Exact Sturm certificate passed: certificate_N6.json
Exact Sturm certificate passed: certificate_N11.json
Exact Sturm certificate passed: certificate_N16.json
Independent Bernstein certificate passed: certificate_N6.json
Independent Bernstein certificate passed: certificate_N11.json
Independent Bernstein certificate passed: certificate_N16.json
exact scanned 1048576 exceptions 0 seconds 3.09
exact scanned 2097152 exceptions 1 seconds 6.16
exact scanned 3145728 exceptions 1 seconds 9.21
{
  "M": 22,
  "nonempty_subsets_checked": 4194303,
  "grid_intervals": 4096,
  "grid_points": 4097,
  "arithmetic": "int64 sums of certified rational cosine upper bounds; integer thresholds",
  "grid": {
    "internal_bits": 96,
    "output_bits": 40,
    "max_internal_width": 9621121,
    "max_output_width": 1
  },
  "ratio_upper_winner": "24548327531/40000000000",
  "exception_sets": [
    {
      "A": [
        1,
        2,
        3,
        4,
        5,
        6,
        7,
        8,
        10,
        11,
        13,
        14,
        15,
        16,
        17,
        21
      ],
      "N": 16,
      "lambda_lower_grid_exact": "1349555785747/549755813888"
    }
  ],
  "winning_set": [
    1,
    2,
    3,
    4,
    5,
    6,
    7,
    8,
    10,
    11,
    13,
    14,
    15,
    16,
    17,
    21
  ],
  "conclusion": "Every other nonempty A subset {1,...,22} has lambda(A)/sqrt(|A|) >= ratio_upper_winner, which strictly exceeds the independently certified ratio for the winning set. Thus the displayed winning set is the unique minimizer over this finite universe.",
  "seconds": 12.263951063156128
}
Winner Sturm identities and strict bounds verified independently.
Independent rational Taylor cosine upper bounds constructed.
All 4194303 subsets checked; coarse-grid survivors: 1
{
  "nonempty_subsets_checked": 4194303,
  "coarse_grid_points": 513,
  "full_grid_points": 4097,
  "coarse_survivors": 1,
  "exceptions": [
    {
      "A": [
        1,
        2,
        3,
        4,
        5,
        6,
        7,
        8,
        10,
        11,
        13,
        14,
        15,
        16,
        17,
        21
      ],
      "N": 16,
      "lambda_lower_grid": "1349555785747/549755813888"
    }
  ],
  "winner": [
    1,
    2,
    3,
    4,
    5,
    6,
    7,
    8,
    10,
    11,
    13,
    14,
    15,
    16,
    17,
    21
  ],
  "strict_winner_ratio_upper": "24548327531/40000000000",
  "winner_verification": {
    "lambda_lower": "2454832753/1000000000",
    "lambda_upper": "24548327531/10000000000",
    "sturm_variations_at_minus_one": 10,
    "sturm_variations_at_plus_one": 10,
    "checked_sturm_polynomials": 22,
    "independent_chebyshev_method": "closed factorial coefficient formula"
  },
  "cosine_bounds": {
    "pi_lower": "3141592653589793238462643383271/1000000000000000000000000000000",
    "pi_upper": "39269908169872415480783042291/12500000000000000000000000000",
    "cos_taylor_upper_last_index": 12,
    "cos_taylor_lower_last_index": 13,
    "output_scale": 1099511627776
  },
  "enumeration": "11+11 direct subset-sum table, no Gray-code",
  "conclusion": "Unique finite-universe minimizer certified independently; no claim about unbounded frequencies or the Chowla conjecture.",
  "seconds": 2.1520543098449707
}
All included exact checks passed.
