{
  "checks": {
    "A_keller_det": {
      "det": "-1/2",
      "map": [
        "x**3*y**3*z + 3*x**2*y**4/2 + 3*x**2*y**2*z + 7*x*y**3/2 + 3*x*y*z + 2*y**2 + z",
        "3*x**3*y**2*z + 9*x**2*y**3/2 + 6*x**2*y*z + 6*x*y**2 + 3*x*z + y/2",
        "-x**3*z - 3*x**2*y/2 + x"
      ],
      "pass": true
    },
    "B_torus_equivariance": {
      "pass": true,
      "weights_source": [
        1,
        -1,
        -2
      ],
      "weights_target": [
        -2,
        -1,
        1
      ]
    },
    "C_insert_structure": {
      "C0_const": "-c/(108*a**2*c**2 - 36*a*b*c + 8*a + 4*b**3*c - b**2)",
      "C1_coeff_of_x": "(-3*b*c + 1)/(108*a**2*c**2 - 36*a*b*c + 8*a + 4*b**3*c - b**2)",
      "D": "108*a**2*c**2 - 36*a*b*c + 8*a + 4*b**3*c - b**2",
      "fiber_cubic_depressed": true,
      "meaning": "3 preimage x-coords = roots of depressed cubic attached to image => Sym^3 insert map",
      "pass": true,
      "x2_coeff": "0"
    },
    "D_rational_collision": {
      "image": [
        "15/2",
        "8",
        "0"
      ],
      "pass": true,
      "preimages": [
        [
          "1/2",
          "1",
          "1"
        ],
        [
          "-1/2",
          "7",
          "25"
        ]
      ]
    },
    "E_alpoge_same_class": {
      "alpoge_det": "-2",
      "alpoge_torus_weights_match": true,
      "conclusion": "generic-degree-3, torus-(1,-1,-2)-equivariant, non-injective => same graded-gauge orbit as Phi3 (Report II Thm 6): Alpoge IS the affine Sym^3 insert map, up to gauge.",
      "pass": true
    },
    "F_dim2_firewall": {
      "detail": {
        "k1": {
          "coeff_(k+1)/(d-k)": "zoo",
          "is_finite": false,
          "member_constructs": false
        },
        "k2": {
          "coeff_(k+1)/(d-k)": "zoo",
          "is_finite": false,
          "member_constructs": false
        },
        "k3": {
          "coeff_(k+1)/(d-k)": "zoo",
          "is_finite": false,
          "member_constructs": false
        }
      },
      "meaning": "n=2 requires d=k; coefficient (k+1)/(d-k) = zoo (pole). construct(k,k) fails. No dimension-two / generic-degree-2 member exists. Consistent with JC_2 remaining open.",
      "pass": true
    },
    "G_k1_row": {
      "members": [
        {
          "det": "-1/2",
          "is_true_symN_insert_map": true,
          "n_generic_degree": 3,
          "torus_equivariant": true
        },
        {
          "det": "-1/2",
          "is_true_symN_insert_map": false,
          "n_generic_degree": 4,
          "torus_equivariant": true
        },
        {
          "det": "-1/2",
          "is_true_symN_insert_map": false,
          "n_generic_degree": 5,
          "torus_equivariant": true
        },
        {
          "det": "-1/2",
          "is_true_symN_insert_map": false,
          "n_generic_degree": 6,
          "torus_equivariant": true
        },
        {
          "det": "-1/2",
          "is_true_symN_insert_map": false,
          "n_generic_degree": 7,
          "torus_equivariant": true
        }
      ],
      "pass": true,
      "scope_note": "det -1/2 and (1,-1,-2)-equivariant for all n=d+1; but these are maps A^3->A^3. Only n=3 (dim=degree=3) is the genuine Sym^3 insert map; d>2 are dimension-3 power-weighted lifts (Report III), NOT Sym^{d+1} maps."
    }
  },
  "engine": "SymPy",
  "exact_arithmetic": true,
  "sympy_version": "1.14.0"
}