{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R469",
  "content_hash": null,
  "slug": "life6-claim-exact-maximum-1093273",
  "type": "claim",
  "title": "The exact maximum preimage multiplicity is 1,093,273",
  "summary": "The all-dead output attains the value, and an exact output-prefix search proves that no output has more predecessors.",
  "relevance": "This records a result, observation, or conjecture used to assess the target problem.",
  "relevance_source": "derived",
  "body": "For the fixed-dead finite-board map in the question,\n\\[\n\\boxed{P_6=1,093,273}.\n\\]\nThe all-dead output attains this value. A two-row transfer counts its predecessors exactly.\n\nFor the upper bound, encode each predecessor row by a six-bit integer. Before output row \\(i\\) is evaluated, the transfer state is the pair \\((a,b)\\) of predecessor rows \\(i-1\\) and \\(i\\). A choice of the next row \\(c\\) determines the full six-bit output row \\(F(a,b,c)\\). The top predecessor row starts at zero, and the final choice of \\(c\\) is forced to zero by the boundary convention.\n\nThe executable certificate branches over output rows. At every output prefix it stores the exact number of compatible predecessor prefixes in each state. Its suffix bound lets each state choose its own best remaining output suffix, an admissible relaxation of the requirement that every state share one suffix. Every pruned prefix therefore has at most 1,093,273 completions. The search closes all prefixes whose bounds could exceed the incumbent. It visits 1, 64, 293, 361, 235, and 34 nodes at prefix depths zero through five. At the last row it checks 2,176 reachable output branches. No branch improves the all-dead count.\n\nPrefixes whose bound equals the incumbent are pruned. The computation therefore proves the maximum value and supplies one maximizing output, while leaving uniqueness undecided.",
  "status": "established",
  "evidence_grade": "reproduced",
  "scope": {
    "kind": "bounded",
    "statement": "all 2^36 outputs of the synchronous Conway Life map on a 6 by 6 board, with predecessor and output both confined to the board and every exterior cell fixed dead",
    "bounds": {
      "width": {
        "min": 6,
        "max": 6
      },
      "height": {
        "min": 6,
        "max": 6
      },
      "outputs": {
        "min": 68719476736,
        "max": 68719476736
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://doi.org/10.1063/1.5006143",
      "locator": "Exact certificate in life6-artifact-row-transfer-branch-and-bound, reproduced on 2026-07-25"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.1063/1.5006143",
    "locator": "Exact certificate in life6-artifact-row-transfer-branch-and-bound, reproduced on 2026-07-25"
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R467",
      "title": "Executable row-transfer certificate for the full output space",
      "object_type": "artifact",
      "relation": "verifies",
      "direction": "incoming"
    },
    {
      "slug": "R470",
      "title": "An independent-state suffix relaxation gives a rigorous search bound",
      "object_type": "claim",
      "relation": "supports",
      "direction": "incoming"
    },
    {
      "slug": "R468",
      "title": "Published precursor counts use a different finite-window convention",
      "object_type": "attempt",
      "relation": "informs",
      "direction": "incoming"
    },
    {
      "slug": "life-six-preimage-multiplicity",
      "title": "life six preimage multiplicity",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}