{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R455",
  "content_hash": null,
  "slug": "levine-claim-current-interval-2026-07",
  "type": "claim",
  "title": "The checked interval is 7/20 through 81/224, and equality at 7/20 remains open",
  "summary": "As of 2026-07-28, the strongest checked bounds are \\(7/20\\le p^*\\le81/224\\); the exact value of \\(p^*\\) remains unknown.",
  "relevance": "For The value of Levine's two-player coin-index game, record levine-claim-current-interval-2026-07 (“The checked interval is 7/20 through 81/224, and equality at 7/20 remains open”) records a bound, answer, status fact, or structural consequence. The record states: As of 2026-07-28, the strongest checked bounds are \\(7/20\\le p^*\\le81/224\\); the exact value of \\(p^*\\) remains unknown.",
  "relevance_source": "recorded",
  "body": "For the Borel-strategy value \\(p^*\\) in the canonical statement, the checked literature gives\n\\[\n\\frac{7}{20}\\le p^*\\le\\frac{81}{224}=0.361607142857\\ldots.\n\\]\nThe lower endpoint is attained by recursive block strategies. Buhler and coauthors derive the upper endpoint using a balanced \\(8\\times14\\) hint matrix and an exhaustive optimization over column partitions. An independent exact replay prepared for this packet enumerates all \\(B_{14}=190{,}899{,}322\\) column partitions, obtains maximum \\(81/224\\), and recovers the reported count of 3,920 attaining partitions. Bouquet and coauthors, in arXiv:2508.01737v2 dated 2026-04-19, still state \\(p^*=7/20\\) as Conjecture 1. Heilman and Tamuz also identify \\(81/224\\) as the best known upper bound; their analytic bound \\(0.37193\\) is weaker.\n\nThe exact open remainder is to prove \\(p^*\\le7/20\\), or to construct Borel strategies with winning probability greater than \\(7/20\\). The older interval ending at \\(3/8\\) is valid but no longer strongest.",
  "status": "supported",
  "evidence_grade": "sourced",
  "scope": {
    "kind": "universal",
    "statement": "the supremum over all pairs of Borel measurable strategies in the stated two-player fair-bit game"
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://arxiv.org/abs/2508.01737v2",
      "locator": "Bouquet et al., Sections 1.1-1.3 and Conjecture 1, checked 2026-07-28"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://arxiv.org/abs/2508.01737v2",
    "locator": "Bouquet et al., Sections 1.1-1.3 and Conjecture 1, checked 2026-07-28"
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R454",
      "title": "Borel infinite-stack strategies have the same supremum as finite-stack strategies",
      "object_type": "claim",
      "relation": "supports",
      "direction": "incoming"
    },
    {
      "slug": "R458",
      "title": "A symmetric recursive three-block strategy wins with probability exactly 7/20",
      "object_type": "claim",
      "relation": "supports",
      "direction": "incoming"
    },
    {
      "slug": "R456",
      "title": "The truncated strategy is a best response to itself through seven blocks",
      "object_type": "claim",
      "relation": "informs",
      "direction": "incoming"
    },
    {
      "slug": "R451",
      "title": "The 2026 source audit retains the exact two-player value as an open conjecture",
      "object_type": "attempt",
      "relation": "evidences",
      "direction": "incoming"
    },
    {
      "slug": "R453",
      "title": "The published 81/224 matrix optimization has an independent exact replay",
      "object_type": "attempt",
      "relation": "supports",
      "direction": "incoming"
    },
    {
      "slug": "R448",
      "title": "Exact all-partitions certificate for the 81/224 hint-matrix value",
      "object_type": "artifact",
      "relation": "evidences",
      "direction": "incoming"
    },
    {
      "slug": "R457",
      "title": "The three-block strategy is a best response to itself at every depth",
      "object_type": "claim",
      "relation": "informs",
      "direction": "incoming"
    },
    {
      "slug": "R1815",
      "title": "The checked interval is 7/20 through 81/224, and equality at 7/20 remains open",
      "object_type": "claim",
      "relation": "supersedes",
      "direction": "incoming",
      "metadata": {
        "reason": "Preserves the published record identity while attaching the independently reviewed release-300 bibliography."
      }
    },
    {
      "slug": "levine-two-player-seven-twentieths",
      "title": "levine two player seven twentieths",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}