1{
  2  "predictions": {
  3    "telescoping": [
  4      {
  5        "k": 1,
  6        "max_abs_R": 4.163336342344337e-17,
  7        "std_R_over_k": 2.168116284641578e-17
  8      },
  9      {
 10        "k": 2,
 11        "max_abs_R": 8.326672684688674e-17,
 12        "std_R_over_k": 2.16082492855429e-17
 13      },
 14      {
 15        "k": 4,
 16        "max_abs_R": 2.220446049250313e-16,
 17        "std_R_over_k": 2.265712673939649e-17
 18      },
 19      {
 20        "k": 8,
 21        "max_abs_R": 3.3306690738754696e-16,
 22        "std_R_over_k": 2.030441442089029e-17
 23      },
 24      {
 25        "k": 16,
 26        "max_abs_R": 6.661338147750939e-16,
 27        "std_R_over_k": 1.5241093489646827e-17
 28      },
 29      {
 30        "k": 32,
 31        "max_abs_R": 1.3322676295501878e-15,
 32        "std_R_over_k": 1.0857732333752398e-17
 33      }
 34    ],
 35    "variation": [
 36      {
 37        "eps": 0,
 38        "pointwise_var": 0.0,
 39        "pointwise_max_abs": 0.0
 40      },
 41      {
 42        "eps": 0.05,
 43        "pointwise_var": 1.0169118040411112e-06,
 44        "pointwise_max_abs": 0.002878376042883945
 45      },
 46      {
 47        "eps": 0.1,
 48        "pointwise_var": 4.731678692573781e-06,
 49        "pointwise_max_abs": 0.0061871279240460035
 50      },
 51      {
 52        "eps": 0.2,
 53        "pointwise_var": 2.6255435432403586e-05,
 54        "pointwise_max_abs": 0.014493763122599802
 55      },
 56      {
 57        "eps": 0.4,
 58        "pointwise_var": 0.0002316087056906549,
 59        "pointwise_max_abs": 0.04286763431030212
 60      },
 61      {
 62        "eps": 0.6,
 63        "pointwise_var": 0.001572758070888074,
 64        "pointwise_max_abs": 0.1126197056294694
 65      },
 66      {
 67        "eps": 0.8,
 68        "pointwise_var": 0.01854074816350247,
 69        "pointwise_max_abs": 0.38800068937592674
 70      }
 71    ],
 72    "small_eps_loglog_slope_variance_vs_eps": 2.596624544078102,
 73    "scaled_potential": [
 74      {
 75        "alpha": 0,
 76        "1": 0.040152144012692285,
 77        "4": 0.050434239322700435,
 78        "16": 0.03997558433644486,
 79        "64": 0.010303624245433907
 80      },
 81      {
 82        "alpha": 0.25,
 83        "1": 0.030114108009519212,
 84        "4": 0.03782567949202533,
 85        "16": 0.029981688252333646,
 86        "64": 0.007727718184075429
 87      },
 88      {
 89        "alpha": 0.5,
 90        "1": 0.020076072006346136,
 91        "4": 0.02521711966135022,
 92        "16": 0.01998779216822243,
 93        "64": 0.005151812122716953
 94      },
 95      {
 96        "alpha": 0.75,
 97        "1": 0.010038036003173061,
 98        "4": 0.012608559830675109,
 99        "16": 0.009993896084111218,
100        "64": 0.002575906061358477
101      },
102      {
103        "alpha": 1.0,
104        "1": 2.247863622854625e-17,
105        "4": 2.32362243473473e-17,
106        "16": 1.5570951353874558e-17,
107        "64": 8.461428283561199e-18
108      }
109    ],
110    "scaled_residual_ratio_prediction": [
111      {
112        "alpha": 0,
113        "ratio_to_alpha0_k1": 1.0
114      },
115      {
116        "alpha": 0.25,
117        "ratio_to_alpha0_k1": 0.75
118      },
119      {
120        "alpha": 0.5,
121        "ratio_to_alpha0_k1": 0.49999999999999983
122      },
123      {
124        "alpha": 0.75,
125        "ratio_to_alpha0_k1": 0.24999999999999975
126      }
127    ]
128  },
129  "baseline_vs_idea": {
130    "cohomological_mse": 3.4203940963475086e-11,
131    "pointwise_const_mse": 0.0016075948744300505,
132    "fitted_c": 0.029562285993071438,
133    "true_c": 0.02955880224154443,
134    "mse_ratio_coh_over_point": 2.127646803775833e-08
135  },
136  "prediction_statements": [
137    "P1 exact conjugacy predicts R_k=0 for every horizon k.",
138    "P2 pointwise variance is predicted to scale as eps^2 near eps=0.",
139    "P3 imperfect-potential residual per step scales as (1-alpha) and has a 1/k boundary decay."
140  ]
141}