{
 "what": "Detached certificates for Jacobian/Hessian-conjecture counterexamples. Each entry claims: (1) det of the Jacobian of F is identically the stated constant — a polynomial identity over Q; (2) the listed rational points all map to the listed image under F, exactly; (3) the points are pairwise distinct. Together: F is a Keller map that is not injective, so the Jacobian conjecture is false in that dimension (for hessian entries, F = grad Psi and the Hessian conjecture is false). Verify with tools/verify_keller.py — Python stdlib only, no code from this repo.",
 "polynomialFormat": "each polynomial is a list of monomials [e_1, ..., e_n, coeff] — exponents of x_1..x_n, then the exact rational coefficient as a string",
 "generatedBy": "tools/export-keller-certificate.js @ git 04d4b7c",
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 },
 "entries": [
  {
   "id": "keller-0",
   "n": 3,
   "source": "Alpöge 2026-07-19",
   "statement": "the Jacobian conjecture is FALSE in dimension 3: an explicit polynomial map with det J = -2 proved as a polynomial identity, and 3 distinct rational points sharing one image, all decided in exact arithmetic (Alpöge 2026-07-19, audited here)",
   "det": "-2",
   "F": [
    [
     [
      0,
      0,
      1,
      "1"
     ],
     [
      0,
      2,
      0,
      "4"
     ],
     [
      1,
      1,
      1,
      "3"
     ],
     [
      1,
      3,
      0,
      "7"
     ],
     [
      2,
      2,
      1,
      "3"
     ],
     [
      2,
      4,
      0,
      "3"
     ],
     [
      3,
      3,
      1,
      "1"
     ]
    ],
    [
     [
      0,
      1,
      0,
      "1"
     ],
     [
      1,
      0,
      1,
      "3"
     ],
     [
      1,
      2,
      0,
      "12"
     ],
     [
      2,
      1,
      1,
      "6"
     ],
     [
      2,
      3,
      0,
      "9"
     ],
     [
      3,
      2,
      1,
      "3"
     ]
    ],
    [
     [
      1,
      0,
      0,
      "2"
     ],
     [
      2,
      1,
      0,
      "-3"
     ],
     [
      3,
      0,
      1,
      "-1"
     ]
    ]
   ],
   "collisions": [
    [
     "0",
     "0",
     "-1/4"
    ],
    [
     "1",
     "-3/2",
     "13/2"
    ],
    [
     "-1",
     "3/2",
     "13/2"
    ]
   ],
   "image": [
    "-1/4",
    "0",
    "0"
   ],
   "transcription": "P = (1+xy)^3 z + y^2(1+xy)(4+3xy); Q = y + 3x(1+xy)^2 z + 3xy^2(4+3xy); R = 2x - 3x^2y - x^3z; det J = -2; witnesses (0,0,-1/4), (1,-3/2,13/2), (-1,3/2,13/2) -> (-1/4,0,0)",
   "sourcePin": {
    "file": "mengyang2026.pdf",
    "sha256": "090c1735997c598f14741fb8083666b3043cf62bc59da303494166bb5e9c18f5"
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  },
  {
   "id": "keller-1",
   "n": 8,
   "source": "Alpöge 2026-07-19",
   "padded": true,
   "statement": "the STABILIZATION, stated once: adjoining identity coordinates carries the n=3 counterexample to every dimension n >= 3 with det J and the witnesses unchanged — audited here at n=8 as the one padded representative (det J = -2 through the 8x8 symbolic determinant, 3 collisions re-evaluated); identity padding, not new mathematics",
   "det": "-2",
   "F": [
    [
     [
      0,
      0,
      1,
      0,
      0,
      0,
      0,
      0,
      "1"
     ],
     [
      0,
      2,
      0,
      0,
      0,
      0,
      0,
      0,
      "4"
     ],
     [
      1,
      1,
      1,
      0,
      0,
      0,
      0,
      0,
      "3"
     ],
     [
      1,
      3,
      0,
      0,
      0,
      0,
      0,
      0,
      "7"
     ],
     [
      2,
      2,
      1,
      0,
      0,
      0,
      0,
      0,
      "3"
     ],
     [
      2,
      4,
      0,
      0,
      0,
      0,
      0,
      0,
      "3"
     ],
     [
      3,
      3,
      1,
      0,
      0,
      0,
      0,
      0,
      "1"
     ]
    ],
    [
     [
      0,
      1,
      0,
      0,
      0,
      0,
      0,
      0,
      "1"
     ],
     [
      1,
      0,
      1,
      0,
      0,
      0,
      0,
      0,
      "3"
     ],
     [
      1,
      2,
      0,
      0,
      0,
      0,
      0,
      0,
      "12"
     ],
     [
      2,
      1,
      1,
      0,
      0,
      0,
      0,
      0,
      "6"
     ],
     [
      2,
      3,
      0,
      0,
      0,
      0,
      0,
      0,
      "9"
     ],
     [
      3,
      2,
      1,
      0,
      0,
      0,
      0,
      0,
      "3"
     ]
    ],
    [
     [
      1,
      0,
      0,
      0,
      0,
      0,
      0,
      0,
      "2"
     ],
     [
      2,
      1,
      0,
      0,
      0,
      0,
      0,
      0,
      "-3"
     ],
     [
      3,
      0,
      1,
      0,
      0,
      0,
      0,
      0,
      "-1"
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    ],
    [
     [
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    ],
    [
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   ],
   "collisions": [
    [
     "0",
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     "0",
     "0",
     "0",
     "0",
     "0"
    ],
    [
     "1",
     "-3/2",
     "13/2",
     "0",
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    [
     "-1",
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     "0",
     "0",
     "0",
     "0",
     "0"
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   "image": [
    "-1/4",
    "0",
    "0",
    "0",
    "0",
    "0",
    "0",
    "0"
   ],
   "transcription": "P = (1+xy)^3 z + y^2(1+xy)(4+3xy); Q = y + 3x(1+xy)^2 z + 3xy^2(4+3xy); R = 2x - 3x^2y - x^3z; det J = -2; witnesses (0,0,-1/4), (1,-3/2,13/2), (-1,3/2,13/2) -> (-1/4,0,0)",
   "sourcePin": {
    "file": "mengyang2026.pdf",
    "sha256": "090c1735997c598f14741fb8083666b3043cf62bc59da303494166bb5e9c18f5"
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  },
  {
   "id": "keller-2",
   "n": 3,
   "source": "tangent-sweep d=3 (new curve through the published mechanism), generated+certified here",
   "statement": "a NEW instance through the PUBLISHED tangent-sweep mechanism: our own degree-3 curve (coefficients [6, -6, 1]) gives geometric degree 4, det J = -2 proved as a polynomial identity, 2 distinct rational points sharing one image — a new curve, not a new mechanism, and not coordinate-equivalence-checked against Gallagher's members; generated AND decided in exact arithmetic",
   "det": "-2",
   "F": [
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      1,
      "1/2"
     ],
     [
      0,
      2,
      0,
      "9/2"
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      1,
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      1,
      "3"
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     [
      1,
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      "14"
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     [
      2,
      0,
      2,
      "3/8"
     ],
     [
      2,
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      1,
      "9"
     ],
     [
      2,
      4,
      0,
      "43/2"
     ],
     [
      3,
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      2,
      "3/2"
     ],
     [
      3,
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      "14"
     ],
     [
      3,
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      0,
      "18"
     ],
     [
      4,
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      2,
      "9/4"
     ],
     [
      4,
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      1,
      "21/2"
     ],
     [
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     [
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     [
      5,
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      "3"
     ],
     [
      6,
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      "3/8"
     ]
    ],
    [
     [
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     [
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     ]
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    [
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     [
      2,
      1,
      0,
      "-4"
     ],
     [
      3,
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      1,
      "-1"
     ]
    ]
   ],
   "collisions": [
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     "-2/3",
     "5/2",
     "183/8"
    ],
    [
     "2/11",
     "-13/2",
     "297/8"
    ]
   ],
   "image": [
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    "-2",
    "1"
   ],
   "sweep": {
    "d": 3,
    "geometricDegree": 4,
    "p": [
     "6",
     "-6",
     "1"
    ],
    "a": "-4",
    "g0": "2",
    "b": "-1",
    "det": "-2",
    "secant": [
     "1",
     "-1"
    ]
   }
  },
  {
   "id": "keller-3",
   "n": 3,
   "source": "tangent-sweep d=4 (new curve through the published mechanism), generated+certified here",
   "statement": "a NEW instance through the PUBLISHED tangent-sweep mechanism: our own degree-4 curve (coefficients [52/5, -51/5, 0, 1]) gives geometric degree 5, det J = -2 proved as a polynomial identity, 2 distinct rational points sharing one image — a new curve, not a new mechanism, and not coordinate-equivalence-checked against Gallagher's members; generated AND decided in exact arithmetic",
   "det": "-2",
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    "a": "-9/7",
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   "id": "keller-4",
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   "source": "tangent-sweep d=5 (new curve through the published mechanism), generated+certified here",
   "statement": "a NEW instance through the PUBLISHED tangent-sweep mechanism: our own degree-5 curve (coefficients [20, -19, 0, 0, 1]) gives geometric degree 6, det J = -2 proved as a polynomial identity, 2 distinct rational points sharing one image — a new curve, not a new mechanism, and not coordinate-equivalence-checked against Gallagher's members; generated AND decided in exact arithmetic",
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      0,
      4,
      "5/12"
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      2,
      3,
      "-220/7"
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      2,
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      0,
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     [
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     [
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      2,
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     ],
     [
      8,
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      1,
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     ],
     [
      8,
      9,
      0,
      "28561/2401"
     ],
     [
      9,
      2,
      4,
      "10"
     ],
     [
      9,
      4,
      3,
      "240/7"
     ],
     [
      9,
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      2,
      "2886/49"
     ],
     [
      9,
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      1,
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     ]
    ],
    [
     [
      1,
      0,
      0,
      "2"
     ],
     [
      2,
      1,
      0,
      "-13/7"
     ],
     [
      3,
      0,
      1,
      "-1"
     ]
    ]
   ],
   "collisions": [
    [
     "-6/25",
     "31/6",
     "222325/1512"
    ],
    [
     "6/101",
     "-107/6",
     "-5512277/1512"
    ]
   ],
   "image": [
    "-61/12",
    "-19/3",
    "1"
   ],
   "sweep": {
    "d": 5,
    "geometricDegree": 6,
    "p": [
     "20",
     "-19",
     "0",
     "0",
     "1"
    ],
    "a": "-13/7",
    "g0": "2",
    "b": "-1",
    "det": "-2",
    "secant": [
     "1",
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    ]
   }
  },
  {
   "id": "keller-5",
   "n": 5,
   "source": "Meng–Yang arXiv:2607.22198",
   "hessian": true,
   "statement": "the HESSIAN conjecture is FALSE in 5 variables: an explicit degree-14 integer polynomial whose Hessian determinant is 128 as a polynomial identity while its gradient identifies 2 distinct rational points, all decided in exact arithmetic (Meng–Yang arXiv:2607.22198, audited here)",
   "det": "128",
   "F": [
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      "4"
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      "36"
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     [
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     [
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      0,
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     ],
     [
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      2,
      0,
      0,
      0,
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     ],
     [
      3,
      2,
      1,
      0,
      0,
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     ],
     [
      4,
      0,
      0,
      0,
      1,
      "-6"
     ],
     [
      4,
      2,
      0,
      1,
      0,
      "108"
     ],
     [
      4,
      3,
      1,
      0,
      0,
      "60"
     ],
     [
      5,
      1,
      0,
      0,
      1,
      "-12"
     ],
     [
      5,
      3,
      0,
      1,
      0,
      "72"
     ],
     [
      5,
      4,
      1,
      0,
      0,
      "30"
     ],
     [
      6,
      2,
      0,
      0,
      1,
      "-6"
     ],
     [
      6,
      4,
      0,
      1,
      0,
      "18"
     ],
     [
      6,
      5,
      1,
      0,
      0,
      "6"
     ]
    ],
    [
     [
      1,
      0,
      0,
      0,
      0,
      "4"
     ],
     [
      2,
      1,
      0,
      0,
      0,
      "-6"
     ],
     [
      3,
      0,
      0,
      0,
      0,
      "-13"
     ],
     [
      3,
      0,
      1,
      0,
      0,
      "-2"
     ],
     [
      4,
      0,
      0,
      1,
      0,
      "-6"
     ],
     [
      4,
      1,
      1,
      0,
      0,
      "-6"
     ],
     [
      5,
      1,
      0,
      1,
      0,
      "-12"
     ],
     [
      5,
      2,
      1,
      0,
      0,
      "-6"
     ],
     [
      6,
      0,
      0,
      0,
      1,
      "2"
     ],
     [
      6,
      2,
      0,
      1,
      0,
      "-6"
     ],
     [
      6,
      3,
      1,
      0,
      0,
      "-2"
     ]
    ]
   ],
   "collisions": [
    [
     "1",
     "-3/2",
     "0",
     "0",
     "0"
    ],
    [
     "-1",
     "3/2",
     "0",
     "0",
     "0"
    ]
   ],
   "image": [
    "0",
    "0",
    "-1/2",
    "0",
    "0"
   ],
   "transcription": "Psi = A^2 + 13A + 2B (42 monomials, total degree 14) with A, B from the Schur descent of phi = y.F in x3; det Hess Psi = 128; grad Psi identifies (1,-3/2,0,0,0) and (-1,3/2,0,0,0)",
   "sourcePin": {
    "file": "mengyang2026.pdf",
    "sha256": "090c1735997c598f14741fb8083666b3043cf62bc59da303494166bb5e9c18f5"
   }
  },
  {
   "id": "keller-6",
   "n": 3,
   "source": "Gallagher zenodo.21479195 d=2, fiber degree 3",
   "statement": "the Jacobian conjecture is FALSE in dimension 3: Gallagher zenodo.21479195 d=2 reconstructed from its seed — det J = 1 proved as a polynomial identity, generic fiber degree 3, 2 distinct rational points sharing one image, every claim decided in exact arithmetic",
   "det": "1",
   "F": [
    [
     [
      0,
      0,
      1,
      "-2"
     ],
     [
      0,
      2,
      0,
      "4"
     ],
     [
      1,
      1,
      1,
      "-6"
     ],
     [
      1,
      3,
      0,
      "7"
     ],
     [
      2,
      2,
      1,
      "-6"
     ],
     [
      2,
      4,
      0,
      "3"
     ],
     [
      3,
      3,
      1,
      "-2"
     ]
    ],
    [
     [
      0,
      1,
      0,
      "1/2"
     ],
     [
      1,
      0,
      1,
      "-3"
     ],
     [
      1,
      2,
      0,
      "6"
     ],
     [
      2,
      1,
      1,
      "-6"
     ],
     [
      2,
      3,
      0,
      "9/2"
     ],
     [
      3,
      2,
      1,
      "-3"
     ]
    ],
    [
     [
      1,
      0,
      0,
      "1"
     ],
     [
      2,
      1,
      0,
      "-3/2"
     ],
     [
      3,
      0,
      1,
      "1"
     ]
    ]
   ],
   "collisions": [
    [
     "-1/3",
     "4",
     "-54"
    ],
    [
     "1/4",
     "-2",
     "36"
    ]
   ],
   "image": [
    "-4",
    "-4",
    "1"
   ],
   "transcription": "seed gauge p(0)=0, p(1)=-c, INT_0^1 p = 0; q from c q' = w p'; kappa = p'(1)/c; a = -(1+kappa)/(2+kappa); gamma = 1 + a xy + b x^2 z; F = (alpha/x^2, beta/x, x gamma); det J = bc; family seed p_d = 2w - 3w^2 + w(1-w)(w^{d-2} - 6/(d(d+1))), c = 1",
   "sourcePin": {
    "file": "gallagher2026.pdf",
    "sha256": "1782eefa569a094b52f90b18967fb28d8f6ebd3781026f1c3aa46b3cd67a79c1"
   },
   "sweep": {
    "d": 2,
    "geometricDegree": 3,
    "p": [
     "2",
     "-3"
    ],
    "a": "-3/2",
    "b": "1",
    "c": "1",
    "det": "1",
    "secant": [
     "1",
     "2"
    ]
   }
  },
  {
   "id": "keller-7",
   "n": 3,
   "source": "Gallagher zenodo.21479195 d=3, fiber degree 4",
   "statement": "the Jacobian conjecture is FALSE in dimension 3: Gallagher zenodo.21479195 d=3 reconstructed from its seed — det J = 1 proved as a polynomial identity, generic fiber degree 4, 2 distinct rational points sharing one image, every claim decided in exact arithmetic",
   "det": "1",
   "F": [
    [
     [
      0,
      0,
      1,
      "-5/2"
     ],
     [
      0,
      2,
      0,
      "219/50"
     ],
     [
      1,
      1,
      1,
      "-69/10"
     ],
     [
      1,
      3,
      0,
      "173/25"
     ],
     [
      2,
      0,
      2,
      "-3/4"
     ],
     [
      2,
      2,
      1,
      "-18/5"
     ],
     [
      2,
      4,
      0,
      "23/100"
     ],
     [
      3,
      1,
      2,
      "-3"
     ],
     [
      3,
      3,
      1,
      "28/5"
     ],
     [
      3,
      5,
      0,
      "-189/50"
     ],
     [
      4,
      2,
      2,
      "-9/2"
     ],
     [
      4,
      4,
      1,
      "69/10"
     ],
     [
      4,
      6,
      0,
      "-147/100"
     ],
     [
      5,
      3,
      2,
      "-3"
     ],
     [
      5,
      5,
      1,
      "21/10"
     ],
     [
      6,
      4,
      2,
      "-3/4"
     ]
    ],
    [
     [
      0,
      1,
      0,
      "2/5"
     ],
     [
      1,
      0,
      1,
      "-7/2"
     ],
     [
      1,
      2,
      0,
      "307/50"
     ],
     [
      2,
      1,
      1,
      "-31/5"
     ],
     [
      2,
      3,
      0,
      "181/50"
     ],
     [
      3,
      0,
      2,
      "-1"
     ],
     [
      3,
      2,
      1,
      "9/10"
     ],
     [
      3,
      4,
      0,
      "-77/25"
     ],
     [
      4,
      1,
      2,
      "-3"
     ],
     [
      4,
      3,
      1,
      "32/5"
     ],
     [
      4,
      5,
      0,
      "-49/25"
     ],
     [
      5,
      2,
      2,
      "-3"
     ],
     [
      5,
      4,
      1,
      "14/5"
     ],
     [
      6,
      3,
      2,
      "-1"
     ]
    ],
    [
     [
      1,
      0,
      0,
      "1"
     ],
     [
      2,
      1,
      0,
      "-7/5"
     ],
     [
      3,
      0,
      1,
      "1"
     ]
    ]
   ],
   "collisions": [
    [
     "2",
     "1/2",
     "9/40"
    ],
    [
     "2/3",
     "-5/2",
     "-33/8"
    ]
   ],
   "image": [
    "-1/2",
    "-1/2",
    "1"
   ],
   "transcription": "seed gauge p(0)=0, p(1)=-c, INT_0^1 p = 0; q from c q' = w p'; kappa = p'(1)/c; a = -(1+kappa)/(2+kappa); gamma = 1 + a xy + b x^2 z; F = (alpha/x^2, beta/x, x gamma); det J = bc; family seed p_d = 2w - 3w^2 + w(1-w)(w^{d-2} - 6/(d(d+1))), c = 1",
   "sourcePin": {
    "file": "gallagher2026.pdf",
    "sha256": "1782eefa569a094b52f90b18967fb28d8f6ebd3781026f1c3aa46b3cd67a79c1"
   },
   "sweep": {
    "d": 3,
    "geometricDegree": 4,
    "p": [
     "3/2",
     "-3/2",
     "-1"
    ],
    "a": "-7/5",
    "b": "1",
    "c": "1",
    "det": "1",
    "secant": [
     "1",
     "-1"
    ]
   }
  },
  {
   "id": "keller-8",
   "n": 3,
   "source": "Gallagher zenodo.21479195 d=4, fiber degree 5",
   "statement": "the Jacobian conjecture is FALSE in dimension 3: Gallagher zenodo.21479195 d=4 reconstructed from its seed — det J = 1 proved as a polynomial identity, generic fiber degree 5, 2 distinct rational points sharing one image, every claim decided in exact arithmetic",
   "det": "1",
   "F": [
    [
     [
      0,
      0,
      1,
      "-27/10"
     ],
     [
      0,
      2,
      0,
      "10871/2430"
     ],
     [
      1,
      1,
      1,
      "-619/90"
     ],
     [
      1,
      3,
      0,
      "621301/98415"
     ],
     [
      2,
      0,
      2,
      "-33/20"
     ],
     [
      2,
      2,
      1,
      "-292/1215"
     ],
     [
      2,
      4,
      0,
      "-961417/393660"
     ],
     [
      3,
      1,
      2,
      "-257/45"
     ],
     [
      3,
      3,
      1,
      "13498/1215"
     ],
     [
      3,
      5,
      0,
      "-1035289/196830"
     ],
     [
      4,
      0,
      3,
      "-4/5"
     ],
     [
      4,
      2,
      2,
      "-55/18"
     ],
     [
      4,
      4,
      1,
      "965/486"
     ],
     [
      4,
      6,
      0,
      "1082509/393660"
     ],
     [
      5,
      1,
      3,
      "-4"
     ],
     [
      5,
      3,
      2,
      "107/9"
     ],
     [
      5,
      5,
      1,
      "-40427/2430"
     ],
     [
      5,
      7,
      0,
      "569504/98415"
     ],
     [
      6,
      2,
      3,
      "-8"
     ],
     [
      6,
      4,
      2,
      "779/36"
     ],
     [
      6,
      6,
      1,
      "-19388/1215"
     ],
     [
      6,
      8,
      0,
      "202612/98415"
     ],
     [
      7,
      3,
      3,
      "-8"
     ],
     [
      7,
      5,
      2,
      "632/45"
     ],
     [
      7,
      7,
      1,
      "-5476/1215"
     ],
     [
      8,
      4,
      3,
      "-4"
     ],
     [
      8,
      6,
      2,
      "148/45"
     ],
     [
      9,
      5,
      3,
      "-4/5"
     ]
    ],
    [
     [
      0,
      1,
      0,
      "10/27"
     ],
     [
      1,
      0,
      1,
      "-37/10"
     ],
     [
      1,
      2,
      0,
      "44953/7290"
     ],
     [
      2,
      1,
      1,
      "-799/135"
     ],
     [
      2,
      3,
      0,
      "554401/196830"
     ],
     [
      3,
      0,
      2,
      "-2"
     ],
     [
      3,
      2,
      1,
      "10529/2430"
     ],
     [
      3,
      4,
      0,
      "-101786/19683"
     ],
     [
      4,
      1,
      2,
      "-44/9"
     ],
     [
      4,
      3,
      1,
      "2084/243"
     ],
     [
      4,
      5,
      0,
      "-7252/6561"
     ],
     [
      5,
      0,
      3,
      "-1"
     ],
     [
      5,
      2,
      2,
      "13/9"
     ],
     [
      5,
      4,
      1,
      "-539/81"
     ],
     [
      5,
      6,
      0,
      "91723/19683"
     ],
     [
      6,
      1,
      3,
      "-4"
     ],
     [
      6,
      3,
      2,
      "41/3"
     ],
     [
      6,
      5,
      1,
      "-3478/243"
     ],
     [
      6,
      7,
      0,
      "50653/19683"
     ],
     [
      7,
      2,
      3,
      "-6"
     ],
     [
      7,
      4,
      2,
      "121/9"
     ],
     [
      7,
      6,
      1,
      "-1369/243"
     ],
     [
      8,
      3,
      3,
      "-4"
     ],
     [
      8,
      5,
      2,
      "37/9"
     ],
     [
      9,
      4,
      3,
      "-1"
     ]
    ],
    [
     [
      1,
      0,
      0,
      "1"
     ],
     [
      2,
      1,
      0,
      "-37/27"
     ],
     [
      3,
      0,
      1,
      "1"
     ]
    ]
   ],
   "collisions": [
    [
     "-10",
     "11/10",
     "-4367/27000"
    ],
    [
     "10/53",
     "-63/10",
     "225091/3000"
    ]
   ],
   "image": [
    "-11/10",
    "-11/10",
    "1"
   ],
   "transcription": "seed gauge p(0)=0, p(1)=-c, INT_0^1 p = 0; q from c q' = w p'; kappa = p'(1)/c; a = -(1+kappa)/(2+kappa); gamma = 1 + a xy + b x^2 z; F = (alpha/x^2, beta/x, x gamma); det J = bc; family seed p_d = 2w - 3w^2 + w(1-w)(w^{d-2} - 6/(d(d+1))), c = 1",
   "sourcePin": {
    "file": "gallagher2026.pdf",
    "sha256": "1782eefa569a094b52f90b18967fb28d8f6ebd3781026f1c3aa46b3cd67a79c1"
   },
   "sweep": {
    "d": 4,
    "geometricDegree": 5,
    "p": [
     "17/10",
     "-27/10",
     "1",
     "-1"
    ],
    "a": "-37/27",
    "b": "1",
    "c": "1",
    "det": "1",
    "secant": [
     "1",
     "-1"
    ]
   }
  },
  {
   "id": "keller-9",
   "n": 3,
   "source": "Gallagher zenodo.21479195 d=5, fiber degree 6",
   "statement": "the Jacobian conjecture is FALSE in dimension 3: Gallagher zenodo.21479195 d=5 reconstructed from its seed — det J = 1 proved as a polynomial identity, generic fiber degree 6, 2 distinct rational points sharing one image, every claim decided in exact arithmetic",
   "det": "1",
   "F": [
    [
     [
      0,
      0,
      1,
      "-14/5"
     ],
     [
      0,
      2,
      0,
      "4379/980"
     ],
     [
      1,
      1,
      1,
      "-229/35"
     ],
     [
      1,
      3,
      0,
      "113843/20580"
     ],
     [
      10,
      4,
      4,
      "-25/2"
     ],
     [
      10,
      6,
      3,
      "500/21"
     ],
     [
      10,
      8,
      2,
      "-1805/196"
     ],
     [
      11,
      5,
      4,
      "-5"
     ],
     [
      11,
      7,
      3,
      "95/21"
     ],
     [
      12,
      6,
      4,
      "-5/6"
     ],
     [
      2,
      0,
      2,
      "-13/5"
     ],
     [
      2,
      2,
      1,
      "1597/490"
     ],
     [
      2,
      4,
      0,
      "-5564609/1152480"
     ],
     [
      3,
      1,
      2,
      "-269/35"
     ],
     [
      3,
      3,
      1,
      "284399/20580"
     ],
     [
      3,
      5,
      0,
      "-880741/192080"
     ],
     [
      4,
      0,
      3,
      "-38/15"
     ],
     [
      4,
      2,
      2,
      "967/196"
     ],
     [
      4,
      4,
      1,
      "-4731/343"
     ],
     [
      4,
      6,
      0,
      "4324651/384160"
     ],
     [
      5,
      1,
      3,
      "-241/21"
     ],
     [
      5,
      3,
      2,
      "3895/98"
     ],
     [
      5,
      5,
      1,
      "-314061/6860"
     ],
     [
      5,
      7,
      0,
      "1165251/96040"
     ],
     [
      6,
      0,
      4,
      "-5/6"
     ],
     [
      6,
      2,
      3,
      "-104/7"
     ],
     [
      6,
      4,
      2,
      "7393/196"
     ],
     [
      6,
      6,
      1,
      "-54709/3430"
     ],
     [
      6,
      8,
      0,
      "-1390933/384160"
     ],
     [
      7,
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     ],
     [
      7,
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     ],
     [
      7,
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     ],
     [
      7,
      7,
      1,
      "222699/6860"
     ],
     [
      7,
      9,
      0,
      "-994555/115248"
     ],
     [
      8,
      10,
      0,
      "-651605/230496"
     ],
     [
      8,
      2,
      4,
      "-25/2"
     ],
     [
      8,
      4,
      3,
      "934/21"
     ],
     [
      8,
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      2,
      "-63667/980"
     ],
     [
      8,
      8,
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     ],
     [
      9,
      3,
      4,
      "-50/3"
     ],
     [
      9,
      5,
      3,
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     ],
     [
      9,
      7,
      2,
      "-4085/98"
     ],
     [
      9,
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     ]
    ],
    [
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     ],
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      1,
      "-19/5"
     ],
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      1,
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      0,
      "6009/980"
     ],
     [
      10,
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      4,
      "-10"
     ],
     [
      10,
      5,
      3,
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     ],
     [
      10,
      7,
      2,
      "-1083/98"
     ],
     [
      11,
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      4,
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     [
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     ],
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     ],
     [
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     ],
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     ],
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     ],
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     [
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      4,
      0,
      "-259513/38416"
     ],
     [
      4,
      1,
      2,
      "-81/14"
     ],
     [
      4,
      3,
      1,
      "5207/686"
     ],
     [
      4,
      5,
      0,
      "64887/38416"
     ],
     [
      5,
      0,
      3,
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     ],
     [
      5,
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     ],
     [
      5,
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     ],
     [
      5,
      6,
      0,
      "225929/19208"
     ],
     [
      6,
      1,
      3,
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     ],
     [
      6,
      3,
      2,
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     ],
     [
      6,
      5,
      1,
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     ],
     [
      6,
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      0,
      "24187/9604"
     ],
     [
      7,
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     ],
     [
      7,
      2,
      3,
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     ],
     [
      7,
      4,
      2,
      "444/49"
     ],
     [
      7,
      6,
      1,
      "15485/1372"
     ],
     [
      7,
      8,
      0,
      "-267501/38416"
     ],
     [
      8,
      1,
      4,
      "-5"
     ],
     [
      8,
      3,
      3,
      "128/7"
     ],
     [
      8,
      5,
      2,
      "-3837/98"
     ],
     [
      8,
      7,
      1,
      "19133/686"
     ],
     [
      8,
      9,
      0,
      "-130321/38416"
     ],
     [
      9,
      2,
      4,
      "-10"
     ],
     [
      9,
      4,
      3,
      "247/7"
     ],
     [
      9,
      6,
      2,
      "-3819/98"
     ],
     [
      9,
      8,
      1,
      "6859/686"
     ]
    ],
    [
     [
      1,
      0,
      0,
      "1"
     ],
     [
      2,
      1,
      0,
      "-19/14"
     ],
     [
      3,
      0,
      1,
      "1"
     ]
    ]
   ],
   "collisions": [
    [
     "15/4",
     "11/15",
     "5038/23625"
    ],
    [
     "15/28",
     "-43/15",
     "-14318/3375"
    ]
   ],
   "image": [
    "-11/15",
    "-11/15",
    "1"
   ],
   "transcription": "seed gauge p(0)=0, p(1)=-c, INT_0^1 p = 0; q from c q' = w p'; kappa = p'(1)/c; a = -(1+kappa)/(2+kappa); gamma = 1 + a xy + b x^2 z; F = (alpha/x^2, beta/x, x gamma); det J = bc; family seed p_d = 2w - 3w^2 + w(1-w)(w^{d-2} - 6/(d(d+1))), c = 1",
   "sourcePin": {
    "file": "gallagher2026.pdf",
    "sha256": "1782eefa569a094b52f90b18967fb28d8f6ebd3781026f1c3aa46b3cd67a79c1"
   },
   "sweep": {
    "d": 5,
    "geometricDegree": 6,
    "p": [
     "9/5",
     "-14/5",
     "0",
     "1",
     "-1"
    ],
    "a": "-19/14",
    "b": "1",
    "c": "1",
    "det": "1",
    "secant": [
     "1",
     "-1"
    ]
   }
  },
  {
   "id": "keller-10",
   "n": 3,
   "source": "Gallagher zenodo.21479195 distinct member (w-2w^3), not coordinate-equivalent to Alpöge",
   "statement": "the Jacobian conjecture is FALSE in dimension 3: Gallagher zenodo.21479195 distinct member (w-2w^3) reconstructed from its seed — det J = -1 proved as a polynomial identity, generic fiber degree 4, 2 distinct rational points sharing one image, every claim decided in exact arithmetic",
   "det": "-1",
   "F": [
    [
     [
      0,
      0,
      1,
      "3"
     ],
     [
      0,
      2,
      0,
      "29/6"
     ],
     [
      1,
      1,
      1,
      "8"
     ],
     [
      1,
      3,
      0,
      "22/3"
     ],
     [
      2,
      0,
      2,
      "-3/2"
     ],
     [
      2,
      2,
      1,
      "2"
     ],
     [
      2,
      4,
      0,
      "-3/2"
     ],
     [
      3,
      1,
      2,
      "-6"
     ],
     [
      3,
      3,
      1,
      "-12"
     ],
     [
      3,
      5,
      0,
      "-20/3"
     ],
     [
      4,
      2,
      2,
      "-9"
     ],
     [
      4,
      4,
      1,
      "-13"
     ],
     [
      4,
      6,
      0,
      "-8/3"
     ],
     [
      5,
      3,
      2,
      "-6"
     ],
     [
      5,
      5,
      1,
      "-4"
     ],
     [
      6,
      4,
      2,
      "-3/2"
     ]
    ],
    [
     [
      0,
      1,
      0,
      "1/3"
     ],
     [
      1,
      0,
      1,
      "4"
     ],
     [
      1,
      2,
      0,
      "58/9"
     ],
     [
      2,
      1,
      1,
      "20/3"
     ],
     [
      2,
      3,
      0,
      "10/3"
     ],
     [
      3,
      0,
      2,
      "-2"
     ],
     [
      3,
      2,
      1,
      "-4"
     ],
     [
      3,
      4,
      0,
      "-16/3"
     ],
     [
      4,
      1,
      2,
      "-6"
     ],
     [
      4,
      3,
      1,
      "-12"
     ],
     [
      4,
      5,
      0,
      "-32/9"
     ],
     [
      5,
      2,
      2,
      "-6"
     ],
     [
      5,
      4,
      1,
      "-16/3"
     ],
     [
      6,
      3,
      2,
      "-2"
     ]
    ],
    [
     [
      1,
      0,
      0,
      "1"
     ],
     [
      2,
      1,
      0,
      "-4/3"
     ],
     [
      3,
      0,
      1,
      "-1"
     ]
    ]
   ],
   "collisions": [
    [
     "1",
     "0",
     "0"
    ],
    [
     "-1",
     "0",
     "2"
    ]
   ],
   "image": [
    "0",
    "0",
    "1"
   ],
   "transcription": "the paper's genuinely distinct member: p = w - 2w^3, b = -1, giving a = -4/3 and det J = -1",
   "sourcePin": {
    "file": "gallagher2026.pdf",
    "sha256": "1782eefa569a094b52f90b18967fb28d8f6ebd3781026f1c3aa46b3cd67a79c1"
   },
   "sweep": {
    "d": 3,
    "geometricDegree": 4,
    "p": [
     "1",
     "0",
     "-2"
    ],
    "a": "-4/3",
    "b": "-1",
    "c": "1",
    "det": "-1",
    "secant": [
     "1",
     "-1"
    ]
   }
  }
 ]
}