{
  "seed": 20260921,
  "question": "Is the Jacobian conjecture true",
  "repeats": 3,
  "baseline": [
    "Maybe",
    "Maybe",
    "Maybe"
  ],
  "baseline_seconds": [
    0.9219366829993305,
    0.7833913240001493,
    0.7482216200005496
  ],
  "rows": [
    {
      "case": "C02",
      "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) + 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": "-4",
      "certificate": "det J=-4; three distinct rational preimages share (3/2, -1, -2)",
      "probe_target": "No",
      "jev_answers": [
        "Maybe",
        "No",
        "Maybe"
      ],
      "seconds": [
        0.7256743170000846,
        0.818545558000551,
        0.7693762940007218
      ]
    },
    {
      "case": "B01",
      "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",
        "Maybe",
        "Maybe"
      ],
      "seconds": [
        0.7352168630004599,
        0.7965877879996697,
        0.7485773409998728
      ]
    },
    {
      "case": "C03",
      "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 - 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) + 2, 2*x - 3*y*(x - 2)^2 - (x - 2)^3*(2*x^2 + z - 1) - 2)",
      "determinant": "-4",
      "certificate": "det J=-4; three distinct rational preimages share (1/2, 2, 2)",
      "probe_target": "No",
      "jev_answers": [
        "No",
        "Maybe",
        "No"
      ],
      "seconds": [
        0.7422018909992403,
        0.7451740159995097,
        0.7346408470002643
      ]
    },
    {
      "case": "B05",
      "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",
        "Maybe",
        "Maybe"
      ],
      "seconds": [
        0.8196844989997771,
        0.7017473989999417,
        0.7703112670005794
      ]
    },
    {
      "case": "B02",
      "family": "bijection",
      "mapping": "F(x,y,z)=(x + (2*x^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",
        "No",
        "Maybe"
      ],
      "seconds": [
        0.7459295170001496,
        0.7540532329994676,
        0.7865567690005264
      ]
    },
    {
      "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",
        "No",
        "No"
      ],
      "seconds": [
        0.6994803010002215,
        0.7663704790002157,
        0.8328607349994854
      ]
    },
    {
      "case": "B04",
      "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": [
        "No",
        "Maybe",
        "Maybe"
      ],
      "seconds": [
        0.7338926410002387,
        0.7917076889998498,
        0.7521882700002607
      ]
    },
    {
      "case": "B03",
      "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",
        "Maybe",
        "Maybe"
      ],
      "seconds": [
        0.7361801489996651,
        0.7799162490000526,
        0.7376251260002391
      ]
    },
    {
      "case": "C05",
      "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) - 1, 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 (-1/2, -2, 1)",
      "probe_target": "No",
      "jev_answers": [
        "No",
        "No",
        "No"
      ],
      "seconds": [
        0.7717006610000681,
        0.7694203889996061,
        0.7802221070005544
      ]
    },
    {
      "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": [
        "No",
        "Maybe",
        "Maybe"
      ],
      "seconds": [
        0.721951815000466,
        0.7530835190000289,
        0.736215214999902
      ]
    },
    {
      "case": "C04",
      "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) - 2, 2*x - (x - 1)^3*(-x^2 + z + 2) - 3*(x - 1)^2*(y - 1))",
      "determinant": "-2",
      "certificate": "det J=-2; three distinct rational preimages share (-9/4, -1, 2)",
      "probe_target": "No",
      "jev_answers": [
        "No",
        "No",
        "No"
      ],
      "seconds": [
        0.7768281299995579,
        0.7542491180001889,
        0.7298592520000966
      ]
    },
    {
      "case": "C01",
      "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) + 4, 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, 2, -1)",
      "probe_target": "No",
      "jev_answers": [
        "No",
        "No",
        "No"
      ],
      "seconds": [
        0.7942071899997245,
        0.778283275000831,
        0.7304734700001063
      ]
    }
  ]
}