{
  "purpose": "targeted examination of the single specified prefix-floor rounding construction",
  "rounding_samples": [
    {
      "M": 255,
      "N": 127,
      "grid": 4080,
      "j": 1912,
      "theta_over_2pi": "1912/4080",
      "sampled_value": -22.00691085959472,
      "sampled_negative_value_over_sqrt_N": 1.952797560340154,
      "value_at_pi_exact": 7,
      "bits_sha256": "45cd50ff4e098127a67ee7a8020a78d524e27c9220b5b9e6c2f0ec3833ca111c",
      "status": "floating point sampled value; not a global minimum certificate"
    },
    {
      "M": 256,
      "N": 127,
      "grid": 4096,
      "j": 1664,
      "theta_over_2pi": "1664/4096",
      "sampled_value": -13.576649040254768,
      "sampled_negative_value_over_sqrt_N": 1.2047327901928102,
      "value_at_pi_exact": 3,
      "bits_sha256": "acbd81a85f72d67d31327908849c9a0d8426432f1519f5696dfba2bf928d4484",
      "status": "floating point sampled value; not a global minimum certificate"
    },
    {
      "M": 257,
      "N": 128,
      "grid": 4112,
      "j": 1992,
      "theta_over_2pi": "1992/4112",
      "sampled_value": -19.335906227782765,
      "sampled_negative_value_over_sqrt_N": 1.7090688017565485,
      "value_at_pi_exact": -6,
      "bits_sha256": "858f01edcf039ae7be8084d2b6b643cf60b2af92cc70a3bba3b4bba037d634b1",
      "status": "floating point sampled value; not a global minimum certificate"
    },
    {
      "M": 1024,
      "N": 511,
      "grid": 16384,
      "j": 7667,
      "theta_over_2pi": "7667/16384",
      "sampled_value": -34.1709761236897,
      "sampled_negative_value_over_sqrt_N": 1.5116349861011529,
      "value_at_pi_exact": -5,
      "bits_sha256": "a78b7dbaad3253e3249fad7f6de1eecb97643435f6f9ce6eb672ff7ba14b8e13",
      "status": "floating point sampled value; not a global minimum certificate"
    },
    {
      "M": 4096,
      "N": 2047,
      "grid": 65536,
      "j": 26672,
      "theta_over_2pi": "26672/65536",
      "sampled_value": -64.27028113823496,
      "sampled_negative_value_over_sqrt_N": 1.4205328402963484,
      "value_at_pi_exact": -17,
      "bits_sha256": "31498f888741c6128dec9069cd1c1ab35b656148875b78de7a297e422f3726f9",
      "status": "floating point sampled value; not a global minimum certificate"
    },
    {
      "M": 16384,
      "N": 8191,
      "grid": 262144,
      "j": 111392,
      "theta_over_2pi": "111392/262144",
      "sampled_value": -147.7009954085504,
      "sampled_negative_value_over_sqrt_N": 1.6319804774951319,
      "value_at_pi_exact": 43,
      "bits_sha256": "df45deca32d3af1a0422b6ba327e46da713a4e0d75159eb32297a39394b54e3d",
      "status": "floating point sampled value; not a global minimum certificate"
    },
    {
      "M": 65536,
      "N": 32767,
      "grid": 1048576,
      "j": 413070,
      "theta_over_2pi": "413070/1048576",
      "sampled_value": -320.97726525949724,
      "sampled_negative_value_over_sqrt_N": 1.7731926887898208,
      "value_at_pi_exact": -137,
      "bits_sha256": "e1fc7317becb7f1cd52def3fe04eed5cbc54aa7459b052768bef86c572d0be76",
      "status": "floating point sampled value; not a global minimum certificate"
    },
    {
      "M": 262144,
      "N": 131071,
      "grid": 4194304,
      "j": 2023809,
      "theta_over_2pi": "2023809/4194304",
      "sampled_value": -698.2347527467812,
      "sampled_negative_value_over_sqrt_N": 1.92862660920069,
      "value_at_pi_exact": 3,
      "bits_sha256": "7227a186e0c84fb965837050aa69f2ab598ffd7c43c0d5e9a7c67983a8ee1e42",
      "status": "floating point sampled value; not a global minimum certificate"
    }
  ],
  "rational_value_certificate": {
    "M": 262144,
    "N": 131071,
    "j": 2023809,
    "grid": 4194304,
    "Q": "1267650600228229401496703205376",
    "C": "-1260007141646021631229272260010",
    "S": "138996572835493526588070818263",
    "fixed_point_sum": "-885117703419666298886051080502254",
    "sum_frequencies": 11453246122,
    "error_numerator": "68719476732",
    "value_lower": [
      "-442558851709833149443059899989493",
      "633825300114114700748351602688"
    ],
    "value_upper": [
      "-442558851709833149442991180512761",
      "633825300114114700748351602688"
    ],
    "certified_K_over_sqrt_N_greater_than": "241/125",
    "value_upper_decimal_for_display_only": -698.234752746781,
    "proof": "Machin/Taylor rational enclosure; fixed-point Euclidean error <=6a/Q; exact final squared rational comparison",
    "status": "certifies one value, not the continuous minimum and not an asymptotic claim"
  },
  "elementary_checks": {
    "A_1_2_third_moment_exact": "3/4",
    "projected_Fejer4_at_2pi_over_3_exact": "-1/4",
    "A_1_2_exact_negative_minimum": "9/8",
    "A_1_2_uniform_reversal_error_at_least": "7/8"
  }
}