{
  "title": "Overlapping subset probability: control experiment",
  "date_utc": "2026-10-22",
  "seed": 20261022,
  "scope": "Toy subset-sum surrogate; NOT a reproduction or assessment of an ECDLP method.",
  "hypothesis": "Independence formulas can fail when candidate events share latent variables.",
  "models": {
    "uniform": "n independently random residues; subset events overlap in coordinates.",
    "paired": "Each pair of positions shares one residue; stronger dependence.",
    "identical": "All positions share one residue; deliberately extreme dependence."
  },
  "statistics": "Wilson 95% binomial interval across independent simulation trials",
  "limitation": "This is not the zero-minor signature-matrix distribution in Mahalanobis 2026.",
  "elapsed_seconds": 12.54,
  "results": [
    {
      "prime": 1009,
      "positions": 14,
      "subset_size": 3,
      "candidate_sets": 364,
      "dependency_model": "uniform",
      "trials": 1000,
      "positive_cases": 305,
      "observed_success_rate": 0.305,
      "wilson95": [
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      ],
      "independence_formula": 0.3029736798726306,
      "difference_observed_minus_formula": 0.0020263201273694187,
      "independence_inside_ci": true
    },
    {
      "prime": 1009,
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      "subset_size": 3,
      "candidate_sets": 364,
      "dependency_model": "paired",
      "trials": 1000,
      "positive_cases": 81,
      "observed_success_rate": 0.081,
      "wilson95": [
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      ],
      "independence_formula": 0.3029736798726306,
      "difference_observed_minus_formula": -0.22197367987263056,
      "independence_inside_ci": false
    },
    {
      "prime": 1009,
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      "subset_size": 3,
      "candidate_sets": 364,
      "dependency_model": "identical",
      "trials": 1000,
      "positive_cases": 0,
      "observed_success_rate": 0.0,
      "wilson95": [
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      "independence_formula": 0.3029736798726306,
      "difference_observed_minus_formula": -0.3029736798726306,
      "independence_inside_ci": false
    },
    {
      "prime": 4093,
      "positions": 20,
      "subset_size": 3,
      "candidate_sets": 1140,
      "dependency_model": "uniform",
      "trials": 1000,
      "positive_cases": 259,
      "observed_success_rate": 0.259,
      "wilson95": [
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      "independence_formula": 0.24312588942992752,
      "difference_observed_minus_formula": 0.015874110570072486,
      "independence_inside_ci": true
    },
    {
      "prime": 4093,
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      "subset_size": 3,
      "candidate_sets": 1140,
      "dependency_model": "paired",
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      "positive_cases": 51,
      "observed_success_rate": 0.051,
      "wilson95": [
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      "independence_inside_ci": false
    },
    {
      "prime": 4093,
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      "subset_size": 3,
      "candidate_sets": 1140,
      "dependency_model": "identical",
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      "positive_cases": 1,
      "observed_success_rate": 0.001,
      "wilson95": [
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      "independence_formula": 0.24312588942992752,
      "difference_observed_minus_formula": -0.24212588942992752,
      "independence_inside_ci": false
    },
    {
      "prime": 4093,
      "positions": 18,
      "subset_size": 4,
      "candidate_sets": 3060,
      "dependency_model": "uniform",
      "trials": 1000,
      "positive_cases": 518,
      "observed_success_rate": 0.518,
      "wilson95": [
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      "difference_observed_minus_formula": -0.00855012279319789,
      "independence_inside_ci": true
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    {
      "prime": 4093,
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      "subset_size": 4,
      "candidate_sets": 3060,
      "dependency_model": "paired",
      "trials": 1000,
      "positive_cases": 92,
      "observed_success_rate": 0.092,
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    {
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      "dependency_model": "identical",
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    {
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    {
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      "dependency_model": "identical",
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}