{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R458",
  "content_hash": null,
  "slug": "levine-claim-three-block-seven-twentieths",
  "type": "claim",
  "title": "A symmetric recursive three-block strategy wins with probability exactly 7/20",
  "summary": "The first-nonmonochromatic three-block strategy attains \\(7/20\\), giving the current lower bound for the Borel game.",
  "relevance": "For The value of Levine's two-player coin-index game, record levine-claim-three-block-seven-twentieths (“A symmetric recursive three-block strategy wins with probability exactly 7/20”) records a bound, answer, status fact, or structural consequence. The record states: The first-nonmonochromatic three-block strategy attains \\(7/20\\), giving the current lower bound for the Borel game.",
  "relevance_source": "recorded",
  "body": "Number coordinates within a block by \\(0,1,2\\). Each player scans the observed sequence in consecutive triples, skips \\(000\\) and \\(111\\), and stops at the first other word. On that word, use\n\\[\n001\\mapsto1,\\quad010\\mapsto2,\\quad011\\mapsto2,\\quad\n100\\mapsto0,\\quad101\\mapsto1,\\quad110\\mapsto0.\n\\]\nThe selected global coordinate is the start of the stopping block plus this local index. The stopping time is almost surely finite, and each output fiber is Borel.\n\nAmong the \\(36\\) ordered pairs of nonmonochromatic triples, exactly \\(15\\) are winning, so the conditional win probability when the players stop in the same block is \\(5/12\\). A triple is monochromatic with probability \\(1/4\\). The probability that the first nonmonochromatic blocks coincide is\n\\[\n\\sum_{k\\ge0}(1/4)^{2k}(3/4)^2=3/5.\n\\]\nWhen the stopping blocks differ, each selected bit lies in a block that the other player's observed stopping decision did not inspect, and the two selected bits are independent fair bits. The conditional win probability is \\(1/4\\). The total is\n\\[\n\\frac35\\frac5{12}+\\frac25\\frac14=\\frac7{20}.\n\\]\nThe exact replay independently enumerates the \\(36\\) local pairs and checks finite truncations through twelve blocks.",
  "status": "supported",
  "evidence_grade": "executable",
  "scope": {
    "kind": "family",
    "statement": "the symmetric recursive strategy that scans consecutive three-bit blocks",
    "family": "first-nonmonochromatic three-block strategy with the displayed local index map"
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://mathoverflow.net/questions/326669/guessing-each-others-coins",
      "locator": "MathOverflow answer 326787 by mihaild"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://mathoverflow.net/questions/326669/guessing-each-others-coins",
    "locator": "MathOverflow answer 326787 by mihaild"
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R455",
      "title": "The checked interval is 7/20 through 81/224, and equality at 7/20 remains open",
      "object_type": "claim",
      "relation": "supports",
      "direction": "outgoing"
    },
    {
      "slug": "R450",
      "title": "Exact replay of the three-block 7/20 strategy",
      "object_type": "artifact",
      "relation": "evidences",
      "direction": "incoming"
    },
    {
      "slug": "levine-two-player-seven-twentieths",
      "title": "levine two player seven twentieths",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}