{
  "schema_version": 1,
  "study": {
    "generated_on": "2026-09-04",
    "accepted_deals_per_scenario": 200000,
    "batch_size": 100000,
    "base_seed": 2026090406,
    "deck": "standard 52-card deck; two private cards per hand",
    "target": "Button's two-card hand conditional on all upstream folds",
    "dealing_order_note": "All private cards are dealt before action; the simulation order is exchangeable bookkeeping only.",
    "primary_estimator": "whole-deal acceptance/rejection on the product of all upstream fold likelihoods; accepted deals are iid",
    "interval_convention": "95% Wilson intervals for binary events and hand classes; normal 95% Monte Carlo intervals for mean remaining-rank shares",
    "table_size_mapping": {
      "6": {
        "observed_folds_before_button": 3,
        "interpretation": "all 3 seats before the button folded"
      },
      "8": {
        "observed_folds_before_button": 5,
        "interpretation": "all 5 seats before the button folded"
      },
      "9": {
        "observed_folds_before_button": 6,
        "interpretation": "all 6 seats before the button folded"
      }
    },
    "scope_warning": "Synthetic card-composition experiment only; no GTO, population, EV, or product claim is encoded or supported."
  },
  "models": {
    "random_fold_control": {
      "purpose": "Negative control: a fold conveys no card information.",
      "base_formula": "base p(fold | hand) = 0.60 for all 169 hand classes",
      "raw_unconditional_physical_combo_mean": 0.6,
      "multiplicative_normalization_factor": 1.0,
      "actual_probability_rule": "(p(fold | hand) = 0.60 for all 169 hand classes) * 1",
      "target_unconditional_physical_combo_mean": 0.6,
      "unconditional_physical_combo_mean_fold_likelihood": 0.6
    },
    "rank_gradient": {
      "purpose": "Smooth synthetic model in which high-rank hands are less fold-prone.",
      "base_formula": "p = clamp(0.86 - 0.0275 * ((high_rank + low_rank) - 4), 0.18, 0.86)",
      "raw_unconditional_physical_combo_mean": 0.53,
      "multiplicative_normalization_factor": 1.1320754716981132,
      "actual_probability_rule": "(p = clamp(0.86 - 0.0275 * ((high_rank + low_rank) - 4), 0.18, 0.86)) * 1.13207547169811",
      "target_unconditional_physical_combo_mean": 0.6,
      "unconditional_physical_combo_mean_fold_likelihood": 0.6
    },
    "pair_suited_gradient": {
      "purpose": "Smooth synthetic model that additionally makes pairs and suited hands less fold-prone.",
      "base_formula": "score = high + low + 5*pair + 1*suited; p = 0.05 + 0.90/(1 + exp((score - 20)/2.4))",
      "raw_unconditional_physical_combo_mean": 0.6760953795036003,
      "multiplicative_normalization_factor": 0.8874487508560246,
      "actual_probability_rule": "(score = high + low + 5*pair + 1*suited; p = 0.05 + 0.90/(1 + exp((score - 20)/2.4))) * 0.887448750856025",
      "target_unconditional_physical_combo_mean": 0.6,
      "unconditional_physical_combo_mean_fold_likelihood": 0.6000000000000001
    },
    "bucketed_top_band": {
      "purpose": "Discontinuous stress test with an explicitly defined top, middle, and other band.",
      "base_formula": "p=0.08 for pairs 88+, AT+, and suited A8+/KT+/QT+; p=0.38 for remaining pairs, remaining Ax, and remaining suited hands with high>=J and low>=8; p=0.86 otherwise",
      "raw_unconditional_physical_combo_mean": 0.7145701357466063,
      "multiplicative_normalization_factor": 0.8396656534954408,
      "actual_probability_rule": "(p=0.08 for pairs 88+, AT+, and suited A8+/KT+/QT+; p=0.38 for remaining pairs, remaining Ax, and remaining suited hands with high>=J and low>=8; p=0.86 otherwise) * 0.839665653495441",
      "target_unconditional_physical_combo_mean": 0.6,
      "unconditional_physical_combo_mean_fold_likelihood": 0.6
    },
    "inverted_rank_diagnostic": {
      "purpose": "Deliberately non-behavioral direction check in which high-rank hands are more fold-prone.",
      "base_formula": "p = clamp(0.20 + 0.0275 * ((high_rank + low_rank) - 4), 0.20, 0.86)",
      "raw_unconditional_physical_combo_mean": 0.53,
      "multiplicative_normalization_factor": 1.1320754716981132,
      "actual_probability_rule": "(p = clamp(0.20 + 0.0275 * ((high_rank + low_rank) - 4), 0.20, 0.86)) * 1.13207547169811",
      "target_unconditional_physical_combo_mean": 0.6,
      "unconditional_physical_combo_mean_fold_likelihood": 0.6
    }
  },
  "exact_no_information_baselines": {
    "at_least_one_ace": 0.14932126696832582,
    "any_pair": 0.058823529411764705,
    "premium_pair_tt_plus": 0.02262443438914027,
    "ace_king": 0.012066365007541479,
    "both_cards_ten_plus": 0.14328808446455504
  },
  "exact_no_information_rank_share": 0.07692307692307693,
  "one_fold_exact_oracle_file": "one_fold_exact_oracle.json",
  "scenarios": [
    {
      "model": "random_fold_control",
      "table_size": 6,
      "observed_folds_before_button": 3,
      "accepted_deals": 200000,
      "attempted_deals": 923226,
      "rng": "NumPy PCG64DXSM",
      "seed": 2026091012,
      "sampling_method": "whole-deal accept/reject using the product of all upstream fold likelihoods",
      "metrics": {
        "at_least_one_ace": {
          "successes": 29972,
          "estimate": 0.14986,
          "monte_carlo_standard_error": 0.0007981290008513661,
          "ci95": [
            0.14830242158284232,
            0.15143102864273003
          ],
          "exact_no_information_baseline": 0.14932126696832582,
          "absolute_change": 0.0005387330316741767,
          "absolute_change_percentage_points": 0.053873303167417674,
          "relative_change_percent": 0.3607878787878577,
          "event_frequency_translation": {
            "direction": "more",
            "difference_resolved_at_95_percent": false,
            "approximately_one_event_per_target_hands": null
          }
        },
        "any_pair": {
          "successes": 11778,
          "estimate": 0.05889,
          "monte_carlo_standard_error": 0.000526412233425478,
          "ci95": [
            0.057866698470822944,
            0.059930246262718534
          ],
          "exact_no_information_baseline": 0.058823529411764705,
          "absolute_change": 6.647058823529284e-05,
          "absolute_change_percentage_points": 0.006647058823529284,
          "relative_change_percent": 0.11299999999999782,
          "event_frequency_translation": {
            "direction": "more",
            "difference_resolved_at_95_percent": false,
            "approximately_one_event_per_target_hands": null
          }
        },
        "premium_pair_tt_plus": {
          "successes": 4529,
          "estimate": 0.022645,
          "monte_carlo_standard_error": 0.00033265751137619,
          "ci95": [
            0.022002113579427968,
            0.02330622346412057
          ],
          "exact_no_information_baseline": 0.02262443438914027,
          "absolute_change": 2.056561085972755e-05,
          "absolute_change_percentage_points": 0.002056561085972755,
          "relative_change_percent": 0.09089999999999578,
          "event_frequency_translation": {
            "direction": "more",
            "difference_resolved_at_95_percent": false,
            "approximately_one_event_per_target_hands": null
          }
        },
        "ace_king": {
          "successes": 2465,
          "estimate": 0.012325,
          "monte_carlo_standard_error": 0.0002467092861547777,
          "ci95": [
            0.01185073935095917,
            0.012817994123525297
          ],
          "exact_no_information_baseline": 0.012066365007541479,
          "absolute_change": 0.0002586349924585205,
          "absolute_change_percentage_points": 0.025863499245852052,
          "relative_change_percent": 2.1434374999999886,
          "event_frequency_translation": {
            "direction": "more",
            "difference_resolved_at_95_percent": false,
            "approximately_one_event_per_target_hands": null
          }
        },
        "both_cards_ten_plus": {
          "successes": 28614,
          "estimate": 0.14307,
          "monte_carlo_standard_error": 0.0007829462788084506,
          "ci95": [
            0.14154230844307003,
            0.1446114026125464
          ],
          "exact_no_information_baseline": 0.14328808446455504,
          "absolute_change": -0.0002180844645550406,
          "absolute_change_percentage_points": -0.02180844645550406,
          "relative_change_percent": -0.1521999999999915,
          "event_frequency_translation": {
            "direction": "fewer",
            "difference_resolved_at_95_percent": false,
            "approximately_one_event_per_target_hands": null
          }
        }
      },
      "remaining_rank_composition": {
        "2": {
          "expected_share_of_remaining_cards": 0.07690782608697132,
          "monte_carlo_standard_error": 3.0119331898453827e-05,
          "ci95": [
            0.07684879328121194,
            0.0769668588927307
          ],
          "exact_no_information_share": 0.07692307692307693,
          "absolute_change_percentage_points": -0.0015250836105609733,
          "relative_change_percent": -0.019826086937292653
        },
        "3": {
          "expected_share_of_remaining_cards": 0.07687423913044972,
          "monte_carlo_standard_error": 3.0220534958199722e-05,
          "ci95": [
            0.0768150079703381,
            0.07693347029056133
          ],
          "exact_no_information_share": 0.07692307692307693,
          "absolute_change_percentage_points": -0.004883779262720989,
          "relative_change_percent": -0.06348913041537285
        },
        "4": {
          "expected_share_of_remaining_cards": 0.07686271739131911,
          "monte_carlo_standard_error": 3.0222415656493856e-05,
          "ci95": [
            0.07680348254510659,
            0.07692195223753164
          ],
          "exact_no_information_share": 0.07692307692307693,
          "absolute_change_percentage_points": -0.006035953175781428,
          "relative_change_percent": -0.07846739128515856
        },
        "5": {
          "expected_share_of_remaining_cards": 0.07693521739131935,
          "monte_carlo_standard_error": 3.0180199466643285e-05,
          "ci95": [
            0.0768760652873185,
            0.07699436949532021
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          "exact_no_information_share": 0.07692307692307693,
          "absolute_change_percentage_points": 0.0012140468242424762,
          "relative_change_percent": 0.01578260871515219
        },
        "6": {
          "expected_share_of_remaining_cards": 0.07693891304349333,
          "monte_carlo_standard_error": 3.0114622822511426e-05,
          "ci95": [
            0.0768798894673532,
            0.07699793661963346
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          "exact_no_information_share": 0.07692307692307693,
          "absolute_change_percentage_points": 0.0015836120416404253,
          "relative_change_percent": 0.02058695654132553
        },
        "7": {
          "expected_share_of_remaining_cards": 0.07697456521740628,
          "monte_carlo_standard_error": 3.0074923180304405e-05,
          "ci95": [
            0.07691561945113508,
            0.07703351098367749
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          "exact_no_information_share": 0.07692307692307693,
          "absolute_change_percentage_points": 0.005148829432935487,
          "relative_change_percent": 0.06693478262816133
        },
        "8": {
          "expected_share_of_remaining_cards": 0.07687641304349337,
          "monte_carlo_standard_error": 3.021282231202308e-05,
          "ci95": [
            0.0768171969998905,
            0.07693562908709624
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          "exact_no_information_share": 0.07692307692307693,
          "absolute_change_percentage_points": -0.004666387958356111,
          "relative_change_percent": -0.06066304345862944
        },
        "9": {
          "expected_share_of_remaining_cards": 0.07694760869566708,
          "monte_carlo_standard_error": 3.0135349743387794e-05,
          "ci95": [
            0.07688854449550853,
            0.07700667289582563
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          "exact_no_information_share": 0.07692307692307693,
          "absolute_change_percentage_points": 0.0024531772590152823,
          "relative_change_percent": 0.03189130436719867
        },
        "T": {
          "expected_share_of_remaining_cards": 0.07697586956523224,
          "monte_carlo_standard_error": 3.0003146995296405e-05,
          "ci95": [
            0.0769170644776986,
            0.07703467465276588
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          "exact_no_information_share": 0.07692307692307693,
          "absolute_change_percentage_points": 0.005279264215531099,
          "relative_change_percent": 0.06863043480190428
        },
        "J": {
          "expected_share_of_remaining_cards": 0.07694239130436263,
          "monte_carlo_standard_error": 3.0150594809271384e-05,
          "ci95": [
            0.076883297224424,
            0.07700148538430127
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          "exact_no_information_share": 0.07692307692307693,
          "absolute_change_percentage_points": 0.001931438128570384,
          "relative_change_percent": 0.025108695671414993
        },
        "Q": {
          "expected_share_of_remaining_cards": 0.07694043478262359,
          "monte_carlo_standard_error": 3.00936066687311e-05,
          "ci95": [
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            0.07699941716785923
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          "exact_no_information_share": 0.07692307692307693,
          "absolute_change_percentage_points": 0.0017357859546665577,
          "relative_change_percent": 0.02256521741066525
        },
        "K": {
          "expected_share_of_remaining_cards": 0.07691326086958014,
          "monte_carlo_standard_error": 3.0119765643974237e-05,
          "ci95": [
            0.07685422721369517,
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          "exact_no_information_share": 0.07692307692307693,
          "absolute_change_percentage_points": -0.000981605349678616,
          "relative_change_percent": -0.012760869545822007
        },
        "A": {
          "expected_share_of_remaining_cards": 0.07691054347827574,
          "monte_carlo_standard_error": 3.0115822649833334e-05,
          "ci95": [
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          "exact_no_information_share": 0.07692307692307693,
          "absolute_change_percentage_points": -0.0012533444801191007,
          "relative_change_percent": -0.01629347824154831
        }
      },
      "all_169_button_hand_classes": {
        "AA": {
          "shape": "pair",
          "physical_combo_count": 6,
          "count": 899,
          "estimate": 0.004495,
          "ci95_wilson": [
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            0.004797838841702668
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          "exact_no_information_baseline": 0.004524886877828055,
          "absolute_change_percentage_points": -0.002988687782805484,
          "relative_change_percent": -0.6605000000000119
        },
        "AKs": {
          "shape": "suited",
          "physical_combo_count": 4,
          "count": 616,
          "estimate": 0.00308,
          "ci95_wilson": [
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            0.003332579732259995
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          "exact_no_information_baseline": 0.0030165912518853697,
          "absolute_change_percentage_points": 0.006340874811463018,
          "relative_change_percent": 2.1019999999999905
        },
        "AKo": {
          "shape": "offsuit",
          "physical_combo_count": 12,
          "count": 1849,
          "estimate": 0.009245,
          "ci95_wilson": [
            0.008834884174956691,
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          "exact_no_information_baseline": 0.00904977375565611,
          "absolute_change_percentage_points": 0.019522624434389034,
          "relative_change_percent": 2.157249999999988
        },
        "AQs": {
          "shape": "suited",
          "physical_combo_count": 4,
          "count": 588,
          "estimate": 0.00294,
          "ci95_wilson": [
            0.0027120738627441185,
            0.0031870201257261717
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          "exact_no_information_baseline": 0.0030165912518853697,
          "absolute_change_percentage_points": -0.007659125188536975,
          "relative_change_percent": -2.5390000000000073
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        "AQo": {
          "shape": "offsuit",
          "physical_combo_count": 12,
          "count": 1721,
          "estimate": 0.008605,
          "ci95_wilson": [
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          "exact_no_information_baseline": 0.00904977375565611,
          "absolute_change_percentage_points": -0.04447737556561096,
          "relative_change_percent": -4.91475000000001
        },
        "AJs": {
          "shape": "suited",
          "physical_combo_count": 4,
          "count": 596,
          "estimate": 0.00298,
          "ci95_wilson": [
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          "exact_no_information_baseline": 0.0030165912518853697,
          "absolute_change_percentage_points": -0.0036591251885369647,
          "relative_change_percent": -1.2130000000000039
        },
        "AJo": {
          "shape": "offsuit",
          "physical_combo_count": 12,
          "count": 1901,
          "estimate": 0.009505,
          "ci95_wilson": [
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          "exact_no_information_baseline": 0.00904977375565611,
          "absolute_change_percentage_points": 0.045522624434389015,
          "relative_change_percent": 5.0302499999999855
        },
        "ATs": {
          "shape": "suited",
          "physical_combo_count": 4,
          "count": 613,
          "estimate": 0.003065,
          "ci95_wilson": [
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          "exact_no_information_baseline": 0.0030165912518853697,
          "absolute_change_percentage_points": 0.004840874811463036,
          "relative_change_percent": 1.6047499999999963
        },
        "ATo": {
          "shape": "offsuit",
          "physical_combo_count": 12,
          "count": 1791,
          "estimate": 0.008955,
          "ci95_wilson": [
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          "exact_no_information_baseline": 0.00904977375565611,
          "absolute_change_percentage_points": -0.009477375565610999,
          "relative_change_percent": -1.0472500000000153
        },
        "A9s": {
          "shape": "suited",
          "physical_combo_count": 4,
          "count": 647,
          "estimate": 0.003235,
          "ci95_wilson": [
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          "exact_no_information_baseline": 0.0030165912518853697,
          "absolute_change_percentage_points": 0.02184087481146304,
          "relative_change_percent": 7.240249999999997
        },
        "A9o": {
          "shape": "offsuit",
          "physical_combo_count": 12,
          "count": 1814,
          "estimate": 0.00907,
          "ci95_wilson": [
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          "relative_change_percent": 0.22349999999999035
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        "A8s": {
          "shape": "suited",
          "physical_combo_count": 4,
          "count": 595,
          "estimate": 0.002975,
          "ci95_wilson": [
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          "relative_change_percent": -1.378749999999997
        },
        "A8o": {
          "shape": "offsuit",
          "physical_combo_count": 12,
          "count": 1786,
          "estimate": 0.00893,
          "ci95_wilson": [
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          "exact_no_information_baseline": 0.00904977375565611,
          "absolute_change_percentage_points": -0.011977375565610897,
          "relative_change_percent": -1.323500000000004
        },
        "A7s": {
          "shape": "suited",
          "physical_combo_count": 4,
          "count": 605,
          "estimate": 0.003025,
          "ci95_wilson": [
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          "exact_no_information_baseline": 0.0030165912518853697,
          "absolute_change_percentage_points": 0.0008408748114630254,
          "relative_change_percent": 0.2787499999999929
        },
        "A7o": {
          "shape": "offsuit",
          "physical_combo_count": 12,
          "count": 1893,
          "estimate": 0.009465,
          "ci95_wilson": [
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          "exact_no_information_baseline": 0.00904977375565611,
          "absolute_change_percentage_points": 0.041522624434389005,
          "relative_change_percent": 4.588249999999984
        },
        "A6s": {
          "shape": "suited",
          "physical_combo_count": 4,
          "count": 598,
          "estimate": 0.00299,
          "ci95_wilson": [
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          "exact_no_information_baseline": 0.0030165912518853697,
          "absolute_change_percentage_points": -0.002659125188536962,
          "relative_change_percent": -0.881500000000003
        },
        "A6o": {
          "shape": "offsuit",
          "physical_combo_count": 12,
          "count": 1807,
          "estimate": 0.009035,
          "ci95_wilson": [
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            0.009459225790838353
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          "exact_no_information_baseline": 0.00904977375565611,
          "absolute_change_percentage_points": -0.0014773755656109777,
          "relative_change_percent": -0.16325000000001302
        },
        "A5s": {
          "shape": "suited",
          "physical_combo_count": 4,
          "count": 637,
          "estimate": 0.003185,
          "ci95_wilson": [
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            0.0034416662910783254
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          "exact_no_information_baseline": 0.0030165912518853697,
          "absolute_change_percentage_points": 0.016840874811463024,
          "relative_change_percent": 5.582749999999992
        },
        "A5o": {
          "shape": "offsuit",
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