{
  "status": "PASS",
  "checks": 2875,
  "groups": {
    "average matrix idempotence": 129,
    "average and joint kernel dimension": 129,
    "average columns fixed": 129,
    "all joint kernel vectors retained": 129,
    "C3 fixed monomial count": 19,
    "cyclotomic average selector": 190,
    "C3 invariant normal form": 321,
    "one-sided normal form upper": 176,
    "one-sided normal form lower": 145,
    "formal relation reduction preserves image": 729,
    "invariant module linearity": 144,
    "sixth binomial average": 1,
    "nondiagonal basis-covariant dimensions": 52,
    "Molien versus module rational identity": 1,
    "relation versus module rational identity": 1,
    "four-way all tested Hilbert coefficients": 121,
    "modular norm polynomial invariant": 1,
    "modular generators complete degree piece": 25,
    "modular basis fixed": 169,
    "modular averaging rejected": 1,
    "transfer not projection": 1,
    "modular fourth-degree example": 1,
    "prime-field integer dimension": 57,
    "split prime-field orbit fibres": 195,
    "average is not multiplicative": 1,
    "finite prefix is not series equality": 1,
    "invalid input rejected": 7
  },
  "records": {
    "C3_degree6_basis": [
      [
        "1",
        "0",
        "0",
        "0",
        "0",
        "0",
        "0"
      ],
      [
        "0",
        "0",
        "0",
        "1",
        "0",
        "0",
        "0"
      ],
      [
        "0",
        "0",
        "0",
        "0",
        "0",
        "0",
        "1"
      ]
    ],
    "averaged_sixth": [
      {
        "exponents": [
          0,
          6
        ],
        "coefficient": "1"
      },
      {
        "exponents": [
          3,
          3
        ],
        "coefficient": "20"
      },
      {
        "exponents": [
          6,
          0
        ],
        "coefficient": "1"
      }
    ],
    "rational_order3_degree3_basis": [
      [
        "0",
        "-1",
        "-1",
        "0"
      ],
      [
        "1/3",
        "1",
        "0",
        "-1/3"
      ]
    ],
    "Hilbert_first_19": [
      1,
      0,
      1,
      2,
      1,
      2,
      3,
      2,
      3,
      4,
      3,
      4,
      5,
      4,
      5,
      6,
      5,
      6,
      7
    ],
    "modular_ring": "F2[y, x^2+x*y]; averaging denominator rejected; degree d dimension floor(d/2)+1"
  },
  "scope": "Exact two-variable finite certificates; arbitrary-characteristic finite generation and full presentations are proved in the articles, not inferred from the tested degree ranges."
}