{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R465",
  "content_hash": null,
  "slug": "life-sbu-claim-koynnos-singleton-simulation",
  "type": "claim",
  "title": "The 6 by 3 Köynnös agar passes the singleton test",
  "summary": "Salo and Törmä give a 6 by 3 periodic Life fixed point whose only global predecessor is itself; it therefore strongly simulates the constant map on a singleton SFT.",
  "relevance": "For Strong block universality of Conway's Game of Life, record life-sbu-claim-koynnos-singleton-simulation (“The 6 by 3 Köynnös agar passes the singleton test”) records a bound, answer, status fact, or structural consequence. The record states: Salo and Törmä give a 6 by 3 periodic Life fixed point whose only global predecessor is itself; it therefore strongly simulates the constant map on a singleton SFT.",
  "relevance_source": "recorded",
  "body": "Repeat the following three rows periodically:\n\\[\nP=\\begin{pmatrix}\n1&1&1&0&0&0\\\\\n0&1&0&1&1&1\\\\\n0&0&0&0&1&0\n\\end{pmatrix}.\n\\]\nSalo and Törmä call the resulting configuration Köynnös. Every live cell has three live neighbors and every dead cell has four, so \\(g(x^P)=x^P\\). Their finite forcing certificate proves the stronger statement\n\\[\ng^{-1}(x^P)=\\{x^P\\}.\n\\]\nTake a singleton alphabet \\(D=\\{d\\}\\), a singleton SFT \\(Y=\\{y\\}\\), and its only block map. Set \\((m,n)=(6,3)\\), \\(\\tau(d)=P\\), and map the only Life predecessor to \\(y\\). This is a strong simulation of that target.\n\nThe example closes the easiest singleton specialization. Any obstruction to full strong universality must use a richer target or another invariant. The binary full-shift test in this packet does so.",
  "status": "supported",
  "evidence_grade": "sourced",
  "scope": {
    "kind": "family",
    "statement": "the singleton SFT and its constant block map, strongly simulated by the explicit 6 by 3 Köynnös macrotile",
    "family": "constant block maps with a singleton domain and codomain"
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://doi.org/10.4230/LIPIcs.ICALP.2022.131",
      "locator": "Ville Salo and Ilkka Törmä, What Can Oracles Teach Us About the Ultimate Fate of Life?, ICALP 2022, Section 3.1, Köynnös, especially Lemma KoynnosForcing and the following global forcing argument"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.4230/LIPIcs.ICALP.2022.131",
    "locator": "Ville Salo and Ilkka Törmä, What Can Oracles Teach Us About the Ultimate Fate of Life?, ICALP 2022, Section 3.1, Köynnös, especially Lemma KoynnosForcing and the following global forcing argument"
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R466",
      "title": "Strong universality forces exact periodic-point counts in a periodic Life fiber",
      "object_type": "claim",
      "relation": "informs",
      "direction": "outgoing"
    },
    {
      "slug": "life-strong-block-universality",
      "title": "life strong block universality",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}