{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R442",
  "content_hash": null,
  "slug": "lat7-claim-first-complete-return",
  "type": "claim",
  "title": "The complete initial state first returns after 64,165,598 updates",
  "summary": "Exact cycle detection gives transient length zero and cycle length 64,165,598 for the full labeled state.",
  "relevance": "This records a result, observation, or conjecture used to assess the target problem.",
  "relevance_source": "derived",
  "body": "Number cells by \\(7y+x\\), with \\(0\\leq x,y<7\\). Store the 49 colors in bits 0 through 48, the ant's cell in bits 49 through 54, and its direction in bits 55 and 56, using north, east, south, west as 0, 1, 2, 3. The all-white state at cell 0 facing north is therefore the integer 0. Each transition reads the bit at the current cell, turns, toggles that bit, moves with coordinates reduced modulo 7, and repacks the state.\n\nFloyd cycle detection on this exact 57-bit transition map gives\n\\[\n\\mu=0,\\qquad \\lambda=64{,}165{,}598.\n\\]\nThus the initial state belongs to the eventual cycle and the orbit has no transient. A second direct replay starts from state 0, asserts that every state after steps 1 through 64,165,597 is nonzero, and reaches state 0 after step 64,165,598. This proves both recurrence of the complete state and minimality of the positive return time:\n\\[\n\\boxed{64{,}165{,}598}.\n\\]\nSince the map is deterministic, these checks also imply that all states at times 0 through 64,165,597 are distinct. The cycle contains the labeled colors, position, and direction, with no quotient by translation or rotation.",
  "status": "established",
  "evidence_grade": "reproduced",
  "scope": {
    "kind": "bounded",
    "statement": "Langton's ant on the labeled 7 by 7 square torus, starting all white at (0,0) facing north under the stated white-right and black-left rule",
    "bounds": {
      "torus_width": {
        "min": 7,
        "max": 7
      },
      "torus_height": {
        "min": 7,
        "max": 7
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://math.stackexchange.com/questions/3296855/what-is-the-period-of-langtons-ant-on-a-torus",
      "locator": "Independent 2019 answer by user Stack Overflown, square-torus table at row 7 and column 7; exact reproduction in lat7-artifact-full-state-cycle-certificate"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://math.stackexchange.com/questions/3296855/what-is-the-period-of-langtons-ant-on-a-torus",
    "locator": "Independent 2019 answer by user Stack Overflown, square-torus table at row 7 and column 7; exact reproduction in lat7-artifact-full-state-cycle-certificate"
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R440",
      "title": "Bit-packed cycle detection and direct replay certificate",
      "object_type": "artifact",
      "relation": "verifies",
      "direction": "incoming"
    },
    {
      "slug": "R441",
      "title": "A public finite-torus table already records the same period",
      "object_type": "attempt",
      "relation": "supports",
      "direction": "incoming"
    },
    {
      "slug": "langton-ant-seven-torus-return",
      "title": "langton ant seven torus return",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}