{
  "study_id": "gtogecko-mystery-bounty-draw-timing-2026-09-01",
  "run_date": "2026-09-01",
  "method": "Exact enumeration of all 720 ordered labeled three-envelope prefixes; no simulation",
  "pool_input": {
    "schema": "envelope_id,payout_usd",
    "sha256": "b773e42615cf338c940f881e41b89e0037723f6214785a3cd8fd93ce3391cf29"
  },
  "scope": {
    "model": "One earned draw from a fixed prize pool without replacement; every intervening external reveal and the selected draw are conditionally uniform and exchangeable among the same remaining envelopes, the pool changes only through those draws, and redemption is forced by the final available position",
    "timing_rule": "Bounded and predictable: each stop decision is committed before the selected envelope is revealed, using only earlier reveals and independent private randomness",
    "excludes": [
      "The probability of earning a bounty token",
      "Tournament hand strategy, coverage, chips, and ICM",
      "Event rules that force redemption earlier than the modeled final-position deadline, token expiry, event closure, nested draws, and non-uniform procedures",
      "Biased or non-exchangeable external removals and pool changes caused by anything other than the modeled draws",
      "Adaptive joint timing of multiple accumulated tokens"
    ]
  },
  "synthetic_pool": {
    "prize_count": 10,
    "total_usd": 3000,
    "mean_usd": {
      "fraction": "300/1",
      "decimal": 300.0
    },
    "median_usd": {
      "fraction": "200/1",
      "decimal": 200.0
    },
    "mode_usd": 100,
    "jackpot_usd": 1000,
    "jackpot_probability": {
      "fraction": "1/10",
      "decimal": 0.1
    },
    "envelopes": [
      {
        "envelope_id": "E01",
        "payout_usd": 100
      },
      {
        "envelope_id": "E02",
        "payout_usd": 100
      },
      {
        "envelope_id": "E03",
        "payout_usd": 100
      },
      {
        "envelope_id": "E04",
        "payout_usd": 100
      },
      {
        "envelope_id": "E05",
        "payout_usd": 200
      },
      {
        "envelope_id": "E06",
        "payout_usd": 200
      },
      {
        "envelope_id": "E07",
        "payout_usd": 300
      },
      {
        "envelope_id": "E08",
        "payout_usd": 400
      },
      {
        "envelope_id": "E09",
        "payout_usd": 500
      },
      {
        "envelope_id": "E10",
        "payout_usd": 1000
      }
    ],
    "groups": [
      {
        "payout_usd": 100,
        "count": 4,
        "pool_value_usd": 400,
        "single_draw_probability_fraction": "2/5",
        "single_draw_probability_pct": 40.0
      },
      {
        "payout_usd": 200,
        "count": 2,
        "pool_value_usd": 400,
        "single_draw_probability_fraction": "1/5",
        "single_draw_probability_pct": 20.0
      },
      {
        "payout_usd": 300,
        "count": 1,
        "pool_value_usd": 300,
        "single_draw_probability_fraction": "1/10",
        "single_draw_probability_pct": 10.0
      },
      {
        "payout_usd": 400,
        "count": 1,
        "pool_value_usd": 400,
        "single_draw_probability_fraction": "1/10",
        "single_draw_probability_pct": 10.0
      },
      {
        "payout_usd": 500,
        "count": 1,
        "pool_value_usd": 500,
        "single_draw_probability_fraction": "1/10",
        "single_draw_probability_pct": 10.0
      },
      {
        "payout_usd": 1000,
        "count": 1,
        "pool_value_usd": 1000,
        "single_draw_probability_fraction": "1/10",
        "single_draw_probability_pct": 10.0
      }
    ]
  },
  "conditional_after_one_external_reveal": {
    "rows": [
      {
        "revealed_payout_usd": 100,
        "reveal_probability_fraction": "2/5",
        "reveal_probability_pct": 40.0,
        "remaining_prizes": 9,
        "remaining_total_usd": 2900,
        "next_draw_ev_fraction_usd": "2900/9",
        "next_draw_ev_usd": 322.222222,
        "change_from_initial_ev_fraction_usd": "200/9",
        "change_from_initial_ev_usd": 22.222222,
        "next_draw_jackpot_probability_fraction": "1/9",
        "next_draw_jackpot_probability_pct": 11.111111
      },
      {
        "revealed_payout_usd": 200,
        "reveal_probability_fraction": "1/5",
        "reveal_probability_pct": 20.0,
        "remaining_prizes": 9,
        "remaining_total_usd": 2800,
        "next_draw_ev_fraction_usd": "2800/9",
        "next_draw_ev_usd": 311.111111,
        "change_from_initial_ev_fraction_usd": "100/9",
        "change_from_initial_ev_usd": 11.111111,
        "next_draw_jackpot_probability_fraction": "1/9",
        "next_draw_jackpot_probability_pct": 11.111111
      },
      {
        "revealed_payout_usd": 300,
        "reveal_probability_fraction": "1/10",
        "reveal_probability_pct": 10.0,
        "remaining_prizes": 9,
        "remaining_total_usd": 2700,
        "next_draw_ev_fraction_usd": "300/1",
        "next_draw_ev_usd": 300.0,
        "change_from_initial_ev_fraction_usd": "0/1",
        "change_from_initial_ev_usd": 0.0,
        "next_draw_jackpot_probability_fraction": "1/9",
        "next_draw_jackpot_probability_pct": 11.111111
      },
      {
        "revealed_payout_usd": 400,
        "reveal_probability_fraction": "1/10",
        "reveal_probability_pct": 10.0,
        "remaining_prizes": 9,
        "remaining_total_usd": 2600,
        "next_draw_ev_fraction_usd": "2600/9",
        "next_draw_ev_usd": 288.888889,
        "change_from_initial_ev_fraction_usd": "-100/9",
        "change_from_initial_ev_usd": -11.111111,
        "next_draw_jackpot_probability_fraction": "1/9",
        "next_draw_jackpot_probability_pct": 11.111111
      },
      {
        "revealed_payout_usd": 500,
        "reveal_probability_fraction": "1/10",
        "reveal_probability_pct": 10.0,
        "remaining_prizes": 9,
        "remaining_total_usd": 2500,
        "next_draw_ev_fraction_usd": "2500/9",
        "next_draw_ev_usd": 277.777778,
        "change_from_initial_ev_fraction_usd": "-200/9",
        "change_from_initial_ev_usd": -22.222222,
        "next_draw_jackpot_probability_fraction": "1/9",
        "next_draw_jackpot_probability_pct": 11.111111
      },
      {
        "revealed_payout_usd": 1000,
        "reveal_probability_fraction": "1/10",
        "reveal_probability_pct": 10.0,
        "remaining_prizes": 9,
        "remaining_total_usd": 2000,
        "next_draw_ev_fraction_usd": "2000/9",
        "next_draw_ev_usd": 222.222222,
        "change_from_initial_ev_fraction_usd": "-700/9",
        "change_from_initial_ev_usd": -77.777778,
        "next_draw_jackpot_probability_fraction": "0/1",
        "next_draw_jackpot_probability_pct": 0.0
      }
    ],
    "weighted_next_draw_ev_usd": {
      "fraction": "300/1",
      "decimal": 300.0
    },
    "weighted_next_draw_jackpot_probability": {
      "fraction": "1/10",
      "decimal": 0.1
    }
  },
  "policy_enumeration": {
    "ordered_labeled_prefixes": 720,
    "prefix_formula": "10 \u00d7 9 \u00d7 8",
    "prefix_length": 3,
    "policies": [
      {
        "key": "redeem_now",
        "description": "Redeem before another envelope is revealed",
        "labeled_envelope_distribution": [
          {
            "policy": "redeem_now",
            "policy_description": "Redeem before another envelope is revealed",
            "envelope_id": "E01",
            "payout_usd": 100,
            "selected_prefixes": 72,
            "total_prefixes": 720,
            "selection_probability_fraction": "1/10",
            "selection_probability_pct": 10.0
          },
          {
            "policy": "redeem_now",
            "policy_description": "Redeem before another envelope is revealed",
            "envelope_id": "E02",
            "payout_usd": 100,
            "selected_prefixes": 72,
            "total_prefixes": 720,
            "selection_probability_fraction": "1/10",
            "selection_probability_pct": 10.0
          },
          {
            "policy": "redeem_now",
            "policy_description": "Redeem before another envelope is revealed",
            "envelope_id": "E03",
            "payout_usd": 100,
            "selected_prefixes": 72,
            "total_prefixes": 720,
            "selection_probability_fraction": "1/10",
            "selection_probability_pct": 10.0
          },
          {
            "policy": "redeem_now",
            "policy_description": "Redeem before another envelope is revealed",
            "envelope_id": "E04",
            "payout_usd": 100,
            "selected_prefixes": 72,
            "total_prefixes": 720,
            "selection_probability_fraction": "1/10",
            "selection_probability_pct": 10.0
          },
          {
            "policy": "redeem_now",
            "policy_description": "Redeem before another envelope is revealed",
            "envelope_id": "E05",
            "payout_usd": 200,
            "selected_prefixes": 72,
            "total_prefixes": 720,
            "selection_probability_fraction": "1/10",
            "selection_probability_pct": 10.0
          },
          {
            "policy": "redeem_now",
            "policy_description": "Redeem before another envelope is revealed",
            "envelope_id": "E06",
            "payout_usd": 200,
            "selected_prefixes": 72,
            "total_prefixes": 720,
            "selection_probability_fraction": "1/10",
            "selection_probability_pct": 10.0
          },
          {
            "policy": "redeem_now",
            "policy_description": "Redeem before another envelope is revealed",
            "envelope_id": "E07",
            "payout_usd": 300,
            "selected_prefixes": 72,
            "total_prefixes": 720,
            "selection_probability_fraction": "1/10",
            "selection_probability_pct": 10.0
          },
          {
            "policy": "redeem_now",
            "policy_description": "Redeem before another envelope is revealed",
            "envelope_id": "E08",
            "payout_usd": 400,
            "selected_prefixes": 72,
            "total_prefixes": 720,
            "selection_probability_fraction": "1/10",
            "selection_probability_pct": 10.0
          },
          {
            "policy": "redeem_now",
            "policy_description": "Redeem before another envelope is revealed",
            "envelope_id": "E09",
            "payout_usd": 500,
            "selected_prefixes": 72,
            "total_prefixes": 720,
            "selection_probability_fraction": "1/10",
            "selection_probability_pct": 10.0
          },
          {
            "policy": "redeem_now",
            "policy_description": "Redeem before another envelope is revealed",
            "envelope_id": "E10",
            "payout_usd": 1000,
            "selected_prefixes": 72,
            "total_prefixes": 720,
            "selection_probability_fraction": "1/10",
            "selection_probability_pct": 10.0
          }
        ],
        "payout_distribution": [
          {
            "policy": "redeem_now",
            "payout_usd": 100,
            "selected_prefixes": 288,
            "total_prefixes": 720,
            "probability_fraction": "2/5",
            "probability_pct": 40.0,
            "original_envelope_count": 4,
            "policy_ev_fraction_usd": "300/1",
            "policy_ev_usd": 300.0
          },
          {
            "policy": "redeem_now",
            "payout_usd": 200,
            "selected_prefixes": 144,
            "total_prefixes": 720,
            "probability_fraction": "1/5",
            "probability_pct": 20.0,
            "original_envelope_count": 2,
            "policy_ev_fraction_usd": "300/1",
            "policy_ev_usd": 300.0
          },
          {
            "policy": "redeem_now",
            "payout_usd": 300,
            "selected_prefixes": 72,
            "total_prefixes": 720,
            "probability_fraction": "1/10",
            "probability_pct": 10.0,
            "original_envelope_count": 1,
            "policy_ev_fraction_usd": "300/1",
            "policy_ev_usd": 300.0
          },
          {
            "policy": "redeem_now",
            "payout_usd": 400,
            "selected_prefixes": 72,
            "total_prefixes": 720,
            "probability_fraction": "1/10",
            "probability_pct": 10.0,
            "original_envelope_count": 1,
            "policy_ev_fraction_usd": "300/1",
            "policy_ev_usd": 300.0
          },
          {
            "policy": "redeem_now",
            "payout_usd": 500,
            "selected_prefixes": 72,
            "total_prefixes": 720,
            "probability_fraction": "1/10",
            "probability_pct": 10.0,
            "original_envelope_count": 1,
            "policy_ev_fraction_usd": "300/1",
            "policy_ev_usd": 300.0
          },
          {
            "policy": "redeem_now",
            "payout_usd": 1000,
            "selected_prefixes": 72,
            "total_prefixes": 720,
            "probability_fraction": "1/10",
            "probability_pct": 10.0,
            "original_envelope_count": 1,
            "policy_ev_fraction_usd": "300/1",
            "policy_ev_usd": 300.0
          }
        ]
      },
      {
        "key": "wait_exactly_one",
        "description": "Let one other envelope be revealed, then redeem",
        "labeled_envelope_distribution": [
          {
            "policy": "wait_exactly_one",
            "policy_description": "Let one other envelope be revealed, then redeem",
            "envelope_id": "E01",
            "payout_usd": 100,
            "selected_prefixes": 72,
            "total_prefixes": 720,
            "selection_probability_fraction": "1/10",
            "selection_probability_pct": 10.0
          },
          {
            "policy": "wait_exactly_one",
            "policy_description": "Let one other envelope be revealed, then redeem",
            "envelope_id": "E02",
            "payout_usd": 100,
            "selected_prefixes": 72,
            "total_prefixes": 720,
            "selection_probability_fraction": "1/10",
            "selection_probability_pct": 10.0
          },
          {
            "policy": "wait_exactly_one",
            "policy_description": "Let one other envelope be revealed, then redeem",
            "envelope_id": "E03",
            "payout_usd": 100,
            "selected_prefixes": 72,
            "total_prefixes": 720,
            "selection_probability_fraction": "1/10",
            "selection_probability_pct": 10.0
          },
          {
            "policy": "wait_exactly_one",
            "policy_description": "Let one other envelope be revealed, then redeem",
            "envelope_id": "E04",
            "payout_usd": 100,
            "selected_prefixes": 72,
            "total_prefixes": 720,
            "selection_probability_fraction": "1/10",
            "selection_probability_pct": 10.0
          },
          {
            "policy": "wait_exactly_one",
            "policy_description": "Let one other envelope be revealed, then redeem",
            "envelope_id": "E05",
            "payout_usd": 200,
            "selected_prefixes": 72,
            "total_prefixes": 720,
            "selection_probability_fraction": "1/10",
            "selection_probability_pct": 10.0
          },
          {
            "policy": "wait_exactly_one",
            "policy_description": "Let one other envelope be revealed, then redeem",
            "envelope_id": "E06",
            "payout_usd": 200,
            "selected_prefixes": 72,
            "total_prefixes": 720,
            "selection_probability_fraction": "1/10",
            "selection_probability_pct": 10.0
          },
          {
            "policy": "wait_exactly_one",
            "policy_description": "Let one other envelope be revealed, then redeem",
            "envelope_id": "E07",
            "payout_usd": 300,
            "selected_prefixes": 72,
            "total_prefixes": 720,
            "selection_probability_fraction": "1/10",
            "selection_probability_pct": 10.0
          },
          {
            "policy": "wait_exactly_one",
            "policy_description": "Let one other envelope be revealed, then redeem",
            "envelope_id": "E08",
            "payout_usd": 400,
            "selected_prefixes": 72,
            "total_prefixes": 720,
            "selection_probability_fraction": "1/10",
            "selection_probability_pct": 10.0
          },
          {
            "policy": "wait_exactly_one",
            "policy_description": "Let one other envelope be revealed, then redeem",
            "envelope_id": "E09",
            "payout_usd": 500,
            "selected_prefixes": 72,
            "total_prefixes": 720,
            "selection_probability_fraction": "1/10",
            "selection_probability_pct": 10.0
          },
          {
            "policy": "wait_exactly_one",
            "policy_description": "Let one other envelope be revealed, then redeem",
            "envelope_id": "E10",
            "payout_usd": 1000,
            "selected_prefixes": 72,
            "total_prefixes": 720,
            "selection_probability_fraction": "1/10",
            "selection_probability_pct": 10.0
          }
        ],
        "payout_distribution": [
          {
            "policy": "wait_exactly_one",
            "payout_usd": 100,
            "selected_prefixes": 288,
            "total_prefixes": 720,
            "probability_fraction": "2/5",
            "probability_pct": 40.0,
            "original_envelope_count": 4,
            "policy_ev_fraction_usd": "300/1",
            "policy_ev_usd": 300.0
          },
          {
            "policy": "wait_exactly_one",
            "payout_usd": 200,
            "selected_prefixes": 144,
            "total_prefixes": 720,
            "probability_fraction": "1/5",
            "probability_pct": 20.0,
            "original_envelope_count": 2,
            "policy_ev_fraction_usd": "300/1",
            "policy_ev_usd": 300.0
          },
          {
            "policy": "wait_exactly_one",
            "payout_usd": 300,
            "selected_prefixes": 72,
            "total_prefixes": 720,
            "probability_fraction": "1/10",
            "probability_pct": 10.0,
            "original_envelope_count": 1,
            "policy_ev_fraction_usd": "300/1",
            "policy_ev_usd": 300.0
          },
          {
            "policy": "wait_exactly_one",
            "payout_usd": 400,
            "selected_prefixes": 72,
            "total_prefixes": 720,
            "probability_fraction": "1/10",
            "probability_pct": 10.0,
            "original_envelope_count": 1,
            "policy_ev_fraction_usd": "300/1",
            "policy_ev_usd": 300.0
          },
          {
            "policy": "wait_exactly_one",
            "payout_usd": 500,
            "selected_prefixes": 72,
            "total_prefixes": 720,
            "probability_fraction": "1/10",
            "probability_pct": 10.0,
            "original_envelope_count": 1,
            "policy_ev_fraction_usd": "300/1",
            "policy_ev_usd": 300.0
          },
          {
            "policy": "wait_exactly_one",
            "payout_usd": 1000,
            "selected_prefixes": 72,
            "total_prefixes": 720,
            "probability_fraction": "1/10",
            "probability_pct": 10.0,
            "original_envelope_count": 1,
            "policy_ev_fraction_usd": "300/1",
            "policy_ev_usd": 300.0
          }
        ]
      },
      {
        "key": "adaptive_threshold",
        "description": "After one reveal, redeem next at $300 or less; above $300, wait once more",
        "labeled_envelope_distribution": [
          {
            "policy": "adaptive_threshold",
            "policy_description": "After one reveal, redeem next at $300 or less; above $300, wait once more",
            "envelope_id": "E01",
            "payout_usd": 100,
            "selected_prefixes": 72,
            "total_prefixes": 720,
            "selection_probability_fraction": "1/10",
            "selection_probability_pct": 10.0
          },
          {
            "policy": "adaptive_threshold",
            "policy_description": "After one reveal, redeem next at $300 or less; above $300, wait once more",
            "envelope_id": "E02",
            "payout_usd": 100,
            "selected_prefixes": 72,
            "total_prefixes": 720,
            "selection_probability_fraction": "1/10",
            "selection_probability_pct": 10.0
          },
          {
            "policy": "adaptive_threshold",
            "policy_description": "After one reveal, redeem next at $300 or less; above $300, wait once more",
            "envelope_id": "E03",
            "payout_usd": 100,
            "selected_prefixes": 72,
            "total_prefixes": 720,
            "selection_probability_fraction": "1/10",
            "selection_probability_pct": 10.0
          },
          {
            "policy": "adaptive_threshold",
            "policy_description": "After one reveal, redeem next at $300 or less; above $300, wait once more",
            "envelope_id": "E04",
            "payout_usd": 100,
            "selected_prefixes": 72,
            "total_prefixes": 720,
            "selection_probability_fraction": "1/10",
            "selection_probability_pct": 10.0
          },
          {
            "policy": "adaptive_threshold",
            "policy_description": "After one reveal, redeem next at $300 or less; above $300, wait once more",
            "envelope_id": "E05",
            "payout_usd": 200,
            "selected_prefixes": 72,
            "total_prefixes": 720,
            "selection_probability_fraction": "1/10",
            "selection_probability_pct": 10.0
          },
          {
            "policy": "adaptive_threshold",
            "policy_description": "After one reveal, redeem next at $300 or less; above $300, wait once more",
            "envelope_id": "E06",
            "payout_usd": 200,
            "selected_prefixes": 72,
            "total_prefixes": 720,
            "selection_probability_fraction": "1/10",
            "selection_probability_pct": 10.0
          },
          {
            "policy": "adaptive_threshold",
            "policy_description": "After one reveal, redeem next at $300 or less; above $300, wait once more",
            "envelope_id": "E07",
            "payout_usd": 300,
            "selected_prefixes": 72,
            "total_prefixes": 720,
            "selection_probability_fraction": "1/10",
            "selection_probability_pct": 10.0
          },
          {
            "policy": "adaptive_threshold",
            "policy_description": "After one reveal, redeem next at $300 or less; above $300, wait once more",
            "envelope_id": "E08",
            "payout_usd": 400,
            "selected_prefixes": 72,
            "total_prefixes": 720,
            "selection_probability_fraction": "1/10",
            "selection_probability_pct": 10.0
          },
          {
            "policy": "adaptive_threshold",
            "policy_description": "After one reveal, redeem next at $300 or less; above $300, wait once more",
            "envelope_id": "E09",
            "payout_usd": 500,
            "selected_prefixes": 72,
            "total_prefixes": 720,
            "selection_probability_fraction": "1/10",
            "selection_probability_pct": 10.0
          },
          {
            "policy": "adaptive_threshold",
            "policy_description": "After one reveal, redeem next at $300 or less; above $300, wait once more",
            "envelope_id": "E10",
            "payout_usd": 1000,
            "selected_prefixes": 72,
            "total_prefixes": 720,
            "selection_probability_fraction": "1/10",
            "selection_probability_pct": 10.0
          }
        ],
        "payout_distribution": [
          {
            "policy": "adaptive_threshold",
            "payout_usd": 100,
            "selected_prefixes": 288,
            "total_prefixes": 720,
            "probability_fraction": "2/5",
            "probability_pct": 40.0,
            "original_envelope_count": 4,
            "policy_ev_fraction_usd": "300/1",
            "policy_ev_usd": 300.0
          },
          {
            "policy": "adaptive_threshold",
            "payout_usd": 200,
            "selected_prefixes": 144,
            "total_prefixes": 720,
            "probability_fraction": "1/5",
            "probability_pct": 20.0,
            "original_envelope_count": 2,
            "policy_ev_fraction_usd": "300/1",
            "policy_ev_usd": 300.0
          },
          {
            "policy": "adaptive_threshold",
            "payout_usd": 300,
            "selected_prefixes": 72,
            "total_prefixes": 720,
            "probability_fraction": "1/10",
            "probability_pct": 10.0,
            "original_envelope_count": 1,
            "policy_ev_fraction_usd": "300/1",
            "policy_ev_usd": 300.0
          },
          {
            "policy": "adaptive_threshold",
            "payout_usd": 400,
            "selected_prefixes": 72,
            "total_prefixes": 720,
            "probability_fraction": "1/10",
            "probability_pct": 10.0,
            "original_envelope_count": 1,
            "policy_ev_fraction_usd": "300/1",
            "policy_ev_usd": 300.0
          },
          {
            "policy": "adaptive_threshold",
            "payout_usd": 500,
            "selected_prefixes": 72,
            "total_prefixes": 720,
            "probability_fraction": "1/10",
            "probability_pct": 10.0,
            "original_envelope_count": 1,
            "policy_ev_fraction_usd": "300/1",
            "policy_ev_usd": 300.0
          },
          {
            "policy": "adaptive_threshold",
            "payout_usd": 1000,
            "selected_prefixes": 72,
            "total_prefixes": 720,
            "probability_fraction": "1/10",
            "probability_pct": 10.0,
            "original_envelope_count": 1,
            "policy_ev_fraction_usd": "300/1",
            "policy_ev_usd": 300.0
          }
        ]
      }
    ]
  },
  "draw_k_now": {
    "formula_expected_total": "k \u00d7 current mean",
    "formula_at_least_one_threshold": "1 - C(N-J,k) / C(N,k)",
    "rows": [
      {
        "draws_now": 1,
        "expected_total_fraction_usd": "300/1",
        "expected_total_usd": 300.0,
        "prize_ge_1000_count": 1,
        "probability_ge_1000_fraction": "1/10",
        "probability_ge_1000_pct": 10.0,
        "prize_ge_400_count": 3,
        "probability_ge_400_fraction": "3/10",
        "probability_ge_400_pct": 30.0
      },
      {
        "draws_now": 2,
        "expected_total_fraction_usd": "600/1",
        "expected_total_usd": 600.0,
        "prize_ge_1000_count": 1,
        "probability_ge_1000_fraction": "1/5",
        "probability_ge_1000_pct": 20.0,
        "prize_ge_400_count": 3,
        "probability_ge_400_fraction": "8/15",
        "probability_ge_400_pct": 53.333333
      },
      {
        "draws_now": 3,
        "expected_total_fraction_usd": "900/1",
        "expected_total_usd": 900.0,
        "prize_ge_1000_count": 1,
        "probability_ge_1000_fraction": "3/10",
        "probability_ge_1000_pct": 30.0,
        "prize_ge_400_count": 3,
        "probability_ge_400_fraction": "17/24",
        "probability_ge_400_pct": 70.833333
      },
      {
        "draws_now": 4,
        "expected_total_fraction_usd": "1200/1",
        "expected_total_usd": 1200.0,
        "prize_ge_1000_count": 1,
        "probability_ge_1000_fraction": "2/5",
        "probability_ge_1000_pct": 40.0,
        "prize_ge_400_count": 3,
        "probability_ge_400_fraction": "5/6",
        "probability_ge_400_pct": 83.333333
      }
    ]
  },
  "theorem": {
    "statement": "When every intervening external reveal and the selected draw are conditionally uniform and exchangeable from the same fixed pool, every bounded predictable single-draw timing rule with redemption forced by the final available position has the original ex-ante marginal payout distribution.",
    "proof_outline": "For each payout class, its share of the remaining pool is a bounded martingale. Conditional on any admissible draw time, the next prize has that remaining share; optional sampling returns the initial share for every class."
  }
}