{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R456",
  "content_hash": null,
  "slug": "levine-claim-self-best-response-through-seven-blocks",
  "type": "claim",
  "title": "The truncated strategy is a best response to itself through seven blocks",
  "summary": "For every truncation with \\(1\\le m\\le7\\), exact enumeration finds no unilateral deterministic improvement over the symmetric three-block strategy.",
  "relevance": "For The value of Levine's two-player coin-index game, record levine-claim-self-best-response-through-seven-blocks (“The truncated strategy is a best response to itself through seven blocks”) records a bound, answer, status fact, or structural consequence. The record states: For every truncation with \\(1\\le m\\le7\\), exact enumeration finds no unilateral deterministic improvement over the symmetric three-block strategy.",
  "relevance_source": "recorded",
  "body": "Fix one player's \\((3m+1)\\)-hat truncation of the three-block strategy. For every observed string of the other player, a deterministic best response can choose the coordinate with the greatest exact winning count. The computation exhausts all observed strings, evaluates every coordinate, and sums these pointwise maxima.\n\nFor each \\(m=1,\\ldots,7\\), the best-response total equals the total achieved by using the same truncated strategy on both sides. The best-response excess is exactly zero. The checked probabilities run from \\(11/32\\) for \\(m=1\\) to \\(187904819/536870912\\) for \\(m=7\\). Direct enumeration of all pairs of bit strings supplies a second implementation for \\(m\\le3\\).\n\nThis is a bounded unilateral-optimality result for one fixed opponent strategy. It gives no global upper bound on \\(V_{3m+1}\\), and it does not prove that the infinite recursive strategy is optimal.",
  "status": "observed",
  "evidence_grade": "computational",
  "scope": {
    "kind": "bounded",
    "statement": "the finite first-nonmonochromatic three-block strategy with its fallback coordinate, for 1 through 7 blocks",
    "bounds": {
      "blocks": {
        "min": 1,
        "max": 7
      },
      "finite_hat_count": {
        "min": 4,
        "max": 22
      },
      "observed_strings_per_instance": {
        "min": 16,
        "max": 4194304
      },
      "observed_strings_total": {
        "min": 4793488,
        "max": 4793488
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "locator": "Exact run of tools/levine_three_block_best_response.py --max-blocks 7 on 2026-07-28"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": null,
    "locator": "Exact run of tools/levine_three_block_best_response.py --max-blocks 7 on 2026-07-28"
  },
  "models": [],
  "relations": [
    {
      "slug": "R449",
      "title": "Exact finite best-response enumerator",
      "object_type": "artifact",
      "relation": "evidences",
      "direction": "incoming"
    },
    {
      "slug": "R455",
      "title": "The checked interval is 7/20 through 81/224, and equality at 7/20 remains open",
      "object_type": "claim",
      "relation": "informs",
      "direction": "outgoing"
    },
    {
      "slug": "R457",
      "title": "The three-block strategy is a best response to itself at every depth",
      "object_type": "claim",
      "relation": "supports",
      "direction": "outgoing"
    },
    {
      "slug": "levine-two-player-seven-twentieths",
      "title": "levine two player seven twentieths",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}