{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R743",
  "content_hash": null,
  "slug": "ssclt-claim-all-n-status-unresolved",
  "type": "claim",
  "title": "The all-n monotonicity claim remains unresolved in this audit",
  "summary": "Exact computation proves the claim through n=1000; the located asymptotic theorem gives convergence without a termwise inequality or an effective threshold.",
  "relevance": "For Eventual monotonicity in a signed subset-sum local limit, record ssclt-claim-all-n-status-unresolved (“The all-n monotonicity claim remains unresolved in this audit”) records a bound, answer, status fact, or structural consequence. The record states: Exact computation proves the claim through n=1000; the located asymptotic theorem gives convergence without a termwise inequality or an effective threshold.",
  "relevance_source": "recorded",
  "body": "Write\n\\[\nA_n=\\sigma_n\\Pr(S_n=0),\\qquad \\sigma_n^2=\\sum_{k=1}^n k^2.\n\\]\nThe targeted search located the exact coefficient sequence, a constructive growth bound, and the proved first-order asymptotic. It found no paper proving that \\(A_n\\) increases at every consecutive admissible index after 16.\n\nSullivan proves\n\\[\nC_n\\sim \\sqrt{\\frac6\\pi}\\,2^n n^{-3/2}\n\\]\nfor \\(n\\equiv0,3\\pmod4\\), where \\(C_n\\) is the zero-sum sign count. Since \\(\\sigma_n\\sim n^{3/2}/\\sqrt3\\), this yields \\(A_n\\to\\sqrt{2/\\pi}\\). Convergence to the limit permits occasional decreases and supplies no finite threshold. A formal local Edgeworth calculation gives the first correction\n\\[\nA_n=\\sqrt{\\frac2\\pi}\\left(1-\\frac9{20n}+O(n^{-2})\\right),\n\\]\nwhich predicts eventual increase. The displayed error order is still the same size as a consecutive difference. A proof for all \\(n\\ge16\\) needs a sharper effective remainder, or a direct coefficient inequality, joined to the finite computation.\n\nThe companion artifact establishes all 492 requested comparisons whose larger endpoint is at most 1000. The infinite tail remains open in this record.",
  "status": "reported",
  "evidence_grade": "sourced",
  "scope": {
    "kind": "bounded",
    "statement": "the literature and computational status of strict increase of sigma_n Pr(S_n=0) over every admissible n at least 16, checked on 2026-07-24",
    "bounds": {
      "starting_n": {
        "min": 16,
        "max": 16
      }
    },
    "exhaustive": false
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://cs.uwaterloo.ca/journals/JIS/VOL16/Sullivan/sullivan8.html",
      "locator": "Blair D. Sullivan, On a Conjecture of Andrica and Tomescu, Journal of Integer Sequences 16 (2013), Article 13.3.1, Theorem 4 and its proof; targeted 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": "Blair D. Sullivan, On a Conjecture of Andrica and Tomescu, Journal of Integer Sequences 16 (2013), Article 13.3.1, Theorem 4 and its proof; targeted search performed 2026-07-24"
  },
  "models": [],
  "relations": [
    {
      "slug": "R745",
      "title": "All 492 comparisons through n=1000 are strict increases",
      "object_type": "claim",
      "relation": "supports",
      "direction": "incoming"
    },
    {
      "slug": "R742",
      "title": "Published growth and asymptotic results stop short of the requested comparison",
      "object_type": "attempt",
      "relation": "informs",
      "direction": "incoming"
    },
    {
      "slug": "R741",
      "title": "Bit-packed exact subset-sum computation through n=1000",
      "object_type": "artifact",
      "relation": "tests",
      "direction": "incoming"
    },
    {
      "slug": "signed-subset-sum-local-clt-monotone",
      "title": "signed subset sum local clt monotone",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}