{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R757",
  "content_hash": null,
  "slug": "ssp-claim-zero-based-ancestor",
  "type": "claim",
  "title": "Schwartz proved the zero-based ancestor in 1978",
  "summary": "For every \\(m\\geq0\\), the set \\(\\{0,\\ldots,m\\}\\) has a permutation whose termwise sums are squares.",
  "relevance": "This records a result, observation, or conjecture used to assess the target problem.",
  "relevance_source": "derived",
  "body": "Benjamin Schwartz introduced square permutations in Mathematics Magazine and proved existence on every zero-based initial interval. The indexing changes the answer: the zero-based problem has no exceptional orders, while the one-based candidate has six. Both use the same reflection construction. This source also separates the entry from the square-sum sequencing puzzle, which asks for a Hamiltonian path rather than a bipartite perfect matching.",
  "status": "established",
  "evidence_grade": "sourced",
  "scope": {
    "kind": "universal",
    "statement": "every finite initial interval indexed from zero"
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://doi.org/10.1080/0025570X.1978.11976678",
      "locator": "Benjamin L. Schwartz, Square Permutations, Mathematics Magazine 51 (1978), pages 64-66"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.1080/0025570X.1978.11976678",
    "locator": "Benjamin L. Schwartz, Square Permutations, Mathematics Magazine 51 (1978), pages 64-66"
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R756",
      "title": "The six listed orders are the exact exception set",
      "object_type": "claim",
      "relation": "informs",
      "direction": "outgoing"
    },
    {
      "slug": "square-sum-permutation-existence",
      "title": "square sum permutation existence",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}