{
  "artifact": "Explicit degree-3 Keller counterexample to the Jacobian Conjecture",
  "date": "2026-07-21",
  "provenance": "Constructed from the Alpoge/Claude-Fable-5 dim-3 counterexample (2026-07-20) by verified elementary-factorization degree reduction (BCW-style splitting).",
  "dimension": 22,
  "variables": [
    "x",
    "y",
    "z",
    "a1",
    "a2",
    "a3",
    "a4",
    "a5",
    "a6",
    "a7",
    "a8",
    "a9",
    "a10",
    "a11",
    "a12",
    "a13",
    "a14",
    "a15",
    "a16",
    "a17",
    "a18",
    "a19"
  ],
  "components": [
    "a10*a9/2 + a10*x**2/2 + a9*x*z/2 - 3*x**2*y/2 + x",
    "3*a11*a9 + 3*a11*x**2 + 9*a12*a9 + 9*a12*x**2 + 3*a13*a6 + 3*a13*y**2 - 3*a2*a3 + 3*a2*a9*x + 3*a3*a6*z - 9*a4*a6 - 9*a4*y**2 + 9*a6*a9*y - 6*a8*a9 - 6*a8*x**2 - 6*a9*y*z + 12*x*y**2 + 3*x*z + y",
    "-a1*a2 + a1*a6*z - a14*a9 - a14*x**2 + 3*a15*a9 + 3*a15*x**2 + 3*a16*a9 + 3*a16*x**2 - 7*a17*a6 - 7*a17*y**2 + 3*a18*a6 + 3*a18*y**2 + a19*a6 + a19*y**2 + a2*a3*y + a3*a7 - 3*a4*a5 + 3*a4*a6*y - 3*a4*a8 - 3*a4*y*z + 3*a5*a9*y - 7*a6*x*y - a7*a9*x + 3*a8*a9*y + 3*x*y*z + 4*y**2 + z",
    "a1 - a17*a9 - a17*x**2 - a9*x*y",
    "a2 + y**2*z",
    "a3 + x**3",
    "a4 + x**2*y",
    "a5 + y**3",
    "a6 + y**2",
    "a2*y + a7",
    "a8 + y*z",
    "a9 + x**2",
    "a10 + x*z",
    "a11 + a2*x",
    "a12 + a6*y",
    "a13 + a3*z",
    "a14 + a7*x",
    "a15 + a5*y",
    "a16 + a8*y",
    "a17 + x*y",
    "a18 + a4*y",
    "a1*z + a19"
  ],
  "degree_of_H_per_component": [
    3,
    3,
    3,
    3,
    3,
    3,
    3,
    3,
    2,
    2,
    2,
    2,
    2,
    2,
    2,
    2,
    2,
    2,
    2,
    2,
    2,
    2
  ],
  "total_monomials_in_H": 68,
  "max_degree": 3,
  "collision_points": [
    [
      "0",
      "0",
      "-1/4",
      "0",
      "0",
      "0",
      "0",
      "0",
      "0",
      "0",
      "0",
      "0",
      "0",
      "0",
      "0",
      "0",
      "0",
      "0",
      "0",
      "0",
      "0",
      "0"
    ],
    [
      "1",
      "-3/2",
      "13/2",
      "3/2",
      "-117/8",
      "-1",
      "3/2",
      "27/8",
      "-9/4",
      "-351/16",
      "39/4",
      "-1",
      "-13/2",
      "117/8",
      "-27/8",
      "13/2",
      "351/16",
      "81/16",
      "117/8",
      "3/2",
      "9/4",
      "-39/4"
    ],
    [
      "-1",
      "3/2",
      "13/2",
      "3/2",
      "-117/8",
      "1",
      "-3/2",
      "-27/8",
      "-9/4",
      "351/16",
      "-39/4",
      "-1",
      "13/2",
      "-117/8",
      "27/8",
      "-13/2",
      "351/16",
      "81/16",
      "117/8",
      "3/2",
      "9/4",
      "-39/4"
    ]
  ],
  "collision_image": [
    "0",
    "0",
    "-1/4",
    "0",
    "0",
    "0",
    "0",
    "0",
    "0",
    "0",
    "0",
    "0",
    "0",
    "0",
    "0",
    "0",
    "0",
    "0",
    "0",
    "0",
    "0",
    "0"
  ],
  "elimination_steps": [
    {
      "step": 1,
      "component": 2,
      "coeff": "1",
      "monomial": "x**3*y**3*z",
      "degree": 7,
      "u": "x**3*y",
      "v": "y**2*z",
      "aux_u": "a1",
      "aux_v": "a2",
      "new_aux_created": [
        "a1",
        "a2"
      ],
      "dimension_after": 5,
      "max_degree_after": 6
    },
    {
      "step": 2,
      "component": 1,
      "coeff": "3",
      "monomial": "x**3*y**2*z",
      "degree": 6,
      "u": "x**3",
      "v": "y**2*z",
      "aux_u": "a3",
      "aux_v": "a2",
      "new_aux_created": [
        "a3"
      ],
      "dimension_after": 6,
      "max_degree_after": 6
    },
    {
      "step": 3,
      "component": 2,
      "coeff": "3",
      "monomial": "x**2*y**4",
      "degree": 6,
      "u": "x**2*y",
      "v": "y**3",
      "aux_u": "a4",
      "aux_v": "a5",
      "new_aux_created": [
        "a4",
        "a5"
      ],
      "dimension_after": 8,
      "max_degree_after": 5
    },
    {
      "step": 4,
      "component": 1,
      "coeff": "9",
      "monomial": "x**2*y**3",
      "degree": 5,
      "u": "x**2*y",
      "v": "y**2",
      "aux_u": "a4",
      "aux_v": "a6",
      "new_aux_created": [
        "a6"
      ],
      "dimension_after": 9,
      "max_degree_after": 5
    },
    {
      "step": 5,
      "component": 2,
      "coeff": "-1",
      "monomial": "a2*x**3*y",
      "degree": 5,
      "u": "x**3",
      "v": "a2*y",
      "aux_u": "a3",
      "aux_v": "a7",
      "new_aux_created": [
        "a7"
      ],
      "dimension_after": 10,
      "max_degree_after": 5
    },
    {
      "step": 6,
      "component": 2,
      "coeff": "3",
      "monomial": "x**2*y**2*z",
      "degree": 5,
      "u": "x**2*y",
      "v": "y*z",
      "aux_u": "a4",
      "aux_v": "a8",
      "new_aux_created": [
        "a8"
      ],
      "dimension_after": 11,
      "max_degree_after": 4
    },
    {
      "step": 7,
      "component": 0,
      "coeff": "-1/2",
      "monomial": "x**3*z",
      "degree": 4,
      "u": "x**2",
      "v": "x*z",
      "aux_u": "a9",
      "aux_v": "a10",
      "new_aux_created": [
        "a9",
        "a10"
      ],
      "dimension_after": 13,
      "max_degree_after": 4
    },
    {
      "step": 8,
      "component": 1,
      "coeff": "-3",
      "monomial": "a2*x**3",
      "degree": 4,
      "u": "x**2",
      "v": "a2*x",
      "aux_u": "a9",
      "aux_v": "a11",
      "new_aux_created": [
        "a11"
      ],
      "dimension_after": 14,
      "max_degree_after": 4
    },
    {
      "step": 9,
      "component": 1,
      "coeff": "6",
      "monomial": "x**2*y*z",
      "degree": 4,
      "u": "x**2",
      "v": "y*z",
      "aux_u": "a9",
      "aux_v": "a8",
      "new_aux_created": [],
      "dimension_after": 14,
      "max_degree_after": 4
    },
    {
      "step": 10,
      "component": 1,
      "coeff": "-9",
      "monomial": "a6*x**2*y",
      "degree": 4,
      "u": "x**2",
      "v": "a6*y",
      "aux_u": "a9",
      "aux_v": "a12",
      "new_aux_created": [
        "a12"
      ],
      "dimension_after": 15,
      "max_degree_after": 4
    },
    {
      "step": 11,
      "component": 1,
      "coeff": "-3",
      "monomial": "a3*y**2*z",
      "degree": 4,
      "u": "y**2",
      "v": "a3*z",
      "aux_u": "a6",
      "aux_v": "a13",
      "new_aux_created": [
        "a13"
      ],
      "dimension_after": 16,
      "max_degree_after": 4
    },
    {
      "step": 12,
      "component": 2,
      "coeff": "1",
      "monomial": "a7*x**3",
      "degree": 4,
      "u": "x**2",
      "v": "a7*x",
      "aux_u": "a9",
      "aux_v": "a14",
      "new_aux_created": [
        "a14"
      ],
      "dimension_after": 17,
      "max_degree_after": 4
    },
    {
      "step": 13,
      "component": 2,
      "coeff": "-3",
      "monomial": "a5*x**2*y",
      "degree": 4,
      "u": "x**2",
      "v": "a5*y",
      "aux_u": "a9",
      "aux_v": "a15",
      "new_aux_created": [
        "a15"
      ],
      "dimension_after": 18,
      "max_degree_after": 4
    },
    {
      "step": 14,
      "component": 2,
      "coeff": "-3",
      "monomial": "a8*x**2*y",
      "degree": 4,
      "u": "x**2",
      "v": "a8*y",
      "aux_u": "a9",
      "aux_v": "a16",
      "new_aux_created": [
        "a16"
      ],
      "dimension_after": 19,
      "max_degree_after": 4
    },
    {
      "step": 15,
      "component": 2,
      "coeff": "7",
      "monomial": "x*y**3",
      "degree": 4,
      "u": "x*y",
      "v": "y**2",
      "aux_u": "a17",
      "aux_v": "a6",
      "new_aux_created": [
        "a17"
      ],
      "dimension_after": 20,
      "max_degree_after": 4
    },
    {
      "step": 16,
      "component": 2,
      "coeff": "-3",
      "monomial": "a4*y**3",
      "degree": 4,
      "u": "y**2",
      "v": "a4*y",
      "aux_u": "a6",
      "aux_v": "a18",
      "new_aux_created": [
        "a18"
      ],
      "dimension_after": 21,
      "max_degree_after": 4
    },
    {
      "step": 17,
      "component": 2,
      "coeff": "-1",
      "monomial": "a1*y**2*z",
      "degree": 4,
      "u": "y**2",
      "v": "a1*z",
      "aux_u": "a6",
      "aux_v": "a19",
      "new_aux_created": [
        "a19"
      ],
      "dimension_after": 22,
      "max_degree_after": 4
    },
    {
      "step": 18,
      "component": 3,
      "coeff": "1",
      "monomial": "x**3*y",
      "degree": 4,
      "u": "x**2",
      "v": "x*y",
      "aux_u": "a9",
      "aux_v": "a17",
      "new_aux_created": [],
      "dimension_after": 22,
      "max_degree_after": 3
    }
  ],
  "det_proof": "chain rule over machine-verified elementary factorization (every Phi unitriangular, every composition identity checked symbolically) + 30 exact random-rational point confirmations",
  "sharpness": "Wang (1980): every Keller map of degree <= 2 is invertible. Hence degree 3 is the minimal possible degree for any counterexample.",
  "elapsed_sec": 7.01
}