{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R196",
  "content_hash": null,
  "slug": "dbm-claim-rank-smith-form",
  "type": "claim",
  "title": "Rank and Smith normal form follow from the determinant formula",
  "summary": "Even orders are unimodular; odd orders have rank \\(n-1\\) and one zero Smith invariant.",
  "relevance": "This records a result, observation, or conjecture used to assess the target problem.",
  "relevance_source": "derived",
  "body": "At even order, the determinant is a unit, so \\(A_n\\) has full rank and Smith normal form equal to the identity. At odd order \\(n=2m+1\\), the kernel argument gives rank at most \\(n-1\\). The leading \\(2m\\times2m\\) principal minor is \\(A_{2m}\\), whose determinant is a unit. Therefore the rank is exactly \\(n-1\\), and the Smith normal form is \\(\\operatorname{diag}(1,\\ldots,1,0)\\).",
  "status": "established",
  "evidence_grade": "sourced",
  "scope": {
    "kind": "universal",
    "statement": "every positive matrix order n"
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://arxiv.org/abs/2210.12913",
      "locator": "Immediate corollary of the exact determinant and unimodular-minor argument"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://arxiv.org/abs/2210.12913",
    "locator": "Immediate corollary of the exact determinant and unimodular-minor argument"
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R194",
      "title": "Exact determinant of the disjoint-binary-support matrix",
      "object_type": "claim",
      "relation": "uses",
      "direction": "outgoing"
    },
    {
      "slug": "disjoint-binary-support-determinant",
      "title": "disjoint binary support determinant",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}