{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R132",
  "content_hash": null,
  "slug": "cds-claim-binary-maxima-through-twenty-six",
  "type": "claim",
  "title": "Exact binary maxima through length twenty-six",
  "summary": "Two exhaustive binary programs agree on total, short-square, long-square, primitive, and nonprimitive maxima for every length through 26.",
  "relevance": "For The three-halves bound for distinct squares in circular words, record cds-claim-binary-maxima-through-twenty-six (“Exact binary maxima through length twenty-six”) records a bound, answer, status fact, or structural consequence. The record states: Two exhaustive binary programs agree on total, short-square, long-square, primitive, and nonprimitive maxima for every length through 26.",
  "relevance_source": "recorded",
  "body": "Fixing the first bit to 0 identifies each binary word with its complement and still covers every binary word. Two independent programs exhaust this space through n=26. The total maxima for n=1,...,26 are\n[0,1,1,2,3,4,4,6,6,9,8,10,11,13,13,16,15,18,17,19,21,22,22,25,24,28].\n\nThe maxima restricted to primitive words are\n[0,0,1,2,3,3,4,6,6,7,8,10,11,11,13,15,15,17,17,19,21,20,22,24,24,26].\nFor nonprimitive words they are undefined at n=1 and then\n[1,1,2,2,4,3,6,4,9,5,9,6,13,9,16,8,18,9,19,13,22,11,25,14,28].\n\nThe maxima restricted to square factors of total length at most n/2 are\n[0,0,0,2,2,2,2,4,4,5,5,9,8,10,9,12,12,13,13,16,16,17,18,19,20,20].\nThe complementary long-square maxima are\n[0,1,1,2,2,3,3,4,4,6,7,6,7,9,11,10,11,14,13,13,17,16,17,20,19,20].\n\nAt n=16 and n=24, the exact total maxima 16 and 25 are attained by rotations of the k=0 and k=1 words in Amit and Gawrychowski's 2017 family. Their Lemma 1 gives 10k+16-(k mod 2) squares at length 8k+16. The region beyond n=14 is binary-only and makes no claim about larger alphabets.",
  "status": "supported",
  "evidence_grade": "computational",
  "scope": {
    "kind": "bounded",
    "statement": "every binary word of lengths 1 through 26, with complements identified",
    "bounds": {
      "word_length": {
        "min": 1,
        "max": 26
      },
      "alphabet_size": {
        "min": 2,
        "max": 2
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "locator": "Two independent exhaustive binary C++17 enumerations executed 2026-07-28; source-family comparison against arXiv:1708.00639, Section 3, Lemma 1"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": null,
    "locator": "Two independent exhaustive binary C++17 enumerations executed 2026-07-28; source-family comparison against arXiv:1708.00639, Section 3, Lemma 1"
  },
  "models": [],
  "relations": [
    {
      "slug": "R125",
      "title": "Packed binary enumeration through length twenty-six",
      "object_type": "artifact",
      "relation": "evidences",
      "direction": "incoming"
    },
    {
      "slug": "R124",
      "title": "Direct-string independent binary replay",
      "object_type": "artifact",
      "relation": "evidences",
      "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"
    }
  ]
}