{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R200",
  "content_hash": null,
  "slug": "dclp-claim-exact-decrease-through-5000",
  "type": "claim",
  "title": "Strict decrease holds exactly through n=5000",
  "summary": "Exact integer arithmetic proves \\(q_n<q_{n-1}\\) for every \\(31\\le n\\le5000\\); proving the same inequality for every \\(n\\ge5001\\) remains open.",
  "relevance": "For Eventual decrease for distinct cycle lengths in random permutations, record dclp-claim-exact-decrease-through-5000 (“Strict decrease holds exactly through n=5000”) records a bound, answer, status fact, or structural consequence. The record states: Exact integer arithmetic proves \\(q_n<q_{n-1}\\) for every \\(31\\le n\\le5000\\); proving the same inequality for every \\(n\\ge5001\\) remains open.",
  "relevance_source": "recorded",
  "body": "Set \\(D=5000!\\) and compute the coefficients of the finite product through degree 5000 after scaling by \\(D\\). Every scaled coefficient is an integer. Indeed, a term indexed by a set \\(A\\) of distinct positive integers with sum at most 5000 has denominator \\(\\prod_{a\\in A}a\\), which divides \\(5000!\\).\n\nThe descending update in the executable artifact constructs \\(c_n=Dq_n\\) exactly. Direct integer comparisons give\n\\[\nc_n<c_{n-1}\\qquad(31\\leq n\\leq5000).\n\\]\nAcross the whole prefix, the last index satisfying \\(c_n\\geq c_{n-1}\\) is \\(n=30\\). Thus \\(q_{30}>q_{29}\\), followed by all 4,970 strict decreases certified here.",
  "status": "established",
  "evidence_grade": "reproduced",
  "scope": {
    "kind": "bounded",
    "statement": "every comparison q_n < q_(n-1) for 31 <= n <= 5000",
    "bounds": {
      "n": {
        "min": 31,
        "max": 5000
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://oeis.org/A007838",
      "locator": "Exact computation in dclp-artifact-integer-prefix-certificate, reproduced 2026-07-24"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://oeis.org/A007838",
    "locator": "Exact computation in dclp-artifact-integer-prefix-certificate, reproduced 2026-07-24"
  },
  "models": [],
  "relations": [
    {
      "slug": "dclp-problem-eventual-strict-decrease",
      "title": "Does the distinct-cycle-length probability decrease after n=30?",
      "object_type": "problem",
      "relation": "supports",
      "direction": "outgoing"
    },
    {
      "slug": "R197",
      "title": "Exact scaled-integer coefficient certificate",
      "object_type": "artifact",
      "relation": "verifies",
      "direction": "incoming"
    },
    {
      "slug": "distinct-cycle-length-probability-decreasing",
      "title": "distinct cycle length probability decreasing",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}