{
  "seed": 20260921,
  "question": "Is the Jacobian conjecture true",
  "repeats": 1,
  "baseline": [],
  "baseline_seconds": [],
  "completed_calls": 500,
  "total_calls": 500,
  "complete": true,
  "rows": [
    {
      "case": "B13",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-6*x^2 + y)*(z + 4*(-6*x^2 + y)^2)^2, -6*x^2 + y, z + 4*(-6*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7720165069995346
      ]
    },
    {
      "case": "B224",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (3*x^2 - y)*(z - 6*(-3*x^2 + y)^2)^2, -3*x^2 + y, z - 6*(-3*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        1.0463053559997206
      ]
    },
    {
      "case": "C249",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(2*(y + 1)^2*((x + 1)*(y + 1) + 1)*(3*(x + 1)*(y + 1) + 4) + 2*(z + 2)*((x + 1)*(y + 1) + 1)^3 + 1, y + 3*(x + 1)*(y + 1)^2*(3*(x + 1)*(y + 1) + 4) + 3*(x + 1)*(z + 2)*((x + 1)*(y + 1) + 1)^2 + 3, 2*x - (x + 1)^3*(z + 2) - 3*(x + 1)^2*(y + 1) + 4)",
      "determinant": "-4",
      "certificate": "det J=-4; three distinct rational preimages share (1/2, 2, 2)",
      "probe_target": "No",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.8106546570006685
      ]
    },
    {
      "case": "C228",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(2*(y - 1)^2*(x*(y - 1) + 1)*(3*x*(y - 1) + 4) + 2*(z + 1)*(x*(y - 1) + 1)^3 + 2, 3*x*(y - 1)^2*(3*x*(y - 1) + 4) + 3*x*(z + 1)*(x*(y - 1) + 1)^2 + y, -x^3*(z + 1) - 3*x^2*(y - 1) + 2*x)",
      "determinant": "-4",
      "certificate": "det J=-4; three distinct rational preimages share (3/2, 1, 0)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.8045738939999865
      ]
    },
    {
      "case": "B126",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (4*x^2 + 2*y)*(z - 6*(2*x^2 + y)^2)^2, 2*x^2 + y, z - 6*(2*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7661065600004804
      ]
    },
    {
      "case": "B154",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (4*x^2 + y)*(z + 2*(4*x^2 + y)^2)^2, 4*x^2 + y, z + 2*(4*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7555858220002847
      ]
    },
    {
      "case": "C81",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(2*(y - 1)^2*((x - 2)*(y - 1) + 1)*(3*(x - 2)*(y - 1) + 4) + 2*((x - 2)*(y - 1) + 1)^3*(2*x^2 + z - 2) - 1, y + 3*(x - 2)*(y - 1)^2*(3*(x - 2)*(y - 1) + 4) + 3*(x - 2)*((x - 2)*(y - 1) + 1)^2*(2*x^2 + z - 2) - 1, 2*x - (x - 2)^3*(2*x^2 + z - 2) - 3*(x - 2)^2*(y - 1) - 6)",
      "determinant": "-4",
      "certificate": "det J=-4; three distinct rational preimages share (-3/2, 0, -2)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.8280635729997812
      ]
    },
    {
      "case": "B176",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-15*x^2 - 5*y)*(z + 4*(3*x^2 + y)^2)^2, 3*x^2 + y, z + 4*(3*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.8792228669999531
      ]
    },
    {
      "case": "C43",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-(-x^2 + z)*((x - 2)*(y + 2) + 1)^3 - (y + 2)^2*((x - 2)*(y + 2) + 1)*(3*(x - 2)*(y + 2) + 4) - 2, y + 3*(x - 2)*(-x^2 + z)*((x - 2)*(y + 2) + 1)^2 + 3*(x - 2)*(y + 2)^2*(3*(x - 2)*(y + 2) + 4) + 1, 2*x - (x - 2)^3*(-x^2 + z) - 3*(x - 2)^2*(y + 2) - 6)",
      "determinant": "2",
      "certificate": "det J=2; three distinct rational preimages share (-7/4, -1, -2)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7350384219998887
      ]
    },
    {
      "case": "C181",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(2*(y + 1)^2*((x + 2)*(y + 1) + 1)*(3*(x + 2)*(y + 1) + 4) + 2*(z - 1)*((x + 2)*(y + 1) + 1)^3 - 1, y + 3*(x + 2)*(y + 1)^2*(3*(x + 2)*(y + 1) + 4) + 3*(x + 2)*(z - 1)*((x + 2)*(y + 1) + 1)^2 + 1, 2*x - (x + 2)^3*(z - 1) - 3*(x + 2)^2*(y + 1) + 2)",
      "determinant": "-4",
      "certificate": "det J=-4; three distinct rational preimages share (-3/2, 0, -2)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7200828300001376
      ]
    },
    {
      "case": "B94",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-8*x^2 + 4*y)*(z + (-2*x^2 + y)^2)^2, -2*x^2 + y, z + (-2*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7688457150006798
      ]
    },
    {
      "case": "B235",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-24*x^2 + 4*y)*(z + 2*(-6*x^2 + y)^2)^2, -6*x^2 + y, z + 2*(-6*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7570045909997134
      ]
    },
    {
      "case": "B142",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-18*x^2 - 3*y)*(z + (6*x^2 + y)^2)^2, 6*x^2 + y, z + (6*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7921119309994538
      ]
    },
    {
      "case": "B229",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-20*x^2 - 5*y)*(z - 4*(4*x^2 + y)^2)^2, 4*x^2 + y, z - 4*(4*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.8544934009996723
      ]
    },
    {
      "case": "B109",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (24*x^2 - 6*y)*(z - 6*(-4*x^2 + y)^2)^2, -4*x^2 + y, z - 6*(-4*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.8071127769999293
      ]
    },
    {
      "case": "B249",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-4*x^2 - 2*y)*(z + 5*(2*x^2 + y)^2)^2, 2*x^2 + y, z + 5*(2*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.8204390819992113
      ]
    },
    {
      "case": "B184",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (20*x^2 + 4*y)*(z - 2*(5*x^2 + y)^2)^2, 5*x^2 + y, z - 2*(5*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.6970015460001378
      ]
    },
    {
      "case": "B28",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (6*x^2 - y)*(z - 4*(-6*x^2 + y)^2)^2, -6*x^2 + y, z - 4*(-6*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7540215959998022
      ]
    },
    {
      "case": "C106",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-2*(y - 2)^2*(x*(y - 2) + 1)*(3*x*(y - 2) + 4) - 2*(x*(y - 2) + 1)^3*(2*x^2 + z - 1) + 2, 3*x*(y - 2)^2*(3*x*(y - 2) + 4) + 3*x*(x*(y - 2) + 1)^2*(2*x^2 + z - 1) + y - 2, -x^3*(2*x^2 + z - 1) - 3*x^2*(y - 2) + 2*x)",
      "determinant": "4",
      "certificate": "det J=4; three distinct rational preimages share (5/2, 0, 0)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7203394349999144
      ]
    },
    {
      "case": "C64",
      "family": "counterexample",
      "mapping": "F(x,y,z)=((2*x^2 + z)*(x*(y - 1) + 1)^3 + (y - 1)^2*(x*(y - 1) + 1)*(3*x*(y - 1) + 4) + 1, 3*x*(2*x^2 + z)*(x*(y - 1) + 1)^2 + 3*x*(y - 1)^2*(3*x*(y - 1) + 4) + y, -x^3*(2*x^2 + z) - 3*x^2*(y - 1) + 2*x + 1)",
      "determinant": "-2",
      "certificate": "det J=-2; three distinct rational preimages share (3/4, 1, 1)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.8049338009996063
      ]
    },
    {
      "case": "B203",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-15*x^2 + 3*y)*(z + 5*(-5*x^2 + y)^2)^2, -5*x^2 + y, z + 5*(-5*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7797167609996905
      ]
    },
    {
      "case": "C193",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-(y - 2)^2*((x - 1)*(y - 2) + 1)*(3*(x - 1)*(y - 2) + 4) - (z - 1)*((x - 1)*(y - 2) + 1)^3 - 2, y + 3*(x - 1)*(y - 2)^2*(3*(x - 1)*(y - 2) + 4) + 3*(x - 1)*(z - 1)*((x - 1)*(y - 2) + 1)^2, 2*x - (x - 1)^3*(z - 1) - 3*(x - 1)^2*(y - 2) - 2)",
      "determinant": "2",
      "certificate": "det J=2; three distinct rational preimages share (-7/4, 2, 0)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.8466603320002832
      ]
    },
    {
      "case": "C156",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-2*(y - 2)^2*((x + 2)*(y - 2) + 1)*(3*(x + 2)*(y - 2) + 4) - 2*((x + 2)*(y - 2) + 1)^3*(x^2 + z - 2) - 2, y + 3*(x + 2)*(y - 2)^2*(3*(x + 2)*(y - 2) + 4) + 3*(x + 2)*((x + 2)*(y - 2) + 1)^2*(x^2 + z - 2) - 4, 2*x - (x + 2)^3*(x^2 + z - 2) - 3*(x + 2)^2*(y - 2) + 2)",
      "determinant": "4",
      "certificate": "det J=4; three distinct rational preimages share (-3/2, -2, -2)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.8045249240003614
      ]
    },
    {
      "case": "C55",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(2*y^2*(y*(x + 2) + 1)*(3*y*(x + 2) + 4) + 2*(z + 1)*(y*(x + 2) + 1)^3 - 2, 3*y^2*(x + 2)*(3*y*(x + 2) + 4) + y + 3*(x + 2)*(z + 1)*(y*(x + 2) + 1)^2 - 1, 2*x - 3*y*(x + 2)^2 - (x + 2)^3*(z + 1) + 3)",
      "determinant": "-4",
      "certificate": "det J=-4; three distinct rational preimages share (-5/2, -1, -1)",
      "probe_target": "No",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.8759373309994771
      ]
    },
    {
      "case": "B32",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (20*x^2 - 5*y)*(z - 4*(-4*x^2 + y)^2)^2, -4*x^2 + y, z - 4*(-4*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7817925150002338
      ]
    },
    {
      "case": "B147",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (5*x^2 - y)*(z + 4*(-5*x^2 + y)^2)^2, -5*x^2 + y, z + 4*(-5*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.90720897600022
      ]
    },
    {
      "case": "C144",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-(y - 1)^2*((x - 1)*(y - 1) + 1)*(3*(x - 1)*(y - 1) + 4) - ((x - 1)*(y - 1) + 1)^3*(2*x^2 + z + 2), y + 3*(x - 1)*(y - 1)^2*(3*(x - 1)*(y - 1) + 4) + 3*(x - 1)*((x - 1)*(y - 1) + 1)^2*(2*x^2 + z + 2), 2*x - (x - 1)^3*(2*x^2 + z + 2) - 3*(x - 1)^2*(y - 1) - 4)",
      "determinant": "2",
      "certificate": "det J=2; three distinct rational preimages share (1/4, 1, -2)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7558403240000189
      ]
    },
    {
      "case": "B42",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (15*x^2 + 5*y)*(z - 2*(3*x^2 + y)^2)^2, 3*x^2 + y, z - 2*(3*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7438868619992718
      ]
    },
    {
      "case": "C25",
      "family": "counterexample",
      "mapping": "F(x,y,z)=((y + 1)^2*((x - 1)*(y + 1) + 1)*(3*(x - 1)*(y + 1) + 4) + (z + 1)*((x - 1)*(y + 1) + 1)^3 + 2, y + 3*(x - 1)*(y + 1)^2*(3*(x - 1)*(y + 1) + 4) + 3*(x - 1)*(z + 1)*((x - 1)*(y + 1) + 1)^2 - 1, 2*x - (x - 1)^3*(z + 1) - 3*(x - 1)^2*(y + 1) - 3)",
      "determinant": "-2",
      "certificate": "det J=-2; three distinct rational preimages share (7/4, -2, -1)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.8255383659998188
      ]
    },
    {
      "case": "C58",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(y^2*(y*(x + 2) + 1)*(3*y*(x + 2) + 4) + (y*(x + 2) + 1)^3*(x^2 + z + 1) + 1, 3*y^2*(x + 2)*(3*y*(x + 2) + 4) + y + 3*(x + 2)*(y*(x + 2) + 1)^2*(x^2 + z + 1) + 2, 2*x - 3*y*(x + 2)^2 - (x + 2)^3*(x^2 + z + 1) + 5)",
      "determinant": "-2",
      "certificate": "det J=-2; three distinct rational preimages share (3/4, 2, 1)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7174211390001801
      ]
    },
    {
      "case": "B128",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (10*x^2 - 5*y)*(z - 6*(-2*x^2 + y)^2)^2, -2*x^2 + y, z - 6*(-2*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7860499709995565
      ]
    },
    {
      "case": "B185",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (6*x^2 + 2*y)*(z + 6*(3*x^2 + y)^2)^2, 3*x^2 + y, z + 6*(3*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7607078500004718
      ]
    },
    {
      "case": "C133",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(2*(-x^2 + z)*((x + 1)*(y + 2) + 1)^3 + 2*(y + 2)^2*((x + 1)*(y + 2) + 1)*(3*(x + 1)*(y + 2) + 4) - 1, y + 3*(x + 1)*(-x^2 + z)*((x + 1)*(y + 2) + 1)^2 + 3*(x + 1)*(y + 2)^2*(3*(x + 1)*(y + 2) + 4) + 3, 2*x - (x + 1)^3*(-x^2 + z) - 3*(x + 1)^2*(y + 2))",
      "determinant": "-4",
      "certificate": "det J=-4; three distinct rational preimages share (-3/2, 1, -2)",
      "probe_target": "No",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7068495869998515
      ]
    },
    {
      "case": "C152",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-(y - 2)^2*((x + 2)*(y - 2) + 1)*(3*(x + 2)*(y - 2) + 4) - ((x + 2)*(y - 2) + 1)^3*(x^2 + z + 1) - 1, y + 3*(x + 2)*(y - 2)^2*(3*(x + 2)*(y - 2) + 4) + 3*(x + 2)*((x + 2)*(y - 2) + 1)^2*(x^2 + z + 1) - 3, 2*x - (x + 2)^3*(x^2 + z + 1) - 3*(x + 2)^2*(y - 2) + 2)",
      "determinant": "2",
      "certificate": "det J=2; three distinct rational preimages share (-3/4, -1, -2)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7311853049995989
      ]
    },
    {
      "case": "B125",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-20*x^2 + 5*y)*(z - 5*(-4*x^2 + y)^2)^2, -4*x^2 + y, z - 5*(-4*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7808047959997566
      ]
    },
    {
      "case": "B95",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (25*x^2 - 5*y)*(z + (-5*x^2 + y)^2)^2, -5*x^2 + y, z + (-5*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7082471059993622
      ]
    },
    {
      "case": "C14",
      "family": "counterexample",
      "mapping": "F(x,y,z)=((y - 2)^2*(x*(y - 2) + 1)*(3*x*(y - 2) + 4) + (x*(y - 2) + 1)^3*(2*x^2 + z + 2) + 1, 3*x*(y - 2)^2*(3*x*(y - 2) + 4) + 3*x*(x*(y - 2) + 1)^2*(2*x^2 + z + 2) + y - 3, -x^3*(2*x^2 + z + 2) - 3*x^2*(y - 2) + 2*x + 2)",
      "determinant": "-2",
      "certificate": "det J=-2; three distinct rational preimages share (3/4, -1, 2)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7567277770003784
      ]
    },
    {
      "case": "C85",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(2*(y + 1)^2*(x*(y + 1) + 1)*(3*x*(y + 1) + 4) + 2*(x*(y + 1) + 1)^3*(-x^2 + z - 1) - 2, 3*x*(y + 1)^2*(3*x*(y + 1) + 4) + 3*x*(x*(y + 1) + 1)^2*(-x^2 + z - 1) + y + 3, -x^3*(-x^2 + z - 1) - 3*x^2*(y + 1) + 2*x)",
      "determinant": "-4",
      "certificate": "det J=-4; three distinct rational preimages share (-5/2, 2, 0)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7823713580000913
      ]
    },
    {
      "case": "B159",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (6*x^2 + 3*y)*(z + (2*x^2 + y)^2)^2, 2*x^2 + y, z + (2*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7687910519998695
      ]
    },
    {
      "case": "B190",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (25*x^2 - 5*y)*(z + 2*(-5*x^2 + y)^2)^2, -5*x^2 + y, z + 2*(-5*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.704330741999911
      ]
    },
    {
      "case": "C231",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(2*(y - 1)^2*((x + 2)*(y - 1) + 1)*(3*(x + 2)*(y - 1) + 4) + 2*((x + 2)*(y - 1) + 1)^3*(2*x^2 + z + 2), y + 3*(x + 2)*(y - 1)^2*(3*(x + 2)*(y - 1) + 4) + 3*(x + 2)*((x + 2)*(y - 1) + 1)^2*(2*x^2 + z + 2) - 3, 2*x - (x + 2)^3*(2*x^2 + z + 2) - 3*(x + 2)^2*(y - 1) + 5)",
      "determinant": "-4",
      "certificate": "det J=-4; three distinct rational preimages share (-1/2, -2, 1)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.759613403000003
      ]
    },
    {
      "case": "B62",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-3*x^2 + y)*(z - 6*(-3*x^2 + y)^2)^2, -3*x^2 + y, z - 6*(-3*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7146814549996634
      ]
    },
    {
      "case": "C63",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-2*(y + 2)^2*((x + 1)*(y + 2) + 1)*(3*(x + 1)*(y + 2) + 4) - 2*((x + 1)*(y + 2) + 1)^3*(-x^2 + z + 1) - 2, y + 3*(x + 1)*(y + 2)^2*(3*(x + 1)*(y + 2) + 4) + 3*(x + 1)*((x + 1)*(y + 2) + 1)^2*(-x^2 + z + 1) + 2, 2*x - (x + 1)^3*(-x^2 + z + 1) - 3*(x + 1)^2*(y + 2) + 1)",
      "determinant": "4",
      "certificate": "det J=4; three distinct rational preimages share (-3/2, 0, -1)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7660487129996909
      ]
    },
    {
      "case": "C127",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(2*(y - 1)^2*((x + 1)*(y - 1) + 1)*(3*(x + 1)*(y - 1) + 4) + 2*((x + 1)*(y - 1) + 1)^3*(-2*x^2 + z + 2) + 2, y + 3*(x + 1)*(y - 1)^2*(3*(x + 1)*(y - 1) + 4) + 3*(x + 1)*((x + 1)*(y - 1) + 1)^2*(-2*x^2 + z + 2) - 2, 2*x - (x + 1)^3*(-2*x^2 + z + 2) - 3*(x + 1)^2*(y - 1) + 3)",
      "determinant": "-4",
      "certificate": "det J=-4; three distinct rational preimages share (3/2, -1, 1)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7368847520001509
      ]
    },
    {
      "case": "B208",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (15*x^2 + 3*y)*(z + 2*(5*x^2 + y)^2)^2, 5*x^2 + y, z + 2*(5*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.753256066999711
      ]
    },
    {
      "case": "C67",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-2*(-2*x^2 + z)*((x - 2)*(y - 1) + 1)^3 - 2*(y - 1)^2*((x - 2)*(y - 1) + 1)*(3*(x - 2)*(y - 1) + 4) + 2, y + 3*(x - 2)*(-2*x^2 + z)*((x - 2)*(y - 1) + 1)^2 + 3*(x - 2)*(y - 1)^2*(3*(x - 2)*(y - 1) + 4) - 1, 2*x - (x - 2)^3*(-2*x^2 + z) - 3*(x - 2)^2*(y - 1) - 2)",
      "determinant": "4",
      "certificate": "det J=4; three distinct rational preimages share (5/2, 0, 2)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7182841690000714
      ]
    },
    {
      "case": "B27",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (12*x^2 - 4*y)*(z + (-3*x^2 + y)^2)^2, -3*x^2 + y, z + (-3*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7717920420000155
      ]
    },
    {
      "case": "B140",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (8*x^2 + 4*y)*(z - 4*(2*x^2 + y)^2)^2, 2*x^2 + y, z - 4*(2*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7626605369996469
      ]
    },
    {
      "case": "B133",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-6*x^2 + 2*y)*(z - (-3*x^2 + y)^2)^2, -3*x^2 + y, z - (-3*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7100137789993823
      ]
    },
    {
      "case": "B151",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (4*x^2 + 4*y)*(z + (x^2 + y)^2)^2, x^2 + y, z + (x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.691042637999999
      ]
    },
    {
      "case": "C214",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(2*(y - 2)^2*((x + 1)*(y - 2) + 1)*(3*(x + 1)*(y - 2) + 4) + 2*((x + 1)*(y - 2) + 1)^3*(x^2 + z + 2) + 2, y + 3*(x + 1)*(y - 2)^2*(3*(x + 1)*(y - 2) + 4) + 3*(x + 1)*((x + 1)*(y - 2) + 1)^2*(x^2 + z + 2) - 4, 2*x - (x + 1)^3*(x^2 + z + 2) - 3*(x + 1)^2*(y - 2) + 3)",
      "determinant": "-4",
      "certificate": "det J=-4; three distinct rational preimages share (3/2, -2, 1)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.815343281999958
      ]
    },
    {
      "case": "C145",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-y^2*(y*(x + 2) + 1)*(3*y*(x + 2) + 4) - (y*(x + 2) + 1)^3*(2*x^2 + z + 1), 3*y^2*(x + 2)*(3*y*(x + 2) + 4) + y + 3*(x + 2)*(y*(x + 2) + 1)^2*(2*x^2 + z + 1) - 1, 2*x - 3*y*(x + 2)^2 - (x + 2)^3*(2*x^2 + z + 1) + 4)",
      "determinant": "2",
      "certificate": "det J=2; three distinct rational preimages share (1/4, -1, 0)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7603448979998575
      ]
    },
    {
      "case": "C68",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-2*(y - 2)^2*(x*(y - 2) + 1)*(3*x*(y - 2) + 4) - 2*(x*(y - 2) + 1)^3*(x^2 + z + 2) - 2, 3*x*(y - 2)^2*(3*x*(y - 2) + 4) + 3*x*(x*(y - 2) + 1)^2*(x^2 + z + 2) + y - 2, -x^3*(x^2 + z + 2) - 3*x^2*(y - 2) + 2*x - 1)",
      "determinant": "4",
      "certificate": "det J=4; three distinct rational preimages share (-3/2, 0, -1)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7964421409997158
      ]
    },
    {
      "case": "B179",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-12*x^2 - 4*y)*(z + 3*(3*x^2 + y)^2)^2, 3*x^2 + y, z + 3*(3*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.8121280569994269
      ]
    },
    {
      "case": "B39",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-6*x^2 + 6*y)*(z - 5*(-x^2 + y)^2)^2, -x^2 + y, z - 5*(-x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7645964129997083
      ]
    },
    {
      "case": "C18",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-2*(y + 1)^2*((x + 2)*(y + 1) + 1)*(3*(x + 2)*(y + 1) + 4) - 2*((x + 2)*(y + 1) + 1)^3*(-x^2 + z - 1) - 1, y + 3*(x + 2)*(y + 1)^2*(3*(x + 2)*(y + 1) + 4) + 3*(x + 2)*((x + 2)*(y + 1) + 1)^2*(-x^2 + z - 1) + 1, 2*x - (x + 2)^3*(-x^2 + z - 1) - 3*(x + 2)^2*(y + 1) + 6)",
      "determinant": "4",
      "certificate": "det J=4; three distinct rational preimages share (-1/2, 0, 2)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7542206540001644
      ]
    },
    {
      "case": "B160",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-9*x^2 - 3*y)*(z - 2*(3*x^2 + y)^2)^2, 3*x^2 + y, z - 2*(3*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7971162599997115
      ]
    },
    {
      "case": "B232",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-12*x^2 - 3*y)*(z + 2*(4*x^2 + y)^2)^2, 4*x^2 + y, z + 2*(4*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7252334910008358
      ]
    },
    {
      "case": "B31",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-3*x^2 + y)*(z - 4*(-3*x^2 + y)^2)^2, -3*x^2 + y, z - 4*(-3*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7855283959997905
      ]
    },
    {
      "case": "C76",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(2*y^2*(y*(x - 2) + 1)*(3*y*(x - 2) + 4) + 2*(y*(x - 2) + 1)^3*(x^2 + z - 1) + 1, 3*y^2*(x - 2)*(3*y*(x - 2) + 4) + y + 3*(x - 2)*(y*(x - 2) + 1)^2*(x^2 + z - 1), 2*x - 3*y*(x - 2)^2 - (x - 2)^3*(x^2 + z - 1) - 4)",
      "determinant": "-4",
      "certificate": "det J=-4; three distinct rational preimages share (1/2, 0, 0)",
      "probe_target": "No",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.730462376000105
      ]
    },
    {
      "case": "C163",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(2*(y - 1)^2*(x*(y - 1) + 1)*(3*x*(y - 1) + 4) + 2*(x*(y - 1) + 1)^3*(2*x^2 + z - 2), 3*x*(y - 1)^2*(3*x*(y - 1) + 4) + 3*x*(x*(y - 1) + 1)^2*(2*x^2 + z - 2) + y - 1, -x^3*(2*x^2 + z - 2) - 3*x^2*(y - 1) + 2*x - 2)",
      "determinant": "-4",
      "certificate": "det J=-4; three distinct rational preimages share (-1/2, 0, -2)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.9165178900002502
      ]
    },
    {
      "case": "B88",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-36*x^2 - 6*y)*(z - 4*(6*x^2 + y)^2)^2, 6*x^2 + y, z - 4*(6*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.75374481200015
      ]
    },
    {
      "case": "B210",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-24*x^2 - 4*y)*(z + 5*(6*x^2 + y)^2)^2, 6*x^2 + y, z + 5*(6*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.8074255559995436
      ]
    },
    {
      "case": "C19",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-2*(y - 1)^2*(x*(y - 1) + 1)*(3*x*(y - 1) + 4) - 2*(x*(y - 1) + 1)^3*(-2*x^2 + z + 1), 3*x*(y - 1)^2*(3*x*(y - 1) + 4) + 3*x*(x*(y - 1) + 1)^2*(-2*x^2 + z + 1) + y - 2, -x^3*(-2*x^2 + z + 1) - 3*x^2*(y - 1) + 2*x)",
      "determinant": "4",
      "certificate": "det J=4; three distinct rational preimages share (1/2, -1, 0)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.859785994000049
      ]
    },
    {
      "case": "C36",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(y^2*(y*(x + 1) + 1)*(3*y*(x + 1) + 4) + (y*(x + 1) + 1)^3*(-x^2 + z - 2) - 2, 3*y^2*(x + 1)*(3*y*(x + 1) + 4) + y + 3*(x + 1)*(y*(x + 1) + 1)^2*(-x^2 + z - 2) + 2, 2*x - 3*y*(x + 1)^2 - (x + 1)^3*(-x^2 + z - 2))",
      "determinant": "-2",
      "certificate": "det J=-2; three distinct rational preimages share (-9/4, 2, -2)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7541516560004311
      ]
    },
    {
      "case": "B200",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (x^2 - y)*(z + 4*(-x^2 + y)^2)^2, -x^2 + y, z + 4*(-x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7181491680003091
      ]
    },
    {
      "case": "C183",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-y^2*(y*(x + 2) + 1)*(3*y*(x + 2) + 4) - z*(y*(x + 2) + 1)^3 + 1, 3*y^2*(x + 2)*(3*y*(x + 2) + 4) + y + 3*z*(x + 2)*(y*(x + 2) + 1)^2 - 2, 2*x - 3*y*(x + 2)^2 - z*(x + 2)^3 + 2)",
      "determinant": "2",
      "certificate": "det J=2; three distinct rational preimages share (5/4, -2, -2)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.8589319419997992
      ]
    },
    {
      "case": "C27",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-(x^2 + z)*(x*(y - 1) + 1)^3 - (y - 1)^2*(x*(y - 1) + 1)*(3*x*(y - 1) + 4) + 2, 3*x*(x^2 + z)*(x*(y - 1) + 1)^2 + 3*x*(y - 1)^2*(3*x*(y - 1) + 4) + y - 3, -x^3*(x^2 + z) - 3*x^2*(y - 1) + 2*x - 1)",
      "determinant": "2",
      "certificate": "det J=2; three distinct rational preimages share (9/4, -2, -1)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.8292736060002426
      ]
    },
    {
      "case": "B144",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-24*x^2 - 6*y)*(z - 5*(4*x^2 + y)^2)^2, 4*x^2 + y, z - 5*(4*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.6990350530004434
      ]
    },
    {
      "case": "B135",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (3*x^2 - y)*(z + 2*(-3*x^2 + y)^2)^2, -3*x^2 + y, z + 2*(-3*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7390246759996444
      ]
    },
    {
      "case": "C221",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-2*(y - 1)^2*((x + 2)*(y - 1) + 1)*(3*(x + 2)*(y - 1) + 4) - 2*((x + 2)*(y - 1) + 1)^3*(-x^2 + z + 2) - 1, y + 3*(x + 2)*(y - 1)^2*(3*(x + 2)*(y - 1) + 4) + 3*(x + 2)*((x + 2)*(y - 1) + 1)^2*(-x^2 + z + 2) - 3, 2*x - (x + 2)^3*(-x^2 + z + 2) - 3*(x + 2)^2*(y - 1) + 5)",
      "determinant": "4",
      "certificate": "det J=4; three distinct rational preimages share (-1/2, -2, 1)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7510574839998299
      ]
    },
    {
      "case": "C83",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(2*y^2*(y*(x + 2) + 1)*(3*y*(x + 2) + 4) + 2*(y*(x + 2) + 1)^3*(-x^2 + z - 1) - 1, 3*y^2*(x + 2)*(3*y*(x + 2) + 4) + y + 3*(x + 2)*(y*(x + 2) + 1)^2*(-x^2 + z - 1) - 1, 2*x - 3*y*(x + 2)^2 - (x + 2)^3*(-x^2 + z - 1) + 5)",
      "determinant": "-4",
      "certificate": "det J=-4; three distinct rational preimages share (-3/2, -1, 1)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.78309002900005
      ]
    },
    {
      "case": "B207",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-15*x^2 + 3*y)*(z + 3*(-5*x^2 + y)^2)^2, -5*x^2 + y, z + 3*(-5*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.8045382670006802
      ]
    },
    {
      "case": "B104",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-20*x^2 - 4*y)*(z - 5*(5*x^2 + y)^2)^2, 5*x^2 + y, z - 5*(5*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7478025650007112
      ]
    },
    {
      "case": "C33",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-z*((x - 1)*(y - 1) + 1)^3 - (y - 1)^2*((x - 1)*(y - 1) + 1)*(3*(x - 1)*(y - 1) + 4) + 2, y + 3*z*(x - 1)*((x - 1)*(y - 1) + 1)^2 + 3*(x - 1)*(y - 1)^2*(3*(x - 1)*(y - 1) + 4), 2*x - z*(x - 1)^3 - 3*(x - 1)^2*(y - 1) - 1)",
      "determinant": "2",
      "certificate": "det J=2; three distinct rational preimages share (9/4, 1, 1)",
      "probe_target": "No",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7716132010000365
      ]
    },
    {
      "case": "B180",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (25*x^2 + 5*y)*(z - 5*(5*x^2 + y)^2)^2, 5*x^2 + y, z - 5*(5*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7498258030000216
      ]
    },
    {
      "case": "B157",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (20*x^2 - 5*y)*(z + 2*(-4*x^2 + y)^2)^2, -4*x^2 + y, z + 2*(-4*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7437426749993392
      ]
    },
    {
      "case": "B09",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (15*x^2 + 5*y)*(z - 3*(3*x^2 + y)^2)^2, 3*x^2 + y, z - 3*(3*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.8092432029998236
      ]
    },
    {
      "case": "B60",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-16*x^2 - 4*y)*(z + 2*(4*x^2 + y)^2)^2, 4*x^2 + y, z + 2*(4*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7714226680000138
      ]
    },
    {
      "case": "B21",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (x^2 + y)*(z + 6*(x^2 + y)^2)^2, x^2 + y, z + 6*(x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.8179373270004362
      ]
    },
    {
      "case": "C06",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-2*(y - 2)^2*((x + 1)*(y - 2) + 1)*(3*(x + 1)*(y - 2) + 4) - 2*((x + 1)*(y - 2) + 1)^3*(x^2 + z + 2) - 2, y + 3*(x + 1)*(y - 2)^2*(3*(x + 1)*(y - 2) + 4) + 3*(x + 1)*((x + 1)*(y - 2) + 1)^2*(x^2 + z + 2), 2*x - (x + 1)^3*(x^2 + z + 2) - 3*(x + 1)^2*(y - 2) + 3)",
      "determinant": "4",
      "certificate": "det J=4; three distinct rational preimages share (-3/2, 2, 1)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.8196366860001945
      ]
    },
    {
      "case": "B17",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-8*x^2 - 4*y)*(z - 3*(2*x^2 + y)^2)^2, 2*x^2 + y, z - 3*(2*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7460658090003562
      ]
    },
    {
      "case": "B233",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-4*x^2 + y)*(z - (-4*x^2 + y)^2)^2, -4*x^2 + y, z - (-4*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.6935001819992976
      ]
    },
    {
      "case": "C248",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-(y - 2)^2*((x + 2)*(y - 2) + 1)*(3*(x + 2)*(y - 2) + 4) - ((x + 2)*(y - 2) + 1)^3*(x^2 + z + 2) - 1, y + 3*(x + 2)*(y - 2)^2*(3*(x + 2)*(y - 2) + 4) + 3*(x + 2)*((x + 2)*(y - 2) + 1)^2*(x^2 + z + 2) - 4, 2*x - (x + 2)^3*(x^2 + z + 2) - 3*(x + 2)^2*(y - 2) + 5)",
      "determinant": "2",
      "certificate": "det J=2; three distinct rational preimages share (-3/4, -2, 1)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7316463099996326
      ]
    },
    {
      "case": "C202",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(y^2*(y*(x - 1) + 1)*(3*y*(x - 1) + 4) + (y*(x - 1) + 1)^3*(-x^2 + z + 1) - 1, 3*y^2*(x - 1)*(3*y*(x - 1) + 4) + y + 3*(x - 1)*(y*(x - 1) + 1)^2*(-x^2 + z + 1) + 1, 2*x - 3*y*(x - 1)^2 - (x - 1)^3*(-x^2 + z + 1) - 1)",
      "determinant": "-2",
      "certificate": "det J=-2; three distinct rational preimages share (-5/4, 1, 1)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7718651580007645
      ]
    },
    {
      "case": "B129",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-15*x^2 - 5*y)*(z - 6*(3*x^2 + y)^2)^2, 3*x^2 + y, z - 6*(3*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7552166590003253
      ]
    },
    {
      "case": "C35",
      "family": "counterexample",
      "mapping": "F(x,y,z)=((y - 1)^2*((x - 2)*(y - 1) + 1)*(3*(x - 2)*(y - 1) + 4) + ((x - 2)*(y - 1) + 1)^3*(2*x^2 + z - 2) - 2, y + 3*(x - 2)*(y - 1)^2*(3*(x - 2)*(y - 1) + 4) + 3*(x - 2)*((x - 2)*(y - 1) + 1)^2*(2*x^2 + z - 2) - 1, 2*x - (x - 2)^3*(2*x^2 + z - 2) - 3*(x - 2)^2*(y - 1) - 4)",
      "determinant": "-2",
      "certificate": "det J=-2; three distinct rational preimages share (-9/4, 0, 0)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7382874109998738
      ]
    },
    {
      "case": "B102",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-9*x^2 - 3*y)*(z - 4*(3*x^2 + y)^2)^2, 3*x^2 + y, z - 4*(3*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7991571809998277
      ]
    },
    {
      "case": "B198",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (18*x^2 + 3*y)*(z - 6*(6*x^2 + y)^2)^2, 6*x^2 + y, z - 6*(6*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7538700450004399
      ]
    },
    {
      "case": "C218",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-y^2*(y*(x + 2) + 1)*(3*y*(x + 2) + 4) - (y*(x + 2) + 1)^3*(x^2 + z - 1) + 1, 3*y^2*(x + 2)*(3*y*(x + 2) + 4) + y + 3*(x + 2)*(y*(x + 2) + 1)^2*(x^2 + z - 1), 2*x - 3*y*(x + 2)^2 - (x + 2)^3*(x^2 + z - 1) + 4)",
      "determinant": "2",
      "certificate": "det J=2; three distinct rational preimages share (5/4, 0, 0)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7459995259996504
      ]
    },
    {
      "case": "C233",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(2*(y - 1)^2*((x - 1)*(y - 1) + 1)*(3*(x - 1)*(y - 1) + 4) + 2*((x - 1)*(y - 1) + 1)^3*(2*x^2 + z + 1) - 1, y + 3*(x - 1)*(y - 1)^2*(3*(x - 1)*(y - 1) + 4) + 3*(x - 1)*((x - 1)*(y - 1) + 1)^2*(2*x^2 + z + 1) - 3, 2*x - (x - 1)^3*(2*x^2 + z + 1) - 3*(x - 1)^2*(y - 1))",
      "determinant": "-4",
      "certificate": "det J=-4; three distinct rational preimages share (-3/2, -2, 2)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7646186550000493
      ]
    },
    {
      "case": "C203",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(y^2*(y*(x - 1) + 1)*(3*y*(x - 1) + 4) + (y*(x - 1) + 1)^3*(-2*x^2 + z - 2), 3*y^2*(x - 1)*(3*y*(x - 1) + 4) + y + 3*(x - 1)*(y*(x - 1) + 1)^2*(-2*x^2 + z - 2), 2*x - 3*y*(x - 1)^2 - (x - 1)^3*(-2*x^2 + z - 2) - 1)",
      "determinant": "-2",
      "certificate": "det J=-2; three distinct rational preimages share (-1/4, 0, 1)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7044964799997615
      ]
    },
    {
      "case": "C192",
      "family": "counterexample",
      "mapping": "F(x,y,z)=((y + 2)^2*((x - 1)*(y + 2) + 1)*(3*(x - 1)*(y + 2) + 4) + ((x - 1)*(y + 2) + 1)^3*(2*x^2 + z + 1) + 1, y + 3*(x - 1)*(y + 2)^2*(3*(x - 1)*(y + 2) + 4) + 3*(x - 1)*((x - 1)*(y + 2) + 1)^2*(2*x^2 + z + 1), 2*x - (x - 1)^3*(2*x^2 + z + 1) - 3*(x - 1)^2*(y + 2) - 1)",
      "determinant": "-2",
      "certificate": "det J=-2; three distinct rational preimages share (3/4, -2, 1)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7644917999996323
      ]
    },
    {
      "case": "C237",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-2*(y - 2)^2*((x - 2)*(y - 2) + 1)*(3*(x - 2)*(y - 2) + 4) - 2*(z - 2)*((x - 2)*(y - 2) + 1)^3 + 1, y + 3*(x - 2)*(y - 2)^2*(3*(x - 2)*(y - 2) + 4) + 3*(x - 2)*(z - 2)*((x - 2)*(y - 2) + 1)^2 - 2, 2*x - (x - 2)^3*(z - 2) - 3*(x - 2)^2*(y - 2) - 6)",
      "determinant": "4",
      "certificate": "det J=4; three distinct rational preimages share (3/2, 0, -2)",
      "probe_target": "No",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7527776419992733
      ]
    },
    {
      "case": "B18",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-16*x^2 - 4*y)*(z + 6*(4*x^2 + y)^2)^2, 4*x^2 + y, z + 6*(4*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7828675479995582
      ]
    },
    {
      "case": "B244",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (20*x^2 + 5*y)*(z + 6*(4*x^2 + y)^2)^2, 4*x^2 + y, z + 6*(4*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7195312900003046
      ]
    },
    {
      "case": "B214",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-9*x^2 + 3*y)*(z + 2*(-3*x^2 + y)^2)^2, -3*x^2 + y, z + 2*(-3*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.8071149819998027
      ]
    },
    {
      "case": "B92",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-12*x^2 - 6*y)*(z + 3*(2*x^2 + y)^2)^2, 2*x^2 + y, z + 3*(2*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.6977840870004002
      ]
    },
    {
      "case": "C159",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-2*z*((x - 2)*(y - 2) + 1)^3 - 2*(y - 2)^2*((x - 2)*(y - 2) + 1)*(3*(x - 2)*(y - 2) + 4), y + 3*z*(x - 2)*((x - 2)*(y - 2) + 1)^2 + 3*(x - 2)*(y - 2)^2*(3*(x - 2)*(y - 2) + 4) - 2, 2*x - z*(x - 2)^3 - 3*(x - 2)^2*(y - 2) - 2)",
      "determinant": "4",
      "certificate": "det J=4; three distinct rational preimages share (1/2, 0, 2)",
      "probe_target": "No",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.8045941130003484
      ]
    },
    {
      "case": "B139",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (10*x^2 - 5*y)*(z + (-2*x^2 + y)^2)^2, -2*x^2 + y, z + (-2*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.8103689089994077
      ]
    },
    {
      "case": "C199",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(2*z*((x + 1)*(y - 1) + 1)^3 + 2*(y - 1)^2*((x + 1)*(y - 1) + 1)*(3*(x + 1)*(y - 1) + 4) + 2, y + 3*z*(x + 1)*((x + 1)*(y - 1) + 1)^2 + 3*(x + 1)*(y - 1)^2*(3*(x + 1)*(y - 1) + 4) - 2, 2*x - z*(x + 1)^3 - 3*(x + 1)^2*(y - 1) + 2)",
      "determinant": "-4",
      "certificate": "det J=-4; three distinct rational preimages share (3/2, -1, 0)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7727663569994547
      ]
    },
    {
      "case": "C165",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-(y + 1)^2*((x + 2)*(y + 1) + 1)*(3*(x + 2)*(y + 1) + 4) - ((x + 2)*(y + 1) + 1)^3*(x^2 + z - 2) + 1, y + 3*(x + 2)*(y + 1)^2*(3*(x + 2)*(y + 1) + 4) + 3*(x + 2)*((x + 2)*(y + 1) + 1)^2*(x^2 + z - 2) + 1, 2*x - (x + 2)^3*(x^2 + z - 2) - 3*(x + 2)^2*(y + 1) + 2)",
      "determinant": "2",
      "certificate": "det J=2; three distinct rational preimages share (5/4, 0, -2)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7363646089997928
      ]
    },
    {
      "case": "C170",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(2*(-x^2 + z)*((x + 2)*(y + 1) + 1)^3 + 2*(y + 1)^2*((x + 2)*(y + 1) + 1)*(3*(x + 2)*(y + 1) + 4), y + 3*(x + 2)*(-x^2 + z)*((x + 2)*(y + 1) + 1)^2 + 3*(x + 2)*(y + 1)^2*(3*(x + 2)*(y + 1) + 4), 2*x - (x + 2)^3*(-x^2 + z) - 3*(x + 2)^2*(y + 1) + 2)",
      "determinant": "-4",
      "certificate": "det J=-4; three distinct rational preimages share (-1/2, -1, -2)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7521854850001546
      ]
    },
    {
      "case": "C212",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-(y - 2)^2*((x + 2)*(y - 2) + 1)*(3*(x + 2)*(y - 2) + 4) - (z + 1)*((x + 2)*(y - 2) + 1)^3 + 1, y + 3*(x + 2)*(y - 2)^2*(3*(x + 2)*(y - 2) + 4) + 3*(x + 2)*(z + 1)*((x + 2)*(y - 2) + 1)^2 - 1, 2*x - (x + 2)^3*(z + 1) - 3*(x + 2)^2*(y - 2) + 5)",
      "determinant": "2",
      "certificate": "det J=2; three distinct rational preimages share (5/4, 1, 1)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7012766049992933
      ]
    },
    {
      "case": "B209",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (36*x^2 - 6*y)*(z + 4*(-6*x^2 + y)^2)^2, -6*x^2 + y, z + 4*(-6*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7384984319996875
      ]
    },
    {
      "case": "B226",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-30*x^2 + 5*y)*(z + (-6*x^2 + y)^2)^2, -6*x^2 + y, z + (-6*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7621235229999002
      ]
    },
    {
      "case": "B134",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (3*x^2 + y)*(z - 5*(3*x^2 + y)^2)^2, 3*x^2 + y, z - 5*(3*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7241038989996014
      ]
    },
    {
      "case": "B155",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (12*x^2 + 2*y)*(z + 2*(6*x^2 + y)^2)^2, 6*x^2 + y, z + 2*(6*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7596385699998791
      ]
    },
    {
      "case": "C54",
      "family": "counterexample",
      "mapping": "F(x,y,z)=((y + 2)^2*((x + 1)*(y + 2) + 1)*(3*(x + 1)*(y + 2) + 4) + (z - 1)*((x + 1)*(y + 2) + 1)^3 + 2, y + 3*(x + 1)*(y + 2)^2*(3*(x + 1)*(y + 2) + 4) + 3*(x + 1)*(z - 1)*((x + 1)*(y + 2) + 1)^2 + 3, 2*x - (x + 1)^3*(z - 1) - 3*(x + 1)^2*(y + 2))",
      "determinant": "-2",
      "certificate": "det J=-2; three distinct rational preimages share (7/4, 1, -2)",
      "probe_target": "No",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7388903080000091
      ]
    },
    {
      "case": "C236",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-y^2*(y*(x - 2) + 1)*(3*y*(x - 2) + 4) - (y*(x - 2) + 1)^3*(2*x^2 + z + 2), 3*y^2*(x - 2)*(3*y*(x - 2) + 4) + y + 3*(x - 2)*(y*(x - 2) + 1)^2*(2*x^2 + z + 2) + 1, 2*x - 3*y*(x - 2)^2 - (x - 2)^3*(2*x^2 + z + 2) - 6)",
      "determinant": "2",
      "certificate": "det J=2; three distinct rational preimages share (1/4, 1, -2)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7190286479999486
      ]
    },
    {
      "case": "B205",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-24*x^2 + 4*y)*(z + 6*(-6*x^2 + y)^2)^2, -6*x^2 + y, z + 6*(-6*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7190303509996738
      ]
    },
    {
      "case": "C90",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(y^2*(y*(x + 1) + 1)*(3*y*(x + 1) + 4) + (y*(x + 1) + 1)^3*(-x^2 + z + 2) + 1, 3*y^2*(x + 1)*(3*y*(x + 1) + 4) + y + 3*(x + 1)*(y*(x + 1) + 1)^2*(-x^2 + z + 2) + 2, 2*x - 3*y*(x + 1)^2 - (x + 1)^3*(-x^2 + z + 2))",
      "determinant": "-2",
      "certificate": "det J=-2; three distinct rational preimages share (3/4, 2, -2)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7293458180001835
      ]
    },
    {
      "case": "C222",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-2*(y - 2)^2*((x + 2)*(y - 2) + 1)*(3*(x + 2)*(y - 2) + 4) - 2*(z - 2)*((x + 2)*(y - 2) + 1)^3 - 1, y + 3*(x + 2)*(y - 2)^2*(3*(x + 2)*(y - 2) + 4) + 3*(x + 2)*(z - 2)*((x + 2)*(y - 2) + 1)^2 - 4, 2*x - (x + 2)^3*(z - 2) - 3*(x + 2)^2*(y - 2) + 3)",
      "determinant": "4",
      "certificate": "det J=4; three distinct rational preimages share (-1/2, -2, -1)",
      "probe_target": "No",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.722659993999514
      ]
    },
    {
      "case": "C21",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-(y + 2)^2*((x - 1)*(y + 2) + 1)*(3*(x - 1)*(y + 2) + 4) - (z - 1)*((x - 1)*(y + 2) + 1)^3 + 1, y + 3*(x - 1)*(y + 2)^2*(3*(x - 1)*(y + 2) + 4) + 3*(x - 1)*(z - 1)*((x - 1)*(y + 2) + 1)^2, 2*x - (x - 1)^3*(z - 1) - 3*(x - 1)^2*(y + 2) - 4)",
      "determinant": "2",
      "certificate": "det J=2; three distinct rational preimages share (5/4, -2, -2)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7047890510002617
      ]
    },
    {
      "case": "B239",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-10*x^2 + 5*y)*(z - 5*(-2*x^2 + y)^2)^2, -2*x^2 + y, z - 5*(-2*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.718805563999922
      ]
    },
    {
      "case": "B91",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (12*x^2 + 6*y)*(z + (2*x^2 + y)^2)^2, 2*x^2 + y, z + (2*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.760031787999651
      ]
    },
    {
      "case": "C82",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(2*(y - 2)^2*((x - 2)*(y - 2) + 1)*(3*(x - 2)*(y - 2) + 4) + 2*((x - 2)*(y - 2) + 1)^3*(-x^2 + z + 2) + 1, y + 3*(x - 2)*(y - 2)^2*(3*(x - 2)*(y - 2) + 4) + 3*(x - 2)*((x - 2)*(y - 2) + 1)^2*(-x^2 + z + 2), 2*x - (x - 2)^3*(-x^2 + z + 2) - 3*(x - 2)^2*(y - 2) - 2)",
      "determinant": "-4",
      "certificate": "det J=-4; three distinct rational preimages share (1/2, 2, 2)",
      "probe_target": "No",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7356623879995823
      ]
    },
    {
      "case": "C49",
      "family": "counterexample",
      "mapping": "F(x,y,z)=((x^2 + z)*((x + 1)*(y - 1) + 1)^3 + (y - 1)^2*((x + 1)*(y - 1) + 1)*(3*(x + 1)*(y - 1) + 4) - 1, y + 3*(x + 1)*(x^2 + z)*((x + 1)*(y - 1) + 1)^2 + 3*(x + 1)*(y - 1)^2*(3*(x + 1)*(y - 1) + 4), 2*x - (x + 1)^3*(x^2 + z) - 3*(x + 1)^2*(y - 1) + 1)",
      "determinant": "-2",
      "certificate": "det J=-2; three distinct rational preimages share (-5/4, 1, -1)",
      "probe_target": "No",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7371034280004096
      ]
    },
    {
      "case": "C44",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-(y - 1)^2*((x + 2)*(y - 1) + 1)*(3*(x + 2)*(y - 1) + 4) - ((x + 2)*(y - 1) + 1)^3*(-x^2 + z - 1) - 2, y + 3*(x + 2)*(y - 1)^2*(3*(x + 2)*(y - 1) + 4) + 3*(x + 2)*((x + 2)*(y - 1) + 1)^2*(-x^2 + z - 1) - 1, 2*x - (x + 2)^3*(-x^2 + z - 1) - 3*(x + 2)^2*(y - 1) + 2)",
      "determinant": "2",
      "certificate": "det J=2; three distinct rational preimages share (-7/4, 0, -2)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7540648260001035
      ]
    },
    {
      "case": "B90",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-4*x^2 + 4*y)*(z + 4*(-x^2 + y)^2)^2, -x^2 + y, z + 4*(-x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7047184399998514
      ]
    },
    {
      "case": "C180",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-(y + 2)^2*((x - 1)*(y + 2) + 1)*(3*(x - 1)*(y + 2) + 4) - ((x - 1)*(y + 2) + 1)^3*(-2*x^2 + z - 2) + 1, y + 3*(x - 1)*(y + 2)^2*(3*(x - 1)*(y + 2) + 4) + 3*(x - 1)*((x - 1)*(y + 2) + 1)^2*(-2*x^2 + z - 2) + 3, 2*x - (x - 1)^3*(-2*x^2 + z - 2) - 3*(x - 1)^2*(y + 2) - 1)",
      "determinant": "2",
      "certificate": "det J=2; three distinct rational preimages share (5/4, 1, 1)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7987449470001593
      ]
    },
    {
      "case": "C167",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(2*z*((x + 2)*(y - 1) + 1)^3 + 2*(y - 1)^2*((x + 2)*(y - 1) + 1)*(3*(x + 2)*(y - 1) + 4) - 1, y + 3*z*(x + 2)*((x + 2)*(y - 1) + 1)^2 + 3*(x + 2)*(y - 1)^2*(3*(x + 2)*(y - 1) + 4) - 3, 2*x - z*(x + 2)^3 - 3*(x + 2)^2*(y - 1) + 3)",
      "determinant": "-4",
      "certificate": "det J=-4; three distinct rational preimages share (-3/2, -2, -1)",
      "probe_target": "No",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7780531590005921
      ]
    },
    {
      "case": "B193",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-3*x^2 - 3*y)*(z - 2*(x^2 + y)^2)^2, x^2 + y, z - 2*(x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7666213250004148
      ]
    },
    {
      "case": "B74",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-30*x^2 - 5*y)*(z + 6*(6*x^2 + y)^2)^2, 6*x^2 + y, z + 6*(6*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7576941280003666
      ]
    },
    {
      "case": "B93",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-8*x^2 + 4*y)*(z - 3*(-2*x^2 + y)^2)^2, -2*x^2 + y, z - 3*(-2*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7957526449999932
      ]
    },
    {
      "case": "B08",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-30*x^2 + 5*y)*(z + 3*(-6*x^2 + y)^2)^2, -6*x^2 + y, z + 3*(-6*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7182339760001923
      ]
    },
    {
      "case": "B34",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-2*x^2 - 2*y)*(z + 4*(x^2 + y)^2)^2, x^2 + y, z + 4*(x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.8064753470007417
      ]
    },
    {
      "case": "B222",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (30*x^2 - 6*y)*(z + 2*(-5*x^2 + y)^2)^2, -5*x^2 + y, z + 2*(-5*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7608522579994315
      ]
    },
    {
      "case": "C74",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-(y - 1)^2*((x + 2)*(y - 1) + 1)*(3*(x + 2)*(y - 1) + 4) - ((x + 2)*(y - 1) + 1)^3*(x^2 + z + 2), y + 3*(x + 2)*(y - 1)^2*(3*(x + 2)*(y - 1) + 4) + 3*(x + 2)*((x + 2)*(y - 1) + 1)^2*(x^2 + z + 2) + 1, 2*x - (x + 2)^3*(x^2 + z + 2) - 3*(x + 2)^2*(y - 1) + 2)",
      "determinant": "2",
      "certificate": "det J=2; three distinct rational preimages share (1/4, 2, -2)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7609954529998504
      ]
    },
    {
      "case": "B158",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-4*x^2 + 4*y)*(z - 2*(-x^2 + y)^2)^2, -x^2 + y, z - 2*(-x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7510214379999525
      ]
    },
    {
      "case": "B161",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-8*x^2 - 4*y)*(z + 4*(2*x^2 + y)^2)^2, 2*x^2 + y, z + 4*(2*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.785169531000065
      ]
    },
    {
      "case": "C223",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-2*(x^2 + z)*((x + 2)*(y + 1) + 1)^3 - 2*(y + 1)^2*((x + 2)*(y + 1) + 1)*(3*(x + 2)*(y + 1) + 4) + 2, y + 3*(x + 2)*(x^2 + z)*((x + 2)*(y + 1) + 1)^2 + 3*(x + 2)*(y + 1)^2*(3*(x + 2)*(y + 1) + 4) + 1, 2*x - (x + 2)^3*(x^2 + z) - 3*(x + 2)^2*(y + 1) + 3)",
      "determinant": "4",
      "certificate": "det J=4; three distinct rational preimages share (5/2, 0, -1)",
      "probe_target": "No",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.8389743549996638
      ]
    },
    {
      "case": "C62",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-2*(y + 1)^2*((x + 2)*(y + 1) + 1)*(3*(x + 2)*(y + 1) + 4) - 2*((x + 2)*(y + 1) + 1)^3*(2*x^2 + z - 2) - 1, y + 3*(x + 2)*(y + 1)^2*(3*(x + 2)*(y + 1) + 4) + 3*(x + 2)*((x + 2)*(y + 1) + 1)^2*(2*x^2 + z - 2) + 3, 2*x - (x + 2)^3*(2*x^2 + z - 2) - 3*(x + 2)^2*(y + 1) + 4)",
      "determinant": "4",
      "certificate": "det J=4; three distinct rational preimages share (-1/2, 2, 0)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7668884390004678
      ]
    },
    {
      "case": "B80",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (30*x^2 - 6*y)*(z + 3*(-5*x^2 + y)^2)^2, -5*x^2 + y, z + 3*(-5*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7678628040002877
      ]
    },
    {
      "case": "C29",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(2*(y - 1)^2*((x - 1)*(y - 1) + 1)*(3*(x - 1)*(y - 1) + 4) + 2*((x - 1)*(y - 1) + 1)^3*(-2*x^2 + z + 2) + 1, y + 3*(x - 1)*(y - 1)^2*(3*(x - 1)*(y - 1) + 4) + 3*(x - 1)*((x - 1)*(y - 1) + 1)^2*(-2*x^2 + z + 2) - 2, 2*x - (x - 1)^3*(-2*x^2 + z + 2) - 3*(x - 1)^2*(y - 1) - 1)",
      "determinant": "-4",
      "certificate": "det J=-4; three distinct rational preimages share (1/2, -1, 1)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7646685559993784
      ]
    },
    {
      "case": "C73",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(2*(y + 2)^2*(x*(y + 2) + 1)*(3*x*(y + 2) + 4) + 2*(x*(y + 2) + 1)^3*(x^2 + z - 1) - 2, 3*x*(y + 2)^2*(3*x*(y + 2) + 4) + 3*x*(x*(y + 2) + 1)^2*(x^2 + z - 1) + y + 3, -x^3*(x^2 + z - 1) - 3*x^2*(y + 2) + 2*x - 2)",
      "determinant": "-4",
      "certificate": "det J=-4; three distinct rational preimages share (-5/2, 1, -2)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7767042010000296
      ]
    },
    {
      "case": "C79",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(2*(y - 2)^2*((x + 2)*(y - 2) + 1)*(3*(x + 2)*(y - 2) + 4) + 2*((x + 2)*(y - 2) + 1)^3*(-2*x^2 + z + 2), y + 3*(x + 2)*(y - 2)^2*(3*(x + 2)*(y - 2) + 4) + 3*(x + 2)*((x + 2)*(y - 2) + 1)^2*(-2*x^2 + z + 2) - 1, 2*x - (x + 2)^3*(-2*x^2 + z + 2) - 3*(x + 2)^2*(y - 2) + 5)",
      "determinant": "-4",
      "certificate": "det J=-4; three distinct rational preimages share (-1/2, 1, 1)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7682288910000352
      ]
    },
    {
      "case": "B86",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (16*x^2 - 4*y)*(z - 2*(-4*x^2 + y)^2)^2, -4*x^2 + y, z - 2*(-4*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7878344470000229
      ]
    },
    {
      "case": "B245",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-36*x^2 + 6*y)*(z - 2*(-6*x^2 + y)^2)^2, -6*x^2 + y, z - 2*(-6*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7762764789995344
      ]
    },
    {
      "case": "C185",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(2*(y - 2)^2*((x - 1)*(y - 2) + 1)*(3*(x - 1)*(y - 2) + 4) + 2*((x - 1)*(y - 2) + 1)^3*(-2*x^2 + z - 2) - 2, y + 3*(x - 1)*(y - 2)^2*(3*(x - 1)*(y - 2) + 4) + 3*(x - 1)*((x - 1)*(y - 2) + 1)^2*(-2*x^2 + z - 2) - 3, 2*x - (x - 1)^3*(-2*x^2 + z - 2) - 3*(x - 1)^2*(y - 2))",
      "determinant": "-4",
      "certificate": "det J=-4; three distinct rational preimages share (-5/2, -1, 2)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7183757279999554
      ]
    },
    {
      "case": "C42",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(2*(y - 2)^2*((x - 2)*(y - 2) + 1)*(3*(x - 2)*(y - 2) + 4) + 2*((x - 2)*(y - 2) + 1)^3*(-2*x^2 + z - 1) - 2, y + 3*(x - 2)*(y - 2)^2*(3*(x - 2)*(y - 2) + 4) + 3*(x - 2)*((x - 2)*(y - 2) + 1)^2*(-2*x^2 + z - 1) - 2, 2*x - (x - 2)^3*(-2*x^2 + z - 1) - 3*(x - 2)^2*(y - 2) - 4)",
      "determinant": "-4",
      "certificate": "det J=-4; three distinct rational preimages share (-5/2, 0, 0)",
      "probe_target": "No",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7272755630001484
      ]
    },
    {
      "case": "C117",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-2*y^2*(x*y + 1)*(3*x*y + 4) - 2*(x*y + 1)^3*(-2*x^2 + z + 2) - 1, 3*x*y^2*(3*x*y + 4) + 3*x*(x*y + 1)^2*(-2*x^2 + z + 2) + y - 2, -x^3*(-2*x^2 + z + 2) - 3*x^2*y + 2*x)",
      "determinant": "4",
      "certificate": "det J=4; three distinct rational preimages share (-1/2, -2, 0)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7783790140001656
      ]
    },
    {
      "case": "C197",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-2*(y + 2)^2*(x*(y + 2) + 1)*(3*x*(y + 2) + 4) - 2*(x*(y + 2) + 1)^3*(-2*x^2 + z - 2) + 1, 3*x*(y + 2)^2*(3*x*(y + 2) + 4) + 3*x*(x*(y + 2) + 1)^2*(-2*x^2 + z - 2) + y + 3, -x^3*(-2*x^2 + z - 2) - 3*x^2*(y + 2) + 2*x + 1)",
      "determinant": "4",
      "certificate": "det J=4; three distinct rational preimages share (3/2, 1, 1)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.8320334079999157
      ]
    },
    {
      "case": "B131",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-8*x^2 - 4*y)*(z + 6*(2*x^2 + y)^2)^2, 2*x^2 + y, z + 6*(2*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7362722379994011
      ]
    },
    {
      "case": "B202",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-20*x^2 + 5*y)*(z + 6*(-4*x^2 + y)^2)^2, -4*x^2 + y, z + 6*(-4*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.761785666999458
      ]
    },
    {
      "case": "B116",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (5*x^2 - y)*(z - (-5*x^2 + y)^2)^2, -5*x^2 + y, z - (-5*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7355395710001176
      ]
    },
    {
      "case": "B197",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (25*x^2 - 5*y)*(z - 3*(-5*x^2 + y)^2)^2, -5*x^2 + y, z - 3*(-5*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7136545530001968
      ]
    },
    {
      "case": "B50",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (24*x^2 - 4*y)*(z + 2*(-6*x^2 + y)^2)^2, -6*x^2 + y, z + 2*(-6*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7050292570002057
      ]
    },
    {
      "case": "C121",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-2*z*(x*(y - 2) + 1)^3 - 2*(y - 2)^2*(x*(y - 2) + 1)*(3*x*(y - 2) + 4), 3*x*z*(x*(y - 2) + 1)^2 + 3*x*(y - 2)^2*(3*x*(y - 2) + 4) + y - 4, -x^3*z - 3*x^2*(y - 2) + 2*x - 2)",
      "determinant": "4",
      "certificate": "det J=4; three distinct rational preimages share (1/2, -2, -2)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7786040999999386
      ]
    },
    {
      "case": "B04",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (3*x^2 - y)*(z + 6*(-3*x^2 + y)^2)^2, -3*x^2 + y, z + 6*(-3*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7389710970001033
      ]
    },
    {
      "case": "B143",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-36*x^2 - 6*y)*(z - 3*(6*x^2 + y)^2)^2, 6*x^2 + y, z - 3*(6*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.756638132000262
      ]
    },
    {
      "case": "B69",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (3*x^2 + 3*y)*(z + (x^2 + y)^2)^2, x^2 + y, z + (x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.8111746009999479
      ]
    },
    {
      "case": "C216",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-2*(y - 2)^2*((x + 1)*(y - 2) + 1)*(3*(x + 1)*(y - 2) + 4) - 2*((x + 1)*(y - 2) + 1)^3*(-x^2 + z + 1) - 2, y + 3*(x + 1)*(y - 2)^2*(3*(x + 1)*(y - 2) + 4) + 3*(x + 1)*((x + 1)*(y - 2) + 1)^2*(-x^2 + z + 1) - 3, 2*x - (x + 1)^3*(-x^2 + z + 1) - 3*(x + 1)^2*(y - 2) + 2)",
      "determinant": "4",
      "certificate": "det J=4; three distinct rational preimages share (-3/2, -1, 0)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7522071439998399
      ]
    },
    {
      "case": "B164",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-4*x^2 - 4*y)*(z - (x^2 + y)^2)^2, x^2 + y, z - (x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.757124742999622
      ]
    },
    {
      "case": "B192",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-2*x^2 - 2*y)*(z + (x^2 + y)^2)^2, x^2 + y, z + (x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7371355469995251
      ]
    },
    {
      "case": "C124",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-y^2*(y*(x - 1) + 1)*(3*y*(x - 1) + 4) - (y*(x - 1) + 1)^3*(x^2 + z - 1) - 1, 3*y^2*(x - 1)*(3*y*(x - 1) + 4) + y + 3*(x - 1)*(y*(x - 1) + 1)^2*(x^2 + z - 1) - 1, 2*x - 3*y*(x - 1)^2 - (x - 1)^3*(x^2 + z - 1) - 2)",
      "determinant": "2",
      "certificate": "det J=2; three distinct rational preimages share (-3/4, -1, 0)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7511513389999891
      ]
    },
    {
      "case": "C173",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-2*(x^2 + z)*((x + 1)*(y + 1) + 1)^3 - 2*(y + 1)^2*((x + 1)*(y + 1) + 1)*(3*(x + 1)*(y + 1) + 4) - 1, y + 3*(x + 1)*(x^2 + z)*((x + 1)*(y + 1) + 1)^2 + 3*(x + 1)*(y + 1)^2*(3*(x + 1)*(y + 1) + 4) + 3, 2*x - (x + 1)^3*(x^2 + z) - 3*(x + 1)^2*(y + 1) + 4)",
      "determinant": "4",
      "certificate": "det J=4; three distinct rational preimages share (-1/2, 2, 2)",
      "probe_target": "No",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7305651659999057
      ]
    },
    {
      "case": "C220",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-y^2*(y*(x + 1) + 1)*(3*y*(x + 1) + 4) - (y*(x + 1) + 1)^3*(x^2 + z + 2) - 1, 3*y^2*(x + 1)*(3*y*(x + 1) + 4) + y + 3*(x + 1)*(y*(x + 1) + 1)^2*(x^2 + z + 2), 2*x - 3*y*(x + 1)^2 - (x + 1)^3*(x^2 + z + 2) + 4)",
      "determinant": "2",
      "certificate": "det J=2; three distinct rational preimages share (-3/4, 0, 2)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7553989660000298
      ]
    },
    {
      "case": "B15",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (24*x^2 + 6*y)*(z + (4*x^2 + y)^2)^2, 4*x^2 + y, z + (4*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7801565560002928
      ]
    },
    {
      "case": "C171",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-(y + 2)^2*(x*(y + 2) + 1)*(3*x*(y + 2) + 4) - (x*(y + 2) + 1)^3*(-x^2 + z - 2) - 1, 3*x*(y + 2)^2*(3*x*(y + 2) + 4) + 3*x*(x*(y + 2) + 1)^2*(-x^2 + z - 2) + y, -x^3*(-x^2 + z - 2) - 3*x^2*(y + 2) + 2*x + 1)",
      "determinant": "2",
      "certificate": "det J=2; three distinct rational preimages share (-3/4, -2, 1)",
      "probe_target": "No",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7223119099999167
      ]
    },
    {
      "case": "B89",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-5*x^2 - y)*(z - 2*(5*x^2 + y)^2)^2, 5*x^2 + y, z - 2*(5*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.720806461000393
      ]
    },
    {
      "case": "B169",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-4*x^2 + 2*y)*(z - 3*(-2*x^2 + y)^2)^2, -2*x^2 + y, z - 3*(-2*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7545029370003249
      ]
    },
    {
      "case": "B247",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-6*x^2 - y)*(z + (6*x^2 + y)^2)^2, 6*x^2 + y, z + (6*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7512084749996575
      ]
    },
    {
      "case": "C46",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-(y - 1)^2*((x - 2)*(y - 1) + 1)*(3*(x - 2)*(y - 1) + 4) - ((x - 2)*(y - 1) + 1)^3*(-2*x^2 + z - 1) - 2, y + 3*(x - 2)*(y - 1)^2*(3*(x - 2)*(y - 1) + 4) + 3*(x - 2)*((x - 2)*(y - 1) + 1)^2*(-2*x^2 + z - 1), 2*x - (x - 2)^3*(-2*x^2 + z - 1) - 3*(x - 2)^2*(y - 1) - 3)",
      "determinant": "2",
      "certificate": "det J=2; three distinct rational preimages share (-7/4, 1, 1)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7458441689996107
      ]
    },
    {
      "case": "C47",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-2*(-x^2 + z)*(x*(y + 1) + 1)^3 - 2*(y + 1)^2*(x*(y + 1) + 1)*(3*x*(y + 1) + 4) + 2, 3*x*(-x^2 + z)*(x*(y + 1) + 1)^2 + 3*x*(y + 1)^2*(3*x*(y + 1) + 4) + y + 2, -x^3*(-x^2 + z) - 3*x^2*(y + 1) + 2*x)",
      "determinant": "4",
      "certificate": "det J=4; three distinct rational preimages share (5/2, 1, 0)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7511204219999854
      ]
    },
    {
      "case": "B10",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (2*x^2 + 2*y)*(z - 2*(x^2 + y)^2)^2, x^2 + y, z - 2*(x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7939907509999102
      ]
    },
    {
      "case": "B153",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-16*x^2 + 4*y)*(z - 6*(-4*x^2 + y)^2)^2, -4*x^2 + y, z - 6*(-4*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7452002059999359
      ]
    },
    {
      "case": "C51",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-2*y^2*(y*(x + 1) + 1)*(3*y*(x + 1) + 4) - 2*(z + 1)*(y*(x + 1) + 1)^3 + 2, 3*y^2*(x + 1)*(3*y*(x + 1) + 4) + y + 3*(x + 1)*(z + 1)*(y*(x + 1) + 1)^2, 2*x - 3*y*(x + 1)^2 - (x + 1)^3*(z + 1))",
      "determinant": "4",
      "certificate": "det J=4; three distinct rational preimages share (5/2, 0, -2)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7240128900002674
      ]
    },
    {
      "case": "C229",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(2*(y + 2)^2*(x*(y + 2) + 1)*(3*x*(y + 2) + 4) + 2*(x*(y + 2) + 1)^3*(-2*x^2 + z - 1) + 1, 3*x*(y + 2)^2*(3*x*(y + 2) + 4) + 3*x*(x*(y + 2) + 1)^2*(-2*x^2 + z - 1) + y, -x^3*(-2*x^2 + z - 1) - 3*x^2*(y + 2) + 2*x + 1)",
      "determinant": "-4",
      "certificate": "det J=-4; three distinct rational preimages share (1/2, -2, 1)",
      "probe_target": "No",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.9090150629999698
      ]
    },
    {
      "case": "B22",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (18*x^2 - 3*y)*(z - 3*(-6*x^2 + y)^2)^2, -6*x^2 + y, z - 3*(-6*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.6890105849997781
      ]
    },
    {
      "case": "C103",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(2*(y - 1)^2*((x - 1)*(y - 1) + 1)*(3*(x - 1)*(y - 1) + 4) + 2*((x - 1)*(y - 1) + 1)^3*(x^2 + z + 2) + 2, y + 3*(x - 1)*(y - 1)^2*(3*(x - 1)*(y - 1) + 4) + 3*(x - 1)*((x - 1)*(y - 1) + 1)^2*(x^2 + z + 2) - 3, 2*x - (x - 1)^3*(x^2 + z + 2) - 3*(x - 1)^2*(y - 1))",
      "determinant": "-4",
      "certificate": "det J=-4; three distinct rational preimages share (3/2, -2, 2)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7393290499994691
      ]
    },
    {
      "case": "C196",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(2*(y - 1)^2*((x + 1)*(y - 1) + 1)*(3*(x + 1)*(y - 1) + 4) + 2*((x + 1)*(y - 1) + 1)^3*(x^2 + z + 2) + 1, y + 3*(x + 1)*(y - 1)^2*(3*(x + 1)*(y - 1) + 4) + 3*(x + 1)*((x + 1)*(y - 1) + 1)^2*(x^2 + z + 2) - 2, 2*x - (x + 1)^3*(x^2 + z + 2) - 3*(x + 1)^2*(y - 1) + 2)",
      "determinant": "-4",
      "certificate": "det J=-4; three distinct rational preimages share (1/2, -1, 0)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7892353930001264
      ]
    },
    {
      "case": "B149",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-15*x^2 - 3*y)*(z + 2*(5*x^2 + y)^2)^2, 5*x^2 + y, z + 2*(5*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7364632319995508
      ]
    },
    {
      "case": "C04",
      "family": "counterexample",
      "mapping": "F(x,y,z)=((y + 2)^2*((x - 2)*(y + 2) + 1)*(3*(x - 2)*(y + 2) + 4) + (z + 1)*((x - 2)*(y + 2) + 1)^3 - 2, y + 3*(x - 2)*(y + 2)^2*(3*(x - 2)*(y + 2) + 4) + 3*(x - 2)*(z + 1)*((x - 2)*(y + 2) + 1)^2 + 4, 2*x - (x - 2)^3*(z + 1) - 3*(x - 2)^2*(y + 2) - 2)",
      "determinant": "-2",
      "certificate": "det J=-2; three distinct rational preimages share (-9/4, 2, 2)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7345200019999538
      ]
    },
    {
      "case": "C195",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-2*(y - 2)^2*((x - 1)*(y - 2) + 1)*(3*(x - 1)*(y - 2) + 4) - 2*((x - 1)*(y - 2) + 1)^3*(2*x^2 + z + 1) + 1, y + 3*(x - 1)*(y - 2)^2*(3*(x - 1)*(y - 2) + 4) + 3*(x - 1)*((x - 1)*(y - 2) + 1)^2*(2*x^2 + z + 1) - 2, 2*x - (x - 1)^3*(2*x^2 + z + 1) - 3*(x - 1)^2*(y - 2) - 1)",
      "determinant": "4",
      "certificate": "det J=4; three distinct rational preimages share (3/2, 0, 1)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7489561930005948
      ]
    },
    {
      "case": "B77",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-9*x^2 + 3*y)*(z - 3*(-3*x^2 + y)^2)^2, -3*x^2 + y, z - 3*(-3*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.8996079270000337
      ]
    },
    {
      "case": "C93",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-(y - 2)^2*((x - 1)*(y - 2) + 1)*(3*(x - 1)*(y - 2) + 4) - (z - 2)*((x - 1)*(y - 2) + 1)^3 + 1, y + 3*(x - 1)*(y - 2)^2*(3*(x - 1)*(y - 2) + 4) + 3*(x - 1)*(z - 2)*((x - 1)*(y - 2) + 1)^2 - 1, 2*x - (x - 1)^3*(z - 2) - 3*(x - 1)^2*(y - 2) - 2)",
      "determinant": "2",
      "certificate": "det J=2; three distinct rational preimages share (5/4, 1, 0)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.801468302000103
      ]
    },
    {
      "case": "C60",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-(y + 2)^2*((x + 2)*(y + 2) + 1)*(3*(x + 2)*(y + 2) + 4) - ((x + 2)*(y + 2) + 1)^3*(x^2 + z - 2) + 1, y + 3*(x + 2)*(y + 2)^2*(3*(x + 2)*(y + 2) + 4) + 3*(x + 2)*((x + 2)*(y + 2) + 1)^2*(x^2 + z - 2) + 1, 2*x - (x + 2)^3*(x^2 + z - 2) - 3*(x + 2)^2*(y + 2) + 6)",
      "determinant": "2",
      "certificate": "det J=2; three distinct rational preimages share (5/4, -1, 2)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.8604407370003173
      ]
    },
    {
      "case": "C38",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-(y + 1)^2*((x + 1)*(y + 1) + 1)*(3*(x + 1)*(y + 1) + 4) - ((x + 1)*(y + 1) + 1)^3*(-2*x^2 + z - 2) - 2, y + 3*(x + 1)*(y + 1)^2*(3*(x + 1)*(y + 1) + 4) + 3*(x + 1)*((x + 1)*(y + 1) + 1)^2*(-2*x^2 + z - 2) + 3, 2*x - (x + 1)^3*(-2*x^2 + z - 2) - 3*(x + 1)^2*(y + 1) + 2)",
      "determinant": "2",
      "certificate": "det J=2; three distinct rational preimages share (-7/4, 2, 0)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.782078907000141
      ]
    },
    {
      "case": "C177",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-y^2*(y*(x + 1) + 1)*(3*y*(x + 1) + 4) - (y*(x + 1) + 1)^3*(2*x^2 + z + 2) + 2, 3*y^2*(x + 1)*(3*y*(x + 1) + 4) + y + 3*(x + 1)*(y*(x + 1) + 1)^2*(2*x^2 + z + 2), 2*x - 3*y*(x + 1)^2 - (x + 1)^3*(2*x^2 + z + 2) + 3)",
      "determinant": "2",
      "certificate": "det J=2; three distinct rational preimages share (9/4, 0, 1)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.743661797000641
      ]
    },
    {
      "case": "C116",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-(y + 1)^2*(x*(y + 1) + 1)*(3*x*(y + 1) + 4) - (z - 1)*(x*(y + 1) + 1)^3 + 2, 3*x*(y + 1)^2*(3*x*(y + 1) + 4) + 3*x*(z - 1)*(x*(y + 1) + 1)^2 + y - 1, -x^3*(z - 1) - 3*x^2*(y + 1) + 2*x - 1)",
      "determinant": "2",
      "certificate": "det J=2; three distinct rational preimages share (9/4, -2, -1)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7950025159998404
      ]
    },
    {
      "case": "C78",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(2*(y - 1)^2*(x*(y - 1) + 1)*(3*x*(y - 1) + 4) + 2*(x*(y - 1) + 1)^3*(2*x^2 + z + 1) + 2, 3*x*(y - 1)^2*(3*x*(y - 1) + 4) + 3*x*(x*(y - 1) + 1)^2*(2*x^2 + z + 1) + y - 2, -x^3*(2*x^2 + z + 1) - 3*x^2*(y - 1) + 2*x - 1)",
      "determinant": "-4",
      "certificate": "det J=-4; three distinct rational preimages share (3/2, -1, -1)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7921336310000697
      ]
    },
    {
      "case": "C92",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-2*(y - 2)^2*(x*(y - 2) + 1)*(3*x*(y - 2) + 4) - 2*(z + 2)*(x*(y - 2) + 1)^3 + 1, 3*x*(y - 2)^2*(3*x*(y - 2) + 4) + 3*x*(z + 2)*(x*(y - 2) + 1)^2 + y - 4, -x^3*(z + 2) - 3*x^2*(y - 2) + 2*x - 2)",
      "determinant": "4",
      "certificate": "det J=4; three distinct rational preimages share (3/2, -2, -2)",
      "probe_target": "No",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7541498729997329
      ]
    },
    {
      "case": "C80",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(2*(y + 1)^2*((x + 2)*(y + 1) + 1)*(3*(x + 2)*(y + 1) + 4) + 2*((x + 2)*(y + 1) + 1)^3*(x^2 + z - 1) + 2, y + 3*(x + 2)*(y + 1)^2*(3*(x + 2)*(y + 1) + 4) + 3*(x + 2)*((x + 2)*(y + 1) + 1)^2*(x^2 + z - 1) + 2, 2*x - (x + 2)^3*(x^2 + z - 1) - 3*(x + 2)^2*(y + 1) + 2)",
      "determinant": "-4",
      "certificate": "det J=-4; three distinct rational preimages share (3/2, 1, -2)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7456993699997838
      ]
    },
    {
      "case": "B177",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (6*x^2 + 2*y)*(z - 3*(3*x^2 + y)^2)^2, 3*x^2 + y, z - 3*(3*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7321291739999651
      ]
    },
    {
      "case": "C91",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(2*(y + 2)^2*((x - 2)*(y + 2) + 1)*(3*(x - 2)*(y + 2) + 4) + 2*((x - 2)*(y + 2) + 1)^3*(x^2 + z - 2), y + 3*(x - 2)*(y + 2)^2*(3*(x - 2)*(y + 2) + 4) + 3*(x - 2)*((x - 2)*(y + 2) + 1)^2*(x^2 + z - 2) + 3, 2*x - (x - 2)^3*(x^2 + z - 2) - 3*(x - 2)^2*(y + 2) - 2)",
      "determinant": "-4",
      "certificate": "det J=-4; three distinct rational preimages share (-1/2, 1, 2)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.8107013139997434
      ]
    },
    {
      "case": "C01",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(2*z*((x + 2)*(y - 2) + 1)^3 + 2*(y - 2)^2*((x + 2)*(y - 2) + 1)*(3*(x + 2)*(y - 2) + 4) - 2, y + 3*z*(x + 2)*((x + 2)*(y - 2) + 1)^2 + 3*(x + 2)*(y - 2)^2*(3*(x + 2)*(y - 2) + 4), 2*x - z*(x + 2)^3 - 3*(x + 2)^2*(y - 2) + 5)",
      "determinant": "-4",
      "certificate": "det J=-4; three distinct rational preimages share (-5/2, 2, 1)",
      "probe_target": "No",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7419758329997421
      ]
    },
    {
      "case": "B152",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (16*x^2 - 4*y)*(z + 3*(-4*x^2 + y)^2)^2, -4*x^2 + y, z + 3*(-4*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7664483540002038
      ]
    },
    {
      "case": "B97",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-8*x^2 + 2*y)*(z - 3*(-4*x^2 + y)^2)^2, -4*x^2 + y, z - 3*(-4*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7286525240006085
      ]
    },
    {
      "case": "C136",
      "family": "counterexample",
      "mapping": "F(x,y,z)=((y - 1)^2*((x + 2)*(y - 1) + 1)*(3*(x + 2)*(y - 1) + 4) + ((x + 2)*(y - 1) + 1)^3*(2*x^2 + z + 1), y + 3*(x + 2)*(y - 1)^2*(3*(x + 2)*(y - 1) + 4) + 3*(x + 2)*((x + 2)*(y - 1) + 1)^2*(2*x^2 + z + 1) - 1, 2*x - (x + 2)^3*(2*x^2 + z + 1) - 3*(x + 2)^2*(y - 1) + 6)",
      "determinant": "-2",
      "certificate": "det J=-2; three distinct rational preimages share (-1/4, 0, 2)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7851581699997041
      ]
    },
    {
      "case": "C02",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-(y + 2)^2*((x - 1)*(y + 2) + 1)*(3*(x - 1)*(y + 2) + 4) - ((x - 1)*(y + 2) + 1)^3*(x^2 + z + 2) + 2, y + 3*(x - 1)*(y + 2)^2*(3*(x - 1)*(y + 2) + 4) + 3*(x - 1)*((x - 1)*(y + 2) + 1)^2*(x^2 + z + 2) + 3, 2*x - (x - 1)^3*(x^2 + z + 2) - 3*(x - 1)^2*(y + 2) - 2)",
      "determinant": "2",
      "certificate": "det J=2; three distinct rational preimages share (9/4, 1, 0)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7484410289998777
      ]
    },
    {
      "case": "C98",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-2*y^2*(y*(x + 2) + 1)*(3*y*(x + 2) + 4) - 2*(y*(x + 2) + 1)^3*(x^2 + z - 1) - 1, 3*y^2*(x + 2)*(3*y*(x + 2) + 4) + y + 3*(x + 2)*(y*(x + 2) + 1)^2*(x^2 + z - 1) + 1, 2*x - 3*y*(x + 2)^2 - (x + 2)^3*(x^2 + z - 1) + 5)",
      "determinant": "4",
      "certificate": "det J=4; three distinct rational preimages share (-1/2, 1, 1)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7346242260000508
      ]
    },
    {
      "case": "B243",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-4*x^2 + 4*y)*(z - 3*(-x^2 + y)^2)^2, -x^2 + y, z - 3*(-x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7160911569999371
      ]
    },
    {
      "case": "C153",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-2*y^2*(y*(x + 1) + 1)*(3*y*(x + 1) + 4) - 2*(-x^2 + z)*(y*(x + 1) + 1)^3 + 2, 3*y^2*(x + 1)*(3*y*(x + 1) + 4) + y + 3*(x + 1)*(-x^2 + z)*(y*(x + 1) + 1)^2 + 2, 2*x - 3*y*(x + 1)^2 - (x + 1)^3*(-x^2 + z) + 4)",
      "determinant": "4",
      "certificate": "det J=4; three distinct rational preimages share (5/2, 2, 2)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.738262104999194
      ]
    },
    {
      "case": "C110",
      "family": "counterexample",
      "mapping": "F(x,y,z)=((y - 2)^2*((x - 1)*(y - 2) + 1)*(3*(x - 1)*(y - 2) + 4) + ((x - 1)*(y - 2) + 1)^3*(-2*x^2 + z + 1) - 2, y + 3*(x - 1)*(y - 2)^2*(3*(x - 1)*(y - 2) + 4) + 3*(x - 1)*((x - 1)*(y - 2) + 1)^2*(-2*x^2 + z + 1) - 4, 2*x - (x - 1)^3*(-2*x^2 + z + 1) - 3*(x - 1)^2*(y - 2))",
      "determinant": "-2",
      "certificate": "det J=-2; three distinct rational preimages share (-9/4, -2, 2)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7456999119995089
      ]
    },
    {
      "case": "B12",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-12*x^2 - 4*y)*(z + 5*(3*x^2 + y)^2)^2, 3*x^2 + y, z + 5*(3*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7627953580004032
      ]
    },
    {
      "case": "B00",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x+(y+x^2)*(z+(y+x^2)^2)^2, y+x^2, z+(y+x^2)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7431322960001125
      ]
    },
    {
      "case": "B173",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-12*x^2 - 6*y)*(z - 5*(2*x^2 + y)^2)^2, 2*x^2 + y, z - 5*(2*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7100309110001035
      ]
    },
    {
      "case": "C241",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-(y + 1)^2*((x + 2)*(y + 1) + 1)*(3*(x + 2)*(y + 1) + 4) - ((x + 2)*(y + 1) + 1)^3*(x^2 + z + 2) + 1, y + 3*(x + 2)*(y + 1)^2*(3*(x + 2)*(y + 1) + 4) + 3*(x + 2)*((x + 2)*(y + 1) + 1)^2*(x^2 + z + 2), 2*x - (x + 2)^3*(x^2 + z + 2) - 3*(x + 2)^2*(y + 1) + 5)",
      "determinant": "2",
      "certificate": "det J=2; three distinct rational preimages share (5/4, -1, 1)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7671358970001165
      ]
    },
    {
      "case": "C166",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(2*(y - 2)^2*((x - 1)*(y - 2) + 1)*(3*(x - 1)*(y - 2) + 4) + 2*((x - 1)*(y - 2) + 1)^3*(-x^2 + z - 2) + 2, y + 3*(x - 1)*(y - 2)^2*(3*(x - 1)*(y - 2) + 4) + 3*(x - 1)*((x - 1)*(y - 2) + 1)^2*(-x^2 + z - 2), 2*x - (x - 1)^3*(-x^2 + z - 2) - 3*(x - 1)^2*(y - 2) - 1)",
      "determinant": "-4",
      "certificate": "det J=-4; three distinct rational preimages share (3/2, 2, 1)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.8012573519999933
      ]
    },
    {
      "case": "C178",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-(y - 2)^2*(x*(y - 2) + 1)*(3*x*(y - 2) + 4) - (x*(y - 2) + 1)^3*(x^2 + z + 2) + 2, 3*x*(y - 2)^2*(3*x*(y - 2) + 4) + 3*x*(x*(y - 2) + 1)^2*(x^2 + z + 2) + y, -x^3*(x^2 + z + 2) - 3*x^2*(y - 2) + 2*x - 1)",
      "determinant": "2",
      "certificate": "det J=2; three distinct rational preimages share (9/4, 2, -1)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7790786690002278
      ]
    },
    {
      "case": "C247",
      "family": "counterexample",
      "mapping": "F(x,y,z)=((x^2 + z)*((x + 2)*(y - 1) + 1)^3 + (y - 1)^2*((x + 2)*(y - 1) + 1)*(3*(x + 2)*(y - 1) + 4) + 2, y + 3*(x + 2)*(x^2 + z)*((x + 2)*(y - 1) + 1)^2 + 3*(x + 2)*(y - 1)^2*(3*(x + 2)*(y - 1) + 4) - 2, 2*x - (x + 2)^3*(x^2 + z) - 3*(x + 2)^2*(y - 1) + 3)",
      "determinant": "-2",
      "certificate": "det J=-2; three distinct rational preimages share (7/4, -1, -1)",
      "probe_target": "No",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7443871000004947
      ]
    },
    {
      "case": "B120",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-12*x^2 - 2*y)*(z - 6*(6*x^2 + y)^2)^2, 6*x^2 + y, z - 6*(6*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7201680280004439
      ]
    },
    {
      "case": "C184",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(2*y^2*(x*y + 1)*(3*x*y + 4) + 2*(z + 2)*(x*y + 1)^3 - 2, 3*x*y^2*(3*x*y + 4) + 3*x*(z + 2)*(x*y + 1)^2 + y + 2, -x^3*(z + 2) - 3*x^2*y + 2*x - 2)",
      "determinant": "-4",
      "certificate": "det J=-4; three distinct rational preimages share (-5/2, 2, -2)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7532171060001929
      ]
    },
    {
      "case": "C12",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(2*(-x^2 + z)*((x - 1)*(y + 1) + 1)^3 + 2*(y + 1)^2*((x - 1)*(y + 1) + 1)*(3*(x - 1)*(y + 1) + 4) - 1, y + 3*(x - 1)*(-x^2 + z)*((x - 1)*(y + 1) + 1)^2 + 3*(x - 1)*(y + 1)^2*(3*(x - 1)*(y + 1) + 4) + 2, 2*x - (x - 1)^3*(-x^2 + z) - 3*(x - 1)^2*(y + 1) - 3)",
      "determinant": "-4",
      "certificate": "det J=-4; three distinct rational preimages share (-3/2, 1, -1)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7255422129992439
      ]
    },
    {
      "case": "B206",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-2*x^2 + y)*(z - 4*(-2*x^2 + y)^2)^2, -2*x^2 + y, z - 4*(-2*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7111328119999598
      ]
    },
    {
      "case": "C143",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-(y + 1)^2*((x - 2)*(y + 1) + 1)*(3*(x - 2)*(y + 1) + 4) - ((x - 2)*(y + 1) + 1)^3*(-2*x^2 + z + 1) + 2, y + 3*(x - 2)*(y + 1)^2*(3*(x - 2)*(y + 1) + 4) + 3*(x - 2)*((x - 2)*(y + 1) + 1)^2*(-2*x^2 + z + 1), 2*x - (x - 2)^3*(-2*x^2 + z + 1) - 3*(x - 2)^2*(y + 1) - 2)",
      "determinant": "2",
      "certificate": "det J=2; three distinct rational preimages share (9/4, -1, 2)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7140858120001212
      ]
    },
    {
      "case": "C204",
      "family": "counterexample",
      "mapping": "F(x,y,z)=((y - 1)^2*((x - 1)*(y - 1) + 1)*(3*(x - 1)*(y - 1) + 4) + ((x - 1)*(y - 1) + 1)^3*(x^2 + z - 1), y + 3*(x - 1)*(y - 1)^2*(3*(x - 1)*(y - 1) + 4) + 3*(x - 1)*((x - 1)*(y - 1) + 1)^2*(x^2 + z - 1) - 1, 2*x - (x - 1)^3*(x^2 + z - 1) - 3*(x - 1)^2*(y - 1) - 4)",
      "determinant": "-2",
      "certificate": "det J=-2; three distinct rational preimages share (-1/4, 0, -2)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7458846439994886
      ]
    },
    {
      "case": "B150",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (12*x^2 + 6*y)*(z + 5*(2*x^2 + y)^2)^2, 2*x^2 + y, z + 5*(2*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7540357509997193
      ]
    },
    {
      "case": "B49",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-10*x^2 + 5*y)*(z + 6*(-2*x^2 + y)^2)^2, -2*x^2 + y, z + 6*(-2*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7057452620001641
      ]
    },
    {
      "case": "C48",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(y^2*(y*(x + 2) + 1)*(3*y*(x + 2) + 4) + (y*(x + 2) + 1)^3*(-x^2 + z + 2) + 1, 3*y^2*(x + 2)*(3*y*(x + 2) + 4) + y + 3*(x + 2)*(y*(x + 2) + 1)^2*(-x^2 + z + 2) - 1, 2*x - 3*y*(x + 2)^2 - (x + 2)^3*(-x^2 + z + 2) + 4)",
      "determinant": "-2",
      "certificate": "det J=-2; three distinct rational preimages share (3/4, -1, 0)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7503551540003173
      ]
    },
    {
      "case": "B30",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-12*x^2 + 3*y)*(z + 2*(-4*x^2 + y)^2)^2, -4*x^2 + y, z + 2*(-4*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7264615770000091
      ]
    },
    {
      "case": "B83",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-24*x^2 + 6*y)*(z + 6*(-4*x^2 + y)^2)^2, -4*x^2 + y, z + 6*(-4*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7457668450006167
      ]
    },
    {
      "case": "B123",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (24*x^2 + 6*y)*(z + 2*(4*x^2 + y)^2)^2, 4*x^2 + y, z + 2*(4*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7347647659998984
      ]
    },
    {
      "case": "B196",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-12*x^2 + 2*y)*(z - (-6*x^2 + y)^2)^2, -6*x^2 + y, z - (-6*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7464004690000365
      ]
    },
    {
      "case": "C28",
      "family": "counterexample",
      "mapping": "F(x,y,z)=((y + 2)^2*((x + 2)*(y + 2) + 1)*(3*(x + 2)*(y + 2) + 4) + ((x + 2)*(y + 2) + 1)^3*(-x^2 + z - 2) + 2, y + 3*(x + 2)*(y + 2)^2*(3*(x + 2)*(y + 2) + 4) + 3*(x + 2)*((x + 2)*(y + 2) + 1)^2*(-x^2 + z - 2), 2*x - (x + 2)^3*(-x^2 + z - 2) - 3*(x + 2)^2*(y + 2) + 3)",
      "determinant": "-2",
      "certificate": "det J=-2; three distinct rational preimages share (7/4, -2, -1)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.8181765610006551
      ]
    },
    {
      "case": "C99",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-y^2*(y*(x + 2) + 1)*(3*y*(x + 2) + 4) - (z + 2)*(y*(x + 2) + 1)^3 - 1, 3*y^2*(x + 2)*(3*y*(x + 2) + 4) + y + 3*(x + 2)*(z + 2)*(y*(x + 2) + 1)^2 - 1, 2*x - 3*y*(x + 2)^2 - (x + 2)^3*(z + 2) + 5)",
      "determinant": "2",
      "certificate": "det J=2; three distinct rational preimages share (-3/4, -1, 1)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7828651340005308
      ]
    },
    {
      "case": "C66",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(y^2*(y*(x - 2) + 1)*(3*y*(x - 2) + 4) + (y*(x - 2) + 1)^3*(-2*x^2 + z + 2) - 2, 3*y^2*(x - 2)*(3*y*(x - 2) + 4) + y + 3*(x - 2)*(y*(x - 2) + 1)^2*(-2*x^2 + z + 2), 2*x - 3*y*(x - 2)^2 - (x - 2)^3*(-2*x^2 + z + 2) - 6)",
      "determinant": "-2",
      "certificate": "det J=-2; three distinct rational preimages share (-9/4, 0, -2)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7885895469999014
      ]
    },
    {
      "case": "B05",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (x^2 - y)*(z - 5*(-x^2 + y)^2)^2, -x^2 + y, z - 5*(-x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7622614789997897
      ]
    },
    {
      "case": "C206",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(2*(y + 1)^2*(x*(y + 1) + 1)*(3*x*(y + 1) + 4) + 2*(z + 1)*(x*(y + 1) + 1)^3, 3*x*(y + 1)^2*(3*x*(y + 1) + 4) + 3*x*(z + 1)*(x*(y + 1) + 1)^2 + y + 3, -x^3*(z + 1) - 3*x^2*(y + 1) + 2*x + 1)",
      "determinant": "-4",
      "certificate": "det J=-4; three distinct rational preimages share (-1/2, 2, 1)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7895822660002523
      ]
    },
    {
      "case": "B81",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-25*x^2 + 5*y)*(z - 6*(-5*x^2 + y)^2)^2, -5*x^2 + y, z - 6*(-5*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.765103683000234
      ]
    },
    {
      "case": "C142",
      "family": "counterexample",
      "mapping": "F(x,y,z)=((y - 1)^2*((x - 2)*(y - 1) + 1)*(3*(x - 2)*(y - 1) + 4) + (z + 2)*((x - 2)*(y - 1) + 1)^3, y + 3*(x - 2)*(y - 1)^2*(3*(x - 2)*(y - 1) + 4) + 3*(x - 2)*(z + 2)*((x - 2)*(y - 1) + 1)^2 + 1, 2*x - (x - 2)^3*(z + 2) - 3*(x - 2)^2*(y - 1) - 2)",
      "determinant": "-2",
      "certificate": "det J=-2; three distinct rational preimages share (-1/4, 2, 2)",
      "probe_target": "No",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7250597800002652
      ]
    },
    {
      "case": "C61",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-2*(y + 2)^2*((x - 1)*(y + 2) + 1)*(3*(x - 1)*(y + 2) + 4) - 2*((x - 1)*(y + 2) + 1)^3*(-x^2 + z + 2) - 2, y + 3*(x - 1)*(y + 2)^2*(3*(x - 1)*(y + 2) + 4) + 3*(x - 1)*((x - 1)*(y + 2) + 1)^2*(-x^2 + z + 2) + 2, 2*x - (x - 1)^3*(-x^2 + z + 2) - 3*(x - 1)^2*(y + 2) - 3)",
      "determinant": "4",
      "certificate": "det J=4; three distinct rational preimages share (-3/2, 0, -1)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7992550629996913
      ]
    },
    {
      "case": "B138",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (4*x^2 + 2*y)*(z + 2*(2*x^2 + y)^2)^2, 2*x^2 + y, z + 2*(2*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7662003840005127
      ]
    },
    {
      "case": "B175",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-2*x^2 + y)*(z - 5*(-2*x^2 + y)^2)^2, -2*x^2 + y, z - 5*(-2*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7534991380007341
      ]
    },
    {
      "case": "C208",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-2*(y - 2)^2*(x*(y - 2) + 1)*(3*x*(y - 2) + 4) - 2*(x*(y - 2) + 1)^3*(x^2 + z + 2) - 1, 3*x*(y - 2)^2*(3*x*(y - 2) + 4) + 3*x*(x*(y - 2) + 1)^2*(x^2 + z + 2) + y - 2, -x^3*(x^2 + z + 2) - 3*x^2*(y - 2) + 2*x + 1)",
      "determinant": "4",
      "certificate": "det J=4; three distinct rational preimages share (-1/2, 0, 1)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7426891150007577
      ]
    },
    {
      "case": "B82",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (10*x^2 - 2*y)*(z - 6*(-5*x^2 + y)^2)^2, -5*x^2 + y, z - 6*(-5*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7262139570002546
      ]
    },
    {
      "case": "B240",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-18*x^2 + 3*y)*(z + 6*(-6*x^2 + y)^2)^2, -6*x^2 + y, z + 6*(-6*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.8568239979995269
      ]
    },
    {
      "case": "C245",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-(-x^2 + z)*((x - 2)*(y - 1) + 1)^3 - (y - 1)^2*((x - 2)*(y - 1) + 1)*(3*(x - 2)*(y - 1) + 4) + 2, y + 3*(x - 2)*(-x^2 + z)*((x - 2)*(y - 1) + 1)^2 + 3*(x - 2)*(y - 1)^2*(3*(x - 2)*(y - 1) + 4) - 2, 2*x - (x - 2)^3*(-x^2 + z) - 3*(x - 2)^2*(y - 1) - 3)",
      "determinant": "2",
      "certificate": "det J=2; three distinct rational preimages share (9/4, -1, 1)",
      "probe_target": "No",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.8068722120005987
      ]
    },
    {
      "case": "B23",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (12*x^2 + 2*y)*(z - 6*(6*x^2 + y)^2)^2, 6*x^2 + y, z - 6*(6*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7594209870003397
      ]
    },
    {
      "case": "B106",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-5*x^2 + 5*y)*(z - 4*(-x^2 + y)^2)^2, -x^2 + y, z - 4*(-x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7701593270003286
      ]
    },
    {
      "case": "B167",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (10*x^2 - 5*y)*(z + 2*(-2*x^2 + y)^2)^2, -2*x^2 + y, z + 2*(-2*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7271886230000746
      ]
    },
    {
      "case": "C108",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(2*(2*x^2 + z)*((x + 2)*(y - 2) + 1)^3 + 2*(y - 2)^2*((x + 2)*(y - 2) + 1)*(3*(x + 2)*(y - 2) + 4) + 1, y + 3*(x + 2)*(2*x^2 + z)*((x + 2)*(y - 2) + 1)^2 + 3*(x + 2)*(y - 2)^2*(3*(x + 2)*(y - 2) + 4) - 4, 2*x - (x + 2)^3*(2*x^2 + z) - 3*(x + 2)^2*(y - 2) + 5)",
      "determinant": "-4",
      "certificate": "det J=-4; three distinct rational preimages share (1/2, -2, 1)",
      "probe_target": "No",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.721966299000087
      ]
    },
    {
      "case": "B204",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-2*x^2 - 2*y)*(z + 6*(x^2 + y)^2)^2, x^2 + y, z + 6*(x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.72315308799989
      ]
    },
    {
      "case": "C97",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-(y + 2)^2*((x - 2)*(y + 2) + 1)*(3*(x - 2)*(y + 2) + 4) - ((x - 2)*(y + 2) + 1)^3*(-2*x^2 + z - 2), y + 3*(x - 2)*(y + 2)^2*(3*(x - 2)*(y + 2) + 4) + 3*(x - 2)*((x - 2)*(y + 2) + 1)^2*(-2*x^2 + z - 2) + 2, 2*x - (x - 2)^3*(-2*x^2 + z - 2) - 3*(x - 2)^2*(y + 2) - 3)",
      "determinant": "2",
      "certificate": "det J=2; three distinct rational preimages share (1/4, 0, 1)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7100346580000405
      ]
    },
    {
      "case": "C11",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-2*y^2*(y*(x + 1) + 1)*(3*y*(x + 1) + 4) - 2*(y*(x + 1) + 1)^3*(-x^2 + z + 2), 3*y^2*(x + 1)*(3*y*(x + 1) + 4) + y + 3*(x + 1)*(y*(x + 1) + 1)^2*(-x^2 + z + 2) + 1, 2*x - 3*y*(x + 1)^2 - (x + 1)^3*(-x^2 + z + 2) + 4)",
      "determinant": "4",
      "certificate": "det J=4; three distinct rational preimages share (1/2, 1, 2)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.778493735999291
      ]
    },
    {
      "case": "B101",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-25*x^2 - 5*y)*(z + 4*(5*x^2 + y)^2)^2, 5*x^2 + y, z + 4*(5*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.6873459109992837
      ]
    },
    {
      "case": "B218",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (30*x^2 - 5*y)*(z + 6*(-6*x^2 + y)^2)^2, -6*x^2 + y, z + 6*(-6*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7694208299999445
      ]
    },
    {
      "case": "C120",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-(y + 1)^2*((x - 1)*(y + 1) + 1)*(3*(x - 1)*(y + 1) + 4) - ((x - 1)*(y + 1) + 1)^3*(-x^2 + z + 1) - 1, y + 3*(x - 1)*(y + 1)^2*(3*(x - 1)*(y + 1) + 4) + 3*(x - 1)*((x - 1)*(y + 1) + 1)^2*(-x^2 + z + 1), 2*x - (x - 1)^3*(-x^2 + z + 1) - 3*(x - 1)^2*(y + 1))",
      "determinant": "2",
      "certificate": "det J=2; three distinct rational preimages share (-3/4, -1, 2)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7265931400006593
      ]
    },
    {
      "case": "C123",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(2*(y - 2)^2*((x + 1)*(y - 2) + 1)*(3*(x + 1)*(y - 2) + 4) + 2*((x + 1)*(y - 2) + 1)^3*(-x^2 + z - 2) + 2, y + 3*(x + 1)*(y - 2)^2*(3*(x + 1)*(y - 2) + 4) + 3*(x + 1)*((x + 1)*(y - 2) + 1)^2*(-x^2 + z - 2) - 1, 2*x - (x + 1)^3*(-x^2 + z - 2) - 3*(x + 1)^2*(y - 2) + 2)",
      "determinant": "-4",
      "certificate": "det J=-4; three distinct rational preimages share (3/2, 1, 0)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7836876270002904
      ]
    },
    {
      "case": "B75",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (4*x^2 + 4*y)*(z + 5*(x^2 + y)^2)^2, x^2 + y, z + 5*(x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7405537589993401
      ]
    },
    {
      "case": "B79",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (15*x^2 + 5*y)*(z + 3*(3*x^2 + y)^2)^2, 3*x^2 + y, z + 3*(3*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7374359740006184
      ]
    },
    {
      "case": "B43",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-25*x^2 + 5*y)*(z - 3*(-5*x^2 + y)^2)^2, -5*x^2 + y, z - 3*(-5*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.822945355000229
      ]
    },
    {
      "case": "B145",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-6*x^2 - 2*y)*(z - 5*(3*x^2 + y)^2)^2, 3*x^2 + y, z - 5*(3*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7285964100001365
      ]
    },
    {
      "case": "B03",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (4*x^2 - y)*(z + (-4*x^2 + y)^2)^2, -4*x^2 + y, z + (-4*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.8326585060003708
      ]
    },
    {
      "case": "C147",
      "family": "counterexample",
      "mapping": "F(x,y,z)=((y - 1)^2*((x - 1)*(y - 1) + 1)*(3*(x - 1)*(y - 1) + 4) + (z - 2)*((x - 1)*(y - 1) + 1)^3 + 2, y + 3*(x - 1)*(y - 1)^2*(3*(x - 1)*(y - 1) + 4) + 3*(x - 1)*(z - 2)*((x - 1)*(y - 1) + 1)^2 - 1, 2*x - (x - 1)^3*(z - 2) - 3*(x - 1)^2*(y - 1) - 2)",
      "determinant": "-2",
      "certificate": "det J=-2; three distinct rational preimages share (7/4, 0, 0)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.8070335409993277
      ]
    },
    {
      "case": "C114",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-2*(y - 1)^2*((x + 2)*(y - 1) + 1)*(3*(x + 2)*(y - 1) + 4) - 2*((x + 2)*(y - 1) + 1)^3*(-2*x^2 + z + 1) - 2, y + 3*(x + 2)*(y - 1)^2*(3*(x + 2)*(y - 1) + 4) + 3*(x + 2)*((x + 2)*(y - 1) + 1)^2*(-2*x^2 + z + 1) + 1, 2*x - (x + 2)^3*(-2*x^2 + z + 1) - 3*(x + 2)^2*(y - 1) + 5)",
      "determinant": "4",
      "certificate": "det J=4; three distinct rational preimages share (-3/2, 2, 1)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7770259369999621
      ]
    },
    {
      "case": "C232",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(y^2*(y*(x - 2) + 1)*(3*y*(x - 2) + 4) + (y*(x - 2) + 1)^3*(-2*x^2 + z + 2) + 1, 3*y^2*(x - 2)*(3*y*(x - 2) + 4) + y + 3*(x - 2)*(y*(x - 2) + 1)^2*(-2*x^2 + z + 2) + 2, 2*x - 3*y*(x - 2)^2 - (x - 2)^3*(-2*x^2 + z + 2) - 3)",
      "determinant": "-2",
      "certificate": "det J=-2; three distinct rational preimages share (3/4, 2, 1)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7866128760006177
      ]
    },
    {
      "case": "C37",
      "family": "counterexample",
      "mapping": "F(x,y,z)=((y - 1)^2*((x - 2)*(y - 1) + 1)*(3*(x - 2)*(y - 1) + 4) + ((x - 2)*(y - 1) + 1)^3*(2*x^2 + z + 2) - 1, y + 3*(x - 2)*(y - 1)^2*(3*(x - 2)*(y - 1) + 4) + 3*(x - 2)*((x - 2)*(y - 1) + 1)^2*(2*x^2 + z + 2), 2*x - (x - 2)^3*(2*x^2 + z + 2) - 3*(x - 2)^2*(y - 1) - 4)",
      "determinant": "-2",
      "certificate": "det J=-2; three distinct rational preimages share (-5/4, 1, 0)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7208456640000804
      ]
    },
    {
      "case": "B141",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-2*x^2 - 2*y)*(z + 2*(x^2 + y)^2)^2, x^2 + y, z + 2*(x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7589725859997998
      ]
    },
    {
      "case": "B26",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (10*x^2 + 5*y)*(z - 2*(2*x^2 + y)^2)^2, 2*x^2 + y, z - 2*(2*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7181828020002285
      ]
    },
    {
      "case": "B117",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-6*x^2 - 6*y)*(z - 5*(x^2 + y)^2)^2, x^2 + y, z - 5*(x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7461799809998411
      ]
    },
    {
      "case": "C115",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(2*(y + 2)^2*((x - 1)*(y + 2) + 1)*(3*(x - 1)*(y + 2) + 4) + 2*(z - 2)*((x - 1)*(y + 2) + 1)^3, y + 3*(x - 1)*(y + 2)^2*(3*(x - 1)*(y + 2) + 4) + 3*(x - 1)*(z - 2)*((x - 1)*(y + 2) + 1)^2 + 3, 2*x - (x - 1)^3*(z - 2) - 3*(x - 1)^2*(y + 2) - 4)",
      "determinant": "-4",
      "certificate": "det J=-4; three distinct rational preimages share (-1/2, 1, -2)",
      "probe_target": "No",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7462265069998466
      ]
    },
    {
      "case": "C70",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(2*y^2*(x*y + 1)*(3*x*y + 4) + 2*(z + 2)*(x*y + 1)^3 - 1, 3*x*y^2*(3*x*y + 4) + 3*x*(z + 2)*(x*y + 1)^2 + y + 2, -x^3*(z + 2) - 3*x^2*y + 2*x + 1)",
      "determinant": "-4",
      "certificate": "det J=-4; three distinct rational preimages share (-3/2, 2, 1)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7737546479993398
      ]
    },
    {
      "case": "B78",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (6*x^2 + 2*y)*(z + 3*(3*x^2 + y)^2)^2, 3*x^2 + y, z + 3*(3*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7711622850001731
      ]
    },
    {
      "case": "C243",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-2*(-x^2 + z)*((x - 1)*(y - 2) + 1)^3 - 2*(y - 2)^2*((x - 1)*(y - 2) + 1)*(3*(x - 1)*(y - 2) + 4) + 1, y + 3*(x - 1)*(-x^2 + z)*((x - 1)*(y - 2) + 1)^2 + 3*(x - 1)*(y - 2)^2*(3*(x - 1)*(y - 2) + 4), 2*x - (x - 1)^3*(-x^2 + z) - 3*(x - 1)^2*(y - 2) - 1)",
      "determinant": "4",
      "certificate": "det J=4; three distinct rational preimages share (3/2, 2, 1)",
      "probe_target": "No",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.8594496120003896
      ]
    },
    {
      "case": "B41",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (8*x^2 + 4*y)*(z + 2*(2*x^2 + y)^2)^2, 2*x^2 + y, z + 2*(2*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7254489800006922
      ]
    },
    {
      "case": "C150",
      "family": "counterexample",
      "mapping": "F(x,y,z)=((y - 2)^2*((x - 1)*(y - 2) + 1)*(3*(x - 1)*(y - 2) + 4) + ((x - 1)*(y - 2) + 1)^3*(-2*x^2 + z - 1) - 2, y + 3*(x - 1)*(y - 2)^2*(3*(x - 1)*(y - 2) + 4) + 3*(x - 1)*((x - 1)*(y - 2) + 1)^2*(-2*x^2 + z - 1), 2*x - (x - 1)^3*(-2*x^2 + z - 1) - 3*(x - 1)^2*(y - 2) - 2)",
      "determinant": "-2",
      "certificate": "det J=-2; three distinct rational preimages share (-9/4, 2, 0)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7823068190000413
      ]
    },
    {
      "case": "C137",
      "family": "counterexample",
      "mapping": "F(x,y,z)=((y + 1)^2*((x - 2)*(y + 1) + 1)*(3*(x - 2)*(y + 1) + 4) + ((x - 2)*(y + 1) + 1)^3*(-2*x^2 + z + 2) - 2, y + 3*(x - 2)*(y + 1)^2*(3*(x - 2)*(y + 1) + 4) + 3*(x - 2)*((x - 2)*(y + 1) + 1)^2*(-2*x^2 + z + 2) + 3, 2*x - (x - 2)^3*(-2*x^2 + z + 2) - 3*(x - 2)^2*(y + 1) - 3)",
      "determinant": "-2",
      "certificate": "det J=-2; three distinct rational preimages share (-9/4, 2, 1)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.8657794860000649
      ]
    },
    {
      "case": "C242",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-2*(y - 2)^2*((x + 2)*(y - 2) + 1)*(3*(x + 2)*(y - 2) + 4) - 2*((x + 2)*(y - 2) + 1)^3*(x^2 + z - 2) - 2, y + 3*(x + 2)*(y - 2)^2*(3*(x + 2)*(y - 2) + 4) + 3*(x + 2)*((x + 2)*(y - 2) + 1)^2*(x^2 + z - 2) - 2, 2*x - (x + 2)^3*(x^2 + z - 2) - 3*(x + 2)^2*(y - 2) + 3)",
      "determinant": "4",
      "certificate": "det J=4; three distinct rational preimages share (-3/2, 0, -1)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7474547219999295
      ]
    },
    {
      "case": "B216",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (8*x^2 + 2*y)*(z - 4*(4*x^2 + y)^2)^2, 4*x^2 + y, z - 4*(4*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7728832140001032
      ]
    },
    {
      "case": "C100",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-2*(y - 2)^2*((x - 1)*(y - 2) + 1)*(3*(x - 1)*(y - 2) + 4) - 2*((x - 1)*(y - 2) + 1)^3*(-x^2 + z + 2) + 1, y + 3*(x - 1)*(y - 2)^2*(3*(x - 1)*(y - 2) + 4) + 3*(x - 1)*((x - 1)*(y - 2) + 1)^2*(-x^2 + z + 2) - 4, 2*x - (x - 1)^3*(-x^2 + z + 2) - 3*(x - 1)^2*(y - 2) - 2)",
      "determinant": "4",
      "certificate": "det J=4; three distinct rational preimages share (3/2, -2, 0)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.8232044039996254
      ]
    },
    {
      "case": "B53",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (20*x^2 + 5*y)*(z - 6*(4*x^2 + y)^2)^2, 4*x^2 + y, z - 6*(4*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7604851189998953
      ]
    },
    {
      "case": "B112",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (10*x^2 - 5*y)*(z - 2*(-2*x^2 + y)^2)^2, -2*x^2 + y, z - 2*(-2*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7363472569995793
      ]
    },
    {
      "case": "C129",
      "family": "counterexample",
      "mapping": "F(x,y,z)=((x^2 + z)*((x + 1)*(y - 2) + 1)^3 + (y - 2)^2*((x + 1)*(y - 2) + 1)*(3*(x + 1)*(y - 2) + 4) + 1, y + 3*(x + 1)*(x^2 + z)*((x + 1)*(y - 2) + 1)^2 + 3*(x + 1)*(y - 2)^2*(3*(x + 1)*(y - 2) + 4) - 1, 2*x - (x + 1)^3*(x^2 + z) - 3*(x + 1)^2*(y - 2) + 2)",
      "determinant": "-2",
      "certificate": "det J=-2; three distinct rational preimages share (3/4, 1, 0)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7621556320000309
      ]
    },
    {
      "case": "C50",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(2*(x^2 + z)*((x + 2)*(y + 1) + 1)^3 + 2*(y + 1)^2*((x + 2)*(y + 1) + 1)*(3*(x + 2)*(y + 1) + 4), y + 3*(x + 2)*(x^2 + z)*((x + 2)*(y + 1) + 1)^2 + 3*(x + 2)*(y + 1)^2*(3*(x + 2)*(y + 1) + 4) + 2, 2*x - (x + 2)^3*(x^2 + z) - 3*(x + 2)^2*(y + 1) + 4)",
      "determinant": "-4",
      "certificate": "det J=-4; three distinct rational preimages share (-1/2, 1, 0)",
      "probe_target": "No",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7227403830002004
      ]
    },
    {
      "case": "C95",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(2*(y - 2)^2*((x + 1)*(y - 2) + 1)*(3*(x + 1)*(y - 2) + 4) + 2*((x + 1)*(y - 2) + 1)^3*(-2*x^2 + z + 1) + 1, y + 3*(x + 1)*(y - 2)^2*(3*(x + 1)*(y - 2) + 4) + 3*(x + 1)*((x + 1)*(y - 2) + 1)^2*(-2*x^2 + z + 1) - 1, 2*x - (x + 1)^3*(-2*x^2 + z + 1) - 3*(x + 1)^2*(y - 2) + 2)",
      "determinant": "-4",
      "certificate": "det J=-4; three distinct rational preimages share (1/2, 1, 0)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7713505550000264
      ]
    },
    {
      "case": "B225",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-12*x^2 + 6*y)*(z + 5*(-2*x^2 + y)^2)^2, -2*x^2 + y, z + 5*(-2*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7660779580000963
      ]
    },
    {
      "case": "C198",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(y^2*(y*(x - 2) + 1)*(3*y*(x - 2) + 4) + (y*(x - 2) + 1)^3*(2*x^2 + z + 1) - 1, 3*y^2*(x - 2)*(3*y*(x - 2) + 4) + y + 3*(x - 2)*(y*(x - 2) + 1)^2*(2*x^2 + z + 1) - 1, 2*x - 3*y*(x - 2)^2 - (x - 2)^3*(2*x^2 + z + 1) - 5)",
      "determinant": "-2",
      "certificate": "det J=-2; three distinct rational preimages share (-5/4, -1, -1)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7767472010000347
      ]
    },
    {
      "case": "C45",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(2*(y - 2)^2*(x*(y - 2) + 1)*(3*x*(y - 2) + 4) + 2*(x*(y - 2) + 1)^3*(x^2 + z + 2) - 1, 3*x*(y - 2)^2*(3*x*(y - 2) + 4) + 3*x*(x*(y - 2) + 1)^2*(x^2 + z + 2) + y - 2, -x^3*(x^2 + z + 2) - 3*x^2*(y - 2) + 2*x + 2)",
      "determinant": "-4",
      "certificate": "det J=-4; three distinct rational preimages share (-3/2, 0, 2)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7699487479994787
      ]
    },
    {
      "case": "C53",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-2*(y - 1)^2*((x - 1)*(y - 1) + 1)*(3*(x - 1)*(y - 1) + 4) - 2*((x - 1)*(y - 1) + 1)^3*(-2*x^2 + z + 2), y + 3*(x - 1)*(y - 1)^2*(3*(x - 1)*(y - 1) + 4) + 3*(x - 1)*((x - 1)*(y - 1) + 1)^2*(-2*x^2 + z + 2) - 3, 2*x - (x - 1)^3*(-2*x^2 + z + 2) - 3*(x - 1)^2*(y - 1) - 1)",
      "determinant": "4",
      "certificate": "det J=4; three distinct rational preimages share (1/2, -2, 1)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7860682869995799
      ]
    },
    {
      "case": "B114",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (x^2 + y)*(z + 3*(x^2 + y)^2)^2, x^2 + y, z + 3*(x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7337085400004071
      ]
    },
    {
      "case": "B246",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (6*x^2 + y)*(z + 2*(6*x^2 + y)^2)^2, 6*x^2 + y, z + 2*(6*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7564750600004118
      ]
    },
    {
      "case": "B189",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (4*x^2 - 4*y)*(z - 2*(-x^2 + y)^2)^2, -x^2 + y, z - 2*(-x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7568039289999433
      ]
    },
    {
      "case": "B231",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (5*x^2 + 5*y)*(z - 4*(x^2 + y)^2)^2, x^2 + y, z - 4*(x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7265746850007417
      ]
    },
    {
      "case": "B98",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-2*x^2 + 2*y)*(z - 5*(-x^2 + y)^2)^2, -x^2 + y, z - 5*(-x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.8576579830005358
      ]
    },
    {
      "case": "C168",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(2*(y + 1)^2*((x + 2)*(y + 1) + 1)*(3*(x + 2)*(y + 1) + 4) + 2*((x + 2)*(y + 1) + 1)^3*(-2*x^2 + z - 2) + 1, y + 3*(x + 2)*(y + 1)^2*(3*(x + 2)*(y + 1) + 4) + 3*(x + 2)*((x + 2)*(y + 1) + 1)^2*(-2*x^2 + z - 2) + 3, 2*x - (x + 2)^3*(-2*x^2 + z - 2) - 3*(x + 2)^2*(y + 1) + 3)",
      "determinant": "-4",
      "certificate": "det J=-4; three distinct rational preimages share (1/2, 2, -1)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7651865769994401
      ]
    },
    {
      "case": "C135",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-y^2*(x*y + 1)*(3*x*y + 4) - (z - 1)*(x*y + 1)^3 - 2, 3*x*y^2*(3*x*y + 4) + 3*x*(z - 1)*(x*y + 1)^2 + y + 1, -x^3*(z - 1) - 3*x^2*y + 2*x - 2)",
      "determinant": "2",
      "certificate": "det J=2; three distinct rational preimages share (-7/4, 1, -2)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.8018801759999405
      ]
    },
    {
      "case": "C96",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(2*y^2*(y*(x - 2) + 1)*(3*y*(x - 2) + 4) + 2*(y*(x - 2) + 1)^3*(-2*x^2 + z - 2) + 2, 3*y^2*(x - 2)*(3*y*(x - 2) + 4) + y + 3*(x - 2)*(y*(x - 2) + 1)^2*(-2*x^2 + z - 2) - 1, 2*x - 3*y*(x - 2)^2 - (x - 2)^3*(-2*x^2 + z - 2) - 6)",
      "determinant": "-4",
      "certificate": "det J=-4; three distinct rational preimages share (3/2, -1, -2)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7821908949999852
      ]
    },
    {
      "case": "C111",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-(y + 1)^2*(x*(y + 1) + 1)*(3*x*(y + 1) + 4) - (x*(y + 1) + 1)^3*(-2*x^2 + z - 2), 3*x*(y + 1)^2*(3*x*(y + 1) + 4) + 3*x*(x*(y + 1) + 1)^2*(-2*x^2 + z - 2) + y + 2, -x^3*(-2*x^2 + z - 2) - 3*x^2*(y + 1) + 2*x - 2)",
      "determinant": "2",
      "certificate": "det J=2; three distinct rational preimages share (1/4, 1, -2)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7823403910006164
      ]
    },
    {
      "case": "C10",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-2*(y + 1)^2*((x - 1)*(y + 1) + 1)*(3*(x - 1)*(y + 1) + 4) - 2*((x - 1)*(y + 1) + 1)^3*(2*x^2 + z - 1) + 1, y + 3*(x - 1)*(y + 1)^2*(3*(x - 1)*(y + 1) + 4) + 3*(x - 1)*((x - 1)*(y + 1) + 1)^2*(2*x^2 + z - 1), 2*x - (x - 1)^3*(2*x^2 + z - 1) - 3*(x - 1)^2*(y + 1) - 4)",
      "determinant": "4",
      "certificate": "det J=4; three distinct rational preimages share (3/2, -1, -2)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7300831550001021
      ]
    },
    {
      "case": "B122",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-9*x^2 - 3*y)*(z + 2*(3*x^2 + y)^2)^2, 3*x^2 + y, z + 2*(3*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.8639125689996945
      ]
    },
    {
      "case": "B188",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-4*x^2 - 2*y)*(z - 6*(2*x^2 + y)^2)^2, 2*x^2 + y, z - 6*(2*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.810480785999971
      ]
    },
    {
      "case": "C174",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-2*(y - 1)^2*((x - 1)*(y - 1) + 1)*(3*(x - 1)*(y - 1) + 4) - 2*(z + 2)*((x - 1)*(y - 1) + 1)^3 + 1, y + 3*(x - 1)*(y - 1)^2*(3*(x - 1)*(y - 1) + 4) + 3*(x - 1)*(z + 2)*((x - 1)*(y - 1) + 1)^2 - 2, 2*x - (x - 1)^3*(z + 2) - 3*(x - 1)^2*(y - 1) - 4)",
      "determinant": "4",
      "certificate": "det J=4; three distinct rational preimages share (3/2, -1, -2)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.788799666000159
      ]
    },
    {
      "case": "B58",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (5*x^2 + y)*(z - 6*(5*x^2 + y)^2)^2, 5*x^2 + y, z - 6*(5*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7446339369998896
      ]
    },
    {
      "case": "C200",
      "family": "counterexample",
      "mapping": "F(x,y,z)=((-x^2 + z)*((x - 1)*(y - 1) + 1)^3 + (y - 1)^2*((x - 1)*(y - 1) + 1)*(3*(x - 1)*(y - 1) + 4) - 2, y + 3*(x - 1)*(-x^2 + z)*((x - 1)*(y - 1) + 1)^2 + 3*(x - 1)*(y - 1)^2*(3*(x - 1)*(y - 1) + 4) - 3, 2*x - (x - 1)^3*(-x^2 + z) - 3*(x - 1)^2*(y - 1) - 3)",
      "determinant": "-2",
      "certificate": "det J=-2; three distinct rational preimages share (-9/4, -2, -1)",
      "probe_target": "No",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7723327639996569
      ]
    },
    {
      "case": "B172",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-24*x^2 + 4*y)*(z - 3*(-6*x^2 + y)^2)^2, -6*x^2 + y, z - 3*(-6*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.8886060689992519
      ]
    },
    {
      "case": "C238",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(2*(y - 1)^2*((x - 2)*(y - 1) + 1)*(3*(x - 2)*(y - 1) + 4) + 2*((x - 2)*(y - 1) + 1)^3*(-x^2 + z + 2) - 1, y + 3*(x - 2)*(y - 1)^2*(3*(x - 2)*(y - 1) + 4) + 3*(x - 2)*((x - 2)*(y - 1) + 1)^2*(-x^2 + z + 2) - 1, 2*x - (x - 2)^3*(-x^2 + z + 2) - 3*(x - 2)^2*(y - 1) - 3)",
      "determinant": "-4",
      "certificate": "det J=-4; three distinct rational preimages share (-3/2, 0, 1)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7985411500003465
      ]
    },
    {
      "case": "C119",
      "family": "counterexample",
      "mapping": "F(x,y,z)=((y - 2)^2*((x - 2)*(y - 2) + 1)*(3*(x - 2)*(y - 2) + 4) + ((x - 2)*(y - 2) + 1)^3*(x^2 + z - 1) - 2, y + 3*(x - 2)*(y - 2)^2*(3*(x - 2)*(y - 2) + 4) + 3*(x - 2)*((x - 2)*(y - 2) + 1)^2*(x^2 + z - 1) - 1, 2*x - (x - 2)^3*(x^2 + z - 1) - 3*(x - 2)^2*(y - 2) - 2)",
      "determinant": "-2",
      "certificate": "det J=-2; three distinct rational preimages share (-9/4, 1, 2)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.8380573869999353
      ]
    },
    {
      "case": "C59",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(2*(y - 1)^2*((x + 1)*(y - 1) + 1)*(3*(x + 1)*(y - 1) + 4) + 2*((x + 1)*(y - 1) + 1)^3*(-x^2 + z - 2) - 2, y + 3*(x + 1)*(y - 1)^2*(3*(x + 1)*(y - 1) + 4) + 3*(x + 1)*((x + 1)*(y - 1) + 1)^2*(-x^2 + z - 2) - 3, 2*x - (x + 1)^3*(-x^2 + z - 2) - 3*(x + 1)^2*(y - 1) + 3)",
      "determinant": "-4",
      "certificate": "det J=-4; three distinct rational preimages share (-5/2, -2, 1)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7839999050002007
      ]
    },
    {
      "case": "B24",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (12*x^2 + 2*y)*(z - 5*(6*x^2 + y)^2)^2, 6*x^2 + y, z - 5*(6*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7453731360001257
      ]
    },
    {
      "case": "B213",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (6*x^2 - 3*y)*(z - (-2*x^2 + y)^2)^2, -2*x^2 + y, z - (-2*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.715410066000004
      ]
    },
    {
      "case": "B20",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (4*x^2 - y)*(z + 3*(-4*x^2 + y)^2)^2, -4*x^2 + y, z + 3*(-4*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7506627350003328
      ]
    },
    {
      "case": "C138",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(2*(y + 1)^2*((x + 2)*(y + 1) + 1)*(3*(x + 2)*(y + 1) + 4) + 2*((x + 2)*(y + 1) + 1)^3*(-2*x^2 + z + 2) - 2, y + 3*(x + 2)*(y + 1)^2*(3*(x + 2)*(y + 1) + 4) + 3*(x + 2)*((x + 2)*(y + 1) + 1)^2*(-2*x^2 + z + 2) + 3, 2*x - (x + 2)^3*(-2*x^2 + z + 2) - 3*(x + 2)^2*(y + 1) + 3)",
      "determinant": "-4",
      "certificate": "det J=-4; three distinct rational preimages share (-5/2, 2, -1)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7039481039992097
      ]
    },
    {
      "case": "C113",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-(y - 2)^2*((x - 2)*(y - 2) + 1)*(3*(x - 2)*(y - 2) + 4) - ((x - 2)*(y - 2) + 1)^3*(x^2 + z - 1) + 2, y + 3*(x - 2)*(y - 2)^2*(3*(x - 2)*(y - 2) + 4) + 3*(x - 2)*((x - 2)*(y - 2) + 1)^2*(x^2 + z - 1) - 4, 2*x - (x - 2)^3*(x^2 + z - 1) - 3*(x - 2)^2*(y - 2) - 4)",
      "determinant": "2",
      "certificate": "det J=2; three distinct rational preimages share (9/4, -2, 0)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.8006134789993666
      ]
    },
    {
      "case": "C52",
      "family": "counterexample",
      "mapping": "F(x,y,z)=((y - 1)^2*((x - 1)*(y - 1) + 1)*(3*(x - 1)*(y - 1) + 4) + ((x - 1)*(y - 1) + 1)^3*(2*x^2 + z - 1), y + 3*(x - 1)*(y - 1)^2*(3*(x - 1)*(y - 1) + 4) + 3*(x - 1)*((x - 1)*(y - 1) + 1)^2*(2*x^2 + z - 1) - 2, 2*x - (x - 1)^3*(2*x^2 + z - 1) - 3*(x - 1)^2*(y - 1) - 2)",
      "determinant": "-2",
      "certificate": "det J=-2; three distinct rational preimages share (-1/4, -1, 0)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.8168195870002819
      ]
    },
    {
      "case": "B242",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-4*x^2 - 2*y)*(z + (2*x^2 + y)^2)^2, 2*x^2 + y, z + (2*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7332044960003259
      ]
    },
    {
      "case": "C26",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-y^2*(y*(x + 2) + 1)*(3*y*(x + 2) + 4) - (y*(x + 2) + 1)^3*(x^2 + z - 2) + 2, 3*y^2*(x + 2)*(3*y*(x + 2) + 4) + y + 3*(x + 2)*(y*(x + 2) + 1)^2*(x^2 + z - 2), 2*x - 3*y*(x + 2)^2 - (x + 2)^3*(x^2 + z - 2) + 3)",
      "determinant": "2",
      "certificate": "det J=2; three distinct rational preimages share (9/4, 0, -1)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7047195630002534
      ]
    },
    {
      "case": "B215",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-12*x^2 - 6*y)*(z - 4*(2*x^2 + y)^2)^2, 2*x^2 + y, z - 4*(2*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7308438840000235
      ]
    },
    {
      "case": "C15",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-y^2*(y*(x - 1) + 1)*(3*y*(x - 1) + 4) - (y*(x - 1) + 1)^3*(-2*x^2 + z - 1) + 1, 3*y^2*(x - 1)*(3*y*(x - 1) + 4) + y + 3*(x - 1)*(y*(x - 1) + 1)^2*(-2*x^2 + z - 1) - 1, 2*x - 3*y*(x - 1)^2 - (x - 1)^3*(-2*x^2 + z - 1) - 3)",
      "determinant": "2",
      "certificate": "det J=2; three distinct rational preimages share (5/4, -1, -1)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7267515930006994
      ]
    },
    {
      "case": "B38",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-16*x^2 - 4*y)*(z + 5*(4*x^2 + y)^2)^2, 4*x^2 + y, z + 5*(4*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7521595070002149
      ]
    },
    {
      "case": "B178",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-10*x^2 - 5*y)*(z + (2*x^2 + y)^2)^2, 2*x^2 + y, z + (2*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7978292320003675
      ]
    },
    {
      "case": "B36",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (18*x^2 - 3*y)*(z - 4*(-6*x^2 + y)^2)^2, -6*x^2 + y, z - 4*(-6*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7287101909996636
      ]
    },
    {
      "case": "C239",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(2*(y - 2)^2*((x - 1)*(y - 2) + 1)*(3*(x - 1)*(y - 2) + 4) + 2*(z - 1)*((x - 1)*(y - 2) + 1)^3 + 1, y + 3*(x - 1)*(y - 2)^2*(3*(x - 1)*(y - 2) + 4) + 3*(x - 1)*(z - 1)*((x - 1)*(y - 2) + 1)^2 - 2, 2*x - (x - 1)^3*(z - 1) - 3*(x - 1)^2*(y - 2))",
      "determinant": "-4",
      "certificate": "det J=-4; three distinct rational preimages share (1/2, 0, 2)",
      "probe_target": "No",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7778687479994915
      ]
    },
    {
      "case": "B217",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (2*x^2 + 2*y)*(z + 4*(x^2 + y)^2)^2, x^2 + y, z + 4*(x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.740610834999643
      ]
    },
    {
      "case": "B168",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-10*x^2 - 2*y)*(z + (5*x^2 + y)^2)^2, 5*x^2 + y, z + (5*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7739716500000213
      ]
    },
    {
      "case": "C246",
      "family": "counterexample",
      "mapping": "F(x,y,z)=((y - 1)^2*((x - 2)*(y - 1) + 1)*(3*(x - 2)*(y - 1) + 4) + ((x - 2)*(y - 1) + 1)^3*(x^2 + z - 2) - 1, y + 3*(x - 2)*(y - 1)^2*(3*(x - 2)*(y - 1) + 4) + 3*(x - 2)*((x - 2)*(y - 1) + 1)^2*(x^2 + z - 2) - 1, 2*x - (x - 2)^3*(x^2 + z - 2) - 3*(x - 2)^2*(y - 1) - 6)",
      "determinant": "-2",
      "certificate": "det J=-2; three distinct rational preimages share (-5/4, 0, -2)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7449782450003113
      ]
    },
    {
      "case": "B220",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-3*x^2 + y)*(z + (-3*x^2 + y)^2)^2, -3*x^2 + y, z + (-3*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7672963750001145
      ]
    },
    {
      "case": "B85",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-8*x^2 + 2*y)*(z + 5*(-4*x^2 + y)^2)^2, -4*x^2 + y, z + 5*(-4*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7452121980004449
      ]
    },
    {
      "case": "C158",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-(-x^2 + z)*(x*(y - 2) + 1)^3 - (y - 2)^2*(x*(y - 2) + 1)*(3*x*(y - 2) + 4), 3*x*(-x^2 + z)*(x*(y - 2) + 1)^2 + 3*x*(y - 2)^2*(3*x*(y - 2) + 4) + y - 2, -x^3*(-x^2 + z) - 3*x^2*(y - 2) + 2*x + 2)",
      "determinant": "2",
      "certificate": "det J=2; three distinct rational preimages share (1/4, 0, 2)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7644701999997778
      ]
    },
    {
      "case": "B100",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (24*x^2 + 6*y)*(z + 4*(4*x^2 + y)^2)^2, 4*x^2 + y, z + 4*(4*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.8026523359994826
      ]
    },
    {
      "case": "C240",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-2*(y - 1)^2*((x + 1)*(y - 1) + 1)*(3*(x + 1)*(y - 1) + 4) - 2*((x + 1)*(y - 1) + 1)^3*(2*x^2 + z - 2) + 2, y + 3*(x + 1)*(y - 1)^2*(3*(x + 1)*(y - 1) + 4) + 3*(x + 1)*((x + 1)*(y - 1) + 1)^2*(2*x^2 + z - 2), 2*x - (x + 1)^3*(2*x^2 + z - 2) - 3*(x + 1)^2*(y - 1) + 2)",
      "determinant": "4",
      "certificate": "det J=4; three distinct rational preimages share (5/2, 1, 0)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7718979279998166
      ]
    },
    {
      "case": "B07",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-3*x^2 - 3*y)*(z + 4*(x^2 + y)^2)^2, x^2 + y, z + 4*(x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.737944485999833
      ]
    },
    {
      "case": "C139",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-(y + 1)^2*((x + 2)*(y + 1) + 1)*(3*(x + 2)*(y + 1) + 4) - ((x + 2)*(y + 1) + 1)^3*(2*x^2 + z - 2) + 1, y + 3*(x + 2)*(y + 1)^2*(3*(x + 2)*(y + 1) + 4) + 3*(x + 2)*((x + 2)*(y + 1) + 1)^2*(2*x^2 + z - 2) + 2, 2*x - (x + 2)^3*(2*x^2 + z - 2) - 3*(x + 2)^2*(y + 1) + 5)",
      "determinant": "2",
      "certificate": "det J=2; three distinct rational preimages share (5/4, 1, 1)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7357564730000377
      ]
    },
    {
      "case": "C84",
      "family": "counterexample",
      "mapping": "F(x,y,z)=((y + 1)^2*(x*(y + 1) + 1)*(3*x*(y + 1) + 4) + (z + 1)*(x*(y + 1) + 1)^3 + 2, 3*x*(y + 1)^2*(3*x*(y + 1) + 4) + 3*x*(z + 1)*(x*(y + 1) + 1)^2 + y - 1, -x^3*(z + 1) - 3*x^2*(y + 1) + 2*x)",
      "determinant": "-2",
      "certificate": "det J=-2; three distinct rational preimages share (7/4, -2, 0)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.8528448979996028
      ]
    },
    {
      "case": "C07",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(2*(y - 1)^2*((x - 2)*(y - 1) + 1)*(3*(x - 2)*(y - 1) + 4) + 2*((x - 2)*(y - 1) + 1)^3*(-2*x^2 + z - 1) + 1, y + 3*(x - 2)*(y - 1)^2*(3*(x - 2)*(y - 1) + 4) + 3*(x - 2)*((x - 2)*(y - 1) + 1)^2*(-2*x^2 + z - 1) + 1, 2*x - (x - 2)^3*(-2*x^2 + z - 1) - 3*(x - 2)^2*(y - 1) - 4)",
      "determinant": "-4",
      "certificate": "det J=-4; three distinct rational preimages share (1/2, 2, 0)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7217822490001709
      ]
    },
    {
      "case": "C149",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-(2*x^2 + z)*(x*(y - 1) + 1)^3 - (y - 1)^2*(x*(y - 1) + 1)*(3*x*(y - 1) + 4) - 2, 3*x*(2*x^2 + z)*(x*(y - 1) + 1)^2 + 3*x*(y - 1)^2*(3*x*(y - 1) + 4) + y - 2, -x^3*(2*x^2 + z) - 3*x^2*(y - 1) + 2*x)",
      "determinant": "2",
      "certificate": "det J=2; three distinct rational preimages share (-7/4, -1, 0)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7638600299997051
      ]
    },
    {
      "case": "B234",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (8*x^2 - 2*y)*(z + 3*(-4*x^2 + y)^2)^2, -4*x^2 + y, z + 3*(-4*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7395727920002173
      ]
    },
    {
      "case": "C161",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(2*y^2*(y*(x - 2) + 1)*(3*y*(x - 2) + 4) + 2*(x^2 + z)*(y*(x - 2) + 1)^3 + 2, 3*y^2*(x - 2)*(3*y*(x - 2) + 4) + y + 3*(x - 2)*(x^2 + z)*(y*(x - 2) + 1)^2 - 1, 2*x - 3*y*(x - 2)^2 - (x - 2)^3*(x^2 + z) - 6)",
      "determinant": "-4",
      "certificate": "det J=-4; three distinct rational preimages share (3/2, -1, -2)",
      "probe_target": "No",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7401398619995234
      ]
    },
    {
      "case": "C201",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-2*(y + 2)^2*((x - 1)*(y + 2) + 1)*(3*(x - 1)*(y + 2) + 4) - 2*((x - 1)*(y + 2) + 1)^3*(2*x^2 + z + 1) + 2, y + 3*(x - 1)*(y + 2)^2*(3*(x - 1)*(y + 2) + 4) + 3*(x - 1)*((x - 1)*(y + 2) + 1)^2*(2*x^2 + z + 1) + 3, 2*x - (x - 1)^3*(2*x^2 + z + 1) - 3*(x - 1)^2*(y + 2) - 1)",
      "determinant": "4",
      "certificate": "det J=4; three distinct rational preimages share (5/2, 1, 1)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7387747840002703
      ]
    },
    {
      "case": "B165",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-18*x^2 + 3*y)*(z - 5*(-6*x^2 + y)^2)^2, -6*x^2 + y, z - 5*(-6*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7430294260002483
      ]
    },
    {
      "case": "C69",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-(y + 1)^2*((x - 1)*(y + 1) + 1)*(3*(x - 1)*(y + 1) + 4) - (z + 1)*((x - 1)*(y + 1) + 1)^3 - 1, y + 3*(x - 1)*(y + 1)^2*(3*(x - 1)*(y + 1) + 4) + 3*(x - 1)*(z + 1)*((x - 1)*(y + 1) + 1)^2 + 2, 2*x - (x - 1)^3*(z + 1) - 3*(x - 1)^2*(y + 1) - 3)",
      "determinant": "2",
      "certificate": "det J=2; three distinct rational preimages share (-3/4, 1, -1)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7742237070006013
      ]
    },
    {
      "case": "C107",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-(x^2 + z)*((x - 1)*(y - 2) + 1)^3 - (y - 2)^2*((x - 1)*(y - 2) + 1)*(3*(x - 1)*(y - 2) + 4) - 1, y + 3*(x - 1)*(x^2 + z)*((x - 1)*(y - 2) + 1)^2 + 3*(x - 1)*(y - 2)^2*(3*(x - 1)*(y - 2) + 4) - 2, 2*x - (x - 1)^3*(x^2 + z) - 3*(x - 1)^2*(y - 2) - 2)",
      "determinant": "2",
      "certificate": "det J=2; three distinct rational preimages share (-3/4, 0, 0)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7621348439997746
      ]
    },
    {
      "case": "B19",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (30*x^2 + 5*y)*(z - 6*(6*x^2 + y)^2)^2, 6*x^2 + y, z - 6*(6*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.707600239000385
      ]
    },
    {
      "case": "C56",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(2*(y - 2)^2*((x - 1)*(y - 2) + 1)*(3*(x - 1)*(y - 2) + 4) + 2*((x - 1)*(y - 2) + 1)^3*(-2*x^2 + z - 2) + 2, y + 3*(x - 1)*(y - 2)^2*(3*(x - 1)*(y - 2) + 4) + 3*(x - 1)*((x - 1)*(y - 2) + 1)^2*(-2*x^2 + z - 2) - 3, 2*x - (x - 1)^3*(-2*x^2 + z - 2) - 3*(x - 1)^2*(y - 2))",
      "determinant": "-4",
      "certificate": "det J=-4; three distinct rational preimages share (3/2, -1, 2)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7618075989994395
      ]
    },
    {
      "case": "C146",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-2*(2*x^2 + z)*((x - 1)*(y + 1) + 1)^3 - 2*(y + 1)^2*((x - 1)*(y + 1) + 1)*(3*(x - 1)*(y + 1) + 4) - 2, y + 3*(x - 1)*(2*x^2 + z)*((x - 1)*(y + 1) + 1)^2 + 3*(x - 1)*(y + 1)^2*(3*(x - 1)*(y + 1) + 4) + 2, 2*x - (x - 1)^3*(2*x^2 + z) - 3*(x - 1)^2*(y + 1) - 4)",
      "determinant": "4",
      "certificate": "det J=4; three distinct rational preimages share (-3/2, 1, -2)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.761380977999579
      ]
    },
    {
      "case": "B121",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-3*x^2 - 3*y)*(z - 6*(x^2 + y)^2)^2, x^2 + y, z - 6*(x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7243463679997149
      ]
    },
    {
      "case": "B59",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-12*x^2 + 6*y)*(z - 2*(-2*x^2 + y)^2)^2, -2*x^2 + y, z - 2*(-2*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.722867038000004
      ]
    },
    {
      "case": "B67",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (12*x^2 + 2*y)*(z + 5*(6*x^2 + y)^2)^2, 6*x^2 + y, z + 5*(6*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7329977829995187
      ]
    },
    {
      "case": "B156",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-6*x^2 + 3*y)*(z + (-2*x^2 + y)^2)^2, -2*x^2 + y, z + (-2*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7064849440002945
      ]
    },
    {
      "case": "B199",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-30*x^2 + 5*y)*(z - 2*(-6*x^2 + y)^2)^2, -6*x^2 + y, z - 2*(-6*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7664110549994803
      ]
    },
    {
      "case": "B72",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-5*x^2 - 5*y)*(z + 3*(x^2 + y)^2)^2, x^2 + y, z + 3*(x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.721929681999427
      ]
    },
    {
      "case": "B84",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (8*x^2 + 4*y)*(z + 3*(2*x^2 + y)^2)^2, 2*x^2 + y, z + 3*(2*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.733510742999897
      ]
    },
    {
      "case": "C22",
      "family": "counterexample",
      "mapping": "F(x,y,z)=((y + 1)^2*((x + 1)*(y + 1) + 1)*(3*(x + 1)*(y + 1) + 4) + ((x + 1)*(y + 1) + 1)^3*(x^2 + z + 2) + 2, y + 3*(x + 1)*(y + 1)^2*(3*(x + 1)*(y + 1) + 4) + 3*(x + 1)*((x + 1)*(y + 1) + 1)^2*(x^2 + z + 2) - 1, 2*x - (x + 1)^3*(x^2 + z + 2) - 3*(x + 1)^2*(y + 1) + 3)",
      "determinant": "-2",
      "certificate": "det J=-2; three distinct rational preimages share (7/4, -2, 1)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7374980300000971
      ]
    },
    {
      "case": "C104",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(2*(y - 1)^2*((x - 1)*(y - 1) + 1)*(3*(x - 1)*(y - 1) + 4) + 2*((x - 1)*(y - 1) + 1)^3*(2*x^2 + z + 1) - 2, y + 3*(x - 1)*(y - 1)^2*(3*(x - 1)*(y - 1) + 4) + 3*(x - 1)*((x - 1)*(y - 1) + 1)^2*(2*x^2 + z + 1) - 3, 2*x - (x - 1)^3*(2*x^2 + z + 1) - 3*(x - 1)^2*(y - 1) - 1)",
      "determinant": "-4",
      "certificate": "det J=-4; three distinct rational preimages share (-5/2, -2, 1)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.8011859489997732
      ]
    },
    {
      "case": "C122",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(2*(y + 1)^2*((x + 1)*(y + 1) + 1)*(3*(x + 1)*(y + 1) + 4) + 2*((x + 1)*(y + 1) + 1)^3*(2*x^2 + z + 2) + 2, y + 3*(x + 1)*(y + 1)^2*(3*(x + 1)*(y + 1) + 4) + 3*(x + 1)*((x + 1)*(y + 1) + 1)^2*(2*x^2 + z + 2) + 2, 2*x - (x + 1)^3*(2*x^2 + z + 2) - 3*(x + 1)^2*(y + 1))",
      "determinant": "-4",
      "certificate": "det J=-4; three distinct rational preimages share (3/2, 1, -2)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7790668280003956
      ]
    },
    {
      "case": "B223",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-12*x^2 + 3*y)*(z - 2*(-4*x^2 + y)^2)^2, -4*x^2 + y, z - 2*(-4*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.8058335480000096
      ]
    },
    {
      "case": "B51",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (12*x^2 + 4*y)*(z - 2*(3*x^2 + y)^2)^2, 3*x^2 + y, z - 2*(3*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.8032737270004873
      ]
    },
    {
      "case": "B136",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-20*x^2 - 4*y)*(z + 4*(5*x^2 + y)^2)^2, 5*x^2 + y, z + 4*(5*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.757853594000153
      ]
    },
    {
      "case": "C32",
      "family": "counterexample",
      "mapping": "F(x,y,z)=((y - 2)^2*((x + 2)*(y - 2) + 1)*(3*(x + 2)*(y - 2) + 4) + ((x + 2)*(y - 2) + 1)^3*(x^2 + z + 1) - 1, y + 3*(x + 2)*(y - 2)^2*(3*(x + 2)*(y - 2) + 4) + 3*(x + 2)*((x + 2)*(y - 2) + 1)^2*(x^2 + z + 1), 2*x - (x + 2)^3*(x^2 + z + 1) - 3*(x + 2)^2*(y - 2) + 3)",
      "determinant": "-2",
      "certificate": "det J=-2; three distinct rational preimages share (-5/4, 2, -1)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7235584190002555
      ]
    },
    {
      "case": "C30",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-2*y^2*(y*(x - 1) + 1)*(3*y*(x - 1) + 4) - 2*(y*(x - 1) + 1)^3*(-x^2 + z + 2) - 2, 3*y^2*(x - 1)*(3*y*(x - 1) + 4) + y + 3*(x - 1)*(y*(x - 1) + 1)^2*(-x^2 + z + 2) + 1, 2*x - 3*y*(x - 1)^2 - (x - 1)^3*(-x^2 + z + 2) - 4)",
      "determinant": "4",
      "certificate": "det J=4; three distinct rational preimages share (-3/2, 1, -2)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7352487130001464
      ]
    },
    {
      "case": "B111",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-36*x^2 + 6*y)*(z + 6*(-6*x^2 + y)^2)^2, -6*x^2 + y, z + 6*(-6*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7273793260001185
      ]
    },
    {
      "case": "C182",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-(-2*x^2 + z)*(x*(y + 1) + 1)^3 - (y + 1)^2*(x*(y + 1) + 1)*(3*x*(y + 1) + 4) - 1, 3*x*(-2*x^2 + z)*(x*(y + 1) + 1)^2 + 3*x*(y + 1)^2*(3*x*(y + 1) + 4) + y + 3, -x^3*(-2*x^2 + z) - 3*x^2*(y + 1) + 2*x - 1)",
      "determinant": "2",
      "certificate": "det J=2; three distinct rational preimages share (-3/4, 2, -1)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7867396510000617
      ]
    },
    {
      "case": "B76",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (12*x^2 + 6*y)*(z - (2*x^2 + y)^2)^2, 2*x^2 + y, z - (2*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.8606846500006213
      ]
    },
    {
      "case": "C230",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(2*(-2*x^2 + z)*((x + 1)*(y + 2) + 1)^3 + 2*(y + 2)^2*((x + 1)*(y + 2) + 1)*(3*(x + 1)*(y + 2) + 4) + 2, y + 3*(x + 1)*(-2*x^2 + z)*((x + 1)*(y + 2) + 1)^2 + 3*(x + 1)*(y + 2)^2*(3*(x + 1)*(y + 2) + 4) + 4, 2*x - (x + 1)^3*(-2*x^2 + z) - 3*(x + 1)^2*(y + 2) + 4)",
      "determinant": "-4",
      "certificate": "det J=-4; three distinct rational preimages share (3/2, 2, 2)",
      "probe_target": "No",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7193738180003493
      ]
    },
    {
      "case": "C17",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-z*((x - 1)*(y + 2) + 1)^3 - (y + 2)^2*((x - 1)*(y + 2) + 1)*(3*(x - 1)*(y + 2) + 4) + 1, y + 3*z*(x - 1)*((x - 1)*(y + 2) + 1)^2 + 3*(x - 1)*(y + 2)^2*(3*(x - 1)*(y + 2) + 4) + 2, 2*x - z*(x - 1)^3 - 3*(x - 1)^2*(y + 2) - 4)",
      "determinant": "2",
      "certificate": "det J=2; three distinct rational preimages share (5/4, 0, -2)",
      "probe_target": "No",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7445379990003858
      ]
    },
    {
      "case": "C13",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(2*z*((x - 1)*(y + 2) + 1)^3 + 2*(y + 2)^2*((x - 1)*(y + 2) + 1)*(3*(x - 1)*(y + 2) + 4) - 2, y + 3*z*(x - 1)*((x - 1)*(y + 2) + 1)^2 + 3*(x - 1)*(y + 2)^2*(3*(x - 1)*(y + 2) + 4) + 2, 2*x - z*(x - 1)^3 - 3*(x - 1)^2*(y + 2) - 4)",
      "determinant": "-4",
      "certificate": "det J=-4; three distinct rational preimages share (-5/2, 0, -2)",
      "probe_target": "No",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.8119291789998897
      ]
    },
    {
      "case": "B248",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-20*x^2 + 5*y)*(z - 3*(-4*x^2 + y)^2)^2, -4*x^2 + y, z - 3*(-4*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7180193389995111
      ]
    },
    {
      "case": "B132",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (6*x^2 - 6*y)*(z + 4*(-x^2 + y)^2)^2, -x^2 + y, z + 4*(-x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7152221669994105
      ]
    },
    {
      "case": "C40",
      "family": "counterexample",
      "mapping": "F(x,y,z)=((y + 2)^2*((x - 2)*(y + 2) + 1)*(3*(x - 2)*(y + 2) + 4) + ((x - 2)*(y + 2) + 1)^3*(2*x^2 + z + 1) + 1, y + 3*(x - 2)*(y + 2)^2*(3*(x - 2)*(y + 2) + 4) + 3*(x - 2)*((x - 2)*(y + 2) + 1)^2*(2*x^2 + z + 1), 2*x - (x - 2)^3*(2*x^2 + z + 1) - 3*(x - 2)^2*(y + 2) - 5)",
      "determinant": "-2",
      "certificate": "det J=-2; three distinct rational preimages share (3/4, -2, -1)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7244108679997225
      ]
    },
    {
      "case": "B238",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-24*x^2 - 4*y)*(z - 6*(6*x^2 + y)^2)^2, 6*x^2 + y, z - 6*(6*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7774897369999962
      ]
    },
    {
      "case": "B47",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (4*x^2 - 2*y)*(z + 3*(-2*x^2 + y)^2)^2, -2*x^2 + y, z + 3*(-2*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7618379949999508
      ]
    },
    {
      "case": "C39",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(2*y^2*(y*(x - 2) + 1)*(3*y*(x - 2) + 4) + 2*(y*(x - 2) + 1)^3*(-x^2 + z + 1) - 1, 3*y^2*(x - 2)*(3*y*(x - 2) + 4) + y + 3*(x - 2)*(y*(x - 2) + 1)^2*(-x^2 + z + 1) - 1, 2*x - 3*y*(x - 2)^2 - (x - 2)^3*(-x^2 + z + 1) - 2)",
      "determinant": "-4",
      "certificate": "det J=-4; three distinct rational preimages share (-3/2, -1, 2)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7859869260000778
      ]
    },
    {
      "case": "B181",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (3*x^2 - y)*(z + 3*(-3*x^2 + y)^2)^2, -3*x^2 + y, z + 3*(-3*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7251013669992972
      ]
    },
    {
      "case": "B113",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (15*x^2 + 5*y)*(z + 2*(3*x^2 + y)^2)^2, 3*x^2 + y, z + 2*(3*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7584959839996372
      ]
    },
    {
      "case": "C179",
      "family": "counterexample",
      "mapping": "F(x,y,z)=((y - 2)^2*(x*(y - 2) + 1)*(3*x*(y - 2) + 4) + (x*(y - 2) + 1)^3*(-2*x^2 + z - 1), 3*x*(y - 2)^2*(3*x*(y - 2) + 4) + 3*x*(x*(y - 2) + 1)^2*(-2*x^2 + z - 1) + y - 1, -x^3*(-2*x^2 + z - 1) - 3*x^2*(y - 2) + 2*x - 1)",
      "determinant": "-2",
      "certificate": "det J=-2; three distinct rational preimages share (-1/4, 1, -1)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7569476759999816
      ]
    },
    {
      "case": "B56",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-9*x^2 - 3*y)*(z + 3*(3*x^2 + y)^2)^2, 3*x^2 + y, z + 3*(3*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7398250090000147
      ]
    },
    {
      "case": "C112",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(2*(y + 1)^2*((x + 2)*(y + 1) + 1)*(3*(x + 2)*(y + 1) + 4) + 2*(z - 1)*((x + 2)*(y + 1) + 1)^3 - 2, y + 3*(x + 2)*(y + 1)^2*(3*(x + 2)*(y + 1) + 4) + 3*(x + 2)*(z - 1)*((x + 2)*(y + 1) + 1)^2 - 1, 2*x - (x + 2)^3*(z - 1) - 3*(x + 2)^2*(y + 1) + 2)",
      "determinant": "-4",
      "certificate": "det J=-4; three distinct rational preimages share (-5/2, -2, -2)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7108717190003517
      ]
    },
    {
      "case": "C175",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(2*(y - 1)^2*((x + 1)*(y - 1) + 1)*(3*(x + 1)*(y - 1) + 4) + 2*((x + 1)*(y - 1) + 1)^3*(x^2 + z - 2), y + 3*(x + 1)*(y - 1)^2*(3*(x + 1)*(y - 1) + 4) + 3*(x + 1)*((x + 1)*(y - 1) + 1)^2*(x^2 + z - 2) - 1, 2*x - (x + 1)^3*(x^2 + z - 2) - 3*(x + 1)^2*(y - 1))",
      "determinant": "-4",
      "certificate": "det J=-4; three distinct rational preimages share (-1/2, 0, -2)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7126396639996528
      ]
    },
    {
      "case": "C125",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-(-x^2 + z)*((x - 2)*(y + 2) + 1)^3 - (y + 2)^2*((x - 2)*(y + 2) + 1)*(3*(x - 2)*(y + 2) + 4) + 2, y + 3*(x - 2)*(-x^2 + z)*((x - 2)*(y + 2) + 1)^2 + 3*(x - 2)*(y + 2)^2*(3*(x - 2)*(y + 2) + 4) + 3, 2*x - (x - 2)^3*(-x^2 + z) - 3*(x - 2)^2*(y + 2) - 3)",
      "determinant": "2",
      "certificate": "det J=2; three distinct rational preimages share (9/4, 1, 1)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7648456060005628
      ]
    },
    {
      "case": "C188",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-(y + 2)^2*(x*(y + 2) + 1)*(3*x*(y + 2) + 4) - (x*(y + 2) + 1)^3*(2*x^2 + z - 1) - 2, 3*x*(y + 2)^2*(3*x*(y + 2) + 4) + 3*x*(x*(y + 2) + 1)^2*(2*x^2 + z - 1) + y + 2, -x^3*(2*x^2 + z - 1) - 3*x^2*(y + 2) + 2*x + 1)",
      "determinant": "2",
      "certificate": "det J=2; three distinct rational preimages share (-7/4, 0, 1)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.8284805969997251
      ]
    },
    {
      "case": "B48",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (15*x^2 - 5*y)*(z + 5*(-3*x^2 + y)^2)^2, -3*x^2 + y, z + 5*(-3*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7405361470000571
      ]
    },
    {
      "case": "B45",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (8*x^2 + 2*y)*(z + (4*x^2 + y)^2)^2, 4*x^2 + y, z + (4*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7409733369995593
      ]
    },
    {
      "case": "B124",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-4*x^2 - y)*(z + 3*(4*x^2 + y)^2)^2, 4*x^2 + y, z + 3*(4*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7578847519998817
      ]
    },
    {
      "case": "C57",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-2*y^2*(y*(x + 2) + 1)*(3*y*(x + 2) + 4) - 2*(y*(x + 2) + 1)^3*(-2*x^2 + z + 2), 3*y^2*(x + 2)*(3*y*(x + 2) + 4) + y + 3*(x + 2)*(y*(x + 2) + 1)^2*(-2*x^2 + z + 2) - 2, 2*x - 3*y*(x + 2)^2 - (x + 2)^3*(-2*x^2 + z + 2) + 3)",
      "determinant": "4",
      "certificate": "det J=4; three distinct rational preimages share (1/2, -2, -1)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7622256829999969
      ]
    },
    {
      "case": "C03",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-2*(y - 2)^2*((x - 1)*(y - 2) + 1)*(3*(x - 1)*(y - 2) + 4) - 2*((x - 1)*(y - 2) + 1)^3*(-2*x^2 + z - 2) + 1, y + 3*(x - 1)*(y - 2)^2*(3*(x - 1)*(y - 2) + 4) + 3*(x - 1)*((x - 1)*(y - 2) + 1)^2*(-2*x^2 + z - 2), 2*x - (x - 1)^3*(-2*x^2 + z - 2) - 3*(x - 1)^2*(y - 2) - 1)",
      "determinant": "4",
      "certificate": "det J=4; three distinct rational preimages share (3/2, 2, 1)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7024535359996662
      ]
    },
    {
      "case": "C24",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-y^2*(y*(x - 1) + 1)*(3*y*(x - 1) + 4) - (y*(x - 1) + 1)^3*(2*x^2 + z + 1) - 1, 3*y^2*(x - 1)*(3*y*(x - 1) + 4) + y + 3*(x - 1)*(y*(x - 1) + 1)^2*(2*x^2 + z + 1) - 1, 2*x - 3*y*(x - 1)^2 - (x - 1)^3*(2*x^2 + z + 1) - 2)",
      "determinant": "2",
      "certificate": "det J=2; three distinct rational preimages share (-3/4, -1, 0)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7453882649997468
      ]
    },
    {
      "case": "C41",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-2*y^2*(x*y + 1)*(3*x*y + 4) - 2*(z - 1)*(x*y + 1)^3 + 2, 3*x*y^2*(3*x*y + 4) + 3*x*(z - 1)*(x*y + 1)^2 + y - 2, -x^3*(z - 1) - 3*x^2*y + 2*x + 1)",
      "determinant": "4",
      "certificate": "det J=4; three distinct rational preimages share (5/2, -2, 1)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7456675819994416
      ]
    },
    {
      "case": "B110",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-24*x^2 + 6*y)*(z + (-4*x^2 + y)^2)^2, -4*x^2 + y, z + (-4*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7814524370005529
      ]
    },
    {
      "case": "B119",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (4*x^2 - 2*y)*(z - 5*(-2*x^2 + y)^2)^2, -2*x^2 + y, z - 5*(-2*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7717209820002608
      ]
    },
    {
      "case": "C211",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-(y + 2)^2*((x + 2)*(y + 2) + 1)*(3*(x + 2)*(y + 2) + 4) - ((x + 2)*(y + 2) + 1)^3*(-x^2 + z - 1) + 1, y + 3*(x + 2)*(y + 2)^2*(3*(x + 2)*(y + 2) + 4) + 3*(x + 2)*((x + 2)*(y + 2) + 1)^2*(-x^2 + z - 1) + 4, 2*x - (x + 2)^3*(-x^2 + z - 1) - 3*(x + 2)^2*(y + 2) + 4)",
      "determinant": "2",
      "certificate": "det J=2; three distinct rational preimages share (5/4, 2, 0)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7119268340002236
      ]
    },
    {
      "case": "C164",
      "family": "counterexample",
      "mapping": "F(x,y,z)=((-2*x^2 + z)*((x - 1)*(y - 2) + 1)^3 + (y - 2)^2*((x - 1)*(y - 2) + 1)*(3*(x - 1)*(y - 2) + 4) - 2, y + 3*(x - 1)*(-2*x^2 + z)*((x - 1)*(y - 2) + 1)^2 + 3*(x - 1)*(y - 2)^2*(3*(x - 1)*(y - 2) + 4), 2*x - (x - 1)^3*(-2*x^2 + z) - 3*(x - 1)^2*(y - 2))",
      "determinant": "-2",
      "certificate": "det J=-2; three distinct rational preimages share (-9/4, 2, 2)",
      "probe_target": "No",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7518201090006187
      ]
    },
    {
      "case": "B65",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-8*x^2 - 2*y)*(z + 2*(4*x^2 + y)^2)^2, 4*x^2 + y, z + 2*(4*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7434629090003
      ]
    },
    {
      "case": "C157",
      "family": "counterexample",
      "mapping": "F(x,y,z)=((y - 1)^2*((x + 2)*(y - 1) + 1)*(3*(x + 2)*(y - 1) + 4) + ((x + 2)*(y - 1) + 1)^3*(-2*x^2 + z + 1) - 1, y + 3*(x + 2)*(y - 1)^2*(3*(x + 2)*(y - 1) + 4) + 3*(x + 2)*((x + 2)*(y - 1) + 1)^2*(-2*x^2 + z + 1) - 3, 2*x - (x + 2)^3*(-2*x^2 + z + 1) - 3*(x + 2)^2*(y - 1) + 4)",
      "determinant": "-2",
      "certificate": "det J=-2; three distinct rational preimages share (-5/4, -2, 0)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7300754409998262
      ]
    },
    {
      "case": "B166",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (24*x^2 + 4*y)*(z + (6*x^2 + y)^2)^2, 6*x^2 + y, z + (6*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7786872950000543
      ]
    },
    {
      "case": "C08",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-z*((x - 2)*(y - 1) + 1)^3 - (y - 1)^2*((x - 2)*(y - 1) + 1)*(3*(x - 2)*(y - 1) + 4) + 1, y + 3*z*(x - 2)*((x - 2)*(y - 1) + 1)^2 + 3*(x - 2)*(y - 1)^2*(3*(x - 2)*(y - 1) + 4) + 1, 2*x - z*(x - 2)^3 - 3*(x - 2)^2*(y - 1) - 2)",
      "determinant": "2",
      "certificate": "det J=2; three distinct rational preimages share (5/4, 2, 2)",
      "probe_target": "No",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.719265436000569
      ]
    },
    {
      "case": "C134",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-2*(y + 1)^2*((x - 2)*(y + 1) + 1)*(3*(x - 2)*(y + 1) + 4) - 2*((x - 2)*(y + 1) + 1)^3*(-x^2 + z + 1) - 1, y + 3*(x - 2)*(y + 1)^2*(3*(x - 2)*(y + 1) + 4) + 3*(x - 2)*((x - 2)*(y + 1) + 1)^2*(-x^2 + z + 1) - 1, 2*x - (x - 2)^3*(-x^2 + z + 1) - 3*(x - 2)^2*(y + 1) - 5)",
      "determinant": "4",
      "certificate": "det J=4; three distinct rational preimages share (-1/2, -2, -1)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.788172764000592
      ]
    },
    {
      "case": "B127",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (3*x^2 - 3*y)*(z - 3*(-x^2 + y)^2)^2, -x^2 + y, z - 3*(-x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7256528979996801
      ]
    },
    {
      "case": "B182",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-10*x^2 - 2*y)*(z - 5*(5*x^2 + y)^2)^2, 5*x^2 + y, z - 5*(5*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7804659589992298
      ]
    },
    {
      "case": "B01",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (9*x^2 - 3*y)*(z - 6*(-3*x^2 + y)^2)^2, -3*x^2 + y, z - 6*(-3*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.8332919599997695
      ]
    },
    {
      "case": "B105",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (5*x^2 + 5*y)*(z - 3*(x^2 + y)^2)^2, x^2 + y, z - 3*(x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7789174999998068
      ]
    },
    {
      "case": "C205",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-(y + 2)^2*(x*(y + 2) + 1)*(3*x*(y + 2) + 4) - (x*(y + 2) + 1)^3*(-x^2 + z + 1) - 2, 3*x*(y + 2)^2*(3*x*(y + 2) + 4) + 3*x*(x*(y + 2) + 1)^2*(-x^2 + z + 1) + y + 1, -x^3*(-x^2 + z + 1) - 3*x^2*(y + 2) + 2*x)",
      "determinant": "2",
      "certificate": "det J=2; three distinct rational preimages share (-7/4, -1, 0)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.8150709780002217
      ]
    },
    {
      "case": "C23",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-(y + 1)^2*((x + 2)*(y + 1) + 1)*(3*(x + 2)*(y + 1) + 4) - (z - 2)*((x + 2)*(y + 1) + 1)^3 - 2, y + 3*(x + 2)*(y + 1)^2*(3*(x + 2)*(y + 1) + 4) + 3*(x + 2)*(z - 2)*((x + 2)*(y + 1) + 1)^2 - 1, 2*x - (x + 2)^3*(z - 2) - 3*(x + 2)^2*(y + 1) + 4)",
      "determinant": "2",
      "certificate": "det J=2; three distinct rational preimages share (-7/4, -2, 0)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.739411022999775
      ]
    },
    {
      "case": "C213",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(2*z*((x - 1)*(y + 1) + 1)^3 + 2*(y + 1)^2*((x - 1)*(y + 1) + 1)*(3*(x - 1)*(y + 1) + 4) + 2, y + 3*z*(x - 1)*((x - 1)*(y + 1) + 1)^2 + 3*(x - 1)*(y + 1)^2*(3*(x - 1)*(y + 1) + 4) - 1, 2*x - z*(x - 1)^3 - 3*(x - 1)^2*(y + 1) - 3)",
      "determinant": "-4",
      "certificate": "det J=-4; three distinct rational preimages share (3/2, -2, -1)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7141461639994304
      ]
    },
    {
      "case": "B29",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-6*x^2 - 6*y)*(z - 6*(x^2 + y)^2)^2, x^2 + y, z - 6*(x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7550436089995856
      ]
    },
    {
      "case": "B66",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-20*x^2 - 4*y)*(z + (5*x^2 + y)^2)^2, 5*x^2 + y, z + (5*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7705936729998939
      ]
    },
    {
      "case": "B186",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-12*x^2 - 6*y)*(z + 2*(2*x^2 + y)^2)^2, 2*x^2 + y, z + 2*(2*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.709031571000196
      ]
    },
    {
      "case": "C34",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(2*(y + 2)^2*((x - 1)*(y + 2) + 1)*(3*(x - 1)*(y + 2) + 4) + 2*((x - 1)*(y + 2) + 1)^3*(-2*x^2 + z - 1), y + 3*(x - 1)*(y + 2)^2*(3*(x - 1)*(y + 2) + 4) + 3*(x - 1)*((x - 1)*(y + 2) + 1)^2*(-2*x^2 + z - 1) + 4, 2*x - (x - 1)^3*(-2*x^2 + z - 1) - 3*(x - 1)^2*(y + 2) - 3)",
      "determinant": "-4",
      "certificate": "det J=-4; three distinct rational preimages share (-1/2, 2, -1)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7588467639998271
      ]
    },
    {
      "case": "B191",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (30*x^2 + 5*y)*(z + 5*(6*x^2 + y)^2)^2, 6*x^2 + y, z + 5*(6*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.6895597630000339
      ]
    },
    {
      "case": "B146",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (2*x^2 + 2*y)*(z + 3*(x^2 + y)^2)^2, x^2 + y, z + 3*(x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7942409459992632
      ]
    },
    {
      "case": "C132",
      "family": "counterexample",
      "mapping": "F(x,y,z)=((y + 1)^2*((x - 1)*(y + 1) + 1)*(3*(x - 1)*(y + 1) + 4) + ((x - 1)*(y + 1) + 1)^3*(-2*x^2 + z - 2) + 1, y + 3*(x - 1)*(y + 1)^2*(3*(x - 1)*(y + 1) + 4) + 3*(x - 1)*((x - 1)*(y + 1) + 1)^2*(-2*x^2 + z - 2) + 1, 2*x - (x - 1)^3*(-2*x^2 + z - 2) - 3*(x - 1)^2*(y + 1) - 1)",
      "determinant": "-2",
      "certificate": "det J=-2; three distinct rational preimages share (3/4, 0, 1)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.8212456270002804
      ]
    },
    {
      "case": "C00",
      "family": "counterexample",
      "mapping": "F(x,y,z)=((1+xy)^3z+y^2(1+xy)(4+3xy), y+3x(1+xy)^2z+3xy^2(4+3xy), 2x-3x^2y-x^3z)",
      "determinant": "-2",
      "certificate": "det J=-2; three distinct rational preimages share (-1/4, 0, 0)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.8033490369998617
      ]
    },
    {
      "case": "B71",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (4*x^2 + y)*(z - 4*(4*x^2 + y)^2)^2, 4*x^2 + y, z - 4*(4*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7723296290005237
      ]
    },
    {
      "case": "B237",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (x^2 + y)*(z + 5*(x^2 + y)^2)^2, x^2 + y, z + 5*(x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7447512459993959
      ]
    },
    {
      "case": "B35",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-4*x^2 - 4*y)*(z - 2*(x^2 + y)^2)^2, x^2 + y, z - 2*(x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.8200093169998581
      ]
    },
    {
      "case": "C225",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-2*(y + 1)^2*((x - 2)*(y + 1) + 1)*(3*(x - 2)*(y + 1) + 4) - 2*((x - 2)*(y + 1) + 1)^3*(x^2 + z - 1), y + 3*(x - 2)*(y + 1)^2*(3*(x - 2)*(y + 1) + 4) + 3*(x - 2)*((x - 2)*(y + 1) + 1)^2*(x^2 + z - 1) - 1, 2*x - (x - 2)^3*(x^2 + z - 1) - 3*(x - 2)^2*(y + 1) - 3)",
      "determinant": "4",
      "certificate": "det J=4; three distinct rational preimages share (1/2, -2, 1)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7421051940000325
      ]
    },
    {
      "case": "C186",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-2*(y - 2)^2*(x*(y - 2) + 1)*(3*x*(y - 2) + 4) - 2*(x*(y - 2) + 1)^3*(-2*x^2 + z - 1) + 2, 3*x*(y - 2)^2*(3*x*(y - 2) + 4) + 3*x*(x*(y - 2) + 1)^2*(-2*x^2 + z - 1) + y - 2, -x^3*(-2*x^2 + z - 1) - 3*x^2*(y - 2) + 2*x - 1)",
      "determinant": "4",
      "certificate": "det J=4; three distinct rational preimages share (5/2, 0, -1)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7690225540000029
      ]
    },
    {
      "case": "C16",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-(2*x^2 + z)*((x - 1)*(y + 1) + 1)^3 - (y + 1)^2*((x - 1)*(y + 1) + 1)*(3*(x - 1)*(y + 1) + 4) + 1, y + 3*(x - 1)*(2*x^2 + z)*((x - 1)*(y + 1) + 1)^2 + 3*(x - 1)*(y + 1)^2*(3*(x - 1)*(y + 1) + 4) + 2, 2*x - (x - 1)^3*(2*x^2 + z) - 3*(x - 1)^2*(y + 1) - 1)",
      "determinant": "2",
      "certificate": "det J=2; three distinct rational preimages share (5/4, 1, 1)",
      "probe_target": "No",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7028327090001767
      ]
    },
    {
      "case": "C102",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-2*(y + 2)^2*(x*(y + 2) + 1)*(3*x*(y + 2) + 4) - 2*(x*(y + 2) + 1)^3*(x^2 + z + 2) + 1, 3*x*(y + 2)^2*(3*x*(y + 2) + 4) + 3*x*(x*(y + 2) + 1)^2*(x^2 + z + 2) + y, -x^3*(x^2 + z + 2) - 3*x^2*(y + 2) + 2*x)",
      "determinant": "4",
      "certificate": "det J=4; three distinct rational preimages share (3/2, -2, 0)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7470960190003098
      ]
    },
    {
      "case": "B54",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (30*x^2 + 6*y)*(z - 5*(5*x^2 + y)^2)^2, 5*x^2 + y, z - 5*(5*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7697374469998977
      ]
    },
    {
      "case": "B219",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-12*x^2 + 4*y)*(z - 5*(-3*x^2 + y)^2)^2, -3*x^2 + y, z - 5*(-3*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7801275230003739
      ]
    },
    {
      "case": "C31",
      "family": "counterexample",
      "mapping": "F(x,y,z)=((y + 1)^2*((x + 1)*(y + 1) + 1)*(3*(x + 1)*(y + 1) + 4) + (z + 2)*((x + 1)*(y + 1) + 1)^3 + 2, y + 3*(x + 1)*(y + 1)^2*(3*(x + 1)*(y + 1) + 4) + 3*(x + 1)*(z + 2)*((x + 1)*(y + 1) + 1)^2, 2*x - (x + 1)^3*(z + 2) - 3*(x + 1)^2*(y + 1))",
      "determinant": "-2",
      "certificate": "det J=-2; three distinct rational preimages share (7/4, -1, -2)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7214313389995368
      ]
    },
    {
      "case": "B115",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (12*x^2 - 4*y)*(z + 6*(-3*x^2 + y)^2)^2, -3*x^2 + y, z + 6*(-3*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7932563519998439
      ]
    },
    {
      "case": "C89",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-(y - 1)^2*((x - 2)*(y - 1) + 1)*(3*(x - 2)*(y - 1) + 4) - ((x - 2)*(y - 1) + 1)^3*(-x^2 + z + 2), y + 3*(x - 2)*(y - 1)^2*(3*(x - 2)*(y - 1) + 4) + 3*(x - 2)*((x - 2)*(y - 1) + 1)^2*(-x^2 + z + 2) + 1, 2*x - (x - 2)^3*(-x^2 + z + 2) - 3*(x - 2)^2*(y - 1) - 3)",
      "determinant": "2",
      "certificate": "det J=2; three distinct rational preimages share (1/4, 2, 1)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.8166482409997116
      ]
    },
    {
      "case": "B241",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (5*x^2 + 5*y)*(z - 2*(x^2 + y)^2)^2, x^2 + y, z - 2*(x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7212110520003989
      ]
    },
    {
      "case": "C191",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-(y + 2)^2*((x + 2)*(y + 2) + 1)*(3*(x + 2)*(y + 2) + 4) - ((x + 2)*(y + 2) + 1)^3*(-x^2 + z + 1), y + 3*(x + 2)*(y + 2)^2*(3*(x + 2)*(y + 2) + 4) + 3*(x + 2)*((x + 2)*(y + 2) + 1)^2*(-x^2 + z + 1) + 4, 2*x - (x + 2)^3*(-x^2 + z + 1) - 3*(x + 2)^2*(y + 2) + 3)",
      "determinant": "2",
      "certificate": "det J=2; three distinct rational preimages share (1/4, 2, -1)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7230393270001514
      ]
    },
    {
      "case": "B87",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-36*x^2 - 6*y)*(z - (6*x^2 + y)^2)^2, 6*x^2 + y, z - (6*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7444313119995059
      ]
    },
    {
      "case": "C209",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-2*(y + 2)^2*(x*(y + 2) + 1)*(3*x*(y + 2) + 4) - 2*(x*(y + 2) + 1)^3*(x^2 + z + 2) - 1, 3*x*(y + 2)^2*(3*x*(y + 2) + 4) + 3*x*(x*(y + 2) + 1)^2*(x^2 + z + 2) + y, -x^3*(x^2 + z + 2) - 3*x^2*(y + 2) + 2*x)",
      "determinant": "4",
      "certificate": "det J=4; three distinct rational preimages share (-1/2, -2, 0)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7544479060006779
      ]
    },
    {
      "case": "B06",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (10*x^2 - 5*y)*(z + 6*(-2*x^2 + y)^2)^2, -2*x^2 + y, z + 6*(-2*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7988123430004634
      ]
    },
    {
      "case": "C210",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(2*z*((x - 1)*(y - 2) + 1)^3 + 2*(y - 2)^2*((x - 1)*(y - 2) + 1)*(3*(x - 1)*(y - 2) + 4) + 1, y + 3*z*(x - 1)*((x - 1)*(y - 2) + 1)^2 + 3*(x - 1)*(y - 2)^2*(3*(x - 1)*(y - 2) + 4) - 2, 2*x - z*(x - 1)^3 - 3*(x - 1)^2*(y - 2) - 3)",
      "determinant": "-4",
      "certificate": "det J=-4; three distinct rational preimages share (1/2, 0, -1)",
      "probe_target": "No",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7090186559998983
      ]
    },
    {
      "case": "C65",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(2*(y + 2)^2*((x + 2)*(y + 2) + 1)*(3*(x + 2)*(y + 2) + 4) + 2*((x + 2)*(y + 2) + 1)^3*(x^2 + z - 1) - 1, y + 3*(x + 2)*(y + 2)^2*(3*(x + 2)*(y + 2) + 4) + 3*(x + 2)*((x + 2)*(y + 2) + 1)^2*(x^2 + z - 1) + 1, 2*x - (x + 2)^3*(x^2 + z - 1) - 3*(x + 2)^2*(y + 2) + 5)",
      "determinant": "-4",
      "certificate": "det J=-4; three distinct rational preimages share (-3/2, -1, 1)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7956900300005145
      ]
    },
    {
      "case": "C88",
      "family": "counterexample",
      "mapping": "F(x,y,z)=((2*x^2 + z)*((x - 2)*(y + 1) + 1)^3 + (y + 1)^2*((x - 2)*(y + 1) + 1)*(3*(x - 2)*(y + 1) + 4) + 1, y + 3*(x - 2)*(2*x^2 + z)*((x - 2)*(y + 1) + 1)^2 + 3*(x - 2)*(y + 1)^2*(3*(x - 2)*(y + 1) + 4) + 2, 2*x - (x - 2)^3*(2*x^2 + z) - 3*(x - 2)^2*(y + 1) - 3)",
      "determinant": "-2",
      "certificate": "det J=-2; three distinct rational preimages share (3/4, 1, 1)",
      "probe_target": "No",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7295837500005291
      ]
    },
    {
      "case": "C162",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-2*(y + 1)^2*((x - 1)*(y + 1) + 1)*(3*(x - 1)*(y + 1) + 4) - 2*((x - 1)*(y + 1) + 1)^3*(-2*x^2 + z + 2), y + 3*(x - 1)*(y + 1)^2*(3*(x - 1)*(y + 1) + 4) + 3*(x - 1)*((x - 1)*(y + 1) + 1)^2*(-2*x^2 + z + 2) + 2, 2*x - (x - 1)^3*(-2*x^2 + z + 2) - 3*(x - 1)^2*(y + 1))",
      "determinant": "4",
      "certificate": "det J=4; three distinct rational preimages share (1/2, 1, 2)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7608357989993237
      ]
    },
    {
      "case": "B171",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (15*x^2 + 5*y)*(z - 5*(3*x^2 + y)^2)^2, 3*x^2 + y, z - 5*(3*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7428589889996147
      ]
    },
    {
      "case": "C71",
      "family": "counterexample",
      "mapping": "F(x,y,z)=((y - 2)^2*((x + 1)*(y - 2) + 1)*(3*(x + 1)*(y - 2) + 4) + ((x + 1)*(y - 2) + 1)^3*(x^2 + z - 2) - 2, y + 3*(x + 1)*(y - 2)^2*(3*(x + 1)*(y - 2) + 4) + 3*(x + 1)*((x + 1)*(y - 2) + 1)^2*(x^2 + z - 2) - 3, 2*x - (x + 1)^3*(x^2 + z - 2) - 3*(x + 1)^2*(y - 2) + 4)",
      "determinant": "-2",
      "certificate": "det J=-2; three distinct rational preimages share (-9/4, -1, 2)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7852141149996896
      ]
    },
    {
      "case": "C151",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-2*(y - 1)^2*((x + 2)*(y - 1) + 1)*(3*(x + 2)*(y - 1) + 4) - 2*(z + 1)*((x + 2)*(y - 1) + 1)^3 + 2, y + 3*(x + 2)*(y - 1)^2*(3*(x + 2)*(y - 1) + 4) + 3*(x + 2)*(z + 1)*((x + 2)*(y - 1) + 1)^2 - 1, 2*x - (x + 2)^3*(z + 1) - 3*(x + 2)^2*(y - 1) + 2)",
      "determinant": "4",
      "certificate": "det J=4; three distinct rational preimages share (5/2, 0, -2)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.727898959000413
      ]
    },
    {
      "case": "B68",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-5*x^2 - y)*(z + (5*x^2 + y)^2)^2, 5*x^2 + y, z + (5*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7510517259997869
      ]
    },
    {
      "case": "C189",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(2*(y - 2)^2*((x + 1)*(y - 2) + 1)*(3*(x + 1)*(y - 2) + 4) + 2*((x + 1)*(y - 2) + 1)^3*(-x^2 + z - 2) - 1, y + 3*(x + 1)*(y - 2)^2*(3*(x + 1)*(y - 2) + 4) + 3*(x + 1)*((x + 1)*(y - 2) + 1)^2*(-x^2 + z - 2) - 3, 2*x - (x + 1)^3*(-x^2 + z - 2) - 3*(x + 1)^2*(y - 2) + 3)",
      "determinant": "-4",
      "certificate": "det J=-4; three distinct rational preimages share (-3/2, -1, 1)",
      "probe_target": "No",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7657392610008174
      ]
    },
    {
      "case": "C75",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-2*(y - 2)^2*((x + 2)*(y - 2) + 1)*(3*(x + 2)*(y - 2) + 4) - 2*((x + 2)*(y - 2) + 1)^3*(2*x^2 + z + 2) + 1, y + 3*(x + 2)*(y - 2)^2*(3*(x + 2)*(y - 2) + 4) + 3*(x + 2)*((x + 2)*(y - 2) + 1)^2*(2*x^2 + z + 2) - 3, 2*x - (x + 2)^3*(2*x^2 + z + 2) - 3*(x + 2)^2*(y - 2) + 3)",
      "determinant": "4",
      "certificate": "det J=4; three distinct rational preimages share (3/2, -1, -1)",
      "probe_target": "No",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7436983349998627
      ]
    },
    {
      "case": "C154",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-(y - 1)^2*((x - 2)*(y - 1) + 1)*(3*(x - 2)*(y - 1) + 4) - ((x - 2)*(y - 1) + 1)^3*(x^2 + z + 1) + 1, y + 3*(x - 2)*(y - 1)^2*(3*(x - 2)*(y - 1) + 4) + 3*(x - 2)*((x - 2)*(y - 1) + 1)^2*(x^2 + z + 1) - 2, 2*x - (x - 2)^3*(x^2 + z + 1) - 3*(x - 2)^2*(y - 1) - 4)",
      "determinant": "2",
      "certificate": "det J=2; three distinct rational preimages share (5/4, -1, 0)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7665609129999211
      ]
    },
    {
      "case": "B61",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (30*x^2 + 6*y)*(z - 6*(5*x^2 + y)^2)^2, 5*x^2 + y, z - 6*(5*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7731522929998391
      ]
    },
    {
      "case": "B33",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-3*x^2 + 3*y)*(z - 4*(-x^2 + y)^2)^2, -x^2 + y, z - 4*(-x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7355266780004968
      ]
    },
    {
      "case": "C72",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-y^2*(y*(x - 1) + 1)*(3*y*(x - 1) + 4) - (y*(x - 1) + 1)^3*(-x^2 + z - 1) - 2, 3*y^2*(x - 1)*(3*y*(x - 1) + 4) + y + 3*(x - 1)*(y*(x - 1) + 1)^2*(-x^2 + z - 1), 2*x - 3*y*(x - 1)^2 - (x - 1)^3*(-x^2 + z - 1) - 4)",
      "determinant": "2",
      "certificate": "det J=2; three distinct rational preimages share (-7/4, 0, -2)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.8477618739998434
      ]
    },
    {
      "case": "B107",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (30*x^2 + 5*y)*(z - 5*(6*x^2 + y)^2)^2, 6*x^2 + y, z - 5*(6*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.8749046900002213
      ]
    },
    {
      "case": "B108",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-12*x^2 - 3*y)*(z - 4*(4*x^2 + y)^2)^2, 4*x^2 + y, z - 4*(4*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7564068849997057
      ]
    },
    {
      "case": "C87",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-2*y^2*(y*(x + 1) + 1)*(3*y*(x + 1) + 4) - 2*(z + 1)*(y*(x + 1) + 1)^3 - 2, 3*y^2*(x + 1)*(3*y*(x + 1) + 4) + y + 3*(x + 1)*(z + 1)*(y*(x + 1) + 1)^2 - 2, 2*x - 3*y*(x + 1)^2 - (x + 1)^3*(z + 1) + 4)",
      "determinant": "4",
      "certificate": "det J=4; three distinct rational preimages share (-3/2, -2, 2)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7209634719993119
      ]
    },
    {
      "case": "C94",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-(y + 2)^2*((x - 1)*(y + 2) + 1)*(3*(x - 1)*(y + 2) + 4) - ((x - 1)*(y + 2) + 1)^3*(-x^2 + z + 2) + 2, y + 3*(x - 1)*(y + 2)^2*(3*(x - 1)*(y + 2) + 4) + 3*(x - 1)*((x - 1)*(y + 2) + 1)^2*(-x^2 + z + 2) + 3, 2*x - (x - 1)^3*(-x^2 + z + 2) - 3*(x - 1)^2*(y + 2))",
      "determinant": "2",
      "certificate": "det J=2; three distinct rational preimages share (9/4, 1, 2)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7284262160001163
      ]
    },
    {
      "case": "C109",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(2*(y - 2)^2*((x + 1)*(y - 2) + 1)*(3*(x + 1)*(y - 2) + 4) + 2*((x + 1)*(y - 2) + 1)^3*(-2*x^2 + z - 2) - 2, y + 3*(x + 1)*(y - 2)^2*(3*(x + 1)*(y - 2) + 4) + 3*(x + 1)*((x + 1)*(y - 2) + 1)^2*(-2*x^2 + z - 2) - 2, 2*x - (x + 1)^3*(-2*x^2 + z - 2) - 3*(x + 1)^2*(y - 2) + 1)",
      "determinant": "-4",
      "certificate": "det J=-4; three distinct rational preimages share (-5/2, 0, -1)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7281874840000455
      ]
    },
    {
      "case": "B162",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (20*x^2 + 5*y)*(z + 5*(4*x^2 + y)^2)^2, 4*x^2 + y, z + 5*(4*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7002384430006714
      ]
    },
    {
      "case": "B130",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-2*x^2 + y)*(z + (-2*x^2 + y)^2)^2, -2*x^2 + y, z + (-2*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7333825559999241
      ]
    },
    {
      "case": "B201",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-4*x^2 - y)*(z - 2*(4*x^2 + y)^2)^2, 4*x^2 + y, z - 2*(4*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7881639690003794
      ]
    },
    {
      "case": "B16",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (12*x^2 + 4*y)*(z + 4*(3*x^2 + y)^2)^2, 3*x^2 + y, z + 4*(3*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7335691429998406
      ]
    },
    {
      "case": "C09",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(y^2*(y*(x - 1) + 1)*(3*y*(x - 1) + 4) + z*(y*(x - 1) + 1)^3 - 2, 3*y^2*(x - 1)*(3*y*(x - 1) + 4) + y + 3*z*(x - 1)*(y*(x - 1) + 1)^2 + 2, 2*x - 3*y*(x - 1)^2 - z*(x - 1)^3 - 3)",
      "determinant": "-2",
      "certificate": "det J=-2; three distinct rational preimages share (-9/4, 2, -1)",
      "probe_target": "No",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7377323939999769
      ]
    },
    {
      "case": "B236",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-15*x^2 + 3*y)*(z - 4*(-5*x^2 + y)^2)^2, -5*x^2 + y, z - 4*(-5*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7469267549995493
      ]
    },
    {
      "case": "B99",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (20*x^2 + 5*y)*(z + 4*(4*x^2 + y)^2)^2, 4*x^2 + y, z + 4*(4*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7386372599994502
      ]
    },
    {
      "case": "C226",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(2*(y - 2)^2*((x + 2)*(y - 2) + 1)*(3*(x + 2)*(y - 2) + 4) + 2*((x + 2)*(y - 2) + 1)^3*(-2*x^2 + z - 2), y + 3*(x + 2)*(y - 2)^2*(3*(x + 2)*(y - 2) + 4) + 3*(x + 2)*((x + 2)*(y - 2) + 1)^2*(-2*x^2 + z - 2) - 3, 2*x - (x + 2)^3*(-2*x^2 + z - 2) - 3*(x + 2)^2*(y - 2) + 3)",
      "determinant": "-4",
      "certificate": "det J=-4; three distinct rational preimages share (-1/2, -1, -1)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.8210664949992861
      ]
    },
    {
      "case": "C169",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(y^2*(y*(x - 1) + 1)*(3*y*(x - 1) + 4) + (y*(x - 1) + 1)^3*(-2*x^2 + z + 2), 3*y^2*(x - 1)*(3*y*(x - 1) + 4) + y + 3*(x - 1)*(y*(x - 1) + 1)^2*(-2*x^2 + z + 2) - 1, 2*x - 3*y*(x - 1)^2 - (x - 1)^3*(-2*x^2 + z + 2) - 4)",
      "determinant": "-2",
      "certificate": "det J=-2; three distinct rational preimages share (-1/4, -1, -2)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.8104339100000288
      ]
    },
    {
      "case": "C131",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(2*(y + 2)^2*((x - 2)*(y + 2) + 1)*(3*(x - 2)*(y + 2) + 4) + 2*((x - 2)*(y + 2) + 1)^3*(x^2 + z + 1) - 1, y + 3*(x - 2)*(y + 2)^2*(3*(x - 2)*(y + 2) + 4) + 3*(x - 2)*((x - 2)*(y + 2) + 1)^2*(x^2 + z + 1) + 1, 2*x - (x - 2)^3*(x^2 + z + 1) - 3*(x - 2)^2*(y + 2) - 4)",
      "determinant": "-4",
      "certificate": "det J=-4; three distinct rational preimages share (-3/2, -1, 0)",
      "probe_target": "No",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7611879209998733
      ]
    },
    {
      "case": "C130",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-2*(-x^2 + z)*((x - 1)*(y + 2) + 1)^3 - 2*(y + 2)^2*((x - 1)*(y + 2) + 1)*(3*(x - 1)*(y + 2) + 4) + 2, y + 3*(x - 1)*(-x^2 + z)*((x - 1)*(y + 2) + 1)^2 + 3*(x - 1)*(y + 2)^2*(3*(x - 1)*(y + 2) + 4) + 4, 2*x - (x - 1)^3*(-x^2 + z) - 3*(x - 1)^2*(y + 2) - 1)",
      "determinant": "4",
      "certificate": "det J=4; three distinct rational preimages share (5/2, 2, 1)",
      "probe_target": "No",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.8022561120005776
      ]
    },
    {
      "case": "B11",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (12*x^2 + 4*y)*(z + 6*(3*x^2 + y)^2)^2, 3*x^2 + y, z + 6*(3*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.710754511000232
      ]
    },
    {
      "case": "B73",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (2*x^2 - 2*y)*(z + 4*(-x^2 + y)^2)^2, -x^2 + y, z + 4*(-x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7718059180006094
      ]
    },
    {
      "case": "C140",
      "family": "counterexample",
      "mapping": "F(x,y,z)=((y - 1)^2*(x*(y - 1) + 1)*(3*x*(y - 1) + 4) + (x*(y - 1) + 1)^3*(-2*x^2 + z - 2), 3*x*(y - 1)^2*(3*x*(y - 1) + 4) + 3*x*(x*(y - 1) + 1)^2*(-2*x^2 + z - 2) + y, -x^3*(-2*x^2 + z - 2) - 3*x^2*(y - 1) + 2*x + 2)",
      "determinant": "-2",
      "certificate": "det J=-2; three distinct rational preimages share (-1/4, 1, 2)",
      "probe_target": "No",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7707705110005918
      ]
    },
    {
      "case": "C141",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-(y + 2)^2*((x - 2)*(y + 2) + 1)*(3*(x - 2)*(y + 2) + 4) - (z + 2)*((x - 2)*(y + 2) + 1)^3 - 1, y + 3*(x - 2)*(y + 2)^2*(3*(x - 2)*(y + 2) + 4) + 3*(x - 2)*(z + 2)*((x - 2)*(y + 2) + 1)^2 + 3, 2*x - (x - 2)^3*(z + 2) - 3*(x - 2)^2*(y + 2) - 6)",
      "determinant": "2",
      "certificate": "det J=2; three distinct rational preimages share (-3/4, 1, -2)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7951902029999474
      ]
    },
    {
      "case": "B195",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-6*x^2 - 3*y)*(z - 3*(2*x^2 + y)^2)^2, 2*x^2 + y, z - 3*(2*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.8073216849998062
      ]
    },
    {
      "case": "C101",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(2*(y - 1)^2*((x - 2)*(y - 1) + 1)*(3*(x - 2)*(y - 1) + 4) + 2*((x - 2)*(y - 1) + 1)^3*(-x^2 + z - 1) + 2, y + 3*(x - 2)*(y - 1)^2*(3*(x - 2)*(y - 1) + 4) + 3*(x - 2)*((x - 2)*(y - 1) + 1)^2*(-x^2 + z - 1) - 2, 2*x - (x - 2)^3*(-x^2 + z - 1) - 3*(x - 2)^2*(y - 1) - 4)",
      "determinant": "-4",
      "certificate": "det J=-4; three distinct rational preimages share (3/2, -1, 0)",
      "probe_target": "No",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7841156300000875
      ]
    },
    {
      "case": "C126",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-2*(y - 2)^2*((x - 2)*(y - 2) + 1)*(3*(x - 2)*(y - 2) + 4) - 2*((x - 2)*(y - 2) + 1)^3*(2*x^2 + z + 2), y + 3*(x - 2)*(y - 2)^2*(3*(x - 2)*(y - 2) + 4) + 3*(x - 2)*((x - 2)*(y - 2) + 1)^2*(2*x^2 + z + 2) - 3, 2*x - (x - 2)^3*(2*x^2 + z + 2) - 3*(x - 2)^2*(y - 2) - 6)",
      "determinant": "4",
      "certificate": "det J=4; three distinct rational preimages share (1/2, -1, -2)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.8032291649997205
      ]
    },
    {
      "case": "B221",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (6*x^2 - 3*y)*(z - 6*(-2*x^2 + y)^2)^2, -2*x^2 + y, z - 6*(-2*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7213878679995105
      ]
    },
    {
      "case": "C05",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(2*y^2*(y*(x + 2) + 1)*(3*y*(x + 2) + 4) + 2*(y*(x + 2) + 1)^3*(2*x^2 + z - 2) - 2, 3*y^2*(x + 2)*(3*y*(x + 2) + 4) + y + 3*(x + 2)*(y*(x + 2) + 1)^2*(2*x^2 + z - 2) - 1, 2*x - 3*y*(x + 2)^2 - (x + 2)^3*(2*x^2 + z - 2) + 5)",
      "determinant": "-4",
      "certificate": "det J=-4; three distinct rational preimages share (-5/2, -1, 1)",
      "probe_target": "No",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7886460919999081
      ]
    },
    {
      "case": "B174",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-18*x^2 - 6*y)*(z + 5*(3*x^2 + y)^2)^2, 3*x^2 + y, z + 5*(3*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7841438519999429
      ]
    },
    {
      "case": "C207",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-(y + 2)^2*((x + 1)*(y + 2) + 1)*(3*(x + 1)*(y + 2) + 4) - ((x + 1)*(y + 2) + 1)^3*(-2*x^2 + z - 1) - 1, y + 3*(x + 1)*(y + 2)^2*(3*(x + 1)*(y + 2) + 4) + 3*(x + 1)*((x + 1)*(y + 2) + 1)^2*(-2*x^2 + z - 1) + 3, 2*x - (x + 1)^3*(-2*x^2 + z - 1) - 3*(x + 1)^2*(y + 2) + 3)",
      "determinant": "2",
      "certificate": "det J=2; three distinct rational preimages share (-3/4, 1, 1)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7823591860005763
      ]
    },
    {
      "case": "B163",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-2*x^2 + 2*y)*(z + 2*(-x^2 + y)^2)^2, -x^2 + y, z + 2*(-x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.8256089890001022
      ]
    },
    {
      "case": "C176",
      "family": "counterexample",
      "mapping": "F(x,y,z)=((y + 2)^2*((x + 1)*(y + 2) + 1)*(3*(x + 1)*(y + 2) + 4) + ((x + 1)*(y + 2) + 1)^3*(-x^2 + z + 1), y + 3*(x + 1)*(y + 2)^2*(3*(x + 1)*(y + 2) + 4) + 3*(x + 1)*((x + 1)*(y + 2) + 1)^2*(-x^2 + z + 1) + 3, 2*x - (x + 1)^3*(-x^2 + z + 1) - 3*(x + 1)^2*(y + 2) + 4)",
      "determinant": "-2",
      "certificate": "det J=-2; three distinct rational preimages share (-1/4, 1, 2)",
      "probe_target": "No",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7481382989999474
      ]
    },
    {
      "case": "B57",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-6*x^2 + 2*y)*(z - 2*(-3*x^2 + y)^2)^2, -3*x^2 + y, z - 2*(-3*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7321368689999872
      ]
    },
    {
      "case": "B02",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-30*x^2 + 6*y)*(z - 6*(-5*x^2 + y)^2)^2, -5*x^2 + y, z - 6*(-5*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.777873577999344
      ]
    },
    {
      "case": "B64",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-4*x^2 - 2*y)*(z - 3*(2*x^2 + y)^2)^2, 2*x^2 + y, z - 3*(2*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7827697069997157
      ]
    },
    {
      "case": "B14",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (18*x^2 - 3*y)*(z - 2*(-6*x^2 + y)^2)^2, -6*x^2 + y, z - 2*(-6*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7895496759992966
      ]
    },
    {
      "case": "C20",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(2*(y - 2)^2*((x + 1)*(y - 2) + 1)*(3*(x + 1)*(y - 2) + 4) + 2*((x + 1)*(y - 2) + 1)^3*(-x^2 + z + 2) + 1, y + 3*(x + 1)*(y - 2)^2*(3*(x + 1)*(y - 2) + 4) + 3*(x + 1)*((x + 1)*(y - 2) + 1)^2*(-x^2 + z + 2) - 1, 2*x - (x + 1)^3*(-x^2 + z + 2) - 3*(x + 1)^2*(y - 2) + 1)",
      "determinant": "-4",
      "certificate": "det J=-4; three distinct rational preimages share (1/2, 1, -1)",
      "probe_target": "No",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.757102304
      ]
    },
    {
      "case": "C235",
      "family": "counterexample",
      "mapping": "F(x,y,z)=((y + 2)^2*((x - 2)*(y + 2) + 1)*(3*(x - 2)*(y + 2) + 4) + ((x - 2)*(y + 2) + 1)^3*(2*x^2 + z + 2) + 2, y + 3*(x - 2)*(y + 2)^2*(3*(x - 2)*(y + 2) + 4) + 3*(x - 2)*((x - 2)*(y + 2) + 1)^2*(2*x^2 + z + 2), 2*x - (x - 2)^3*(2*x^2 + z + 2) - 3*(x - 2)^2*(y + 2) - 6)",
      "determinant": "-2",
      "certificate": "det J=-2; three distinct rational preimages share (7/4, -2, -2)",
      "probe_target": "No",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7397833330005597
      ]
    },
    {
      "case": "B187",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (2*x^2 + y)*(z + 5*(2*x^2 + y)^2)^2, 2*x^2 + y, z + 5*(2*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.8453152120000595
      ]
    },
    {
      "case": "C227",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-2*(y - 2)^2*((x + 1)*(y - 2) + 1)*(3*(x + 1)*(y - 2) + 4) - 2*(z + 1)*((x + 1)*(y - 2) + 1)^3 + 2, y + 3*(x + 1)*(y - 2)^2*(3*(x + 1)*(y - 2) + 4) + 3*(x + 1)*(z + 1)*((x + 1)*(y - 2) + 1)^2 - 1, 2*x - (x + 1)^3*(z + 1) - 3*(x + 1)^2*(y - 2))",
      "determinant": "4",
      "certificate": "det J=4; three distinct rational preimages share (5/2, 1, -2)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.836935799999992
      ]
    },
    {
      "case": "B194",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (24*x^2 - 4*y)*(z - 5*(-6*x^2 + y)^2)^2, -6*x^2 + y, z - 5*(-6*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.9339981210005135
      ]
    },
    {
      "case": "B46",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-12*x^2 + 3*y)*(z + 5*(-4*x^2 + y)^2)^2, -4*x^2 + y, z + 5*(-4*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7823687449999852
      ]
    },
    {
      "case": "B183",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (3*x^2 + y)*(z + 4*(3*x^2 + y)^2)^2, 3*x^2 + y, z + 4*(3*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.727325808000387
      ]
    },
    {
      "case": "B212",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (36*x^2 - 6*y)*(z + (-6*x^2 + y)^2)^2, -6*x^2 + y, z + (-6*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7896998739997798
      ]
    },
    {
      "case": "C160",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-2*(y - 2)^2*(x*(y - 2) + 1)*(3*x*(y - 2) + 4) - 2*(x*(y - 2) + 1)^3*(2*x^2 + z + 2), 3*x*(y - 2)^2*(3*x*(y - 2) + 4) + 3*x*(x*(y - 2) + 1)^2*(2*x^2 + z + 2) + y - 2, -x^3*(2*x^2 + z + 2) - 3*x^2*(y - 2) + 2*x)",
      "determinant": "4",
      "certificate": "det J=4; three distinct rational preimages share (1/2, 0, 0)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.775653866000539
      ]
    },
    {
      "case": "C190",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-2*y^2*(y*(x + 1) + 1)*(3*y*(x + 1) + 4) - 2*(z + 1)*(y*(x + 1) + 1)^3 + 2, 3*y^2*(x + 1)*(3*y*(x + 1) + 4) + y + 3*(x + 1)*(z + 1)*(y*(x + 1) + 1)^2 - 1, 2*x - 3*y*(x + 1)^2 - (x + 1)^3*(z + 1) + 3)",
      "determinant": "4",
      "certificate": "det J=4; three distinct rational preimages share (5/2, -1, 1)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7467454089992316
      ]
    },
    {
      "case": "C219",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-(y - 2)^2*(x*(y - 2) + 1)*(3*x*(y - 2) + 4) - (x*(y - 2) + 1)^3*(-x^2 + z - 1) + 2, 3*x*(y - 2)^2*(3*x*(y - 2) + 4) + 3*x*(x*(y - 2) + 1)^2*(-x^2 + z - 1) + y - 3, -x^3*(-x^2 + z - 1) - 3*x^2*(y - 2) + 2*x - 2)",
      "determinant": "2",
      "certificate": "det J=2; three distinct rational preimages share (9/4, -1, -2)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7613889429994742
      ]
    },
    {
      "case": "C187",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(2*(y - 2)^2*((x - 1)*(y - 2) + 1)*(3*(x - 1)*(y - 2) + 4) + 2*((x - 1)*(y - 2) + 1)^3*(2*x^2 + z - 2) + 2, y + 3*(x - 1)*(y - 2)^2*(3*(x - 1)*(y - 2) + 4) + 3*(x - 1)*((x - 1)*(y - 2) + 1)^2*(2*x^2 + z - 2), 2*x - (x - 1)^3*(2*x^2 + z - 2) - 3*(x - 1)^2*(y - 2) - 2)",
      "determinant": "-4",
      "certificate": "det J=-4; three distinct rational preimages share (3/2, 2, 0)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.8685161369994603
      ]
    },
    {
      "case": "B103",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-9*x^2 + 3*y)*(z + (-3*x^2 + y)^2)^2, -3*x^2 + y, z + (-3*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7470981819997178
      ]
    },
    {
      "case": "B63",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-36*x^2 + 6*y)*(z - 5*(-6*x^2 + y)^2)^2, -6*x^2 + y, z - 5*(-6*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7460562429996571
      ]
    },
    {
      "case": "B44",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-25*x^2 + 5*y)*(z + 2*(-5*x^2 + y)^2)^2, -5*x^2 + y, z + 2*(-5*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7551316429999133
      ]
    },
    {
      "case": "C128",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-2*(y + 1)^2*((x + 2)*(y + 1) + 1)*(3*(x + 2)*(y + 1) + 4) - 2*((x + 2)*(y + 1) + 1)^3*(-x^2 + z - 2) + 2, y + 3*(x + 2)*(y + 1)^2*(3*(x + 2)*(y + 1) + 4) + 3*(x + 2)*((x + 2)*(y + 1) + 1)^2*(-x^2 + z - 2) + 2, 2*x - (x + 2)^3*(-x^2 + z - 2) - 3*(x + 2)^2*(y + 1) + 2)",
      "determinant": "4",
      "certificate": "det J=4; three distinct rational preimages share (5/2, 1, -2)",
      "probe_target": "No",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7231782050002948
      ]
    },
    {
      "case": "C224",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-(y - 2)^2*((x + 1)*(y - 2) + 1)*(3*(x + 1)*(y - 2) + 4) - (z - 2)*((x + 1)*(y - 2) + 1)^3 + 1, y + 3*(x + 1)*(y - 2)^2*(3*(x + 1)*(y - 2) + 4) + 3*(x + 1)*(z - 2)*((x + 1)*(y - 2) + 1)^2 - 3, 2*x - (x + 1)^3*(z - 2) - 3*(x + 1)^2*(y - 2) + 2)",
      "determinant": "2",
      "certificate": "det J=2; three distinct rational preimages share (5/4, -1, 0)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.8262144609998359
      ]
    },
    {
      "case": "B137",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-12*x^2 - 6*y)*(z - 3*(2*x^2 + y)^2)^2, 2*x^2 + y, z - 3*(2*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7235469880006349
      ]
    },
    {
      "case": "C105",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-2*(y + 1)^2*((x + 2)*(y + 1) + 1)*(3*(x + 2)*(y + 1) + 4) - 2*((x + 2)*(y + 1) + 1)^3*(2*x^2 + z - 1) - 1, y + 3*(x + 2)*(y + 1)^2*(3*(x + 2)*(y + 1) + 4) + 3*(x + 2)*((x + 2)*(y + 1) + 1)^2*(2*x^2 + z - 1) + 3, 2*x - (x + 2)^3*(2*x^2 + z - 1) - 3*(x + 2)^2*(y + 1) + 4)",
      "determinant": "4",
      "certificate": "det J=4; three distinct rational preimages share (-1/2, 2, 0)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7358409110001958
      ]
    },
    {
      "case": "B52",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (30*x^2 + 5*y)*(z + 2*(6*x^2 + y)^2)^2, 6*x^2 + y, z + 2*(6*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7366804860002958
      ]
    },
    {
      "case": "B118",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-6*x^2 - 2*y)*(z + (3*x^2 + y)^2)^2, 3*x^2 + y, z + (3*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7757390540000415
      ]
    },
    {
      "case": "B96",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (18*x^2 - 6*y)*(z + 4*(-3*x^2 + y)^2)^2, -3*x^2 + y, z + 4*(-3*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7535660030007421
      ]
    },
    {
      "case": "C172",
      "family": "counterexample",
      "mapping": "F(x,y,z)=((y - 2)^2*(x*(y - 2) + 1)*(3*x*(y - 2) + 4) + (x*(y - 2) + 1)^3*(2*x^2 + z + 1) + 2, 3*x*(y - 2)^2*(3*x*(y - 2) + 4) + 3*x*(x*(y - 2) + 1)^2*(2*x^2 + z + 1) + y - 2, -x^3*(2*x^2 + z + 1) - 3*x^2*(y - 2) + 2*x + 2)",
      "determinant": "-2",
      "certificate": "det J=-2; three distinct rational preimages share (7/4, 0, 2)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7660432149996268
      ]
    },
    {
      "case": "B170",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (25*x^2 + 5*y)*(z + (5*x^2 + y)^2)^2, 5*x^2 + y, z + (5*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7283476310003607
      ]
    },
    {
      "case": "C86",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-(y - 2)^2*((x - 2)*(y - 2) + 1)*(3*(x - 2)*(y - 2) + 4) - ((x - 2)*(y - 2) + 1)^3*(-2*x^2 + z - 1) + 2, y + 3*(x - 2)*(y - 2)^2*(3*(x - 2)*(y - 2) + 4) + 3*(x - 2)*((x - 2)*(y - 2) + 1)^2*(-2*x^2 + z - 1) - 4, 2*x - (x - 2)^3*(-2*x^2 + z - 1) - 3*(x - 2)^2*(y - 2) - 4)",
      "determinant": "2",
      "certificate": "det J=2; three distinct rational preimages share (9/4, -2, 0)",
      "probe_target": "No",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7019716839995453
      ]
    },
    {
      "case": "B55",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-36*x^2 + 6*y)*(z - 4*(-6*x^2 + y)^2)^2, -6*x^2 + y, z - 4*(-6*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7579951170000641
      ]
    },
    {
      "case": "C194",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-(y + 2)^2*((x + 2)*(y + 2) + 1)*(3*(x + 2)*(y + 2) + 4) - (z + 1)*((x + 2)*(y + 2) + 1)^3, y + 3*(x + 2)*(y + 2)^2*(3*(x + 2)*(y + 2) + 4) + 3*(x + 2)*(z + 1)*((x + 2)*(y + 2) + 1)^2 + 1, 2*x - (x + 2)^3*(z + 1) - 3*(x + 2)^2*(y + 2) + 3)",
      "determinant": "2",
      "certificate": "det J=2; three distinct rational preimages share (1/4, -1, -1)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7452211560002979
      ]
    },
    {
      "case": "C215",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-(y + 2)^2*((x + 1)*(y + 2) + 1)*(3*(x + 1)*(y + 2) + 4) - ((x + 1)*(y + 2) + 1)^3*(x^2 + z + 1) + 1, y + 3*(x + 1)*(y + 2)^2*(3*(x + 1)*(y + 2) + 4) + 3*(x + 1)*((x + 1)*(y + 2) + 1)^2*(x^2 + z + 1) + 1, 2*x - (x + 1)^3*(x^2 + z + 1) - 3*(x + 1)^2*(y + 2) + 1)",
      "determinant": "2",
      "certificate": "det J=2; three distinct rational preimages share (5/4, -1, -1)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7242997820003438
      ]
    },
    {
      "case": "C148",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(y^2*(y*(x + 1) + 1)*(3*y*(x + 1) + 4) + (-x^2 + z)*(y*(x + 1) + 1)^3 - 2, 3*y^2*(x + 1)*(3*y*(x + 1) + 4) + y + 3*(x + 1)*(-x^2 + z)*(y*(x + 1) + 1)^2 - 2, 2*x - 3*y*(x + 1)^2 - (x + 1)^3*(-x^2 + z))",
      "determinant": "-2",
      "certificate": "det J=-2; three distinct rational preimages share (-9/4, -2, -2)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7890997029999198
      ]
    },
    {
      "case": "B25",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (20*x^2 + 4*y)*(z - 6*(5*x^2 + y)^2)^2, 5*x^2 + y, z - 6*(5*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7714106670000547
      ]
    },
    {
      "case": "C118",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(2*(y + 2)^2*(x*(y + 2) + 1)*(3*x*(y + 2) + 4) + 2*(x*(y + 2) + 1)^3*(x^2 + z + 2), 3*x*(y + 2)^2*(3*x*(y + 2) + 4) + 3*x*(x*(y + 2) + 1)^2*(x^2 + z + 2) + y + 3, -x^3*(x^2 + z + 2) - 3*x^2*(y + 2) + 2*x - 2)",
      "determinant": "-4",
      "certificate": "det J=-4; three distinct rational preimages share (-1/2, 1, -2)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7370503610000014
      ]
    },
    {
      "case": "C217",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(2*y^2*(x*y + 1)*(3*x*y + 4) + 2*(x*y + 1)^3*(x^2 + z + 2) - 1, 3*x*y^2*(3*x*y + 4) + 3*x*(x*y + 1)^2*(x^2 + z + 2) + y - 2, -x^3*(x^2 + z + 2) - 3*x^2*y + 2*x - 2)",
      "determinant": "-4",
      "certificate": "det J=-4; three distinct rational preimages share (-3/2, -2, -2)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7011516050006321
      ]
    },
    {
      "case": "B37",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-30*x^2 + 5*y)*(z + 6*(-6*x^2 + y)^2)^2, -6*x^2 + y, z + 6*(-6*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7466929629999868
      ]
    },
    {
      "case": "B211",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (6*x^2 + 6*y)*(z + 3*(x^2 + y)^2)^2, x^2 + y, z + 3*(x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7306264419994477
      ]
    },
    {
      "case": "C155",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(-(y - 1)^2*(x*(y - 1) + 1)*(3*x*(y - 1) + 4) - (x*(y - 1) + 1)^3*(-x^2 + z + 1) + 1, 3*x*(y - 1)^2*(3*x*(y - 1) + 4) + 3*x*(x*(y - 1) + 1)^2*(-x^2 + z + 1) + y - 1, -x^3*(-x^2 + z + 1) - 3*x^2*(y - 1) + 2*x + 1)",
      "determinant": "2",
      "certificate": "det J=2; three distinct rational preimages share (5/4, 0, 1)",
      "probe_target": "No",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7550078639997082
      ]
    },
    {
      "case": "B230",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (12*x^2 + 3*y)*(z - 6*(4*x^2 + y)^2)^2, 4*x^2 + y, z - 6*(4*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7219467949998943
      ]
    },
    {
      "case": "B228",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-2*x^2 - y)*(z + 5*(2*x^2 + y)^2)^2, 2*x^2 + y, z + 5*(2*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7313229709998268
      ]
    },
    {
      "case": "B40",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-6*x^2 - 2*y)*(z - (3*x^2 + y)^2)^2, 3*x^2 + y, z - (3*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7680842249992565
      ]
    },
    {
      "case": "C234",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(2*(y - 1)^2*((x + 1)*(y - 1) + 1)*(3*(x + 1)*(y - 1) + 4) + 2*(z - 1)*((x + 1)*(y - 1) + 1)^3 + 2, y + 3*(x + 1)*(y - 1)^2*(3*(x + 1)*(y - 1) + 4) + 3*(x + 1)*(z - 1)*((x + 1)*(y - 1) + 1)^2 - 1, 2*x - (x + 1)^3*(z - 1) - 3*(x + 1)^2*(y - 1) + 4)",
      "determinant": "-4",
      "certificate": "det J=-4; three distinct rational preimages share (3/2, 0, 2)",
      "probe_target": "No",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7328521439994802
      ]
    },
    {
      "case": "B148",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-10*x^2 + 5*y)*(z - 2*(-2*x^2 + y)^2)^2, -2*x^2 + y, z - 2*(-2*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.7148747239998556
      ]
    },
    {
      "case": "B227",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-x^2 + y)*(z + 2*(-x^2 + y)^2)^2, -x^2 + y, z + 2*(-x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.8059190170006332
      ]
    },
    {
      "case": "C244",
      "family": "counterexample",
      "mapping": "F(x,y,z)=(2*(y + 1)^2*((x + 2)*(y + 1) + 1)*(3*(x + 2)*(y + 1) + 4) + 2*(z + 2)*((x + 2)*(y + 1) + 1)^3, y + 3*(x + 2)*(y + 1)^2*(3*(x + 2)*(y + 1) + 4) + 3*(x + 2)*(z + 2)*((x + 2)*(y + 1) + 1)^2 - 1, 2*x - (x + 2)^3*(z + 2) - 3*(x + 2)^2*(y + 1) + 6)",
      "determinant": "-4",
      "certificate": "det J=-4; three distinct rational preimages share (-1/2, -2, 2)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7648884250002084
      ]
    },
    {
      "case": "B70",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (-30*x^2 - 5*y)*(z + 5*(6*x^2 + y)^2)^2, 6*x^2 + y, z + 5*(6*x^2 + y)^2)",
      "determinant": "1",
      "certificate": "Explicit inverse of three polynomial shears verified",
      "probe_target": "Maybe",
      "jev_answers": [
        "Maybe"
      ],
      "seconds": [
        0.8419238689994017
      ]
    },
    {
      "case": "C77",
      "family": "counterexample",
      "mapping": "F(x,y,z)=((-2*x^2 + z)*((x - 1)*(y - 1) + 1)^3 + (y - 1)^2*((x - 1)*(y - 1) + 1)*(3*(x - 1)*(y - 1) + 4) - 1, y + 3*(x - 1)*(-2*x^2 + z)*((x - 1)*(y - 1) + 1)^2 + 3*(x - 1)*(y - 1)^2*(3*(x - 1)*(y - 1) + 4) + 1, 2*x - (x - 1)^3*(-2*x^2 + z) - 3*(x - 1)^2*(y - 1) - 3)",
      "determinant": "-2",
      "certificate": "det J=-2; three distinct rational preimages share (-5/4, 2, -1)",
      "probe_target": "No",
      "jev_answers": [
        "No"
      ],
      "seconds": [
        0.7906502440000622
      ]
    }
  ]
}