{
  "status": "PASS",
  "checks": 4985,
  "groups": {
    "terminal boundary Laurent coefficients": 1,
    "terminal boundary canonical factor": 1,
    "root reflection preserves spectrum but not canonicality": 1,
    "terminal complex factor coefficients": 1,
    "strict factor roots outside circle": 1,
    "terminal exact spectral readings": 1,
    "legal finite prefix": 1,
    "zero-extension and full-polynomial rejection": 1,
    "terminal unique complex moment recovery": 1,
    "forced next moment sign": 1,
    "actual outward perturbation negative direction": 3,
    "terminal maximal prescribed atom": 2,
    "terminal second extremal node": 1,
    "all root families yield verified nonnegative spectra": 81,
    "coefficient identity at rational circle diagnostics": 486,
    "exact Parseval constant coefficient": 81,
    "nonzero polynomial square has all finite Toeplitz blocks PD": 243,
    "generated atoms exact Toeplitz rank": 840,
    "full moment witness verified": 840,
    "singular reconstruction and unique weights": 420,
    "all recovered spectral lines annihilated": 420,
    "rotation with fixed negative exponent moment convention": 840,
    "exact full complex extension disk": 216,
    "extension boundary rank": 162,
    "negative PSD witness reproduced": 54,
    "all rational prescribed nodes attain bound": 135,
    "increased prescribed mass rejected": 135,
    "zero moment table has zero measure": 5,
    "zero Laurent polynomial separated": 1,
    "invalid or false certificate rejected": 10
  },
  "boundary_factor": [
    {
      "re": "1",
      "im": "0"
    },
    {
      "re": "-3/2",
      "im": "0"
    },
    {
      "re": "1/2",
      "im": "0"
    }
  ],
  "boundary_coefficients": [
    {
      "re": "7/2",
      "im": "0"
    },
    {
      "re": "-9/4",
      "im": "0"
    },
    {
      "re": "1/2",
      "im": "0"
    }
  ],
  "strict_factor": [
    {
      "re": "1",
      "im": "0"
    },
    {
      "re": "-1/2",
      "im": "1/3"
    },
    {
      "re": "0",
      "im": "-1/6"
    }
  ],
  "strict_coefficients": [
    {
      "re": "25/18",
      "im": "0"
    },
    {
      "re": "-5/9",
      "im": "5/12"
    },
    {
      "re": "0",
      "im": "-1/6"
    }
  ],
  "unique_moments": [
    {
      "re": "1",
      "im": "0"
    },
    {
      "re": "2/3",
      "im": "-1/3"
    },
    {
      "re": "1/3",
      "im": "0"
    }
  ],
  "unique_recovery": {
    "nodes": [
      {
        "re": "1",
        "im": "0"
      },
      {
        "re": "0",
        "im": "1"
      }
    ],
    "weights": [
      "2/3",
      "1/3"
    ],
    "kernel": [
      {
        "re": "0",
        "im": "1"
      },
      {
        "re": "-1",
        "im": "-1"
      },
      {
        "re": "1",
        "im": "0"
      }
    ]
  },
  "prescribed_atom_extremizers": [
    {
      "node": {
        "re": "1",
        "im": "0"
      },
      "weight": "7/8",
      "residual_moments": [
        {
          "re": "1/8",
          "im": "0"
        },
        {
          "re": "-1/8",
          "im": "0"
        }
      ],
      "kernel": [
        {
          "re": "1",
          "im": "0"
        },
        {
          "re": "1",
          "im": "0"
        }
      ],
      "remainder": {
        "nodes": [
          {
            "re": "-1",
            "im": "0"
          }
        ],
        "weights": [
          "1/8"
        ],
        "kernel": [
          {
            "re": "1",
            "im": "0"
          },
          {
            "re": "1",
            "im": "0"
          }
        ]
      }
    },
    {
      "node": {
        "re": "0",
        "im": "1"
      },
      "weight": "7/32",
      "residual_moments": [
        {
          "re": "25/32",
          "im": "0"
        },
        {
          "re": "3/4",
          "im": "7/32"
        }
      ],
      "kernel": [
        {
          "re": "-24/25",
          "im": "7/25"
        },
        {
          "re": "1",
          "im": "0"
        }
      ],
      "remainder": {
        "nodes": [
          {
            "re": "24/25",
            "im": "-7/25"
          }
        ],
        "weights": [
          "25/32"
        ],
        "kernel": [
          {
            "re": "-24/25",
            "im": "7/25"
          },
          {
            "re": "1",
            "im": "0"
          }
        ]
      }
    }
  ],
  "scope": "Exact supplied-factor and finite-moment certificates; general algebraic root isolation, noisy rank decisions and entropy selection are outside this checker"
}