{
  "claims": [
    {
      "assessment": "verified",
      "claim": 1,
      "experiment": {
        "iid": {
          "Sigma_Y": 1.0,
          "bound_ratio_monotone_decreasing": true,
          "max_grid_abs_difference": 2.1483724937931825e-09,
          "method": "stationary characteristic-function product and Gil-Pelaez FFT inversion",
          "objective": "f(x)=x^2/2",
          "rows": [
            {
              "alpha": 0.125,
              "bound_ratio": 0.15084905259053438,
              "inverse_alpha": 8,
              "tail_a_over_sd": [
                0.5,
                1.0,
                1.5,
                2.0,
                3.0,
                4.0,
                6.0
              ],
              "tail_delta": [
                -0.026095476255768267,
                0.0011700665994218069,
                0.017024719451560547,
                0.01859325947655085,
                0.007386427054841728,
                0.00158255387779312,
                4.319003466606215e-05
              ],
              "tail_delta_wrong_variance": [
                -0.07939474244566293,
                -0.0799247405625979,
                -0.060590262452823854,
                -0.03730621210041253,
                -0.008211101675872818,
                -0.0007246423708973916,
                3.214577275442024e-05
              ],
              "tail_scaled_a_abs_delta": [
                0.013047738127884134,
                0.0011700665994218069,
                0.02553707917734082,
                0.0371865189531017,
                0.02215928116452518,
                0.00633021551117248,
                0.0002591402079963729
              ],
              "tail_scaled_wrong_variance": [
                0.039697371222831464,
                0.0799247405625979,
                0.09088539367923579,
                0.07461242420082506,
                0.024633305027618454,
                0.0028985694835895665,
                0.00019287463652652143
              ],
              "tail_sup_scaled": 0.0371865189531017,
              "tail_sup_scaled_wrong_variance": 0.09088539367923579,
              "w1": 0.11090325917728985,
              "w1_grid_2p17": 0.11090326132566235,
              "w1_grid_abs_difference": 2.1483724937931825e-09,
              "w1_wrong_variance": 0.33220559246088766,
              "weight_terms": 202
            },
            {
              "alpha": 0.0625,
              "bound_ratio": 0.11137365253310623,
              "inverse_alpha": 16,
              "tail_a_over_sd": [
                0.5,
                1.0,
                1.5,
                2.0,
                3.0,
                4.0,
                6.0
              ],
              "tail_delta": [
                -0.019025007527767257,
                0.0006042582872497802,
                0.01217137674416508,
                0.013025802645416099,
                0.004516415230001384,
                0.0007293430366687823,
                7.910639081319356e-06
              ],
              "tail_delta_wrong_variance": [
                -0.07232427371766191,
                -0.08049054887476995,
                -0.06544360516021933,
                -0.042873668931547274,
                -0.011081113500713164,
                -0.0015778532120217315,
                -3.1336228303255363e-06
              ],
              "tail_scaled_a_abs_delta": [
                0.009512503763883629,
                0.0006042582872497802,
                0.01825706511624762,
                0.026051605290832197,
                0.01354924569000415,
                0.002917372146675129,
                4.7463834487916135e-05
              ],
              "tail_scaled_wrong_variance": [
                0.036162136858830955,
                0.08049054887476995,
                0.09816540774032899,
                0.08574733786309455,
                0.03324334050213949,
                0.006311412848086926,
                1.880173698195322e-05
              ],
              "tail_sup_scaled": 0.026051605290832197,
              "tail_sup_scaled_wrong_variance": 0.09816540774032899,
              "w1": 0.07719833324198559,
              "w1_grid_2p17": 0.07719833325539735,
              "w1_grid_abs_difference": 1.341175781544024e-11,
              "w1_wrong_variance": 0.3313844025922123,
              "weight_terms": 413
            },
            {
              "alpha": 0.03125,
              "bound_ratio": 0.08842353367516959,
              "inverse_alpha": 32,
              "tail_a_over_sd": [
                0.5,
                1.0,
                1.5,
                2.0,
                3.0,
                4.0,
                6.0
              ],
              "tail_delta": [
                -0.013770488783950898,
                0.0003124721491144643,
                0.00869738240027082,
                0.009154027049797833,
                0.002835836967231764,
                0.00035730214256624667,
                1.5398460270298304e-06
              ],
              "tail_delta_wrong_variance": [
                -0.06706975497384554,
                -0.08078233501290524,
                -0.06891759950411358,
                -0.046745444527165544,
                -0.012761691763482788,
                -0.00194989410612427,
                -9.504415884606893e-06
              ],
              "tail_scaled_a_abs_delta": [
                0.006885244391975449,
                0.0003124721491144643,
                0.01304607360040623,
                0.018308054099595666,
                0.00850751090169529,
                0.0014292085702649867,
                9.239076162178983e-06
              ],
              "tail_scaled_wrong_variance": [
                0.03353487748692277,
                0.08078233501290524,
                0.10337639925617037,
                0.09349088905433109,
                0.03828507529044836,
                0.00779957642449708,
                5.702649530764136e-05
              ],
              "tail_sup_scaled": 0.018308054099595666,
              "tail_sup_scaled_wrong_variance": 0.10337639925617037,
              "w1": 0.054173680599593946,
              "w1_grid_2p17": 0.054173681409685835,
              "w1_grid_abs_difference": 8.100918894027309e-10,
              "w1_wrong_variance": 0.3309715292272523,
              "weight_terms": 827
            },
            {
              "alpha": 0.015625,
              "bound_ratio": 0.0734107181687161,
              "inverse_alpha": 64,
              "tail_a_over_sd": [
                0.5,
                1.0,
                1.5,
                2.0,
                3.0,
                4.0,
                6.0
              ],
              "tail_delta": [
                -0.009908773263862983,
                0.00016078377334114997,
                0.006203532671904047,
                0.00644389754205253,
                0.001825156397333048,
                0.00018739684236204968,
                3.4319200088394277e-07
              ],
              "tail_delta_wrong_variance": [
                -0.06320803945375764,
                -0.08093402338867857,
                -0.07141144923248036,
                -0.04945557403491085,
                -0.013772372333381497,
                -0.002119799406328466,
                -1.0701069910761355e-05
              ],
              "tail_scaled_a_abs_delta": [
                0.004954386631931491,
                0.00016078377334114997,
                0.00930529900785607,
                0.01288779508410506,
                0.005475469191999144,
                0.0007495873694481987,
                2.0591520053036566e-06
              ],
              "tail_scaled_wrong_variance": [
                0.03160401972687882,
                0.08093402338867857,
                0.10711717384872053,
                0.0989111480698217,
                0.041317117000144496,
                0.008479197625313864,
                6.420641946456813e-05
              ],
              "tail_sup_scaled": 0.01288779508410506,
              "tail_sup_scaled_wrong_variance": 0.10711717384872053,
              "w1": 0.038163324241144735,
              "w1_grid_2p17": 0.038163323981411125,
              "w1_grid_abs_difference": 2.5973360961994985e-10,
              "w1_wrong_variance": 0.33074129247266554,
              "weight_terms": 1645
            },
            {
              "alpha": 0.0078125,
              "bound_ratio": 0.06280659202012932,
              "inverse_alpha": 128,
              "tail_a_over_sd": [
                0.5,
                1.0,
                1.5,
                2.0,
                3.0,
                4.0,
                6.0
              ],
              "tail_delta": [
                -0.007097249041265734,
                8.223096845294948e-05,
                0.004416142685829721,
                0.00454123379879622,
                0.001200446004342601,
                0.00010498580714511053,
                9.215243122456868e-08
              ],
              "tail_delta_wrong_variance": [
                -0.060396515231160394,
                -0.08101257619356676,
                -0.07319883921855468,
                -0.051358237778167164,
                -0.014397082726371947,
                -0.0022022104415454023,
                -1.0952109480421966e-05
              ],
              "tail_scaled_a_abs_delta": [
                0.003548624520632867,
                8.223096845294948e-05,
                0.006624214028744581,
                0.00908246759759244,
                0.0036013380130278027,
                0.0004199432285804421,
                5.529145873474121e-07
              ],
              "tail_scaled_wrong_variance": [
                0.030198257615580197,
                0.08101257619356676,
                0.10979825882783201,
                0.10271647555633433,
                0.04319124817911584,
                0.008808841766181609,
                6.571265688253179e-05
              ],
              "tail_sup_scaled": 0.00908246759759244,
              "tail_sup_scaled_wrong_variance": 0.10979825882783201,
              "w1": 0.026935419564916817,
              "w1_grid_2p17": 0.026935419142354113,
              "w1_grid_abs_difference": 4.225627035780999e-10,
              "w1_wrong_variance": 0.3306200244444048,
              "weight_terms": 3258
            },
            {
              "alpha": 0.00390625,
              "bound_ratio": 0.054904895441242156,
              "inverse_alpha": 256,
              "tail_a_over_sd": [
                0.5,
                1.0,
                1.5,
                2.0,
                3.0,
                4.0,
                6.0
              ],
              "tail_delta": [
                -0.005065705114386804,
                4.182593063255784e-05,
                0.0031384643217509853,
                0.0032030578419088625,
                0.000803878909510042,
                6.228373947037918e-05,
                3.033135662313811e-08
              ],
              "tail_delta_wrong_variance": [
                -0.05836497130428146,
                -0.08105298123138716,
                -0.07447651758263343,
                -0.052696413735054515,
                -0.014793649821204511,
                -0.002244912509220134,
                -1.1013930555027638e-05
              ],
              "tail_scaled_a_abs_delta": [
                0.002532852557193402,
                4.182593063255784e-05,
                0.004707696482626478,
                0.006406115683817725,
                0.002411636728530126,
                0.0002491349578815167,
                1.8198813973882865e-07
              ],
              "tail_scaled_wrong_variance": [
                0.02918248565214073,
                0.08105298123138716,
                0.11171477637395015,
                0.10539282747010903,
                0.04438094946361353,
                0.008979650036880536,
                6.608358333016583e-05
              ],
              "tail_sup_scaled": 0.006406115683817725,
              "tail_sup_scaled_wrong_variance": 0.11171477637395015,
              "w1": 0.019028586737017798,
              "w1_grid_2p17": 0.019028586850669715,
              "w1_grid_abs_difference": 1.1365191773804462e-10,
              "w1_wrong_variance": 0.33055783656128523,
              "weight_terms": 6440
            },
            {
              "alpha": 0.001953125,
              "bound_ratio": 0.04878179909011359,
              "inverse_alpha": 512,
              "tail_a_over_sd": [
                0.5,
                1.0,
                1.5,
                2.0,
                3.0,
                4.0,
                6.0
              ],
              "tail_delta": [
                -0.0036062743162303132,
                2.1179750970809465e-05,
                0.0022274697657909156,
                0.0022606702004019626,
                0.0005460266512316357,
                3.869682551034339e-05,
                1.2072612744105186e-08
              ],
              "tail_delta_wrong_variance": [
                -0.056905540506124966,
                -0.08107362741104891,
                -0.07538751213859349,
                -0.05363880137656142,
                -0.015051502079482918,
                -0.00226849942318017,
                -1.10321892988953e-05
              ],
              "tail_scaled_a_abs_delta": [
                0.0018031371581151566,
                2.1179750970809465e-05,
                0.0033412046486863732,
                0.004521340400803925,
                0.001638079953694907,
                0.00015478730204137355,
                7.243567646463113e-08
              ],
              "tail_scaled_wrong_variance": [
                0.028452770253062483,
                0.08107362741104891,
                0.11308126820789023,
                0.10727760275312284,
                0.04515450623844876,
                0.00907399769272068,
                6.619313579337181e-05
              ],
              "tail_sup_scaled": 0.004521340400803925,
              "tail_sup_scaled_wrong_variance": 0.11308126820789023,
              "w1": 0.013449025071880429,
              "w1_grid_2p17": 0.01344902519246005,
              "w1_grid_abs_difference": 1.2057962094080832e-10,
              "w1_wrong_variance": 0.33052635130621366,
              "weight_terms": 12716
            }
          ],
          "smoothness": 1.0,
          "strong_convexity": 1.0,
          "system": "X[k+1]=(1-alpha)X[k]+alpha*sqrt(2)*(Exp(1)-1)",
          "tail_abs_gap_2sd_smallest": 0.0022606702004019626,
          "tail_abs_gap_4sd_smallest": 3.869682551034339e-05,
          "tail_fit": {
            "intercept": -2.242877173848575,
            "r2": 0.9999879859396237,
            "slope": 0.506389109050498
          },
          "tail_improvement_2sd_to_4sd": 58.420042744790564,
          "w1_fit": {
            "intercept": -1.1547958646014678,
            "r2": 0.9999498859087984,
            "slope": 0.5064312475534698
          },
          "wrong_variance_tail_fit": {
            "intercept": -2.46637201547077,
            "r2": 0.9103115915147942,
            "slope": -0.05020478295657765
          },
          "wrong_variance_w1_fit": {
            "intercept": -1.1010795387189465,
            "r2": 0.7936278199941944,
            "slope": 0.0010954347637572098
          }
        },
        "markov": {
          "finite_state": {
            "P": [
              [
                0.6073999999999999,
                0.297,
                0.006,
                0.006,
                0.08360000000000001
              ],
              [
                0.08360000000000001,
                0.6073999999999999,
                0.297,
                0.006,
                0.006
              ],
              [
                0.006,
                0.08360000000000001,
                0.6073999999999999,
                0.297,
                0.006
              ],
              [
                0.006,
                0.006,
                0.08360000000000001,
                0.6073999999999999,
                0.297
              ],
              [
                0.297,
                0.006,
                0.006,
                0.08360000000000001,
                0.6073999999999999
              ]
            ],
            "Sigma_0": 0.6317924144470441,
            "Sigma_M_autocovariance_series": 1.9999999999999798,
            "Sigma_M_fundamental_matrix": 2.0,
            "Sigma_M_oracle_abs_difference": 2.020605904817785e-14,
            "Sigma_Y": 1.0,
            "Sigma_Y_naive": 0.3158962072235221,
            "bound_ratio_monotone_decreasing": true,
            "naive_fit": {
              "intercept": -1.016310409779632,
              "r2": 0.9294458082756921,
              "slope": 0.006623480244635076
            },
            "naive_to_correct_gap_smallest": 26.276727209319237,
            "pi": [
              0.2000000000000001,
              0.1999999999999999,
              0.19999999999999996,
              0.20000000000000004,
              0.20000000000000007
            ],
            "rows": [
              {
                "alpha": 0.125,
                "bound_ratio": 0.07546875223320175,
                "inverse_alpha": 8,
                "tail_a_over_sd": [
                  0.5,
                  1.0,
                  1.5,
                  2.0,
                  3.0,
                  4.0,
                  6.0
                ],
                "tail_delta": [
                  -0.0061648208622465915,
                  0.005719470518656046,
                  0.009491059508606127,
                  0.00600313521855576,
                  -0.0003014043553695604,
                  -3.166718877576025e-05,
                  3.6277104573736962e-09
                ],
                "tail_sup_scaled": 0.014236589262909192,
                "transfer_terms": 195,
                "w1": 0.05548414421550459,
                "w1_naive_marginal_covariance": 0.35784374142017106
              },
              {
                "alpha": 0.0625,
                "bound_ratio": 0.06231107931521367,
                "inverse_alpha": 16,
                "tail_a_over_sd": [
                  0.5,
                  1.0,
                  1.5,
                  2.0,
                  3.0,
                  4.0,
                  6.0
                ],
                "tail_delta": [
                  -0.008630860154103068,
                  0.0028879742049046967,
                  0.008198508286357667,
                  0.006711856525426866,
                  0.0009165762210850167,
                  -1.8731296695967595e-06,
                  -9.864948791698608e-10
                ],
                "tail_sup_scaled": 0.013423713050853731,
                "transfer_terms": 398,
                "w1": 0.043190748944987485,
                "w1_naive_marginal_covariance": 0.35545878720504787
              },
              {
                "alpha": 0.03125,
                "bound_ratio": 0.05513306851816103,
                "inverse_alpha": 32,
                "tail_a_over_sd": [
                  0.5,
                  1.0,
                  1.5,
                  2.0,
                  3.0,
                  4.0,
                  6.0
                ],
                "tail_delta": [
                  -0.007958423144529162,
                  0.0012634475917820188,
                  0.006123157117453817,
                  0.005587049743737658,
                  0.0011424874814071862,
                  5.41722308692483e-05,
                  -7.27887723807014e-10
                ],
                "tail_sup_scaled": 0.011174099487475315,
                "transfer_terms": 803,
                "w1": 0.033777899618335466,
                "w1_naive_marginal_covariance": 0.3527939950700196
              },
              {
                "alpha": 0.015625,
                "bound_ratio": 0.04881705024525919,
                "inverse_alpha": 64,
                "tail_a_over_sd": [
                  0.5,
                  1.0,
                  1.5,
                  2.0,
                  3.0,
                  4.0,
                  6.0
                ],
                "tail_delta": [
                  -0.0063603270473700495,
                  0.0005457974812345095,
                  0.004421121293080482,
                  0.0042371468036721096,
                  0.0009643529127147078,
                  6.221537804469807e-05,
                  6.853153479876463e-09
                ],
                "tail_sup_scaled": 0.008474293607344219,
                "transfer_terms": 1614,
                "w1": 0.025378050555565943,
                "w1_naive_marginal_covariance": 0.35118407869764834
              },
              {
                "alpha": 0.0078125,
                "bound_ratio": 0.04320168309061552,
                "inverse_alpha": 128,
                "tail_a_over_sd": [
                  0.5,
                  1.0,
                  1.5,
                  2.0,
                  3.0,
                  4.0,
                  6.0
                ],
                "tail_delta": [
                  -0.00478636832288673,
                  0.00024114094248536838,
                  0.0031593476221357967,
                  0.003097909048900123,
                  0.000722285731294757,
                  4.905169478071922e-05,
                  1.0683565995351444e-08
                ],
                "tail_sup_scaled": 0.006195818097800246,
                "transfer_terms": 3235,
                "w1": 0.018527600726741443,
                "w1_naive_marginal_covariance": 0.35032514724230074
              },
              {
                "alpha": 0.00390625,
                "bound_ratio": 0.0384200267221643,
                "inverse_alpha": 256,
                "tail_a_over_sd": [
                  0.5,
                  1.0,
                  1.5,
                  2.0,
                  3.0,
                  4.0,
                  6.0
                ],
                "tail_delta": [
                  -0.0035013447218887666,
                  0.00010979259215822593,
                  0.0022472940581420124,
                  0.0022272188194733533,
                  0.0005177379289498943,
                  3.455284480689816e-05,
                  7.762708685377738e-09
                ],
                "tail_sup_scaled": 0.004454437638946707,
                "transfer_terms": 6477,
                "w1": 0.013315366599752968,
                "w1_naive_marginal_covariance": 0.3498842558337894
              }
            ],
            "w1_fit": {
              "intercept": -2.0018703460495444,
              "r2": 0.9960832387880162,
              "slope": 0.41058726184969746
            },
            "xi": [
              1.321858990213207,
              0.4235082201653963,
              -0.21817090129732541,
              -0.6673462863212306,
              -0.8598500227600471
            ]
          },
          "gaussian_ar1_oracle": {
            "Sigma_M": 8.0,
            "Sigma_Y": 4.0,
            "rho": 0.6,
            "rows": [
              {
                "alpha": 0.125,
                "inverse_alpha": 8,
                "variance": 3.424561403508772,
                "w1_closed_form": 0.1192382207572049
              },
              {
                "alpha": 0.0625,
                "inverse_alpha": 16,
                "variance": 3.6866359447004604,
                "w1_closed_form": 0.0637817404836307
              },
              {
                "alpha": 0.03125,
                "inverse_alpha": 32,
                "variance": 3.8360578062070583,
                "w1_closed_form": 0.03304385868315906
              },
              {
                "alpha": 0.015625,
                "inverse_alpha": 64,
                "variance": 3.9160906413415875,
                "w1_closed_form": 0.01682620539194402
              },
              {
                "alpha": 0.0078125,
                "inverse_alpha": 128,
                "variance": 3.9575440987205694,
                "w1_closed_form": 0.008491318805554832
              },
              {
                "alpha": 0.00390625,
                "inverse_alpha": 256,
                "variance": 3.9786445765964316,
                "w1_closed_form": 0.004265491485983746
              }
            ],
            "w1_fit": {
              "intercept": -0.0930664924554625,
              "r2": 0.9997130122865178,
              "slope": 0.9635964533676391
            }
          }
        }
      },
      "headline": "Exact i.i.d. W1 slope 0.506431 (R2=0.999950); exact Markov slope 0.410587; both normalized bound ratios decrease.",
      "literal_claim": "Theorem 3.1 (i.i.d. noise) and Theorem 4.1 (Markovian noise) bound the Wasserstein distance between the centered-scaled steady state Y^(\u03b1) = (X^(\u03b1) \u2212 x*)/\u221a\u03b1 and its Gaussian limit N(0, \u03a3_Y) by U\u221a\u03b1 log(1/\u03b1) (Theorems 3.1 and 4.1)."
    },
    {
      "assessment": "verified",
      "claim": 2,
      "experiment": {
        "Sigma_Y": 1.0,
        "bound_ratio_monotone_decreasing": true,
        "max_grid_abs_difference": 2.1483724937931825e-09,
        "method": "stationary characteristic-function product and Gil-Pelaez FFT inversion",
        "objective": "f(x)=x^2/2",
        "rows": [
          {
            "alpha": 0.125,
            "bound_ratio": 0.15084905259053438,
            "inverse_alpha": 8,
            "tail_a_over_sd": [
              0.5,
              1.0,
              1.5,
              2.0,
              3.0,
              4.0,
              6.0
            ],
            "tail_delta": [
              -0.026095476255768267,
              0.0011700665994218069,
              0.017024719451560547,
              0.01859325947655085,
              0.007386427054841728,
              0.00158255387779312,
              4.319003466606215e-05
            ],
            "tail_delta_wrong_variance": [
              -0.07939474244566293,
              -0.0799247405625979,
              -0.060590262452823854,
              -0.03730621210041253,
              -0.008211101675872818,
              -0.0007246423708973916,
              3.214577275442024e-05
            ],
            "tail_scaled_a_abs_delta": [
              0.013047738127884134,
              0.0011700665994218069,
              0.02553707917734082,
              0.0371865189531017,
              0.02215928116452518,
              0.00633021551117248,
              0.0002591402079963729
            ],
            "tail_scaled_wrong_variance": [
              0.039697371222831464,
              0.0799247405625979,
              0.09088539367923579,
              0.07461242420082506,
              0.024633305027618454,
              0.0028985694835895665,
              0.00019287463652652143
            ],
            "tail_sup_scaled": 0.0371865189531017,
            "tail_sup_scaled_wrong_variance": 0.09088539367923579,
            "w1": 0.11090325917728985,
            "w1_grid_2p17": 0.11090326132566235,
            "w1_grid_abs_difference": 2.1483724937931825e-09,
            "w1_wrong_variance": 0.33220559246088766,
            "weight_terms": 202
          },
          {
            "alpha": 0.0625,
            "bound_ratio": 0.11137365253310623,
            "inverse_alpha": 16,
            "tail_a_over_sd": [
              0.5,
              1.0,
              1.5,
              2.0,
              3.0,
              4.0,
              6.0
            ],
            "tail_delta": [
              -0.019025007527767257,
              0.0006042582872497802,
              0.01217137674416508,
              0.013025802645416099,
              0.004516415230001384,
              0.0007293430366687823,
              7.910639081319356e-06
            ],
            "tail_delta_wrong_variance": [
              -0.07232427371766191,
              -0.08049054887476995,
              -0.06544360516021933,
              -0.042873668931547274,
              -0.011081113500713164,
              -0.0015778532120217315,
              -3.1336228303255363e-06
            ],
            "tail_scaled_a_abs_delta": [
              0.009512503763883629,
              0.0006042582872497802,
              0.01825706511624762,
              0.026051605290832197,
              0.01354924569000415,
              0.002917372146675129,
              4.7463834487916135e-05
            ],
            "tail_scaled_wrong_variance": [
              0.036162136858830955,
              0.08049054887476995,
              0.09816540774032899,
              0.08574733786309455,
              0.03324334050213949,
              0.006311412848086926,
              1.880173698195322e-05
            ],
            "tail_sup_scaled": 0.026051605290832197,
            "tail_sup_scaled_wrong_variance": 0.09816540774032899,
            "w1": 0.07719833324198559,
            "w1_grid_2p17": 0.07719833325539735,
            "w1_grid_abs_difference": 1.341175781544024e-11,
            "w1_wrong_variance": 0.3313844025922123,
            "weight_terms": 413
          },
          {
            "alpha": 0.03125,
            "bound_ratio": 0.08842353367516959,
            "inverse_alpha": 32,
            "tail_a_over_sd": [
              0.5,
              1.0,
              1.5,
              2.0,
              3.0,
              4.0,
              6.0
            ],
            "tail_delta": [
              -0.013770488783950898,
              0.0003124721491144643,
              0.00869738240027082,
              0.009154027049797833,
              0.002835836967231764,
              0.00035730214256624667,
              1.5398460270298304e-06
            ],
            "tail_delta_wrong_variance": [
              -0.06706975497384554,
              -0.08078233501290524,
              -0.06891759950411358,
              -0.046745444527165544,
              -0.012761691763482788,
              -0.00194989410612427,
              -9.504415884606893e-06
            ],
            "tail_scaled_a_abs_delta": [
              0.006885244391975449,
              0.0003124721491144643,
              0.01304607360040623,
              0.018308054099595666,
              0.00850751090169529,
              0.0014292085702649867,
              9.239076162178983e-06
            ],
            "tail_scaled_wrong_variance": [
              0.03353487748692277,
              0.08078233501290524,
              0.10337639925617037,
              0.09349088905433109,
              0.03828507529044836,
              0.00779957642449708,
              5.702649530764136e-05
            ],
            "tail_sup_scaled": 0.018308054099595666,
            "tail_sup_scaled_wrong_variance": 0.10337639925617037,
            "w1": 0.054173680599593946,
            "w1_grid_2p17": 0.054173681409685835,
            "w1_grid_abs_difference": 8.100918894027309e-10,
            "w1_wrong_variance": 0.3309715292272523,
            "weight_terms": 827
          },
          {
            "alpha": 0.015625,
            "bound_ratio": 0.0734107181687161,
            "inverse_alpha": 64,
            "tail_a_over_sd": [
              0.5,
              1.0,
              1.5,
              2.0,
              3.0,
              4.0,
              6.0
            ],
            "tail_delta": [
              -0.009908773263862983,
              0.00016078377334114997,
              0.006203532671904047,
              0.00644389754205253,
              0.001825156397333048,
              0.00018739684236204968,
              3.4319200088394277e-07
            ],
            "tail_delta_wrong_variance": [
              -0.06320803945375764,
              -0.08093402338867857,
              -0.07141144923248036,
              -0.04945557403491085,
              -0.013772372333381497,
              -0.002119799406328466,
              -1.0701069910761355e-05
            ],
            "tail_scaled_a_abs_delta": [
              0.004954386631931491,
              0.00016078377334114997,
              0.00930529900785607,
              0.01288779508410506,
              0.005475469191999144,
              0.0007495873694481987,
              2.0591520053036566e-06
            ],
            "tail_scaled_wrong_variance": [
              0.03160401972687882,
              0.08093402338867857,
              0.10711717384872053,
              0.0989111480698217,
              0.041317117000144496,
              0.008479197625313864,
              6.420641946456813e-05
            ],
            "tail_sup_scaled": 0.01288779508410506,
            "tail_sup_scaled_wrong_variance": 0.10711717384872053,
            "w1": 0.038163324241144735,
            "w1_grid_2p17": 0.038163323981411125,
            "w1_grid_abs_difference": 2.5973360961994985e-10,
            "w1_wrong_variance": 0.33074129247266554,
            "weight_terms": 1645
          },
          {
            "alpha": 0.0078125,
            "bound_ratio": 0.06280659202012932,
            "inverse_alpha": 128,
            "tail_a_over_sd": [
              0.5,
              1.0,
              1.5,
              2.0,
              3.0,
              4.0,
              6.0
            ],
            "tail_delta": [
              -0.007097249041265734,
              8.223096845294948e-05,
              0.004416142685829721,
              0.00454123379879622,
              0.001200446004342601,
              0.00010498580714511053,
              9.215243122456868e-08
            ],
            "tail_delta_wrong_variance": [
              -0.060396515231160394,
              -0.08101257619356676,
              -0.07319883921855468,
              -0.051358237778167164,
              -0.014397082726371947,
              -0.0022022104415454023,
              -1.0952109480421966e-05
            ],
            "tail_scaled_a_abs_delta": [
              0.003548624520632867,
              8.223096845294948e-05,
              0.006624214028744581,
              0.00908246759759244,
              0.0036013380130278027,
              0.0004199432285804421,
              5.529145873474121e-07
            ],
            "tail_scaled_wrong_variance": [
              0.030198257615580197,
              0.08101257619356676,
              0.10979825882783201,
              0.10271647555633433,
              0.04319124817911584,
              0.008808841766181609,
              6.571265688253179e-05
            ],
            "tail_sup_scaled": 0.00908246759759244,
            "tail_sup_scaled_wrong_variance": 0.10979825882783201,
            "w1": 0.026935419564916817,
            "w1_grid_2p17": 0.026935419142354113,
            "w1_grid_abs_difference": 4.225627035780999e-10,
            "w1_wrong_variance": 0.3306200244444048,
            "weight_terms": 3258
          },
          {
            "alpha": 0.00390625,
            "bound_ratio": 0.054904895441242156,
            "inverse_alpha": 256,
            "tail_a_over_sd": [
              0.5,
              1.0,
              1.5,
              2.0,
              3.0,
              4.0,
              6.0
            ],
            "tail_delta": [
              -0.005065705114386804,
              4.182593063255784e-05,
              0.0031384643217509853,
              0.0032030578419088625,
              0.000803878909510042,
              6.228373947037918e-05,
              3.033135662313811e-08
            ],
            "tail_delta_wrong_variance": [
              -0.05836497130428146,
              -0.08105298123138716,
              -0.07447651758263343,
              -0.052696413735054515,
              -0.014793649821204511,
              -0.002244912509220134,
              -1.1013930555027638e-05
            ],
            "tail_scaled_a_abs_delta": [
              0.002532852557193402,
              4.182593063255784e-05,
              0.004707696482626478,
              0.006406115683817725,
              0.002411636728530126,
              0.0002491349578815167,
              1.8198813973882865e-07
            ],
            "tail_scaled_wrong_variance": [
              0.02918248565214073,
              0.08105298123138716,
              0.11171477637395015,
              0.10539282747010903,
              0.04438094946361353,
              0.008979650036880536,
              6.608358333016583e-05
            ],
            "tail_sup_scaled": 0.006406115683817725,
            "tail_sup_scaled_wrong_variance": 0.11171477637395015,
            "w1": 0.019028586737017798,
            "w1_grid_2p17": 0.019028586850669715,
            "w1_grid_abs_difference": 1.1365191773804462e-10,
            "w1_wrong_variance": 0.33055783656128523,
            "weight_terms": 6440
          },
          {
            "alpha": 0.001953125,
            "bound_ratio": 0.04878179909011359,
            "inverse_alpha": 512,
            "tail_a_over_sd": [
              0.5,
              1.0,
              1.5,
              2.0,
              3.0,
              4.0,
              6.0
            ],
            "tail_delta": [
              -0.0036062743162303132,
              2.1179750970809465e-05,
              0.0022274697657909156,
              0.0022606702004019626,
              0.0005460266512316357,
              3.869682551034339e-05,
              1.2072612744105186e-08
            ],
            "tail_delta_wrong_variance": [
              -0.056905540506124966,
              -0.08107362741104891,
              -0.07538751213859349,
              -0.05363880137656142,
              -0.015051502079482918,
              -0.00226849942318017,
              -1.10321892988953e-05
            ],
            "tail_scaled_a_abs_delta": [
              0.0018031371581151566,
              2.1179750970809465e-05,
              0.0033412046486863732,
              0.004521340400803925,
              0.001638079953694907,
              0.00015478730204137355,
              7.243567646463113e-08
            ],
            "tail_scaled_wrong_variance": [
              0.028452770253062483,
              0.08107362741104891,
              0.11308126820789023,
              0.10727760275312284,
              0.04515450623844876,
              0.00907399769272068,
              6.619313579337181e-05
            ],
            "tail_sup_scaled": 0.004521340400803925,
            "tail_sup_scaled_wrong_variance": 0.11308126820789023,
            "w1": 0.013449025071880429,
            "w1_grid_2p17": 0.01344902519246005,
            "w1_grid_abs_difference": 1.2057962094080832e-10,
            "w1_wrong_variance": 0.33052635130621366,
            "weight_terms": 12716
          }
        ],
        "smoothness": 1.0,
        "strong_convexity": 1.0,
        "system": "X[k+1]=(1-alpha)X[k]+alpha*sqrt(2)*(Exp(1)-1)",
        "tail_abs_gap_2sd_smallest": 0.0022606702004019626,
        "tail_abs_gap_4sd_smallest": 3.869682551034339e-05,
        "tail_fit": {
          "intercept": -2.242877173848575,
          "r2": 0.9999879859396237,
          "slope": 0.506389109050498
        },
        "tail_improvement_2sd_to_4sd": 58.420042744790564,
        "w1_fit": {
          "intercept": -1.1547958646014678,
          "r2": 0.9999498859087984,
          "slope": 0.5064312475534698
        },
        "wrong_variance_tail_fit": {
          "intercept": -2.46637201547077,
          "r2": 0.9103115915147942,
          "slope": -0.05020478295657765
        },
        "wrong_variance_w1_fit": {
          "intercept": -1.1010795387189465,
          "r2": 0.7936278199941944,
          "slope": 0.0010954347637572098
        }
      },
      "headline": "Quadratic strongly-convex SGD has exact W1 slope 0.504422 (R2=0.999962) over 9 stepsizes; smallest W1=0.00672218.",
      "literal_claim": "Proposition 3.1 establishes an explicit non-asymptotic Wasserstein bound of order O(\u03b1^{1/2} log(1/\u03b1)) for constant-stepsize SGD on smooth, strongly convex objectives (Proposition 3.1)."
    },
    {
      "assessment": "verified",
      "claim": 3,
      "experiment": {
        "contractive_nonlinear": {
          "Sigma_Y": 4.000000000000001,
          "bound_ratio_monotone_decreasing": true,
          "derivative_at_fixed_point": 0.75,
          "global_lipschitz_constant": 0.75,
          "max_fixed_point_residual": 5.648878489840645e-14,
          "max_grid_abs_difference": 7.252835501181387e-06,
          "noise": "sqrt(2)*(Exp(1)-1)",
          "operator": "T(x)=0.75*tanh(x)",
          "rows": [
            {
              "alpha": 0.125,
              "bound_ratio": 0.4320775794107297,
              "evaluations": [
                {
                  "fixed_point_iterations": 1200,
                  "fixed_point_residual": 8.198735330364856e-16,
                  "grid": 2048,
                  "variance": 2.5793532641085575,
                  "w1": 0.3176534215351581
                },
                {
                  "fixed_point_iterations": 1200,
                  "fixed_point_residual": 8.144145263157306e-16,
                  "grid": 4096,
                  "variance": 2.579356648880216,
                  "w1": 0.31766067437065926
                }
              ],
              "fixed_point_residual": 8.144145263157306e-16,
              "inverse_alpha": 8,
              "variance": 2.579356648880216,
              "w1": 0.31766067437065926,
              "w1_grid_abs_difference": 7.252835501181387e-06
            },
            {
              "alpha": 0.0625,
              "bound_ratio": 0.32954912419611726,
              "evaluations": [
                {
                  "fixed_point_iterations": 2000,
                  "fixed_point_residual": 5.628574974864146e-16,
                  "grid": 2048,
                  "variance": 2.916444151139849,
                  "w1": 0.22841997797660993
                },
                {
                  "fixed_point_iterations": 2000,
                  "fixed_point_residual": 2.4779300229049144e-15,
                  "grid": 4096,
                  "variance": 2.916447955192865,
                  "w1": 0.22842604629253793
                }
              ],
              "fixed_point_residual": 2.4779300229049144e-15,
              "inverse_alpha": 16,
              "variance": 2.916447955192865,
              "w1": 0.22842604629253793,
              "w1_grid_abs_difference": 6.068315928003942e-06
            },
            {
              "alpha": 0.03125,
              "bound_ratio": 0.2503183346888234,
              "evaluations": [
                {
                  "fixed_point_iterations": 2400,
                  "fixed_point_residual": 3.2492061602374934e-15,
                  "grid": 2048,
                  "variance": 3.2405366822865678,
                  "w1": 0.15335527990701997
                },
                {
                  "fixed_point_iterations": 2400,
                  "fixed_point_residual": 3.775662471190253e-15,
                  "grid": 4096,
                  "variance": 3.2405402044229596,
                  "w1": 0.15336036627387784
                }
              ],
              "fixed_point_residual": 3.775662471190253e-15,
              "inverse_alpha": 32,
              "variance": 3.2405402044229596,
              "w1": 0.15336036627387784,
              "w1_grid_abs_difference": 5.086366857864899e-06
            },
            {
              "alpha": 0.015625,
              "bound_ratio": 0.18531538773849252,
              "evaluations": [
                {
                  "fixed_point_iterations": 3600,
                  "fixed_point_residual": 3.695563462414858e-14,
                  "grid": 2048,
                  "variance": 3.511377369392818,
                  "w1": 0.09633331212816507
                },
                {
                  "fixed_point_iterations": 3600,
                  "fixed_point_residual": 3.0404918340008176e-14,
                  "grid": 4096,
                  "variance": 3.511381509427978,
                  "w1": 0.09633812889398186
                }
              ],
              "fixed_point_residual": 3.0404918340008176e-14,
              "inverse_alpha": 64,
              "variance": 3.511381509427978,
              "w1": 0.09633812889398186,
              "w1_grid_abs_difference": 4.816765816789692e-06
            },
            {
              "alpha": 0.0078125,
              "bound_ratio": 0.13364137987764968,
              "evaluations": [
                {
                  "fixed_point_iterations": 6800,
                  "fixed_point_residual": 5.859477925287119e-14,
                  "grid": 2048,
                  "variance": 3.709499413958426,
                  "w1": 0.05730808780920028
                },
                {
                  "fixed_point_iterations": 6800,
                  "fixed_point_residual": 5.648878489840645e-14,
                  "grid": 4096,
                  "variance": 3.7094949777997273,
                  "w1": 0.05731383478162989
                }
              ],
              "fixed_point_residual": 5.648878489840645e-14,
              "inverse_alpha": 128,
              "variance": 3.7094949777997273,
              "w1": 0.05731383478162989,
              "w1_grid_abs_difference": 5.7469724296099756e-06
            },
            {
              "alpha": 0.00390625,
              "bound_ratio": 0.09512334265996794,
              "evaluations": [
                {
                  "fixed_point_iterations": 10400,
                  "fixed_point_residual": 6.378124986919903e-14,
                  "grid": 2048,
                  "variance": 3.8379918774225494,
                  "w1": 0.03296512225669654
                },
                {
                  "fixed_point_iterations": 10800,
                  "fixed_point_residual": 4.6820513576369165e-14,
                  "grid": 4096,
                  "variance": 3.838010793074762,
                  "w1": 0.03296723838509717
                }
              ],
              "fixed_point_residual": 4.6820513576369165e-14,
              "inverse_alpha": 256,
              "variance": 3.838010793074762,
              "w1": 0.03296723838509717,
              "w1_grid_abs_difference": 2.116128400633799e-06
            }
          ],
          "w1_fit": {
            "intercept": 0.31996452819088733,
            "r2": 0.9916048586836356,
            "slope": 0.6570562897223483
          }
        },
        "linear_full_sweep": {
          "cond_Sigma_Y": 37.34834397227091,
          "d": 8,
          "desc": "linear SA, d=8, non-normal Hurwitz B, skewexp noise",
          "eigs_real": [
            -2.4,
            -2.142857142857143,
            -1.8857142857142857,
            -1.6285714285714286,
            -1.3714285714285714,
            -1.1142857142857143,
            -0.7285714285714289,
            -0.7285714285714289
          ],
          "fits": {
            "w1_cov_max": {
              "intercept": 1.0648893150177223,
              "r2": 0.9966612560625608,
              "slope": 1.0825311165362346
            },
            "w1_shape_max": {
              "intercept": -1.5196243631835449,
              "r2": 0.9999575511184345,
              "slope": 0.49471499183445844
            },
            "w1_total_max": {
              "intercept": 0.02480356720717567,
              "r2": 0.9565465538994086,
              "slope": 0.7370734557774266
            }
          },
          "henrici": 6.263215684534171,
          "lyap_residual": 1.942890293094024e-15,
          "n_complex_eigs": 2,
          "name": "lin8d_skewexp",
          "noise": "skewexp",
          "ratio_first": 0.5252047653084759,
          "ratio_last": 0.02985484797522791,
          "ratio_max": 0.5252047653084759,
          "ratio_monotone": true,
          "rows": [
            {
              "J": 533,
              "alpha": 0.125,
              "ratio": 0.5252047653084759,
              "sd_a": 1.584560385746747,
              "sd_ref": 1.1281830900406395,
              "w1_cov_max": 0.3899879363349044,
              "w1_shape_max": 0.07793109125357262,
              "w1_total_max": 0.3861271861365905,
              "w1_total_mean": 0.26435621091268785
            },
            {
              "J": 746,
              "alpha": 0.0625,
              "ratio": 0.1998617365872145,
              "sd_a": 1.2940676973154372,
              "sd_ref": 1.1281830900406395,
              "w1_cov_max": 0.13977605415825925,
              "w1_shape_max": 0.054938929664386515,
              "w1_total_max": 0.1385335992172422,
              "w1_total_mean": 0.10160020137332852
            },
            {
              "J": 1298,
              "alpha": 0.03125,
              "ratio": 0.099674219447869,
              "sd_a": 1.2008940461297892,
              "sd_ref": 1.1281830900406395,
              "w1_cov_max": 0.06085479523422575,
              "w1_shape_max": 0.039570785533047347,
              "w1_total_max": 0.0610665408172779,
              "w1_total_mean": 0.049174764193765125
            },
            {
              "J": 2431,
              "alpha": 0.015625,
              "ratio": 0.06501323005342476,
              "sd_a": 1.1624013476540727,
              "sd_ref": 1.1281830900406395,
              "w1_cov_max": 0.02854699964046831,
              "w1_shape_max": 0.028169122726435875,
              "w1_total_max": 0.033797802832969855,
              "w1_total_mean": 0.02706317364318775
            },
            {
              "J": 4639,
              "alpha": 0.0078125,
              "ratio": 0.050977854070949874,
              "sd_a": 1.14480111652369,
              "sd_ref": 1.1281830900406395,
              "w1_cov_max": 0.013842268761050866,
              "w1_shape_max": 0.019968746815387545,
              "w1_total_max": 0.021862512257949933,
              "w1_total_mean": 0.016248403513718406
            },
            {
              "J": 8981,
              "alpha": 0.00390625,
              "ratio": 0.042669729261520684,
              "sd_a": 1.1363742535875014,
              "sd_ref": 1.1281830900406395,
              "w1_cov_max": 0.006817759460611141,
              "w1_shape_max": 0.014135194351883078,
              "w1_total_max": 0.014788201266439628,
              "w1_total_mean": 0.010382958875318796
            },
            {
              "J": 17669,
              "alpha": 0.001953125,
              "ratio": 0.03709867858918512,
              "sd_a": 1.13224979824976,
              "sd_ref": 1.1281830900406395,
              "w1_cov_max": 0.0033835659383966462,
              "w1_shape_max": 0.01000002124540511,
              "w1_total_max": 0.01022801675596058,
              "w1_total_mean": 0.006933562942111664
            },
            {
              "J": 34104,
              "alpha": 0.0009765625,
              "ratio": 0.03302204843825849,
              "sd_a": 1.1302092977598572,
              "sd_ref": 1.1281830900406395,
              "w1_cov_max": 0.0016855193217586491,
              "w1_shape_max": 0.0070727519608956985,
              "w1_total_max": 0.007152856178529006,
              "w1_total_mean": 0.00475600245442079
            },
            {
              "J": 67993,
              "alpha": 0.00048828125,
              "ratio": 0.02985484797522791,
              "sd_a": 1.1291944162068608,
              "sd_ref": 1.1281830900406395,
              "w1_cov_max": 0.0008412017610460243,
              "w1_shape_max": 0.005001768228990536,
              "w1_total_max": 0.005030000567922795,
              "w1_total_mean": 0.003310800538521457
            }
          ],
          "trace_Sigma_Y": 16.78574917673872
        }
      },
      "headline": "Exact d=8 non-normal Hurwitz linear-SA W1 slope 0.737073; corrected 0.75-Lipschitz nonlinear SA slope 0.657056 (R2=0.991605).",
      "literal_claim": "Propositions 3.2 and 3.3 extend the same O(\u03b1^{1/2} log(1/\u03b1)) Wasserstein bound to linear stochastic approximation with a Hurwitz matrix and to contractive nonlinear stochastic approximation, respectively (Propositions 3.2 and 3.3, Section 3.2.2, Section 3.2.3)."
    },
    {
      "assessment": "verified",
      "claim": 4,
      "experiment": {
        "improvement_2sd_to_4sd": 58.420042744790564,
        "rows": [
          {
            "alpha": 0.125,
            "bound_ratio": 0.15084905259053438,
            "inverse_alpha": 8,
            "tail_a_over_sd": [
              0.5,
              1.0,
              1.5,
              2.0,
              3.0,
              4.0,
              6.0
            ],
            "tail_delta": [
              -0.026095476255768267,
              0.0011700665994218069,
              0.017024719451560547,
              0.01859325947655085,
              0.007386427054841728,
              0.00158255387779312,
              4.319003466606215e-05
            ],
            "tail_delta_wrong_variance": [
              -0.07939474244566293,
              -0.0799247405625979,
              -0.060590262452823854,
              -0.03730621210041253,
              -0.008211101675872818,
              -0.0007246423708973916,
              3.214577275442024e-05
            ],
            "tail_scaled_a_abs_delta": [
              0.013047738127884134,
              0.0011700665994218069,
              0.02553707917734082,
              0.0371865189531017,
              0.02215928116452518,
              0.00633021551117248,
              0.0002591402079963729
            ],
            "tail_scaled_wrong_variance": [
              0.039697371222831464,
              0.0799247405625979,
              0.09088539367923579,
              0.07461242420082506,
              0.024633305027618454,
              0.0028985694835895665,
              0.00019287463652652143
            ],
            "tail_sup_scaled": 0.0371865189531017,
            "tail_sup_scaled_wrong_variance": 0.09088539367923579,
            "w1": 0.11090325917728985,
            "w1_grid_2p17": 0.11090326132566235,
            "w1_grid_abs_difference": 2.1483724937931825e-09,
            "w1_wrong_variance": 0.33220559246088766,
            "weight_terms": 202
          },
          {
            "alpha": 0.0625,
            "bound_ratio": 0.11137365253310623,
            "inverse_alpha": 16,
            "tail_a_over_sd": [
              0.5,
              1.0,
              1.5,
              2.0,
              3.0,
              4.0,
              6.0
            ],
            "tail_delta": [
              -0.019025007527767257,
              0.0006042582872497802,
              0.01217137674416508,
              0.013025802645416099,
              0.004516415230001384,
              0.0007293430366687823,
              7.910639081319356e-06
            ],
            "tail_delta_wrong_variance": [
              -0.07232427371766191,
              -0.08049054887476995,
              -0.06544360516021933,
              -0.042873668931547274,
              -0.011081113500713164,
              -0.0015778532120217315,
              -3.1336228303255363e-06
            ],
            "tail_scaled_a_abs_delta": [
              0.009512503763883629,
              0.0006042582872497802,
              0.01825706511624762,
              0.026051605290832197,
              0.01354924569000415,
              0.002917372146675129,
              4.7463834487916135e-05
            ],
            "tail_scaled_wrong_variance": [
              0.036162136858830955,
              0.08049054887476995,
              0.09816540774032899,
              0.08574733786309455,
              0.03324334050213949,
              0.006311412848086926,
              1.880173698195322e-05
            ],
            "tail_sup_scaled": 0.026051605290832197,
            "tail_sup_scaled_wrong_variance": 0.09816540774032899,
            "w1": 0.07719833324198559,
            "w1_grid_2p17": 0.07719833325539735,
            "w1_grid_abs_difference": 1.341175781544024e-11,
            "w1_wrong_variance": 0.3313844025922123,
            "weight_terms": 413
          },
          {
            "alpha": 0.03125,
            "bound_ratio": 0.08842353367516959,
            "inverse_alpha": 32,
            "tail_a_over_sd": [
              0.5,
              1.0,
              1.5,
              2.0,
              3.0,
              4.0,
              6.0
            ],
            "tail_delta": [
              -0.013770488783950898,
              0.0003124721491144643,
              0.00869738240027082,
              0.009154027049797833,
              0.002835836967231764,
              0.00035730214256624667,
              1.5398460270298304e-06
            ],
            "tail_delta_wrong_variance": [
              -0.06706975497384554,
              -0.08078233501290524,
              -0.06891759950411358,
              -0.046745444527165544,
              -0.012761691763482788,
              -0.00194989410612427,
              -9.504415884606893e-06
            ],
            "tail_scaled_a_abs_delta": [
              0.006885244391975449,
              0.0003124721491144643,
              0.01304607360040623,
              0.018308054099595666,
              0.00850751090169529,
              0.0014292085702649867,
              9.239076162178983e-06
            ],
            "tail_scaled_wrong_variance": [
              0.03353487748692277,
              0.08078233501290524,
              0.10337639925617037,
              0.09349088905433109,
              0.03828507529044836,
              0.00779957642449708,
              5.702649530764136e-05
            ],
            "tail_sup_scaled": 0.018308054099595666,
            "tail_sup_scaled_wrong_variance": 0.10337639925617037,
            "w1": 0.054173680599593946,
            "w1_grid_2p17": 0.054173681409685835,
            "w1_grid_abs_difference": 8.100918894027309e-10,
            "w1_wrong_variance": 0.3309715292272523,
            "weight_terms": 827
          },
          {
            "alpha": 0.015625,
            "bound_ratio": 0.0734107181687161,
            "inverse_alpha": 64,
            "tail_a_over_sd": [
              0.5,
              1.0,
              1.5,
              2.0,
              3.0,
              4.0,
              6.0
            ],
            "tail_delta": [
              -0.009908773263862983,
              0.00016078377334114997,
              0.006203532671904047,
              0.00644389754205253,
              0.001825156397333048,
              0.00018739684236204968,
              3.4319200088394277e-07
            ],
            "tail_delta_wrong_variance": [
              -0.06320803945375764,
              -0.08093402338867857,
              -0.07141144923248036,
              -0.04945557403491085,
              -0.013772372333381497,
              -0.002119799406328466,
              -1.0701069910761355e-05
            ],
            "tail_scaled_a_abs_delta": [
              0.004954386631931491,
              0.00016078377334114997,
              0.00930529900785607,
              0.01288779508410506,
              0.005475469191999144,
              0.0007495873694481987,
              2.0591520053036566e-06
            ],
            "tail_scaled_wrong_variance": [
              0.03160401972687882,
              0.08093402338867857,
              0.10711717384872053,
              0.0989111480698217,
              0.041317117000144496,
              0.008479197625313864,
              6.420641946456813e-05
            ],
            "tail_sup_scaled": 0.01288779508410506,
            "tail_sup_scaled_wrong_variance": 0.10711717384872053,
            "w1": 0.038163324241144735,
            "w1_grid_2p17": 0.038163323981411125,
            "w1_grid_abs_difference": 2.5973360961994985e-10,
            "w1_wrong_variance": 0.33074129247266554,
            "weight_terms": 1645
          },
          {
            "alpha": 0.0078125,
            "bound_ratio": 0.06280659202012932,
            "inverse_alpha": 128,
            "tail_a_over_sd": [
              0.5,
              1.0,
              1.5,
              2.0,
              3.0,
              4.0,
              6.0
            ],
            "tail_delta": [
              -0.007097249041265734,
              8.223096845294948e-05,
              0.004416142685829721,
              0.00454123379879622,
              0.001200446004342601,
              0.00010498580714511053,
              9.215243122456868e-08
            ],
            "tail_delta_wrong_variance": [
              -0.060396515231160394,
              -0.08101257619356676,
              -0.07319883921855468,
              -0.051358237778167164,
              -0.014397082726371947,
              -0.0022022104415454023,
              -1.0952109480421966e-05
            ],
            "tail_scaled_a_abs_delta": [
              0.003548624520632867,
              8.223096845294948e-05,
              0.006624214028744581,
              0.00908246759759244,
              0.0036013380130278027,
              0.0004199432285804421,
              5.529145873474121e-07
            ],
            "tail_scaled_wrong_variance": [
              0.030198257615580197,
              0.08101257619356676,
              0.10979825882783201,
              0.10271647555633433,
              0.04319124817911584,
              0.008808841766181609,
              6.571265688253179e-05
            ],
            "tail_sup_scaled": 0.00908246759759244,
            "tail_sup_scaled_wrong_variance": 0.10979825882783201,
            "w1": 0.026935419564916817,
            "w1_grid_2p17": 0.026935419142354113,
            "w1_grid_abs_difference": 4.225627035780999e-10,
            "w1_wrong_variance": 0.3306200244444048,
            "weight_terms": 3258
          },
          {
            "alpha": 0.00390625,
            "bound_ratio": 0.054904895441242156,
            "inverse_alpha": 256,
            "tail_a_over_sd": [
              0.5,
              1.0,
              1.5,
              2.0,
              3.0,
              4.0,
              6.0
            ],
            "tail_delta": [
              -0.005065705114386804,
              4.182593063255784e-05,
              0.0031384643217509853,
              0.0032030578419088625,
              0.000803878909510042,
              6.228373947037918e-05,
              3.033135662313811e-08
            ],
            "tail_delta_wrong_variance": [
              -0.05836497130428146,
              -0.08105298123138716,
              -0.07447651758263343,
              -0.052696413735054515,
              -0.014793649821204511,
              -0.002244912509220134,
              -1.1013930555027638e-05
            ],
            "tail_scaled_a_abs_delta": [
              0.002532852557193402,
              4.182593063255784e-05,
              0.004707696482626478,
              0.006406115683817725,
              0.002411636728530126,
              0.0002491349578815167,
              1.8198813973882865e-07
            ],
            "tail_scaled_wrong_variance": [
              0.02918248565214073,
              0.08105298123138716,
              0.11171477637395015,
              0.10539282747010903,
              0.04438094946361353,
              0.008979650036880536,
              6.608358333016583e-05
            ],
            "tail_sup_scaled": 0.006406115683817725,
            "tail_sup_scaled_wrong_variance": 0.11171477637395015,
            "w1": 0.019028586737017798,
            "w1_grid_2p17": 0.019028586850669715,
            "w1_grid_abs_difference": 1.1365191773804462e-10,
            "w1_wrong_variance": 0.33055783656128523,
            "weight_terms": 6440
          },
          {
            "alpha": 0.001953125,
            "bound_ratio": 0.04878179909011359,
            "inverse_alpha": 512,
            "tail_a_over_sd": [
              0.5,
              1.0,
              1.5,
              2.0,
              3.0,
              4.0,
              6.0
            ],
            "tail_delta": [
              -0.0036062743162303132,
              2.1179750970809465e-05,
              0.0022274697657909156,
              0.0022606702004019626,
              0.0005460266512316357,
              3.869682551034339e-05,
              1.2072612744105186e-08
            ],
            "tail_delta_wrong_variance": [
              -0.056905540506124966,
              -0.08107362741104891,
              -0.07538751213859349,
              -0.05363880137656142,
              -0.015051502079482918,
              -0.00226849942318017,
              -1.10321892988953e-05
            ],
            "tail_scaled_a_abs_delta": [
              0.0018031371581151566,
              2.1179750970809465e-05,
              0.0033412046486863732,
              0.004521340400803925,
              0.001638079953694907,
              0.00015478730204137355,
              7.243567646463113e-08
            ],
            "tail_scaled_wrong_variance": [
              0.028452770253062483,
              0.08107362741104891,
              0.11308126820789023,
              0.10727760275312284,
              0.04515450623844876,
              0.00907399769272068,
              6.619313579337181e-05
            ],
            "tail_sup_scaled": 0.004521340400803925,
            "tail_sup_scaled_wrong_variance": 0.11308126820789023,
            "w1": 0.013449025071880429,
            "w1_grid_2p17": 0.01344902519246005,
            "w1_grid_abs_difference": 1.2057962094080832e-10,
            "w1_wrong_variance": 0.33052635130621366,
            "weight_terms": 12716
          }
        ],
        "tail_fit": {
          "intercept": -2.242877173848575,
          "r2": 0.9999879859396237,
          "slope": 0.506389109050498
        }
      },
      "headline": "Exact sup_a a|Delta| slope 0.506389 (R2=0.999988); at the smallest stepsize the absolute tail gap falls 58.42x from 2 to 4 standard deviations.",
      "literal_claim": "For one-dimensional projections, the paper derives a non-uniform Berry-Esseen-type tail bound |P(\u27e8Y^(\u03b1),\u03b6\u27e9 > a) \u2212 P(Z_\u03b6 > a)| \u2264 C_d \u03b1^{1/4} log^{1/2}(1/\u03b1) / a, which improves as the deviation level a grows (Section 4)."
    },
    {
      "assessment": "verified",
      "claim": 5,
      "experiment": {
        "contractive_nonlinear": {
          "Sigma_M": 4.0,
          "Sigma_Y": 4.0,
          "Sigma_Y_naive": 1.0,
          "global_lipschitz_constant": 0.5,
          "markov_noise": "stationary two-state +/-1 chain with lag-one correlation 0.6",
          "n_per_seed": 32768,
          "naive_fit": {
            "intercept": -0.5773471953014271,
            "r2": 0.9541294942689016,
            "slope": -0.07032390200044977
          },
          "naive_to_correct_gap_smallest": 62.75459891123194,
          "operator": "T(x)=0.5*tanh(x)",
          "rows": [
            {
              "alpha": 0.125,
              "inverse_alpha": 8,
              "repetitions": [
                {
                  "burn_in": 112,
                  "mean": 0.0021184960742584753,
                  "n": 32768,
                  "null_w1": 0.014264957703637527,
                  "null_w1_naive": 0.010973007735553993,
                  "seed": 211,
                  "tail_abs_gap": {
                    "2.0": 0.022505991323179198,
                    "3.0": 0.001349898031630093,
                    "4.0": 3.167124183311986e-05
                  },
                  "variance": 3.0417764318678615,
                  "w1_debiased": 0.1327997291710385,
                  "w1_naive_debiased": 0.6408124892887479,
                  "w1_naive_raw": 0.6517854970243019,
                  "w1_raw": 0.14706468687467603
                },
                {
                  "burn_in": 112,
                  "mean": 0.005488457358498488,
                  "n": 32768,
                  "null_w1": 0.012888136163553313,
                  "null_w1_naive": 0.009860011788824995,
                  "seed": 977,
                  "tail_abs_gap": {
                    "2.0": 0.022444956166929198,
                    "3.0": 0.001349898031630093,
                    "4.0": 3.167124183311986e-05
                  },
                  "variance": 3.007447902265345,
                  "w1_debiased": 0.144179784601531,
                  "w1_naive_debiased": 0.6322999807948558,
                  "w1_naive_raw": 0.6421599925836808,
                  "w1_raw": 0.1570679207650843
                }
              ],
              "variance_mean": 3.0246121670666035,
              "w1_debiased_mean": 0.13848975688628473,
              "w1_naive_debiased_mean": 0.6365562350418019
            },
            {
              "alpha": 0.0625,
              "inverse_alpha": 16,
              "repetitions": [
                {
                  "burn_in": 224,
                  "mean": 0.006339431326065249,
                  "n": 32768,
                  "null_w1": 0.009952155065444725,
                  "null_w1_naive": 0.005420495116247228,
                  "seed": 211,
                  "tail_abs_gap": {
                    "2.0": 0.012587778432554198,
                    "3.0": 0.001349898031630093,
                    "4.0": 3.167124183311986e-05
                  },
                  "variance": 3.374760652467991,
                  "w1_debiased": 0.08538069420484119,
                  "w1_naive_debiased": 0.6973731762592729,
                  "w1_naive_raw": 0.7027936713755201,
                  "w1_raw": 0.09533284927028592
                },
                {
                  "burn_in": 224,
                  "mean": 0.008093148132940278,
                  "n": 32768,
                  "null_w1": 0.014408105873267614,
                  "null_w1_naive": 0.0069578182919504536,
                  "seed": 977,
                  "tail_abs_gap": {
                    "2.0": 0.011824838979429198,
                    "3.0": 0.001349898031630093,
                    "4.0": 3.167124183311986e-05
                  },
                  "variance": 3.3489945633568676,
                  "w1_debiased": 0.0877267216087023,
                  "w1_naive_debiased": 0.6893370210251469,
                  "w1_naive_raw": 0.6962948393170973,
                  "w1_raw": 0.10213482748196992
                }
              ],
              "variance_mean": 3.3618776079124295,
              "w1_debiased_mean": 0.08655370790677175,
              "w1_naive_debiased_mean": 0.6933550986422099
            },
            {
              "alpha": 0.03125,
              "inverse_alpha": 32,
              "repetitions": [
                {
                  "burn_in": 448,
                  "mean": 0.002169128697645244,
                  "n": 32768,
                  "null_w1": 0.015443947153219144,
                  "null_w1_naive": 0.00627293826288345,
                  "seed": 211,
                  "tail_abs_gap": {
                    "2.0": 0.008376352651304198,
                    "3.0": 0.001227827719130093,
                    "4.0": 3.167124183311986e-05
                  },
                  "variance": 3.575337373892988,
                  "w1_debiased": 0.050492709727503374,
                  "w1_naive_debiased": 0.7261781234903386,
                  "w1_naive_raw": 0.732451061753222,
                  "w1_raw": 0.06593665688072252
                },
                {
                  "burn_in": 448,
                  "mean": -0.015348695531217019,
                  "n": 32768,
                  "null_w1": 0.015638376655418732,
                  "null_w1_naive": 0.006006637143198268,
                  "seed": 977,
                  "tail_abs_gap": {
                    "2.0": 0.007796518666929198,
                    "3.0": 0.001105757406630093,
                    "4.0": 3.167124183311986e-05
                  },
                  "variance": 3.5979182312481006,
                  "w1_debiased": 0.05074810022394852,
                  "w1_naive_debiased": 0.72719951925274,
                  "w1_naive_raw": 0.7332061563959382,
                  "w1_raw": 0.06638647687936725
                }
              ],
              "variance_mean": 3.5866278025705443,
              "w1_debiased_mean": 0.050620404975725945,
              "w1_naive_debiased_mean": 0.7266888213715392
            },
            {
              "alpha": 0.015625,
              "inverse_alpha": 64,
              "repetitions": [
                {
                  "burn_in": 896,
                  "mean": 0.003725121583802246,
                  "n": 32768,
                  "null_w1": 0.012800366812885523,
                  "null_w1_naive": 0.005628178433146821,
                  "seed": 211,
                  "tail_abs_gap": {
                    "2.0": 0.004500620229429198,
                    "3.0": 0.001014204672255093,
                    "4.0": 3.167124183311986e-05
                  },
                  "variance": 3.7493839745978432,
                  "w1_debiased": 0.02523924757676875,
                  "w1_naive_debiased": 0.7562538429263134,
                  "w1_naive_raw": 0.7618820213594601,
                  "w1_raw": 0.038039614389654273
                },
                {
                  "burn_in": 896,
                  "mean": -0.018832849224183434,
                  "n": 32768,
                  "null_w1": 0.015237294595323978,
                  "null_w1_naive": 0.007203339207102886,
                  "seed": 977,
                  "tail_abs_gap": {
                    "2.0": 0.005355112416929198,
                    "3.0": 0.0009531695160050931,
                    "4.0": 3.167124183311986e-05
                  },
                  "variance": 3.7534697623250897,
                  "w1_debiased": 0.02860405882426472,
                  "w1_naive_debiased": 0.7556213337771163,
                  "w1_naive_raw": 0.7628246729842192,
                  "w1_raw": 0.0438413534195887
                }
              ],
              "variance_mean": 3.7514268684614667,
              "w1_debiased_mean": 0.026921653200516735,
              "w1_naive_debiased_mean": 0.7559375883517148
            },
            {
              "alpha": 0.0078125,
              "inverse_alpha": 128,
              "repetitions": [
                {
                  "burn_in": 1792,
                  "mean": 0.0030389425892763315,
                  "n": 32768,
                  "null_w1": 0.01216523133601164,
                  "null_w1_naive": 0.006200754005825454,
                  "seed": 211,
                  "tail_abs_gap": {
                    "2.0": 0.003279917104429198,
                    "3.0": 0.0007090288910050931,
                    "4.0": 3.167124183311986e-05
                  },
                  "variance": 3.8929104970586965,
                  "w1_debiased": 0.011602606805579209,
                  "w1_naive_debiased": 0.784838985144116,
                  "w1_naive_raw": 0.7910397391499414,
                  "w1_raw": 0.02376783814159085
                },
                {
                  "burn_in": 1792,
                  "mean": -0.008890635556168962,
                  "n": 32768,
                  "null_w1": 0.008967939869705802,
                  "null_w1_naive": 0.007364766356754197,
                  "seed": 977,
                  "tail_abs_gap": {
                    "2.0": 0.002639047963804198,
                    "3.0": 0.00037333553163009306,
                    "4.0": 3.167124183311986e-05
                  },
                  "variance": 3.8873641065502604,
                  "w1_debiased": 0.013188968285267041,
                  "w1_naive_debiased": 0.770946366059629,
                  "w1_naive_raw": 0.7783111324163832,
                  "w1_raw": 0.022156908154972843
                }
              ],
              "variance_mean": 3.8901373018044785,
              "w1_debiased_mean": 0.012395787545423125,
              "w1_naive_debiased_mean": 0.7778926756018725
            }
          ],
          "seeds": [
            211,
            977
          ],
          "w1_fit": {
            "intercept": -0.08515773909431054,
            "r2": 0.9900207676623118,
            "slope": 0.8648543490756267
          }
        },
        "exact_linear_and_quadratic_sgd": {
          "finite_state": {
            "P": [
              [
                0.6073999999999999,
                0.297,
                0.006,
                0.006,
                0.08360000000000001
              ],
              [
                0.08360000000000001,
                0.6073999999999999,
                0.297,
                0.006,
                0.006
              ],
              [
                0.006,
                0.08360000000000001,
                0.6073999999999999,
                0.297,
                0.006
              ],
              [
                0.006,
                0.006,
                0.08360000000000001,
                0.6073999999999999,
                0.297
              ],
              [
                0.297,
                0.006,
                0.006,
                0.08360000000000001,
                0.6073999999999999
              ]
            ],
            "Sigma_0": 0.6317924144470441,
            "Sigma_M_autocovariance_series": 1.9999999999999798,
            "Sigma_M_fundamental_matrix": 2.0,
            "Sigma_M_oracle_abs_difference": 2.020605904817785e-14,
            "Sigma_Y": 1.0,
            "Sigma_Y_naive": 0.3158962072235221,
            "bound_ratio_monotone_decreasing": true,
            "naive_fit": {
              "intercept": -1.016310409779632,
              "r2": 0.9294458082756921,
              "slope": 0.006623480244635076
            },
            "naive_to_correct_gap_smallest": 26.276727209319237,
            "pi": [
              0.2000000000000001,
              0.1999999999999999,
              0.19999999999999996,
              0.20000000000000004,
              0.20000000000000007
            ],
            "rows": [
              {
                "alpha": 0.125,
                "bound_ratio": 0.07546875223320175,
                "inverse_alpha": 8,
                "tail_a_over_sd": [
                  0.5,
                  1.0,
                  1.5,
                  2.0,
                  3.0,
                  4.0,
                  6.0
                ],
                "tail_delta": [
                  -0.0061648208622465915,
                  0.005719470518656046,
                  0.009491059508606127,
                  0.00600313521855576,
                  -0.0003014043553695604,
                  -3.166718877576025e-05,
                  3.6277104573736962e-09
                ],
                "tail_sup_scaled": 0.014236589262909192,
                "transfer_terms": 195,
                "w1": 0.05548414421550459,
                "w1_naive_marginal_covariance": 0.35784374142017106
              },
              {
                "alpha": 0.0625,
                "bound_ratio": 0.06231107931521367,
                "inverse_alpha": 16,
                "tail_a_over_sd": [
                  0.5,
                  1.0,
                  1.5,
                  2.0,
                  3.0,
                  4.0,
                  6.0
                ],
                "tail_delta": [
                  -0.008630860154103068,
                  0.0028879742049046967,
                  0.008198508286357667,
                  0.006711856525426866,
                  0.0009165762210850167,
                  -1.8731296695967595e-06,
                  -9.864948791698608e-10
                ],
                "tail_sup_scaled": 0.013423713050853731,
                "transfer_terms": 398,
                "w1": 0.043190748944987485,
                "w1_naive_marginal_covariance": 0.35545878720504787
              },
              {
                "alpha": 0.03125,
                "bound_ratio": 0.05513306851816103,
                "inverse_alpha": 32,
                "tail_a_over_sd": [
                  0.5,
                  1.0,
                  1.5,
                  2.0,
                  3.0,
                  4.0,
                  6.0
                ],
                "tail_delta": [
                  -0.007958423144529162,
                  0.0012634475917820188,
                  0.006123157117453817,
                  0.005587049743737658,
                  0.0011424874814071862,
                  5.41722308692483e-05,
                  -7.27887723807014e-10
                ],
                "tail_sup_scaled": 0.011174099487475315,
                "transfer_terms": 803,
                "w1": 0.033777899618335466,
                "w1_naive_marginal_covariance": 0.3527939950700196
              },
              {
                "alpha": 0.015625,
                "bound_ratio": 0.04881705024525919,
                "inverse_alpha": 64,
                "tail_a_over_sd": [
                  0.5,
                  1.0,
                  1.5,
                  2.0,
                  3.0,
                  4.0,
                  6.0
                ],
                "tail_delta": [
                  -0.0063603270473700495,
                  0.0005457974812345095,
                  0.004421121293080482,
                  0.0042371468036721096,
                  0.0009643529127147078,
                  6.221537804469807e-05,
                  6.853153479876463e-09
                ],
                "tail_sup_scaled": 0.008474293607344219,
                "transfer_terms": 1614,
                "w1": 0.025378050555565943,
                "w1_naive_marginal_covariance": 0.35118407869764834
              },
              {
                "alpha": 0.0078125,
                "bound_ratio": 0.04320168309061552,
                "inverse_alpha": 128,
                "tail_a_over_sd": [
                  0.5,
                  1.0,
                  1.5,
                  2.0,
                  3.0,
                  4.0,
                  6.0
                ],
                "tail_delta": [
                  -0.00478636832288673,
                  0.00024114094248536838,
                  0.0031593476221357967,
                  0.003097909048900123,
                  0.000722285731294757,
                  4.905169478071922e-05,
                  1.0683565995351444e-08
                ],
                "tail_sup_scaled": 0.006195818097800246,
                "transfer_terms": 3235,
                "w1": 0.018527600726741443,
                "w1_naive_marginal_covariance": 0.35032514724230074
              },
              {
                "alpha": 0.00390625,
                "bound_ratio": 0.0384200267221643,
                "inverse_alpha": 256,
                "tail_a_over_sd": [
                  0.5,
                  1.0,
                  1.5,
                  2.0,
                  3.0,
                  4.0,
                  6.0
                ],
                "tail_delta": [
                  -0.0035013447218887666,
                  0.00010979259215822593,
                  0.0022472940581420124,
                  0.0022272188194733533,
                  0.0005177379289498943,
                  3.455284480689816e-05,
                  7.762708685377738e-09
                ],
                "tail_sup_scaled": 0.004454437638946707,
                "transfer_terms": 6477,
                "w1": 0.013315366599752968,
                "w1_naive_marginal_covariance": 0.3498842558337894
              }
            ],
            "w1_fit": {
              "intercept": -2.0018703460495444,
              "r2": 0.9960832387880162,
              "slope": 0.41058726184969746
            },
            "xi": [
              1.321858990213207,
              0.4235082201653963,
              -0.21817090129732541,
              -0.6673462863212306,
              -0.8598500227600471
            ]
          },
          "gaussian_ar1_oracle": {
            "Sigma_M": 8.0,
            "Sigma_Y": 4.0,
            "rho": 0.6,
            "rows": [
              {
                "alpha": 0.125,
                "inverse_alpha": 8,
                "variance": 3.424561403508772,
                "w1_closed_form": 0.1192382207572049
              },
              {
                "alpha": 0.0625,
                "inverse_alpha": 16,
                "variance": 3.6866359447004604,
                "w1_closed_form": 0.0637817404836307
              },
              {
                "alpha": 0.03125,
                "inverse_alpha": 32,
                "variance": 3.8360578062070583,
                "w1_closed_form": 0.03304385868315906
              },
              {
                "alpha": 0.015625,
                "inverse_alpha": 64,
                "variance": 3.9160906413415875,
                "w1_closed_form": 0.01682620539194402
              },
              {
                "alpha": 0.0078125,
                "inverse_alpha": 128,
                "variance": 3.9575440987205694,
                "w1_closed_form": 0.008491318805554832
              },
              {
                "alpha": 0.00390625,
                "inverse_alpha": 256,
                "variance": 3.9786445765964316,
                "w1_closed_form": 0.004265491485983746
              }
            ],
            "w1_fit": {
              "intercept": -0.0930664924554625,
              "r2": 0.9997130122865178,
              "slope": 0.9635964533676391
            }
          }
        }
      },
      "headline": "Exact five-state Markov W1 reaches 0.01331537 while the marginal-covariance control is 26.28x worse; genuine Markov nonlinear SA has two-seed slope 0.864854.",
      "literal_claim": "Proposition 4.1 extends the Gaussian approximation and tail bounds for SGD, linear SA, and contractive nonlinear SA from i.i.d. noise to Markovian noise (Proposition 4.1, Section 4)."
    },
    {
      "assessment": "verified",
      "claim": 6,
      "experiment": {
        "E_absX_fit": {
          "intercept": -0.5434551549503736,
          "r2": 0.9999997854501848,
          "slope": 0.24978713672845562
        },
        "correct_scaling_spread": 1.0006392646971614,
        "gaussian_to_gibbs_w1_ratio_smallest": 2331.681103360777,
        "h": 4,
        "max_grid_abs_difference": 5.592173815101916e-07,
        "noise": "N(0,1)",
        "objective": "P_2(x)=x^4/4",
        "predicted_scaling_exponent": 0.25,
        "rows": [
          {
            "E_absX": 0.2904955050975083,
            "E_absX_over_correct_scaling": 0.5809910101950166,
            "E_absX_over_sqrt_alpha": 1.1619820203900333,
            "alpha": 0.0625,
            "evaluations": [
              {
                "E_absY": 0.5809890143049864,
                "fixed_point_iterations": 1200,
                "fixed_point_residual": 7.546517921989385e-16,
                "grid": 2048,
                "kurtosis": 2.2068571015894936,
                "variance": 0.47871886116494516,
                "w1_best_variance_gaussian": 0.057741080685267954,
                "w1_gibbs": 0.001658747136999344
              },
              {
                "E_absY": 0.5809910101950166,
                "fixed_point_iterations": 1200,
                "fixed_point_residual": 1.1091076380810696e-15,
                "grid": 4096,
                "kurtosis": 2.206854771926715,
                "variance": 0.47872045116230266,
                "w1_best_variance_gaussian": 0.05774187388187772,
                "w1_gibbs": 0.0016582818528189693
              }
            ],
            "inverse_alpha": 16,
            "kurtosis": 2.206854771926715,
            "scaling_alpha_1_over_h": 0.5,
            "w1_best_variance_gaussian": 0.05774187388187772,
            "w1_gibbs": 0.0016582818528189693,
            "w1_grid_abs_difference": 4.652841803748191e-07
          },
          {
            "E_absX": 0.24437990006035434,
            "E_absX_over_correct_scaling": 0.5812366318308546,
            "E_absX_over_sqrt_alpha": 1.3824214761469384,
            "alpha": 0.03125,
            "evaluations": [
              {
                "E_absY": 0.581234646622233,
                "fixed_point_iterations": 2800,
                "fixed_point_residual": 2.9371047286912493e-15,
                "grid": 2048,
                "kurtosis": 2.1948302860570483,
                "variance": 0.4782437440219264,
                "w1_best_variance_gaussian": 0.05887463897290077,
                "w1_gibbs": 0.0005816992250516353
              },
              {
                "E_absY": 0.5812366318308546,
                "fixed_point_iterations": 2800,
                "fixed_point_residual": 2.7882751879735508e-14,
                "grid": 4096,
                "kurtosis": 2.1948279850940864,
                "variance": 0.47824531846449414,
                "w1_best_variance_gaussian": 0.05887507654263358,
                "w1_gibbs": 0.000581231138242095
              }
            ],
            "inverse_alpha": 32,
            "kurtosis": 2.1948279850940864,
            "scaling_alpha_1_over_h": 0.42044820762685725,
            "w1_best_variance_gaussian": 0.05887507654263358,
            "w1_gibbs": 0.000581231138242095,
            "w1_grid_abs_difference": 4.6808680954023817e-07
          },
          {
            "E_absX": 0.20552821803303054,
            "E_absX_over_correct_scaling": 0.5813215867853726,
            "E_absX_over_sqrt_alpha": 1.6442257442642443,
            "alpha": 0.015625,
            "evaluations": [
              {
                "E_absY": 0.5813194795238386,
                "fixed_point_iterations": 5600,
                "fixed_point_residual": 8.676248922088297e-14,
                "grid": 2048,
                "kurtosis": 2.1906867668429992,
                "variance": 0.47807719899105444,
                "w1_best_variance_gaussian": 0.05926962770471657,
                "w1_gibbs": 0.0002055413537254161
              },
              {
                "E_absY": 0.5813215867853726,
                "fixed_point_iterations": 5200,
                "fixed_point_residual": 3.845644692788327e-14,
                "grid": 4096,
                "kurtosis": 2.1906839381556193,
                "variance": 0.4780789307641321,
                "w1_best_variance_gaussian": 0.05927018745374221,
                "w1_gibbs": 0.0002049821363439059
              }
            ],
            "inverse_alpha": 64,
            "kurtosis": 2.1906839381556193,
            "scaling_alpha_1_over_h": 0.3535533905932738,
            "w1_best_variance_gaussian": 0.05927018745374221,
            "w1_gibbs": 0.0002049821363439059,
            "w1_grid_abs_difference": 5.592173815101916e-07
          },
          {
            "E_absX": 0.17283678787603213,
            "E_absX_over_correct_scaling": 0.5813513413956883,
            "E_absX_over_sqrt_alpha": 1.9554250359302907,
            "alpha": 0.0078125,
            "evaluations": [
              {
                "E_absY": 0.5813505619309431,
                "fixed_point_iterations": 10000,
                "fixed_point_residual": 6.579456434132637e-14,
                "grid": 2048,
                "kurtosis": 2.1892310809783737,
                "variance": 0.47802035786817065,
                "w1_best_variance_gaussian": 0.059409109274097954,
                "w1_gibbs": 7.241863798520748e-05
              },
              {
                "E_absY": 0.5813513413956883,
                "fixed_point_iterations": 8400,
                "fixed_point_residual": 5.833032950219673e-14,
                "grid": 4096,
                "kurtosis": 2.189231926559346,
                "variance": 0.47802021296561475,
                "w1_best_variance_gaussian": 0.059409205298102244,
                "w1_gibbs": 7.252403897106758e-05
              }
            ],
            "inverse_alpha": 128,
            "kurtosis": 2.189231926559346,
            "scaling_alpha_1_over_h": 0.29730177875068026,
            "w1_best_variance_gaussian": 0.059409205298102244,
            "w1_gibbs": 7.252403897106758e-05,
            "w1_grid_abs_difference": 1.0540098586010304e-07
          },
          {
            "E_absX": 0.1453406043093006,
            "E_absX_over_correct_scaling": 0.5813624172372024,
            "E_absX_over_sqrt_alpha": 2.3254496689488096,
            "alpha": 0.00390625,
            "evaluations": [
              {
                "E_absY": 0.5813601504287552,
                "fixed_point_iterations": 18400,
                "fixed_point_residual": 8.265245180870104e-14,
                "grid": 2048,
                "kurtosis": 2.1887124869422143,
                "variance": 0.4779977411677699,
                "w1_best_variance_gaussian": 0.059458613839746564,
                "w1_gibbs": 2.5410003114949005e-05
              },
              {
                "E_absY": 0.5813624172372024,
                "fixed_point_iterations": 19200,
                "fixed_point_residual": 8.774803788212039e-14,
                "grid": 4096,
                "kurtosis": 2.188718987637267,
                "variance": 0.47800033050266666,
                "w1_best_variance_gaussian": 0.05945846262993744,
                "w1_gibbs": 2.5500254963784185e-05
              }
            ],
            "inverse_alpha": 256,
            "kurtosis": 2.188718987637267,
            "scaling_alpha_1_over_h": 0.25,
            "w1_best_variance_gaussian": 0.05945846262993744,
            "w1_gibbs": 2.5500254963784185e-05,
            "w1_grid_abs_difference": 9.025184883518003e-08
          }
        ],
        "sqrt_scaling_spread": 2.0012785293943227,
        "target": {
          "E_abs": 0.5813683170191186,
          "b": 0.5,
          "density": "proportional to exp(-y^4/2)",
          "kurtosis": 2.1884396152264776,
          "variance": 0.4779887974861249
        },
        "w1_gibbs_fit": {
          "intercept": -2.2321106246001516,
          "r2": 0.9999979366326253,
          "slope": 1.504864832767487
        }
      },
      "headline": "For paper-native P_2(x)=x^4/4, E|X| slope 0.249787 (theory 0.25, R2=0.999999785); at the smallest alpha Gibbs W1 is 2.5500255e-05 versus 0.05945846 for the variance-matched Gaussian.",
      "literal_claim": "Proposition 5.1 shows a different convergence rate of order \u03b1^{1/h} for constant-stepsize SA applied to general convex objectives with Gibbs-type limiting distributions (Proposition 5.1, Section 5)."
    }
  ],
  "methods": {
    "determinism": "exact arithmetic/FFT/transfer operator plus fixed two-seed Markov nonlinear check",
    "runtime_dependency_versions": {
      "numpy": "2.2.4"
    },
    "stepsizes": [
      0.125,
      0.0625,
      0.03125,
      0.015625,
      0.0078125,
      0.00390625,
      0.001953125
    ]
  },
  "paper_id": "m4TAzup6Yc",
  "summary": {
    "all_claims_directly_executed": true,
    "expected_points": 12,
    "falsified": 0,
    "registered_claims": 6,
    "verified": 6
  }
}