{
  "status": "PASS",
  "checks": 49583,
  "categories": {
    "Rational zero divisor to norm witness": 1,
    "Target matrix exact inverse coordinates": 1,
    "Target matrix round trip": 1,
    "Complete split certificate": 1,
    "Joint kernel recovers rational automorphism": 1,
    "Complete rational conjugator family": 49,
    "Family norm identity": 49,
    "Rational basis associativity": 320,
    "Rational conjugation reverses product": 405,
    "Rational norm multiplication": 405,
    "Four-dimensional left determinant": 405,
    "Four-dimensional left trace": 405,
    "Reduced quadratic identity": 405,
    "Both inverse directions": 352,
    "Nonzero two-sided annihilator": 48,
    "Two singular basis vectors": 1,
    "Three-point grid succeeds despite bad basis": 1,
    "Noncentral repeated-root conjugacy": 1,
    "Scalar and nilpotent branch rejected": 1,
    "Zero denominator branch rejected": 1,
    "Square-parameter fallback gives affine witness": 1,
    "Finite-field split certificate": 20,
    "Finite basis associativity": 1280,
    "All finite elements matrix round trip": 10324,
    "All finite matrix determinants equal reduced norm": 10324,
    "Finite matrix multiplication matches quaternion multiplication": 10324,
    "Class size by discriminant type": 436,
    "Independent full unit orbit equals invariant class": 436,
    "Orbit centralizer exact count": 436,
    "Centralizer size by quadratic algebra type": 436,
    "All finite noncentral targets certified": 10232,
    "Full finite intertwiner space enumerated independently": 436,
    "All finite inner maps recover unique scalar class": 2016,
    "Finite automorphism scalar-fiber count": 20,
    "False norm witness rejected": 1,
    "Wrong matrix sign rejected": 1,
    "Altered conjugator rejected": 1,
    "Dependent kernel basis rejected": 1,
    "Nonzero singular intertwiner rejected": 1,
    "False automorphism images rejected": 1,
    "Zero parameter rejected": 1,
    "Characteristic two rejected": 1,
    "Composite coefficient modulus rejected": 1
  },
  "rational_capstone": {
    "a": 2,
    "b": -1,
    "zero_divisor": [
      "1",
      "1",
      "1",
      "0"
    ],
    "annihilator": [
      "1",
      "-1",
      "-1",
      "0"
    ],
    "invertible_element": [
      "1",
      "1",
      "1",
      "1"
    ],
    "inverse": [
      "-1/2",
      "1/2",
      "1/2",
      "1/2"
    ],
    "norm_witness": [
      "1",
      "1"
    ],
    "zero_divisor_matrix": [
      [
        "2",
        "0"
      ],
      [
        "2",
        "0"
      ]
    ],
    "target_matrix": [
      [
        "3",
        "4"
      ],
      [
        "5",
        "6"
      ]
    ],
    "target_inverse_coordinates": [
      "9/2",
      "7/2",
      "9/2",
      "-3"
    ],
    "split_certificate": {
      "u": "1",
      "v": "1",
      "images": [
        [
          [
            "1",
            "0"
          ],
          [
            "0",
            "1"
          ]
        ],
        [
          [
            "0",
            "2"
          ],
          [
            "1",
            "0"
          ]
        ],
        [
          [
            "1",
            "-2"
          ],
          [
            "1",
            "-1"
          ]
        ]
      ]
    },
    "alpha": [
      "0",
      "1",
      "0",
      "0"
    ],
    "beta": [
      "0",
      "0",
      "0",
      "-1"
    ],
    "conjugacy_certificate": {
      "basis": [
        [
          "1",
          "0",
          "1",
          "0"
        ],
        [
          "0",
          "-1",
          "0",
          "1"
        ]
      ],
      "weights": [
        "0",
        "1"
      ],
      "conjugator": [
        "0",
        "-1",
        "0",
        "1"
      ]
    },
    "joint_automorphism_conjugator": [
      "1",
      "0",
      "1",
      "0"
    ]
  },
  "singular_basis_grid_certificate": {
    "basis": [
      [
        "1",
        "1",
        "0",
        "0"
      ],
      [
        "1",
        "-1",
        "0",
        "0"
      ]
    ],
    "weights": [
      "1",
      "1"
    ],
    "conjugator": [
      "2",
      "0",
      "0",
      "0"
    ]
  },
  "nilpotent_matrix_conjugacy": {
    "alpha": [
      "0",
      "1/2",
      "0",
      "-1/2"
    ],
    "beta": [
      "0",
      "1/2",
      "0",
      "1/2"
    ],
    "certificate": {
      "basis": [
        [
          "0",
          "1",
          "0",
          "0"
        ],
        [
          "-1",
          "0",
          "1",
          "0"
        ]
      ],
      "weights": [
        "1",
        "0"
      ],
      "conjugator": [
        "0",
        "1",
        "0",
        "0"
      ]
    }
  },
  "finite_models": [
    {
      "p": 3,
      "a": 1,
      "b": 1,
      "norm_witness": [
        1,
        0
      ],
      "elements": 81,
      "units": 48,
      "noncentral_classes": 9,
      "automorphisms_recovered": 24
    },
    {
      "p": 3,
      "a": 1,
      "b": 2,
      "norm_witness": [
        0,
        1
      ],
      "elements": 81,
      "units": 48,
      "noncentral_classes": 9,
      "automorphisms_recovered": 24
    },
    {
      "p": 3,
      "a": 2,
      "b": 1,
      "norm_witness": [
        0,
        1
      ],
      "elements": 81,
      "units": 48,
      "noncentral_classes": 9,
      "automorphisms_recovered": 24
    },
    {
      "p": 3,
      "a": 2,
      "b": 2,
      "norm_witness": [
        1,
        1
      ],
      "elements": 81,
      "units": 48,
      "noncentral_classes": 9,
      "automorphisms_recovered": 24
    },
    {
      "p": 5,
      "a": 1,
      "b": 1,
      "norm_witness": [
        0,
        2
      ],
      "elements": 625,
      "units": 480,
      "noncentral_classes": 25,
      "automorphisms_recovered": 120
    },
    {
      "p": 5,
      "a": 1,
      "b": 2,
      "norm_witness": [
        1,
        2
      ],
      "elements": 625,
      "units": 480,
      "noncentral_classes": 25,
      "automorphisms_recovered": 120
    },
    {
      "p": 5,
      "a": 1,
      "b": 3,
      "norm_witness": [
        2,
        1
      ],
      "elements": 625,
      "units": 480,
      "noncentral_classes": 25,
      "automorphisms_recovered": 120
    },
    {
      "p": 5,
      "a": 1,
      "b": 4,
      "norm_witness": [
        0,
        1
      ],
      "elements": 625,
      "units": 480,
      "noncentral_classes": 25,
      "automorphisms_recovered": 120
    },
    {
      "p": 5,
      "a": 2,
      "b": 1,
      "norm_witness": [
        1,
        0
      ],
      "elements": 625,
      "units": 480,
      "noncentral_classes": 25,
      "automorphisms_recovered": 120
    },
    {
      "p": 5,
      "a": 2,
      "b": 2,
      "norm_witness": [
        0,
        2
      ],
      "elements": 625,
      "units": 480,
      "noncentral_classes": 25,
      "automorphisms_recovered": 120
    },
    {
      "p": 5,
      "a": 2,
      "b": 3,
      "norm_witness": [
        0,
        1
      ],
      "elements": 625,
      "units": 480,
      "noncentral_classes": 25,
      "automorphisms_recovered": 120
    },
    {
      "p": 5,
      "a": 2,
      "b": 4,
      "norm_witness": [
        1,
        1
      ],
      "elements": 625,
      "units": 480,
      "noncentral_classes": 25,
      "automorphisms_recovered": 120
    },
    {
      "p": 5,
      "a": 3,
      "b": 1,
      "norm_witness": [
        1,
        0
      ],
      "elements": 625,
      "units": 480,
      "noncentral_classes": 25,
      "automorphisms_recovered": 120
    },
    {
      "p": 5,
      "a": 3,
      "b": 2,
      "norm_witness": [
        0,
        1
      ],
      "elements": 625,
      "units": 480,
      "noncentral_classes": 25,
      "automorphisms_recovered": 120
    },
    {
      "p": 5,
      "a": 3,
      "b": 3,
      "norm_witness": [
        0,
        2
      ],
      "elements": 625,
      "units": 480,
      "noncentral_classes": 25,
      "automorphisms_recovered": 120
    },
    {
      "p": 5,
      "a": 3,
      "b": 4,
      "norm_witness": [
        1,
        2
      ],
      "elements": 625,
      "units": 480,
      "noncentral_classes": 25,
      "automorphisms_recovered": 120
    },
    {
      "p": 5,
      "a": 4,
      "b": 1,
      "norm_witness": [
        0,
        1
      ],
      "elements": 625,
      "units": 480,
      "noncentral_classes": 25,
      "automorphisms_recovered": 120
    },
    {
      "p": 5,
      "a": 4,
      "b": 2,
      "norm_witness": [
        1,
        1
      ],
      "elements": 625,
      "units": 480,
      "noncentral_classes": 25,
      "automorphisms_recovered": 120
    },
    {
      "p": 5,
      "a": 4,
      "b": 3,
      "norm_witness": [
        2,
        2
      ],
      "elements": 625,
      "units": 480,
      "noncentral_classes": 25,
      "automorphisms_recovered": 120
    },
    {
      "p": 5,
      "a": 4,
      "b": 4,
      "norm_witness": [
        0,
        2
      ],
      "elements": 625,
      "units": 480,
      "noncentral_classes": 25,
      "automorphisms_recovered": 120
    }
  ],
  "scope": "Exact finite algebra and complete F3/F5 element, unit-orbit and intertwiner checks; not a proof of arbitrary-field theorems or a decision algorithm for rational conic solvability."
}