{
  "scope": "Exact finite illustration data; theorem-derived curves are not numerical evidence for universal proofs.",
  "coefficient_definition": "C_N,j(n)=(Nn)![t^(Nn)](j! sum_q t^(Nq)/(Nq+j)!)^(-1)",
  "modular_precision_bits": 256,
  "modular_rows": 255,
  "rational_crosscheck_rows": 12,
  "assertion_counts": {
    "complete_residue_permutation": 4,
    "finite_period_repeat": 296,
    "finite_scaled_isometry": 56,
    "first_row_benchmark": 1,
    "independent_rational_inversion": 52,
    "mixed_counterexample_valuation": 1,
    "mixed_counterexample_value": 1,
    "odd_source_denominator": 1024,
    "plotted_original_metric": 4,
    "rational_modular_crosscheck": 52,
    "sampled_address_two_filtration": 340,
    "sampled_address_two_power_order_exact": 30,
    "sampled_factorial_bound": 864,
    "sampled_first_difference_exact": 288,
    "sampled_homogeneous_equality": 432,
    "sampled_mixed_factorial_bound": 1360,
    "sampled_mixed_parity_rule": 680,
    "stencil_exact_valuation": 1,
    "stencil_offsets": 1,
    "upper_half_counterexample": 3
  },
  "total_assertions": 5490,
  "rational_first_rows": {
    "0": [
      "1",
      "-1",
      "69",
      "-33661",
      "60376809"
    ],
    "1": [
      "1",
      "-1/5",
      "121/45",
      "-16593/65",
      "68164277/765"
    ],
    "2": [
      "1",
      "-1/15",
      "13/45",
      "-4009/455",
      "151129/153"
    ],
    "3": [
      "1",
      "-1/35",
      "59/1155",
      "-407/637",
      "15330979/522291"
    ]
  },
  "mixed_stencil": {
    "steps": [
      1,
      2,
      4
    ],
    "offset_weights": [
      [
        0,
        -1
      ],
      [
        1,
        1
      ],
      [
        2,
        1
      ],
      [
        3,
        -1
      ],
      [
        4,
        1
      ],
      [
        5,
        -1
      ],
      [
        6,
        -1
      ],
      [
        7,
        1
      ]
    ],
    "value": "-60924841253875487872",
    "valuation": 7
  },
  "mixed_order_two": [
    {
      "steps": [
        1,
        1
      ],
      "lower": 3,
      "actual": 3,
      "value": "72"
    },
    {
      "steps": [
        1,
        2
      ],
      "lower": 4,
      "actual": 6,
      "value": "-33728"
    },
    {
      "steps": [
        2,
        2
      ],
      "lower": 5,
      "actual": 5,
      "value": "60376672"
    },
    {
      "steps": [
        1,
        4
      ],
      "lower": 5,
      "actual": 6,
      "value": "-288354427328"
    },
    {
      "steps": [
        2,
        4
      ],
      "lower": 6,
      "actual": 6,
      "value": "3019098102225472"
    },
    {
      "steps": [
        3,
        4
      ],
      "lower": 5,
      "actual": 10,
      "value": "-60921822444127689728"
    }
  ],
  "residue_permutations": [
    {
      "j": 0,
      "modulus": 256,
      "B": 1,
      "period": 128,
      "Euler_residues": [
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        39,
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        81,
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        61,
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        161,
        159,
        229,
        35,
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        11,
        241,
        111,
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        151,
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        63,
        133,
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        121,
        55,
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        251,
        225,
        223,
        37,
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        49,
        175,
        245,
        179,
        25,
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        197,
        3,
        105,
        167,
        237,
        235,
        209,
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        149,
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        185,
        119,
        189,
        59,
        33,
        31,
        101,
        163,
        9,
        71,
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        139,
        113,
        239,
        53,
        243,
        89,
        23,
        93,
        219,
        193,
        191,
        5,
        67,
        169,
        231,
        45,
        43,
        17,
        143,
        213,
        147,
        249,
        183,
        253,
        123,
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        95,
        165,
        227,
        73,
        135,
        205,
        203,
        177,
        47,
        117,
        51,
        153,
        87,
        157,
        27
      ],
      "scaled_residues": [
        0,
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        34,
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        19,
        54,
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        40,
        103,
        10,
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        93,
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        114,
        17,
        68,
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        6,
        5,
        120,
        55,
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        31,
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        42,
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        44,
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        2,
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        22,
        21,
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        23,
        58,
        25,
        76,
        43,
        78,
        13
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    },
    {
      "j": 1,
      "modulus": 256,
      "B": 1,
      "period": 128,
      "Euler_residues": [
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        31,
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        53,
        231,
        17,
        195,
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        127,
        41,
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        85,
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        49,
        227,
        173,
        159,
        73,
        187,
        133,
        183,
        225,
        19,
        221,
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        235,
        181,
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        13,
        255,
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        27,
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        23,
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        115,
        61,
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        89,
        75,
        21,
        199,
        241,
        163,
        109,
        95,
        9,
        123,
        69,
        119,
        161,
        211,
        157,
        15,
        185,
        171,
        117,
        39,
        81,
        3,
        205,
        191,
        105,
        219,
        165,
        215
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      "scaled_residues": [
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        25,
        126,
        55,
        12,
        5,
        106,
        67,
        88,
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        22,
        15,
        100,
        29,
        2,
        27,
        48,
        73,
        46,
        103,
        60,
        53,
        26,
        115,
        8,
        97,
        70,
        63,
        20,
        77,
        50,
        75,
        96,
        121,
        94,
        23,
        108,
        101,
        74,
        35,
        56,
        17,
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        111,
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        14,
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        39,
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        40,
        1,
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        52,
        109,
        82,
        107
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    },
    {
      "j": 2,
      "modulus": 256,
      "B": 5,
      "period": 8,
      "Euler_residues": [
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        34,
        194,
        226,
        130,
        162,
        66,
        98
      ],
      "scaled_residues": [
        0,
        1,
        6,
        7,
        4,
        5,
        2,
        3
      ]
    },
    {
      "j": 3,
      "modulus": 256,
      "B": 3,
      "period": 32,
      "Euler_residues": [
        6,
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        118,
        46,
        230,
        158,
        86,
        14,
        198,
        126,
        54,
        238,
        166,
        94,
        22,
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        174,
        102,
        30,
        214,
        142,
        70,
        254,
        182,
        110,
        38,
        222,
        150,
        78
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      "scaled_residues": [
        0,
        23,
        14,
        5,
        28,
        19,
        10,
        1,
        24,
        15,
        6,
        29,
        20,
        11,
        2,
        25,
        16,
        7,
        30,
        21,
        12,
        3,
        26,
        17,
        8,
        31,
        22,
        13,
        4,
        27,
        18,
        9
      ]
    }
  ],
  "upper_half_counterexamples": [
    {
      "N": 16,
      "j": 9,
      "first_valuation": 5,
      "step_two_valuation": 7,
      "naive_prediction": 6,
      "C0": "1",
      "C1": "-1/2042975",
      "C2": "319118897/2215923048625"
    },
    {
      "N": 32,
      "j": 18,
      "first_valuation": 6,
      "step_two_valuation": 6,
      "naive_prediction": 7,
      "C0": "1",
      "C1": "-1/18053528883775",
      "C2": "187347588441574511/33319533660275387449299375"
    },
    {
      "N": 32,
      "j": 19,
      "first_valuation": 5,
      "step_two_valuation": 7,
      "naive_prediction": 6,
      "C0": "1",
      "C1": "-1/48459472266975",
      "C2": "47098222049108039/60351555833951953390194375"
    }
  ],
  "theorem_derived_unit_step_exponents": {
    "orders": [
      1,
      2,
      3,
      4,
      5,
      6,
      7,
      8,
      9,
      10,
      11,
      12,
      13,
      14,
      15,
      16,
      17,
      18,
      19,
      20,
      21,
      22,
      23,
      24,
      25,
      26,
      27,
      28,
      29,
      30,
      31,
      32
    ],
    "Euler_j01": [
      1,
      3,
      4,
      7,
      8,
      10,
      11,
      15,
      16,
      18,
      19,
      22,
      23,
      25,
      26,
      31,
      32,
      34,
      35,
      38,
      39,
      41,
      42,
      46,
      47,
      49,
      50,
      53,
      54,
      56,
      57,
      63
    ],
    "Euler_j2": [
      5,
      7,
      9,
      14,
      16,
      20,
      22,
      28,
      30,
      34,
      36,
      41,
      43,
      47,
      49,
      56,
      58,
      62,
      64,
      69,
      71,
      75,
      77,
      83,
      85,
      89,
      91,
      96,
      98,
      102,
      104,
      112
    ],
    "Euler_j3": [
      3,
      5,
      6,
      9,
      10,
      12,
      13,
      17,
      18,
      20,
      21,
      24,
      25,
      27,
      28,
      33,
      34,
      36,
      37,
      40,
      41,
      43,
      44,
      48,
      49,
      51,
      52,
      55,
      56,
      58,
      59,
      65
    ]
  },
  "known_exactness": "j0/j1 all homogeneous orders; j2 powers of two; no complete higher-order j3 formula claimed"
}