{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R596",
  "content_hash": null,
  "slug": "pp12-claim-five-mols-lower-bound",
  "type": "claim",
  "title": "Five order-12 MOLS are constructed, while eleven would settle the problem",
  "summary": "A 1960 construction proves N(12) at least 5; the universal upper bound is 11, and equality is equivalent to an order-12 plane.",
  "relevance": "For A projective plane of order 12, record pp12-claim-five-mols-lower-bound (“Five order-12 MOLS are constructed, while eleven would settle the problem”) records a bound, answer, status fact, or structural consequence. The record states: A 1960 construction proves N(12) at least 5; the universal upper bound is 11, and equality is equivalent to an order-12 plane.",
  "relevance_source": "recorded",
  "body": "Let \\(N(12)\\) be the largest size of a pairwise mutually orthogonal family of Latin squares of order 12. Bose, Chakravarti, and Knuth explicitly construct two families of five in their 1960 paper, proving\n\\[\nN(12)\\ge5.\n\\]\nThe universal bound gives \\(N(12)\\le11\\). A family of 11 is a complete family, equivalent to an affine plane and hence to a projective plane of order 12.\n\nThus the published five-square construction supplies a concrete partial object. Extending any compatible family to 11 would prove existence. A proof that all complete-family branches fail would prove nonexistence.",
  "status": "established",
  "evidence_grade": "sourced",
  "scope": {
    "kind": "bounded",
    "statement": "the maximum number N(12) of pairwise mutually orthogonal Latin squares of order 12",
    "bounds": {
      "order": {
        "min": 12,
        "max": 12
      },
      "constructed_squares": {
        "min": 5,
        "max": 5
      },
      "complete_set_size": {
        "min": 11,
        "max": 11
      }
    },
    "exhaustive": false
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://doi.org/10.1080/00401706.1960.10489916",
      "locator": "R. C. Bose, I. M. Chakravarti, and D. E. Knuth, On Methods of Constructing Sets of Mutually Orthogonal Latin Squares Using a Computer. I, Technometrics 2(4) (1960), 507-516, abstract, Figure 1, and Table 2"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.1080/00401706.1960.10489916",
    "locator": "R. C. Bose, I. M. Chakravarti, and D. E. Knuth, On Methods of Constructing Sets of Mutually Orthogonal Latin Squares Using a Computer. I, Technometrics 2(4) (1960), 507-516, abstract, Figure 1, and Table 2"
  },
  "models": [],
  "relations": [
    {
      "slug": "R597",
      "title": "Existence at order 12 remains open",
      "object_type": "claim",
      "relation": "informs",
      "direction": "outgoing"
    },
    {
      "slug": "projective-plane-order-12",
      "title": "projective plane order 12",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}