{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R135",
  "content_hash": null,
  "slug": "cds-claim-five-thirds-upper-bound",
  "type": "claim",
  "title": "Five-thirds is the current checked upper bound",
  "summary": "Current checked results give infinitely many lower constructions with \\(3n/2-O(\\sqrt n)\\) distinct circular squares and the universal upper bound \\(s(w)\\le5n/3\\); whether \\(s(w)\\le\\lceil3n/2\\rceil\\) for every finite word remains open.",
  "relevance": "For The three-halves bound for distinct squares in circular words, record cds-claim-five-thirds-upper-bound (“Five-thirds is the current checked upper bound”) records a bound, answer, status fact, or structural consequence. The record states: Current checked results give infinitely many lower constructions with \\(3n/2-O(\\sqrt n)\\) distinct circular squares and the universal upper bound \\(s(w)\\le5n/3\\); whether \\(s(w)\\le\\lceil3n/2\\rceil\\) for every finite word remains open.",
  "relevance_source": "recorded",
  "body": "Let s(w) count the distinct square words occurring among the rotations of a word w of length n, equivalently the distinct square factors of ww whose total length is at most n. Li and Song's Main Theorem gives s(w) <= 5n/3 for every finite word w. Since s(w) is integral, this also gives s(w) <= floor(5n/3). The result leaves the three-halves target open.",
  "status": "reported",
  "evidence_grade": "sourced",
  "scope": {
    "kind": "universal",
    "statement": "the sourced five-thirds upper bound for all finite circular words under the length-at-most-n square convention"
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://arxiv.org/abs/2605.12215v1",
      "locator": "Shuo Li and Yuan Song, A Tighter Upper Bound for the Number of Distinct Squares in Circular Words, arXiv:2605.12215v1, Main Theorem on pages 1-2, proof on pages 6-10, and Conclusion on page 10"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://arxiv.org/abs/2605.12215v1",
    "locator": "Shuo Li and Yuan Song, A Tighter Upper Bound for the Number of Distinct Squares in Circular Words, arXiv:2605.12215v1, Main Theorem on pages 1-2, proof on pages 6-10, and Conclusion on page 10"
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R131",
      "title": "The source conjecture uses the floor-strength convention",
      "object_type": "attempt",
      "relation": "reports",
      "direction": "incoming"
    },
    {
      "slug": "R129",
      "title": "Strengthen the primitive Rauzy split case",
      "object_type": "attempt",
      "relation": "informs",
      "direction": "outgoing"
    },
    {
      "slug": "circular-distinct-squares-three-halves",
      "title": "circular distinct squares three halves",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}