{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R466",
  "content_hash": null,
  "slug": "life-sbu-claim-periodic-fiber-count-test",
  "type": "claim",
  "title": "Strong universality forces exact periodic-point counts in a periodic Life fiber",
  "summary": "For the constant map from any SFT Y to a singleton, strong simulation would make one periodic Life fiber conjugate to Y under blocked shifts; every finite periodic-point count must therefore agree.",
  "relevance": "For Strong block universality of Conway's Game of Life, record life-sbu-claim-periodic-fiber-count-test (“Strong universality forces exact periodic-point counts in a periodic Life fiber”) records a bound, answer, status fact, or structural consequence. The record states: For the constant map from any SFT Y to a singleton, strong simulation would make one periodic Life fiber conjugate to Y under blocked shifts; every finite periodic-point count must therefore agree.",
  "relevance_source": "recorded",
  "body": "Let \\(D=\\{d\\}\\), let \\(Y\\) be a two-dimensional SFT, and let \\(\\phi:Y\\to D^{\\mathbb Z^2}\\) be the constant block map. Suppose Life strongly simulates \\(\\phi\\) with data \\((m,n,\\tau,h)\\). Put\n\\[\nz=\\tau(d^{\\mathbb Z^2}).\n\\]\nThe configuration \\(z\\) is periodic with periods \\((m,0)\\) and \\((0,n)\\). By Definition 5, the domain of the partial map \\(h\\) is exactly \\(g^{-1}(z)\\). Strong simulation makes\n\\[\nh:(g^{-1}(z),m\\mathbb Z\\times n\\mathbb Z)\\longrightarrow Y\n\\]\nbijective. Definition 5 notes that such an \\(h\\) is a conjugacy for the blocked shift actions.\n\nFix positive \\(p,q\\). Conjugacy preserves points fixed by \\((p,0)\\) and \\((0,q)\\). Hence\n\\[\n\\left|\\{x:g(x)=z,\\ \\sigma^{(pm,0)}x=x,\\ \\sigma^{(0,qn)}x=x\\}\\right|\n=|\\operatorname{Fix}_Y(p,q)|. \\tag{1}\n\\]\nFor the binary full shift, the right side is \\(2^{pq}\\). In particular, a candidate macrotile must give fiber size 2 on the \\(m\\)-by-\\(n\\) torus and fiber size 4 after doubling either fundamental period. For singleton \\(Y\\), every count in (1) is 1, and the whole global fiber must contain one configuration.\n\nEquation (1) is a necessary condition. Finite periodic counts alone cannot exclude extra aperiodic preimages, so satisfying the identities would leave a separate global conjugacy problem.",
  "status": "supported",
  "evidence_grade": "self_reported",
  "scope": {
    "kind": "conditional",
    "statement": "periodic-point count identities forced by any strong Life simulation of a constant map from a two-dimensional SFT",
    "conditions": [
      "life-strong-block-universality"
    ]
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "locator": "Direct strengthening of Salo and Törmä's Definition 5 and Theorem 7 constant-map argument, with the fixed-point-count consequence recorded on 2026-07-28"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": null,
    "locator": "Direct strengthening of Salo and Törmä's Definition 5 and Theorem 7 constant-map argument, with the fixed-point-count consequence recorded on 2026-07-28"
  },
  "relations": [
    {
      "slug": "R465",
      "title": "The 6 by 3 Köynnös agar passes the singleton test",
      "object_type": "claim",
      "relation": "informs",
      "direction": "incoming"
    },
    {
      "slug": "R463",
      "title": "The binary full-shift target needs macrotile area greater than 11",
      "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"
    }
  ]
}