{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R742",
  "content_hash": null,
  "slug": "ssclt-attempt-literature-audit",
  "type": "attempt",
  "title": "Published growth and asymptotic results stop short of the requested comparison",
  "summary": "The located papers identify the coefficient and its limit; neither supplies strict normalized monotonicity from n=16.",
  "relevance": "For Eventual monotonicity in a signed subset-sum local limit, record ssclt-attempt-literature-audit (“Published growth and asymptotic results stop short of the requested comparison”) documents a concrete method, search boundary, or failed route. The record states: The located papers identify the coefficient and its limit; neither supplies strict normalized monotonicity from n=16.",
  "relevance_source": "recorded",
  "body": "Andrica and Tomescu identify \\(C_n\\) as the middle coefficient of \\(\\prod_{k=1}^n(1+x^k)\\), derive an integral representation, and prove the constructive bound \\(C_n\\geq6C_{n-4}\\) for \\(n\\geq8\\). Their bound compares indices four apart and concerns the unnormalized count.\n\nSullivan proves the Andrica-Tomescu asymptotic by Laplace's method. The proof separates a neighborhood of zero in the cosine-product integral and shows the remaining integral is lower order. Its conclusion is a first-order equivalence as \\(n\\to\\infty\\). The paper states no effective error bound that decides each adjacent admissible comparison.\n\nOEIS A063865 records the exact sequence and points to both papers. Searches for the sequence identifier, normalized central coefficients, weighted Rademacher local limits, and monotonicity found no direct theorem for the present claim. This is a targeted audit rather than a proof of novelty.",
  "status": "inconclusive",
  "evidence_grade": "sourced",
  "scope": {
    "kind": "bounded",
    "statement": "published results directly concerning the central coefficient of product from k=1 to n of (1+x^k), its asymptotics, and monotonicity",
    "bounds": {
      "search_date": {
        "min": 20260724,
        "max": 20260724
      }
    },
    "exhaustive": false
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "attempt",
    "citation": {
      "url": "https://cs.uwaterloo.ca/journals/JIS/VOL16/Sullivan/sullivan8.html",
      "locator": "Andrica and Tomescu, Journal of Integer Sequences 5 (2002), Article 02.2.4; Sullivan, Journal of Integer Sequences 16 (2013), Article 13.3.1; OEIS A063865; search performed 2026-07-24"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://cs.uwaterloo.ca/journals/JIS/VOL16/Sullivan/sullivan8.html",
    "locator": "Andrica and Tomescu, Journal of Integer Sequences 5 (2002), Article 02.2.4; Sullivan, Journal of Integer Sequences 16 (2013), Article 13.3.1; OEIS A063865; search performed 2026-07-24"
  },
  "models": [],
  "relations": [
    {
      "slug": "R743",
      "title": "The all-n monotonicity claim remains unresolved in this audit",
      "object_type": "claim",
      "relation": "informs",
      "direction": "outgoing"
    },
    {
      "slug": "signed-subset-sum-local-clt-monotone",
      "title": "signed subset sum local clt monotone",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}