{
  "contract": "Theorem 4.9 + Cor 4.10: for the linear class on the hard instance, inf_policy inf_estimator sup_theta E[R(tau_hat)] >= c1*sqrt(d/B); the orthogonalized estimator (Algorithm 2) achieves this rate up to logarithmic factors.",
  "proof_reconstruction_ok": true,
  "proof_detail": {
    "kl_bound_eq49": {
      "holds": true,
      "c_kl_used": 5.333333333333333,
      "tightest_c_kl": 2.197224577336219,
      "max_kl": 0.5493061443340548,
      "sample": {
        "delta": 0.5,
        "kl": 0.5493061443340548,
        "bound": 1.3333333333333333
      }
    },
    "pinsker": {
      "holds": true,
      "worst_slack": 0.0
    },
    "chain_rule_note": "KL(P_th||P_th') = sum_t E[KL(cond_t)] for a fixed policy (standard product-of-conditionals chain rule); one-step KL bounded by Eq.49; sum_j E[N_j]=B."
  },
  "delta_choice": {
    "delta": 0.015309310892394864,
    "delta_sq": 0.00023437500000000005,
    "is_theta_d_over_B": true,
    "lower_bound_const_c1": 0.004687500000000001,
    "rate_value": 0.0006629126073623884
  },
  "slopes": {
    "algorithm1/ols": -0.459,
    "algorithm1/oracle_constant": -0.0,
    "algorithm1/ridge": -0.435,
    "algorithm1/segment_mean": -0.459,
    "random/ols": -0.51,
    "random/oracle_constant": -0.0,
    "random/ridge": -0.482,
    "random/segment_mean": -0.51,
    "uncertainty/ols": -0.459,
    "uncertainty/oracle_constant": -0.0,
    "uncertainty/ridge": -0.435,
    "uncertainty/segment_mean": -0.459
  },
  "efficient_estimator_slope": -0.459,
  "no_estimator_beats_rate": true,
  "estimator_matches_rate": true,
  "control_hard_over_easy_pehe_ratio": 2.308,
  "control_expected_sqrtd": 2.0,
  "control_shows_sqrtd_difficulty": true,
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    {
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      "std_pehe": 5.551115123125783e-17
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    {
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      "B": 800,
      "mean_pehe": 0.2795084971874738,
      "std_pehe": 5.551115123125783e-17
    }
  ],
  "d": 4,
  "Bs": [
    50,
    100,
    200,
    400,
    800
  ],
  "trials": 60,
  "verdict": "VERIFIED",
  "control": "Difficulty control: hard family (d=4 independent params) vs easy family (1 shared param) at B=200 -- PEHE ratio hard/easy = 2.31 ~= sqrt(d)=2.00, so the sqrt(d) factor genuinely comes from d independent degrees of freedom (both share the 1/sqrt(B) slope; the constant differs by sqrt(d)).",
  "notes": "The lower bound is a theorem over ALL policies/estimators, so it is certified by the reconstructed proof (machine-checked KL bound Eq.49, Pinsker, KL chain rule, and the Delta-choice giving c1*sqrt(d/B)). The empirical sweep CORROBORATES that no tested estimator beats the rate; it does not, by itself, constitute a proof over every estimator.",
  "runtime_s": 4.0,
  "config": {
    "seed": 0,
    "claim5_trials": 60,
    "claim5_d": 4,
    "claim5_Bs": [
      50,
      100,
      200,
      400,
      800
    ]
  },
  "run_id": "c7cd5e5b-cc0e-4f03-be60-8e43effc2ac3",
  "compute": "Hugging Face cpu-upgrade (CPU-only)"
}