{
  "checks": [
    {
      "label": "fixed-Pauli-matrix-relation",
      "passed": true
    },
    {
      "label": "free-polynomial-nonzero-does-not-refute-Pauli-relation",
      "passed": true
    },
    {
      "label": "large-source-dimension-refused-without-computation",
      "passed": true
    },
    {
      "label": "depth-cap",
      "passed": true
    },
    {
      "label": "gates-cap",
      "passed": true
    },
    {
      "label": "scalar-bits",
      "passed": true
    },
    {
      "label": "boolean-coefficients",
      "passed": true
    },
    {
      "label": "boolean-budget",
      "passed": true
    }
  ],
  "passed": 8,
  "model_example": {
    "formula": {
      "variables": 1,
      "formula": {
        "add": [
          {
            "mul": [
              {
                "var": 1
              },
              {
                "var": 1
              }
            ]
          },
          {
            "scalar": "-1"
          }
        ]
      },
      "model": {
        "field": "rational_characteristic_zero",
        "algebra": "free_associative",
        "gates": "division_free_tree"
      }
    },
    "pauli_matrix": [
      [
        "0",
        "1"
      ],
      [
        "1",
        "0"
      ]
    ],
    "pauli_square_minus_identity": [
      [
        "0",
        "0"
      ],
      [
        "0",
        "0"
      ]
    ],
    "free_algebra_result": {
      "variables": 1,
      "gates": 5,
      "source_dimension": 50,
      "requested_dimension": 50,
      "completed_dimension": 50,
      "requested_full_source_dimension": true,
      "model": {
        "field": "rational_characteristic_zero",
        "algebra": "free_associative",
        "gates": "division_free_tree"
      },
      "independent_identity_conclusion": "not_an_identity_in_the_declared_free_rational_algebra",
      "source_universal_hitting_acceptance": "not_independently_verified",
      "implementation": "Visible formula tree; structured rational matrix action, not an opaque black-box oracle",
      "budget_scope": "Counts selected rational add/multiply/divide calls only; excludes parsing, negation, bit arithmetic, allocation and wall time",
      "status": "nonzero_polynomial_witness",
      "witness": {
        "row": 0,
        "column": 0,
        "value": "-1",
        "dimension": 50,
        "generator": "[1] for n=s=1; otherwise entry (-1)^(r-q-1)/(r*i^(r-q)) for q<r, zero otherwise"
      },
      "arithmetic_charges": 394
    }
  },
  "scope": "Separate fixed-matrix relation versus free identity, plus rejected resource/schema boundaries."
}