{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R806",
  "content_hash": null,
  "slug": "tsp-claim-sun-near-variant",
  "type": "claim",
  "title": "Sun proved a nearby involution theorem",
  "summary": "Every order admits an involution whose termwise sums are either 2 or triangular.",
  "relevance": "This records a result, observation, or conjecture used to assess the target problem.",
  "relevance_source": "derived",
  "body": "Sun's Corollary 1.2 gives an involution \\(\\sigma\\) on every \\([n]\\) such that each \\(k+\\sigma(k)\\) belongs to \\(\\{2\\}\\cup\\{T_j:j\\geq2\\}\\). Remark 1.2 identifies order 4 as a case where the sum 2 is needed. This result supplies the reflection method and explains the exceptional behavior, though it does not state the candidate's exact one-based classification.",
  "status": "established",
  "evidence_grade": "sourced",
  "scope": {
    "kind": "universal",
    "statement": "every positive integer n for Sun's variant allowing sums 2 or triangular"
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://doi.org/10.1007/s10801-021-01028-8",
      "locator": "Zhi-Wei Sun, Journal of Algebraic Combinatorics 54 (2021), Corollary 1.2 and Remark 1.2, page 895"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.1007/s10801-021-01028-8",
    "locator": "Zhi-Wei Sun, Journal of Algebraic Combinatorics 54 (2021), Corollary 1.2 and Remark 1.2, page 895"
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R805",
      "title": "Orders 1 and 4 are the exact exceptions",
      "object_type": "claim",
      "relation": "supports",
      "direction": "outgoing"
    },
    {
      "slug": "triangular-sum-permutation-existence",
      "title": "triangular sum permutation existence",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}