{
  "seed": 1203,
  "M": 256,
  "predictions": [
    "For every L, min(K)>=0 and sum(K)=1, hence induced sup norm and random error ratios are <=1.",
    "Matched-rank signed truncation has induced sup norm sum(abs(K)); its worst-case ratio equals that value and generally grows with rank.",
    "Fejer Fourier eigenvalue at integer frequency k is max(1-|k|/L,0), giving linear attenuation to the cutoff and zero beyond it."
  ],
  "kernel_sweep": [
    {
      "L": 2,
      "min_fejer_kernel": 2.9292183255114406e-35,
      "mass_error": 1.1102230246251565e-16,
      "fejer_inf_norm": 0.9999999999999999,
      "signed_inf_norm": 1.4360095172636305,
      "signed_worst_case_ratio": 1.4360095172636305,
      "fejer_random_ratio": 0.045108277874436886
    },
    {
      "L": 4,
      "min_fejer_kernel": 5.858436651022879e-35,
      "mass_error": 2.220446049250313e-16,
      "fejer_inf_norm": 0.9999999999999998,
      "signed_inf_norm": 1.7784319428805193,
      "signed_worst_case_ratio": 1.7784319428805193,
      "fejer_random_ratio": 0.048089527403431
    },
    {
      "L": 8,
      "min_fejer_kernel": 1.1716873302045763e-34,
      "mass_error": 2.220446049250313e-16,
      "fejer_inf_norm": 0.9999999999999998,
      "signed_inf_norm": 2.086461979555266,
      "signed_worst_case_ratio": 2.086461979555266,
      "fejer_random_ratio": 0.1350374025518305
    },
    {
      "L": 16,
      "min_fejer_kernel": 1.1811367237062345e-34,
      "mass_error": 6.661338147750939e-16,
      "fejer_inf_norm": 0.9999999999999993,
      "signed_inf_norm": 2.3826342877350895,
      "signed_worst_case_ratio": 2.3826342877350895,
      "fejer_random_ratio": 0.20814128441985832
    },
    {
      "L": 32,
      "min_fejer_kernel": 1.0240906784363769e-34,
      "mass_error": 2.220446049250313e-16,
      "fejer_inf_norm": 1.0000000000000002,
      "signed_inf_norm": 2.656816980720107,
      "signed_worst_case_ratio": 2.6568169807201065,
      "fejer_random_ratio": 0.30104351387825673
    },
    {
      "L": 64,
      "min_fejer_kernel": 6.13608051129234e-35,
      "mass_error": 0.0,
      "fejer_inf_norm": 1.0,
      "signed_inf_norm": 2.797267840356451,
      "signed_worst_case_ratio": 2.797267840356451,
      "fejer_random_ratio": 0.4657958293871679
    }
  ],
  "attenuation_sweep": [
    {
      "L": 2,
      "checks": [
        {
          "k": 0,
          "observed": 1.0,
          "predicted": 1.0,
          "abs_error": 0.0
        },
        {
          "k": 1,
          "observed": 0.5,
          "predicted": 0.5,
          "abs_error": 0.0
        },
        {
          "k": 1,
          "observed": 0.5,
          "predicted": 0.5,
          "abs_error": 0.0
        },
        {
          "k": 2,
          "observed": -1.8022814952877887e-17,
          "predicted": 0.0,
          "abs_error": 1.8022814952877887e-17
        },
        {
          "k": 5,
          "observed": 7.01022896296564e-18,
          "predicted": 0.0,
          "abs_error": 7.01022896296564e-18
        }
      ]
    },
    {
      "L": 4,
      "checks": [
        {
          "k": 0,
          "observed": 0.9999999999999998,
          "predicted": 1.0,
          "abs_error": 2.220446049250313e-16
        },
        {
          "k": 2,
          "observed": 0.4999999999999999,
          "predicted": 0.5,
          "abs_error": 1.1102230246251565e-16
        },
        {
          "k": 3,
          "observed": 0.25,
          "predicted": 0.25,
          "abs_error": 0.0
        },
        {
          "k": 4,
          "observed": 2.2218053890416e-17,
          "predicted": 0.0,
          "abs_error": 2.2218053890416e-17
        },
        {
          "k": 7,
          "observed": -1.1041124689322516e-17,
          "predicted": 0.0,
          "abs_error": 1.1041124689322516e-17
        }
      ]
    },
    {
      "L": 8,
      "checks": [
        {
          "k": 0,
          "observed": 0.9999999999999998,
          "predicted": 1.0,
          "abs_error": 2.220446049250313e-16
        },
        {
          "k": 4,
          "observed": 0.49999999999999994,
          "predicted": 0.5,
          "abs_error": 5.551115123125783e-17
        },
        {
          "k": 7,
          "observed": 0.12500000000000014,
          "predicted": 0.125,
          "abs_error": 1.3877787807814457e-16
        },
        {
          "k": 8,
          "observed": 1.1211633618255831e-16,
          "predicted": 0.0,
          "abs_error": 1.1211633618255831e-16
        },
        {
          "k": 11,
          "observed": 6.938893903907228e-17,
          "predicted": 0.0,
          "abs_error": 6.938893903907228e-17
        }
      ]
    },
    {
      "L": 16,
      "checks": [
        {
          "k": 0,
          "observed": 0.9999999999999993,
          "predicted": 1.0,
          "abs_error": 6.661338147750939e-16
        },
        {
          "k": 8,
          "observed": 0.49999999999999994,
          "predicted": 0.5,
          "abs_error": 5.551115123125783e-17
        },
        {
          "k": 15,
          "observed": 0.06250000000000062,
          "predicted": 0.0625,
          "abs_error": 6.245004513516506e-16
        },
        {
          "k": 16,
          "observed": 6.009045720724373e-16,
          "predicted": 0.0,
          "abs_error": 6.009045720724373e-16
        },
        {
          "k": 19,
          "observed": 4.112946560504012e-16,
          "predicted": 0.0,
          "abs_error": 4.112946560504012e-16
        }
      ]
    },
    {
      "L": 32,
      "checks": [
        {
          "k": 0,
          "observed": 1.0000000000000002,
          "predicted": 1.0,
          "abs_error": 2.220446049250313e-16
        },
        {
          "k": 16,
          "observed": 0.5000000000000004,
          "predicted": 0.5,
          "abs_error": 4.440892098500626e-16
        },
        {
          "k": 31,
          "observed": 0.03124999999999993,
          "predicted": 0.03125,
          "abs_error": 6.938893903907228e-17
        },
        {
          "k": 32,
          "observed": -1.6148247973969785e-16,
          "predicted": 0.0,
          "abs_error": 1.6148247973969785e-16
        },
        {
          "k": 35,
          "observed": -4.163336342344337e-16,
          "predicted": 0.0,
          "abs_error": 4.163336342344337e-16
        }
      ]
    },
    {
      "L": 64,
      "checks": [
        {
          "k": 0,
          "observed": 1.0,
          "predicted": 1.0,
          "abs_error": 0.0
        },
        {
          "k": 32,
          "observed": 0.5000000000000001,
          "predicted": 0.5,
          "abs_error": 1.1102230246251565e-16
        },
        {
          "k": 63,
          "observed": 0.015624999999999917,
          "predicted": 0.015625,
          "abs_error": 8.326672684688674e-17
        },
        {
          "k": 64,
          "observed": -1.6653345369377348e-16,
          "predicted": 0.0,
          "abs_error": 1.6653345369377348e-16
        },
        {
          "k": 67,
          "observed": -3.365363543395006e-16,
          "predicted": 0.0,
          "abs_error": 3.365363543395006e-16
        }
      ]
    }
  ],
  "equivariance_max_errors": [
    2.7755575615628914e-17,
    6.938893903907228e-17,
    1.1102230246251565e-16,
    1.8041124150158794e-16,
    2.220446049250313e-16,
    2.220446049250313e-16
  ],
  "order_violation_max": 0.0,
  "random_and_stack": [
    {
      "L": 2,
      "mean_fejer_ratio": 0.0583947677047889,
      "mean_signed_ratio": 0.08859360989821372,
      "20_layer_signed_norm_ratio": 5.328913391621261
    },
    {
      "L": 4,
      "mean_fejer_ratio": 0.1044140331177652,
      "mean_signed_ratio": 0.18709733702224654,
      "20_layer_signed_norm_ratio": 30.348196518810855
    },
    {
      "L": 8,
      "mean_fejer_ratio": 0.17177002183166398,
      "mean_signed_ratio": 0.3207410370611741,
      "20_layer_signed_norm_ratio": 42.61679913053706
    },
    {
      "L": 16,
      "mean_fejer_ratio": 0.2841416403889107,
      "mean_signed_ratio": 0.5253006736515848,
      "20_layer_signed_norm_ratio": 4.229985449727612
    },
    {
      "L": 32,
      "mean_fejer_ratio": 0.4279204044860396,
      "mean_signed_ratio": 0.7889652829318335,
      "20_layer_signed_norm_ratio": 13.611940109013641
    },
    {
      "L": 64,
      "mean_fejer_ratio": 0.6265712964133795,
      "mean_signed_ratio": 1.0815584398602223,
      "20_layer_signed_norm_ratio": 13.272669290001211
    }
  ],
  "stacked_20_layers": [
    {
      "L": 2,
      "fejer_20_layer_sup_ratio": 0.011489785304145104,
      "signed_20_layer_sup_ratio": 0.050078718741095575
    },
    {
      "L": 4,
      "fejer_20_layer_sup_ratio": 0.041079369759704665,
      "signed_20_layer_sup_ratio": 0.26077283401090773
    },
    {
      "L": 8,
      "fejer_20_layer_sup_ratio": 0.037491544556016734,
      "signed_20_layer_sup_ratio": 0.41650962818785814
    },
    {
      "L": 16,
      "fejer_20_layer_sup_ratio": 0.032817268694412656,
      "signed_20_layer_sup_ratio": 0.5108483451033049
    },
    {
      "L": 32,
      "fejer_20_layer_sup_ratio": 0.07191079728031154,
      "signed_20_layer_sup_ratio": 0.7186433133753423
    },
    {
      "L": 64,
      "fejer_20_layer_sup_ratio": 0.14606585222262247,
      "signed_20_layer_sup_ratio": 1.3934100850081501
    }
  ],
  "summary": {
    "max_fejer_mass_error": 6.661338147750939e-16,
    "min_fejer_kernel": 2.9292183255114406e-35,
    "max_equivariance_error": 2.220446049250313e-16,
    "max_attenuation_error": 6.661338147750939e-16,
    "signed_norms": [
      1.4360095172636305,
      1.7784319428805193,
      2.086461979555266,
      2.3826342877350895,
      2.656816980720107,
      2.797267840356451
    ],
    "fejer_norms": [
      0.9999999999999999,
      0.9999999999999998,
      0.9999999999999998,
      0.9999999999999993,
      1.0000000000000002,
      1.0
    ]
  }
}