{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R809",
  "content_hash": null,
  "slug": "tspu-claim-complete-exceptions",
  "type": "claim",
  "title": "The complete failure set is 2, 4, 6, 8, 9, and 10",
  "summary": "Almkvist's theorem and exact small cases classify every positive order.",
  "relevance": "For Eventual unimodality of ternary subset-sum polynomials, record tspu-claim-complete-exceptions (“The complete failure set is 2, 4, 6, 8, 9, and 10”) records a bound, answer, status fact, or structural consequence. The record states: Almkvist's theorem and exact small cases classify every positive order.",
  "relevance_source": "recorded",
  "body": "Exact convolution shows that \\(P_n\\) is unimodal for \\(n=1,3,5,7\\) and fails for \\(n=2,4,6,8,9,10\\). The descents on the increasing half occur after exponents 2, 8, 18 and 20, 34, 44, and 54, respectively. The corresponding coefficient drops are\n\\[\n2>1,\\quad7>6,\\quad34>33,\\quad36>35,\\quad214>213,\\quad549>547,\\quad1423>1417.\n\\]\nAt \\(n=10\\), the eleven central coefficients are\n\\[\n1367,1379,1404,1408,1423,1417,1423,1408,1404,1379,1367.\n\\]\nAt \\(n=11\\), the central valley has disappeared; the corresponding window is\n\\[\n3608,3657,3684,3715,3723,3735,3723,3715,3684,3657,3608.\n\\]\nAlmkvist's theorem covers every \\(n\\geq11\\). Therefore the six listed orders are all the failures.",
  "status": "established",
  "evidence_grade": "sourced",
  "scope": {
    "kind": "universal",
    "statement": "every positive integer n"
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://doi.org/10.1016/0022-314X(89)90096-6",
      "locator": "Almkvist's all-n result for n at least 11, combined with the exact finite computation in tspu-artifact-sweep-300"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.1016/0022-314X(89)90096-6",
    "locator": "Almkvist's all-n result for n at least 11, combined with the exact finite computation in tspu-artifact-sweep-300"
  },
  "models": [],
  "relations": [
    {
      "slug": "R808",
      "title": "Almkvist's theorem settles every n at least 11",
      "object_type": "claim",
      "relation": "supports",
      "direction": "incoming"
    },
    {
      "slug": "R807",
      "title": "Exact coefficient sweep through n = 300",
      "object_type": "artifact",
      "relation": "evidences",
      "direction": "incoming"
    },
    {
      "slug": "ternary-subset-polynomial-unimodality",
      "title": "ternary subset polynomial unimodality",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}