{
  "ideal": {
    "lower": "2147232289/2147483648",
    "minimum_state": [
      8,
      0
    ],
    "scale": 2147483648,
    "lambda_": 3216,
    "denominator": 321600,
    "state_count": 3240,
    "blocks": 3216,
    "last_nonzero_weight": 12,
    "matrix_accumulator_bound": 690630741196800,
    "weighted_accumulator_bound": 4611686007689969664,
    "weight_sum": 2147483643,
    "scope": "Exact downward integer recurrence, not a floating-point exponential. Stochastic-semigroup soundness uses the manuscript theorem; no Lean execution is claimed."
  },
  "imperfect": {
    "lower": "1073576721/1073741824",
    "minimum_state": [
      8,
      0
    ],
    "scale": 2147483648,
    "lambda_": 3216,
    "denominator": 321600,
    "state_count": 3240,
    "blocks": 3216,
    "last_nonzero_weight": 12,
    "matrix_accumulator_bound": 690630741196800,
    "weighted_accumulator_bound": 4611686007689969664,
    "weight_sum": 2147483643,
    "scope": "Exact downward integer recurrence, not a floating-point exponential. Stochastic-semigroup soundness uses the manuscript theorem; no Lean execution is claimed."
  },
  "endpoint_reduction": "Recovery is monotone; the complementary split payoff is symmetric about 1/2 and increases up to 1/2. Thus theta in [0.48,0.52], eta in [0.9,1] is covered by (0.48,0.9), not by a parameter grid."
}