{
  "claim": "Claim 2: Theorems 2.4 & 2.5 (noise conditions cannot be relaxed)",
  "thm_2_4": {
    "theorem": "2.4",
    "params": {
      "d": 8,
      "k": 2,
      "lambda_k": 1.0,
      "beta": 0.16,
      "two_sqrt_beta": 0.8,
      "eps": 0.01,
      "Delta": 0.19999999999999996
    },
    "assumptions": {
      "cos_theta_k_X0": 0.999800059980007,
      "cos_positive": true,
      "tan_theta_k_X0": 0.020000000000000004,
      "equals_2eps": true
    },
    "noise": {
      "strict_c": 0.03125,
      "relaxed_constant": 8.0,
      "ratio_relaxed_over_strict": 256.0,
      "relaxed_condition_holds": true,
      "complementary_condition_zero": true
    },
    "divergence": {
      "horizon": 4000,
      "min_tan_theta": 0.020000000000000004,
      "max_tan_theta": 0.08268743525391134,
      "min_sin_theta": 0.019996001199600422,
      "never_reaches_eps": true,
      "official_qr_min_tan_theta": 0.011105145563000716,
      "official_qr_never_reaches_eps": true
    },
    "analytic_bound_phi": {
      "phi_min": 0.015,
      "phi_above_eps": true,
      "phi_le_observed_h": true
    }
  },
  "thm_2_5": {
    "theorem": "2.5",
    "params": {
      "d": 8,
      "k": 2,
      "lambda_k": 1.0,
      "beta": 0.16,
      "two_sqrt_beta": 0.8,
      "eps": 0.01,
      "Delta": 0.19999999999999996
    },
    "assumptions": {
      "cos_theta_k_X0": 0.999800059980007,
      "cos_positive": true,
      "tan_theta_k_X0": 0.020000000000000004,
      "equals_2eps": true
    },
    "noise": {
      "strict_c": 0.03125,
      "relaxed_constant": 1.0,
      "ratio_relaxed_over_strict": 32.0,
      "relaxed_condition_holds": true,
      "complementary_condition_zero": true
    },
    "divergence": {
      "horizon": 4000,
      "min_tan_theta": 0.019999999999989325,
      "max_tan_theta": 0.0200000000000001,
      "min_sin_theta": 0.019996001199589317,
      "never_reaches_eps": true,
      "official_qr_min_tan_theta": 0.0,
      "official_qr_never_reaches_eps": false
    },
    "analytic_bound_phi": {
      "phi_min": 0.015,
      "phi_above_eps": true,
      "phi_le_observed_h": true
    }
  },
  "strict_constant_converges": {
    "c": 0.03125,
    "final_tan_theta": 0.0003125000152587904,
    "converges": true
  },
  "verdict": "VERIFIED"
}