{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R108",
  "content_hash": null,
  "slug": "casf4-claim-stronger-cyclic-through-149",
  "type": "claim",
  "title": "The source reports no cyclic counterexample below length 150",
  "summary": "Four-letter circular abelian-square-free words exist at arbitrarily large lengths, and the checked source reports no counterexample below length 150; no threshold \\(N\\) covering every \\(n\\ge N\\) is known, so eventual existence remains open.",
  "relevance": "For Eventual existence of four-letter circular abelian-square-free words, record casf4-claim-stronger-cyclic-through-149 (“The source reports no cyclic counterexample below length 150”) records a bound, answer, status fact, or structural consequence. The record states: Four-letter circular abelian-square-free words exist at arbitrarily large lengths, and the checked source reports no counterexample below length 150; no threshold \\(N\\) covering every \\(n\\ge N\\) is known, so eventual existence remains open.",
  "relevance_source": "recorded",
  "body": "The candidate asks for circular avoidance. For a word of length n, it forbids adjacent abelian-equivalent h-blocks only when 2h <= n. Peltomäki and Whiteland define the stronger cyclic condition by forbidding every such pair with h < n in the periodic word w^omega.\n\nTheir Theorem 1.2 proves A_infinity(4)=2. Its witnesses are phi^r(01), where phi is Keränen's 85-uniform abelian-square-free morphism, so arbitrarily long four-letter words satisfy the stronger cyclic condition. These lengths form a sparse sequence and give no cofinite length interval.\n\nSection 6 reports that their computer experiments found no counterexample to A(4)=2 among lengths below 150. Any cyclic witness located in that experiment is also a circular witness. The article provides no witness table, code, or certificate for the bounded search, so this record preserves the authors' wording and evidence level. The exact mathematical theorem remains the arbitrarily-long result.",
  "status": "reported",
  "evidence_grade": "sourced",
  "scope": {
    "kind": "bounded",
    "statement": "source-reported search with no stronger cyclic counterexample at integer lengths n from 1 through 149",
    "bounds": {
      "n": {
        "min": 1,
        "max": 149
      }
    },
    "exhaustive": false
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://doi.org/10.1016/j.aam.2020.102095",
      "locator": "Peltomäki and Whiteland, Avoiding abelian powers cyclically, Advances in Applied Mathematics 121 (2020), Theorem 1.2, proof in Section 3, and Section 6; arXiv:2006.06307v2"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.1016/j.aam.2020.102095",
    "locator": "Peltomäki and Whiteland, Avoiding abelian powers cyclically, Advances in Applied Mathematics 121 (2020), Theorem 1.2, proof in Section 3, and Section 6; arXiv:2006.06307v2"
  },
  "relations": [
    {
      "slug": "R102",
      "title": "The 2026-07-28 source audit found no later resolution",
      "object_type": "attempt",
      "relation": "supports",
      "direction": "incoming"
    },
    {
      "slug": "R101",
      "title": "Build a certified boundary-profile splice system",
      "object_type": "attempt",
      "relation": "informs",
      "direction": "outgoing"
    },
    {
      "slug": "circular-abelian-square-free-four-eventual",
      "title": "circular abelian square free four eventual",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}