{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R134",
  "content_hash": null,
  "slug": "cds-claim-fibonacci-lower-family",
  "type": "claim",
  "title": "Fibonacci constructions approach three halves",
  "summary": "The CPM 2026 construction gives infinitely many lengths with at least 1.5n minus O(sqrt(n)) distinct circular square factors.",
  "relevance": "For The three-halves bound for distinct squares in circular words, record cds-claim-fibonacci-lower-family (“Fibonacci constructions approach three halves”) records a bound, answer, status fact, or structural consequence. The record states: The CPM 2026 construction gives infinitely many lengths with at least 1.5n minus O(sqrt(n)) distinct circular square factors.",
  "relevance_source": "recorded",
  "body": "Charalampopoulos, Mohamed, Radoszewski, Rytter, Waleń, and Zuba construct words from Fibonacci-word blocks and prove that CS(n) >= 1.5n - O(sqrt(n)) for infinitely many n. Here CS(n) is the maximum number of distinct squares of total length at most n in a circular word of length n. This is the strongest checked lower construction and supplies the source for the conjectured coefficient.",
  "status": "reported",
  "evidence_grade": "sourced",
  "scope": {
    "kind": "family",
    "statement": "the Fibonacci-block circular words W_{k,t} used in the CPM 2026 lower-bound construction",
    "family": "CPM 2026 Fibonacci-block words W_{k,t} with k chosen as |F_t|"
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://doi.org/10.4230/LIPIcs.CPM.2026.6",
      "locator": "Panagiotis Charalampopoulos et al., Improved Bounds on the Maximum Number of Distinct Squares in Circular Words, LIPIcs CPM 2026, DOI 10.4230/LIPIcs.CPM.2026.6, Theorem 12 and its proof on pages 6:6, with the conjecture in Final Remarks on page 6:12"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.4230/LIPIcs.CPM.2026.6",
    "locator": "Panagiotis Charalampopoulos et al., Improved Bounds on the Maximum Number of Distinct Squares in Circular Words, LIPIcs CPM 2026, DOI 10.4230/LIPIcs.CPM.2026.6, Theorem 12 and its proof on pages 6:6, with the conjecture in Final Remarks on page 6:12"
  },
  "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"
    }
  ]
}