{
  "scope": "Checks of routes 27-29, not a general Chowla proof.",
  "27": {
    "exact_identity_checks": [
      {
        "m": 0,
        "length": 1,
        "aperiodic_correlations_checked": 1
      },
      {
        "m": 1,
        "length": 2,
        "aperiodic_correlations_checked": 2
      },
      {
        "m": 2,
        "length": 4,
        "aperiodic_correlations_checked": 4
      },
      {
        "m": 3,
        "length": 8,
        "aperiodic_correlations_checked": 8
      },
      {
        "m": 4,
        "length": 16,
        "aperiodic_correlations_checked": 16
      },
      {
        "m": 5,
        "length": 32,
        "aperiodic_correlations_checked": 32
      },
      {
        "m": 6,
        "length": 64,
        "aperiodic_correlations_checked": 64
      },
      {
        "m": 7,
        "length": 128,
        "aperiodic_correlations_checked": 128
      },
      {
        "m": 8,
        "length": 256,
        "aperiodic_correlations_checked": 256
      }
    ],
    "witnesses": [
      {
        "m": 6,
        "length": 64,
        "N": 35,
        "theta_exact_description": "3*pi/(2*length)",
        "certified_K_lower_bound": {
          "numerator": 362687,
          "denominator": 236544,
          "display_float_only": 1.5332749932359306
        },
        "certified_K_over_sqrtN_strictly_greater_than_integer": 0,
        "negative_witness_float_sanity_only": -5.5853779961111885
      },
      {
        "m": 8,
        "length": 256,
        "N": 135,
        "theta_exact_description": "3*pi/(2*length)",
        "certified_K_lower_bound": {
          "numerator": 15517631,
          "denominator": 946176,
          "display_float_only": 16.400364202854437
        },
        "certified_K_over_sqrtN_strictly_greater_than_integer": 1,
        "negative_witness_float_sanity_only": -24.13974534472176
      },
      {
        "m": 10,
        "length": 1024,
        "N": 527,
        "theta_exact_description": "3*pi/(2*length)",
        "certified_K_lower_bound": {
          "numerator": 326831039,
          "denominator": 3784704,
          "display_float_only": 86.35577286889543
        },
        "certified_K_over_sqrtN_strictly_greater_than_integer": 3,
        "negative_witness_float_sanity_only": -101.92197365507126
      },
      {
        "m": 12,
        "length": 4096,
        "N": 2079,
        "theta_exact_description": "3*pi/(2*length)",
        "certified_K_lower_bound": {
          "numerator": 5899205567,
          "denominator": 15138816,
          "display_float_only": 389.6741704899511
        },
        "certified_K_over_sqrtN_strictly_greater_than_integer": 8,
        "negative_witness_float_sanity_only": -420.4269640102946
      },
      {
        "m": 16,
        "length": 65536,
        "N": 32895,
        "theta_exact_description": "3*pi/(2*length)",
        "certified_K_lower_bound": {
          "numerator": 1639845723071,
          "denominator": 242221056,
          "display_float_only": 6770.037874291986
        },
        "certified_K_over_sqrtN_strictly_greater_than_integer": 37,
        "negative_witness_float_sanity_only": -6894.688636913498
      }
    ]
  },
  "28": {
    "exact_power_sum_checks": [
      {
        "m": 1,
        "length": 2,
        "N_each": 1,
        "power_equalities_checked": 1,
        "first_nonzero_power_difference": -1
      },
      {
        "m": 2,
        "length": 4,
        "N_each": 2,
        "power_equalities_checked": 2,
        "first_nonzero_power_difference": 4
      },
      {
        "m": 3,
        "length": 8,
        "N_each": 4,
        "power_equalities_checked": 3,
        "first_nonzero_power_difference": -48
      },
      {
        "m": 4,
        "length": 16,
        "N_each": 8,
        "power_equalities_checked": 4,
        "first_nonzero_power_difference": 1536
      },
      {
        "m": 5,
        "length": 32,
        "N_each": 16,
        "power_equalities_checked": 5,
        "first_nonzero_power_difference": -122880
      },
      {
        "m": 6,
        "length": 64,
        "N_each": 32,
        "power_equalities_checked": 6,
        "first_nonzero_power_difference": 23592960
      },
      {
        "m": 7,
        "length": 128,
        "N_each": 64,
        "power_equalities_checked": 7,
        "first_nonzero_power_difference": -10569646080
      },
      {
        "m": 8,
        "length": 256,
        "N_each": 128,
        "power_equalities_checked": 8,
        "first_nonzero_power_difference": 10823317585920
      },
      {
        "m": 9,
        "length": 512,
        "N_each": 256,
        "power_equalities_checked": 9,
        "first_nonzero_power_difference": -24936923717959680
      },
      {
        "m": 10,
        "length": 1024,
        "N_each": 512,
        "power_equalities_checked": 10,
        "first_nonzero_power_difference": 127677049435953561600
      }
    ],
    "witnesses": [
      {
        "m": 4,
        "length": 16,
        "N": 8,
        "theta_exact_description": "3*pi/(2*length)",
        "certified_signed_polynomial_modulus_upper_bound": {
          "numerator": 1185921,
          "denominator": 2458624,
          "display_float_only": 0.482351510438359
        },
        "certified_K_lower_bound": {
          "numerator": 273809575,
          "denominator": 162269184,
          "display_float_only": 1.6873787631790889
        },
        "certified_K_over_sqrtN_strictly_greater_than_integer": 0,
        "negative_witness_float_sanity_only": -2.0758437511829917
      },
      {
        "m": 6,
        "length": 64,
        "N": 32,
        "theta_exact_description": "3*pi/(2*length)",
        "certified_signed_polynomial_modulus_upper_bound": {
          "numerator": 1291467969,
          "denominator": 246727835648,
          "display_float_only": 0.0052343829207925395
        },
        "certified_K_lower_bound": {
          "numerator": 114487492852511,
          "denominator": 16284037152768,
          "display_float_only": 7.030657801775535
        },
        "certified_K_over_sqrtN_strictly_greater_than_integer": 1,
        "negative_witness_float_sanity_only": -7.036154794062381
      },
      {
        "m": 8,
        "length": 256,
        "N": 128,
        "theta_exact_description": "3*pi/(2*length)",
        "certified_signed_polynomial_modulus_upper_bound": {
          "numerator": 1406408618241,
          "denominator": 396154108207169536,
          "display_float_only": 3.5501553287073725e-06
        },
        "certified_K_lower_bound": {
          "numerator": 716414565380523388639,
          "denominator": 26146171141673189376,
          "display_float_only": 27.400362427776773
        },
        "certified_K_over_sqrtN_strictly_greater_than_integer": 2,
        "negative_witness_float_sanity_only": -27.41167754229609
      },
      {
        "m": 10,
        "length": 1024,
        "N": 512,
        "theta_exact_description": "3*pi/(2*length)",
        "certified_signed_polynomial_modulus_upper_bound": {
          "numerator": 1531578985264449,
          "denominator": 10177243409102304582828032,
          "display_float_only": 1.504905526672024e-10
        },
        "certified_K_lower_bound": {
          "numerator": 73118212000147885920598401695,
          "denominator": 671698065000752102466650112,
          "display_float_only": 108.85577286882018
        },
        "certified_K_over_sqrtN_strictly_greater_than_integer": 4,
        "negative_witness_float_sanity_only": -108.89958273636252
      },
      {
        "m": 12,
        "length": 4096,
        "N": 2048,
        "theta_exact_description": "3*pi/(2*length)",
        "certified_signed_polynomial_modulus_upper_bound": {
          "numerator": 1667889514952984961,
          "denominator": 4183272368497600387055446701113344,
          "display_float_only": 3.987044992607064e-16
        },
        "certified_K_lower_bound": {
          "numerator": 120011789482875022706761662957531154015,
          "denominator": 276095976320841625545659482273480704,
          "display_float_only": 434.6741704899511
        },
        "certified_K_over_sqrtN_strictly_greater_than_integer": 9,
        "negative_witness_float_sanity_only": -434.8490499993682
      },
      {
        "m": 16,
        "length": 65536,
        "N": 32768,
        "theta_exact_description": "3*pi/(2*length)",
        "certified_signed_polynomial_modulus_upper_bound": {
          "numerator": 1977985201462558877934081,
          "denominator": 2894996550129418232606555038658322563912973079948558336,
          "display_float_only": 6.8324267998666e-31
        },
        "certified_K_lower_bound": {
          "numerator": 1328133223949018717267828026017998333337823176682253295672799,
          "denominator": 191069772308541603352032632551449289218256223276604850176,
          "display_float_only": 6951.037874291986
        },
        "certified_K_over_sqrtN_strictly_greater_than_integer": 38,
        "negative_witness_float_sanity_only": -6953.835563984246
      }
    ]
  },
  "29": {
    "original_nonnegative_identity": "(1+2*cos(x))^2 = 3+4*cos(x)+2*cos(2*x)",
    "fractional_coefficient_power_polynomial": "3^p+2*2^p*cos(x)+2*cos(2*x)",
    "exact_minimum_for_0_le_p_le_1": "3^p-2-4^(p-1)",
    "exact_minimum_p0": {
      "numerator": -5,
      "denominator": 4,
      "display_float_only": -1.25
    },
    "exact_minimum_p1": {
      "numerator": 0,
      "denominator": 1,
      "display_float_only": 0.0
    },
    "exact_minimum_p_half": "sqrt(3)-5/2 < -3/4",
    "all_fractional_p_0_le_p_lt_1_negative": "proved analytically in note, not inferred from samples",
    "float_sanity_samples_only": [
      {
        "p": 0,
        "minimum": -1.25
      },
      {
        "p": 0.125,
        "minimum": -1.150099088310803
      },
      {
        "p": 0.25,
        "minimum": -1.0374793776407814
      },
      {
        "p": 0.5,
        "minimum": -0.7679491924311228
      },
      {
        "p": 0.75,
        "minimum": -0.42759972423177006
      },
      {
        "p": 0.875,
        "minimum": -0.22583978663850746
      },
      {
        "p": 0.99,
        "minimum": -0.019010691976846905
      },
      {
        "p": 1,
        "minimum": 0
      }
    ]
  }
}