{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R744",
  "content_hash": null,
  "slug": "ssclt-claim-exact-coefficient-test",
  "type": "claim",
  "title": "Each monotonicity comparison reduces to an integer inequality",
  "summary": "A central coefficient recurrence gives the exact probability, and squaring clears every square root and power-of-two denominator.",
  "relevance": "For Eventual monotonicity in a signed subset-sum local limit, record ssclt-claim-exact-coefficient-test (“Each monotonicity comparison reduces to an integer inequality”) records a bound, answer, status fact, or structural consequence. The record states: A central coefficient recurrence gives the exact probability, and squaring clears every square root and power-of-two denominator.",
  "relevance_source": "recorded",
  "body": "Let\n\\[\nF_n(x)=\\prod_{k=1}^n(1+x^k)=\\sum_s c_{n,s}x^s,\n\\qquad T_n=\\frac{n(n+1)}2.\n\\]\nChoosing the indices assigned sign \\(+1\\) shows\n\\[\nS_n=0\\quad\\Longleftrightarrow\\quad \\sum_{k:\\varepsilon_k=1}k=\\frac{T_n}{2}.\n\\]\nHence, for admissible \\(n\\),\n\\[\nC_n=c_{n,T_n/2}=[x^{n(n+1)/4}]F_n(x),\n\\qquad \\Pr(S_n=0)=\\frac{C_n}{2^n}.\n\\]\nThe coefficients obey the exact subset-sum recurrence\n\\[\nc_{n,s}=c_{n-1,s}+c_{n-1,s-n},\n\\]\nwith \\(c_{0,0}=1\\). Also\n\\[\nQ_n=\\sigma_n^2=\\frac{n(n+1)(2n+1)}6.\n\\]\nFor consecutive admissible indices \\(a<b\\), all quantities are positive. Squaring and clearing denominators gives\n\\[\nA_b>A_a\n\\quad\\Longleftrightarrow\\quad\nC_b^2Q_b-C_a^2Q_a4^{b-a}>0.\n\\]\nThis equivalence is the comparison used by the executable artifact. Every decision is an integer sign test.",
  "status": "established",
  "evidence_grade": "mathematical_identity",
  "scope": {
    "kind": "universal",
    "statement": "every positive integer n and every consecutive pair of admissible indices"
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://oeis.org/A063865",
      "locator": "A063865 gives the sign-count and central-coefficient interpretations; the recurrence and cleared comparison are derived directly here"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://oeis.org/A063865",
    "locator": "A063865 gives the sign-count and central-coefficient interpretations; the recurrence and cleared comparison are derived directly here"
  },
  "models": [],
  "relations": [
    {
      "slug": "R745",
      "title": "All 492 comparisons through n=1000 are strict increases",
      "object_type": "claim",
      "relation": "supports",
      "direction": "outgoing"
    },
    {
      "slug": "signed-subset-sum-local-clt-monotone",
      "title": "signed subset sum local clt monotone",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}