{
  "seed": 1524,
  "n_states": 12000,
  "base_failure_rate": 0.46058333333333334,
  "critic": {
    "bce": 0.3512498187459485,
    "mean_prediction": 0.4608653265895653,
    "mean_label": 0.46058333333333334
  },
  "predictions": {
    "enrichment": "rho_fail=E_q[y]/E_p[y]",
    "ess": "ESS/B approaches 1/E_q[w^2]",
    "gradient_identity": "E_q[w g]=E_p[g]"
  },
  "verification": [
    {
      "alpha": 0.0,
      "predicted_enrichment": 0.9999999999999994,
      "observed_enrichment": 1.00020354622761,
      "predicted_ess_fraction": 0.9999999999999987,
      "weighted_gradient_cosine": 0.9773100556001814,
      "unweighted_gradient_cosine": 0.9773100556001814,
      "weighted_gradient_l2_error": 1.5865122319069402,
      "exact_identity_l2_error": 2.2216270818478592e-14
    },
    {
      "alpha": 0.5,
      "predicted_enrichment": 1.399414505771332,
      "observed_enrichment": 1.4018681020445087,
      "predicted_ess_fraction": 0.6955684305462502,
      "weighted_gradient_cosine": 0.966974955423025,
      "unweighted_gradient_cosine": 0.9592871973865706,
      "weighted_gradient_l2_error": 1.7740178957343766,
      "exact_identity_l2_error": 2.2023380273754356e-14
    },
    {
      "alpha": 1.0,
      "predicted_enrichment": 1.634298288820409,
      "observed_enrichment": 1.6397344852542066,
      "predicted_ess_fraction": 0.26766598629989596,
      "weighted_gradient_cosine": 0.9166753899563264,
      "unweighted_gradient_cosine": 0.9413258850208939,
      "weighted_gradient_l2_error": 2.769184041820306,
      "exact_identity_l2_error": 2.2727613731326205e-14
    },
    {
      "alpha": 2.0,
      "predicted_enrichment": 1.8312356246940125,
      "observed_enrichment": 1.8383051384114346,
      "predicted_ess_fraction": 0.017082705451833817,
      "weighted_gradient_cosine": 0.8670712367326305,
      "unweighted_gradient_cosine": 0.932892435467256,
      "weighted_gradient_l2_error": 5.776019217121439,
      "exact_identity_l2_error": 2.243090158935384e-14
    },
    {
      "alpha": 4.0,
      "predicted_enrichment": 1.945734722609935,
      "observed_enrichment": 1.9507078885471318,
      "predicted_ess_fraction": 2.8695367384982844e-05,
      "weighted_gradient_cosine": 0.9218421287687226,
      "unweighted_gradient_cosine": 0.9279067839672815,
      "weighted_gradient_l2_error": 5.828636433710799,
      "exact_identity_l2_error": 2.2262688202020137e-14
    }
  ],
  "training": [
    {
      "alpha": 0.5,
      "uniform_mse": 0.1651713471398834,
      "weighted_mse": 0.16512279290884183,
      "unweighted_mse": 0.19735122093968538
    },
    {
      "alpha": 1.0,
      "uniform_mse": 0.1651713471398834,
      "weighted_mse": 0.16516038832137736,
      "unweighted_mse": 0.29406721443241945
    },
    {
      "alpha": 2.0,
      "uniform_mse": 0.1651713471398834,
      "weighted_mse": 0.18004326633012838,
      "unweighted_mse": 0.38504488832931943
    }
  ],
  "notes": "Predictions are finite-population exact; observed values are Monte Carlo.",
  "importance_identity_check": {
    "max_exact_l2_error": 2.2727613731326205e-14,
    "criterion": "exact error should be numerical roundoff; finite-batch cosine degrades as ESS falls"
  }
}