{
  "status": "pass",
  "degree": 12,
  "counts": {
    "all_indices": 1820,
    "even_indices": 1036,
    "odd_indices": 784,
    "C_identity_comparisons": 1820,
    "MGF_comparisons": 1036,
    "interval_integral_comparisons": 1036,
    "lower_bound_comparisons": 1036,
    "axis_equality_comparisons": 25
  },
  "failures": 0,
  "arithmetic": "Python integers and fractions.Fraction only",
  "examples": [
    {
      "alpha": [
        0,
        0,
        0,
        0
      ],
      "C": "1",
      "G": "1",
      "HARDY": "1/2",
      "magnitude_lower_bound": "1/2"
    },
    {
      "alpha": [
        1,
        1,
        0,
        0
      ],
      "C": "0",
      "G": "2",
      "HARDY": "1/120",
      "magnitude_lower_bound": "1/120"
    },
    {
      "alpha": [
        1,
        1,
        1,
        1
      ],
      "C": "48",
      "G": "48",
      "HARDY": "1/1120",
      "magnitude_lower_bound": "1/1120"
    },
    {
      "alpha": [
        2,
        2,
        0,
        0
      ],
      "C": "8",
      "G": "20",
      "HARDY": "1/672",
      "magnitude_lower_bound": "1/1120"
    },
    {
      "alpha": [
        3,
        1,
        0,
        0
      ],
      "C": "0",
      "G": "8",
      "HARDY": "-1/1120",
      "magnitude_lower_bound": "1/1120"
    },
    {
      "alpha": [
        2,
        2,
        2,
        2
      ],
      "C": "3840",
      "G": "6120",
      "HARDY": "17/1774080",
      "magnitude_lower_bound": "1/1774080"
    }
  ],
  "sharpness_witness": {
    "lambda": "5",
    "coefficient_t1_squared": "-1",
    "method": "Direct rational expansion of (1-Q)^5 R"
  },
  "source_sha256": {
    "research/millennium/iterations/0029/check_positive_recurrence.py": "4256ab672ec481028fbb10acf7f49db6c73bfa1abe6af9ae474b47cd64efbb1c",
    "research/millennium/iterations/0024/check_hardy_sign.py": "d78f959a25999f4cb0e990dd2027261c8ec6c557ac26278595b13af4369efe44",
    "research/millennium/iterations/0029/sharp-factorization.md": "bac6ab41c3eaff2fff9e75b7ccaddabc141d121f05c46389d25145005c3287cc"
  },
  "scope": "Finite tests of new recurrence, rational identity, magnitude bounds and boundary cases. The interval formula is inherited from cycle24, not the literal source-integral checker. All-orders validity requires analytic review; no full Lean proof or originality claim."
}