{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R129",
  "content_hash": null,
  "slug": "cds-attempt-rauzy-split-accounting",
  "type": "attempt",
  "title": "Strengthen the primitive Rauzy split case",
  "summary": "Instrument Li and Song's Lemma 13 case and search for the missing independent-cycle or odd-power-class charge needed to improve 5n/3 to 3n/2.",
  "relevance": "For The three-halves bound for distinct squares in circular words, record cds-attempt-rauzy-split-accounting (“Strengthen the primitive Rauzy split case”) documents a concrete method, search boundary, or failed route. The record states: Instrument Li and Song's Lemma 13 case and search for the missing independent-cycle or odd-power-class charge needed to improve 5n/3 to 3n/2.",
  "relevance_source": "recorded",
  "body": "First align the live floor or ceiling convention with the cited conjecture. Then focus the proof search on primitive words. Li and Song's Lemma 9 already gives the three-halves bound for nonprimitive words, and their Lemma 10 gives it when every high-order Rauzy-graph basis contains a circuit of size at least n/2. The five-thirds loss occurs in the split branch handled by Lemma 13, where a primitive circuit root p has n/4 <= |p| <= n/3.\n\nA concrete next experiment should build Gamma_m(ww) for the exact primitive maximizers in this packet and for targeted binary words through a larger bounded range. For every split, record n, m, |p|, |Class_p(ww)|, |O_p(ww)|, the cyclomatic number, and the large circuits forced into every basis. Test whether a second circuit class or unused odd-power count supplies an additional n/6 charge in the Lemma 13 branch. Stop with either a smallest profile refuting each proposed charge or a finite list of inequalities that covers every observed split type. Every basis claim needs a certificate or an independently replayable graph computation.",
  "status": "next_experiment",
  "evidence_grade": "self_reported",
  "scope": {
    "kind": "family",
    "statement": "primitive circular words entering the split branch of Li and Song's Lemma 13",
    "family": "primitive words whose high-order Rauzy circuit splits with a root p satisfying n/4 <= |p| <= n/3"
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "attempt",
    "citation": {
      "url": "https://arxiv.org/abs/2605.12215",
      "locator": "Li and Song, arXiv:2605.12215v1, Lemmas 9, 10, 12, and 13 and the proof of the Main Theorem on pages 6-10"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://arxiv.org/abs/2605.12215",
    "locator": "Li and Song, arXiv:2605.12215v1, Lemmas 9, 10, 12, and 13 and the proof of the Main Theorem on pages 6-10"
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R135",
      "title": "Five-thirds is the current checked upper bound",
      "object_type": "claim",
      "relation": "informs",
      "direction": "incoming"
    },
    {
      "slug": "R134",
      "title": "Fibonacci constructions approach three halves",
      "object_type": "claim",
      "relation": "informs",
      "direction": "incoming"
    },
    {
      "slug": "R133",
      "title": "Exact maxima for every alphabet through length fourteen",
      "object_type": "claim",
      "relation": "informs",
      "direction": "incoming"
    },
    {
      "slug": "R132",
      "title": "Exact binary maxima through length twenty-six",
      "object_type": "claim",
      "relation": "informs",
      "direction": "incoming"
    },
    {
      "slug": "R128",
      "title": "One-letter deletion has unbounded square loss",
      "object_type": "attempt",
      "relation": "informs",
      "direction": "incoming"
    },
    {
      "slug": "R130",
      "title": "The short-square half-length charge fails at twelve",
      "object_type": "attempt",
      "relation": "informs",
      "direction": "incoming"
    },
    {
      "slug": "R131",
      "title": "The source conjecture uses the floor-strength convention",
      "object_type": "attempt",
      "relation": "constrains",
      "direction": "incoming"
    },
    {
      "slug": "circular-distinct-squares-three-halves",
      "title": "circular distinct squares three halves",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}