{
  "certificate": "Dixmier conjecture counterexample witness, Weyl algebra A_3 over Q",
  "date": "2026-07-21",
  "base_map_F": [
    "x**3*y**3*z + 3*x**2*y**4 + 3*x**2*y**2*z + 7*x*y**3 + 3*x*y*z + 4*y**2 + z",
    "3*x**3*y**2*z + 9*x**2*y**3 + 6*x**2*y*z + 12*x*y**2 + 3*x*z + y",
    "-x**3*z - 3*x**2*y + 2*x"
  ],
  "det_J": "-2",
  "collision": {
    "points": [
      [
        "0",
        "0",
        "-1/4"
      ],
      [
        "1",
        "-3/2",
        "13/2"
      ],
      [
        "-1",
        "3/2",
        "13/2"
      ]
    ],
    "image": [
      "-1/4",
      "0",
      "0"
    ]
  },
  "witness_G_rows_are_phi_d_coeffs": [
    [
      "3*x**6*y*z + 9*x**5*y**2 + 3*x**5*z + 3*x**4*y - 4*x**3",
      "3*x**4*y*z + 9*x**3*y**2 + 3*x**3*z + 3*x**2*y - 3*x",
      "-9*x**5*y*z**2 - 45*x**4*y**2*z - 9*x**4*z**2 - 54*x**3*y**3 - 30*x**3*y*z - 27*x**2*y**2 + 9*x**2*z + 21*x*y + 1"
    ],
    [
      "-3*x**6*y**2*z/2 - 9*x**5*y**3/2 - 3*x**5*y*z - 6*x**4*y**2 - 3*x**4*z/2 + x**3*y/2 + 3*x**2/2",
      "-3*x**4*y**2*z/2 - 9*x**3*y**3/2 - 3*x**3*y*z - 6*x**2*y**2 - 3*x**2*z/2 + 1",
      "9*x**5*y**2*z**2/2 + 45*x**4*y**3*z/2 + 9*x**4*y*z**2 + 27*x**3*y**4 + 75*x**3*y**2*z/2 + 9*x**3*z**2/2 + 81*x**2*y**3/2 + 21*x**2*y*z/2 + 3*x*y**2 - 3*x*z - 8*y"
    ],
    [
      "-3*x**6*y**4*z/2 - 9*x**5*y**5/2 - 6*x**5*y**3*z - 15*x**4*y**4 - 9*x**4*y**2*z - 16*x**3*y**3 - 6*x**3*y*z - 9*x**2*y**2/2 - 3*x**2*z/2 + 3*x*y/2 + 1/2",
      "-3*x**4*y**4*z/2 - 9*x**3*y**5/2 - 6*x**3*y**3*z - 15*x**2*y**4 - 9*x**2*y**2*z - 33*x*y**3/2 - 6*x*y*z - 6*y**2 - 3*z/2",
      "9*x**5*y**4*z**2/2 + 45*x**4*y**5*z/2 + 18*x**4*y**3*z**2 + 27*x**3*y**6 + 165*x**3*y**4*z/2 + 27*x**3*y**2*z**2 + 189*x**2*y**5/2 + 108*x**2*y**3*z + 18*x**2*y*z**2 + 111*x*y**4 + 117*x*y**2*z/2 + 9*x*z**2/2 + 89*y**3/2 + 21*y*z/2"
    ]
  ],
  "checks": {
    "keller_detJ_equals_minus2": true,
    "three_point_collision": true,
    "G_polynomial": true,
    "R1_GJt_identity_full_expansion": true,
    "R3_27_commutation_identities_full_expansion": true,
    "pathB_random_rational_spotchecks": true
  },
  "non_automorphism_argument": "A_3 is simple, so phi is injective. By the classical constructive proof of DC_n => JC_n (van den Essen, 'Polynomial Automorphisms and the Jacobian Conjecture', ch.10; Belov-Kanel & Kontsevich), if phi were an automorphism of A_3 then F would be a polynomial automorphism of C^3. F is not injective (verified collision), hence not an automorphism. Therefore phi is an injective non-surjective endomorphism of A_3, contradicting the Dixmier conjecture for A_3.",
  "verified_by": "sympy exact rational arithmetic, dual-path (full expansion + random rational evaluation)",
  "elapsed_sec": 1.02,
  "ALL_CHECKS_PASS": true
}