{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R597",
  "content_hash": null,
  "slug": "pp12-claim-open-status",
  "type": "claim",
  "title": "Existence at order 12 remains open",
  "summary": "The literature and catalogue audit found no accepted construction or complete exclusion for a projective plane of order 12.",
  "relevance": "For A projective plane of order 12, record pp12-claim-open-status (“Existence at order 12 remains open”) records a bound, answer, status fact, or structural consequence. The record states: The literature and catalogue audit found no accepted construction or complete exclusion for a projective plane of order 12.",
  "relevance_source": "recorded",
  "body": "A projective plane of order 12 would have 157 points and 157 lines. Every line would contain 13 points, every point would lie on 13 lines, and each pair of points would determine one line. Equivalently, it would be a symmetric \\(2\\text{-}(157,13,1)\\) design.\n\nAkiyama, Suetake, and Tanaka state in their 2019 primary paper that the order-12 existence question is still unknown. Their exhaustive computation excludes collineation groups of order 9. Their 2023 sequel sharpens the symmetry restriction to collineation-group orders 1, 2, or 3 while leaving the plane itself unresolved. Kharaghani and Suda give a 2023 equivalence with a balancedly multi-splittable quaternary Hadamard matrix of order 144, which supplies another exact target rather than a resolution.\n\nThe research check through 2026-07-25 found no later accepted construction or nonexistence theorem. The answer recorded here is therefore open.",
  "status": "open",
  "evidence_grade": "sourced",
  "scope": {
    "kind": "bounded",
    "statement": "existence of a finite projective plane of order 12, equivalently a symmetric 2-(157,13,1) design",
    "bounds": {
      "order": {
        "min": 12,
        "max": 12
      },
      "points": {
        "min": 157,
        "max": 157
      },
      "lines": {
        "min": 157,
        "max": 157
      }
    },
    "exhaustive": false
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://ajc.maths.uq.edu.au/pdf/74/ajc_v74_p112.pdf",
      "locator": "Kenzi Akiyama, Chihiro Suetake, and Masaki Tanaka, The nonexistence of projective planes of order 12 with a collineation group of order 9, Australasian Journal of Combinatorics 74(1) (2019), 112-160, Introduction and main theorem; status cross-checked against the 2023 sources listed in metadata"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://ajc.maths.uq.edu.au/pdf/74/ajc_v74_p112.pdf",
    "locator": "Kenzi Akiyama, Chihiro Suetake, and Masaki Tanaka, The nonexistence of projective planes of order 12 with a collineation group of order 9, Australasian Journal of Combinatorics 74(1) (2019), 112-160, Introduction and main theorem; status cross-checked against the 2023 sources listed in metadata"
  },
  "models": [],
  "relations": [
    {
      "slug": "R594",
      "title": "Bruck-Ryser gives no obstruction at order 12",
      "object_type": "claim",
      "relation": "informs",
      "direction": "incoming"
    },
    {
      "slug": "R595",
      "title": "Every collineation group has order 1, 2, or 3",
      "object_type": "claim",
      "relation": "constrains",
      "direction": "incoming"
    },
    {
      "slug": "R596",
      "title": "Five order-12 MOLS are constructed, while eleven would settle the problem",
      "object_type": "claim",
      "relation": "informs",
      "direction": "incoming"
    },
    {
      "slug": "R592",
      "title": "Exact OA and 11-MOLS computational formulation",
      "object_type": "artifact",
      "relation": "reformulates",
      "direction": "incoming"
    },
    {
      "slug": "R593",
      "title": "Primary literature and small-plane catalogue audit",
      "object_type": "attempt",
      "relation": "supports",
      "direction": "incoming"
    },
    {
      "slug": "projective-plane-order-12",
      "title": "projective plane order 12",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}