{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R452",
  "content_hash": null,
  "slug": "levine-attempt-extend-best-response-m8-m12",
  "type": "attempt",
  "title": "Stress-test the all-depth response proof on blocks eight through twelve",
  "summary": "Implement a block recurrence or symmetry quotient that independently checks the all-depth response formula for \\(8\\le m\\le12\\) without traversing all \\(2^{3m+1}\\) observed strings.",
  "relevance": "For The value of Levine's two-player coin-index game, record levine-attempt-extend-best-response-m8-m12 (“Stress-test the all-depth response proof on blocks eight through twelve”) documents a concrete method, search boundary, or failed route. The record states: Implement a block recurrence or symmetry quotient that independently checks the all-depth response formula for \\(8\\le m\\le12\\) without traversing all \\(2^{3m+1}\\) observed strings.",
  "relevance_source": "recorded",
  "body": "The blockwise proof covers every depth, while the current independent Gray-code replay is exhaustive through \\(m=7\\). The next experiment is to group observed prefixes by the first nonmonochromatic block, the selected local coordinate, and the score-vector orbit induced by the six nonmonochromatic words.\n\nFor each \\(m=8,\\ldots,12\\), compute the exact maximum-response sum over the quotient states and compare it with\n\\[\n4^{3m+1}\\left(\\frac7{20}-\\frac{1}{10\\cdot16^m}\\right).\n\\]\nThe experiment succeeds if a replayable recurrence covers every observed string and returns equality for all five sizes. A discrepancy should preserve the first observed string and both score vectors for diagnosis. If the quotient still grows beyond a fixed memory limit, record the state count, collision rule, and first uncompleted \\(m\\) before changing the method.",
  "status": "next_experiment",
  "evidence_grade": "self_reported",
  "scope": {
    "kind": "bounded",
    "statement": "the finite first-nonmonochromatic three-block strategy for 8 through 12 blocks",
    "bounds": {
      "blocks": {
        "min": 8,
        "max": 12
      },
      "finite_hat_count": {
        "min": 25,
        "max": 37
      }
    },
    "exhaustive": false
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "attempt",
    "citation": {
      "locator": "Bounded continuation designed from tools/levine_three_block_best_response.py on 2026-07-28"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": null,
    "locator": "Bounded continuation designed from tools/levine_three_block_best_response.py on 2026-07-28"
  },
  "models": [],
  "relations": [
    {
      "slug": "R449",
      "title": "Exact finite best-response enumerator",
      "object_type": "artifact",
      "relation": "uses",
      "direction": "outgoing"
    },
    {
      "slug": "R457",
      "title": "The three-block strategy is a best response to itself at every depth",
      "object_type": "claim",
      "relation": "tests",
      "direction": "outgoing"
    },
    {
      "slug": "levine-two-player-seven-twentieths",
      "title": "levine two player seven twentieths",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}