + 2319*x^2 - 14*x*y^6 + 84*x*y^5*z + 192*x*y^5 - 210*x*y^4*z^2 - 980*x*y^4*z - 1082*x*y^4 + 280*x*y^3*z^3 + 2000*x*y^3*z^2 + 4504*x*y^3*z + 3212*x*y^3 - 210*x*y^2*z^4 - 2040*x*y^2*z^3 - 7026*x*y^2*z^2 - 10216*x*y^2*z - 5284*x*y^2 + 84*x*y*z^5 + 1040*x*y*z^4 + 4868*x*y*z^3 + 10820*x*y*z^2 + 11420*x*y*z + 4538*x*y - 14*x*z^6 - 212*x*z^5 - 1264*x*z^4 - 3816*x*z^3 - 6164*x*z^2 - 4998*x*z - 1578*x - 2*y^7 + 14*y^6*z + 32*y^6 - 42*y^5*z^2 - 196*y^5*z - 216*y^5 + 70*y^4*z^3 + 500*y^4*z^2 + 1124*y^4*z + 798*y^4 - 70*y^3*z^4 - 680*y^3*z^3 - 2338*y^3*z^2 - 3384*y^3*z - 1738*y^3 + 42*y^2*z^5 + 520*y^2*z^4 + 2430*y^2*z^3 + 5376*y^2*z^2 + 5630*y^2*z + 2219*y^2 - 14*y*z^6 - 212*y*z^5 - 1262*y*z^4 - 3792*y*z^3 - 6072*y*z^2 - 4878*y*z - 1530*y + 2*z^7 + 36*z^6 + 262*z^5 + 1002*z^4 + 2180*z^3 + 2677*z^2 + 1711*z + 1, -6*x^7 - 42*x^6*y + 42*x^6*z + 96*x^6 - 126*x^5*y^2 + 252*x^5*y*z + 576*x^5*y - 126*x^5*z^2 - 588*x^5*z - 654*x^5 - 210*x^4*y^3 + 630*x^4*y^2*z + 1440*x^4*y^2 - 630*x^4*y*z^2 - 2940*x^4*y*z - 3264*x^4*y + 210*x^4*z^3 + 1500*x^4*z^2 + 3396*x^4*z + 2454*x^4 - 210*x^3*y^4 + 840*x^3*y^3*z + 1920*x^3*y^3 - 1260*x^3*y^2*z^2 - 5880*x^3*y^2*z - 6516*x^3*y^2 + 840*x^3*y*z^3 + 6000*x^3*y*z^2 + 13560*x^3*y*z + 9756*x^3*y - 210*x^3*z^4 - 2040*x^3*z^3 - 7050*x^3*z^2 - 10344*x^3*z - 5426*x^3 - 126*x^2*y^5 + 630*x^2*y^4*z + 1440*x^2*y^4 - 1260*x^2*y^3*z^2 - 5880*x^2*y^3*z - 6504*x^2*y^3 + 1260*x^2*y^2*z^3 + 9000*x^2*y^2*z^2 + 20304*x^2*y^2*z + 14544*x^2*y^2 - 630*x^2*y*z^4 - 6120*x^2*y*z^3 - 21114*x^2*y*z^2 - 30840*x^2*y*z - 16068*x^2*y + 126*x^2*z^5 + 1560*x^2*z^4 + 7314*x^2*z^3 + 16332*x^2*z^2 + 17376*x^2*z + 6969*x^2 - 42*x*y^6 + 252*x*y^5*z + 576*x*y^5 - 630*x*y^4*z^2 - 2940*x*y^4*z - 3246*x*y^4 + 840*x*y^3*z^3 + 6000*x*y^3*z^2 + 13512*x*y^3*z + 9636*x*y^3 - 630*x*y^2*z^4 - 6120*x*y^2*z^3 - 21078*x*y^2*z^2 - 30648*x*y^2*z - 15858*x*y^2 + 252*x*y*z^5 + 3120*x*y*z^4 + 14604*x*y*z^3 + 32460*x*y*z^2 + 34272*x*y*z + 13638*x*y - 42*x*z^6 - 636*x*z^5 - 3792*x*z^4 - 11448*x*z^3 - 18498*x*z^2 - 15018*x*z - 4762*x - 6*y^7 + 42*y^6*z + 96*y^6 - 126*y^5*z^2 - 588*y^5*z - 648*y^5 + 210*y^4*z^3 + 1500*y^4*z^2 + 3372*y^4*z + 2394*y^4 - 210*y^3*z^4 - 2040*y^3*z^3 - 7014*y^3*z^2 - 10152*y^3*z - 5216*y^3 + 126*y^2*z^5 + 1560*y^2*z^4 + 7290*y^2*z^3 + 16128*y^2*z^2 + 16896*y^2*z + 6669*y^2 - 42*y*z^6 - 636*y*z^5 - 3786*y*z^4 - 11376*y*z^3 - 18222*y*z^2 - 14658*y*z - 4616*y + 6*z^7 + 108*z^6 + 786*z^5 + 3006*z^4 + 6542*z^3 + 8043*z^2 + 5159*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -1, -1, 1 ], [ -1, -1, 2 ], [ -2, -1, 1 ] ], "source_shift": [ 2, 1, 1 ], "target_matrix": [ [ 1, 2, 1 ], [ 1, 1, 1 ], [ 3, 3, 2 ] ], "target_shift": [ -443, -438, -1334 ], "inverse_recipe": { "p": 2, "q": 3, "f": "x**2", "g": "x**3", "h": "y**2*z", "polynomials": { "f": [ [ [ 2, 0, 0 ], 1 ] ], "g": [ [ [ 3, 0, 0 ], 1 ] ], "h": [ [ [ 0, 2, 1 ], 1 ] ] }, "h_multiplier": 2 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0808068029982678 ], "surface_features": [ 3.510749747575177, 4.534939449151762, 3.0523090996473234, 0.0, 3.7108365750877494, 4833 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B046", "pair": 46, "family": "bijection", "mapping": "F(x,y,z)=(-3*x^7 - 21*x^6*y + 21*x^6*z + 48*x^6 - 63*x^5*y^2 + 126*x^5*y*z + 291*x^5*y - 63*x^5*z^2 - 288*x^5*z - 326*x^5 - 105*x^4*y^3 + 315*x^4*y^2*z + 735*x^4*y^2 - 315*x^4*y*z^2 - 1455*x^4*y*z - 1663*x^4*y + 105*x^4*z^3 + 720*x^4*z^2 + 1630*x^4*z + 1222*x^4 - 105*x^3*y^4 + 420*x^3*y^3*z + 990*x^3*y^3 - 630*x^3*y^2*z^2 - 2940*x^3*y^2*z - 3392*x^3*y^2 + 420*x^3*y*z^3 + 2910*x^3*y*z^2 + 6652*x^3*y*z + 5035*x^3*y - 105*x^3*z^4 - 960*x^3*z^3 - 3260*x^3*z^2 - 4888*x^3*z - 2738*x^3 - 63*x^2*y^5 + 315*x^2*y^4*z + 750*x^2*y^4 - 630*x^2*y^3*z^2 - 2970*x^2*y^3*z - 3458*x^2*y^3 + 630*x^2*y^2*z^3 + 4410*x^2*y^2*z^2 + 10176*x^2*y^2*z + 7773*x^2*y^2 - 315*x^2*y*z^4 - 2910*x^2*y*z^3 - 9978*x^2*y*z^2 - 15105*x^2*y*z - 8547*x^2*y + 63*x^2*z^5 + 720*x^2*z^4 + 3260*x^2*z^3 + 7332*x^2*z^2 + 8217*x^2*z + 3673*x^2 - 21*x*y^6 + 126*x*y^5*z + 303*x*y^5 - 315*x*y^4*z^2 - 1500*x*y^4*z - 1762*x*y^4 + 420*x*y^3*z^3 + 2970*x*y^3*z^2 + 6916*x*y^3*z + 5329*x*y^3 - 315*x*y^2*z^4 - 2940*x*y^2*z^3 - 10176*x*y^2*z^2 - 15546*x*y^2*z - 8880*x*y^2 + 126*x*y*z^5 + 1455*x*y*z^4 + 6652*x*y*z^3 + 15105*x*y*z^2 + 17100*x*y*z + 7733*x*y - 21*x*z^6 - 288*x*z^5 - 1630*x*z^4 - 4888*x*z^3 - 8220*x*z^2 - 7361*x*z - 2727*x - 3*y^7 + 21*y^6*z + 51*y^6 - 63*y^5*z^2 - 303*y^5*z - 359*y^5 + 105*y^4*z^3 + 750*y^4*z^2 + 1762*y^4*z + 1369*y^4 - 105*y^3*z^4 - 990*y^3*z^3 - 3458*y^3*z^2 - 5329*y^3*z - 3071*y^3 + 63*y^2*z^5 + 735*y^2*z^4 + 3392*y^2*z^3 + 7773*y^2*z^2 + 8883*y^2*z + 4060*y^2 - 21*y*z^6 - 291*y*z^5 - 1663*y*z^4 - 5035*y*z^3 - 8553*y*z^2 - 7751*y*z - 2911*y + 3*z^7 + 48*z^6 + 326*z^5 + 1222*z^4 + 2741*z^3 + 3688*z^2 + 2741*z + 1, 6*x^7 + 42*x^6*y - 42*x^6*z - 96*x^6 + 126*x^5*y^2 - 252*x^5*y*z - 582*x^5*y + 126*x^5*z^2 + 576*x^5*z + 658*x^5 + 210*x^4*y^3 - 630*x^4*y^2*z - 1470*x^4*y^2 + 630*x^4*y*z^2 + 2910*x^4*y*z + 3356*x^4*y - 210*x^4*z^3 - 1440*x^4*z^2 - 3290*x^4*z - 2510*x^4 + 210*x^3*y^4 - 840*x^3*y^3*z - 1980*x^3*y^3 + 1260*x^3*y^2*z^2 + 5880*x^3*y^2*z + 6844*x^3*y^2 - 840*x^3*y*z^3 - 5820*x^3*y*z^2 - 13424*x^3*y*z - 10334*x^3*y + 210*x^3*z^4 + 1920*x^3*z^3 + 6580*x^3*z^2 + 10040*x^3*z + 5770*x^3 + 126*x^2*y^5 - 630*x^2*y^4*z - 1500*x^2*y^4 + 1260*x^2*y^3*z^2 + 5940*x^2*y^3*z + 6976*x^2*y^3 - 1260*x^2*y^2*z^3 - 8820*x^2*y^2*z^2 - 20532*x^2*y^2*z - 15942*x^2*y^2 + 630*x^2*y*z^4 + 5820*x^2*y*z^3 + 20136*x^2*y*z^2 + 31002*x^2*y*z + 17976*x^2*y - 126*x^2*z^5 - 1440*x^2*z^4 - 6580*x^2*z^3 - 15060*x^2*z^2 - 17316*x^2*z - 8011*x^2 + 42*x*y^6 - 252*x*y^5*z - 606*x*y^5 + 630*x*y^4*z^2 + 3000*x*y^4*z + 3554*x*y^4 - 840*x*y^3*z^3 - 5940*x*y^3*z^2 - 13952*x*y^3*z - 10922*x*y^3 + 630*x*y^2*z^4 + 5880*x*y^2*z^3 + 20532*x*y^2*z^2 + 31884*x*y^2*z + 18642*x*y^2 - 252*x*y*z^5 - 2910*x*y*z^4 - 13424*x*y*z^3 - 31002*x*y*z^2 - 35964*x*y*z - 16796*x*y + 42*x*z^6 + 576*x*z^5 + 3290*x*z^4 + 10040*x*z^3 + 17322*x*z^2 + 16052*x*z + 6223*x + 6*y^7 - 42*y^6*z - 102*y^6 + 126*y^5*z^2 + 606*y^5*z + 724*y^5 - 210*y^4*z^3 - 1500*y^4*z^2 - 3554*y^4*z - 2804*y^4 + 210*y^3*z^4 + 1980*y^3*z^3 + 6976*y^3*z^2 + 10922*y^3*z + 6436*y^3 - 126*y^2*z^5 - 1470*y^2*z^4 - 6844*y^2*z^3 - 15942*y^2*z^2 - 18648*y^2*z - 8785*y^2 + 42*y*z^6 + 582*y*z^5 + 3356*y*z^4 + 10334*y*z^3 + 17988*y*z^2 + 16832*y*z + 6590*y - 6*z^7 - 96*z^6 - 658*z^5 - 2510*z^4 - 5776*z^3 - 8041*z^2 - 6257*z - 1, 9*x^7 + 63*x^6*y - 63*x^6*z - 144*x^6 + 189*x^5*y^2 - 378*x^5*y*z - 873*x^5*y + 189*x^5*z^2 + 864*x^5*z + 986*x^5 + 315*x^4*y^3 - 945*x^4*y^2*z - 2205*x^4*y^2 + 945*x^4*y*z^2 + 4365*x^4*y*z + 5029*x^4*y - 315*x^4*z^3 - 2160*x^4*z^2 - 4930*x^4*z - 3754*x^4 + 315*x^3*y^4 - 1260*x^3*y^3*z - 2970*x^3*y^3 + 1890*x^3*y^2*z^2 + 8820*x^3*y^2*z + 10256*x^3*y^2 - 1260*x^3*y*z^3 - 8730*x^3*y*z^2 - 20116*x^3*y*z - 15457*x^3*y + 315*x^3*z^4 + 2880*x^3*z^3 + 9860*x^3*z^2 + 15016*x^3*z + 8606*x^3 + 189*x^2*y^5 - 945*x^2*y^4*z - 2250*x^2*y^4 + 1890*x^2*y^3*z^2 + 8910*x^2*y^3*z + 10454*x^2*y^3 - 1890*x^2*y^2*z^3 - 13230*x^2*y^2*z^2 - 30768*x^2*y^2*z - 23847*x^2*y^2 + 945*x^2*y*z^4 + 8730*x^2*y*z^3 + 30174*x^2*y*z^2 + 46371*x^2*y*z + 26817*x^2*y - 189*x^2*z^5 - 2160*x^2*z^4 - 9860*x^2*z^3 - 22524*x^2*z^2 - 25827*x^2*z - 11906*x^2 + 63*x*y^6 - 378*x*y^5*z - 909*x*y^5 + 945*x*y^4*z^2 + 4500*x*y^4*z + 5326*x*y^4 - 1260*x*y^3*z^3 - 8910*x*y^3*z^2 - 20908*x*y^3*z - 16339*x*y^3 + 945*x*y^2*z^4 + 8820*x*y^2*z^3 + 30768*x*y^2*z^2 + 47694*x*y^2*z + 27816*x*y^2 - 378*x*y*z^5 - 4365*x*y*z^4 - 20116*x*y*z^3 - 46371*x*y*z^2 - 53652*x*y*z - 24973*x*y + 63*x*z^6 + 864*x*z^5 + 4930*x*z^4 + 15016*x*z^3 + 25836*x*z^2 + 23857*x*z + 9208*x + 9*y^7 - 63*y^6*z - 153*y^6 + 189*y^5*z^2 + 909*y^5*z + 1085*y^5 - 315*y^4*z^3 - 2250*y^4*z^2 - 5326*y^4*z - 4195*y^4 + 315*y^3*z^4 + 2970*y^3*z^3 + 10454*y^3*z^2 + 16339*y^3*z + 9605*y^3 - 189*y^2*z^5 - 2205*y^2*z^4 - 10256*y^2*z^3 - 23847*y^2*z^2 - 27825*y^2*z - 13067*y^2 + 63*y*z^6 + 873*y*z^5 + 5029*y*z^4 + 15457*y*z^3 + 26835*y*z^2 + 25027*y*z + 9759*y - 9*z^7 - 144*z^6 - 986*z^5 - 3754*z^4 - 8615*z^3 - 11951*z^2 - 9258*z - 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -1, -1, 1 ], [ -1, -2, 1 ], [ -1, -1, 2 ] ], "source_shift": [ 2, 2, 1 ], "target_matrix": [ [ 1, 1, 2 ], [ -2, -1, -1 ], [ -3, -2, -2 ] ], "target_shift": [ -861, 2083, 3067 ], "inverse_recipe": { "p": 2, "q": 5, "f": "x**2", "g": "x**5 + x**4 + x**3 + x**2", "h": "y*z", "polynomials": { "f": [ [ [ 2, 0, 0 ], 1 ] ], "g": [ [ [ 5, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ], [ [ 3, 0, 0 ], 1 ], [ [ 4, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ] ] }, "h_multiplier": 3 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "Maybe" ], "seconds": [ 1.0969119619985577 ], "surface_features": [ 3.7729088054837785, 4.729585915915656, 3.3434085938038574, 0.0, 3.9348314186379567, 4918 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C180", "pair": 180, "family": "counterexample", "mapping": "F(x,y,z)=(8*x^7 - 80*x^6*y + 68*x^6*z - 104*x^6 + 336*x^5*y^2 - 588*x^5*y*z + 864*x^5*y + 246*x^5*z^2 - 792*x^5*z + 438*x^5 - 768*x^4*y^3 + 2076*x^4*y^2*z - 2928*x^4*y^2 - 1788*x^4*y*z^2 + 5532*x^4*y*z - 2988*x^4*y + 491*x^4*z^3 - 2502*x^4*z^2 + 2823*x^4*z - 930*x^4 + 1032*x^3*y^4 - 3828*x^3*y^3*z + 5184*x^3*y^3 + 5094*x^3*y^2*z^2 - 15120*x^3*y^2*z + 7974*x^3*y^2 - 2879*x^3*y*z^3 + 14112*x^3*y*z^2 - 15555*x^3*y*z + 4976*x^3*y + 584*x^3*z^4 - 4196*x^3*z^3 + 7245*x^3*z^2 - 4860*x^3*z + 1125*x^3 - 816*x^2*y^5 + 3888*x^2*y^4*z - 5064*x^2*y^4 - 7104*x^2*y^3*z^2 + 20220*x^2*y^3*z - 10416*x^2*y^3 + 6204*x^2*y^2*z^3 - 29178*x^2*y^2*z^2 + 31404*x^2*y^2*z - 9750*x^2*y^2 - 2589*x^2*y*z^4 + 17925*x^2*y*z^3 - 30246*x^2*y*z^2 + 19701*x^2*y*z - 4395*x^2*y + 414*x^2*z^5 - 3939*x^2*z^4 + 9252*x^2*z^3 - 9480*x^2*z^2 + 4473*x^2*z - 793*x^2 + 352*x*y^6 - 2064*x*y^5*z + 2592*x*y^5 + 4848*x*y^4*z^2 - 13248*x*y^4*z + 6672*x*y^4 - 5816*x*y^3*z^3 + 26208*x*y^3*z^2 - 27552*x*y^3*z + 8312*x*y^3 + 3750*x*y^2*z^4 - 24936*x*y^2*z^3 + 41088*x*y^2*z^2 - 25956*x*y^2*z + 5580*x*y^2 - 1233*x*y*z^5 + 11334*x*y*z^4 - 26028*x*y*z^3 + 25902*x*y*z^2 - 11772*x*y*z + 1988*x*y + 162*x*z^6 - 1962*x*z^5 + 5877*x*z^4 - 8177*x*z^3 + 5901*x*z^2 - 2139*x*z + 305*x - 64*y^7 + 448*y^6*z - 544*y^6 - 1296*y^5*z^2 + 3408*y^5*z - 1680*y^5 + 2000*y^4*z^3 - 8640*y^4*z^2 + 8880*y^4*z - 2608*y^4 - 1772*y^3*z^4 + 11288*y^3*z^3 - 18168*y^3*z^2 + 11144*y^3*z - 2312*y^3 + 900*y^2*z^5 - 7962*y^2*z^4 + 17856*y^2*z^3 - 17220*y^2*z^2 + 7536*y^2*z - 1216*y^2 - 243*y*z^6 + 2853*y*z^5 - 8361*y*z^4 + 11306*y*z^3 - 7854*y*z^2 + 2706*y*z - 370*y + 27*z^7 - 405*z^6 + 1485*z^5 - 2630*z^4 + 2582*z^3 - 1437*z^2 + 416*z - 1, 16*x^7 - 160*x^6*y + 136*x^6*z - 196*x^6 + 672*x^5*y^2 - 1176*x^5*y*z + 1620*x^5*y + 492*x^5*z^2 - 1500*x^5*z + 744*x^5 - 1536*x^4*y^3 + 4152*x^4*y^2*z - 5460*x^4*y^2 - 3576*x^4*y*z^2 + 10428*x^4*y*z - 5028*x^4*y + 982*x^4*z^3 - 4761*x^4*z^2 + 4824*x^4*z - 1476*x^4 + 2064*x^3*y^4 - 7656*x^3*y^3*z + 9612*x^3*y^3 + 10188*x^3*y^2*z^2 - 28356*x^3*y^2*z + 13296*x^3*y^2 - 5758*x^3*y*z^3 + 26739*x^3*y*z^2 - 26328*x^3*y*z + 7816*x^3*y + 1168*x^3*z^4 - 8020*x^3*z^3 + 12456*x^3*z^2 - 7758*x^3*z + 1716*x^3 - 1632*x^2*y^5 + 7776*x^2*y^4*z - 9336*x^2*y^4 - 14208*x^2*y^3*z^2 + 37716*x^2*y^3*z - 17220*x^2*y^3 + 12408*x^2*y^2*z^3 - 55026*x^2*y^2*z^2 + 52656*x^2*y^2*z - 15168*x^2*y^2 - 5178*x^2*y*z^4 + 34131*x^2*y*z^3 - 51501*x^2*y*z^2 + 31110*x^2*y*z - 6648*x^2*y + 828*x^2*z^5 - 7560*x^2*z^4 + 16005*x^2*z^3 - 15225*x^2*z^2 + 6852*x^2*z - 1193*x^2 + 704*x*y^6 - 4128*x*y^5*z + 4752*x*y^5 + 9696*x*y^4*z^2 - 24576*x*y^4*z + 10944*x*y^4 - 11632*x*y^3*z^3 + 49176*x*y^3*z^2 - 45792*x*y^3*z + 12820*x*y^3 + 7500*x*y^2*z^4 - 47280*x*y^2*z^3 + 69288*x*y^2*z^2 - 40572*x*y^2*z + 8388*x*y^2 - 2466*x*y*z^5 + 21681*x*y*z^4 - 44592*x*y*z^3 + 41133*x*y*z^2 - 17862*x*y*z + 2983*x*y + 324*x*z^6 - 3780*x*z^5 + 10230*x*z^4 - 13216*x*z^3 + 9084*x*z^2 - 3222*x*z + 458*x - 128*y^7 + 896*y^6*z - 992*y^6 - 2592*y^5*z^2 + 6288*y^5*z - 2736*y^5 + 4000*y^4*z^3 - 16128*y^4*z^2 + 14640*y^4*z - 3992*y^4 - 3544*y^3*z^4 + 21304*y^3*z^3 - 30360*y^3*z^2 + 17260*y^3*z - 3460*y^3 + 1800*y^2*z^5 - 15174*y^2*z^4 + 30288*y^2*z^3 - 27054*y^2*z^2 + 11352*y^2*z - 1826*y^2 - 486*y*z^6 + 5481*y*z^5 - 14415*y*z^4 + 18061*y*z^3 - 11961*y*z^2 + 4059*y*z - 555*y + 54*z^7 - 783*z^6 + 2601*z^5 - 4279*z^4 + 3997*z^3 - 2169*z^2 + 625*z - 1, -8*x^7 + 80*x^6*y - 68*x^6*z + 104*x^6 - 336*x^5*y^2 + 588*x^5*y*z - 864*x^5*y - 246*x^5*z^2 + 792*x^5*z - 438*x^5 + 768*x^4*y^3 - 2076*x^4*y^2*z + 2928*x^4*y^2 + 1788*x^4*y*z^2 - 5532*x^4*y*z + 2988*x^4*y - 491*x^4*z^3 + 2502*x^4*z^2 - 2823*x^4*z + 931*x^4 - 1032*x^3*y^4 + 3828*x^3*y^3*z - 5184*x^3*y^3 - 5094*x^3*y^2*z^2 + 15120*x^3*y^2*z - 7974*x^3*y^2 + 2879*x^3*y*z^3 - 14112*x^3*y*z^2 + 15555*x^3*y*z - 4983*x^3*y - 584*x^3*z^4 + 4196*x^3*z^3 - 7245*x^3*z^2 + 4864*x^3*z - 1135*x^3 + 816*x^2*y^5 - 3888*x^2*y^4*z + 5064*x^2*y^4 + 7104*x^2*y^3*z^2 - 20220*x^2*y^3*z + 10416*x^2*y^3 - 6204*x^2*y^2*z^3 + 29178*x^2*y^2*z^2 - 31404*x^2*y^2*z + 9768*x^2*y^2 + 2589*x^2*y*z^4 - 17925*x^2*y*z^3 + 30246*x^2*y*z^2 - 19722*x^2*y*z + 4446*x^2*y - 414*x^2*z^5 + 3939*x^2*z^4 - 9252*x^2*z^3 + 9486*x^2*z^2 - 4506*x^2*z + 814*x^2 - 352*x*y^6 + 2064*x*y^5*z - 2592*x*y^5 - 4848*x*y^4*z^2 + 13248*x*y^4*z - 6672*x*y^4 + 5816*x*y^3*z^3 - 26208*x*y^3*z^2 + 27552*x*y^3*z - 8332*x*y^3 - 3750*x*y^2*z^4 + 24936*x*y^2*z^3 - 41088*x*y^2*z^2 + 25992*x*y^2*z - 5664*x*y^2 + 1233*x*y*z^5 - 11334*x*y*z^4 + 26028*x*y*z^3 - 25923*x*y*z^2 + 11886*x*y*z - 2057*x*y - 162*x*z^6 + 1962*x*z^5 - 5877*x*z^4 + 8181*x*z^3 - 5937*x*z^2 + 2187*x*z - 319*x + 64*y^7 - 448*y^6*z + 544*y^6 + 1296*y^5*z^2 - 3408*y^5*z + 1680*y^5 - 2000*y^4*z^3 + 8640*y^4*z^2 - 8880*y^4*z + 2616*y^4 + 1772*y^3*z^4 - 11288*y^3*z^3 + 18168*y^3*z^2 - 11164*y^3*z + 2356*y^3 - 900*y^2*z^5 + 7962*y^2*z^4 - 17856*y^2*z^3 + 17238*y^2*z^2 - 7632*y^2*z + 1270*y^2 + 243*y*z^6 - 2853*y*z^5 + 8361*y*z^4 - 11313*y*z^3 + 7917*y*z^2 - 2787*y*z + 391*y - 27*z^7 + 405*z^6 - 1485*z^5 + 2631*z^4 - 2595*z^3 + 1464*z^2 - 433*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -1, 2, -1 ], [ -2, 2, -3 ], [ -1, 1, -1 ] ], "source_shift": [ 1, 1, 1 ], "target_matrix": [ [ -1, -1, -2 ], [ -2, -1, -1 ], [ 1, 1, 1 ] ], "target_shift": [ 51, 75, -53 ], "collision_preimages": [ [ "3", "1/4", "-3/2" ], [ "-49/2", "-11/2", "27/2" ], [ "-47/2", "-15/2", "21/2" ] ], "collision_image": [ "205/4", "151/2", "-213/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0889175140000589 ], "surface_features": [ 3.929337310422716, 4.840658025592327, 3.627109711211139, 0.0, 4.0390850747089875, 5044 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B242", "pair": 242, "family": "bijection", "mapping": "F(x,y,z)=(x^7 + 7*x^6*y - 14*x^6*z - 16*x^6 + 21*x^5*y^2 - 84*x^5*y*z - 95*x^5*y + 84*x^5*z^2 + 189*x^5*z + 105*x^5 + 35*x^4*y^3 - 210*x^4*y^2*z - 235*x^4*y^2 + 420*x^4*y*z^2 + 935*x^4*y*z + 515*x^4*y - 280*x^4*z^3 - 930*x^4*z^2 - 1020*x^4*z - 371*x^4 + 35*x^3*y^4 - 280*x^3*y^3*z - 310*x^3*y^3 + 840*x^3*y^2*z^2 + 1850*x^3*y^2*z + 1010*x^3*y^2 - 1120*x^3*y*z^3 - 3680*x^3*y*z^2 - 4000*x^3*y*z - 1444*x^3*y + 560*x^3*z^4 + 2440*x^3*z^3 + 3960*x^3*z^2 + 2848*x^3*z + 774*x^3 + 21*x^2*y^5 - 210*x^2*y^4*z - 230*x^2*y^4 + 840*x^2*y^3*z^2 + 1830*x^2*y^3*z + 990*x^2*y^3 - 1680*x^2*y^2*z^3 - 5460*x^2*y^2*z^2 - 5880*x^2*y^2*z - 2106*x^2*y^2 + 1680*x^2*y*z^4 + 7240*x^2*y*z^3 + 11640*x^2*y*z^2 + 8304*x^2*y*z + 2240*x^2*y - 672*x^2*z^5 - 3600*x^2*z^4 - 7680*x^2*z^3 - 8184*x^2*z^2 - 4397*x^2*z - 975*x^2 + 7*x*y^6 - 84*x*y^5*z - 91*x*y^5 + 420*x*y^4*z^2 + 905*x*y^4*z + 485*x*y^4 - 1120*x*y^3*z^3 - 3600*x*y^3*z^2 - 3840*x*y^3*z - 1364*x*y^3 + 1680*x*y^2*z^4 + 7160*x*y^2*z^3 + 11400*x*y^2*z^2 + 8064*x*y^2*z + 2158*x*y^2 - 1344*x*y*z^5 - 7120*x*y*z^4 - 15040*x*y*z^3 - 15888*x*y*z^2 - 8466*x*y*z - 1856*x*y + 448*x*z^6 + 2832*x*z^5 + 7440*x*z^4 + 10432*x*z^3 + 8300*x*z^2 + 3612*x*z + 693*x + y^7 - 14*y^6*z - 15*y^6 + 84*y^5*z^2 + 179*y^5*z + 95*y^5 - 280*y^4*z^3 - 890*y^4*z^2 - 940*y^4*z - 331*y^4 + 560*y^3*z^4 + 2360*y^3*z^3 + 3720*y^3*z^2 + 2608*y^3*z + 692*y^3 - 672*y^2*z^5 - 3520*y^2*z^4 - 7360*y^2*z^3 - 7704*y^2*z^2 - 4069*y^2*z - 882*y^2 + 448*y*z^6 + 2800*y*z^5 + 7280*y*z^4 + 10112*y*z^3 + 7972*y*z^2 + 3431*y*z + 646*y - 128*z^7 - 928*z^6 - 2880*z^5 - 4976*z^4 - 5204*z^3 - 3336*z^2 - 1240*z - 1, -2*x^7 - 14*x^6*y + 28*x^6*z + 32*x^6 - 42*x^5*y^2 + 168*x^5*y*z + 190*x^5*y - 168*x^5*z^2 - 378*x^5*z - 212*x^5 - 70*x^4*y^3 + 420*x^4*y^2*z + 470*x^4*y^2 - 840*x^4*y*z^2 - 1870*x^4*y*z - 1040*x^4*y + 560*x^4*z^3 + 1860*x^4*z^2 + 2060*x^4*z + 762*x^4 - 70*x^3*y^4 + 560*x^3*y^3*z + 620*x^3*y^3 - 1680*x^3*y^2*z^2 - 3700*x^3*y^2*z - 2040*x^3*y^2 + 2240*x^3*y*z^3 + 7360*x^3*y*z^2 + 8080*x^3*y*z + 2968*x^3*y - 1120*x^3*z^4 - 4880*x^3*z^3 - 8000*x^3*z^2 - 5856*x^3*z - 1628*x^3 - 42*x^2*y^5 + 420*x^2*y^4*z + 460*x^2*y^4 - 1680*x^2*y^3*z^2 - 3660*x^2*y^3*z - 2000*x^2*y^3 + 3360*x^2*y^2*z^3 + 10920*x^2*y^2*z^2 + 11880*x^2*y^2*z + 4332*x^2*y^2 - 3360*x^2*y*z^4 - 14480*x^2*y*z^3 - 23520*x^2*y*z^2 - 17088*x^2*y*z - 4720*x^2*y + 1344*x^2*z^5 + 7200*x^2*z^4 + 15520*x^2*z^3 + 16848*x^2*z^2 + 9274*x^2*z + 2109*x^2 - 14*x*y^6 + 168*x*y^5*z + 182*x*y^5 - 840*x*y^4*z^2 - 1810*x*y^4*z - 980*x*y^4 + 2240*x*y^3*z^3 + 7200*x*y^3*z^2 + 7760*x*y^3*z + 2808*x*y^3 - 3360*x*y^2*z^4 - 14320*x*y^2*z^3 - 23040*x*y^2*z^2 - 16608*x*y^2*z - 4556*x*y^2 + 2688*x*y*z^5 + 14240*x*y*z^4 + 30400*x*y*z^3 + 32736*x*y*z^2 + 17892*x*y*z + 4030*x*y - 896*x*z^6 - 5664*x*z^5 - 15040*x*z^4 - 21504*x*z^3 - 17560*x*z^2 - 7860*x*z - 1544*x - 2*y^7 + 28*y^6*z + 30*y^6 - 168*y^5*z^2 - 358*y^5*z - 192*y^5 + 560*y^4*z^3 + 1780*y^4*z^2 + 1900*y^4*z + 682*y^4 - 1120*y^3*z^4 - 4720*y^3*z^3 - 7520*y^3*z^2 - 5376*y^3*z - 1464*y^3 + 1344*y^2*z^5 + 7040*y^2*z^4 + 14880*y^2*z^3 + 15888*y^2*z^2 + 8618*y^2*z + 1923*y^2 - 896*y*z^6 - 5600*y*z^5 - 14720*y*z^4 - 20864*y*z^3 - 16904*y*z^2 - 7498*y*z - 1451*y + 256*z^7 + 1856*z^6 + 5824*z^5 + 10272*z^4 + 11048*z^3 + 7308*z^2 + 2797*z + 1, x^7 + 7*x^6*y - 14*x^6*z - 16*x^6 + 21*x^5*y^2 - 84*x^5*y*z - 95*x^5*y + 84*x^5*z^2 + 189*x^5*z + 105*x^5 + 35*x^4*y^3 - 210*x^4*y^2*z - 235*x^4*y^2 + 420*x^4*y*z^2 + 935*x^4*y*z + 515*x^4*y - 280*x^4*z^3 - 930*x^4*z^2 - 1020*x^4*z - 371*x^4 + 35*x^3*y^4 - 280*x^3*y^3*z - 310*x^3*y^3 + 840*x^3*y^2*z^2 + 1850*x^3*y^2*z + 1010*x^3*y^2 - 1120*x^3*y*z^3 - 3680*x^3*y*z^2 - 4000*x^3*y*z - 1444*x^3*y + 560*x^3*z^4 + 2440*x^3*z^3 + 3960*x^3*z^2 + 2848*x^3*z + 774*x^3 + 21*x^2*y^5 - 210*x^2*y^4*z - 230*x^2*y^4 + 840*x^2*y^3*z^2 + 1830*x^2*y^3*z + 990*x^2*y^3 - 1680*x^2*y^2*z^3 - 5460*x^2*y^2*z^2 - 5880*x^2*y^2*z - 2106*x^2*y^2 + 1680*x^2*y*z^4 + 7240*x^2*y*z^3 + 11640*x^2*y*z^2 + 8304*x^2*y*z + 2240*x^2*y - 672*x^2*z^5 - 3600*x^2*z^4 - 7680*x^2*z^3 - 8184*x^2*z^2 - 4397*x^2*z - 976*x^2 + 7*x*y^6 - 84*x*y^5*z - 91*x*y^5 + 420*x*y^4*z^2 + 905*x*y^4*z + 485*x*y^4 - 1120*x*y^3*z^3 - 3600*x*y^3*z^2 - 3840*x*y^3*z - 1364*x*y^3 + 1680*x*y^2*z^4 + 7160*x*y^2*z^3 + 11400*x*y^2*z^2 + 8064*x*y^2*z + 2158*x*y^2 - 1344*x*y*z^5 - 7120*x*y*z^4 - 15040*x*y*z^3 - 15888*x*y*z^2 - 8466*x*y*z - 1858*x*y + 448*x*z^6 + 2832*x*z^5 + 7440*x*z^4 + 10432*x*z^3 + 8300*x*z^2 + 3616*x*z + 699*x + y^7 - 14*y^6*z - 15*y^6 + 84*y^5*z^2 + 179*y^5*z + 95*y^5 - 280*y^4*z^3 - 890*y^4*z^2 - 940*y^4*z - 331*y^4 + 560*y^3*z^4 + 2360*y^3*z^3 + 3720*y^3*z^2 + 2608*y^3*z + 692*y^3 - 672*y^2*z^5 - 3520*y^2*z^4 - 7360*y^2*z^3 - 7704*y^2*z^2 - 4069*y^2*z - 883*y^2 + 448*y*z^6 + 2800*y*z^5 + 7280*y*z^4 + 10112*y*z^3 + 7972*y*z^2 + 3435*y*z + 651*y - 128*z^7 - 928*z^6 - 2880*z^5 - 4976*z^4 - 5204*z^3 - 3340*z^2 - 1249*z - 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -1, -1, 2 ], [ -2, -1, 1 ], [ -1, -1, 1 ] ], "source_shift": [ 2, 1, 2 ], "target_matrix": [ [ -1, -1, -1 ], [ 2, 1, 1 ], [ -1, -2, -1 ] ], "target_shift": [ 211, -490, 216 ], "inverse_recipe": { "p": 2, "q": 5, "f": "x**2", "g": "x**5 + x", "h": "y**2 + y*z + 2*z", "polynomials": { "f": [ [ [ 2, 0, 0 ], 1 ] ], "g": [ [ [ 5, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 0, 1 ], 2 ], [ [ 0, 2, 0 ], 1 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 0.9827504960012448 ], "surface_features": [ 3.5571852716081342, 4.515025612032066, 3.1772478362556233, 0.0, 3.698504083870527, 4852 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B118", "pair": 118, "family": "bijection", "mapping": "F(x,y,z)=(-3*x^7 - 21*x^6*y + 42*x^6*z + 24*x^6 - 63*x^5*y^2 + 252*x^5*y*z + 144*x^5*y - 252*x^5*z^2 - 288*x^5*z - 90*x^5 - 105*x^4*y^3 + 630*x^4*y^2*z + 360*x^4*y^2 - 1260*x^4*y*z^2 - 1440*x^4*y*z - 447*x^4*y + 840*x^4*z^3 + 1440*x^4*z^2 + 891*x^4*z + 199*x^4 - 105*x^3*y^4 + 840*x^3*y^3*z + 480*x^3*y^3 - 2520*x^3*y^2*z^2 - 2880*x^3*y^2*z - 888*x^3*y^2 + 3360*x^3*y*z^3 + 5760*x^3*y*z^2 + 3540*x^3*y*z + 784*x^3*y - 1680*x^3*z^4 - 3840*x^3*z^3 - 3528*x^3*z^2 - 1553*x^3*z - 281*x^3 - 63*x^2*y^5 + 630*x^2*y^4*z + 360*x^2*y^4 - 2520*x^2*y^3*z^2 - 2880*x^2*y^3*z - 882*x^2*y^3 + 5040*x^2*y^2*z^3 + 8640*x^2*y^2*z^2 + 5274*x^2*y^2*z + 1158*x^2*y^2 - 5040*x^2*y*z^4 - 11520*x^2*y*z^3 - 10512*x^2*y*z^2 - 4587*x^2*y*z - 822*x^2*y + 2016*x^2*z^5 + 5760*x^2*z^4 + 6984*x^2*z^3 + 4542*x^2*z^2 + 1611*x^2*z + 257*x^2 - 21*x*y^6 + 252*x*y^5*z + 144*x*y^5 - 1260*x*y^4*z^2 - 1440*x*y^4*z - 438*x*y^4 + 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253*z - 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -1, -1, 2 ], [ -2, -1, 1 ], [ -1, -1, 1 ] ], "source_shift": [ 1, 2, 1 ], "target_matrix": [ [ 1, 1, 1 ], [ -2, -1, -1 ], [ -1, -2, -1 ] ], "target_shift": [ -34, 71, 38 ], "inverse_recipe": { "p": 3, "q": 4, "f": "x**3 + x**2", "g": "x**4 + x**2", "h": "y*z", "polynomials": { "f": [ [ [ 3, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "g": [ [ [ 4, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ] ] }, "h_multiplier": 3 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.1026017740005045 ], "surface_features": [ 3.5303651427054423, 4.443169075796116, 3.2079035303860515, 0.0, 3.6408419912197525, 4889 ], 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"Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ 1, -1, -1 ], [ 1, -1, -2 ], [ 2, -1, -1 ] ], "source_shift": [ 1, 1, 2 ], "target_matrix": [ [ -1, -1, -1 ], [ 1, 1, 2 ], [ -3, -2, -3 ] ], "target_shift": [ 72, -69, 172 ], "collision_preimages": [ [ "-5/4", "-1/4", "0" ], [ "9/2", "2", "5/2" ], [ "13/2", "11", "-5/2" ] ], "collision_image": [ "289/4", "-277/4", "691/4" ] }, "probe_target": "No", "jev_answers": [ "Maybe" ], "seconds": [ 1.0326527889992576 ], "surface_features": [ 3.2536939819474684, 4.2203957782315955, 2.9258275746247424, 0.0, 3.4057112764217417, 4785 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B161", "pair": 161, "family": "bijection", "mapping": "F(x,y,z)=(-11*x^7 + 77*x^6*y + 154*x^6*z - 99*x^6 - 231*x^5*y^2 - 924*x^5*y*z + 594*x^5*y - 924*x^5*z^2 + 1188*x^5*z - 385*x^5 + 385*x^4*y^3 + 2310*x^4*y^2*z - 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23760*y^2*z^4 - 30536*y^2*z^3 + 21012*y^2*z^2 - 7993*y^2*z + 1332*y^2 - 4928*y*z^6 + 19008*y*z^5 - 30448*y*z^4 + 27752*y*z^3 - 15700*y*z^2 + 5152*y*z - 813*y - 1408*z^7 + 6336*z^6 - 12144*z^5 + 13744*z^4 - 10276*z^3 + 4987*z^2 - 1528*z + 1, -22*x^7 + 154*x^6*y + 308*x^6*z - 198*x^6 - 462*x^5*y^2 - 1848*x^5*y*z + 1188*x^5*y - 1848*x^5*z^2 + 2376*x^5*z - 770*x^5 + 770*x^4*y^3 + 4620*x^4*y^2*z - 2970*x^4*y^2 + 9240*x^4*y*z^2 - 11880*x^4*y*z + 3850*x^4*y + 6160*x^4*z^3 - 11880*x^4*z^2 + 7678*x^4*z - 1805*x^4 - 770*x^3*y^4 - 6160*x^3*y^3*z + 3960*x^3*y^3 - 18480*x^3*y^2*z^2 + 23760*x^3*y^2*z - 7700*x^3*y^2 - 24640*x^3*y*z^3 + 47520*x^3*y*z^2 - 30712*x^3*y*z + 7198*x^3*y - 12320*x^3*z^4 + 31680*x^3*z^3 - 30624*x^3*z^2 + 14264*x^3*z - 2797*x^3 + 462*x^2*y^5 + 4620*x^2*y^4*z - 2970*x^2*y^4 + 18480*x^2*y^3*z^2 - 23760*x^2*y^3*z + 7700*x^2*y^3 + 36960*x^2*y^2*z^3 - 71280*x^2*y^2*z^2 + 46068*x^2*y^2*z - 10764*x^2*y^2 + 36960*x^2*y*z^4 - 95040*x^2*y*z^3 + 91872*x^2*y*z^2 - 42660*x^2*y*z + 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"certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ 1, -1, -2 ], [ 3, -2, -3 ], [ 1, -1, -1 ] ], "source_shift": [ 1, 1, 2 ], "target_matrix": [ [ -1, -2, -1 ], [ 1, 1, 1 ], [ -2, -1, -1 ] ], "target_shift": [ 220, -216, 437 ], "inverse_recipe": { "p": 4, "q": 3, "f": "x**4 + x**3 + x", "g": "x**3 + x**2", "h": "y*z + y", "polynomials": { "f": [ [ [ 4, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ], [ [ 3, 0, 0 ], 1 ] ], "g": [ [ [ 3, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 1, 0 ], 1 ] ] }, "h_multiplier": 11 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0123343849991215 ], "surface_features": [ 4.163473544484294, 5.086872138269005, 3.8516863154424277, 0.0, 4.274619646741909, 5117 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B069", "pair": 69, "family": "bijection", "mapping": "F(x,y,z)=(-13*x^7 - 91*x^6*y + 91*x^6*z + 182*x^6 - 273*x^5*y^2 + 546*x^5*y*z + 1066*x^5*y - 273*x^5*z^2 - 1066*x^5*z - 1053*x^5 - 455*x^4*y^3 + 1365*x^4*y^2*z + 2600*x^4*y^2 - 1365*x^4*y*z^2 - 5200*x^4*y*z - 5018*x^4*y + 455*x^4*z^3 + 2600*x^4*z^2 + 5031*x^4*z + 3328*x^4 - 455*x^3*y^4 + 1820*x^3*y^3*z + 3380*x^3*y^3 - 2730*x^3*y^2*z^2 - 10140*x^3*y^2*z - 9555*x^3*y^2 + 1820*x^3*y*z^3 + 10140*x^3*y*z^2 + 19162*x^3*y*z + 12350*x^3*y - 455*x^3*z^4 - 3380*x^3*z^3 - 9607*x^3*z^2 - 12480*x^3*z - 6241*x^3 - 273*x^2*y^5 + 1365*x^2*y^4*z + 2470*x^2*y^4 - 2730*x^2*y^3*z^2 - 9880*x^2*y^3*z - 9087*x^2*y^3 + 2730*x^2*y^2*z^3 + 14820*x^2*y^2*z^2 + 27339*x^2*y^2*z + 17160*x^2*y^2 - 1365*x^2*y*z^4 - 9880*x^2*y*z^3 - 27417*x^2*y*z^2 - 34684*x^2*y*z - 16734*x^2*y + 273*x^2*z^5 + 2470*x^2*z^4 + 9165*x^2*z^3 + 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273*y^2*z^5 + 2340*y^2*z^4 + 8268*y^2*z^3 + 14976*y^2*z^2 + 13731*y^2*z + 4994*y^2 - 91*y*z^6 - 936*y*z^5 - 4147*y*z^4 - 10088*y*z^3 - 14043*y*z^2 - 10404*y*z - 3120*y + 13*z^7 + 156*z^6 + 832*z^5 + 2548*z^4 + 4785*z^3 + 5410*z^2 + 3329*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -1, -1, 1 ], [ -3, -2, 3 ], [ -2, -1, 1 ] ], "source_shift": [ 1, 1, 2 ], "target_matrix": [ [ 1, 1, 2 ], [ 2, 1, 1 ], [ 1, 1, 1 ] ], "target_shift": [ -819, -1660, -827 ], "inverse_recipe": { "p": 3, "q": 2, "f": "x**3 + x**2", "g": "x**2 + x", "h": "y*z**2 + z**2", "polynomials": { "f": [ [ [ 3, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "g": [ [ [ 2, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 2 ], 1 ], [ [ 0, 0, 2 ], 1 ] ] }, "h_multiplier": 13 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0785018390015466 ], "surface_features": [ 3.9434462857657286, 4.841159173215329, 3.637689819118401, 0.0, 4.062043583293927, 5036 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B153", "pair": 153, "family": "bijection", "mapping": "F(x,y,z)=(-512*x^7 + 1792*x^6*y - 1792*x^6*z + 1792*x^6 - 2688*x^5*y^2 + 5376*x^5*y*z - 5376*x^5*y - 2688*x^5*z^2 + 5376*x^5*z - 2880*x^5 + 2240*x^4*y^3 - 6720*x^4*y^2*z + 6720*x^4*y^2 + 6720*x^4*y*z^2 - 13440*x^4*y*z + 7232*x^4*y - 2240*x^4*z^3 + 6720*x^4*z^2 - 7296*x^4*z + 2800*x^4 - 1120*x^3*y^4 + 4480*x^3*y^3*z - 4480*x^3*y^3 - 6720*x^3*y^2*z^2 + 13440*x^3*y^2*z - 7264*x^3*y^2 + 4480*x^3*y*z^3 - 13440*x^3*y*z^2 + 14656*x^3*y*z - 5680*x^3*y - 1120*x^3*z^4 + 4480*x^3*z^3 - 7392*x^3*z^2 + 5808*x^3*z - 1760*x^3 + 336*x^2*y^5 - 1680*x^2*y^4*z + 1680*x^2*y^4 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468*y^3*z^2 - 380*y^3*z + 120*y^3 - 42*y^2*z^5 + 210*y^2*z^4 - 472*y^2*z^3 + 582*y^2*z^2 - 372*y^2*z + 98*y^2 + 14*y*z^6 - 84*y*z^5 + 238*y*z^4 - 396*y*z^3 + 384*y*z^2 - 206*y*z + 46*y - 2*z^7 + 14*z^6 - 48*z^5 + 101*z^4 - 132*z^3 + 108*z^2 - 48*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -2, 1, -1 ], [ -1, 1, -1 ], [ -3, 2, -3 ] ], "source_shift": [ 1, 1, 2 ], "target_matrix": [ [ 2, 1, 1 ], [ 1, 2, 1 ], [ 1, 1, 1 ] ], "target_shift": [ -21, -10, -8 ], "inverse_recipe": { "p": 4, "q": 3, "f": "x**4", "g": "x**3", "h": "y*z", "polynomials": { "f": [ [ [ 4, 0, 0 ], 1 ] ], "g": [ [ [ 3, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ] ] }, "h_multiplier": 2 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0048310720012523 ], "surface_features": [ 3.276849075236172, 4.166015456323775, 2.936513742478893, 0.0, 3.385983821620104, 4798 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C169", "pair": 169, "family": "counterexample", "mapping": "F(x,y,z)=(108*x^7 - 918*x^6*y - 594*x^6*z + 1134*x^6 + 3168*x^5*y^2 + 4230*x^5*y*z - 7893*x^5*y + 1386*x^5*z^2 - 5301*x^5*z + 4266*x^5 - 5720*x^4*y^3 - 11868*x^4*y^2*z + 21594*x^4*y^2 - 8034*x^4*y*z^2 + 30039*x^4*y*z - 24093*x^4*y - 1778*x^4*z^3 + 10227*x^4*z^2 - 16506*x^4*z + 8469*x^4 + 5824*x^3*y^4 + 16720*x^3*y^3*z - 29656*x^3*y^3 + 17592*x^3*y^2*z^2 - 64200*x^3*y^2*z + 51192*x^3*y^2 + 8050*x^3*y*z^3 - 45291*x^3*y*z^2 + 72828*x^3*y*z - 37272*x^3*y + 1354*x^3*z^4 - 10423*x^3*z^3 + 25308*x^3*z^2 - 26004*x^3*z + 9900*x^3 - 3360*x^2*y^5 - 12480*x^2*y^4*z + 21648*x^2*y^4 - 18144*x^2*y^3*z^2 + 64680*x^2*y^3*z - 51144*x^2*y^3 - 12900*x^2*y^2*z^3 + 70938*x^2*y^2*z^2 - 113376*x^2*y^2*z + 57648*x^2*y^2 - 4488*x^2*y*z^4 + 33825*x^2*y*z^3 - 81807*x^2*y*z^2 + 83754*x^2*y*z - 31716*x^2*y - 612*x^2*z^5 + 5919*x^2*z^4 - 19224*x^2*z^3 + 29661*x^2*z^2 - 22572*x^2*z + 6906*x^2 + 1024*x*y^6 + 4704*x*y^5*z - 8016*x*y^5 + 8832*x*y^4*z^2 - 30864*x*y^4*z + 24192*x*y^4 + 8664*x*y^3*z^3 - 46644*x*y^3*z^2 + 73968*x*y^3*z - 37296*x*y^3 + 4680*x*y^2*z^4 - 34536*x*y^2*z^3 + 83028*x*y^2*z^2 - 84432*x*y^2*z + 31632*x*y^2 + 1320*x*y*z^5 - 12516*x*y*z^4 + 40488*x*y*z^3 - 62199*x*y*z^2 + 47004*x*y*z - 14208*x*y + 152*x*z^6 - 1776*x*z^5 + 7236*x*z^4 - 14901*x*z^3 + 16992*x*z^2 - 10362*x*z + 2668*x - 128*y^7 - 704*y^6*z + 1184*y^6 - 1632*y^5*z^2 + 5616*y^5*z - 4368*y^5 - 2064*y^4*z^3 + 10920*y^4*z^2 - 17184*y^4*z + 8592*y^4 - 1536*y^3*z^4 + 11124*y^3*z^3 - 26556*y^3*z^2 + 26792*y^3*z - 9920*y^3 - 672*y^2*z^5 + 6252*y^2*z^4 - 20112*y^2*z^3 + 30690*y^2*z^2 - 22944*y^2*z + 6824*y^2 - 160*y*z^6 + 1836*y*z^5 - 7452*y*z^4 + 15273*y*z^3 - 17277*y*z^2 + 10396*y*z - 2624*y - 16*z^7 + 220*z^6 - 1080*z^5 + 2783*z^4 - 4226*z^3 + 3851*z^2 - 1970*z + 1, 54*x^7 - 459*x^6*y - 297*x^6*z + 648*x^6 + 1584*x^5*y^2 + 2115*x^5*y*z - 4581*x^5*y + 693*x^5*z^2 - 3042*x^5*z + 2889*x^5 - 2860*x^4*y^3 - 5934*x^4*y^2*z + 12750*x^4*y^2 - 4017*x^4*y*z^2 + 17508*x^4*y*z - 16722*x^4*y - 889*x^4*z^3 + 5892*x^4*z^2 - 11268*x^4*z + 6524*x^4 + 2912*x^3*y^4 + 8360*x^3*y^3*z - 17816*x^3*y^3 + 8796*x^3*y^2*z^2 - 38040*x^3*y^2*z + 36396*x^3*y^2 + 4025*x^3*y*z^3 - 26493*x^3*y*z^2 + 50913*x^3*y*z - 29560*x^3*y + 677*x^3*z^4 - 6026*x^3*z^3 + 17406*x^3*z^2 - 20230*x^3*z + 8398*x^3 - 1680*x^2*y^5 - 6240*x^2*y^4*z + 13200*x^2*y^4 - 9072*x^2*y^3*z^2 + 38928*x^2*y^3*z - 37128*x^2*y^3 - 6450*x^2*y^2*z^3 + 42138*x^2*y^2*z^2 - 81042*x^2*y^2*z + 46980*x^2*y^2 - 2244*x^2*y*z^4 + 19842*x^2*y*z^3 - 57549*x^2*y*z^2 + 66993*x^2*y*z - 27777*x^2*y - 306*x^2*z^5 + 3432*x^2*z^4 - 13311*x^2*z^3 + 23286*x^2*z^2 - 19356*x^2*z + 6303*x^2 + 512*x*y^6 + 2352*x*y^5*z - 4944*x*y^5 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15849*x^4*y*z - 13605*x^4*y - 889*x^4*z^3 + 5373*x^4*z^2 - 9258*x^4*z + 4997*x^4 + 2912*x^3*y^4 + 8360*x^3*y^3*z - 15824*x^3*y^3 + 8796*x^3*y^2*z^2 - 34080*x^3*y^2*z + 29196*x^3*y^2 + 4025*x^3*y*z^3 - 23928*x^3*y*z^2 + 41247*x^3*y*z - 22273*x^3*y + 677*x^3*z^4 - 5483*x^3*z^3 + 14238*x^3*z^2 - 15409*x^3*z + 6094*x^3 - 1680*x^2*y^5 - 6240*x^2*y^4*z + 11616*x^2*y^4 - 9072*x^2*y^3*z^2 + 34536*x^2*y^3*z - 29424*x^2*y^3 - 6450*x^2*y^2*z^3 + 37692*x^2*y^2*z^2 - 64806*x^2*y^2*z + 34866*x^2*y^2 - 2244*x^2*y*z^4 + 17889*x^2*y*z^3 - 46452*x^2*y*z^2 + 50238*x^2*y*z - 19806*x^2*y - 306*x^2*z^5 + 3117*x^2*z^4 - 10845*x^2*z^3 + 17646*x^2*z^2 - 13962*x^2*z + 4392*x^2 + 512*x*y^6 + 2352*x*y^5*z - 4320*x*y^5 + 4416*x*y^4*z^2 - 16560*x*y^4*z + 14016*x*y^4 + 4332*x*y^3*z^3 - 24912*x*y^3*z^2 + 42612*x*y^3*z - 22780*x*y^3 + 2340*x*y^2*z^4 - 18360*x*y^2*z^3 + 47544*x*y^2*z^2 - 51204*x*y^2*z + 20004*x*y^2 + 660*x*y*z^5 - 6624*x*y*z^4 + 23040*x*y*z^3 - 37431*x*y*z^2 + 29469*x*y*z - 9177*x*y + 76*x*z^6 - 936*x*z^5 + 4092*x*z^4 - 8897*x*z^3 + 10557*x*z^2 - 6624*x*z + 1739*x - 64*y^7 - 352*y^6*z + 640*y^6 - 816*y^5*z^2 + 3024*y^5*z - 2544*y^5 - 1032*y^4*z^3 + 5856*y^4*z^2 - 9960*y^4*z + 5288*y^4 - 768*y^3*z^4 + 5940*y^3*z^3 - 15312*y^3*z^2 + 16392*y^3*z - 6336*y^3 - 336*y^2*z^5 + 3324*y^2*z^4 - 11532*y^2*z^3 + 18654*y^2*z^2 - 14550*y^2*z + 4462*y^2 - 80*y*z^6 + 972*y*z^5 - 4248*y*z^4 + 9217*y*z^3 - 10869*y*z^2 + 6740*y*z - 1739*y - 8*z^7 + 116*z^6 - 612*z^5 + 1667*z^4 - 2636*z^3 + 2473*z^2 - 1291*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ 1, -2, -1 ], [ 3, -2, -2 ], [ 2, -1, -1 ] ], "source_shift": [ 1, 2, 1 ], "target_matrix": [ [ 2, 1, 1 ], [ 1, 2, 1 ], [ 1, 1, 1 ] ], "target_shift": [ -437, -439, -290 ], "collision_preimages": [ [ "-1/2", "1/4", "0" ], [ "29/2", "-9", "65/2" ], [ "23/2", "-4", "43/2" ] ], "collision_image": [ "-875/2", "-1757/4", "-1161/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0215158460014209 ], "surface_features": [ 4.169912378694507, 5.054521130653083, 3.8608169638645378, 0.0, 4.26716986988785, 5122 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C057", "pair": 57, "family": "counterexample", "mapping": "F(x,y,z)=(8*x^7 - 68*x^6*y - 44*x^6*z + 88*x^6 + 234*x^5*y^2 + 312*x^5*y*z - 636*x^5*y + 102*x^5*z^2 - 408*x^5*z + 378*x^5 - 419*x^4*y^3 - 867*x^4*y^2*z + 1818*x^4*y^2 - 585*x^4*y*z^2 + 2388*x^4*y*z - 2259*x^4*y - 129*x^4*z^3 + 774*x^4*z^2 - 1437*x^4*z + 826*x^4 + 419*x^3*y^4 + 1199*x^3*y^3*z - 2617*x^3*y^3 + 1257*x^3*y^2*z^2 - 5286*x^3*y^2*z + 5103*x^3*y^2 + 573*x^3*y*z^3 - 3513*x^3*y*z^2 + 6651*x^3*y*z - 3918*x^3*y + 96*x^3*z^4 - 768*x^3*z^3 + 2142*x^3*z^2 - 2481*x^3*z + 1025*x^3 - 234*x^2*y^5 - 867*x^2*y^4*z + 1992*x^2*y^4 - 1257*x^2*y^3*z^2 + 5505*x^2*y^3*z - 5400*x^2*y^3 - 891*x^2*y^2*z^3 + 5634*x^2*y^2*z^2 - 10854*x^2*y^2*z + 6510*x^2*y^2 - 309*x^2*y*z^4 + 2529*x^2*y*z^3 - 7179*x^2*y*z^2 + 8496*x^2*y*z - 3642*x^2*y - 42*x^2*z^5 + 420*x^2*z^4 - 1563*x^2*z^3 + 2733*x^2*z^2 - 2298*x^2*z + 766*x^2 + 68*x*y^6 + 312*x*y^5*z - 756*x*y^5 + 585*x*y^4*z^2 - 2688*x*y^4*z + 2655*x*y^4 + 573*x*y^3*z^3 - 3771*x*y^3*z^2 + 7347*x*y^3*z - 4457*x*y^3 + 309*x*y^2*z^4 - 2610*x*y^2*z^3 + 7512*x*y^2*z^2 - 9018*x*y^2*z + 3960*x*y^2 + 87*x*y*z^5 - 891*x*y*z^4 + 3366*x*y*z^3 - 5991*x*y*z^2 + 5190*x*y*z - 1801*x*y + 10*x*z^6 - 120*x*z^5 + 558*x*z^4 - 1307*x*z^3 + 1671*x*z^2 - 1139*x*z + 343*x - 8*y^7 - 44*y^6*z + 112*y^6 - 102*y^5*z^2 + 492*y^5*z - 486*y^5 - 129*y^4*z^3 + 888*y^4*z^2 - 1737*y^4*z + 1056*y^4 - 96*y^3*z^4 + 843*y^3*z^3 - 2445*y^3*z^2 + 2955*y^3*z - 1309*y^3 - 42*y^2*z^5 + 444*y^2*z^4 - 1695*y^2*z^3 + 3048*y^2*z^2 - 2682*y^2*z + 952*y^2 - 10*y*z^6 + 123*y*z^5 - 579*y*z^4 + 1375*y*z^3 - 1797*y*z^2 + 1261*y*z - 383*y - z^7 + 14*z^6 - 78*z^5 + 229*z^4 - 394*z^3 + 409*z^2 - 248*z + 1, -16*x^7 + 136*x^6*y + 88*x^6*z - 176*x^6 - 468*x^5*y^2 - 624*x^5*y*z + 1272*x^5*y - 204*x^5*z^2 + 816*x^5*z - 756*x^5 + 838*x^4*y^3 + 1734*x^4*y^2*z - 3636*x^4*y^2 + 1170*x^4*y*z^2 - 4776*x^4*y*z + 4518*x^4*y + 258*x^4*z^3 - 1548*x^4*z^2 + 2874*x^4*z - 1660*x^4 - 838*x^3*y^4 - 2398*x^3*y^3*z + 5234*x^3*y^3 - 2514*x^3*y^2*z^2 + 10572*x^3*y^2*z - 10206*x^3*y^2 - 1146*x^3*y*z^3 + 7026*x^3*y*z^2 - 13302*x^3*y*z + 7856*x^3*y - 192*x^3*z^4 + 1536*x^3*z^3 - 4284*x^3*z^2 + 4982*x^3*z - 2090*x^3 + 468*x^2*y^5 + 1734*x^2*y^4*z - 3984*x^2*y^4 + 2514*x^2*y^3*z^2 - 11010*x^2*y^3*z + 10800*x^2*y^3 + 1782*x^2*y^2*z^3 - 11268*x^2*y^2*z^2 + 21708*x^2*y^2*z - 13038*x^2*y^2 + 618*x^2*y*z^4 - 5058*x^2*y*z^3 + 14358*x^2*y*z^2 - 17028*x^2*y*z + 7368*x^2*y + 84*x^2*z^5 - 840*x^2*z^4 + 3126*x^2*z^3 - 5484*x^2*z^2 + 4668*x^2*z - 1586*x^2 - 136*x*y^6 - 624*x*y^5*z + 1512*x*y^5 - 1170*x*y^4*z^2 + 5376*x*y^4*z - 5310*x*y^4 - 1146*x*y^3*z^3 + 7542*x*y^3*z^2 - 14694*x*y^3*z + 8921*x*y^3 - 618*x*y^2*z^4 + 5220*x*y^2*z^3 - 15024*x*y^2*z^2 + 18057*x*y^2*z - 7974*x*y^2 - 174*x*y*z^5 + 1782*x*y*z^4 - 6732*x*y*z^3 + 12003*x*y*z^2 - 10476*x*y*z + 3677*x*y - 20*x*z^6 + 240*x*z^5 - 1116*x*z^4 + 2621*x*z^3 - 3384*x*z^2 + 2341*x*z - 710*x + 16*y^7 + 88*y^6*z - 224*y^6 + 204*y^5*z^2 - 984*y^5*z + 972*y^5 + 258*y^4*z^3 - 1776*y^4*z^2 + 3474*y^4*z - 2113*y^4 + 192*y^3*z^4 - 1686*y^3*z^3 + 4890*y^3*z^2 - 5914*y^3*z + 2629*y^3 + 84*y^2*z^5 - 888*y^2*z^4 + 3390*y^2*z^3 - 6102*y^2*z^2 + 5394*y^2*z - 1928*y^2 + 20*y*z^6 - 246*y*z^5 + 1158*y*z^4 - 2754*y*z^3 + 3621*y*z^2 - 2564*y*z + 783*y + 2*z^7 - 28*z^6 + 156*z^5 - 459*z^4 + 796*z^3 - 836*z^2 + 510*z - 1, -8*x^7 + 68*x^6*y + 44*x^6*z - 112*x^6 - 234*x^5*y^2 - 312*x^5*y*z + 792*x^5*y - 102*x^5*z^2 + 516*x^5*z - 546*x^5 + 419*x^4*y^3 + 867*x^4*y^2*z - 2208*x^4*y^2 + 585*x^4*y*z^2 - 2952*x^4*y*z + 3207*x^4*y + 129*x^4*z^3 - 972*x^4*z^2 + 2061*x^4*z - 1284*x^4 - 419*x^3*y^4 - 1199*x^3*y^3*z + 3094*x^3*y^3 - 1257*x^3*y^2*z^2 + 6369*x^3*y^2*z - 7101*x^3*y^2 - 573*x^3*y*z^3 + 4308*x^3*y*z^2 - 9363*x^3*y*z + 5951*x^3*y - 96*x^3*z^4 + 957*x^3*z^3 - 3048*x^3*z^2 + 3818*x^3*z - 1713*x^3 + 234*x^2*y^5 + 867*x^2*y^4*z - 2295*x^2*y^4 + 1257*x^2*y^3*z^2 - 6459*x^2*y^3*z + 7347*x^2*y^3 + 891*x^2*y^2*z^3 - 6732*x^2*y^2*z^2 + 14958*x^2*y^2*z - 9660*x^2*y^2 + 309*x^2*y*z^4 - 3075*x^2*y*z^3 + 10014*x^2*y*z^2 - 12768*x^2*y*z + 5853*x^2*y + 42*x^2*z^5 - 519*x^2*z^4 + 2205*x^2*z^3 - 4161*x^2*z^2 + 3768*x^2*z - 1357*x^2 - 68*x*y^6 - 312*x*y^5*z + 852*x*y^5 - 585*x*y^4*z^2 + 3078*x*y^4*z - 3525*x*y^4 - 573*x*y^3*z^3 + 4392*x*y^3*z^2 - 9891*x*y^3*z + 6453*x*y^3 - 309*x*y^2*z^4 + 3093*x*y^2*z^3 - 10248*x*y^2*z^2 + 13230*x*y^2*z - 6150*x*y^2 - 87*x*y*z^5 + 1074*x*y*z^4 - 4650*x*y*z^3 + 8904*x*y*z^2 - 8202*x*y*z + 3010*x*y - 10*x*z^6 + 147*x*z^5 - 780*x*z^4 + 1968*x*z^3 - 2691*x*z^2 + 1958*x*z - 611*x + 8*y^7 + 44*y^6*z - 124*y^6 + 102*y^5*z^2 - 552*y^5*z + 630*y^5 + 129*y^4*z^3 - 1011*y^4*z^2 + 2283*y^4*z - 1492*y^4 + 96*y^3*z^4 - 975*y^3*z^3 + 3258*y^3*z^2 - 4231*y^3*z + 1971*y^3 + 42*y^2*z^5 - 522*y^2*z^4 + 2289*y^2*z^3 - 4422*y^2*z^2 + 4104*y^2*z - 1519*y^2 + 10*y*z^6 - 147*y*z^5 + 792*y*z^4 - 2021*y*z^3 + 2796*y*z^2 - 2059*y*z + 642*y + z^7 - 17*z^6 + 108*z^5 - 341*z^4 + 624*z^3 - 685*z^2 + 428*z - 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ 2, -1, -1 ], [ 1, -2, -1 ], [ 1, -1, -1 ] ], "source_shift": [ 1, 1, 2 ], "target_matrix": [ [ 1, 1, 2 ], [ -2, -2, -3 ], [ -1, -2, -1 ] ], "target_shift": [ -69, 142, 118 ], "collision_preimages": [ [ "5/4", "-5/4", "19/4" ], [ "-9/2", "7", "-16" ], [ "-13/2", "4", "-15" ] ], "collision_image": [ "-277/4", "285/2", "473/4" ] }, "probe_target": "No", "jev_answers": [ "Maybe" ], "seconds": [ 1.0376794510011678 ], "surface_features": [ 3.422471739610566, 4.336619812907439, 3.0788191830988487, 0.0, 3.541094201246957, 4842 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C068", "pair": 68, "family": "counterexample", "mapping": "F(x,y,z)=(64*x^7 + 352*x^6*y - 512*x^6*z - 736*x^6 + 816*x^5*y^2 - 2352*x^5*y*z - 3456*x^5*y + 1680*x^5*z^2 + 4944*x^5*z + 3264*x^5 + 1032*x^4*y^3 - 4416*x^4*y^2*z - 6648*x^4*y^2 + 6240*x^4*y*z^2 + 18816*x^4*y*z + 12696*x^4*y - 2912*x^4*z^3 - 13200*x^4*z^2 - 17856*x^4*z - 7280*x^4 + 768*x^3*y^4 - 4332*x^3*y^3*z - 6696*x^3*y^3 + 9072*x^3*y^2*z^2 + 28092*x^3*y^2*z + 19380*x^3*y^2 - 8360*x^3*y*z^3 - 38928*x^3*y*z^2 - 53868*x^3*y*z - 22404*x^3*y + 2860*x^3*z^4 + 17816*x^3*z^3 + 37128*x^3*z^2 + 31072*x^3*z + 9132*x^3 + 336*x^2*y^5 - 2340*x^2*y^4*z - 3720*x^2*y^4 + 6450*x^2*y^3*z^2 + 20544*x^2*y^3*z + 14484*x^2*y^3 - 8796*x^2*y^2*z^3 - 42138*x^2*y^2*z^2 - 59604*x^2*y^2*z - 25290*x^2*y^2 + 5934*x^2*y*z^4 + 38040*x^2*y*z^3 + 81042*x^2*y*z^2 + 69228*x^2*y*z + 20790*x^2*y - 1584*x^2*z^5 - 12750*x^2*z^4 - 36396*x^2*z^3 - 46980*x^2*z^2 - 28488*x^2*z - 6628*x^2 + 80*x*y^6 - 660*x*y^5*z - 1080*x*y^5 + 2244*x*y^4*z^2 + 7356*x*y^4*z + 5292*x*y^4 - 4025*x*y^3*z^3 - 19842*x*y^3*z^2 - 28632*x*y^3*z - 12385*x*y^3 + 4017*x*y^2*z^4 + 26493*x*y^2*z^3 + 57549*x*y^2*z^2 + 50121*x*y^2*z + 15381*x*y^2 - 2115*x*y*z^5 - 17508*x*y*z^4 - 50913*x*y*z^3 - 66993*x*y*z^2 - 41529*x*y*z - 9905*x*y + 459*x*z^6 + 4581*x*z^5 + 16722*x*z^4 + 29560*x*z^3 + 27777*x*z^2 + 13422*x*z + 2626*x + 8*y^7 - 76*y^6*z - 128*y^6 + 306*y^5*z^2 + 1032*y^5*z + 756*y^5 - 677*y^4*z^3 - 3432*y^4*z^2 - 5040*y^4*z - 2219*y^4 + 889*y^3*z^4 + 6026*y^3*z^3 + 13311*y^3*z^2 + 11795*y^3*z + 3692*y^3 - 693*y^2*z^5 - 5892*y^2*z^4 - 17406*y^2*z^3 - 23286*y^2*z^2 - 14715*y^2*z - 3592*y^2 + 297*y*z^6 + 3042*y*z^5 + 11268*y*z^4 + 20230*y*z^3 + 19356*y*z^2 + 9567*y*z + 1923*y - 54*z^7 - 648*z^6 - 2889*z^5 - 6524*z^4 - 8398*z^3 - 6303*z^2 - 2573*z - 1, 64*x^7 + 352*x^6*y - 512*x^6*z - 640*x^6 + 816*x^5*y^2 - 2352*x^5*y*z - 3024*x^5*y + 1680*x^5*z^2 + 4320*x^5*z + 2544*x^5 + 1032*x^4*y^3 - 4416*x^4*y^2*z - 5856*x^4*y^2 + 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459*x*z^6 + 4158*x*z^5 + 13605*x*z^4 + 22273*x*z^3 + 19806*x*z^2 + 9177*x*z + 1739*x + 8*y^7 - 76*y^6*z - 116*y^6 + 306*y^5*z^2 + 936*y^5*z + 612*y^5 - 677*y^4*z^3 - 3117*y^4*z^2 - 4092*y^4*z - 1667*y^4 + 889*y^3*z^4 + 5483*y^3*z^3 + 10845*y^3*z^2 + 8897*y^3*z + 2636*y^3 - 693*y^2*z^5 - 5373*y^2*z^4 - 14238*y^2*z^3 - 17646*y^2*z^2 - 10557*y^2*z - 2473*y^2 + 297*y*z^6 + 2781*y*z^5 + 9258*y*z^4 + 15409*y*z^3 + 13962*y*z^2 + 6624*y*z + 1291*y - 54*z^7 - 594*z^6 - 2385*z^5 - 4997*z^4 - 6094*z^3 - 4392*z^2 - 1739*z - 1, -192*x^7 - 1056*x^6*y + 1536*x^6*z + 1920*x^6 - 2448*x^5*y^2 + 7056*x^5*y*z + 9072*x^5*y - 5040*x^5*z^2 - 12960*x^5*z - 7632*x^5 - 3096*x^4*y^3 + 13248*x^4*y^2*z + 17568*x^4*y^2 - 18720*x^4*y*z^2 - 49680*x^4*y*z - 29880*x^4*y + 8736*x^4*z^3 + 34848*x^4*z^2 + 42048*x^4*z + 15872*x^4 - 2304*x^3*y^4 + 12996*x^3*y^3*z + 17820*x^3*y^3 - 27216*x^3*y^2*z^2 - 74736*x^3*y^2*z - 45936*x^3*y^2 + 25080*x^3*y*z^3 + 103608*x^3*y*z^2 + 127836*x^3*y*z + 49196*x^3*y - 8580*x^3*z^4 - 47472*x^3*z^3 - 88272*x^3*z^2 - 68368*x^3*z - 19052*x^3 - 1008*x^2*y^5 + 7020*x^2*y^4*z + 9972*x^2*y^4 - 19350*x^2*y^3*z^2 - 55080*x^2*y^3*z - 34596*x^2*y^3 + 26388*x^2*y^2*z^3 + 113076*x^2*y^2*z^2 + 142632*x^2*y^2*z + 55980*x^2*y^2 - 17802*x^2*y*z^4 - 102240*x^2*y*z^3 - 194418*x^2*y*z^2 - 153660*x^2*y*z - 43734*x^2*y + 4752*x^2*z^5 + 34344*x^2*z^4 + 87588*x^2*z^3 + 104628*x^2*z^2 + 60120*x^2*z + 13452*x^2 - 240*x*y^6 + 1980*x*y^5*z + 2916*x*y^5 - 6732*x*y^4*z^2 - 19872*x*y^4*z - 12744*x*y^4 + 12075*x*y^3*z^3 + 53667*x*y^3*z^2 + 69120*x*y^3*z + 27658*x*y^3 - 12051*x*y^2*z^4 - 71784*x*y^2*z^3 - 139356*x*y^2*z^2 - 112320*x*y^2*z - 32658*x*y^2 + 6345*x*y*z^5 + 47547*x*y*z^4 + 123741*x*y*z^3 + 150747*x*y*z^2 + 88533*x*y*z + 20301*x*y - 1377*x*z^6 - 12474*x*z^5 - 40815*x*z^4 - 66832*x*z^3 - 59493*x*z^2 - 27630*x*z - 5250*x - 24*y^7 + 228*y^6*z + 348*y^6 - 918*y^5*z^2 - 2808*y^5*z - 1836*y^5 + 2031*y^4*z^3 + 9351*y^4*z^2 + 12276*y^4*z + 5002*y^4 - 2667*y^3*z^4 - 16449*y^3*z^3 - 32535*y^3*z^2 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20520*y^4*z^2 + 20970*y^4*z - 7434*y^4 + 13680*y^3*z^4 - 53280*y^3*z^3 + 81378*y^3*z^2 - 57366*y^3*z + 15699*y^3 + 15984*y^2*z^5 - 77760*y^2*z^4 + 157788*y^2*z^3 - 165888*y^2*z^2 + 90054*y^2*z - 20223*y^2 + 10368*y*z^6 - 60480*y*z^5 + 152856*y*z^4 - 213048*y*z^3 + 172107*y*z^2 - 76464*y*z + 14649*y + 2880*z^7 - 19584*z^6 + 59184*z^5 - 102528*z^4 + 109581*z^3 - 72279*z^2 + 27331*z - 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ 1, -1, -2 ], [ 3, -2, -3 ], [ 1, -1, -1 ] ], "source_shift": [ 2, 2, 1 ], "target_matrix": [ [ 1, 1, 1 ], [ -1, -1, -2 ], [ -3, -2, -3 ] ], "target_shift": [ -1528, 1506, 4582 ], "inverse_recipe": { "p": 1, "q": 3, "f": "x", "g": "x**3 + x", "h": "y**2 + y*z**2 + y + z", "polynomials": { "f": [ [ [ 1, 0, 0 ], 1 ] ], "g": [ [ [ 3, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 2 ], 1 ], [ [ 0, 1, 0 ], 1 ], [ [ 0, 0, 1 ], 1 ], [ [ 0, 2, 0 ], 1 ] ] }, "h_multiplier": 3 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.1607857859999058 ], "surface_features": [ 4.53202768305082, 5.54637483545944, 4.172734250411153, 0.0, 4.710790163496065, 5204 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B130", "pair": 130, "family": "bijection", "mapping": "F(x,y,z)=(-5*x^7 - 35*x^6*y + 35*x^6*z + 65*x^6 - 105*x^5*y^2 + 210*x^5*y*z + 380*x^5*y - 105*x^5*z^2 - 380*x^5*z - 345*x^5 - 175*x^4*y^3 + 525*x^4*y^2*z + 925*x^4*y^2 - 525*x^4*y*z^2 - 1850*x^4*y*z - 1650*x^4*y + 175*x^4*z^3 + 925*x^4*z^2 + 1645*x^4*z + 1005*x^4 - 175*x^3*y^4 + 700*x^3*y^3*z + 1200*x^3*y^3 - 1050*x^3*y^2*z^2 - 3600*x^3*y^2*z - 3155*x^3*y^2 + 700*x^3*y*z^3 + 3600*x^3*y*z^2 + 6290*x^3*y*z + 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"g": [ [ [ 3, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "h": [ [ [ 0, 2, 1 ], 1 ], [ [ 0, 0, 2 ], 1 ], [ [ 0, 1, 0 ], 1 ] ] }, "h_multiplier": 5 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.1441179629982798 ], "surface_features": [ 3.446197040226551, 4.329601248356519, 3.146128035678238, 0.0, 3.5569176663683595, 4868 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B085", "pair": 85, "family": "bijection", "mapping": "F(x,y,z)=(-2*x^7 - 14*x^6*y + 28*x^6*z + 20*x^6 - 42*x^5*y^2 + 168*x^5*y*z + 120*x^5*y - 168*x^5*z^2 - 238*x^5*z - 93*x^5 - 70*x^4*y^3 + 420*x^4*y^2*z + 300*x^4*y^2 - 840*x^4*y*z^2 - 1190*x^4*y*z - 465*x^4*y + 560*x^4*z^3 + 1180*x^4*z^2 + 914*x^4*z + 263*x^4 - 70*x^3*y^4 + 560*x^3*y^3*z + 400*x^3*y^3 - 1680*x^3*y^2*z^2 - 2380*x^3*y^2*z - 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{ "p": 5, "q": 2, "f": "x**5 + x**2 + x", "g": "x**2 + x", "h": "y*z + y + z**3 + z**2", "polynomials": { "f": [ [ [ 5, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "g": [ [ [ 2, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 0, 3 ], 1 ], [ [ 0, 1, 0 ], 1 ], [ [ 0, 0, 2 ], 1 ] ] }, "h_multiplier": 2 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "Maybe" ], "seconds": [ 0.9890758500005177 ], "surface_features": [ 3.6263018643844678, 4.625065388577897, 3.291368850451583, 0.0, 3.7846269827405226, 4913 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C170", "pair": 170, "family": "counterexample", "mapping": "F(x,y,z)=(-8*x^7 + 44*x^6*y - 52*x^6*z + 92*x^6 - 102*x^5*y^2 + 240*x^5*y*z - 432*x^5*y - 138*x^5*z^2 + 504*x^5*z - 432*x^5 + 129*x^4*y^3 - 453*x^4*y^2*z + 831*x^4*y^2 + 519*x^4*y*z^2 - 1926*x^4*y*z + 1692*x^4*y - 195*x^4*z^3 + 1095*x^4*z^2 - 1944*x^4*z + 1090*x^4 - 96*x^3*y^4 + 447*x^3*y^3*z - 837*x^3*y^3 - 765*x^3*y^2*z^2 + 2889*x^3*y^2*z - 2604*x^3*y^2 + 573*x^3*y*z^3 - 3267*x^3*y*z^2 + 5940*x^3*y*z - 3429*x^3*y - 159*x^3*z^4 + 1215*x^3*z^3 - 3336*x^3*z^2 + 3893*x^3*z - 1618*x^3 + 42*x^2*y^5 - 243*x^2*y^4*z + 465*x^2*y^4 + 552*x^2*y^3*z^2 - 2124*x^2*y^3*z + 1965*x^2*y^3 - 618*x^2*y^2*z^3 + 3582*x^2*y^2*z^2 - 6672*x^2*y^2*z + 3969*x^2*y^2 + 342*x^2*y*z^4 - 2652*x^2*y*z^3 + 7449*x^2*y*z^2 - 8946*x^2*y*z + 3846*x^2*y - 75*x^2*z^5 + 729*x^2*z^4 - 2742*x^2*z^3 + 4977*x^2*z^2 - 4332*x^2*z + 1432*x^2 - 10*x*y^6 + 69*x*y^5*z - 135*x*y^5 - 195*x*y^4*z^2 + 765*x*y^4*z - 726*x*y^4 + 290*x*y^3*z^3 - 1710*x*y^3*z^2 + 3261*x*y^3*z - 1999*x*y^3 - 240*x*y^2*z^4 + 1890*x*y^2*z^3 - 5427*x*y^2*z^2 + 6705*x*y^2*z - 2985*x*y^2 + 105*x*y*z^5 - 1035*x*y*z^4 + 3975*x*y*z^3 - 7413*x*y*z^2 + 6675*x*y*z - 2294*x*y - 19*x*z^6 + 225*x*z^5 - 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2839*y*z - 802*y + 4*z^7 - 50*z^6 + 270*z^5 - 813*z^4 + 1464*z^3 - 1562*z^2 + 902*z - 1, 8*x^7 - 44*x^6*y + 52*x^6*z - 116*x^6 + 102*x^5*y^2 - 240*x^5*y*z + 540*x^5*y + 138*x^5*z^2 - 636*x^5*z + 612*x^5 - 129*x^4*y^3 + 453*x^4*y^2*z - 1029*x^4*y^2 - 519*x^4*y*z^2 + 2406*x^4*y*z - 2376*x^4*y + 195*x^4*z^3 - 1377*x^4*z^2 + 2748*x^4*z - 1654*x^4 + 96*x^3*y^4 - 447*x^3*y^3*z + 1026*x^3*y^3 + 765*x^3*y^2*z^2 - 3570*x^3*y^2*z + 3621*x^3*y^2 - 573*x^3*y*z^3 + 4062*x^3*y*z^2 - 8310*x^3*y*z + 5157*x^3*y + 159*x^3*z^4 - 1518*x^3*z^3 + 4689*x^3*z^2 - 5885*x^3*z + 2578*x^3 - 42*x^2*y^5 + 243*x^2*y^4*z - 564*x^2*y^4 - 552*x^2*y^3*z^2 + 2595*x^2*y^3*z - 2703*x^2*y^3 + 618*x^2*y^2*z^3 - 4401*x^2*y^2*z^2 + 9228*x^2*y^2*z - 5910*x^2*y^2 - 342*x^2*y*z^4 + 3273*x^2*y*z^3 - 10347*x^2*y*z^2 + 13380*x^2*y*z - 6066*x^2*y + 75*x^2*z^5 - 903*x^2*z^4 + 3822*x^2*z^3 - 7470*x^2*z^2 + 6864*x^2*z - 2374*x^2 + 10*x*y^6 - 69*x*y^5*z + 162*x*y^5 + 195*x*y^4*z^2 - 924*x*y^4*z + 987*x*y^4 - 290*x*y^3*z^3 + 2076*x*y^3*z^2 - 4455*x*y^3*z + 2944*x*y^3 + 240*x*y^2*z^4 - 2304*x*y^2*z^3 + 7443*x*y^2*z^2 - 9912*x*y^2*z + 4656*x*y^2 - 105*x*y*z^5 + 1266*x*y*z^4 - 5469*x*y*z^3 + 10992*x*y*z^2 - 10449*x*y*z + 3755*x*y + 19*x*z^6 - 276*x*z^5 + 1494*x*z^4 - 4024*x*z^3 + 5793*x*z^2 - 4235*x*z + 1217*x - y^7 + 8*y^6*z - 19*y^6 - 27*y^5*z^2 + 129*y^5*z - 141*y^5 + 50*y^4*z^3 - 360*y^4*z^2 + 789*y^4*z - 537*y^4 - 55*y^3*z^4 + 530*y^3*z^3 - 1746*y^3*z^2 + 2390*y^3*z - 1162*y^3 + 36*y^2*z^5 - 435*y^2*z^4 + 1914*y^2*z^3 - 3948*y^2*z^2 + 3876*y^2*z - 1450*y^2 - 13*y*z^6 + 189*y*z^5 - 1041*y*z^4 + 2874*y*z^3 - 4266*y*z^2 + 3239*y*z - 971*y + 2*z^7 - 34*z^6 + 225*z^5 - 779*z^4 + 1552*z^3 - 1789*z^2 + 1098*z - 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -2, 1, -1 ], [ -1, 1, -1 ], [ -1, 1, -2 ] ], "source_shift": [ 2, 1, 1 ], "target_matrix": [ [ 1, 1, 1 ], [ -2, -1, -1 ], [ -1, -2, -1 ] ], "target_shift": [ -155, 212, 270 ], "collision_preimages": [ [ "1", "1/4", "1/4" ], [ "-3/2", "-12", "-8" ], [ "7/2", "-1", "-5" ] ], "collision_image": [ "-621/4", "425/2", "1081/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.093713538000884 ], "surface_features": [ 3.3138268108311246, 4.130526745384164, 2.9934362304976116, 0.0, 3.413758112610727, 4803 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B061", "pair": 61, "family": "bijection", "mapping": "F(x,y,z)=(-12*x^7 + 84*x^6*y - 84*x^6*z + 180*x^6 - 252*x^5*y^2 + 504*x^5*y*z - 1080*x^5*y - 252*x^5*z^2 + 1080*x^5*z - 1164*x^5 + 420*x^4*y^3 - 1260*x^4*y^2*z + 2700*x^4*y^2 + 1260*x^4*y*z^2 - 5400*x^4*y*z + 5832*x^4*y - 420*x^4*z^3 + 2700*x^4*z^2 - 5820*x^4*z + 4244*x^4 - 420*x^3*y^4 + 1680*x^3*y^3*z - 3600*x^3*y^3 - 2520*x^3*y^2*z^2 + 10800*x^3*y^2*z - 11688*x^3*y^2 + 1680*x^3*y*z^3 - 10800*x^3*y*z^2 + 23328*x^3*y*z - 17096*x^3*y - 420*x^3*z^4 + 3600*x^3*z^3 - 11640*x^3*z^2 + 16988*x^3*z - 9463*x^3 + 252*x^2*y^5 - 1260*x^2*y^4*z + 2700*x^2*y^4 + 2520*x^2*y^3*z^2 - 10800*x^2*y^3*z + 11712*x^2*y^3 - 2520*x^2*y^2*z^3 + 16200*x^2*y^2*z^2 - 35064*x^2*y^2*z + 25824*x^2*y^2 + 1260*x^2*y*z^4 - 10800*x^2*y*z^3 + 34992*x^2*y*z^2 - 51324*x^2*y*z + 28821*x^2*y - 252*x^2*z^5 + 2700*x^2*z^4 - 11640*x^2*z^3 + 25500*x^2*z^2 - 28461*x^2*z + 12930*x^2 - 84*x*y^6 + 504*x*y^5*z - 1080*x*y^5 - 1260*x*y^4*z^2 + 5400*x*y^4*z - 5868*x*y^4 + 1680*x*y^3*z^3 - 10800*x*y^3*z^2 + 23424*x*y^3*z - 17336*x*y^3 - 1260*x*y^2*z^4 + 10800*x*y^2*z^3 - 35064*x*y^2*z^2 + 51684*x*y^2*z - 29253*x*y^2 + 504*x*y*z^5 - 5400*x*y*z^4 + 23328*x*y*z^3 - 51360*x*y*z^2 + 57786*x*y*z - 26568*x*y - 84*x*z^6 + 1080*x*z^5 - 5820*x*z^4 + 17012*x*z^3 - 28533*x*z^2 + 26016*x*z - 10038*x + 12*y^7 - 84*y^6*z + 180*y^6 + 252*y^5*z^2 - 1080*y^5*z + 1176*y^5 - 420*y^4*z^3 + 2700*y^4*z^2 - 5868*y^4*z + 4364*y^4 + 420*y^3*z^4 - 3600*y^3*z^3 + 11712*y^3*z^2 - 17348*y^3*z + 9895*y^3 - 252*y^2*z^5 + 2700*y^2*z^4 - 11688*y^2*z^3 + 25860*y^2*z^2 - 29325*y^2*z + 13650*y^2 + 84*y*z^6 - 1080*y*z^5 + 5832*y*z^4 - 17132*y*z^3 + 28965*y*z^2 - 26736*y*z + 10485*y - 12*z^7 + 180*z^6 - 1164*z^5 + 4256*z^4 - 9535*z^3 + 13086*z^2 - 10166*z + 1, 4*x^7 - 28*x^6*y + 28*x^6*z - 60*x^6 + 84*x^5*y^2 - 168*x^5*y*z + 360*x^5*y + 84*x^5*z^2 - 360*x^5*z + 388*x^5 - 140*x^4*y^3 + 420*x^4*y^2*z - 900*x^4*y^2 - 420*x^4*y*z^2 + 1800*x^4*y*z - 1944*x^4*y + 140*x^4*z^3 - 900*x^4*z^2 + 1940*x^4*z - 1414*x^4 + 140*x^3*y^4 - 560*x^3*y^3*z + 1200*x^3*y^3 + 840*x^3*y^2*z^2 - 3600*x^3*y^2*z + 3896*x^3*y^2 - 560*x^3*y*z^3 + 3600*x^3*y*z^2 - 7776*x^3*y*z + 5696*x^3*y + 140*x^3*z^4 - 1200*x^3*z^3 + 3880*x^3*z^2 - 5660*x^3*z + 3150*x^3 - 84*x^2*y^5 + 420*x^2*y^4*z - 900*x^2*y^4 - 840*x^2*y^3*z^2 + 3600*x^2*y^3*z - 3904*x^2*y^3 + 840*x^2*y^2*z^3 - 5400*x^2*y^2*z^2 + 11688*x^2*y^2*z - 8604*x^2*y^2 - 420*x^2*y*z^4 + 3600*x^2*y*z^3 - 11664*x^2*y*z^2 + 17100*x^2*y*z - 9594*x^2*y + 84*x^2*z^5 - 900*x^2*z^4 + 3880*x^2*z^3 - 8496*x^2*z^2 + 9474*x^2*z - 4300*x^2 + 28*x*y^6 - 168*x*y^5*z + 360*x*y^5 + 420*x*y^4*z^2 - 1800*x*y^4*z + 1956*x*y^4 - 560*x*y^3*z^3 + 3600*x*y^3*z^2 - 7808*x*y^3*z + 5776*x*y^3 + 420*x*y^2*z^4 - 3600*x*y^2*z^3 + 11688*x*y^2*z^2 - 17220*x*y^2*z + 9738*x*y^2 - 168*x*y*z^5 + 1800*x*y*z^4 - 7776*x*y*z^3 + 17112*x*y*z^2 - 19236*x*y*z + 8836*x*y + 28*x*z^6 - 360*x*z^5 + 1940*x*z^4 - 5668*x*z^3 + 9498*x*z^2 - 8652*x*z + 3337*x - 4*y^7 + 28*y^6*z - 60*y^6 - 84*y^5*z^2 + 360*y^5*z - 392*y^5 + 140*y^4*z^3 - 900*y^4*z^2 + 1956*y^4*z - 1454*y^4 - 140*y^3*z^4 + 1200*y^3*z^3 - 3904*y^3*z^2 + 5780*y^3*z - 3294*y^3 + 84*y^2*z^5 - 900*y^2*z^4 + 3896*y^2*z^3 - 8616*y^2*z^2 + 9762*y^2*z - 4540*y^2 - 28*y*z^6 + 360*y*z^5 - 1944*y*z^4 + 5708*y*z^3 - 9642*y*z^2 + 8892*y*z - 3487*y + 4*z^7 - 60*z^6 + 388*z^5 - 1418*z^4 + 3174*z^3 - 4352*z^2 + 3379*z - 1, 4*x^7 - 28*x^6*y + 28*x^6*z - 60*x^6 + 84*x^5*y^2 - 168*x^5*y*z + 360*x^5*y + 84*x^5*z^2 - 360*x^5*z + 388*x^5 - 140*x^4*y^3 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360*x*z^5 + 1940*x*z^4 - 5668*x*z^3 + 9495*x*z^2 - 8640*x*z + 3324*x - 4*y^7 + 28*y^6*z - 60*y^6 - 84*y^5*z^2 + 360*y^5*z - 392*y^5 + 140*y^4*z^3 - 900*y^4*z^2 + 1956*y^4*z - 1454*y^4 - 140*y^3*z^4 + 1200*y^3*z^3 - 3904*y^3*z^2 + 5780*y^3*z - 3293*y^3 + 84*y^2*z^5 - 900*y^2*z^4 + 3896*y^2*z^3 - 8616*y^2*z^2 + 9759*y^2*z - 4534*y^2 - 28*y*z^6 + 360*y*z^5 - 1944*y*z^4 + 5708*y*z^3 - 9639*y*z^2 + 8880*y*z - 3473*y + 4*z^7 - 60*z^6 + 388*z^5 - 1418*z^4 + 3173*z^3 - 4346*z^2 + 3366*z - 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -1, 1, -1 ], [ -1, 2, -1 ], [ -1, 1, -2 ] ], "source_shift": [ 2, 2, 1 ], "target_matrix": [ [ 3, 3, 2 ], [ -1, -2, -1 ], [ -1, -1, -1 ] ], "target_shift": [ -3411, 1135, 1125 ], "inverse_recipe": { "p": 3, "q": 4, "f": "x**3", "g": "x**4", "h": "y**2 + y*z + z", "polynomials": { "f": [ [ [ 3, 0, 0 ], 1 ] ], "g": [ [ [ 4, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 0, 1 ], 1 ], [ [ 0, 2, 0 ], 1 ] ] }, "h_multiplier": 4 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0109770039998693 ], "surface_features": [ 3.7439711642995745, 4.76182263324384, 3.288696260590256, 0.0, 3.934404511926432, 4913 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C160", "pair": 160, "family": "counterexample", "mapping": "F(x,y,z)=(x^7 - 8*x^6*y + 10*x^6*z - 19*x^6 + 27*x^5*y^2 - 69*x^5*y*z + 129*x^5*y + 42*x^5*z^2 - 162*x^5*z + 153*x^5 - 50*x^4*y^3 + 195*x^4*y^2*z - 360*x^4*y^2 - 243*x^4*y*z^2 + 924*x^4*y*z - 858*x^4*y + 96*x^4*z^3 - 564*x^4*z^2 + 1080*x^4*z - 678*x^4 + 55*x^3*y^4 - 290*x^3*y^3*z + 530*x^3*y^3 + 552*x^3*y^2*z^2 - 2076*x^3*y^2*z + 1902*x^3*y^2 - 447*x^3*y*z^3 + 2595*x^3*y*z^2 - 4890*x^3*y*z + 3020*x^3*y + 129*x^3*z^4 - 1026*x^3*z^3 + 2985*x^3*z^2 - 3799*x^3*z + 1789*x^3 - 36*x^2*y^5 + 240*x^2*y^4*z - 435*x^2*y^4 - 618*x^2*y^3*z^2 + 2304*x^2*y^3*z - 2088*x^2*y^3 + 765*x^2*y^2*z^3 - 4401*x^2*y^2*z^2 + 8190*x^2*y^2*z - 4992*x^2*y^2 - 453*x^2*y*z^4 + 3570*x^2*y*z^3 - 10236*x^2*y*z^2 + 12819*x^2*y*z - 5946*x^2*y + 102*x^2*z^5 - 1029*x^2*z^4 + 4038*x^2*z^3 - 7800*x^2*z^2 + 7446*x^2*z - 2818*x^2 + 13*x*y^6 - 105*x*y^5*z + 189*x*y^5 + 342*x*y^4*z^2 - 1266*x*y^4*z + 1137*x*y^4 - 573*x*y^3*z^3 + 3273*x*y^3*z^2 - 6030*x*y^3*z + 3636*x*y^3 + 519*x*y^2*z^4 - 4062*x*y^2*z^3 + 11517*x*y^2*z^2 - 14241*x*y^2*z + 6525*x*y^2 - 240*x*y*z^5 + 2406*x*y*z^4 - 9324*x*y*z^3 + 17736*x*y*z^2 - 16677*x*y*z + 6227*x*y + 44*x*z^6 - 540*x*z^5 + 2676*x*z^4 - 6957*x*z^3 + 10071*x*z^2 - 7724*x*z + 2463*x - 2*y^7 + 19*y^6*z - 34*y^6 - 75*y^5*z^2 + 276*y^5*z - 246*y^5 + 159*y^4*z^3 - 903*y^4*z^2 + 1650*y^4*z - 986*y^4 - 195*y^3*z^4 + 1518*y^3*z^3 - 4266*y^3*z^2 + 5221*y^3*z - 2368*y^3 + 138*y^2*z^5 - 1377*y^2*z^4 + 5286*y^2*z^3 - 9936*y^2*z^2 + 9231*y^2*z - 3409*y^2 - 52*y*z^6 + 636*y*z^5 - 3120*y*z^4 + 7997*y*z^3 - 11409*y*z^2 + 8636*y*z - 2722*y + 8*z^7 - 116*z^6 + 696*z^5 - 2278*z^4 + 4432*z^3 - 5146*z^2 + 3324*z - 1, 2*x^7 - 16*x^6*y + 20*x^6*z - 35*x^6 + 54*x^5*y^2 - 138*x^5*y*z + 237*x^5*y + 84*x^5*z^2 - 297*x^5*z + 264*x^5 - 100*x^4*y^3 + 390*x^4*y^2*z - 660*x^4*y^2 - 486*x^4*y*z^2 + 1689*x^4*y*z - 1476*x^4*y + 192*x^4*z^3 - 1029*x^4*z^2 + 1851*x^4*z - 1113*x^4 + 110*x^3*y^4 - 580*x^3*y^3*z + 970*x^3*y^3 + 1104*x^3*y^2*z^2 - 3786*x^3*y^2*z + 3264*x^3*y^2 - 894*x^3*y*z^3 + 4719*x^3*y*z^2 - 8352*x^3*y*z + 4945*x^3*y + 258*x^3*z^4 - 1863*x^3*z^3 + 5082*x^3*z^2 - 6191*x^3*z + 2834*x^3 - 72*x^2*y^5 + 480*x^2*y^4*z - 795*x^2*y^4 - 1236*x^2*y^3*z^2 + 4194*x^2*y^3*z - 3576*x^2*y^3 + 1530*x^2*y^2*z^3 - 7983*x^2*y^2*z^2 + 13950*x^2*y^2*z - 8157*x^2*y^2 - 906*x^2*y*z^4 + 6459*x^2*y*z^3 - 17358*x^2*y*z^2 + 20829*x^2*y*z - 9402*x^2*y + 204*x^2*z^5 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44*x*z^6 + 540*x*z^5 - 2676*x*z^4 + 6977*x*z^3 - 10155*x*z^2 + 7844*x*z - 2517*x + 2*y^7 - 19*y^6*z + 34*y^6 + 75*y^5*z^2 - 276*y^5*z + 246*y^5 - 159*y^4*z^3 + 903*y^4*z^2 - 1650*y^4*z + 988*y^4 + 195*y^3*z^4 - 1518*y^3*z^3 + 4266*y^3*z^2 - 5234*y^3*z + 2384*y^3 - 138*y^2*z^5 + 1377*y^2*z^4 - 5286*y^2*z^3 + 9966*y^2*z^2 - 9306*y^2*z + 3457*y^2 + 52*y*z^6 - 636*y*z^5 + 3120*y*z^4 - 8025*y*z^3 + 11517*y*z^2 - 8780*y*z + 2784*y - 8*z^7 + 116*z^6 - 696*z^5 + 2286*z^4 - 4476*z^3 + 5242*z^2 - 3412*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -1, 1, -2 ], [ -1, 1, -1 ], [ -1, 2, -1 ] ], "source_shift": [ 2, 2, 1 ], "target_matrix": [ [ -1, -1, -2 ], [ -2, -1, -1 ], [ 1, 1, 1 ] ], "target_shift": [ 924, 1397, -952 ], "collision_preimages": [ [ "11/4", "3/4", "0" ], [ "15", "9", "-5/2" ], [ "4", "6", "5/2" ] ], "collision_image": [ "3697/4", "2795/2", "-3809/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0181665920026717 ], "surface_features": [ 3.545410213079223, 4.456487497115823, 3.1389339402569236, 0.0, 3.6754240362687134, 4851 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C225", "pair": 225, "family": "counterexample", "mapping": "F(x,y,z)=(6*x^7 - 60*x^6*y - 40*x^6*z + 71*x^6 + 252*x^5*y^2 + 342*x^5*y*z - 615*x^5*y + 114*x^5*z^2 - 405*x^5*z + 369*x^5 - 576*x^4*y^3 - 1194*x^4*y^2*z + 2175*x^4*y^2 - 810*x^4*y*z^2 + 2913*x^4*y*z - 2676*x^4*y - 180*x^4*z^3 + 960*x^4*z^2 - 1752*x^4*z + 1097*x^4 + 774*x^3*y^4 + 2178*x^3*y^3*z - 4017*x^3*y^3 + 2256*x^3*y^2*z^2 - 8211*x^3*y^2*z + 7599*x^3*y^2 + 1020*x^3*y*z^3 - 5502*x^3*y*z^2 + 10116*x^3*y*z - 6370*x^3*y + 170*x^3*z^4 - 1210*x^3*z^3 + 3318*x^3*z^2 - 4164*x^3*z + 2020*x^3 - 612*x^2*y^5 - 2190*x^2*y^4*z + 4086*x^2*y^4 - 3078*x^2*y^3*z^2 + 11331*x^2*y^3*z - 10554*x^2*y^3 - 2124*x^2*y^2*z^3 + 11583*x^2*y^2*z^2 - 21432*x^2*y^2*z + 13542*x^2*y^2 - 720*x^2*y*z^4 + 5178*x^2*y*z^3 - 14292*x^2*y*z^2 + 18024*x^2*y*z - 8784*x^2*y - 96*x^2*z^5 + 855*x^2*z^4 - 3132*x^2*z^3 + 5910*x^2*z^2 - 5748*x^2*z + 2306*x^2 + 264*x*y^6 + 1152*x*y^5*z - 2172*x*y^5 + 2058*x*y^4*z^2 - 7656*x*y^4*z + 7170*x*y^4 + 1926*x*y^3*z^3 - 10611*x*y^3*z^2 + 19734*x*y^3*z - 12479*x*y^3 + 996*x*y^2*z^4 - 7233*x*y^2*z^3 + 20067*x*y^2*z^2 - 25365*x*y^2*z + 12375*x*y^2 + 270*x*y*z^5 - 2427*x*y*z^4 + 8940*x*y*z^3 - 16938*x*y*z^2 + 16536*x*y*z - 6658*x*y + 30*x*z^6 - 321*x*z^5 + 1473*x*z^4 - 3716*x*z^3 + 5436*x*z^2 - 4372*x*z + 1512*x - 48*y^7 - 248*y^6*z + 472*y^6 - 540*y^5*z^2 + 2028*y^5*z - 1908*y^5 - 642*y^4*z^3 + 3570*y^4*z^2 - 6666*y^4*z + 4210*y^4 - 450*y^3*z^4 + 3297*y^3*z^3 - 9180*y^3*z^2 + 11595*y^3*z - 5638*y^3 - 186*y^2*z^5 + 1686*y^2*z^4 - 6234*y^2*z^3 + 11823*y^2*z^2 - 11538*y^2*z + 4631*y^2 - 42*y*z^6 + 453*y*z^5 - 2088*y*z^4 + 5284*y*z^3 - 7752*y*z^2 + 6250*y*z - 2166*y - 4*z^7 + 50*z^6 - 276*z^5 + 873*z^4 - 1708*z^3 + 2066*z^2 - 1432*z + 1, 9*x^7 - 90*x^6*y - 60*x^6*z + 120*x^6 + 378*x^5*y^2 + 513*x^5*y*z - 1044*x^5*y + 171*x^5*z^2 - 684*x^5*z + 666*x^5 - 864*x^4*y^3 - 1791*x^4*y^2*z + 3708*x^4*y^2 - 1215*x^4*y*z^2 + 4941*x^4*y*z - 4860*x^4*y - 270*x^4*z^3 + 1620*x^4*z^2 - 3159*x^4*z + 2067*x^4 + 1161*x^3*y^4 + 3267*x^3*y^3*z - 6876*x^3*y^3 + 3384*x^3*y^2*z^2 - 13986*x^3*y^2*z + 13887*x^3*y^2 + 1530*x^3*y*z^3 - 9324*x^3*y*z^2 + 18351*x^3*y*z - 12087*x^3*y + 255*x^3*z^4 - 2040*x^3*z^3 + 5976*x^3*z^2 - 7838*x^3*z + 3919*x^3 - 918*x^2*y^5 - 3285*x^2*y^4*z + 7020*x^2*y^4 - 4617*x^2*y^3*z^2 + 19377*x^2*y^3*z - 19404*x^2*y^3 - 3186*x^2*y^2*z^3 + 19710*x^2*y^2*z^2 - 39114*x^2*y^2*z + 25875*x^2*y^2 - 1080*x^2*y*z^4 + 8766*x^2*y*z^3 - 25893*x^2*y*z^2 + 34158*x^2*y*z - 17175*x^2*y - 144*x^2*z^5 + 1440*x^2*z^4 - 5634*x^2*z^3 + 11112*x^2*z^2 - 11142*x^2*z + 4563*x^2 + 396*x*y^6 + 1728*x*y^5*z - 3744*x*y^5 + 3087*x*y^4*z^2 - 13140*x*y^4*z + 13257*x*y^4 + 2889*x*y^3*z^3 - 18126*x*y^3*z^2 + 36225*x*y^3*z - 24003*x*y^3 + 1494*x*y^2*z^4 - 12294*x*y^2*z^3 + 36567*x*y^2*z^2 - 48390*x*y^2*z + 24381*x*y^2 + 405*x*y*z^5 - 4104*x*y*z^4 + 16173*x*y*z^3 - 32055*x*y*z^2 + 32295*x*y*z - 13290*x*y + 45*x*z^6 - 540*x*z^5 + 2646*x*z^4 - 6978*x*z^3 + 10527*x*z^2 - 8646*x*z + 3025*x - 72*y^7 - 372*y^6*z + 816*y^6 - 810*y^5*z^2 + 3492*y^5*z - 3546*y^5 - 963*y^4*z^3 + 6120*y^4*z^2 - 12303*y^4*z + 8148*y^4 - 675*y^3*z^4 + 5625*y^3*z^3 - 16821*y^3*z^2 + 22258*y^3*z - 11183*y^3 - 279*y^2*z^5 + 2862*y^2*z^4 - 11340*y^2*z^3 + 22515*y^2*z^2 - 22695*y^2*z + 9321*y^2 - 63*y*z^6 + 765*y*z^5 - 3771*y*z^4 + 9984*y*z^3 - 15120*y*z^2 + 12465*y*z - 4382*y - 6*z^7 + 84*z^6 - 495*z^5 + 1637*z^4 - 3304*z^3 + 4083*z^2 - 2865*z + 1, 3*x^7 - 30*x^6*y - 20*x^6*z + 40*x^6 + 126*x^5*y^2 + 171*x^5*y*z - 348*x^5*y + 57*x^5*z^2 - 228*x^5*z + 222*x^5 - 288*x^4*y^3 - 597*x^4*y^2*z + 1236*x^4*y^2 - 405*x^4*y*z^2 + 1647*x^4*y*z - 1620*x^4*y - 90*x^4*z^3 + 540*x^4*z^2 - 1053*x^4*z + 688*x^4 + 387*x^3*y^4 + 1089*x^3*y^3*z 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2040*y^4*z^2 - 4101*y^4*z + 2708*y^4 - 225*y^3*z^4 + 1875*y^3*z^3 - 5607*y^3*z^2 + 7402*y^3*z - 3709*y^3 - 93*y^2*z^5 + 954*y^2*z^4 - 3780*y^2*z^3 + 7491*y^2*z^2 - 7533*y^2*z + 3085*y^2 - 21*y*z^6 + 255*y*z^5 - 1257*y*z^4 + 3323*y*z^3 - 5022*y*z^2 + 4130*y*z - 1450*y - 2*z^7 + 28*z^6 - 165*z^5 + 545*z^4 - 1098*z^3 + 1354*z^2 - 949*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ 1, -2, -1 ], [ 1, -1, -1 ], [ 3, -3, -2 ] ], "source_shift": [ 1, 2, 1 ], "target_matrix": [ [ 2, 1, 1 ], [ 3, 3, 2 ], [ 1, 1, 1 ] ], "target_shift": [ -437, -877, -290 ], "collision_preimages": [ [ "7/4", "-1", "19/4" ], [ "9", "-7/2", "16" ], [ "8", "3/2", "7" ] ], "collision_image": [ "-875/2", "-3511/4", "-1161/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.024201598000218 ], "surface_features": [ 3.7118283107859056, 4.684755622108624, 3.3469394626989906, 0.0, 3.8548858328755986, 4947 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B014", "pair": 14, "family": "bijection", "mapping": "F(x,y,z)=(2187*x^7 - 10206*x^6*y - 10206*x^6*z + 10206*x^6 + 20412*x^5*y^2 + 40824*x^5*y*z - 40824*x^5*y + 20412*x^5*z^2 - 40824*x^5*z + 20493*x^5 - 22680*x^4*y^3 - 68040*x^4*y^2*z + 68040*x^4*y^2 - 68040*x^4*y*z^2 + 136080*x^4*y*z - 68418*x^4*y - 22680*x^4*z^3 + 68040*x^4*z^2 - 68337*x^4*z + 23436*x^4 + 15120*x^3*y^4 + 60480*x^3*y^3*z - 60480*x^3*y^3 + 90720*x^3*y^2*z^2 - 181440*x^3*y^2*z + 91368*x^3*y^2 + 60480*x^3*y*z^3 - 181440*x^3*y*z^2 + 182520*x^3*y*z - 62775*x^3*y + 15120*x^3*z^4 - 60480*x^3*z^3 + 91152*x^3*z^2 - 62559*x^3*z + 16659*x^3 - 6048*x^2*y^5 - 30240*x^2*y^4*z + 30240*x^2*y^4 - 60480*x^2*y^3*z^2 + 120960*x^2*y^3*z - 61008*x^2*y^3 - 60480*x^2*y^2*z^3 + 181440*x^2*y^2*z^2 - 182808*x^2*y^2*z + 63054*x^2*y^2 - 30240*x^2*y*z^4 + 120960*x^2*y*z^3 - 182592*x^2*y*z^2 + 125676*x^2*y*z - 33588*x^2*y - 6048*x^2*z^5 + 30240*x^2*z^4 - 60792*x^2*z^3 + 62622*x^2*z^2 - 33372*x^2*z + 7355*x^2 + 1344*x*y^6 + 8064*x*y^5*z - 8064*x*y^5 + 20160*x*y^4*z^2 - 40320*x*y^4*z + 20368*x*y^4 + 26880*x*y^3*z^3 - 80640*x*y^3*z^2 + 81376*x*y^3*z - 28148*x*y^3 + 20160*x*y^2*z^4 - 80640*x*y^2*z^3 + 121920*x*y^2*z^2 - 84156*x*y^2*z + 22572*x*y^2 + 8064*x*y*z^5 - 40320*x*y*z^4 + 81184*x*y*z^3 - 83868*x*y*z^2 + 44856*x*y*z - 9929*x*y + 1344*x*z^6 - 8064*x*z^5 + 20272*x*z^4 - 27860*x*z^3 + 22284*x*z^2 - 9828*x*z + 1871*x - 128*y^7 - 896*y^6*z + 896*y^6 - 2688*y^5*z^2 + 5376*y^5*z - 2720*y^5 - 4480*y^4*z^3 + 13440*y^4*z^2 - 13584*y^4*z + 4712*y^4 - 4480*y^3*z^4 + 17920*y^3*z^3 - 27136*y^3*z^2 + 18784*y^3*z - 5056*y^3 - 2688*y^2*z^5 + 13440*y^2*z^4 - 27104*y^2*z^3 + 28080*y^2*z^2 - 15072*y^2*z + 3350*y^2 - 896*y*z^6 + 5376*y*z^5 - 13536*y*z^4 + 18656*y*z^3 - 14976*y*z^2 + 6633*y*z - 1263*y - 128*z^7 + 896*z^6 - 2704*z^5 + 4648*z^4 - 4960*z^3 + 3283*z^2 - 1248*z + 1, 2187*x^7 - 10206*x^6*y - 10206*x^6*z + 10206*x^6 + 20412*x^5*y^2 + 40824*x^5*y*z - 40824*x^5*y + 20412*x^5*z^2 - 40824*x^5*z + 20493*x^5 - 22680*x^4*y^3 - 68040*x^4*y^2*z + 68040*x^4*y^2 - 68040*x^4*y*z^2 + 136080*x^4*y*z - 68418*x^4*y - 22680*x^4*z^3 + 68040*x^4*z^2 - 68337*x^4*z + 23436*x^4 + 15120*x^3*y^4 + 60480*x^3*y^3*z - 60480*x^3*y^3 + 90720*x^3*y^2*z^2 - 181440*x^3*y^2*z + 91368*x^3*y^2 + 60480*x^3*y*z^3 - 181440*x^3*y*z^2 + 182520*x^3*y*z - 62775*x^3*y + 15120*x^3*z^4 - 60480*x^3*z^3 + 91152*x^3*z^2 - 62559*x^3*z + 16713*x^3 - 6048*x^2*y^5 - 30240*x^2*y^4*z + 30240*x^2*y^4 - 60480*x^2*y^3*z^2 + 120960*x^2*y^3*z - 61008*x^2*y^3 - 60480*x^2*y^2*z^3 + 181440*x^2*y^2*z^2 - 182808*x^2*y^2*z + 63054*x^2*y^2 - 30240*x^2*y*z^4 + 120960*x^2*y*z^3 - 182592*x^2*y*z^2 + 125676*x^2*y*z - 33696*x^2*y - 6048*x^2*z^5 + 30240*x^2*z^4 - 60792*x^2*z^3 + 62622*x^2*z^2 - 33480*x^2*z + 7463*x^2 + 1344*x*y^6 + 8064*x*y^5*z - 8064*x*y^5 + 20160*x*y^4*z^2 - 40320*x*y^4*z + 20368*x*y^4 + 26880*x*y^3*z^3 - 80640*x*y^3*z^2 + 81376*x*y^3*z - 28148*x*y^3 + 20160*x*y^2*z^4 - 80640*x*y^2*z^3 + 121920*x*y^2*z^2 - 84156*x*y^2*z + 22644*x*y^2 + 8064*x*y*z^5 - 40320*x*y*z^4 + 81184*x*y*z^3 - 83868*x*y*z^2 + 45000*x*y*z - 10073*x*y + 1344*x*z^6 - 8064*x*z^5 + 20272*x*z^4 - 27860*x*z^3 + 22356*x*z^2 - 9972*x*z + 1945*x - 128*y^7 - 896*y^6*z + 896*y^6 - 2688*y^5*z^2 + 5376*y^5*z - 2720*y^5 - 4480*y^4*z^3 + 13440*y^4*z^2 - 13584*y^4*z + 4712*y^4 - 4480*y^3*z^4 + 17920*y^3*z^3 - 27136*y^3*z^2 + 18784*y^3*z - 5072*y^3 - 2688*y^2*z^5 + 13440*y^2*z^4 - 27104*y^2*z^3 + 28080*y^2*z^2 - 15120*y^2*z + 3398*y^2 - 896*y*z^6 + 5376*y*z^5 - 13536*y*z^4 + 18656*y*z^3 - 15024*y*z^2 + 6729*y*z - 1315*y - 128*z^7 + 896*z^6 - 2704*z^5 + 4648*z^4 - 4976*z^3 + 3331*z^2 - 1298*z + 1, 4374*x^7 - 20412*x^6*y - 20412*x^6*z + 20412*x^6 + 40824*x^5*y^2 + 81648*x^5*y*z - 81648*x^5*y + 40824*x^5*z^2 - 81648*x^5*z + 40986*x^5 - 45360*x^4*y^3 - 136080*x^4*y^2*z + 136080*x^4*y^2 - 136080*x^4*y*z^2 + 272160*x^4*y*z - 136836*x^4*y - 45360*x^4*z^3 + 136080*x^4*z^2 - 136674*x^4*z + 46953*x^4 + 30240*x^3*y^4 + 120960*x^3*y^3*z - 120960*x^3*y^3 + 181440*x^3*y^2*z^2 - 362880*x^3*y^2*z + 182736*x^3*y^2 + 120960*x^3*y*z^3 - 362880*x^3*y*z^2 + 365040*x^3*y*z - 125766*x^3*y + 30240*x^3*z^4 - 120960*x^3*z^3 + 182304*x^3*z^2 - 125334*x^3*z + 33642*x^3 - 12096*x^2*y^5 - 60480*x^2*y^4*z + 60480*x^2*y^4 - 120960*x^2*y^3*z^2 + 241920*x^2*y^3*z - 122016*x^2*y^3 - 120960*x^2*y^2*z^3 + 362880*x^2*y^2*z^2 - 365616*x^2*y^2*z + 126324*x^2*y^2 - 60480*x^2*y*z^4 + 241920*x^2*y*z^3 - 365184*x^2*y*z^2 + 251784*x^2*y*z - 67824*x^2*y - 12096*x^2*z^5 + 60480*x^2*z^4 - 121584*x^2*z^3 + 125460*x^2*z^2 - 67392*x^2*z + 15142*x^2 + 2688*x*y^6 + 16128*x*y^5*z - 16128*x*y^5 + 40320*x*y^4*z^2 - 80640*x*y^4*z + 40736*x*y^4 + 53760*x*y^3*z^3 - 161280*x*y^3*z^2 + 162752*x*y^3*z - 56392*x*y^3 + 40320*x*y^2*z^4 - 161280*x*y^2*z^3 + 243840*x*y^2*z^2 - 168600*x*y^2*z + 45576*x*y^2 + 16128*x*y*z^5 - 80640*x*y*z^4 + 162368*x*y*z^3 - 168024*x*y*z^2 + 90576*x*y*z - 20434*x*y + 2688*x*z^6 - 16128*x*z^5 + 40544*x*z^4 - 55816*x*z^3 + 45000*x*z^2 - 20232*x*z + 3991*x - 256*y^7 - 1792*y^6*z + 1792*y^6 - 5376*y^5*z^2 + 10752*y^5*z - 5440*y^5 - 8960*y^4*z^3 + 26880*y^4*z^2 - 27168*y^4*z + 9440*y^4 - 8960*y^3*z^4 + 35840*y^3*z^3 - 54272*y^3*z^2 + 37632*y^3*z - 10208*y^3 - 5376*y^2*z^5 + 26880*y^2*z^4 - 54208*y^2*z^3 + 56256*y^2*z^2 - 30432*y^2*z + 6892*y^2 - 1792*y*z^6 + 10752*y*z^5 - 27072*y*z^4 + 37376*y*z^3 - 30240*y*z^2 + 13650*y*z - 2697*y - 256*z^7 + 1792*z^6 - 5408*z^5 + 9312*z^4 - 10016*z^3 + 6758*z^2 - 2663*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ 3, -2, -2 ], [ 2, -1, -1 ], [ 1, -2, -1 ] ], "source_shift": [ 2, 1, 1 ], "target_matrix": [ [ 1, 1, 2 ], [ 1, 1, 1 ], [ 2, 3, 2 ] ], "target_shift": [ -212, -230, -480 ], "inverse_recipe": { "p": 4, "q": 3, "f": "x**4 + x", "g": "x**3", "h": "y*z + 3*y", "polynomials": { "f": [ [ [ 4, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "g": [ [ [ 3, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 1, 0 ], 3 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0225367079983698 ], "surface_features": [ 4.731672666070346, 5.563025192968972, 4.446692466371527, 0.0, 4.82792673849781, 5347 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B124", "pair": 124, "family": "bijection", "mapping": "F(x,y,z)=(2*x^7 + 14*x^6*y - 28*x^6*z - 28*x^6 + 42*x^5*y^2 - 168*x^5*y*z - 168*x^5*y + 168*x^5*z^2 + 336*x^5*z + 172*x^5 + 70*x^4*y^3 - 420*x^4*y^2*z - 420*x^4*y^2 + 840*x^4*y*z^2 + 1680*x^4*y*z + 860*x^4*y - 560*x^4*z^3 - 1680*x^4*z^2 - 1718*x^4*z - 596*x^4 + 70*x^3*y^4 - 560*x^3*y^3*z - 560*x^3*y^3 + 1680*x^3*y^2*z^2 + 3360*x^3*y^2*z + 1720*x^3*y^2 - 2240*x^3*y*z^3 - 6720*x^3*y*z^2 - 6872*x^3*y*z - 2386*x^3*y + 1120*x^3*z^4 + 4480*x^3*z^3 + 6864*x^3*z^2 + 4750*x^3*z + 1249*x^3 + 42*x^2*y^5 - 420*x^2*y^4*z - 420*x^2*y^4 + 1680*x^2*y^3*z^2 + 3360*x^2*y^3*z + 1720*x^2*y^3 - 3360*x^2*y^2*z^3 - 10080*x^2*y^2*z^2 - 10308*x^2*y^2*z - 3582*x^2*y^2 + 3360*x^2*y*z^4 + 13440*x^2*y*z^3 + 20592*x^2*y*z^2 + 14262*x^2*y*z + 3759*x^2*y - 1344*x^2*z^5 - 6720*x^2*z^4 - 13712*x^2*z^3 - 14196*x^2*z^2 - 7434*x^2*z - 1578*x^2 + 14*x*y^6 - 168*x*y^5*z - 168*x*y^5 + 840*x*y^4*z^2 + 1680*x*y^4*z + 860*x*y^4 - 2240*x*y^3*z^3 - 6720*x*y^3*z^2 - 6872*x*y^3*z - 2390*x*y^3 + 3360*x*y^2*z^4 + 13440*x*y^2*z^3 + 20592*x*y^2*z^2 + 14274*x*y^2*z + 3771*x*y^2 - 2688*x*y*z^5 - 13440*x*y*z^4 - 27424*x*y*z^3 - 28416*x*y*z^2 - 14916*x*y*z - 3184*x*y + 896*x*z^6 + 5376*x*z^5 + 13696*x*z^4 + 18856*x*z^3 + 14748*x*z^2 + 6218*x*z + 1114*x + 2*y^7 - 28*y^6*z - 28*y^6 + 168*y^5*z^2 + 336*y^5*z + 172*y^5 - 560*y^4*z^3 - 1680*y^4*z^2 - 1718*y^4*z - 598*y^4 + 1120*y^3*z^4 + 4480*y^3*z^3 + 6864*y^3*z^2 + 4762*y^3*z + 1261*y^3 - 1344*y^2*z^5 - 6720*y^2*z^4 - 13712*y^2*z^3 - 14220*y^2*z^2 - 7482*y^2*z - 1606*y^2 + 896*y*z^6 + 5376*y*z^5 + 13696*y*z^4 + 18872*y*z^3 + 14796*y*z^2 + 6272*y*z + 1132*y - 256*z^7 - 1792*z^6 - 5472*z^5 - 9392*z^4 - 9752*z^3 - 6126*z^2 - 2173*z - 1, -2*x^7 - 14*x^6*y + 28*x^6*z + 28*x^6 - 42*x^5*y^2 + 168*x^5*y*z + 168*x^5*y - 168*x^5*z^2 - 336*x^5*z - 172*x^5 - 70*x^4*y^3 + 420*x^4*y^2*z + 420*x^4*y^2 - 840*x^4*y*z^2 - 1680*x^4*y*z - 860*x^4*y + 560*x^4*z^3 + 1680*x^4*z^2 + 1718*x^4*z + 598*x^4 - 70*x^3*y^4 + 560*x^3*y^3*z + 560*x^3*y^3 - 1680*x^3*y^2*z^2 - 3360*x^3*y^2*z - 1720*x^3*y^2 + 2240*x^3*y*z^3 + 6720*x^3*y*z^2 + 6872*x^3*y*z + 2394*x^3*y - 1120*x^3*z^4 - 4480*x^3*z^3 - 6864*x^3*z^2 - 4766*x^3*z - 1263*x^3 - 42*x^2*y^5 + 420*x^2*y^4*z + 420*x^2*y^4 - 1680*x^2*y^3*z^2 - 3360*x^2*y^3*z - 1720*x^2*y^3 + 3360*x^2*y^2*z^3 + 10080*x^2*y^2*z^2 + 10308*x^2*y^2*z + 3594*x^2*y^2 - 3360*x^2*y*z^4 - 13440*x^2*y*z^3 - 20592*x^2*y*z^2 - 14310*x^2*y*z - 3801*x^2*y + 1344*x^2*z^5 + 6720*x^2*z^4 + 13712*x^2*z^3 + 14244*x^2*z^2 + 7518*x^2*z + 1614*x^2 - 14*x*y^6 + 168*x*y^5*z + 168*x*y^5 - 840*x*y^4*z^2 - 1680*x*y^4*z - 860*x*y^4 + 2240*x*y^3*z^3 + 6720*x*y^3*z^2 + 6872*x*y^3*z + 2398*x*y^3 - 3360*x*y^2*z^4 - 13440*x*y^2*z^3 - 20592*x*y^2*z^2 - 14322*x*y^2*z - 3813*x*y^2 + 2688*x*y*z^5 + 13440*x*y*z^4 + 27424*x*y*z^3 + 28512*x*y*z^2 + 15084*x*y*z + 3256*x*y - 896*x*z^6 - 5376*x*z^5 - 13696*x*z^4 - 18920*x*z^3 - 14916*x*z^2 - 6362*x*z - 1152*x - 2*y^7 + 28*y^6*z + 28*y^6 - 168*y^5*z^2 - 336*y^5*z - 172*y^5 + 560*y^4*z^3 + 1680*y^4*z^2 + 1718*y^4*z + 600*y^4 - 1120*y^3*z^4 - 4480*y^3*z^3 - 6864*y^3*z^2 - 4778*y^3*z - 1275*y^3 + 1344*y^2*z^5 + 6720*y^2*z^4 + 13712*y^2*z^3 + 14268*y^2*z^2 + 7566*y^2*z + 1642*y^2 - 896*y*z^6 - 5376*y*z^5 - 13696*y*z^4 - 18936*y*z^3 - 14964*y*z^2 - 6416*y*z - 1172*y + 256*z^7 + 1792*z^6 + 5472*z^5 + 9424*z^4 + 9864*z^3 + 6270*z^2 + 2249*z + 1, x^7 + 7*x^6*y - 14*x^6*z - 14*x^6 + 21*x^5*y^2 - 84*x^5*y*z - 84*x^5*y + 84*x^5*z^2 + 168*x^5*z + 86*x^5 + 35*x^4*y^3 - 210*x^4*y^2*z - 210*x^4*y^2 + 420*x^4*y*z^2 + 840*x^4*y*z + 430*x^4*y - 280*x^4*z^3 - 840*x^4*z^2 - 859*x^4*z - 298*x^4 + 35*x^3*y^4 - 280*x^3*y^3*z - 280*x^3*y^3 + 840*x^3*y^2*z^2 + 1680*x^3*y^2*z + 860*x^3*y^2 - 1120*x^3*y*z^3 - 3360*x^3*y*z^2 - 3436*x^3*y*z - 1193*x^3*y + 560*x^3*z^4 + 2240*x^3*z^3 + 3432*x^3*z^2 + 2375*x^3*z + 625*x^3 + 21*x^2*y^5 - 210*x^2*y^4*z - 210*x^2*y^4 + 840*x^2*y^3*z^2 + 1680*x^2*y^3*z + 860*x^2*y^3 - 1680*x^2*y^2*z^3 - 5040*x^2*y^2*z^2 - 5154*x^2*y^2*z - 1791*x^2*y^2 + 1680*x^2*y*z^4 + 6720*x^2*y*z^3 + 10296*x^2*y*z^2 + 7131*x^2*y*z + 1881*x^2*y - 672*x^2*z^5 - 3360*x^2*z^4 - 6856*x^2*z^3 - 7098*x^2*z^2 - 3720*x^2*z - 792*x^2 + 7*x*y^6 - 84*x*y^5*z - 84*x*y^5 + 420*x*y^4*z^2 + 840*x*y^4*z + 430*x*y^4 - 1120*x*y^3*z^3 - 3360*x*y^3*z^2 - 3436*x*y^3*z - 1195*x*y^3 + 1680*x*y^2*z^4 + 6720*x*y^2*z^3 + 10296*x*y^2*z^2 + 7137*x*y^2*z + 1887*x*y^2 - 1344*x*y*z^5 - 6720*x*y*z^4 - 13712*x*y*z^3 - 14208*x*y*z^2 - 7464*x*y*z - 1598*x*y + 448*x*z^6 + 2688*x*z^5 + 6848*x*z^4 + 9428*x*z^3 + 7380*x*z^2 + 3121*x*z + 564*x + y^7 - 14*y^6*z - 14*y^6 + 84*y^5*z^2 + 168*y^5*z + 86*y^5 - 280*y^4*z^3 - 840*y^4*z^2 - 859*y^4*z - 299*y^4 + 560*y^3*z^4 + 2240*y^3*z^3 + 3432*y^3*z^2 + 2381*y^3*z + 631*y^3 - 672*y^2*z^5 - 3360*y^2*z^4 - 6856*y^2*z^3 - 7110*y^2*z^2 - 3744*y^2*z - 806*y^2 + 448*y*z^6 + 2688*y*z^5 + 6848*y*z^4 + 9436*y*z^3 + 7404*y*z^2 + 3148*y*z + 573*y - 128*z^7 - 896*z^6 - 2736*z^5 - 4696*z^4 - 4880*z^3 - 3075*z^2 - 1100*z - 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -1, -1, 2 ], [ -1, -1, 1 ], [ -1, -2, 1 ] ], "source_shift": [ 2, 1, 1 ], "target_matrix": [ [ -2, -3, -2 ], [ 2, 1, 1 ], [ -1, -2, -1 ] ], "target_shift": [ 342, -354, 176 ], "inverse_recipe": { "p": 3, "q": 4, "f": "x**3 + x", "g": "x**4", "h": "y*z", "polynomials": { "f": [ [ [ 3, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "g": [ [ [ 4, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0153034909999405 ], "surface_features": [ 3.6081205503698213, 4.455027682356781, 3.225309281725863, 0.0, 3.7307006210328835, 4892 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C076", "pair": 76, "family": "counterexample", "mapping": "F(x,y,z)=(128*x^7 - 896*x^6*y - 704*x^6*z + 1504*x^6 + 2592*x^5*y^2 + 4128*x^5*y*z - 9168*x^5*y + 1632*x^5*z^2 - 7056*x^5*z + 6672*x^5 - 4000*x^4*y^3 - 9696*x^4*y^2*z + 22560*x^4*y^2 - 7776*x^4*y*z^2 + 35040*x^4*y*z - 33792*x^4*y - 2064*x^4*z^3 + 13560*x^4*z^2 - 25776*x^4*z + 14720*x^4 + 3544*x^3*y^4 + 11632*x^3*y^3*z - 28568*x^3*y^3 + 14208*x^3*y^2*z^2 - 67224*x^3*y^2*z + 66024*x^3*y^2 + 7656*x^3*y*z^3 - 52548*x^3*y*z^2 + 101760*x^3*y*z - 58816*x^3*y + 1536*x^3*z^4 - 13644*x^3*z^3 + 39084*x^3*z^2 - 44824*x^3*z + 18208*x^3 - 1800*x^2*y^5 - 7500*x^2*y^4*z + 19542*x^2*y^4 - 12408*x^2*y^3*z^2 + 62016*x^2*y^3*z - 61800*x^2*y^3 - 10188*x^2*y^2*z^3 + 73530*x^2*y^2*z^2 - 144672*x^2*y^2*z + 84432*x^2*y^2 - 4152*x^2*y*z^4 + 38604*x^2*y*z^3 - 112452*x^2*y*z^2 + 130344*x^2*y*z - 53568*x^2*y - 672*x^2*z^5 + 7572*x^2*z^4 - 29028*x^2*z^3 + 50106*x^2*z^2 - 40944*x^2*z + 13040*x^2 + 486*x*y^6 + 2466*x*y^5*z - 6813*x*y^5 + 5178*x*y^4*z^2 - 27393*x*y^4*z + 27525*x*y^4 + 5758*x*y^3*z^3 - 43863*x*y^3*z^2 + 87240*x*y^3*z - 51220*x*y^3 + 3576*x*y^2*z^4 - 34971*x*y^2*z^3 + 103179*x*y^2*z^2 - 120486*x*y^2*z + 49956*x*y^2 + 1176*x*y*z^5 - 13884*x*y*z^4 + 53988*x*y*z^3 - 93999*x*y*z^2 + 77568*x*y*z - 25020*x*y + 160*x*z^6 - 2196*x*z^5 + 10548*x*z^4 - 24331*x*z^3 + 29961*x*z^2 - 19228*x*z + 5124*x - 54*y^7 - 324*y^6*z + 945*y^6 - 828*y^5*z^2 + 4626*y^5*z - 4662*y^5 - 1168*y^4*z^3 + 9390*y^4*z^2 - 18777*y^4*z + 11038*y^4 - 982*y^3*z^4 + 10118*y^3*z^3 - 30084*y^3*z^2 + 35242*y^3*z - 14660*y^3 - 492*y^2*z^5 + 6105*y^2*z^4 - 23973*y^2*z^3 + 41943*y^2*z^2 - 34794*y^2*z + 11310*y^2 - 136*y*z^6 + 1956*y*z^5 - 9504*y*z^4 + 22062*y*z^3 - 27354*y*z^2 + 17718*y*z - 4768*y - 16*z^7 + 260*z^6 - 1500*z^5 + 4329*z^4 - 7127*z^3 + 6896*z^2 - 3698*z + 1, 64*x^7 - 448*x^6*y - 352*x^6*z + 704*x^6 + 1296*x^5*y^2 + 2064*x^5*y*z - 4320*x^5*y + 816*x^5*z^2 - 3312*x^5*z + 2928*x^5 - 2000*x^4*y^3 - 4848*x^4*y^2*z + 10704*x^4*y^2 - 3888*x^4*y*z^2 + 16560*x^4*y*z - 14928*x^4*y - 1032*x^4*z^3 + 6384*x^4*z^2 - 11352*x^4*z + 6200*x^4 + 1772*x^3*y^4 + 5816*x^3*y^3*z - 13648*x^3*y^3 + 7104*x^3*y^2*z^2 - 31992*x^3*y^2*z + 29376*x^3*y^2 + 3828*x^3*y*z^3 - 24912*x^3*y*z^2 + 45132*x^3*y*z - 24948*x^3*y + 768*x^3*z^4 - 6444*x^3*z^3 + 17280*x^3*z^2 - 18960*x^3*z + 7480*x^3 - 900*x^2*y^5 - 3750*x^2*y^4*z + 9396*x^2*y^4 - 6204*x^2*y^3*z^2 + 29712*x^2*y^3*z - 27708*x^2*y^3 - 5094*x^2*y^2*z^3 + 35100*x^2*y^2*z^2 - 64650*x^2*y^2*z + 36102*x^2*y^2 - 2076*x^2*y*z^4 + 18360*x^2*y*z^3 - 50088*x^2*y*z^2 + 55560*x^2*y*z - 22176*x^2*y - 336*x^2*z^5 + 3588*x^2*z^4 - 12888*x^2*z^3 + 21294*x^2*z^2 - 16902*x^2*z + 5278*x^2 + 243*x*y^6 + 1233*x*y^5*z - 3294*x*y^5 + 2589*x*y^4*z^2 - 13203*x*y^4*z + 12441*x*y^4 + 2879*x*y^3*z^3 - 21072*x*y^3*z^2 + 39297*x*y^3*z - 22099*x*y^3 + 1788*x*y^2*z^4 - 16743*x*y^2*z^3 + 46320*x*y^2*z^2 - 51804*x*y^2*z + 20868*x*y^2 + 588*x*y*z^5 - 6624*x*y*z^4 + 24156*x*y*z^3 - 40281*x*y*z^2 + 32295*x*y*z - 10215*x*y + 80*x*z^6 - 1044*x*z^5 + 4704*x*z^4 - 10393*x*z^3 + 12435*x*z^2 - 7826*x*z + 2057*x - 27*y^7 - 162*y^6*z + 459*y^6 - 414*y^5*z^2 + 2241*y^5*z - 2124*y^5 - 584*y^4*z^3 + 4536*y^4*z^2 - 8526*y^4*z + 4807*y^4 - 491*y^3*z^4 + 4873*y^3*z^3 - 13614*y^3*z^2 + 15292*y^3*z - 6185*y^3 - 246*y^2*z^5 + 2931*y^2*z^4 - 10812*y^2*z^3 + 18135*y^2*z^2 - 14625*y^2*z + 4665*y^2 - 68*y*z^6 + 936*y*z^5 - 4272*y*z^4 + 9506*y*z^3 - 11457*y*z^2 + 7281*y*z - 1935*y - 8*z^7 + 124*z^6 - 672*z^5 + 1859*z^4 - 2975*z^3 + 2824*z^2 - 1495*z + 1, 64*x^7 - 448*x^6*y - 352*x^6*z + 800*x^6 + 1296*x^5*y^2 + 2064*x^5*y*z - 4848*x^5*y + 816*x^5*z^2 - 3744*x^5*z + 3744*x^5 - 2000*x^4*y^3 - 4848*x^4*y^2*z + 11856*x^4*y^2 - 3888*x^4*y*z^2 + 18480*x^4*y*z - 18864*x^4*y - 1032*x^4*z^3 + 7176*x^4*z^2 - 14424*x^4*z + 8528*x^4 + 1772*x^3*y^4 + 5816*x^3*y^3*z - 14920*x^3*y^3 + 7104*x^3*y^2*z^2 - 35232*x^3*y^2*z + 36648*x^3*y^2 + 3828*x^3*y*z^3 - 27636*x^3*y*z^2 + 56628*x^3*y*z - 33888*x^3*y + 768*x^3*z^4 - 7200*x^3*z^3 + 21804*x^3*z^2 - 25884*x^3*z + 10780*x^3 - 900*x^2*y^5 - 3750*x^2*y^4*z + 10146*x^2*y^4 - 6204*x^2*y^3*z^2 + 32304*x^2*y^3*z - 34092*x^2*y^3 - 5094*x^2*y^2*z^3 + 38430*x^2*y^2*z^2 - 80022*x^2*y^2*z + 48348*x^2*y^2 - 2076*x^2*y*z^4 + 20244*x^2*y*z^3 - 62364*x^2*y*z^2 + 74820*x^2*y*z - 31500*x^2*y - 336*x^2*z^5 + 3984*x^2*z^4 - 16140*x^2*z^3 + 28830*x^2*z^2 - 24138*x^2*z + 7840*x^2 + 243*x*y^6 + 1233*x*y^5*z - 3519*x*y^5 + 2589*x*y^4*z^2 - 14190*x*y^4*z + 15084*x*y^4 + 2879*x*y^3*z^3 - 22791*x*y^3*z^2 + 47943*x*y^3*z - 29128*x*y^3 + 1788*x*y^2*z^4 - 18228*x*y^2*z^3 + 56859*x*y^2*z^2 - 68703*x*y^2*z + 29157*x*y^2 + 588*x*y*z^5 - 7260*x*y*z^4 + 29832*x*y*z^3 - 53739*x*y*z^2 + 45399*x*y*z - 14910*x*y + 80*x*z^6 - 1152*x*z^5 + 5844*x*z^4 - 13945*x*z^3 + 17583*x*z^2 - 11495*x*z + 3106*x - 27*y^7 - 162*y^6*z + 486*y^6 - 414*y^5*z^2 + 2385*y^5*z - 2538*y^5 - 584*y^4*z^3 + 4854*y^4*z^2 - 10251*y^4*z + 6232*y^4 - 491*y^3*z^4 + 5245*y^3*z^3 - 16470*y^3*z^2 + 19954*y^3*z - 8489*y^3 - 246*y^2*z^5 + 3174*y^2*z^4 - 13161*y^2*z^3 + 23814*y^2*z^2 - 20208*y^2*z + 6678*y^2 - 68*y*z^6 + 1020*y*z^5 - 5232*y*z^4 + 12560*y*z^3 - 15933*y*z^2 + 10497*y*z - 2859*y - 8*z^7 + 136*z^6 - 828*z^5 + 2471*z^4 - 4163*z^3 + 4099*z^2 - 2226*z + 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ 2, -1, -1 ], [ 2, -3, -2 ], [ 1, -1, -1 ] ], "source_shift": [ 1, 2, 2 ], "target_matrix": [ [ 2, 3, 3 ], [ 1, 1, 1 ], [ 1, 2, 1 ] ], "target_shift": [ -857, -343, -519 ], "collision_preimages": [ [ "5/4", "-5/2", "6" ], [ "-9/2", "25/2", "-43/2" ], [ "-13/2", "19/2", "-41/2" ] ], "collision_image": [ "-1715/2", "-1373/4", "-2077/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.0133215179994295 ], "surface_features": [ 4.220091445579893, 5.1603844853542, 3.856608068436936, 0.0, 4.354596607356965, 5130 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C042", "pair": 42, "family": "counterexample", "mapping": "F(x,y,z)=(8*x^7 + 44*x^6*y - 52*x^6*z - 104*x^6 + 102*x^5*y^2 - 240*x^5*y*z - 480*x^5*y + 138*x^5*z^2 + 564*x^5*z + 546*x^5 + 129*x^4*y^3 - 453*x^4*y^2*z - 906*x^4*y^2 + 519*x^4*y*z^2 + 2112*x^4*y*z + 2061*x^4*y - 195*x^4*z^3 - 1206*x^4*z^2 - 2379*x^4*z - 1550*x^4 + 96*x^3*y^4 - 447*x^3*y^3*z - 894*x^3*y^3 + 765*x^3*y^2*z^2 + 3099*x^3*y^2*z + 3048*x^3*y^2 - 573*x^3*y*z^3 - 3516*x^3*y*z^2 - 6957*x^3*y*z - 4591*x^3*y + 159*x^3*z^4 + 1311*x^3*z^3 + 3909*x^3*z^2 + 5170*x^3*z + 2593*x^3 + 42*x^2*y^5 - 243*x^2*y^4*z - 486*x^2*y^4 + 552*x^2*y^3*z^2 + 2226*x^2*y^3*z + 2205*x^2*y^3 - 618*x^2*y^2*z^3 - 3762*x^2*y^2*z^2 - 7464*x^2*y^2*z - 4977*x^2*y^2 + 342*x^2*y*z^4 + 2790*x^2*y*z^3 + 8313*x^2*y*z^2 + 11070*x^2*y*z + 5613*x^2*y - 75*x^2*z^5 - 768*x^2*z^4 - 3054*x^2*z^3 - 6093*x^2*z^2 - 6171*x^2*z - 2539*x^2 + 10*x*y^6 - 69*x*y^5*z - 138*x*y^5 + 195*x*y^4*z^2 + 783*x*y^4*z + 780*x*y^4 - 290*x*y^3*z^3 - 1752*x*y^3*z^2 - 3483*x*y^3*z - 2338*x*y^3 + 240*x*y^2*z^4 + 1938*x*y^2*z^3 + 5769*x*y^2*z^2 + 7713*x*y^2*z + 3933*x*y^2 - 105*x*y*z^5 - 1062*x*y*z^4 - 4209*x*y*z^3 - 8412*x*y*z^2 - 8547*x*y*z - 3539*x*y + 19*x*z^6 + 231*x*z^5 + 1143*x*z^4 + 3037*x*z^3 + 4614*x*z^2 + 3812*x*z + 1344*x + y^7 - 8*y^6*z - 16*y^6 + 27*y^5*z^2 + 108*y^5*z + 108*y^5 - 50*y^4*z^3 - 300*y^4*z^2 - 597*y^4*z - 402*y^4 + 55*y^3*z^4 + 440*y^3*z^3 + 1308*y^3*z^2 + 1751*y^3*z + 894*y^3 - 36*y^2*z^5 - 360*y^2*z^4 - 1422*y^2*z^3 - 2841*y^2*z^2 - 2886*y^2*z - 1195*y^2 + 13*y*z^6 + 156*y*z^5 + 768*y*z^4 + 2037*y*z^3 + 3090*y*z^2 + 2549*y*z + 896*y - 2*z^7 - 28*z^6 - 165*z^5 - 545*z^4 - 1098*z^3 - 1354*z^2 - 949*z - 1, -24*x^7 - 132*x^6*y + 156*x^6*z + 288*x^6 - 306*x^5*y^2 + 720*x^5*y*z + 1332*x^5*y - 414*x^5*z^2 - 1560*x^5*z - 1446*x^5 - 387*x^4*y^3 + 1359*x^4*y^2*z + 2520*x^4*y^2 - 1557*x^4*y*z^2 - 5856*x^4*y*z - 5475*x^4*y + 585*x^4*z^3 + 3336*x^4*z^2 + 6297*x^4*z + 3992*x^4 - 288*x^3*y^4 + 1341*x^3*y^3*z + 2493*x^3*y^3 - 2295*x^3*y^2*z^2 - 8616*x^3*y^2*z - 8124*x^3*y^2 + 1719*x^3*y*z^3 + 9753*x^3*y*z^2 + 18483*x^3*y*z + 11864*x^3*y - 477*x^3*z^4 - 3630*x^3*z^3 - 10359*x^3*z^2 - 13327*x^3*z - 6557*x^3 - 126*x^2*y^5 + 729*x^2*y^4*z + 1359*x^2*y^4 - 1656*x^2*y^3*z^2 - 6207*x^2*y^3*z - 5898*x^2*y^3 + 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441*z^5 + 1418*z^4 + 2806*z^3 + 3420*z^2 + 2381*z + 1, 8*x^7 + 44*x^6*y - 52*x^6*z - 128*x^6 + 102*x^5*y^2 - 240*x^5*y*z - 588*x^5*y + 138*x^5*z^2 + 696*x^5*z + 738*x^5 + 129*x^4*y^3 - 453*x^4*y^2*z - 1104*x^4*y^2 + 519*x^4*y*z^2 + 2592*x^4*y*z + 2769*x^4*y - 195*x^4*z^3 - 1488*x^4*z^2 - 3219*x^4*z - 2216*x^4 + 96*x^3*y^4 - 447*x^3*y^3*z - 1083*x^3*y^3 + 765*x^3*y^2*z^2 + 3780*x^3*y^2*z + 4068*x^3*y^2 - 573*x^3*y*z^3 - 4311*x^3*y*z^2 - 9345*x^3*y*z - 6520*x^3*y + 159*x^3*z^4 + 1614*x^3*z^3 + 5277*x^3*z^2 + 7381*x^3*z + 3847*x^3 + 42*x^2*y^5 - 243*x^2*y^4*z - 585*x^2*y^4 + 552*x^2*y^3*z^2 + 2697*x^2*y^3*z + 2922*x^2*y^3 - 618*x^2*y^2*z^3 - 4581*x^2*y^2*z^2 - 9948*x^2*y^2*z - 7017*x^2*y^2 + 342*x^2*y*z^4 + 3411*x^2*y*z^3 + 11130*x^2*y*z^2 + 15675*x^2*y*z + 8262*x^2*y - 75*x^2*z^5 - 942*x^2*z^4 - 4104*x^2*z^3 - 8658*x^2*z^2 - 9111*x^2*z - 3856*x^2 + 10*x*y^6 - 69*x*y^5*z - 165*x*y^5 + 195*x*y^4*z^2 + 942*x*y^4*z + 1026*x*y^4 - 290*x*y^3*z^3 - 2118*x*y^3*z^2 - 4605*x*y^3*z - 3271*x*y^3 + 240*x*y^2*z^4 + 2352*x*y^2*z^3 + 7659*x*y^2*z^2 + 10830*x*y^2*z + 5739*x*y^2 - 105*x*y*z^5 - 1293*x*y*z^4 - 5607*x*y*z^3 - 11847*x*y*z^2 - 12501*x*y*z - 5318*x*y + 19*x*z^6 + 282*x*z^5 + 1527*x*z^4 + 4288*x*z^3 + 6762*x*z^2 + 5735*x*z + 2056*x + y^7 - 8*y^6*z - 19*y^6 + 27*y^5*z^2 + 129*y^5*z + 141*y^5 - 50*y^4*z^3 - 360*y^4*z^2 - 783*y^4*z - 558*y^4 + 55*y^3*z^4 + 530*y^3*z^3 + 1722*y^3*z^2 + 2438*y^3*z + 1293*y^3 - 36*y^2*z^5 - 435*y^2*z^4 - 1878*y^2*z^3 - 3966*y^2*z^2 - 4182*y^2*z - 1777*y^2 + 13*y*z^6 + 189*y*z^5 + 1017*y*z^4 + 2850*y*z^3 + 4485*y*z^2 + 3794*y*z + 1353*y - 2*z^7 - 34*z^6 - 219*z^5 - 764*z^4 - 1596*z^3 - 2017*z^2 - 1433*z - 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -2, -1, 1 ], [ -1, -1, 1 ], [ -1, -1, 2 ] ], "source_shift": [ 1, 2, 1 ], "target_matrix": [ [ -1, -1, -1 ], [ 3, 2, 2 ], [ -1, -2, -1 ] ], "target_shift": [ 290, -728, 439 ], "collision_preimages": [ [ "-1", "15/4", "3/4" ], [ "-7/2", "16", "9" ], [ "3/2", "5", "6" ] ], "collision_image": [ "1161/4", "-2915/4", "1757/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 0.9888530339994759 ], "surface_features": [ 3.482072432128786, 4.457170099784619, 3.1259689630925562, 0.0, 3.6307801441621965, 4842 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B044", "pair": 44, "family": "bijection", "mapping": "F(x,y,z)=(-6*x^7 - 42*x^6*y + 81*x^6*z + 81*x^6 - 126*x^5*y^2 + 486*x^5*y*z + 486*x^5*y - 468*x^5*z^2 - 936*x^5*z - 504*x^5 - 210*x^4*y^3 + 1215*x^4*y^2*z + 1215*x^4*y^2 - 2340*x^4*y*z^2 - 4680*x^4*y*z - 2508*x^4*y + 1500*x^4*z^3 + 4500*x^4*z^2 + 4794*x^4*z + 1782*x^4 - 210*x^3*y^4 + 1620*x^3*y^3*z + 1620*x^3*y^3 - 4680*x^3*y^2*z^2 - 9360*x^3*y^2*z - 4992*x^3*y^2 + 6000*x^3*y*z^3 + 18000*x^3*y*z^2 + 19086*x^3*y*z + 7038*x^3*y - 2880*x^3*z^4 - 11520*x^3*z^3 - 18216*x^3*z^2 - 13302*x^3*z - 3782*x^3 - 126*x^2*y^5 + 1215*x^2*y^4*z + 1215*x^2*y^4 - 4680*x^2*y^3*z^2 - 9360*x^2*y^3*z - 4968*x^2*y^3 + 9000*x^2*y^2*z^3 + 27000*x^2*y^2*z^2 + 28494*x^2*y^2*z + 10422*x^2*y^2 - 8640*x^2*y*z^4 - 34560*x^2*y*z^3 - 54396*x^2*y*z^2 - 39402*x^2*y*z - 11058*x^2*y + 3312*x^2*z^5 + 16560*x^2*z^4 + 34560*x^2*z^3 + 37188*x^2*z^2 + 20541*x^2*z + 4629*x^2 - 42*x*y^6 + 486*x*y^5*z + 486*x*y^5 - 2340*x*y^4*z^2 - 4680*x*y^4*z - 2472*x*y^4 + 6000*x*y^3*z^3 + 18000*x*y^3*z^2 + 18906*x*y^3*z + 6858*x*y^3 - 8640*x*y^2*z^4 - 34560*x*y^2*z^3 - 54144*x*y^2*z^2 - 38898*x*y^2*z - 10776*x*y^2 + 6624*x*y*z^5 + 33120*x*y*z^4 + 68808*x*y*z^3 + 73440*x*y*z^2 + 40068*x*y*z + 8880*x*y - 2112*x*z^6 - 12672*x*z^5 - 32736*x*z^4 - 46152*x*z^3 - 37236*x*z^2 - 16194*x*z - 2943*x - 6*y^7 + 81*y^6*z + 81*y^6 - 468*y^5*z^2 - 936*y^5*z - 492*y^5 + 1500*y^4*z^3 + 4500*y^4*z^2 + 4704*y^4*z + 1692*y^4 - 2880*y^3*z^4 - 11520*y^3*z^3 - 17964*y^3*z^2 - 12798*y^3*z - 3500*y^3 + 3312*y^2*z^5 + 16560*y^2*z^4 + 34248*y^2*z^3 + 36252*y^2*z^2 + 19536*y^2*z + 4260*y^2 - 2112*y*z^6 - 12672*y*z^5 - 32592*y*z^4 - 45576*y*z^3 - 36336*y*z^2 - 15564*y*z - 2779*y + 576*z^7 + 4032*z^6 + 12384*z^5 + 21456*z^4 + 22516*z^3 + 14232*z^2 + 4989*z + 1, -6*x^7 - 42*x^6*y + 81*x^6*z + 81*x^6 - 126*x^5*y^2 + 486*x^5*y*z + 486*x^5*y - 468*x^5*z^2 - 936*x^5*z - 504*x^5 - 210*x^4*y^3 + 1215*x^4*y^2*z + 1215*x^4*y^2 - 2340*x^4*y*z^2 - 4680*x^4*y*z - 2508*x^4*y + 1500*x^4*z^3 + 4500*x^4*z^2 + 4794*x^4*z + 1782*x^4 - 210*x^3*y^4 + 1620*x^3*y^3*z + 1620*x^3*y^3 - 4680*x^3*y^2*z^2 - 9360*x^3*y^2*z - 4992*x^3*y^2 + 6000*x^3*y*z^3 + 18000*x^3*y*z^2 + 19086*x^3*y*z + 7038*x^3*y - 2880*x^3*z^4 - 11520*x^3*z^3 - 18216*x^3*z^2 - 13302*x^3*z - 3781*x^3 - 126*x^2*y^5 + 1215*x^2*y^4*z + 1215*x^2*y^4 - 4680*x^2*y^3*z^2 - 9360*x^2*y^3*z - 4968*x^2*y^3 + 9000*x^2*y^2*z^3 + 27000*x^2*y^2*z^2 + 28494*x^2*y^2*z + 10422*x^2*y^2 - 8640*x^2*y*z^4 - 34560*x^2*y*z^3 - 54396*x^2*y*z^2 - 39402*x^2*y*z - 11055*x^2*y + 3312*x^2*z^5 + 16560*x^2*z^4 + 34560*x^2*z^3 + 37188*x^2*z^2 + 20535*x^2*z + 4623*x^2 - 42*x*y^6 + 486*x*y^5*z + 486*x*y^5 - 2340*x*y^4*z^2 - 4680*x*y^4*z - 2472*x*y^4 + 6000*x*y^3*z^3 + 18000*x*y^3*z^2 + 18906*x*y^3*z + 6858*x*y^3 - 8640*x*y^2*z^4 - 34560*x*y^2*z^3 - 54144*x*y^2*z^2 - 38898*x*y^2*z - 10773*x*y^2 + 6624*x*y*z^5 + 33120*x*y*z^4 + 68808*x*y*z^3 + 73440*x*y*z^2 + 40056*x*y*z + 8868*x*y - 2112*x*z^6 - 12672*x*z^5 - 32736*x*z^4 - 46152*x*z^3 - 37224*x*z^2 - 16170*x*z - 2924*x - 6*y^7 + 81*y^6*z + 81*y^6 - 468*y^5*z^2 - 936*y^5*z - 492*y^5 + 1500*y^4*z^3 + 4500*y^4*z^2 + 4704*y^4*z + 1692*y^4 - 2880*y^3*z^4 - 11520*y^3*z^3 - 17964*y^3*z^2 - 12798*y^3*z - 3499*y^3 + 3312*y^2*z^5 + 16560*y^2*z^4 + 34248*y^2*z^3 + 36252*y^2*z^2 + 19530*y^2*z + 4254*y^2 - 2112*y*z^6 - 12672*y*z^5 - 32592*y*z^4 - 45576*y*z^3 - 36324*y*z^2 - 15540*y*z - 2761*y + 576*z^7 + 4032*z^6 + 12384*z^5 + 21456*z^4 + 22508*z^3 + 14208*z^2 + 4957*z + 1, -6*x^7 - 42*x^6*y + 81*x^6*z + 81*x^6 - 126*x^5*y^2 + 486*x^5*y*z + 486*x^5*y - 468*x^5*z^2 - 936*x^5*z - 504*x^5 - 210*x^4*y^3 + 1215*x^4*y^2*z + 1215*x^4*y^2 - 2340*x^4*y*z^2 - 4680*x^4*y*z - 2508*x^4*y + 1500*x^4*z^3 + 4500*x^4*z^2 + 4794*x^4*z + 1782*x^4 - 210*x^3*y^4 + 1620*x^3*y^3*z + 1620*x^3*y^3 - 4680*x^3*y^2*z^2 - 9360*x^3*y^2*z - 4992*x^3*y^2 + 6000*x^3*y*z^3 + 18000*x^3*y*z^2 + 19086*x^3*y*z + 7038*x^3*y - 2880*x^3*z^4 - 11520*x^3*z^3 - 18216*x^3*z^2 - 13302*x^3*z - 3781*x^3 - 126*x^2*y^5 + 1215*x^2*y^4*z + 1215*x^2*y^4 - 4680*x^2*y^3*z^2 - 9360*x^2*y^3*z - 4968*x^2*y^3 + 9000*x^2*y^2*z^3 + 27000*x^2*y^2*z^2 + 28494*x^2*y^2*z + 10422*x^2*y^2 - 8640*x^2*y*z^4 - 34560*x^2*y*z^3 - 54396*x^2*y*z^2 - 39402*x^2*y*z - 11055*x^2*y + 3312*x^2*z^5 + 16560*x^2*z^4 + 34560*x^2*z^3 + 37188*x^2*z^2 + 20535*x^2*z + 4623*x^2 - 42*x*y^6 + 486*x*y^5*z + 486*x*y^5 - 2340*x*y^4*z^2 - 4680*x*y^4*z - 2472*x*y^4 + 6000*x*y^3*z^3 + 18000*x*y^3*z^2 + 18906*x*y^3*z + 6858*x*y^3 - 8640*x*y^2*z^4 - 34560*x*y^2*z^3 - 54144*x*y^2*z^2 - 38898*x*y^2*z - 10773*x*y^2 + 6624*x*y*z^5 + 33120*x*y*z^4 + 68808*x*y*z^3 + 73440*x*y*z^2 + 40056*x*y*z + 8868*x*y - 2112*x*z^6 - 12672*x*z^5 - 32736*x*z^4 - 46152*x*z^3 - 37224*x*z^2 - 16170*x*z - 2928*x - 6*y^7 + 81*y^6*z + 81*y^6 - 468*y^5*z^2 - 936*y^5*z - 492*y^5 + 1500*y^4*z^3 + 4500*y^4*z^2 + 4704*y^4*z + 1692*y^4 - 2880*y^3*z^4 - 11520*y^3*z^3 - 17964*y^3*z^2 - 12798*y^3*z - 3499*y^3 + 3312*y^2*z^5 + 16560*y^2*z^4 + 34248*y^2*z^3 + 36252*y^2*z^2 + 19530*y^2*z + 4254*y^2 - 2112*y*z^6 - 12672*y*z^5 - 32592*y*z^4 - 45576*y*z^3 - 36324*y*z^2 - 15540*y*z - 2765*y + 576*z^7 + 4032*z^6 + 12384*z^5 + 21456*z^4 + 22508*z^3 + 14208*z^2 + 4963*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -1, -1, 2 ], [ -3, -2, 2 ], [ -1, -1, 1 ] ], "source_shift": [ 2, 1, 1 ], "target_matrix": [ [ 1, 2, 1 ], [ 1, 1, 2 ], [ 1, 1, 1 ] ], "target_shift": [ -742, -727, -733 ], "inverse_recipe": { "p": 3, "q": 1, "f": "x**3", "g": "x", "h": "y**2*z", "polynomials": { "f": [ [ [ 3, 0, 0 ], 1 ] ], "g": [ [ [ 1, 0, 0 ], 1 ] ], "h": [ [ [ 0, 2, 1 ], 1 ] ] }, "h_multiplier": 3 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.067911398000433 ], "surface_features": [ 4.098304462986223, 4.865932668193186, 3.6961815871685237, 0.0, 4.188403774706915, 5082 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B060", "pair": 60, "family": "bijection", "mapping": "F(x,y,z)=(-256*x^7 + 896*x^6*y - 896*x^6*z + 2048*x^6 - 1344*x^5*y^2 + 2688*x^5*y*z - 6144*x^5*y - 1344*x^5*z^2 + 6144*x^5*z - 7136*x^5 + 1120*x^4*y^3 - 3360*x^4*y^2*z + 7680*x^4*y^2 + 3360*x^4*y*z^2 - 15360*x^4*y*z + 17856*x^4*y - 1120*x^4*z^3 + 7680*x^4*z^2 - 17856*x^4*z 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840*x^2*y^2*z^3 + 5760*x^2*y^2*z^2 - 13416*x^2*y^2*z + 10590*x^2*y^2 + 420*x^2*y*z^4 - 3840*x^2*y*z^3 + 13416*x^2*y*z^2 - 21192*x^2*y*z + 12752*x^2*y - 84*x^2*z^5 + 960*x^2*z^4 - 4472*x^2*z^3 + 10602*x^2*z^2 - 12780*x^2*z + 6254*x^2 - 14*x*y^6 + 84*x*y^5*z - 192*x*y^5 - 210*x*y^4*z^2 + 960*x*y^4*z - 1119*x*y^4 + 280*x*y^3*z^3 - 1920*x*y^3*z^2 + 4476*x*y^3*z - 3541*x*y^3 - 210*x*y^2*z^4 + 1920*x*y^2*z^3 - 6714*x*y^2*z^2 + 10629*x*y^2*z - 6422*x*y^2 + 84*x*y*z^5 - 960*x*y*z^4 + 4476*x*y*z^3 - 10635*x*y*z^2 + 12872*x*y*z - 6340*x*y - 14*x*z^6 + 192*x*z^5 - 1119*x*z^4 + 3547*x*z^3 - 6450*x*z^2 + 6373*x*z - 2667*x + y^7 - 7*y^6*z + 16*y^6 + 21*y^5*z^2 - 96*y^5*z + 112*y^5 - 35*y^4*z^3 + 240*y^4*z^2 - 560*y^4*z + 444*y^4 + 35*y^3*z^4 - 320*y^3*z^3 + 1120*y^3*z^2 - 1777*y^3*z + 1078*y^3 - 21*y^2*z^5 + 240*y^2*z^4 - 1120*y^2*z^3 + 2667*y^2*z^2 - 3241*y^2*z + 1607*y^2 + 7*y*z^6 - 96*y*z^5 + 560*y*z^4 - 1779*y*z^3 + 3248*y*z^2 - 3231*y*z + 1366*y - z^7 + 16*z^6 - 112*z^5 + 445*z^4 - 1085*z^3 + 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7098*x*y^3 - 420*x*y^2*z^4 + 3840*x*y^2*z^3 - 13428*x*y^2*z^2 + 21306*x*y^2*z - 12934*x*y^2 + 168*x*y*z^5 - 1920*x*y*z^4 + 8952*x*y*z^3 - 21318*x*y*z^2 + 25924*x*y*z - 12852*x*y - 28*x*z^6 + 384*x*z^5 - 2238*x*z^4 + 7110*x*z^3 - 12990*x*z^2 + 12918*x*z - 5447*x + 2*y^7 - 14*y^6*z + 32*y^6 + 42*y^5*z^2 - 192*y^5*z + 224*y^5 - 70*y^4*z^3 + 480*y^4*z^2 - 1120*y^4*z + 890*y^4 + 70*y^3*z^4 - 640*y^3*z^3 + 2240*y^3*z^2 - 3562*y^3*z + 2171*y^3 - 42*y^2*z^5 + 480*y^2*z^4 - 2240*y^2*z^3 + 5346*y^2*z^2 - 6527*y^2*z + 3257*y^2 + 14*y*z^6 - 192*y*z^5 + 1120*y*z^4 - 3566*y*z^3 + 6541*y*z^2 - 6548*y*z + 2789*y - 2*z^7 + 32*z^6 - 224*z^5 + 892*z^4 - 2185*z^3 + 3291*z^2 - 2817*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -2, 1, -1 ], [ -1, 1, -1 ], [ -1, 1, -2 ] ], "source_shift": [ 2, 2, 2 ], "target_matrix": [ [ 2, 3, 2 ], [ 1, 1, 1 ], [ 2, 1, 1 ] ], "target_shift": [ -1037, -511, -1053 ], "inverse_recipe": { "p": 3, "q": 4, "f": "x**3 + x**2", "g": "x**4 + x**2", "h": "y**2 + y*z + y + z", "polynomials": { "f": [ [ [ 3, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "g": [ [ [ 4, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 0, 1 ], 1 ], [ [ 0, 1, 0 ], 1 ], [ [ 0, 2, 0 ], 1 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.1779899489993113 ], "surface_features": [ 3.754509721901987, 4.628184508073413, 3.3498600821923312, 0.0, 3.888257806457232, 4934 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C119", "pair": 119, "family": "counterexample", "mapping": "F(x,y,z)=(16*x^7 - 128*x^6*y + 104*x^6*z - 164*x^6 + 432*x^5*y^2 - 708*x^5*y*z + 1128*x^5*y + 288*x^5*z^2 - 912*x^5*z + 684*x^5 - 796*x^4*y^3 + 1974*x^4*y^2*z - 3177*x^4*y^2 - 1620*x^4*y*z^2 + 5184*x^4*y*z - 3912*x^4*y + 440*x^4*z^3 - 2100*x^4*z^2 + 3168*x^4*z - 1554*x^4 + 863*x^3*y^4 - 2879*x^3*y^3*z + 4679*x^3*y^3 + 3576*x^3*y^2*z^2 - 11562*x^3*y^2*z + 8781*x^3*y^2 - 1960*x^3*y*z^3 + 9456*x^3*y*z^2 - 14364*x^3*y*z + 7067*x^3*y + 400*x^3*z^4 - 2560*x^3*z^3 + 5832*x^3*z^2 - 5752*x^3*z + 2104*x^3 - 549*x^2*y^5 + 2310*x^2*y^4*z - 3789*x^2*y^4 - 3861*x^2*y^3*z^2 + 12609*x^2*y^3*z - 9642*x^2*y^3 + 3204*x^2*y^2*z^3 - 15624*x^2*y^2*z^2 + 23910*x^2*y^2*z - 11799*x^2*y^2 - 1320*x^2*y*z^4 + 8544*x^2*y*z^3 - 19620*x^2*y*z^2 + 19425*x^2*y*z - 7101*x^2*y + 216*x^2*z^5 - 1740*x^2*z^4 + 5328*x^2*z^3 - 7932*x^2*z^2 + 5826*x^2*z - 1714*x^2 + 189*x*y^6 - 963*x*y^5*z + 1593*x*y^5 + 2031*x*y^4*z^2 - 6696*x*y^4*z + 5160*x*y^4 - 2269*x*y^3*z^3 + 11181*x*y^3*z^2 - 17253*x*y^3*z + 8544*x*y^3 + 1416*x*y^2*z^4 - 9270*x*y^2*z^3 + 21477*x*y^2*z^2 - 21354*x*y^2*z + 7797*x*y^2 - 468*x*y*z^5 + 3816*x*y*z^4 - 11796*x*y*z^3 + 17649*x*y*z^2 - 12963*x*y*z + 3794*x*y + 64*x*z^6 - 624*x*z^5 + 2412*x*z^4 - 4824*x*z^3 + 5340*x*z^2 - 3146*x*z + 785*x - 27*y^7 + 162*y^6*z - 270*y^6 - 414*y^5*z^2 + 1377*y^5*z - 1071*y^5 + 584*y^4*z^3 - 2907*y^4*z^2 + 4530*y^4*z - 2255*y^4 - 491*y^3*z^4 + 3251*y^3*z^3 - 7611*y^3*z^2 + 7609*y^3*z - 2776*y^3 + 246*y^2*z^5 - 2031*y^2*z^4 + 6348*y^2*z^3 - 9555*y^2*z^2 + 7017*y^2*z - 2038*y^2 - 68*y*z^6 + 672*y*z^5 - 2628*y*z^4 + 5291*y*z^3 - 5862*y*z^2 + 3434*y*z - 847*y + 8*z^7 - 92*z^6 + 432*z^5 - 1090*z^4 + 1618*z^3 - 1432*z^2 + 712*z - 1, -16*x^7 + 128*x^6*y - 104*x^6*z + 212*x^6 - 432*x^5*y^2 + 708*x^5*y*z - 1464*x^5*y - 288*x^5*z^2 + 1176*x^5*z - 984*x^5 + 796*x^4*y^3 - 1974*x^4*y^2*z + 4137*x^4*y^2 + 1620*x^4*y*z^2 - 6708*x^4*y*z + 5652*x^4*y - 440*x^4*z^3 + 2700*x^4*z^2 - 4548*x^4*z + 2382*x^4 - 863*x^3*y^4 + 2879*x^3*y^3*z - 6107*x^3*y^3 - 3576*x^3*y^2*z^2 + 15000*x^3*y^2*z - 12732*x^3*y^2 + 1960*x^3*y*z^3 - 12192*x^3*y*z^2 + 20700*x^3*y*z - 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276696*y^2*z^3 - 192834*y^2*z^2 - 69528*y^2*z - 10415*y^2 - 30618*y*z^6 - 122472*y*z^5 - 207441*y*z^4 - 192645*y*z^3 - 104121*y*z^2 - 31173*y*z - 4061*y - 6561*z^7 - 30618*z^6 - 62208*z^5 - 72171*z^4 - 51975*z^3 - 23325*z^2 - 6079*z - 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ 2, 2, 3 ], [ 1, 1, 2 ], [ 2, 1, 1 ] ], "source_shift": [ 2, 1, 2 ], "target_matrix": [ [ -2, -1, -1 ], [ 1, 1, 1 ], [ -3, -2, -3 ] ], "target_shift": [ 486, -243, 710 ], "inverse_recipe": { "p": 4, "q": 3, "f": "x**4 + x**2", "g": "x**3", "h": "y*z + y + z", "polynomials": { "f": [ [ [ 4, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "g": [ [ [ 3, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 0, 1 ], 1 ], [ [ 0, 1, 0 ], 1 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0668570000016189 ], "surface_features": [ 4.911681510033761, 5.744217855417277, 4.613376634694834, 0.0, 5.0197864798706, 5409 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B057", "pair": 57, "family": "bijection", "mapping": "F(x,y,z)=(256*x^7 - 896*x^6*y - 896*x^6*z + 1024*x^6 + 1344*x^5*y^2 + 2688*x^5*y*z - 3264*x^5*y + 1344*x^5*z^2 - 3136*x^5*z + 1840*x^5 - 1120*x^4*y^3 - 3360*x^4*y^2*z + 4320*x^4*y^2 - 3360*x^4*y*z^2 + 8320*x^4*y*z - 5144*x^4*y - 1120*x^4*z^3 + 4000*x^4*z^2 - 4792*x^4*z + 1960*x^4 + 560*x^3*y^4 + 2240*x^3*y^3*z - 3040*x^3*y^3 + 3360*x^3*y^2*z^2 - 8800*x^3*y^2*z + 5724*x^3*y^2 + 2240*x^3*y*z^3 - 8480*x^3*y*z^2 + 10696*x^3*y*z - 4584*x^3*y + 560*x^3*z^4 - 2720*x^3*z^3 + 4988*x^3*z^2 - 4168*x^3*z + 1326*x^3 - 168*x^2*y^5 - 840*x^2*y^4*z + 1200*x^2*y^4 - 1680*x^2*y^3*z^2 + 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input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0094006619983702 ], "surface_features": [ 3.414294929218696, 4.330251389917909, 3.0791812460476247, 0.0, 3.5268809749013195, 4850 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C198", "pair": 198, "family": "counterexample", "mapping": "F(x,y,z)=(-4*x^7 - 42*x^6*y - 40*x^6*z - 38*x^6 - 186*x^5*y^2 - 366*x^5*y*z - 339*x^5*y - 168*x^5*z^2 - 330*x^5*z - 162*x^5 - 450*x^4*y^3 - 1374*x^4*y^2*z - 1242*x^4*y^2 - 1302*x^4*y*z^2 - 2487*x^4*y*z - 1194*x^4*y - 384*x^4*z^3 - 1176*x^4*z^2 - 1185*x^4*z - 398*x^4 - 642*x^3*y^4 - 2706*x^3*y^3*z - 2391*x^3*y^3 - 3978*x^3*y^2*z^2 - 7392*x^3*y^2*z - 3468*x^3*y^2 - 2418*x^3*y*z^3 - 7185*x^3*y*z^2 - 7080*x^3*y*z - 2323*x^3*y - 516*x^3*z^4 - 2202*x^3*z^3 - 3420*x^3*z^2 - 2354*x^3*z - 608*x^3 - 540*x^2*y^5 - 2946*x^2*y^4*z - 2550*x^2*y^4 - 5982*x^2*y^3*z^2 - 10827*x^2*y^3*z - 4962*x^2*y^3 - 5634*x^2*y^2*z^3 - 16236*x^2*y^2*z^2 - 15627*x^2*y^2*z - 5004*x^2*y^2 - 2472*x^2*y*z^4 - 10221*x^2*y*z^3 - 15528*x^2*y*z^2 - 10443*x^2*y*z - 2628*x^2*y - 408*x^2*z^5 - 2286*x^2*z^4 - 4869*x^2*z^3 - 5154*x^2*z^2 - 2727*x^2*z - 578*x^2 - 248*x*y^6 - 1680*x*y^5*z - 1428*x*y^5 - 4422*x*y^4*z^2 - 7812*x*y^4*z - 3498*x*y^4 - 5750*x*y^3*z^3 - 16077*x*y^3*z^2 - 15102*x*y^3*z - 4717*x*y^3 - 3900*x*y^2*z^4 - 15600*x*y^2*z^3 - 23148*x*y^2*z^2 - 15192*x*y^2*z - 3720*x*y^2 - 1320*x*y*z^5 - 7164*x*y*z^4 - 14934*x*y*z^3 - 15450*x*y*z^2 - 7968*x*y*z - 1639*x*y - 176*x*z^6 - 1248*x*z^5 - 3420*x*z^4 - 4952*x*z^3 - 4026*x*z^2 - 1748*x*z - 318*x - 48*y^7 - 392*y^6*z - 328*y^6 - 1284*y^5*z^2 - 2220*y^5*z - 972*y^5 - 2166*y^4*z^3 - 5886*y^4*z^2 - 5394*y^4*z - 1642*y^4 - 2024*y^3*z^4 - 7825*y^3*z^3 - 11328*y^3*z^2 - 7255*y^3*z - 1728*y^3 - 1056*y^2*z^5 - 5538*y^2*z^4 - 11277*y^2*z^3 - 11376*y^2*z^2 - 5709*y^2*z - 1142*y^2 - 288*y*z^6 - 1980*y*z^5 - 5316*y*z^4 - 7532*y*z^3 - 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1353*x^3*y^3*z - 1422*x^3*y^3 - 1989*x^3*y^2*z^2 - 4311*x^3*y^2*z - 2226*x^3*y^2 - 1209*x^3*y*z^3 - 4104*x^3*y*z^2 - 4452*x^3*y*z - 1541*x^3*y - 258*x^3*z^4 - 1233*x^3*z^3 - 2106*x^3*z^2 - 1537*x^3*z - 406*x^3 - 270*x^2*y^5 - 1473*x^2*y^4*z - 1530*x^2*y^4 - 2991*x^2*y^3*z^2 - 6375*x^2*y^3*z - 3216*x^2*y^3 - 2817*x^2*y^2*z^3 - 9360*x^2*y^2*z^2 - 9924*x^2*y^2*z - 3342*x^2*y^2 - 1236*x^2*y*z^4 - 5769*x^2*y*z^3 - 9654*x^2*y*z^2 - 6879*x^2*y*z - 1761*x^2*y - 204*x^2*z^5 - 1266*x^2*z^4 - 2964*x^2*z^3 - 3333*x^2*z^2 - 1815*x^2*z - 385*x^2 - 124*x*y^6 - 840*x*y^5*z - 864*x*y^5 - 2211*x*y^4*z^2 - 4644*x*y^4*z - 2289*x*y^4 - 2875*x*y^3*z^3 - 9360*x*y^3*z^2 - 9687*x*y^3*z - 3167*x*y^3 - 1950*x*y^2*z^4 - 8883*x*y^2*z^3 - 14535*x*y^2*z^2 - 10092*x*y^2*z - 2496*x*y^2 - 660*x*y*z^5 - 3996*x*y*z^4 - 9177*x*y*z^3 - 10086*x*y*z^2 - 5334*x*y*z - 1091*x*y - 88*x*z^6 - 684*x*z^5 - 2058*x*z^4 - 3169*x*z^3 - 2664*x*z^2 - 1165*x*z - 212*x - 24*y^7 - 196*y^6*z - 200*y^6 - 642*y^5*z^2 - 1332*y^5*z - 642*y^5 - 1083*y^4*z^3 - 3462*y^4*z^2 - 3495*y^4*z - 1106*y^4 - 1012*y^3*z^4 - 4499*y^3*z^3 - 7185*y^3*z^2 - 4856*y^3*z - 1158*y^3 - 528*y^2*z^5 - 3114*y^2*z^4 - 6999*y^2*z^3 - 7497*y^2*z^2 - 3837*y^2*z - 757*y^2 - 144*y*z^6 - 1092*y*z^5 - 3228*y*z^4 - 4867*y*z^3 - 3981*y*z^2 - 1675*y*z - 295*y - 16*z^7 - 152*z^6 - 564*z^5 - 1114*z^4 - 1284*z^3 - 871*z^2 - 323*z - 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ 1, 2, 1 ], [ 1, 1, 2 ], [ 2, 3, 2 ] ], "source_shift": [ 1, 1, 1 ], "target_matrix": [ [ -2, -1, -1 ], [ 1, 1, 1 ], [ -1, -1, -2 ] ], "target_shift": [ 75, -53, 51 ], "collision_preimages": [ [ "5/4", "-3/4", "-3/4" ], [ "19", "-11/2", "-8" ], [ "24", "-19/2", "-7" ] ], "collision_image": [ "151/2", "-213/4", "205/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 0.9805547110008774 ], "surface_features": [ 3.4828906471100867, 4.36451347369151, 3.188365926063148, 0.0, 3.5651755783516617, 4910 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "C102", "pair": 102, "family": "counterexample", "mapping": "F(x,y,z)=(-x^7 - 8*x^6*y + 10*x^6*z + 19*x^6 - 27*x^5*y^2 + 69*x^5*y*z + 129*x^5*y - 42*x^5*z^2 - 162*x^5*z - 153*x^5 - 50*x^4*y^3 + 195*x^4*y^2*z + 360*x^4*y^2 - 243*x^4*y*z^2 - 924*x^4*y*z - 858*x^4*y + 96*x^4*z^3 + 564*x^4*z^2 + 1080*x^4*z + 678*x^4 - 55*x^3*y^4 + 290*x^3*y^3*z + 530*x^3*y^3 - 552*x^3*y^2*z^2 - 2076*x^3*y^2*z - 1902*x^3*y^2 + 447*x^3*y*z^3 + 2595*x^3*y*z^2 + 4890*x^3*y*z + 3020*x^3*y - 129*x^3*z^4 - 1026*x^3*z^3 - 2985*x^3*z^2 - 3799*x^3*z - 1789*x^3 - 36*x^2*y^5 + 240*x^2*y^4*z + 435*x^2*y^4 - 618*x^2*y^3*z^2 - 2304*x^2*y^3*z - 2088*x^2*y^3 + 765*x^2*y^2*z^3 + 4401*x^2*y^2*z^2 + 8190*x^2*y^2*z + 4992*x^2*y^2 - 453*x^2*y*z^4 - 3570*x^2*y*z^3 - 10236*x^2*y*z^2 - 12819*x^2*y*z - 5946*x^2*y + 102*x^2*z^5 + 1029*x^2*z^4 + 4038*x^2*z^3 + 7800*x^2*z^2 + 7446*x^2*z + 2818*x^2 - 13*x*y^6 + 105*x*y^5*z + 189*x*y^5 - 342*x*y^4*z^2 - 1266*x*y^4*z - 1137*x*y^4 + 573*x*y^3*z^3 + 3273*x*y^3*z^2 + 6030*x*y^3*z + 3636*x*y^3 - 519*x*y^2*z^4 - 4062*x*y^2*z^3 - 11517*x*y^2*z^2 - 14241*x*y^2*z - 6525*x*y^2 + 240*x*y*z^5 + 2406*x*y*z^4 + 9324*x*y*z^3 + 17736*x*y*z^2 + 16677*x*y*z + 6227*x*y - 44*x*z^6 - 540*x*z^5 - 2676*x*z^4 - 6957*x*z^3 - 10071*x*z^2 - 7724*x*z - 2463*x - 2*y^7 + 19*y^6*z + 34*y^6 - 75*y^5*z^2 - 276*y^5*z - 246*y^5 + 159*y^4*z^3 + 903*y^4*z^2 + 1650*y^4*z + 986*y^4 - 195*y^3*z^4 - 1518*y^3*z^3 - 4266*y^3*z^2 - 5221*y^3*z - 2368*y^3 + 138*y^2*z^5 + 1377*y^2*z^4 + 5286*y^2*z^3 + 9936*y^2*z^2 + 9231*y^2*z + 3409*y^2 - 52*y*z^6 - 636*y*z^5 - 3120*y*z^4 - 7997*y*z^3 - 11409*y*z^2 - 8636*y*z - 2722*y + 8*z^7 + 116*z^6 + 696*z^5 + 2278*z^4 + 4432*z^3 + 5146*z^2 + 3324*z + 1, x^7 + 8*x^6*y - 10*x^6*z - 19*x^6 + 27*x^5*y^2 - 69*x^5*y*z - 129*x^5*y + 42*x^5*z^2 + 162*x^5*z + 153*x^5 + 50*x^4*y^3 - 195*x^4*y^2*z - 360*x^4*y^2 + 243*x^4*y*z^2 + 924*x^4*y*z + 858*x^4*y - 96*x^4*z^3 - 564*x^4*z^2 - 1080*x^4*z - 679*x^4 + 55*x^3*y^4 - 290*x^3*y^3*z - 530*x^3*y^3 + 552*x^3*y^2*z^2 + 2076*x^3*y^2*z + 1902*x^3*y^2 - 447*x^3*y*z^3 - 2595*x^3*y*z^2 - 4890*x^3*y*z - 3025*x^3*y + 129*x^3*z^4 + 1026*x^3*z^3 + 2985*x^3*z^2 + 3806*x^3*z + 1799*x^3 + 36*x^2*y^5 - 240*x^2*y^4*z - 435*x^2*y^4 + 618*x^2*y^3*z^2 + 2304*x^2*y^3*z + 2088*x^2*y^3 - 765*x^2*y^2*z^3 - 4401*x^2*y^2*z^2 - 8190*x^2*y^2*z - 5001*x^2*y^2 + 453*x^2*y*z^4 + 3570*x^2*y*z^3 + 10236*x^2*y*z^2 + 12846*x^2*y*z + 5982*x^2*y - 102*x^2*z^5 - 1029*x^2*z^4 - 4038*x^2*z^3 - 7818*x^2*z^2 - 7497*x^2*z - 2854*x^2 + 13*x*y^6 - 105*x*y^5*z - 189*x*y^5 + 342*x*y^4*z^2 + 1266*x*y^4*z + 1137*x*y^4 - 573*x*y^3*z^3 - 3273*x*y^3*z^2 - 6030*x*y^3*z - 3643*x*y^3 + 519*x*y^2*z^4 + 4062*x*y^2*z^3 + 11517*x*y^2*z^2 + 14274*x*y^2*z + 6567*x*y^2 - 240*x*y*z^5 - 2406*x*y*z^4 - 9324*x*y*z^3 - 17784*x*y*z^2 - 16803*x*y*z - 6311*x*y + 44*x*z^6 + 540*x*z^5 + 2676*x*z^4 + 6977*x*z^3 + 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2688*x^2*y^3 - 765*x^2*y^2*z^3 - 5220*x^2*y^2*z^2 - 10620*x^2*y^2*z - 6855*x^2*y^2 + 453*x^2*y*z^4 + 4251*x^2*y*z^3 + 13350*x^2*y*z^2 + 17736*x^2*y*z + 8580*x^2*y - 102*x^2*z^5 - 1227*x^2*z^4 - 5283*x^2*z^3 - 10848*x^2*z^2 - 10827*x^2*z - 4237*x^2 + 13*x*y^6 - 105*x*y^5*z - 222*x*y^5 + 342*x*y^4*z^2 + 1497*x*y^4*z + 1467*x*y^4 - 573*x*y^3*z^3 - 3894*x*y^3*z^2 - 7842*x*y^3*z - 5005*x*y^3 + 519*x*y^2*z^4 + 4857*x*y^2*z^3 + 15081*x*y^2*z^2 + 19770*x*y^2*z + 9441*x*y^2 - 240*x*y*z^5 - 2886*x*y*z^4 - 12270*x*y*z^3 - 24792*x*y*z^2 - 24351*x*y*z - 9392*x*y + 44*x*z^6 + 648*x*z^5 + 3528*x*z^4 + 9761*x*z^3 + 14799*x*z^2 + 11765*x*z + 3847*x + 2*y^7 - 19*y^6*z - 40*y^6 + 75*y^5*z^2 + 327*y^5*z + 318*y^5 - 159*y^4*z^3 - 1077*y^4*z^2 - 2151*y^4*z - 1360*y^4 + 195*y^3*z^4 + 1821*y^3*z^3 + 5604*y^3*z^2 + 7268*y^3*z + 3434*y^3 - 138*y^2*z^5 - 1659*y^2*z^4 - 6987*y^2*z^3 - 13944*y^2*z^2 - 13524*y^2*z - 5155*y^2 + 52*y*z^6 + 768*y*z^5 + 4140*y*z^4 + 11289*y*z^3 + 16857*y*z^2 + 13211*y*z + 4264*y - 8*z^7 - 140*z^6 - 924*z^5 - 3222*z^4 - 6576*z^3 - 7936*z^2 - 5273*z - 1)", "determinant": "-2", "certificate": "Expanded det J=-2 checked; three distinct rational preimages evaluated to the same image.", "certificate_data": { "source_matrix": [ [ -1, -1, 2 ], [ -1, -1, 1 ], [ -1, -2, 1 ] ], "source_shift": [ 2, 2, 1 ], "target_matrix": [ [ 1, 1, 2 ], [ -1, -1, -1 ], [ -1, -2, -1 ] ], "target_shift": [ -924, 952, 1488 ], "collision_preimages": [ [ "11/4", "-3/4", "0" ], [ "15", "-9", "5/2" ], [ "4", "-6", "-5/2" ] ], "collision_image": [ "-3697/4", "3809/4", "5953/4" ] }, "probe_target": "No", "jev_answers": [ "No" ], "seconds": [ 1.01743571499901 ], "surface_features": [ 3.5121798420560766, 4.394311563231226, 3.0557604646877348, 0.0, 3.642185279573408, 4829 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B209", "pair": 209, "family": "bijection", "mapping": "F(x,y,z)=(-2*x^7 - 14*x^6*y - 14*x^6*z - 34*x^6 - 42*x^5*y^2 - 84*x^5*y*z - 202*x^5*y - 42*x^5*z^2 - 202*x^5*z - 230*x^5 - 70*x^4*y^3 - 210*x^4*y^2*z - 500*x^4*y^2 - 210*x^4*y*z^2 - 1000*x^4*y*z - 1130*x^4*y - 70*x^4*z^3 - 500*x^4*z^2 - 1130*x^4*z - 822*x^4 - 70*x^3*y^4 - 280*x^3*y^3*z - 660*x^3*y^3 - 420*x^3*y^2*z^2 - 1980*x^3*y^2*z - 2220*x^3*y^2 - 280*x^3*y*z^3 - 1980*x^3*y*z^2 - 4440*x^3*y*z - 3208*x^3*y - 70*x^3*z^4 - 660*x^3*z^3 - 2220*x^3*z^2 - 3208*x^3*z - 1706*x^3 - 42*x^2*y^5 - 210*x^2*y^4*z - 490*x^2*y^4 - 420*x^2*y^3*z^2 - 1960*x^2*y^3*z - 2180*x^2*y^3 - 420*x^2*y^2*z^3 - 2940*x^2*y^2*z^2 - 6540*x^2*y^2*z - 4692*x^2*y^2 - 210*x^2*y*z^4 - 1960*x^2*y*z^3 - 6540*x^2*y*z^2 - 9384*x^2*y*z - 4958*x^2*y - 42*x^2*z^5 - 490*x^2*z^4 - 2180*x^2*z^3 - 4692*x^2*z^2 - 4956*x^2*z - 2099*x^2 - 14*x*y^6 - 84*x*y^5*z - 194*x*y^5 - 210*x*y^4*z^2 - 970*x*y^4*z - 1070*x*y^4 - 280*x*y^3*z^3 - 1940*x*y^3*z^2 - 4280*x*y^3*z - 3048*x*y^3 - 210*x*y^2*z^4 - 1940*x*y^2*z^3 - 6420*x*y^2*z^2 - 9144*x*y^2*z - 4798*x*y^2 - 84*x*y*z^5 - 970*x*y*z^4 - 4280*x*y*z^3 - 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"source_shift": [ 2, 1, 2 ], "target_matrix": [ [ -1, -1, -2 ], [ -2, -1, -1 ], [ 1, 1, 1 ] ], "target_shift": [ 436, 1106, -516 ], "inverse_recipe": { "p": 2, "q": 5, "f": "x**2 + x", "g": "x**5 + x**2 + x", "h": "y*z + 2*y", "polynomials": { "f": [ [ [ 2, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "g": [ [ [ 5, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ], [ [ 0, 1, 0 ], 2 ] ] }, "h_multiplier": 2 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.023463986999559 ], "surface_features": [ 3.377396820846521, 4.314372900009328, 3.0, 0.0, 3.523207005134884, 4794 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" }, { "case": "B072", "pair": 72, "family": "bijection", "mapping": "F(x,y,z)=(-x^7 - 14*x^6*y + 7*x^6*z + 14*x^6 - 84*x^5*y^2 + 84*x^5*y*z + 168*x^5*y - 21*x^5*z^2 - 84*x^5*z - 88*x^5 - 280*x^4*y^3 + 420*x^4*y^2*z + 840*x^4*y^2 - 210*x^4*y*z^2 - 840*x^4*y*z - 877*x^4*y + 35*x^4*z^3 + 210*x^4*z^2 + 439*x^4*z + 320*x^4 - 560*x^3*y^4 + 1120*x^3*y^3*z + 2240*x^3*y^3 - 840*x^3*y^2*z^2 - 3360*x^3*y^2*z - 3496*x^3*y^2 + 280*x^3*y*z^3 + 1680*x^3*y*z^2 + 3500*x^3*y*z + 2535*x^3*y - 35*x^3*z^4 - 280*x^3*z^3 - 876*x^3*z^2 - 1272*x^3*z - 720*x^3 - 672*x^2*y^5 + 1680*x^2*y^4*z + 3360*x^2*y^4 - 1680*x^2*y^3*z^2 - 6720*x^2*y^3*z - 6968*x^2*y^3 + 840*x^2*y^2*z^3 + 5040*x^2*y^2*z^2 + 10464*x^2*y^2*z + 7530*x^2*y^2 - 210*x^2*y*z^4 - 1680*x^2*y*z^3 - 5238*x^2*y*z^2 - 7557*x^2*y*z - 4239*x^2*y + 21*x^2*z^5 + 210*x^2*z^4 + 874*x^2*z^3 + 1896*x^2*z^2 + 2135*x^2*z + 996*x^2 - 448*x*y^6 + 1344*x*y^5*z + 2688*x*y^5 - 1680*x*y^4*z^2 - 6720*x*y^4*z - 6944*x*y^4 + 1120*x*y^3*z^3 + 6720*x*y^3*z^2 + 13904*x*y^3*z + 9940*x*y^3 - 420*x*y^2*z^4 - 3360*x*y^2*z^3 - 10440*x*y^2*z^2 - 14964*x*y^2*z - 8316*x*y^2 + 84*x*y*z^5 + 840*x*y*z^4 + 3484*x*y*z^3 + 7509*x*y*z^2 + 8378*x*y*z + 3855*x*y - 7*x*z^6 - 84*x*z^5 - 436*x*z^4 - 1256*x*z^3 - 2110*x*z^2 - 1954*x*z - 772*x - 128*y^7 + 448*y^6*z + 896*y^6 - 672*y^5*z^2 - 2688*y^5*z - 2768*y^5 + 560*y^4*z^3 + 3360*y^4*z^2 + 6928*y^4*z + 4920*y^4 - 280*y^3*z^4 - 2240*y^3*z^3 - 6936*y^3*z^2 - 9876*y^3*z - 5436*y^3 + 84*y^2*z^5 + 840*y^2*z^4 + 3472*y^2*z^3 + 7434*y^2*z^2 + 8216*y^2*z + 3729*y^2 - 14*y*z^6 - 168*y*z^5 - 869*y*z^4 - 2487*y*z^3 - 4139*y*z^2 - 3780*y*z - 1469*y + z^7 + 14*z^6 + 87*z^5 + 312*z^4 + 695*z^3 + 958*z^2 + 751*z + 1, -x^7 - 14*x^6*y + 7*x^6*z + 14*x^6 - 84*x^5*y^2 + 84*x^5*y*z + 168*x^5*y - 21*x^5*z^2 - 84*x^5*z - 88*x^5 - 280*x^4*y^3 + 420*x^4*y^2*z + 840*x^4*y^2 - 210*x^4*y*z^2 - 840*x^4*y*z - 877*x^4*y + 35*x^4*z^3 + 210*x^4*z^2 + 439*x^4*z + 320*x^4 - 560*x^3*y^4 + 1120*x^3*y^3*z + 2240*x^3*y^3 - 840*x^3*y^2*z^2 - 3360*x^3*y^2*z - 3496*x^3*y^2 + 280*x^3*y*z^3 + 1680*x^3*y*z^2 + 3500*x^3*y*z + 2535*x^3*y - 35*x^3*z^4 - 280*x^3*z^3 - 876*x^3*z^2 - 1272*x^3*z - 718*x^3 - 672*x^2*y^5 + 1680*x^2*y^4*z + 3360*x^2*y^4 - 1680*x^2*y^3*z^2 - 6720*x^2*y^3*z - 6968*x^2*y^3 + 840*x^2*y^2*z^3 + 5040*x^2*y^2*z^2 + 10464*x^2*y^2*z + 7530*x^2*y^2 - 210*x^2*y*z^4 - 1680*x^2*y*z^3 - 5238*x^2*y*z^2 - 7557*x^2*y*z - 4227*x^2*y + 21*x^2*z^5 + 210*x^2*z^4 + 874*x^2*z^3 + 1896*x^2*z^2 + 2129*x^2*z + 984*x^2 - 448*x*y^6 + 1344*x*y^5*z + 2688*x*y^5 - 1680*x*y^4*z^2 - 6720*x*y^4*z - 6944*x*y^4 + 1120*x*y^3*z^3 + 6720*x*y^3*z^2 + 13904*x*y^3*z + 9940*x*y^3 - 420*x*y^2*z^4 - 3360*x*y^2*z^3 - 10440*x*y^2*z^2 - 14964*x*y^2*z - 8292*x*y^2 + 84*x*y*z^5 + 840*x*y*z^4 + 3484*x*y*z^3 + 7509*x*y*z^2 + 8354*x*y*z + 3807*x*y - 7*x*z^6 - 84*x*z^5 - 436*x*z^4 - 1256*x*z^3 - 2104*x*z^2 - 1930*x*z - 742*x - 128*y^7 + 448*y^6*z + 896*y^6 - 672*y^5*z^2 - 2688*y^5*z - 2768*y^5 + 560*y^4*z^3 + 3360*y^4*z^2 + 6928*y^4*z + 4920*y^4 - 280*y^3*z^4 - 2240*y^3*z^3 - 6936*y^3*z^2 - 9876*y^3*z - 5420*y^3 + 84*y^2*z^5 + 840*y^2*z^4 + 3472*y^2*z^3 + 7434*y^2*z^2 + 8192*y^2*z + 3681*y^2 - 14*y*z^6 - 168*y*z^5 - 869*y*z^4 - 2487*y*z^3 - 4127*y*z^2 - 3732*y*z - 1415*y + z^7 + 14*z^6 + 87*z^5 + 312*z^4 + 693*z^3 + 946*z^2 + 723*z + 1, -2*x^7 - 28*x^6*y + 14*x^6*z + 28*x^6 - 168*x^5*y^2 + 168*x^5*y*z + 336*x^5*y - 42*x^5*z^2 - 168*x^5*z - 176*x^5 - 560*x^4*y^3 + 840*x^4*y^2*z + 1680*x^4*y^2 - 420*x^4*y*z^2 - 1680*x^4*y*z - 1754*x^4*y + 70*x^4*z^3 + 420*x^4*z^2 + 878*x^4*z + 639*x^4 - 1120*x^3*y^4 + 2240*x^3*y^3*z + 4480*x^3*y^3 - 1680*x^3*y^2*z^2 - 6720*x^3*y^2*z - 6992*x^3*y^2 + 560*x^3*y*z^3 + 3360*x^3*y*z^2 + 7000*x^3*y*z + 5062*x^3*y - 70*x^3*z^4 - 560*x^3*z^3 - 1752*x^3*z^2 - 2540*x^3*z - 1434*x^3 - 1344*x^2*y^5 + 3360*x^2*y^4*z + 6720*x^2*y^4 - 3360*x^2*y^3*z^2 - 13440*x^2*y^3*z - 13936*x^2*y^3 + 1680*x^2*y^2*z^3 + 10080*x^2*y^2*z^2 + 20928*x^2*y^2*z + 15036*x^2*y^2 - 420*x^2*y*z^4 - 3360*x^2*y*z^3 - 10476*x^2*y*z^2 - 15090*x^2*y*z - 8442*x^2*y + 42*x^2*z^5 + 420*x^2*z^4 + 1748*x^2*z^3 + 3786*x^2*z^2 + 4252*x^2*z + 1979*x^2 - 896*x*y^6 + 2688*x*y^5*z + 5376*x*y^5 - 3360*x*y^4*z^2 - 13440*x*y^4*z - 13888*x*y^4 + 2240*x*y^3*z^3 + 13440*x*y^3*z^2 + 27808*x*y^3*z + 19848*x*y^3 - 840*x*y^2*z^4 - 6720*x*y^2*z^3 - 20880*x*y^2*z^2 - 29880*x*y^2*z - 16560*x*y^2 + 168*x*y*z^5 + 1680*x*y*z^4 + 6968*x*y*z^3 + 14994*x*y*z^2 + 16684*x*y*z + 7658*x*y - 14*x*z^6 - 168*x*z^5 - 872*x*z^4 - 2508*x*z^3 - 4202*x*z^2 - 3882*x*z - 1536*x - 256*y^7 + 896*y^6*z + 1792*y^6 - 1344*y^5*z^2 - 5376*y^5*z - 5536*y^5 + 1120*y^4*z^3 + 6720*y^4*z^2 + 13856*y^4*z + 9824*y^4 - 560*y^3*z^4 - 4480*y^3*z^3 - 13872*y^3*z^2 - 19720*y^3*z - 10824*y^3 + 168*y^2*z^5 + 1680*y^2*z^4 + 6944*y^2*z^3 + 14844*y^2*z^2 + 16360*y^2*z + 7406*y^2 - 28*y*z^6 - 336*y*z^5 - 1738*y*z^4 - 4966*y*z^3 - 8242*y*z^2 - 7508*y*z - 2917*y + 2*z^7 + 28*z^6 + 174*z^5 + 623*z^4 + 1384*z^3 + 1903*z^2 + 1492*z + 1)", "determinant": "-2", "certificate": "Composition of explicitly invertible polynomial shears, z scaling by -2, and unimodular affine maps; expanded det J=-2 checked.", "certificate_data": { "source_matrix": [ [ -1, -2, 1 ], [ -1, -1, 1 ], [ -2, -1, 1 ] ], "source_shift": [ 2, 1, 1 ], "target_matrix": [ [ 1, 1, 1 ], [ 1, 1, 2 ], [ 2, 1, 1 ] ], "target_shift": [ -255, -233, -510 ], "inverse_recipe": { "p": 4, "q": 3, "f": "x**4 + x**2 + x", "g": "x**3 + x", "h": "y*z", "polynomials": { "f": [ [ [ 4, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ], [ [ 2, 0, 0 ], 1 ] ], "g": [ [ [ 3, 0, 0 ], 1 ], [ [ 1, 0, 0 ], 1 ] ], "h": [ [ [ 0, 1, 1 ], 1 ] ] }, "h_multiplier": 1 }, "inverse_order": "(u,v,w)=target_inverse*(output-target_shift); Z=-w/2; Y=v; X=u-k*h(Y,Z); input=source_inverse*((X,Y-f(X),Z-g(X))-source_shift)." }, "probe_target": "Maybe", "jev_answers": [ "No" ], "seconds": [ 1.0369783329988422 ], "surface_features": [ 3.5299761590913663, 4.4753805931433615, 3.170408411725318, 0.0, 3.664456965031444, 4842 ], "component_degrees": [ 7, 7, 7 ], "component_terms": [ 120, 120, 120 ], "coefficient_domain": "Z" } ] }