{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R805",
  "content_hash": null,
  "slug": "tsp-claim-exact-classification",
  "type": "claim",
  "title": "Orders 1 and 4 are the exact exceptions",
  "summary": "A triangular-sum permutation exists for every positive order except \\(1\\) and \\(4\\).",
  "relevance": "This records a result, observation, or conjecture used to assess the target problem.",
  "relevance_source": "derived",
  "body": "Let \\(T_k=k(k+1)/2\\). If a working permutation is known on \\([m]\\), and \\(T_k=m+n+1\\) with \\(m<n\\), keep that permutation on \\([m]\\) and set \\(\\sigma(i)=T_k-i\\) on \\([m+1,n]\\). The latter interval maps to itself and every new sum equals \\(T_k\\).\n\nWorking base orders are \\(0,2,3,5,10\\). At order 10, one witness is \\((1\\,2\\,4\\,6\\,9)(5\\,10)(7\\,8)\\), with 3 fixed. For \\(n\\geq6\\), take the least triangular number above \\(n\\) and set \\(m=T_k-n-1\\). This gives \\(0\\leq m<n\\). If \\(m\\) is 1 or 4, the next triangular number gives remainder \\(k+2\\) or \\(k+5\\), respectively; these are earlier admissible orders, apart from the base case \\(n=10\\). Strong induction finishes the construction.\n\nOrder 1 fails because its sole sum is 2. At order 4, left vertices 1 and 4 both have the sole neighbor 2, so Hall's condition fails. The proof is an editorial strengthening of Sun's reflection construction and should receive independent review before the problem state changes.",
  "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://arxiv.org/abs/1811.10503",
      "locator": "Zhi-Wei Sun, On permutations of {1,...,n} and related topics, Theorem 1.1, pages 894-897; strengthened argument recorded here"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://arxiv.org/abs/1811.10503",
    "locator": "Zhi-Wei Sun, On permutations of {1,...,n} and related topics, Theorem 1.1, pages 894-897; strengthened argument recorded here"
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R806",
      "title": "Sun proved a nearby involution theorem",
      "object_type": "claim",
      "relation": "supports",
      "direction": "incoming"
    },
    {
      "slug": "R804",
      "title": "Exact triangular matching sweep through order 260",
      "object_type": "artifact",
      "relation": "tests",
      "direction": "incoming"
    },
    {
      "slug": "triangular-sum-permutation-existence",
      "title": "triangular sum permutation existence",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}