{
  "seed": 3148,
  "distribution": "Exp(rate=1) truncated to [0,8]",
  "utility": "log1p(x/0.5)/log1p(8/0.5)",
  "theory_Cw": 0.4142702640796977,
  "results": [
    {
      "n": 4,
      "weighted_left": {
        "utility_gap": 0.09324517108844732,
        "n_gap": 0.3729806843537893,
        "mse": 0.3908160360658155,
        "upward_fraction": 0.0,
        "mean_abs_error": 0.3801211825816941
      },
      "uniform_left": {
        "utility_gap": 0.24507217705691614,
        "n_gap": 0.9802887082276646,
        "mse": 0.7492558589454571,
        "upward_fraction": 0.0,
        "mean_abs_error": 0.6878339440679532
      },
      "mse_lloyd": {
        "utility_gap": -0.008290065900775563,
        "n_gap": -0.03316026360310225,
        "mse": 0.10127602204511336,
        "upward_fraction": 0.5408575,
        "mean_abs_error": 0.24150891658722082
      }
    },
    {
      "n": 8,
      "weighted_left": {
        "utility_gap": 0.04854488937884897,
        "n_gap": 0.38835911503079174,
        "mse": 0.14985366883315587,
        "upward_fraction": 0.0,
        "mean_abs_error": 0.2168465353561867
      },
      "uniform_left": {
        "utility_gap": 0.15027493297280364,
        "n_gap": 1.2021994637824291,
        "mse": 0.2538252095920833,
        "upward_fraction": 0.0,
        "mean_abs_error": 0.4177264440679533
      },
      "mse_lloyd": {
        "utility_gap": -0.001907189540334921,
        "n_gap": -0.015257516322679368,
        "mse": 0.02770827889938997,
        "upward_fraction": 0.5183075,
        "mean_abs_error": 0.12492912127391771
      }
    },
    {
      "n": 16,
      "weighted_left": {
        "utility_gap": 0.024879950363311396,
        "n_gap": 0.39807920581298234,
        "mse": 0.05295840149474882,
        "upward_fraction": 0.0,
        "mean_abs_error": 0.11828682619482793
      },
      "uniform_left": {
        "utility_gap": 0.08026878503984931,
        "n_gap": 1.284300560637589,
        "mse": 0.0731415326034113,
        "upward_fraction": 0.0,
        "mean_abs_error": 0.22928644406795334
      },
      "mse_lloyd": {
        "utility_gap": 0.0005029697334268081,
        "n_gap": 0.00804751573482893,
        "mse": 0.009560218923305799,
        "upward_fraction": 0.4940625,
        "mean_abs_error": 0.061487157693296955
      }
    },
    {
      "n": 32,
      "weighted_left": {
        "utility_gap": 0.012638084995028224,
        "n_gap": 0.40441871984090316,
        "mse": 0.017530396427408047,
        "upward_fraction": 0.0,
        "mean_abs_error": 0.06271684156830157
      },
      "uniform_left": {
        "utility_gap": 0.04072462813716193,
        "n_gap": 1.3031881003891819,
        "mse": 0.01953784513814554,
        "upward_fraction": 0.0,
        "mean_abs_error": 0.11979019406795328
      },
      "mse_lloyd": {
        "utility_gap": 0.00032317258009456824,
        "n_gap": 0.010341522563026184,
        "mse": 0.004382087953092174,
        "upward_fraction": 0.490755,
        "mean_abs_error": 0.0348433373494
      }
    },
    {
      "n": 64,
      "weighted_left": {
        "utility_gap": 0.006394305394357603,
        "n_gap": 0.4092355452388866,
        "mse": 0.005331562075579124,
        "upward_fraction": 0.0,
        "mean_abs_error": 0.032584814299362025
      },
      "uniform_left": {
        "utility_gap": 0.02039501107767134,
        "n_gap": 1.3052807089709657,
        "mse": 0.005045088429382813,
        "upward_fraction": 0.0,
        "mean_abs_error": 0.06121581906795333
      },
      "mse_lloyd": {
        "utility_gap": 0.00012352114489887618,
        "n_gap": 0.007905353273528076,
        "mse": 0.0020636113079323046,
        "upward_fraction": 0.495115,
        "mean_abs_error": 0.019787217980367658
      }
    }
  ],
  "scaling": [
    {
      "n": 4,
      "weighted_n_gap": 0.3729806843537893,
      "uniform_n_gap": 0.9802887082276646,
      "theory_Cw": 0.4142702640796977
    },
    {
      "n": 8,
      "weighted_n_gap": 0.38835911503079174,
      "uniform_n_gap": 1.2021994637824291,
      "theory_Cw": 0.4142702640796977
    },
    {
      "n": 16,
      "weighted_n_gap": 0.39807920581298234,
      "uniform_n_gap": 1.284300560637589,
      "theory_Cw": 0.4142702640796977
    },
    {
      "n": 32,
      "weighted_n_gap": 0.40441871984090316,
      "uniform_n_gap": 1.3031881003891819,
      "theory_Cw": 0.4142702640796977
    },
    {
      "n": 64,
      "weighted_n_gap": 0.4092355452388866,
      "uniform_n_gap": 1.3052807089709657,
      "theory_Cw": 0.4142702640796977
    }
  ],
  "allocation_check": {
    "weighted_g_mass_cv": 1.938530743037839e-09,
    "weighted_g_mass_min": 0.028445062872180432,
    "weighted_g_mass_max": 0.028445063122261804
  },
  "interpretation": {
    "claimed_signal": "weighted left-edge utility gap scales approximately as Cw/n and is lower than simple uniform left-edge",
    "conservative_rule": "weighted and uniform left-edge have zero upward rounding by construction",
    "mse_control": "Lloyd-Max minimizes squared error, not the monotone left-edge utility gap"
  }
}