{
  "toy": [
    {
      "p": "1/3",
      "split_raf": "505585/531441",
      "projected_or_raf": "505585/531441",
      "homogeneous_quotient_raf": "5137/6561"
    },
    {
      "p": "1/2",
      "split_raf": "4077/4096",
      "projected_or_raf": "4077/4096",
      "homogeneous_quotient_raf": "239/256"
    }
  ],
  "projection": {
    "configurations_checked": 4096,
    "successful_witness_lifts": 4077
  },
  "history_identity": {
    "configurations_checked": 256,
    "distinct_histories": 6,
    "exact_count_polynomials_equal": true
  },
  "seed_cutoffs": [
    {
      "a0": "1/10",
      "m": 10,
      "k": 53667,
      "L": 536670,
      "exact_integer_inequality": true,
      "scope": "Sufficient supplied-seed cutoff; not a convergence rate or practical survival estimate."
    },
    {
      "a0": "1/2",
      "m": 10,
      "k": 2082,
      "L": 20820,
      "exact_integer_inequality": true,
      "scope": "Sufficient supplied-seed cutoff; not a convergence rate or practical survival estimate."
    }
  ],
  "finite_catalytic": [
    {
      "n": 6,
      "lambda": "0.05",
      "p": "3/5020",
      "trials": 256,
      "rafs": 37,
      "gateway_samples": 236,
      "food_only_rafs": 3,
      "raf_estimate": 0.14453125,
      "raf_wilson95": [
        0.10671067600593546,
        0.1928622219218325
      ],
      "gateway_probability_exact_formula_float": 0.9227667067590918,
      "conditional_estimate": 0.15677966101694915,
      "conditional_wilson95": [
        0.11593449631313285,
        0.20861931187099986
      ]
    },
    {
      "n": 6,
      "lambda": "0.1",
      "p": "3/2510",
      "trials": 256,
      "rafs": 59,
      "gateway_samples": 253,
      "food_only_rafs": 6,
      "raf_estimate": 0.23046875,
      "raf_wilson95": [
        0.183093576056734,
        0.28581334654783735
      ],
      "gateway_probability_exact_formula_float": 0.9940441486168119,
      "conditional_estimate": 0.233201581027668,
      "conditional_wilson95": [
        0.18532260421177646,
        0.28906131877377994
      ]
    },
    {
      "n": 6,
      "lambda": "0.25",
      "p": "3/1004",
      "trials": 256,
      "rafs": 245,
      "gateway_samples": 256,
      "food_only_rafs": 0,
      "raf_estimate": 0.95703125,
      "raf_wilson95": [
        0.9247089445363343,
        0.9758401866993924
      ],
      "gateway_probability_exact_formula_float": 0.9999972937853071,
      "conditional_estimate": 0.95703125,
      "conditional_wilson95": [
        0.9247089445363343,
        0.9758401866993924
      ]
    },
    {
      "n": 6,
      "lambda": "0.5",
      "p": "3/502",
      "trials": 256,
      "rafs": 256,
      "gateway_samples": 256,
      "food_only_rafs": 0,
      "raf_estimate": 1.0,
      "raf_wilson95": [
        0.9852161435741286,
        1.0
      ],
      "gateway_probability_exact_formula_float": 0.9999999999929529,
      "conditional_estimate": 1.0,
      "conditional_wilson95": [
        0.9852161435741286,
        1.0
      ]
    },
    {
      "n": 6,
      "lambda": "1",
      "p": "3/251",
      "trials": 256,
      "rafs": 256,
      "gateway_samples": 256,
      "food_only_rafs": 0,
      "raf_estimate": 1.0,
      "raf_wilson95": [
        0.9852161435741286,
        1.0
      ],
      "gateway_probability_exact_formula_float": 1.0,
      "conditional_estimate": 1.0,
      "conditional_wilson95": [
        0.9852161435741286,
        1.0
      ]
    }
  ],
  "finite_escape": [
    {
      "K": 2,
      "cap": 4,
      "openness": "1/10",
      "hits": 237,
      "trials": 256,
      "estimate": 0.92578125,
      "wilson95": [
        0.886999229454155,
        0.9519738928081888
      ],
      "exact_E2": "957608841724783796485705566799/1000000000000000000000000000000",
      "scope": "Estimate of finite escape probability e_K, an upper bound on infinite survival; interval is sampling uncertainty, not a certified survival bracket."
    },
    {
      "K": 3,
      "cap": 6,
      "openness": "1/10",
      "hits": 236,
      "trials": 256,
      "estimate": 0.921875,
      "wilson95": [
        0.8824167617091012,
        0.9488593594315697
      ],
      "exact_E2": null,
      "scope": "Estimate of finite escape probability e_K, an upper bound on infinite survival; interval is sampling uncertainty, not a certified survival bracket."
    },
    {
      "K": 4,
      "cap": 8,
      "openness": "1/10",
      "hits": 236,
      "trials": 256,
      "estimate": 0.921875,
      "wilson95": [
        0.8824167617091012,
        0.9488593594315697
      ],
      "exact_E2": null,
      "scope": "Estimate of finite escape probability e_K, an upper bound on infinite survival; interval is sampling uncertainty, not a certified survival bracket."
    }
  ],
  "scope": "The theorem identifies the limit as S_qt(1-exp(-lambda)); these computations do not evaluate that unknown infinite profile or prove a finite-size convergence rate. Lean not rerun."
}