{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R642",
  "content_hash": null,
  "slug": "r30n24-claim-maximum-cycle-185040",
  "type": "claim",
  "title": "The maximum cycle length at width 24 is 185,040",
  "summary": "Two complete traversals of all 16,777,216 states find one cycle of length 185,040, represented by state 449.",
  "relevance": "This records a result, observation, or conjecture used to assess the target problem.",
  "relevance_source": "derived",
  "body": "Number the cells modulo 24 and store cell \\(i\\) in bit \\(i\\) of an unsigned integer. In this convention the left neighbor of cell \\(i\\) is cell \\(i+1\\), the right neighbor is cell \\(i-1\\), and one synchronous update is\n\\[\nF(x)_i=x_{i+1}\\mathbin{\\mathtt{xor}}(x_i\\mathbin{\\mathtt{or}}x_{i-1}).\n\\]\n\nAn exhaustive traversal of all \\(2^{24}=16{,}777{,}216\\) states gives maximum directed-cycle length\n\\[\n\\boxed{185{,}040}.\n\\]\nThere is exactly one cycle of that length. Its least integer state is 449, which is hexadecimal `0001c1` and binary `000000000000000111000001` when bits 23 through 0 are printed left to right. Starting at 449 and applying \\(F\\) returns to 449 after exactly 185,040 updates.\n\nThe executable certificate uses two full-state methods. The first follows each unprocessed orbit and marks its current path. The second constructs every successor and indegree, removes all transient vertices with Kahn's algorithm, and scans the vertices that remain. Both methods produce the same sorted list of pairs `(least state on cycle, cycle length)`. Since every state belongs to the scan, a longer cycle cannot be omitted.\n\nThe value also appears in the published width table of Gage, Laub, and McGarry and in OEIS A334497.",
  "status": "established",
  "evidence_grade": "computational",
  "scope": {
    "kind": "bounded",
    "statement": "the complete functional graph of cyclic Rule 30 on 24 binary cells",
    "bounds": {
      "ring_width": {
        "min": 24,
        "max": 24
      },
      "states": {
        "min": 16777216,
        "max": 16777216
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://oeis.org/A334497",
      "locator": "Term a(24)=185040; Gage, Laub, and McGarry (2005), page 7, Table 2; complete reproduction in r30n24-artifact-dual-exhaustive-scan"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://oeis.org/A334497",
    "locator": "Term a(24)=185040; Gage, Laub, and McGarry (2005), page 7, Table 2; complete reproduction in r30n24-artifact-dual-exhaustive-scan"
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R639",
      "title": "Dual exhaustive C11 verifier for cyclic Rule 30",
      "object_type": "artifact",
      "relation": "evidences",
      "direction": "incoming"
    },
    {
      "slug": "R640",
      "title": "The width-24 maximum was published in 2005",
      "object_type": "attempt",
      "relation": "corroborates",
      "direction": "incoming"
    },
    {
      "slug": "rule-30-cycle-length-24",
      "title": "rule 30 cycle length 24",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}