{
  "arithmetic": "integers and fractions.Fraction; no floating-point certificate",
  "date_utc": "2026-10-18",
  "84": {
    "status": "passed",
    "exact_cases": [
      {
        "A": [
          1
        ],
        "S2": 1
      },
      {
        "A": [
          1,
          2
        ],
        "S2": 5
      },
      {
        "A": [
          2,
          3
        ],
        "S2": 13
      },
      {
        "A": [
          1,
          4,
          16
        ],
        "S2": 273
      },
      {
        "A": [
          1,
          2,
          3,
          5,
          7,
          11
        ],
        "S2": 209
      },
      {
        "A": [
          1
        ],
        "S2": 1
      },
      {
        "A": [
          1,
          3
        ],
        "S2": 10
      },
      {
        "A": [
          1,
          3,
          5
        ],
        "S2": 35
      },
      {
        "A": [
          1,
          3,
          5,
          7
        ],
        "S2": 84
      },
      {
        "A": [
          1,
          3,
          5,
          7,
          9
        ],
        "S2": 165
      },
      {
        "A": [
          1,
          3,
          5,
          7,
          9,
          11
        ],
        "S2": 286
      },
      {
        "A": [
          1,
          3,
          5,
          7,
          9,
          11,
          13
        ],
        "S2": 455
      },
      {
        "A": [
          1,
          3,
          5,
          7,
          9,
          11,
          13,
          15
        ],
        "S2": 680
      },
      {
        "A": [
          1,
          3,
          5,
          7,
          9,
          11,
          13,
          15,
          17
        ],
        "S2": 969
      },
      {
        "A": [
          1,
          3,
          5,
          7,
          9,
          11,
          13,
          15,
          17,
          19
        ],
        "S2": 1330
      },
      {
        "A": [
          1,
          3,
          5,
          7,
          9,
          11,
          13,
          15,
          17,
          19,
          21
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        "S2": 1771
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      {
        "A": [
          1,
          3,
          5,
          7,
          9,
          11,
          13,
          15,
          17,
          19,
          21,
          23
        ],
        "S2": 2300
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    ],
    "general_certificate": "mu{f <= -K/2} >= sqrt(K/S2)/pi; analytic Taylor proof",
    "dilation_note": "local width shrinks; full level-set measure is invariant"
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        "A": [
          1,
          2
        ],
        "S2": 5,
        "epsilon": "1/7",
        "tau": "1/35"
      },
      {
        "A": [
          2,
          3
        ],
        "S2": 13,
        "epsilon": "1/7",
        "tau": "1/91"
      },
      {
        "A": [
          3,
          4
        ],
        "S2": 25,
        "epsilon": "1/7",
        "tau": "1/175"
      },
      {
        "A": [
          7,
          8
        ],
        "S2": 113,
        "epsilon": "1/7",
        "tau": "1/791"
      },
      {
        "A": [
          31,
          32
        ],
        "S2": 1985,
        "epsilon": "1/7",
        "tau": "1/13895"
      },
      {
        "A": [
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          128
        ],
        "S2": 32513,
        "epsilon": "1/7",
        "tau": "1/227591"
      },
      {
        "A": [
          1024,
          1025
        ],
        "S2": 2099201,
        "epsilon": "1/7",
        "tau": "1/14694407"
      }
    ],
    "gaussian_characteristic_series_coefficients": [
      "1",
      "-18/7",
      "162/49",
      "-972/343",
      "4374/2401",
      "-78732/84035",
      "236196/588245",
      "-4251528/28824005",
      "9565938/201768035",
      "-19131876/1412376245",
      "172186884/49433168575",
      "-3099363912/3806353980275",
      "4649045868/26644477861925"
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    "global_claim": "0 <= K-K_tau <= tau*S2; analytic proof, no exponential rounding"
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    "status": "passed",
    "identical_intensity": {
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      "-1": 1,
      "1": 1
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    "homodyne_difference": "4*R*Re(S), exact multivariate expansion",
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    ],
    "K_2_3_interval": [
      "4412277660168379/2700000000000000",
      "220613883008419/135000000000000"
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    "status": "passed",
    "exhaustive_domain": "nonempty subsets of {1,...,18}",
    "sets_checked": 262143,
    "independent_autocorrelations_checked": 4095,
    "nonpositive_B_count": 0,
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        2,
        3,
        4,
        5,
        7,
        9,
        11,
        13
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      "B": "21/20",
      "Z": "5/128",
      "ratio_squared": "49/400",
      "N": 9
    },
    "minimum_normalized_B": "7/20",
    "minimum_by_size": [
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        "A": [
          1
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        "B": "1/2",
        "Z": "1",
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        "N": 1
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      {
        "A": [
          1,
          2
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        "B": "3/4",
        "Z": "1",
        "ratio_squared": "9/32",
        "N": 2
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      {
        "A": [
          1,
          2,
          3
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        "B": "5/6",
        "Z": "3/4",
        "ratio_squared": "25/108",
        "N": 3
      },
      {
        "A": [
          1,
          2,
          3,
          5
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        "B": "7/8",
        "Z": "1/2",
        "ratio_squared": "49/256",
        "N": 4
      },
      {
        "A": [
          1,
          2,
          3,
          5,
          7
        ],
        "B": "9/10",
        "Z": "5/16",
        "ratio_squared": "81/500",
        "N": 5
      },
      {
        "A": [
          1,
          2,
          3,
          4,
          5,
          7
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        "B": "11/12",
        "Z": "3/16",
        "ratio_squared": "121/864",
        "N": 6
      },
      {
        "A": [
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          3,
          4,
          5,
          7,
          8
        ],
        "B": "11/10",
        "Z": "5/32",
        "ratio_squared": "121/700",
        "N": 7
      },
      {
        "A": [
          1,
          2,
          3,
          4,
          5,
          7,
          9,
          11
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        "B": "21/20",
        "Z": "5/64",
        "ratio_squared": "441/3200",
        "N": 8
      },
      {
        "A": [
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          2,
          3,
          4,
          5,
          7,
          9,
          11,
          13
        ],
        "B": "21/20",
        "Z": "5/128",
        "ratio_squared": "49/400",
        "N": 9
      },
      {
        "A": [
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          3,
          4,
          5,
          7,
          9,
          11,
          13,
          17
        ],
        "B": "9/8",
        "Z": "3/128",
        "ratio_squared": "81/640",
        "N": 10
      },
      {
        "A": [
          1,
          2,
          3,
          4,
          5,
          7,
          10,
          11,
          13,
          15,
          17
        ],
        "B": "69/56",
        "Z": "7/256",
        "ratio_squared": "4761/34496",
        "N": 11
      },
      {
        "A": [
          1,
          2,
          3,
          4,
          5,
          6,
          7,
          8,
          9,
          11,
          13,
          17
        ],
        "B": "13/10",
        "Z": "5/512",
        "ratio_squared": "169/1200",
        "N": 12
      },
      {
        "A": [
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          2,
          3,
          4,
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          6,
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          9,
          10,
          11,
          13,
          17
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        "B": "13/9",
        "Z": "27/4096",
        "ratio_squared": "13/81",
        "N": 13
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      {
        "A": [
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          3,
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          5,
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          10,
          11,
          13,
          15,
          16,
          17
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        "B": "43/32",
        "Z": "1/256",
        "ratio_squared": "1849/14336",
        "N": 14
      },
      {
        "A": [
          1,
          2,
          3,
          4,
          5,
          6,
          7,
          8,
          9,
          10,
          11,
          13,
          15,
          16,
          17
        ],
        "B": "229/132",
        "Z": "33/8192",
        "ratio_squared": "52441/261360",
        "N": 15
      },
      {
        "A": [
          1,
          2,
          3,
          4,
          5,
          6,
          7,
          8,
          10,
          11,
          12,
          14,
          15,
          16,
          17,
          18
        ],
        "B": "485/264",
        "Z": "33/8192",
        "ratio_squared": "235225/1115136",
        "N": 16
      },
      {
        "A": [
          1,
          2,
          3,
          4,
          5,
          6,
          7,
          8,
          9,
          10,
          11,
          12,
          14,
          15,
          16,
          17,
          18
        ],
        "B": "269/114",
        "Z": "57/16384",
        "ratio_squared": "72361/220932",
        "N": 17
      },
      {
        "A": [
          1,
          2,
          3,
          4,
          5,
          6,
          7,
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          9,
          10,
          11,
          12,
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          14,
          15,
          16,
          17,
          18
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        "B": "2927/808",
        "Z": "101/32768",
        "ratio_squared": "8567329/11751552",
        "N": 18
      }
    ],
    "traversal_sha256": "01f7822be05210417dec12738883d5dc15afd8ae924d109aaeba413f4a86e6b7",
    "intervals": [
      {
        "N": 1,
        "B": "1/2",
        "Z": "1"
      },
      {
        "N": 2,
        "B": "3/4",
        "Z": "1"
      },
      {
        "N": 3,
        "B": "5/6",
        "Z": "3/4"
      },
      {
        "N": 4,
        "B": "1",
        "Z": "5/8"
      },
      {
        "N": 5,
        "B": "13/12",
        "Z": "3/8"
      },
      {
        "N": 6,
        "B": "6/5",
        "Z": "5/16"
      },
      {
        "N": 7,
        "B": "11/8",
        "Z": "3/16"
      },
      {
        "N": 8,
        "B": "3/2",
        "Z": "17/128"
      },
      {
        "N": 9,
        "B": "37/22",
        "Z": "11/128"
      },
      {
        "N": 10,
        "B": "29/16",
        "Z": "1/16"
      },
      {
        "N": 11,
        "B": "82/39",
        "Z": "39/1024"
      },
      {
        "N": 12,
        "B": "64/29",
        "Z": "29/1024"
      },
      {
        "N": 13,
        "B": "373/148",
        "Z": "37/2048"
      },
      {
        "N": 14,
        "B": "565/208",
        "Z": "13/1024"
      },
      {
        "N": 15,
        "B": "423/143",
        "Z": "143/16384"
      },
      {
        "N": 16,
        "B": "644/205",
        "Z": "205/32768"
      },
      {
        "N": 17,
        "B": "961/278",
        "Z": "139/32768"
      },
      {
        "N": 18,
        "B": "2927/808",
        "Z": "101/32768"
      },
      {
        "N": 19,
        "B": "549/140",
        "Z": "35/16384"
      },
      {
        "N": 20,
        "B": "2221/540",
        "Z": "405/262144"
      },
      {
        "N": 21,
        "B": "10053/2308",
        "Z": "577/524288"
      },
      {
        "N": 22,
        "B": "3811/838",
        "Z": "419/524288"
      },
      {
        "N": 23,
        "B": "22985/4784",
        "Z": "299/524288"
      },
      {
        "N": 24,
        "B": "34811/6980",
        "Z": "1745/4194304"
      },
      {
        "N": 25,
        "B": "3284/629",
        "Z": "629/2097152"
      },
      {
        "N": 26,
        "B": "9934/1835",
        "Z": "1835/8388608"
      },
      {
        "N": 27,
        "B": "59977/10648",
        "Z": "1331/8388608"
      },
      {
        "N": 28,
        "B": "22658/3891",
        "Z": "3891/33554432"
      },
      {
        "N": 29,
        "B": "68366/11319",
        "Z": "11319/134217728"
      },
      {
        "N": 30,
        "B": "412871/66288",
        "Z": "4143/67108864"
      }
    ],
    "powers4": [
      {
        "N": 1,
        "A": [
          1
        ],
        "B": "1/2"
      },
      {
        "N": 2,
        "A": [
          1,
          4
        ],
        "B": "1"
      },
      {
        "N": 3,
        "A": [
          1,
          4,
          16
        ],
        "B": "3/2"
      },
      {
        "N": 4,
        "A": [
          1,
          4,
          16,
          64
        ],
        "B": "2"
      },
      {
        "N": 5,
        "A": [
          1,
          4,
          16,
          64,
          256
        ],
        "B": "5/2"
      },
      {
        "N": 6,
        "A": [
          1,
          4,
          16,
          64,
          256,
          1024
        ],
        "B": "3"
      },
      {
        "N": 7,
        "A": [
          1,
          4,
          16,
          64,
          256,
          1024,
          4096
        ],
        "B": "7/2"
      }
    ],
    "scope": "Finite domain only. No unrestricted positive lower bound established."
  },
  "88": {
    "status": "passed",
    "rational_measure_checks": [
      {
        "N": 2,
        "one_arc_length_over_pi_lower": "1/20"
      },
      {
        "N": 3,
        "one_arc_length_over_pi_lower": "3/28"
      },
      {
        "N": 4,
        "one_arc_length_over_pi_lower": "5/36"
      },
      {
        "N": 5,
        "one_arc_length_over_pi_lower": "7/44"
      },
      {
        "N": 6,
        "one_arc_length_over_pi_lower": "9/52"
      },
      {
        "N": 7,
        "one_arc_length_over_pi_lower": "11/60"
      },
      {
        "N": 8,
        "one_arc_length_over_pi_lower": "13/68"
      },
      {
        "N": 9,
        "one_arc_length_over_pi_lower": "15/76"
      },
      {
        "N": 10,
        "one_arc_length_over_pi_lower": "17/84"
      },
      {
        "N": 11,
        "one_arc_length_over_pi_lower": "19/92"
      },
      {
        "N": 12,
        "one_arc_length_over_pi_lower": "21/100"
      },
      {
        "N": 13,
        "one_arc_length_over_pi_lower": "23/108"
      },
      {
        "N": 14,
        "one_arc_length_over_pi_lower": "25/116"
      },
      {
        "N": 15,
        "one_arc_length_over_pi_lower": "27/124"
      },
      {
        "N": 16,
        "one_arc_length_over_pi_lower": "29/132"
      },
      {
        "N": 17,
        "one_arc_length_over_pi_lower": "31/140"
      },
      {
        "N": 18,
        "one_arc_length_over_pi_lower": "33/148"
      },
      {
        "N": 19,
        "one_arc_length_over_pi_lower": "35/156"
      },
      {
        "N": 20,
        "one_arc_length_over_pi_lower": "37/164"
      },
      {
        "N": 21,
        "one_arc_length_over_pi_lower": "39/172"
      },
      {
        "N": 22,
        "one_arc_length_over_pi_lower": "41/180"
      },
      {
        "N": 23,
        "one_arc_length_over_pi_lower": "43/188"
      },
      {
        "N": 24,
        "one_arc_length_over_pi_lower": "45/196"
      },
      {
        "N": 25,
        "one_arc_length_over_pi_lower": "47/204"
      },
      {
        "N": 26,
        "one_arc_length_over_pi_lower": "49/212"
      },
      {
        "N": 27,
        "one_arc_length_over_pi_lower": "51/220"
      },
      {
        "N": 28,
        "one_arc_length_over_pi_lower": "53/228"
      },
      {
        "N": 29,
        "one_arc_length_over_pi_lower": "55/236"
      },
      {
        "N": 30,
        "one_arc_length_over_pi_lower": "57/244"
      },
      {
        "N": 31,
        "one_arc_length_over_pi_lower": "59/252"
      },
      {
        "N": 32,
        "one_arc_length_over_pi_lower": "61/260"
      },
      {
        "N": 33,
        "one_arc_length_over_pi_lower": "63/268"
      },
      {
        "N": 34,
        "one_arc_length_over_pi_lower": "65/276"
      },
      {
        "N": 35,
        "one_arc_length_over_pi_lower": "67/284"
      },
      {
        "N": 36,
        "one_arc_length_over_pi_lower": "69/292"
      },
      {
        "N": 37,
        "one_arc_length_over_pi_lower": "71/300"
      },
      {
        "N": 38,
        "one_arc_length_over_pi_lower": "73/308"
      },
      {
        "N": 39,
        "one_arc_length_over_pi_lower": "75/316"
      },
      {
        "N": 40,
        "one_arc_length_over_pi_lower": "77/324"
      },
      {
        "N": 41,
        "one_arc_length_over_pi_lower": "79/332"
      },
      {
        "N": 42,
        "one_arc_length_over_pi_lower": "81/340"
      },
      {
        "N": 43,
        "one_arc_length_over_pi_lower": "83/348"
      },
      {
        "N": 44,
        "one_arc_length_over_pi_lower": "85/356"
      },
      {
        "N": 45,
        "one_arc_length_over_pi_lower": "87/364"
      },
      {
        "N": 46,
        "one_arc_length_over_pi_lower": "89/372"
      },
      {
        "N": 47,
        "one_arc_length_over_pi_lower": "91/380"
      },
      {
        "N": 48,
        "one_arc_length_over_pi_lower": "93/388"
      },
      {
        "N": 49,
        "one_arc_length_over_pi_lower": "95/396"
      },
      {
        "N": 50,
        "one_arc_length_over_pi_lower": "97/404"
      },
      {
        "N": 51,
        "one_arc_length_over_pi_lower": "99/412"
      },
      {
        "N": 52,
        "one_arc_length_over_pi_lower": "101/420"
      },
      {
        "N": 53,
        "one_arc_length_over_pi_lower": "103/428"
      },
      {
        "N": 54,
        "one_arc_length_over_pi_lower": "105/436"
      },
      {
        "N": 55,
        "one_arc_length_over_pi_lower": "107/444"
      },
      {
        "N": 56,
        "one_arc_length_over_pi_lower": "109/452"
      },
      {
        "N": 57,
        "one_arc_length_over_pi_lower": "111/460"
      },
      {
        "N": 58,
        "one_arc_length_over_pi_lower": "113/468"
      },
      {
        "N": 59,
        "one_arc_length_over_pi_lower": "115/476"
      },
      {
        "N": 60,
        "one_arc_length_over_pi_lower": "117/484"
      },
      {
        "N": 61,
        "one_arc_length_over_pi_lower": "119/492"
      },
      {
        "N": 62,
        "one_arc_length_over_pi_lower": "121/500"
      },
      {
        "N": 63,
        "one_arc_length_over_pi_lower": "123/508"
      },
      {
        "N": 64,
        "one_arc_length_over_pi_lower": "125/516"
      },
      {
        "N": 100,
        "one_arc_length_over_pi_lower": "197/804"
      },
      {
        "N": 1000,
        "one_arc_length_over_pi_lower": "1997/8004"
      },
      {
        "N": 10000,
        "one_arc_length_over_pi_lower": "19997/80004"
      }
    ],
    "normalization_floor": "1/40",
    "all_beta_upper_bound": "20+10 log(2N+1)",
    "logistic_algebra_checks": [
      {
        "base": 2,
        "mean_zero_values": [
          -1,
          1
        ],
        "B": "1/3"
      },
      {
        "base": 2,
        "mean_zero_values": [
          -2,
          1,
          1
        ],
        "B": "7/11"
      },
      {
        "base": 2,
        "mean_zero_values": [
          -5,
          1,
          1,
          1,
          1,
          1
        ],
        "B": "35/29"
      },
      {
        "base": 2,
        "mean_zero_values": [
          -3,
          -1,
          2,
          2
        ],
        "B": "57/44"
      },
      {
        "base": 3,
        "mean_zero_values": [
          -1,
          1
        ],
        "B": "1/2"
      },
      {
        "base": 3,
        "mean_zero_values": [
          -2,
          1,
          1
        ],
        "B": "13/14"
      },
      {
        "base": 3,
        "mean_zero_values": [
          -5,
          1,
          1,
          1,
          1,
          1
        ],
        "B": "455/274"
      },
      {
        "base": 3,
        "mean_zero_values": [
          -3,
          -1,
          2,
          2
        ],
        "B": "227/134"
      },
      {
        "base": 5,
        "mean_zero_values": [
          -1,
          1
        ],
        "B": "2/3"
      },
      {
        "base": 5,
        "mean_zero_values": [
          -2,
          1,
          1
        ],
        "B": "124/101"
      },
      {
        "base": 5,
        "mean_zero_values": [
          -5,
          1,
          1,
          1,
          1,
          1
        ],
        "B": "434/191"
      },
      {
        "base": 5,
        "mean_zero_values": [
          -3,
          -1,
          2,
          2
        ],
        "B": "1497/779"
      }
    ],
    "cosine_one_mode": "E w=1/2 by symmetry; limit B=2/pi by analytic integration",
    "scope": "Analytic all-beta obstruction, no floating exponential quadrature."
  },
  "89": {
    "status": "passed",
    "Fejer_coefficient_checks": [
      {
        "N": 1,
        "weighted_K": "1/2",
        "sum_weights_squared": "1/4",
        "RMS_squared": "1/8",
        "integer_K": "1",
        "integer_squared_amplitudes": 1,
        "normalized_ratio_squared": "1"
      },
      {
        "N": 2,
        "weighted_K": "1/2",
        "sum_weights_squared": "5/9",
        "RMS_squared": "5/18",
        "integer_K": "3/2",
        "integer_squared_amplitudes": 5,
        "normalized_ratio_squared": "9/20"
      },
      {
        "N": 3,
        "weighted_K": "1/2",
        "sum_weights_squared": "7/8",
        "RMS_squared": "7/16",
        "integer_K": "2",
        "integer_squared_amplitudes": 14,
        "normalized_ratio_squared": "2/7"
      },
      {
        "N": 4,
        "weighted_K": "1/2",
        "sum_weights_squared": "6/5",
        "RMS_squared": "3/5",
        "integer_K": "5/2",
        "integer_squared_amplitudes": 30,
        "normalized_ratio_squared": "5/24"
      },
      {
        "N": 5,
        "weighted_K": "1/2",
        "sum_weights_squared": "55/36",
        "RMS_squared": "55/72",
        "integer_K": "3",
        "integer_squared_amplitudes": 55,
        "normalized_ratio_squared": "9/55"
      },
      {
        "N": 6,
        "weighted_K": "1/2",
        "sum_weights_squared": "13/7",
        "RMS_squared": "13/14",
        "integer_K": "7/2",
        "integer_squared_amplitudes": 91,
        "normalized_ratio_squared": "7/52"
      },
      {
        "N": 7,
        "weighted_K": "1/2",
        "sum_weights_squared": "35/16",
        "RMS_squared": "35/32",
        "integer_K": "4",
        "integer_squared_amplitudes": 140,
        "normalized_ratio_squared": "4/35"
      },
      {
        "N": 8,
        "weighted_K": "1/2",
        "sum_weights_squared": "68/27",
        "RMS_squared": "34/27",
        "integer_K": "9/2",
        "integer_squared_amplitudes": 204,
        "normalized_ratio_squared": "27/272"
      },
      {
        "N": 9,
        "weighted_K": "1/2",
        "sum_weights_squared": "57/20",
        "RMS_squared": "57/40",
        "integer_K": "5",
        "integer_squared_amplitudes": 285,
        "normalized_ratio_squared": "5/57"
      },
      {
        "N": 10,
        "weighted_K": "1/2",
        "sum_weights_squared": "35/11",
        "RMS_squared": "35/22",
        "integer_K": "11/2",
        "integer_squared_amplitudes": 385,
        "normalized_ratio_squared": "11/140"
      },
      {
        "N": 11,
        "weighted_K": "1/2",
        "sum_weights_squared": "253/72",
        "RMS_squared": "253/144",
        "integer_K": "6",
        "integer_squared_amplitudes": 506,
        "normalized_ratio_squared": "18/253"
      },
      {
        "N": 12,
        "weighted_K": "1/2",
        "sum_weights_squared": "50/13",
        "RMS_squared": "25/13",
        "integer_K": "13/2",
        "integer_squared_amplitudes": 650,
        "normalized_ratio_squared": "13/200"
      },
      {
        "N": 13,
        "weighted_K": "1/2",
        "sum_weights_squared": "117/28",
        "RMS_squared": "117/56",
        "integer_K": "7",
        "integer_squared_amplitudes": 819,
        "normalized_ratio_squared": "7/117"
      },
      {
        "N": 14,
        "weighted_K": "1/2",
        "sum_weights_squared": "203/45",
        "RMS_squared": "203/90",
        "integer_K": "15/2",
        "integer_squared_amplitudes": 1015,
        "normalized_ratio_squared": "45/812"
      },
      {
        "N": 15,
        "weighted_K": "1/2",
        "sum_weights_squared": "155/32",
        "RMS_squared": "155/64",
        "integer_K": "8",
        "integer_squared_amplitudes": 1240,
        "normalized_ratio_squared": "8/155"
      },
      {
        "N": 16,
        "weighted_K": "1/2",
        "sum_weights_squared": "88/17",
        "RMS_squared": "44/17",
        "integer_K": "17/2",
        "integer_squared_amplitudes": 1496,
        "normalized_ratio_squared": "17/352"
      },
      {
        "N": 17,
        "weighted_K": "1/2",
        "sum_weights_squared": "595/108",
        "RMS_squared": "595/216",
        "integer_K": "9",
        "integer_squared_amplitudes": 1785,
        "normalized_ratio_squared": "27/595"
      },
      {
        "N": 18,
        "weighted_K": "1/2",
        "sum_weights_squared": "111/19",
        "RMS_squared": "111/38",
        "integer_K": "19/2",
        "integer_squared_amplitudes": 2109,
        "normalized_ratio_squared": "19/444"
      },
      {
        "N": 19,
        "weighted_K": "1/2",
        "sum_weights_squared": "247/40",
        "RMS_squared": "247/80",
        "integer_K": "10",
        "integer_squared_amplitudes": 2470,
        "normalized_ratio_squared": "10/247"
      },
      {
        "N": 20,
        "weighted_K": "1/2",
        "sum_weights_squared": "410/63",
        "RMS_squared": "205/63",
        "integer_K": "21/2",
        "integer_squared_amplitudes": 2870,
        "normalized_ratio_squared": "63/1640"
      },
      {
        "N": 21,
        "weighted_K": "1/2",
        "sum_weights_squared": "301/44",
        "RMS_squared": "301/88",
        "integer_K": "11",
        "integer_squared_amplitudes": 3311,
        "normalized_ratio_squared": "11/301"
      },
      {
        "N": 22,
        "weighted_K": "1/2",
        "sum_weights_squared": "165/23",
        "RMS_squared": "165/46",
        "integer_K": "23/2",
        "integer_squared_amplitudes": 3795,
        "normalized_ratio_squared": "23/660"
      },
      {
        "N": 23,
        "weighted_K": "1/2",
        "sum_weights_squared": "1081/144",
        "RMS_squared": "1081/288",
        "integer_K": "12",
        "integer_squared_amplitudes": 4324,
        "normalized_ratio_squared": "36/1081"
      },
      {
        "N": 24,
        "weighted_K": "1/2",
        "sum_weights_squared": "196/25",
        "RMS_squared": "98/25",
        "integer_K": "25/2",
        "integer_squared_amplitudes": 4900,
        "normalized_ratio_squared": "25/784"
      },
      {
        "N": 25,
        "weighted_K": "1/2",
        "sum_weights_squared": "425/52",
        "RMS_squared": "425/104",
        "integer_K": "13",
        "integer_squared_amplitudes": 5525,
        "normalized_ratio_squared": "13/425"
      },
      {
        "N": 26,
        "weighted_K": "1/2",
        "sum_weights_squared": "689/81",
        "RMS_squared": "689/162",
        "integer_K": "27/2",
        "integer_squared_amplitudes": 6201,
        "normalized_ratio_squared": "81/2756"
      },
      {
        "N": 27,
        "weighted_K": "1/2",
        "sum_weights_squared": "495/56",
        "RMS_squared": "495/112",
        "integer_K": "14",
        "integer_squared_amplitudes": 6930,
        "normalized_ratio_squared": "14/495"
      },
      {
        "N": 28,
        "weighted_K": "1/2",
        "sum_weights_squared": "266/29",
        "RMS_squared": "133/29",
        "integer_K": "29/2",
        "integer_squared_amplitudes": 7714,
        "normalized_ratio_squared": "29/1064"
      },
      {
        "N": 29,
        "weighted_K": "1/2",
        "sum_weights_squared": "1711/180",
        "RMS_squared": "1711/360",
        "integer_K": "15",
        "integer_squared_amplitudes": 8555,
        "normalized_ratio_squared": "45/1711"
      },
      {
        "N": 30,
        "weighted_K": "1/2",
        "sum_weights_squared": "305/31",
        "RMS_squared": "305/62",
        "integer_K": "31/2",
        "integer_squared_amplitudes": 9455,
        "normalized_ratio_squared": "31/1220"
      },
      {
        "N": 31,
        "weighted_K": "1/2",
        "sum_weights_squared": "651/64",
        "RMS_squared": "651/128",
        "integer_K": "16",
        "integer_squared_amplitudes": 10416,
        "normalized_ratio_squared": "16/651"
      },
      {
        "N": 32,
        "weighted_K": "1/2",
        "sum_weights_squared": "1040/99",
        "RMS_squared": "520/99",
        "integer_K": "33/2",
        "integer_squared_amplitudes": 11440,
        "normalized_ratio_squared": "99/4160"
      }
    ],
    "root_certificate": "(1-z)*sum(z^j,j=0..N)=1-z^(N+1), exact polynomial identity",
    "scope": "Counterexample for unequal positive amplitudes, including integer ones; not for Chowla unit amplitudes."
  }
}