{
  "paper": {
    "title": "Bregman meets L\u00e9vy: Stochastic mirror descent with heavy-tailed noise in continuous and discrete time",
    "source": "arXiv:2606.03769v1",
    "orid": "69IOkVkTQX",
    "geometry": "X=[-1,1], h(x)=x^2/2, mirror(y)=clip(eta*y,-1,1)",
    "noise": "centered symmetric Pareto / compound-Poisson jumps with alpha=p+0.12; Brownian tame component where stated"
  },
  "claim_1": {
    "rows": [
      {
        "p": 1.25,
        "seeds": 8,
        "horizons": [
          256,
          512,
          1024,
          2048,
          4096
        ],
        "gap_at_max_T": 0.0027975895691645996,
        "gap_se": 7.535740222972052e-05,
        "measured_slope": -0.37776974920457357,
        "slope_se": 0.019167750380046526,
        "theory_slope": -0.2,
        "paper_R0_bound_at_max_T": 0.12211924043813976,
        "gap_over_paper_bound": 0.022908671550260218,
        "bound_pass": true
      },
      {
        "p": 1.5,
        "seeds": 8,
        "horizons": [
          256,
          512,
          1024,
          2048,
          4096
        ],
        "gap_at_max_T": 0.00035007881854882776,
        "gap_se": 1.9804117526887215e-05,
        "measured_slope": -0.6439671830292583,
        "slope_se": 0.025713989326114472,
        "theory_slope": -0.3333333333333333,
        "paper_R0_bound_at_max_T": 0.035355711961319064,
        "gap_over_paper_bound": 0.009901619826856596,
        "bound_pass": true
      },
      {
        "p": 1.75,
        "seeds": 8,
        "horizons": [
          256,
          512,
          1024,
          2048,
          4096
        ],
        "gap_at_max_T": 7.857721818242748e-05,
        "gap_se": 7.915933425194222e-06,
        "measured_slope": -0.8160305880222549,
        "slope_se": 0.04288091043475578,
        "theory_slope": -0.42857142857142855,
        "paper_R0_bound_at_max_T": 0.015032066676439535,
        "gap_over_paper_bound": 0.005227306389319391,
        "bound_pass": true
      },
      {
        "p": 2.0,
        "seeds": 8,
        "horizons": [
          256,
          512,
          1024,
          2048,
          4096
        ],
        "gap_at_max_T": 2.6736924457419256e-05,
        "gap_se": 4.088291516479964e-06,
        "measured_slope": -0.9303635689782251,
        "slope_se": 0.0686837330855816,
        "theory_slope": -0.5,
        "paper_R0_bound_at_max_T": 0.008144422535135941,
        "gap_over_paper_bound": 0.003282850851864967,
        "bound_pass": true
      }
    ],
    "baseline": [
      0.0003378313353985914,
      0.0
    ],
    "destructive": [
      1.3496873426608316,
      0.0
    ]
  },
  "claim_2": {
    "rows": [
      {
        "treatment": "tame",
        "p": 1.5,
        "eta": 0.004,
        "seeds": 8,
        "stationary_mse": 0.00014929900449299393,
        "mse_se": 3.7622850557952236e-05,
        "slope": 1.0368699653429303,
        "slope_se": 0.11660538616094784,
        "theory_exponent": 1.0,
        "paper_radius2_bound": 0.00025,
        "mse_over_radius_bound": 0.5971960179719757,
        "bound_pass": true
      },
      {
        "treatment": "tame",
        "p": 1.5,
        "eta": 0.008,
        "seeds": 8,
        "stationary_mse": 0.0002666660062656201,
        "mse_se": 4.039227466934179e-05,
        "slope": 1.0368699653429303,
        "slope_se": 0.11660538616094784,
        "theory_exponent": 1.0,
        "paper_radius2_bound": 0.0005,
        "mse_over_radius_bound": 0.5333320125312402,
        "bound_pass": true
      },
      {
        "treatment": "tame",
        "p": 1.5,
        "eta": 0.016,
        "seeds": 8,
        "stationary_mse": 0.0005072655632009312,
        "mse_se": 4.787226606085312e-05,
        "slope": 1.0368699653429303,
        "slope_se": 0.11660538616094784,
        "theory_exponent": 1.0,
        "paper_radius2_bound": 0.001,
        "mse_over_radius_bound": 0.5072655632009312,
        "bound_pass": true
      },
      {
        "treatment": "tame",
        "p": 1.5,
        "eta": 0.032,
        "seeds": 8,
        "stationary_mse": 0.0010150335607430642,
        "mse_se": 9.4003822213678e-05,
        "slope": 1.0368699653429303,
        "slope_se": 0.11660538616094784,
        "theory_exponent": 1.0,
        "paper_radius2_bound": 0.002,
        "mse_over_radius_bound": 0.5075167803715321,
        "bound_pass": true
      },
      {
        "treatment": "tame",
        "p": 1.5,
        "eta": 0.064,
        "seeds": 8,
        "stationary_mse": 0.0020234008070470298,
        "mse_se": 0.00016287197719308278,
        "slope": 1.0368699653429303,
        "slope_se": 0.11660538616094784,
        "theory_exponent": 1.0,
        "paper_radius2_bound": 0.004,
        "mse_over_radius_bound": 0.5058502017617574,
        "bound_pass": true
      },
      {
        "treatment": "heavy",
        "p": 1.25,
        "eta": 0.004,
        "seeds": 8,
        "stationary_mse": 0.00017476583663295398,
        "mse_se": 6.767571663681561e-05,
        "slope": 0.8030460481197103,
        "slope_se": 0.1800764623141729,
        "theory_exponent": 0.25,
        "paper_radius2_bound": 0.05810778124857912,
        "mse_over_radius_bound": 0.0030076150367077978,
        "bound_pass": true
      },
      {
        "treatment": "heavy",
        "p": 1.25,
        "eta": 0.008,
        "seeds": 8,
        "stationary_mse": 0.0002619893025607717,
        "mse_se": 0.00012062527045458774,
        "slope": 0.8030460481197103,
        "slope_se": 0.1800764623141729,
        "theory_exponent": 0.25,
        "paper_radius2_bound": 0.06910218689783199,
        "mse_over_radius_bound": 0.0037913315673804146,
        "bound_pass": true
      },
      {
        "treatment": "heavy",
        "p": 1.25,
        "eta": 0.016,
        "seeds": 8,
        "stationary_mse": 0.000473011218946514,
        "mse_se": 0.00026194727931930245,
        "slope": 0.8030460481197103,
        "slope_se": 0.1800764623141729,
        "theory_exponent": 0.25,
        "paper_radius2_bound": 0.08217681232114961,
        "mse_over_radius_bound": 0.005756018097878645,
        "bound_pass": true
      },
      {
        "treatment": "heavy",
        "p": 1.25,
        "eta": 0.032,
        "seeds": 8,
        "stationary_mse": 0.0008820029655063091,
        "mse_se": 0.0005310178887407616,
        "slope": 0.8030460481197103,
        "slope_se": 0.1800764623141729,
        "theory_exponent": 0.25,
        "paper_radius2_bound": 0.09772524990055437,
        "mse_over_radius_bound": 0.009025333436382503,
        "bound_pass": true
      },
      {
        "treatment": "heavy",
        "p": 1.25,
        "eta": 0.064,
        "seeds": 8,
        "stationary_mse": 0.0016401994437966144,
        "mse_se": 0.0010597170138130425,
        "slope": 0.8030460481197103,
        "slope_se": 0.1800764623141729,
        "theory_exponent": 0.25,
        "paper_radius2_bound": 0.11621556249715824,
        "mse_over_radius_bound": 0.014113423439625149,
        "bound_pass": true
      },
      {
        "treatment": "heavy",
        "p": 1.5,
        "eta": 0.004,
        "seeds": 8,
        "stationary_mse": 2.6636070530708478e-05,
        "mse_se": 8.299501598672298e-06,
        "slope": 0.9612324664087314,
        "slope_se": 0.12858925872082533,
        "theory_exponent": 0.5,
        "paper_radius2_bound": 0.005464415796770956,
        "mse_over_radius_bound": 0.00487445895798199,
        "bound_pass": true
      },
      {
        "treatment": "heavy",
        "p": 1.5,
        "eta": 0.008,
        "seeds": 8,
        "stationary_mse": 4.7251903797643956e-05,
        "mse_se": 1.6747638001254604e-05,
        "slope": 0.9612324664087314,
        "slope_se": 0.12858925872082533,
        "theory_exponent": 0.5,
        "paper_radius2_bound": 0.007727850930239268,
        "mse_over_radius_bound": 0.006114494731354885,
        "bound_pass": true
      },
      {
        "treatment": "heavy",
        "p": 1.5,
        "eta": 0.016,
        "seeds": 8,
        "stationary_mse": 9.371465785293939e-05,
        "mse_se": 3.7850566947734126e-05,
        "slope": 0.9612324664087314,
        "slope_se": 0.12858925872082533,
        "theory_exponent": 0.5,
        "paper_radius2_bound": 0.010928831593541912,
        "mse_over_radius_bound": 0.008574993314776435,
        "bound_pass": true
      },
      {
        "treatment": "heavy",
        "p": 1.5,
        "eta": 0.032,
        "seeds": 8,
        "stationary_mse": 0.00018598671949029006,
        "mse_se": 8.086916626820479e-05,
        "slope": 0.9612324664087314,
        "slope_se": 0.12858925872082533,
        "theory_exponent": 0.5,
        "paper_radius2_bound": 0.015455701860478536,
        "mse_over_radius_bound": 0.012033534366101675,
        "bound_pass": true
      },
      {
        "treatment": "heavy",
        "p": 1.5,
        "eta": 0.064,
        "seeds": 8,
        "stationary_mse": 0.0003608037705263764,
        "mse_se": 0.00016584022244962081,
        "slope": 0.9612324664087314,
        "slope_se": 0.12858925872082533,
        "theory_exponent": 0.5,
        "paper_radius2_bound": 0.021857663187083824,
        "mse_over_radius_bound": 0.016506969086227997,
        "bound_pass": true
      },
      {
        "treatment": "heavy",
        "p": 1.75,
        "eta": 0.004,
        "seeds": 8,
        "stationary_mse": 7.469037938573843e-06,
        "mse_se": 1.7734414066451914e-06,
        "slope": 1.0758020845653231,
        "slope_se": 0.09429872740415576,
        "theory_exponent": 0.75,
        "paper_radius2_bound": 0.0005016321095481787,
        "mse_over_radius_bound": 0.014889473373826137,
        "bound_pass": true
      },
      {
        "treatment": "heavy",
        "p": 1.75,
        "eta": 0.008,
        "seeds": 8,
        "stationary_mse": 1.5371063148690825e-05,
        "mse_se": 3.4654114915303694e-06,
        "slope": 1.0758020845653231,
        "slope_se": 0.09429872740415576,
        "theory_exponent": 0.75,
        "paper_radius2_bound": 0.0008436412853904444,
        "mse_over_radius_bound": 0.018219903903324227,
        "bound_pass": true
      },
      {
        "treatment": "heavy",
        "p": 1.75,
        "eta": 0.016,
        "seeds": 8,
        "stationary_mse": 3.337590187591608e-05,
        "mse_se": 8.286589651469961e-06,
        "slope": 1.0758020845653231,
        "slope_se": 0.09429872740415576,
        "theory_exponent": 0.75,
        "paper_radius2_bound": 0.001418829865289721,
        "mse_over_radius_bound": 0.02352354055438551,
        "bound_pass": true
      },
      {
        "treatment": "heavy",
        "p": 1.75,
        "eta": 0.032,
        "seeds": 8,
        "stationary_mse": 7.028415940920038e-05,
        "mse_se": 1.9269793422117813e-05,
        "slope": 1.0758020845653231,
        "slope_se": 0.09429872740415576,
        "theory_exponent": 0.75,
        "paper_radius2_bound": 0.002386177895154074,
        "mse_over_radius_bound": 0.02945470224660772,
        "bound_pass": true
      },
      {
        "treatment": "heavy",
        "p": 1.75,
        "eta": 0.064,
        "seeds": 8,
        "stationary_mse": 0.0001409088008229936,
        "mse_se": 4.140548424767764e-05,
        "slope": 1.0758020845653231,
        "slope_se": 0.09429872740415576,
        "theory_exponent": 0.75,
        "paper_radius2_bound": 0.00401305687638543,
        "mse_over_radius_bound": 0.03511258503515418,
        "bound_pass": true
      },
      {
        "treatment": "bounded-control",
        "p": 1.25,
        "eta": 0.004,
        "seeds": 8,
        "stationary_mse": 3.386847057887935e-06,
        "mse_se": 4.387272295379809e-07,
        "slope": 1.170718408582711,
        "slope_se": 0.06361377815110031,
        "theory_exponent": 1.0,
        "paper_radius2_bound": 0.05810778124857912,
        "mse_over_radius_bound": 5.828560280764036e-05,
        "bound_pass": true
      },
      {
        "treatment": "bounded-control",
        "p": 1.25,
        "eta": 0.008,
        "seeds": 8,
        "stationary_mse": 8.46900815286997e-06,
        "mse_se": 7.335720018321259e-07,
        "slope": 1.170718408582711,
        "slope_se": 0.06361377815110031,
        "theory_exponent": 1.0,
        "paper_radius2_bound": 0.06910218689783199,
        "mse_over_radius_bound": 0.00012255774430686907,
        "bound_pass": true
      },
      {
        "treatment": "bounded-control",
        "p": 1.25,
        "eta": 0.016,
        "seeds": 8,
        "stationary_mse": 1.9548381224532743e-05,
        "mse_se": 8.105673779282874e-07,
        "slope": 1.170718408582711,
        "slope_se": 0.06361377815110031,
        "theory_exponent": 1.0,
        "paper_radius2_bound": 0.08217681232114961,
        "mse_over_radius_bound": 0.00023788196052357255,
        "bound_pass": true
      },
      {
        "treatment": "bounded-control",
        "p": 1.25,
        "eta": 0.032,
        "seeds": 8,
        "stationary_mse": 4.153629462439247e-05,
        "mse_se": 2.262732013317742e-06,
        "slope": 1.170718408582711,
        "slope_se": 0.06361377815110031,
        "theory_exponent": 1.0,
        "paper_radius2_bound": 0.09772524990055437,
        "mse_over_radius_bound": 0.00042503134723789385,
        "bound_pass": true
      },
      {
        "treatment": "bounded-control",
        "p": 1.25,
        "eta": 0.064,
        "seeds": 8,
        "stationary_mse": 8.413195210145845e-05,
        "mse_se": 5.537671168253215e-06,
        "slope": 1.170718408582711,
        "slope_se": 0.06361377815110031,
        "theory_exponent": 1.0,
        "paper_radius2_bound": 0.11621556249715824,
        "mse_over_radius_bound": 0.0007239301716025827,
        "bound_pass": true
      },
      {
        "treatment": "bounded-control",
        "p": 1.5,
        "eta": 0.004,
        "seeds": 8,
        "stationary_mse": 2.5460791273237015e-06,
        "mse_se": 2.9724486048579667e-07,
        "slope": 1.1923593010844815,
        "slope_se": 0.06454791953287653,
        "theory_exponent": 1.0,
        "paper_radius2_bound": 0.005464415796770956,
        "mse_over_radius_bound": 0.00046593802924518223,
        "bound_pass": true
      },
      {
        "treatment": "bounded-control",
        "p": 1.5,
        "eta": 0.008,
        "seeds": 8,
        "stationary_mse": 6.6046367547084715e-06,
        "mse_se": 5.255760969689046e-07,
        "slope": 1.1923593010844815,
        "slope_se": 0.06454791953287653,
        "theory_exponent": 1.0,
        "paper_radius2_bound": 0.007727850930239268,
        "mse_over_radius_bound": 0.0008546537471193146,
        "bound_pass": true
      },
      {
        "treatment": "bounded-control",
        "p": 1.5,
        "eta": 0.016,
        "seeds": 8,
        "stationary_mse": 1.5572976904900848e-05,
        "mse_se": 6.129022967410445e-07,
        "slope": 1.1923593010844815,
        "slope_se": 0.06454791953287653,
        "theory_exponent": 1.0,
        "paper_radius2_bound": 0.010928831593541912,
        "mse_over_radius_bound": 0.0014249443567328152,
        "bound_pass": true
      },
      {
        "treatment": "bounded-control",
        "p": 1.5,
        "eta": 0.032,
        "seeds": 8,
        "stationary_mse": 3.348206405434443e-05,
        "mse_se": 1.9955619825243574e-06,
        "slope": 1.1923593010844815,
        "slope_se": 0.06454791953287653,
        "theory_exponent": 1.0,
        "paper_radius2_bound": 0.015455701860478536,
        "mse_over_radius_bound": 0.00216632439966772,
        "bound_pass": true
      },
      {
        "treatment": "bounded-control",
        "p": 1.5,
        "eta": 0.064,
        "seeds": 8,
        "stationary_mse": 6.813730613131983e-05,
        "mse_se": 4.83994917962275e-06,
        "slope": 1.1923593010844815,
        "slope_se": 0.06454791953287653,
        "theory_exponent": 1.0,
        "paper_radius2_bound": 0.021857663187083824,
        "mse_over_radius_bound": 0.00311731888025357,
        "bound_pass": true
      },
      {
        "treatment": "bounded-control",
        "p": 1.75,
        "eta": 0.004,
        "seeds": 8,
        "stationary_mse": 2.0092585150399113e-06,
        "mse_se": 2.4566595467664723e-07,
        "slope": 1.2057123500508078,
        "slope_se": 0.06579312961374312,
        "theory_exponent": 1.0,
        "paper_radius2_bound": 0.0005016321095481787,
        "mse_over_radius_bound": 0.0040054423885457725,
        "bound_pass": true
      },
      {
        "treatment": "bounded-control",
        "p": 1.75,
        "eta": 0.008,
        "seeds": 8,
        "stationary_mse": 5.313153634821394e-06,
        "mse_se": 4.39608935263333e-07,
        "slope": 1.2057123500508078,
        "slope_se": 0.06579312961374312,
        "theory_exponent": 1.0,
        "paper_radius2_bound": 0.0008436412853904444,
        "mse_over_radius_bound": 0.006297882437513025,
        "bound_pass": true
      },
      {
        "treatment": "bounded-control",
        "p": 1.75,
        "eta": 0.016,
        "seeds": 8,
        "stationary_mse": 1.2637064884574077e-05,
        "mse_se": 5.302540525572664e-07,
        "slope": 1.2057123500508078,
        "slope_se": 0.06579312961374312,
        "theory_exponent": 1.0,
        "paper_radius2_bound": 0.001418829865289721,
        "mse_over_radius_bound": 0.008906680916244756,
        "bound_pass": true
      },
      {
        "treatment": "bounded-control",
        "p": 1.75,
        "eta": 0.032,
        "seeds": 8,
        "stationary_mse": 2.7332033053486355e-05,
        "mse_se": 1.661470436378566e-06,
        "slope": 1.2057123500508078,
        "slope_se": 0.06579312961374312,
        "theory_exponent": 1.0,
        "paper_radius2_bound": 0.002386177895154074,
        "mse_over_radius_bound": 0.011454314914656245,
        "bound_pass": true
      },
      {
        "treatment": "bounded-control",
        "p": 1.75,
        "eta": 0.064,
        "seeds": 8,
        "stationary_mse": 5.567109721144275e-05,
        "mse_se": 3.976220630479608e-06,
        "slope": 1.2057123500508078,
        "slope_se": 0.06579312961374312,
        "theory_exponent": 1.0,
        "paper_radius2_bound": 0.00401305687638543,
        "mse_over_radius_bound": 0.013872491451351132,
        "bound_pass": true
      }
    ],
    "baseline": [
      0.0,
      0.0
    ]
  },
  "claim_3": {
    "rows": [
      {
        "p": 1.5,
        "delta": 0.5,
        "seeds": 8,
        "mean_hit_time": 1.1750000000000003,
        "hit_se": 0.025000000000000022,
        "censored": 0,
        "paper_bound": 15.288238079999976,
        "mean_over_bound": 0.07685646925770546,
        "measured_delta_slope": -4.8372629435235535,
        "theory_bound_slope": -6.0
      },
      {
        "p": 1.5,
        "delta": 0.35,
        "seeds": 8,
        "mean_hit_time": 7.925000000000001,
        "hit_se": 0.2561737691489901,
        "censored": 0,
        "paper_bound": 129.9478795399874,
        "mean_over_bound": 0.06098598936784751,
        "measured_delta_slope": -4.8372629435235535,
        "theory_bound_slope": -6.0
      },
      {
        "p": 1.5,
        "delta": 0.25,
        "seeds": 8,
        "mean_hit_time": 39.025000000000006,
        "hit_se": 0.3730903758761792,
        "censored": 0,
        "paper_bound": 978.4472371199985,
        "mean_over_bound": 0.03988462384018558,
        "measured_delta_slope": -4.8372629435235535,
        "theory_bound_slope": -6.0
      },
      {
        "p": 1.5,
        "delta": 0.18,
        "seeds": 8,
        "mean_hit_time": 186.22500000000002,
        "hit_se": 0.7610495572375224,
        "censored": 0,
        "paper_bound": 7023.319615912199,
        "mean_over_bound": 0.026515239257812538,
        "measured_delta_slope": -4.8372629435235535,
        "theory_bound_slope": -6.0
      },
      {
        "p": 1.5,
        "delta": 0.13,
        "seeds": 8,
        "mean_hit_time": 814.6,
        "hit_se": 3.50957872929827,
        "censored": 0,
        "paper_bound": 49489.98810601364,
        "mean_over_bound": 0.016459894842872597,
        "measured_delta_slope": -4.8372629435235535,
        "theory_bound_slope": -6.0
      },
      {
        "p": 1.75,
        "delta": 0.5,
        "seeds": 8,
        "mean_hit_time": 0.2,
        "hit_se": 0.0,
        "censored": 0,
        "paper_bound": 3.2139349063010916,
        "mean_over_bound": 0.062229013913097396,
        "measured_delta_slope": -3.632913212690609,
        "theory_bound_slope": -4.666666666666667
      },
      {
        "p": 1.75,
        "delta": 0.35,
        "seeds": 8,
        "mean_hit_time": 1.0,
        "hit_se": 0.03779644730092273,
        "censored": 0,
        "paper_bound": 16.979030550633492,
        "mean_over_bound": 0.05889617767150373,
        "measured_delta_slope": -3.632913212690609,
        "theory_bound_slope": -4.666666666666667
      },
      {
        "p": 1.75,
        "delta": 0.25,
        "seeds": 8,
        "mean_hit_time": 3.4250000000000003,
        "hit_se": 0.11611263005006327,
        "censored": 0,
        "paper_bound": 81.62885841951473,
        "mean_over_bound": 0.04195820040013199,
        "measured_delta_slope": -3.632913212690609,
        "theory_bound_slope": -4.666666666666667
      },
      {
        "p": 1.75,
        "delta": 0.18,
        "seeds": 8,
        "mean_hit_time": 10.325000000000001,
        "hit_se": 0.42583950699093875,
        "censored": 0,
        "paper_bound": 378.1163058596349,
        "mean_over_bound": 0.027306412973982848,
        "measured_delta_slope": -3.632913212690609,
        "theory_bound_slope": -4.666666666666667
      },
      {
        "p": 1.75,
        "delta": 0.13,
        "seeds": 8,
        "mean_hit_time": 27.550000000000004,
        "hit_se": 0.3849860850917675,
        "censored": 0,
        "paper_bound": 1726.4796060424853,
        "mean_over_bound": 0.015957327212889217,
        "measured_delta_slope": -3.632913212690609,
        "theory_bound_slope": -4.666666666666667
      }
    ],
    "baseline": [
      40.6,
      39.0,
      38.6,
      37.6,
      39.400000000000006,
      39.6,
      39.800000000000004,
      37.6
    ],
    "destructive_censored": 8
  },
  "claim_4": {
    "rows": [
      {
        "p": 1.25,
        "T": 512,
        "seeds": 8,
        "sda_gap": 0.06273554919866123,
        "sda_se": 0.00037013150912261884,
        "lmf_gap": 0.005718644000595717,
        "lmf_se": 0.00025930587776265405,
        "sda_slope": -0.2676957755301662,
        "sda_slope_se": 0.018201976853914177,
        "lmf_slope": -0.366548481131481,
        "lmf_slope_se": 0.034754686990722046,
        "theory_slope": -0.2,
        "sda_paper_bound": 0.6810555664229504,
        "sda_gap_over_bound": 0.09211516987984072,
        "sda_bound_pass": true,
        "lmf_paper_bound": 0.18505837196131789,
        "lmf_gap_over_bound": 0.030901838917025953,
        "lmf_bound_pass": true
      },
      {
        "p": 1.25,
        "T": 1024,
        "seeds": 8,
        "sda_gap": 0.05248535029442655,
        "sda_se": 0.0003664977483775579,
        "lmf_gap": 0.004463997687288207,
        "lmf_se": 0.00021439783309429945,
        "sda_slope": -0.2676957755301662,
        "sda_slope_se": 0.018201976853914177,
        "lmf_slope": -0.366548481131481,
        "lmf_slope_se": 0.034754686990722046,
        "theory_slope": -0.2,
        "sda_paper_bound": 0.5927630353030516,
        "sda_gap_over_bound": 0.08854356153904448,
        "sda_bound_pass": true,
        "lmf_paper_bound": 0.16110266995358896,
        "lmf_gap_over_bound": 0.027709023621856867,
        "lmf_bound_pass": true
      },
      {
        "p": 1.25,
        "T": 2048,
        "seeds": 8,
        "sda_gap": 0.044140012874287315,
        "sda_se": 0.0005941957843441603,
        "lmf_gap": 0.003599440584931757,
        "lmf_se": 0.00010533454774557434,
        "sda_slope": -0.2676957755301662,
        "sda_slope_se": 0.018201976853914177,
        "lmf_slope": -0.366548481131481,
        "lmf_slope_se": 0.034754686990722046,
        "theory_slope": -0.2,
        "sda_paper_bound": 0.5159650584429876,
        "sda_gap_over_bound": 0.08554845362491661,
        "sda_bound_pass": true,
        "lmf_paper_bound": 0.14024802007660644,
        "lmf_gap_over_bound": 0.02566482281151396,
        "lmf_bound_pass": true
      },
      {
        "p": 1.25,
        "T": 4096,
        "seeds": 8,
        "sda_gap": 0.0370380464912295,
        "sda_se": 0.001467599889439019,
        "lmf_gap": 0.002781289717521668,
        "lmf_se": 8.713466275181362e-05,
        "sda_slope": -0.2676957755301662,
        "sda_slope_se": 0.018201976853914177,
        "lmf_slope": -0.366548481131481,
        "lmf_slope_se": 0.034754686990722046,
        "theory_slope": -0.2,
        "sda_paper_bound": 0.44914110434805815,
        "sda_gap_over_bound": 0.08246416578814655,
        "sda_bound_pass": true,
        "lmf_paper_bound": 0.12209299287885586,
        "lmf_gap_over_bound": 0.02278009287790449,
        "lmf_bound_pass": true
      },
      {
        "p": 1.25,
        "T": 8192,
        "seeds": 8,
        "sda_gap": 0.029784548576652997,
        "sda_se": 0.0013475960805069023,
        "lmf_gap": 0.002062079719786912,
        "lmf_se": 0.00015556582567764153,
        "sda_slope": -0.2676957755301662,
        "sda_slope_se": 0.018201976853914177,
        "lmf_slope": -0.366548481131481,
        "lmf_slope_se": 0.034754686990722046,
        "theory_slope": -0.2,
        "sda_paper_bound": 0.3909837574293442,
        "sda_gap_over_bound": 0.07617848059080932,
        "sda_bound_pass": true,
        "lmf_paper_bound": 0.10628812372519765,
        "lmf_gap_over_bound": 0.019400847879470624,
        "lmf_bound_pass": true
      },
      {
        "p": 1.5,
        "T": 512,
        "seeds": 8,
        "sda_gap": 0.020299189970658348,
        "sda_se": 0.0001350993052377635,
        "lmf_gap": 0.0012196309094500268,
        "lmf_se": 0.00010124192639684171,
        "sda_slope": -0.5860486846836914,
        "sda_slope_se": 0.01571147844420545,
        "lmf_slope": -0.6461334464274452,
        "lmf_slope_se": 0.047829592773955466,
        "theory_slope": -0.3333333333333333,
        "sda_paper_bound": 0.2885552660332561,
        "sda_gap_over_bound": 0.07034766770923827,
        "sda_bound_pass": true,
        "lmf_paper_bound": 0.07060000000000001,
        "lmf_gap_over_bound": 0.017275225346317657,
        "lmf_bound_pass": true
      },
      {
        "p": 1.5,
        "T": 1024,
        "seeds": 8,
        "sda_gap": 0.013804600306821398,
        "sda_se": 0.00014340029094809246,
        "lmf_gap": 0.0007893202151434956,
        "lmf_se": 4.815502192397839e-05,
        "sda_slope": -0.5860486846836914,
        "sda_slope_se": 0.01571147844420545,
        "lmf_slope": -0.6461334464274452,
        "lmf_slope_se": 0.047829592773955466,
        "theory_slope": -0.3333333333333333,
        "sda_paper_bound": 0.2289232123349109,
        "sda_gap_over_bound": 0.060302317820988355,
        "sda_bound_pass": true,
        "lmf_paper_bound": 0.05603525713447745,
        "lmf_gap_over_bound": 0.014086135328142211,
        "lmf_bound_pass": true
      },
      {
        "p": 1.5,
        "T": 2048,
        "seeds": 8,
        "sda_gap": 0.009311845934982011,
        "sda_se": 0.00014303413305610552,
        "lmf_gap": 0.0005268383778813218,
        "lmf_se": 2.615453238909645e-05,
        "sda_slope": -0.5860486846836914,
        "sda_slope_se": 0.01571147844420545,
        "lmf_slope": -0.6461334464274452,
        "lmf_slope_se": 0.047829592773955466,
        "theory_slope": -0.3333333333333333,
        "sda_paper_bound": 0.18164484699460537,
        "sda_gap_over_bound": 0.05126402476618872,
        "sda_bound_pass": true,
        "lmf_paper_bound": 0.04447521306128902,
        "lmf_gap_over_bound": 0.011845662822465014,
        "lmf_bound_pass": true
      },
      {
        "p": 1.5,
        "T": 4096,
        "seeds": 8,
        "sda_gap": 0.0062041119932328935,
        "sda_se": 0.00022987555612491003,
        "lmf_gap": 0.00033187104436155487,
        "lmf_se": 1.6491064682524838e-05,
        "sda_slope": -0.5860486846836914,
        "sda_slope_se": 0.01571147844420545,
        "lmf_slope": -0.6461334464274452,
        "lmf_slope_se": 0.047829592773955466,
        "theory_slope": -0.3333333333333333,
        "sda_paper_bound": 0.14414579707912803,
        "sda_gap_over_bound": 0.04304053339707977,
        "sda_bound_pass": true,
        "lmf_paper_bound": 0.035300000000000005,
        "lmf_gap_over_bound": 0.009401446015908069,
        "lmf_bound_pass": true
      },
      {
        "p": 1.5,
        "T": 8192,
        "seeds": 8,
        "sda_gap": 0.00399279074445409,
        "sda_se": 0.0001268877262009361,
        "lmf_gap": 0.00020107692314433383,
        "lmf_se": 1.7484666345610177e-05,
        "sda_slope": -0.5860486846836914,
        "sda_slope_se": 0.01571147844420545,
        "lmf_slope": -0.6461334464274452,
        "lmf_slope_se": 0.047829592773955466,
        "theory_slope": -0.3333333333333333,
        "sda_paper_bound": 0.11439568819870544,
        "sda_gap_over_bound": 0.03490333252350043,
        "sda_bound_pass": true,
        "lmf_paper_bound": 0.028017628567238724,
        "lmf_gap_over_bound": 0.007176800229961466,
        "lmf_bound_pass": true
      },
      {
        "p": 1.75,
        "T": 512,
        "seeds": 8,
        "sda_gap": 0.007561183833494433,
        "sda_se": 6.82066036829144e-05,
        "lmf_gap": 0.0003855622117886156,
        "lmf_se": 5.161907756358701e-05,
        "sda_slope": -0.8167287045566036,
        "sda_slope_se": 0.014409603100413768,
        "lmf_slope": -0.853328489837801,
        "lmf_slope_se": 0.07336203210294678,
        "theory_slope": -0.42857142857142855,
        "sda_paper_bound": 0.1579294649488694,
        "sda_gap_over_bound": 0.047876967328056304,
        "sda_bound_pass": true,
        "lmf_paper_bound": 0.03640708556210963,
        "lmf_gap_over_bound": 0.010590306964584012,
        "lmf_bound_pass": true
      },
      {
        "p": 1.75,
        "T": 1024,
        "seeds": 8,
        "sda_gap": 0.004402023411794142,
        "sda_se": 6.526116958336639e-05,
        "lmf_gap": 0.00021470374620362524,
        "lmf_se": 2.042049541653712e-05,
        "sda_slope": -0.8167287045566036,
        "sda_slope_se": 0.014409603100413768,
        "lmf_slope": -0.853328489837801,
        "lmf_slope_se": 0.07336203210294678,
        "theory_slope": -0.42857142857142855,
        "sda_paper_bound": 0.11725571281659952,
        "sda_gap_over_bound": 0.037542080518323036,
        "sda_bound_pass": true,
        "lmf_paper_bound": 0.02705036061470758,
        "lmf_gap_over_bound": 0.007937186097507641,
        "lmf_bound_pass": true
      },
      {
        "p": 1.75,
        "T": 2048,
        "seeds": 8,
        "sda_gap": 0.002507430185174823,
        "sda_se": 4.524585108297467e-05,
        "lmf_gap": 0.00012530583710447163,
        "lmf_se": 1.064206793552962e-05,
        "sda_slope": -0.8167287045566036,
        "sda_slope_se": 0.014409603100413768,
        "lmf_slope": -0.853328489837801,
        "lmf_slope_se": 0.07336203210294678,
        "theory_slope": -0.42857142857142855,
        "sda_paper_bound": 0.08707794546525582,
        "sda_gap_over_bound": 0.02879523824060927,
        "sda_bound_pass": true,
        "lmf_paper_bound": 0.020098340696275252,
        "lmf_gap_over_bound": 0.006234635933288467,
        "lmf_bound_pass": true
      },
      {
        "p": 1.75,
        "T": 4096,
        "seeds": 8,
        "sda_gap": 0.0014176658045538316,
        "sda_se": 4.055653734780267e-05,
        "lmf_gap": 6.829678301285914e-05,
        "lmf_se": 5.958577468930608e-06,
        "sda_slope": -0.8167287045566036,
        "sda_slope_se": 0.014409603100413768,
        "lmf_slope": -0.853328489837801,
        "lmf_slope_se": 0.07336203210294678,
        "theory_slope": -0.42857142857142855,
        "sda_paper_bound": 0.06467730766466505,
        "sda_gap_over_bound": 0.021919060266145565,
        "sda_bound_pass": true,
        "lmf_paper_bound": 0.014933009747896872,
        "lmf_gap_over_bound": 0.004573544393652987,
        "lmf_bound_pass": true
      },
      {
        "p": 1.75,
        "T": 8192,
        "seeds": 8,
        "sda_gap": 0.0007881026054856755,
        "sda_se": 2.0255505859212637e-05,
        "lmf_gap": 3.5226367570175975e-05,
        "lmf_se": 4.260407316694122e-06,
        "sda_slope": -0.8167287045566036,
        "sda_slope_se": 0.014409603100413768,
        "lmf_slope": -0.853328489837801,
        "lmf_slope_se": 0.07336203210294678,
        "theory_slope": -0.42857142857142855,
        "sda_paper_bound": 0.048044376327768146,
        "sda_gap_over_bound": 0.01640363900467945,
        "sda_bound_pass": true,
        "lmf_paper_bound": 0.011095183602500565,
        "lmf_gap_over_bound": 0.0031749242583274484,
        "lmf_bound_pass": true
      },
      {
        "p": 2.0,
        "T": 512,
        "seeds": 8,
        "sda_gap": 0.0030867723109672075,
        "sda_se": 0.00011794200437935845,
        "lmf_gap": 0.00016251347907582744,
        "lmf_se": 3.2047746472274625e-05,
        "sda_slope": -0.9309027617644667,
        "sda_slope_se": 0.024641881808545537,
        "lmf_slope": -1.0230989279420224,
        "lmf_slope_se": 0.11455613134941793,
        "theory_slope": -0.5,
        "sda_paper_bound": 0.10110788271683502,
        "sda_gap_over_bound": 0.030529492142685754,
        "sda_bound_pass": true,
        "lmf_paper_bound": 0.0225967757041181,
        "lmf_gap_over_bound": 0.007191887957989092,
        "lmf_bound_pass": true
      },
      {
        "p": 2.0,
        "T": 1024,
        "seeds": 8,
        "sda_gap": 0.0015847846314102551,
        "sda_se": 6.398889735758811e-05,
        "lmf_gap": 7.966563315752725e-05,
        "lmf_se": 1.1643850041373132e-05,
        "sda_slope": -0.9309027617644667,
        "sda_slope_se": 0.024641881808545537,
        "lmf_slope": -1.0230989279420224,
        "lmf_slope_se": 0.11455613134941793,
        "theory_slope": -0.5,
        "sda_paper_bound": 0.07142125852272728,
        "sda_gap_over_bound": 0.02218925659096239,
        "sda_bound_pass": true,
        "lmf_paper_bound": 0.015978333333333334,
        "lmf_gap_over_bound": 0.004985853749297627,
        "lmf_bound_pass": true
      },
      {
        "p": 2.0,
        "T": 2048,
        "seeds": 8,
        "sda_gap": 0.0008161060532538181,
        "sda_se": 2.1256955109896918e-05,
        "lmf_gap": 4.2051562210437934e-05,
        "lmf_se": 5.891467102667199e-06,
        "sda_slope": -0.9309027617644667,
        "sda_slope_se": 0.024641881808545537,
        "lmf_slope": -1.0230989279420224,
        "lmf_slope_se": 0.11455613134941793,
        "theory_slope": -0.5,
        "sda_paper_bound": 0.05046605073341751,
        "sda_gap_over_bound": 0.01617138732659757,
        "sda_bound_pass": true,
        "lmf_paper_bound": 0.01129838785205905,
        "lmf_gap_over_bound": 0.0037219081838099884,
        "lmf_bound_pass": true
      },
      {
        "p": 2.0,
        "T": 4096,
        "seeds": 8,
        "sda_gap": 0.0004341538580646733,
        "sda_se": 1.0594666112589563e-05,
        "lmf_gap": 2.0493742766596674e-05,
        "lmf_se": 3.085560054155993e-06,
        "sda_slope": -0.9309027617644667,
        "sda_slope_se": 0.024641881808545537,
        "lmf_slope": -1.0230989279420224,
        "lmf_slope_se": 0.11455613134941793,
        "theory_slope": -0.5,
        "sda_paper_bound": 0.03566668394886364,
        "sda_gap_over_bound": 0.012172532178408631,
        "sda_bound_pass": true,
        "lmf_paper_bound": 0.007989166666666667,
        "lmf_gap_over_bound": 0.002565191542705331,
        "lmf_bound_pass": true
      },
      {
        "p": 2.0,
        "T": 8192,
        "seeds": 8,
        "sda_gap": 0.00023416948415784977,
        "sda_se": 1.054267847585875e-05,
        "lmf_gap": 9.181136220669485e-06,
        "lmf_se": 1.726781423025942e-06,
        "sda_slope": -0.9309027617644667,
        "sda_slope_se": 0.024641881808545537,
        "lmf_slope": -1.0230989279420224,
        "lmf_slope_se": 0.11455613134941793,
        "theory_slope": -0.5,
        "sda_paper_bound": 0.025211052710458755,
        "sda_gap_over_bound": 0.00928836597373441,
        "sda_bound_pass": true,
        "lmf_paper_bound": 0.005649193926029525,
        "lmf_gap_over_bound": 0.0016252117277061415,
        "lmf_bound_pass": true
      }
    ],
    "baseline": [
      0.003939990777114324,
      0.0
    ],
    "destructive": [
      1.3468851594615545,
      0.0
    ]
  },
  "claim_5": {
    "rows": [
      {
        "p": 1.5,
        "eps": 0.08,
        "seeds": 20,
        "eta_mean": 0.19509221155311723,
        "sigma_p": 0.07014805770653947,
        "iterations_to_mean_mse_eps": 40.0,
        "iteration_se": 0.0,
        "log_corrected_ratio": 0.1013568898531113,
        "max_mean_mse": 0.0011930736004510086,
        "measured_eps_slope": -0.9660220671769074,
        "theory_power_slope": -2.0
      },
      {
        "p": 1.5,
        "eps": 0.05,
        "seeds": 20,
        "eta_mean": 0.07620789513793644,
        "sigma_p": 0.07014805770653947,
        "iterations_to_mean_mse_eps": 44.0,
        "iteration_se": 0.21275139927797618,
        "log_corrected_ratio": 0.03671890207648675,
        "max_mean_mse": 0.0008852798018670184,
        "measured_eps_slope": -0.9660220671769074,
        "theory_power_slope": -2.0
      },
      {
        "p": 1.5,
        "eps": 0.03,
        "seeds": 20,
        "eta_mean": 0.02743484224965711,
        "sigma_p": 0.07014805770653947,
        "iterations_to_mean_mse_eps": 76.0,
        "iteration_se": 1.0647460779577054,
        "log_corrected_ratio": 0.01950630846628178,
        "max_mean_mse": 0.00033664730177849133,
        "measured_eps_slope": -0.9660220671769074,
        "theory_power_slope": -2.0
      },
      {
        "p": 1.5,
        "eps": 0.02,
        "seeds": 20,
        "eta_mean": 0.012193263222069827,
        "sigma_p": 0.07014805770653947,
        "iterations_to_mean_mse_eps": 152.0,
        "iteration_se": 1.7605322161808135,
        "log_corrected_ratio": 0.015541830893028151,
        "max_mean_mse": 0.00024188293800586833,
        "measured_eps_slope": -0.9660220671769074,
        "theory_power_slope": -2.0
      },
      {
        "p": 1.75,
        "eps": 0.08,
        "seeds": 20,
        "eta_mean": 0.1714285714285714,
        "sigma_p": 0.033699430083239904,
        "iterations_to_mean_mse_eps": 40.0,
        "iteration_se": 0.0,
        "log_corrected_ratio": 0.5459169989332092,
        "max_mean_mse": 0.0005737974650562938,
        "measured_eps_slope": -0.1504183539919776,
        "theory_power_slope": -1.3333333333333333
      },
      {
        "p": 1.75,
        "eps": 0.05,
        "seeds": 20,
        "eta_mean": 0.1714285714285714,
        "sigma_p": 0.033699430083239904,
        "iterations_to_mean_mse_eps": 40.0,
        "iteration_se": 0.0,
        "log_corrected_ratio": 0.24595198517321656,
        "max_mean_mse": 0.0005737974650562938,
        "measured_eps_slope": -0.1504183539919776,
        "theory_power_slope": -1.3333333333333333
      },
      {
        "p": 1.75,
        "eps": 0.03,
        "seeds": 20,
        "eta_mean": 0.1284565312197629,
        "sigma_p": 0.033699430083239904,
        "iterations_to_mean_mse_eps": 42.0,
        "iteration_se": 0.13377121081198773,
        "log_corrected_ratio": 0.1116511710955674,
        "max_mean_mse": 0.000516434333314606,
        "measured_eps_slope": -0.1504183539919776,
        "theory_power_slope": -1.3333333333333333
      },
      {
        "p": 1.75,
        "eps": 0.02,
        "seeds": 20,
        "eta_mean": 0.07481141082758229,
        "sigma_p": 0.033699430083239904,
        "iterations_to_mean_mse_eps": 50.0,
        "iteration_se": 0.20390916450326857,
        "log_corrected_ratio": 0.06938654534568188,
        "max_mean_mse": 0.00021137691982808332,
        "measured_eps_slope": -0.1504183539919776,
        "theory_power_slope": -1.3333333333333333
      }
    ],
    "destructive": [
      {
        "p": 1.5,
        "eps": 0.05,
        "seeds": 20,
        "control_final_mse": 2.8899999999999997,
        "control_final_mse_se": 1.0188105198175e-16,
        "control_reached": 0
      },
      {
        "p": 1.5,
        "eps": 0.02,
        "seeds": 20,
        "control_final_mse": 2.8899999999999997,
        "control_final_mse_se": 1.0188105198175e-16,
        "control_reached": 0
      },
      {
        "p": 1.75,
        "eps": 0.05,
        "seeds": 20,
        "control_final_mse": 2.8899999999999997,
        "control_final_mse_se": 1.0188105198175e-16,
        "control_reached": 0
      },
      {
        "p": 1.75,
        "eps": 0.02,
        "seeds": 20,
        "control_final_mse": 2.8899999999999997,
        "control_final_mse_se": 1.0188105198175e-16,
        "control_reached": 0
      }
    ],
    "baseline": [
      2.7733391199176196e-30,
      0.0
    ]
  },
  "claim_6": {
    "rows": [
      {
        "function": "Huber(a=1.0)",
        "seeds": 12,
        "trials": 12,
        "margin_mean": 0.10717406311828749,
        "margin_se": 0.03336110366071947,
        "pass_rate": 1.0,
        "destructive_margin_mean": -0.2528259368818276,
        "destructive_margin_se": 0.03336110366067248,
        "destructive_pass_rate": 0.0,
        "max_jump_note": "unbounded Pareto large jumps, alpha=1.45"
      },
      {
        "function": "Huber(a=1.5)",
        "seeds": 12,
        "trials": 12,
        "margin_mean": 0.17434671657397904,
        "margin_se": 0.020135557496682843,
        "pass_rate": 1.0,
        "destructive_margin_mean": -0.18565328342610443,
        "destructive_margin_se": 0.020135557496646556,
        "destructive_pass_rate": 0.0,
        "max_jump_note": "unbounded Pareto large jumps, alpha=1.45"
      },
      {
        "function": "Huber(a=2.0)",
        "seeds": 12,
        "trials": 12,
        "margin_mean": 0.20395777275784496,
        "margin_se": 0.010385763662758767,
        "pass_rate": 1.0,
        "destructive_margin_mean": -0.1560422272422015,
        "destructive_margin_se": 0.010385763662717389,
        "destructive_pass_rate": 0.0,
        "max_jump_note": "unbounded Pareto large jumps, alpha=1.45"
      }
    ],
    "quadratic_baseline": [
      0.0031226959787806342,
      0.0017117577613460437
    ]
  }
}