1{
  2  "seed": 3148,
  3  "distribution": "Exp(rate=1) truncated to [0,8]",
  4  "utility": "log1p(x/0.5)/log1p(8/0.5)",
  5  "theory_Cw": 0.4142702640796977,
  6  "results": [
  7    {
  8      "n": 4,
  9      "weighted_left": {
 10        "utility_gap": 0.09324517108844732,
 11        "n_gap": 0.3729806843537893,
 12        "mse": 0.3908160360658155,
 13        "upward_fraction": 0.0,
 14        "mean_abs_error": 0.3801211825816941
 15      },
 16      "uniform_left": {
 17        "utility_gap": 0.24507217705691614,
 18        "n_gap": 0.9802887082276646,
 19        "mse": 0.7492558589454571,
 20        "upward_fraction": 0.0,
 21        "mean_abs_error": 0.6878339440679532
 22      },
 23      "mse_lloyd": {
 24        "utility_gap": -0.008290065900775563,
 25        "n_gap": -0.03316026360310225,
 26        "mse": 0.10127602204511336,
 27        "upward_fraction": 0.5408575,
 28        "mean_abs_error": 0.24150891658722082
 29      }
 30    },
 31    {
 32      "n": 8,
 33      "weighted_left": {
 34        "utility_gap": 0.04854488937884897,
 35        "n_gap": 0.38835911503079174,
 36        "mse": 0.14985366883315587,
 37        "upward_fraction": 0.0,
 38        "mean_abs_error": 0.2168465353561867
 39      },
 40      "uniform_left": {
 41        "utility_gap": 0.15027493297280364,
 42        "n_gap": 1.2021994637824291,
 43        "mse": 0.2538252095920833,
 44        "upward_fraction": 0.0,
 45        "mean_abs_error": 0.4177264440679533
 46      },
 47      "mse_lloyd": {
 48        "utility_gap": -0.001907189540334921,
 49        "n_gap": -0.015257516322679368,
 50        "mse": 0.02770827889938997,
 51        "upward_fraction": 0.5183075,
 52        "mean_abs_error": 0.12492912127391771
 53      }
 54    },
 55    {
 56      "n": 16,
 57      "weighted_left": {
 58        "utility_gap": 0.024879950363311396,
 59        "n_gap": 0.39807920581298234,
 60        "mse": 0.05295840149474882,
 61        "upward_fraction": 0.0,
 62        "mean_abs_error": 0.11828682619482793
 63      },
 64      "uniform_left": {
 65        "utility_gap": 0.08026878503984931,
 66        "n_gap": 1.284300560637589,
 67        "mse": 0.0731415326034113,
 68        "upward_fraction": 0.0,
 69        "mean_abs_error": 0.22928644406795334
 70      },
 71      "mse_lloyd": {
 72        "utility_gap": 0.0005029697334268081,
 73        "n_gap": 0.00804751573482893,
 74        "mse": 0.009560218923305799,
 75        "upward_fraction": 0.4940625,
 76        "mean_abs_error": 0.061487157693296955
 77      }
 78    },
 79    {
 80      "n": 32,
 81      "weighted_left": {
 82        "utility_gap": 0.012638084995028224,
 83        "n_gap": 0.40441871984090316,
 84        "mse": 0.017530396427408047,
 85        "upward_fraction": 0.0,
 86        "mean_abs_error": 0.06271684156830157
 87      },
 88      "uniform_left": {
 89        "utility_gap": 0.04072462813716193,
 90        "n_gap": 1.3031881003891819,
 91        "mse": 0.01953784513814554,
 92        "upward_fraction": 0.0,
 93        "mean_abs_error": 0.11979019406795328
 94      },
 95      "mse_lloyd": {
 96        "utility_gap": 0.00032317258009456824,
 97        "n_gap": 0.010341522563026184,
 98        "mse": 0.004382087953092174,
 99        "upward_fraction": 0.490755,
100        "mean_abs_error": 0.0348433373494
101      }
102    },
103    {
104      "n": 64,
105      "weighted_left": {
106        "utility_gap": 0.006394305394357603,
107        "n_gap": 0.4092355452388866,
108        "mse": 0.005331562075579124,
109        "upward_fraction": 0.0,
110        "mean_abs_error": 0.032584814299362025
111      },
112      "uniform_left": {
113        "utility_gap": 0.02039501107767134,
114        "n_gap": 1.3052807089709657,
115        "mse": 0.005045088429382813,
116        "upward_fraction": 0.0,
117        "mean_abs_error": 0.06121581906795333
118      },
119      "mse_lloyd": {
120        "utility_gap": 0.00012352114489887618,
121        "n_gap": 0.007905353273528076,
122        "mse": 0.0020636113079323046,
123        "upward_fraction": 0.495115,
124        "mean_abs_error": 0.019787217980367658
125      }
126    }
127  ],
128  "scaling": [
129    {
130      "n": 4,
131      "weighted_n_gap": 0.3729806843537893,
132      "uniform_n_gap": 0.9802887082276646,
133      "theory_Cw": 0.4142702640796977
134    },
135    {
136      "n": 8,
137      "weighted_n_gap": 0.38835911503079174,
138      "uniform_n_gap": 1.2021994637824291,
139      "theory_Cw": 0.4142702640796977
140    },
141    {
142      "n": 16,
143      "weighted_n_gap": 0.39807920581298234,
144      "uniform_n_gap": 1.284300560637589,
145      "theory_Cw": 0.4142702640796977
146    },
147    {
148      "n": 32,
149      "weighted_n_gap": 0.40441871984090316,
150      "uniform_n_gap": 1.3031881003891819,
151      "theory_Cw": 0.4142702640796977
152    },
153    {
154      "n": 64,
155      "weighted_n_gap": 0.4092355452388866,
156      "uniform_n_gap": 1.3052807089709657,
157      "theory_Cw": 0.4142702640796977
158    }
159  ],
160  "allocation_check": {
161    "weighted_g_mass_cv": 1.938530743037839e-09,
162    "weighted_g_mass_min": 0.028445062872180432,
163    "weighted_g_mass_max": 0.028445063122261804
164  },
165  "interpretation": {
166    "claimed_signal": "weighted left-edge utility gap scales approximately as Cw/n and is lower than simple uniform left-edge",
167    "conservative_rule": "weighted and uniform left-edge have zero upward rounding by construction",
168    "mse_control": "Lloyd-Max minimizes squared error, not the monotone left-edge utility gap"
169  }
170}