{
  "source_commit": "fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb",
  "table_schedules": [
    {
      "dimensions": [
        2,
        2
      ],
      "total": 4,
      "d": 19,
      "b": 104,
      "k": 1,
      "outer_transitions_per_trial": "62101027900121962164502179867705203357788099423710577916829159743859001415324892259769363698834778635903675761580564301185329671566110873524987698742788585160840566129573505631886473936231614854163265019182135183589951974127446473016640883292717561166434911025",
      "outer_transition_decimal_digits": 260,
      "outer_trial_cap": 260642,
      "exact_sampler_accuracy_bits": 1017696497792,
      "exact_sampler_correction_branch_probability": "2^(-508848248896)",
      "exact_sampler_outer_transition_decimal_digits": 280,
      "interpretation": "Literal declared general-routine schedules, not measured runtimes or a lower bound for every possible implementation; zero-total/one-dimensional shortcuts are separate."
    },
    {
      "dimensions": [
        3,
        3
      ],
      "total": 9,
      "d": 26,
      "b": 121,
      "k": 20,
      "outer_transitions_per_trial": "196384776806900557797202577026861116590162216825466174925171573279294148863887285169740812842804280964579203928638336695107767793067092438641214284240828261580984966858693196086524315288755819947901766462420178318680108413067547625785654839715525238032923961258771309296830484199826784256",
      "outer_transition_decimal_digits": 288,
      "outer_trial_cap": 9596496,
      "exact_sampler_accuracy_bits": 9045671752832,
      "exact_sampler_correction_branch_probability": "2^(-4522835876416)",
      "exact_sampler_outer_transition_decimal_digits": 309,
      "interpretation": "Literal declared general-routine schedules, not measured runtimes or a lower bound for every possible implementation; zero-total/one-dimensional shortcuts are separate."
    }
  ],
  "matching_backend_guards": [
    {
      "branch_index": 1,
      "a": 1001,
      "exact_required_log2_M_lower_bound": 402406008006002400400000,
      "exact_required_log2_master_size_lower_bound": 201203004003001200200000,
      "log10_master_size_lower_bound_decimal_approximation": "6.05681394E+22",
      "necessary_conditions": [
        "P_star <= b",
        "Lambda >= 2^(1000*a^2)",
        "P_star=Lambda^(400*a^2)",
        "b=M^(1/a^2)",
        "M=2^u",
        "M<=S^2"
      ],
      "interpretation": "Necessary guard condition for the literal backend, not a sufficient threshold, no-input-time estimate, or impossibility theorem for a different matching algorithm. Early structural completions and the deterministic fallback remain distinct."
    },
    {
      "branch_index": 2,
      "a": 1002,
      "exact_required_log2_M_lower_bound": 404824064096076825600000,
      "exact_required_log2_master_size_lower_bound": 202412032048038412800000,
      "log10_master_size_lower_bound_decimal_approximation": "6.09320931E+22",
      "necessary_conditions": [
        "P_star <= b",
        "Lambda >= 2^(1000*a^2)",
        "P_star=Lambda^(400*a^2)",
        "b=M^(1/a^2)",
        "M=2^u",
        "M<=S^2"
      ],
      "interpretation": "Necessary guard condition for the literal backend, not a sufficient threshold, no-input-time estimate, or impossibility theorem for a different matching algorithm. Early structural completions and the deterministic fallback remain distinct."
    }
  ],
  "computations": "exact integer formulas; only log10 illustration is Decimal approximation",
  "source_paths": {
    "table_parameters": "preprints/Exact-Uniform-Sampling-of-Contingency-Tables-with-Arbitrary-Margins-September-24-2026/build/sections/model-and-graph.tex",
    "table_schedule": "preprints/Exact-Uniform-Sampling-of-Contingency-Tables-with-Arbitrary-Margins-September-24-2026/build/sections/algorithms.tex",
    "matching_guards": "preprints/Almost-Linear-Time-Maximum-Cardinality-Matching-in-Sparse-General-Graphs-September-24-2026/build/sections/07-backend-quality.tex",
    "matching_branch": "preprints/Almost-Linear-Time-Maximum-Cardinality-Matching-in-Sparse-General-Graphs-September-24-2026/build/sections/09-uniformization.tex"
  },
  "source_proof_verification": "not_run"
}