{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R463",
  "content_hash": null,
  "slug": "life-sbu-claim-binary-full-shift-area-eleven-exclusion",
  "type": "claim",
  "title": "The binary full-shift target needs macrotile area greater than 11",
  "summary": "Exact finite-torus enumeration rules out every positive macrotile dimension with area at most 11 for a strong Life simulation of the constant map from the binary full shift to a singleton.",
  "relevance": "For Strong block universality of Conway's Game of Life, record life-sbu-claim-binary-full-shift-area-eleven-exclusion (“The binary full-shift target needs macrotile area greater than 11”) records a bound, answer, status fact, or structural consequence. The record states: Exact finite-torus enumeration rules out every positive macrotile dimension with area at most 11 for a strong Life simulation of the constant map from the binary full shift to a singleton.",
  "relevance_source": "recorded",
  "body": "Apply the periodic-fiber count test to \\(Y=\\{0,1\\}^{\\mathbb Z^2}\\) and its constant map to a singleton. For each positive pair \\((m,n)\\) with \\(mn\\leq11\\), rotation symmetry lets us assume \\(m\\leq n\\). The replay enumerates all \\(2^{mn}\\) periodic Life inputs and retains every output with exactly two predecessors, as required by (1) at \\((p,q)=(1,1)\\). It then counts the predecessors of the same output after doubling the width. Strong simulation would require exactly four. When a torus dimension is 1 or 2, distinct neighbor positions in the infinite periodic configuration can share a residue class. The evaluator retains all eight Moore-neighborhood offsets with their multiplicities.\n\nThe base-candidate count and doubled-fiber histogram are given below. A histogram entry \\(a:b\\) says that \\(b\\) candidate outputs have \\(a\\) predecessors after the width is doubled.\n\nFor \\(1\\times1\\), the data is \\(1\\to\\{2:1\\}\\). For \\(1\\times2\\), it is \\(1\\to\\{12:1\\}\\). The \\(1\\times3\\), \\(1\\times4\\), and \\(1\\times5\\) tori have no base candidates.\n\nThe next rows are \\(1\\times6:6\\to\\{12:3,16:3\\}\\), \\(1\\times7:21\\to\\{22:7,38:7,44:7\\}\\), and \\(1\\times8:20\\to\\{60:12,264:8\\}\\).\n\nFor \\(1\\times9\\), the data is \\(54\\to\\{10:9,12:9,16:9,42:9,68:9,154:9\\}\\). For \\(1\\times10\\), it is \\(110\\to\\{22:30,38:20,44:10,54:10,86:10,128:10,188:20\\}\\). For \\(1\\times11\\), it is \\(154\\to\\{50:11,60:33,68:11,70:11,172:33,232:11,264:11,320:11,344:22\\}\\).\n\nThe \\(2\\times2\\), \\(2\\times3\\), and \\(3\\times3\\) tori have no base candidates. The remaining rows are \\(2\\times4:12\\to\\{26:4,36:8\\}\\) and \\(2\\times5:35\\to\\{6:5,26:20,70:10\\}\\).\n\nNo row contains a candidate with doubled fiber size 4. Therefore no strong simulation of this required target can use \\(mn\\leq11\\). This excludes a finite parameter range. It leaves every larger macrotile and the full universality question open.",
  "status": "supported",
  "evidence_grade": "computational",
  "scope": {
    "kind": "bounded",
    "statement": "all positive macrotile dimensions with area at most 11 for the constant map from the binary full shift to a singleton",
    "bounds": {
      "macrotile_area": {
        "min": 1,
        "max": 11
      },
      "simulated_sft_alphabet_size": {
        "min": 2,
        "max": 2
      },
      "constant_codomain_alphabet_size": {
        "min": 1,
        "max": 1
      },
      "tested_horizontal_period_multiplier": {
        "min": 1,
        "max": 2
      },
      "tested_vertical_period_multiplier": {
        "min": 1,
        "max": 1
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "partial",
    "kind": "claim",
    "command": "python3 tools/life_strong_block_universality_replay.py",
    "runtime": "CPython 3.9.6, standard library only",
    "citation": {
      "locator": "Exact Python replay in tools/life_strong_block_universality_replay.py and independent C++17 replay in tools/life_strong_block_universality_crosscheck.cpp, executed 2026-07-28; theoretical reduction in life-sbu-claim-periodic-fiber-count-test"
    },
    "missing": [
      "source",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": null,
    "locator": "Exact Python replay in tools/life_strong_block_universality_replay.py and independent C++17 replay in tools/life_strong_block_universality_crosscheck.cpp, executed 2026-07-28; theoretical reduction in life-sbu-claim-periodic-fiber-count-test"
  },
  "relations": [
    {
      "slug": "R466",
      "title": "Strong universality forces exact periodic-point counts in a periodic Life fiber",
      "object_type": "claim",
      "relation": "supports",
      "direction": "incoming"
    },
    {
      "slug": "R460",
      "title": "Exact finite-torus fiber and certificate replay",
      "object_type": "artifact",
      "relation": "evidences",
      "direction": "incoming"
    },
    {
      "slug": "R459",
      "title": "Independent C++ area-11 cross-check",
      "object_type": "artifact",
      "relation": "evidences",
      "direction": "incoming"
    },
    {
      "slug": "R464",
      "title": "Semiweak universality is proved and strong universality remains open",
      "object_type": "claim",
      "relation": "supports",
      "direction": "outgoing"
    },
    {
      "slug": "life-strong-block-universality",
      "title": "life strong block universality",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}