{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R756",
  "content_hash": null,
  "slug": "ssp-claim-exact-classification",
  "type": "claim",
  "title": "The six listed orders are the exact exception set",
  "summary": "A square-sum permutation exists exactly when \\(n\\notin\\{1,2,4,6,7,11\\}\\).",
  "relevance": "This records a result, observation, or conjecture used to assess the target problem.",
  "relevance_source": "derived",
  "body": "Robert Israel gave a complete construction and obstruction argument in 2012. Suppose a working permutation is known on \\([r]\\). Choose a square \\(q\\) and set \\(r=q-n-1\\), with \\(0\\leq r<n\\). Keep the permutation on \\([r]\\), and set \\(\\sigma(i)=q-i\\) on \\([r+1,n]\\). The latter interval maps to itself and every new sum is \\(q\\).\n\nStrong induction uses seven base witnesses at \\(n=3,5,8,9,10,13,17\\). For \\(n\\geq12\\), the square immediately above \\(n\\), or the following square in the six exceptional remainder cases, reduces the order to an earlier admissible one.\n\nHall's theorem proves failure at the six exceptions. Deficient left sets are \\(\\{1\\}\\) for \\(n=1,2\\), \\(\\{4\\}\\) for \\(n=4\\), \\(\\{1,6\\}\\) for \\(n=6,7\\), and \\(\\{4,11\\}\\) for \\(n=11\\). Their neighbor sets have sizes \\(0,0,0,1,1,1\\), respectively. The candidate's proposed classification is therefore a solved theorem.",
  "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://mathoverflow.net/a/117234",
      "locator": "Robert Israel, MathOverflow answer 117234, 26 December 2012; construction, base permutations, and Hall obstructions"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://mathoverflow.net/a/117234",
    "locator": "Robert Israel, MathOverflow answer 117234, 26 December 2012; construction, base permutations, and Hall obstructions"
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R757",
      "title": "Schwartz proved the zero-based ancestor in 1978",
      "object_type": "claim",
      "relation": "informs",
      "direction": "incoming"
    },
    {
      "slug": "R755",
      "title": "Exact perfect-matching sweep through order 260",
      "object_type": "artifact",
      "relation": "tests",
      "direction": "incoming"
    },
    {
      "slug": "square-sum-permutation-existence",
      "title": "square sum permutation existence",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}