{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R535",
  "content_hash": null,
  "slug": "mols10-claim-published-near-triple",
  "type": "claim",
  "title": "The published near triple contains exactly two orthogonal pairs",
  "summary": "All three arrays are Latin; the A-B and A-C overlays have 100 distinct pairs, while the B-C overlay has 91.",
  "relevance": "For Three mutually orthogonal Latin squares of order ten, record mols10-claim-published-near-triple (“The published near triple contains exactly two orthogonal pairs”) records a bound, answer, status fact, or structural consequence. The record states: All three arrays are Latin; the A-B and A-C overlays have 100 distinct pairs, while the B-C overlay has 91.",
  "relevance_source": "recorded",
  "body": "Egan and Wanless publish three order-10 Latin squares \\(A,B,C\\) in their Section 8 and on Wanless's companion-data page. The public file uses letters `a` through `j` for symbols 0 through 9 and has SHA-256 digest `04db7b4b17a2321b754b694b6b3f894dd5181808a4580dbd6b3a42587dfb488a`.\n\nThe executable replay checks every row and column of all three squares. It then checks all 100 cells in each overlay. The \\((A,B)\\) and \\((A,C)\\) overlays each contain all 100 ordered symbol pairs. The \\((B,C)\\) overlay contains 91 distinct pairs. Its nine repeated pairs are `bi`, `cj`, `dh`, `ei`, `fj`, `gh`, `hi`, `ij`, and `jh`, each occurring twice. Its nine missing pairs are `bj`, `ch`, `di`, `ej`, `fh`, `gi`, `hj`, `ih`, and `ji`.\n\nThus the file supplies two explicit 2 MOLS(10) witnesses sharing \\(A\\). The collisions in the third overlay keep it outside the acceptance condition for a triple.",
  "status": "supported",
  "evidence_grade": "computational",
  "scope": {
    "kind": "bounded",
    "statement": "the first three complete Latin squares in the public neartripleMOLS10.txt file",
    "bounds": {
      "order": {
        "min": 10,
        "max": 10
      },
      "square_count": {
        "min": 3,
        "max": 3
      },
      "cells_checked": {
        "min": 300,
        "max": 300
      },
      "pair_overlays_checked": {
        "min": 3,
        "max": 3
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://users.monash.edu.au/~iwanless/data/MOLS/neartripleMOLS10.txt",
      "locator": "Judith Egan and Ian M. Wanless, Enumeration of MOLS of small order, Mathematics of Computation 85(298) (2016), 799-824, DOI 10.1090/mcom/3010, Section 8 (Order 10), displayed squares A, B, C and the following paragraph; public companion file neartripleMOLS10.txt"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://users.monash.edu.au/~iwanless/data/MOLS/neartripleMOLS10.txt",
    "locator": "Judith Egan and Ian M. Wanless, Enumeration of MOLS of small order, Mathematics of Computation 85(298) (2016), 799-824, DOI 10.1090/mcom/3010, Section 8 (Order 10), displayed squares A, B, C and the following paragraph; public companion file neartripleMOLS10.txt"
  },
  "relations": [
    {
      "slug": "R534",
      "title": "Existence of three MOLS of order 10 remains open",
      "object_type": "claim",
      "relation": "informs",
      "direction": "outgoing"
    },
    {
      "slug": "R533",
      "title": "Exact replay of the order-10 near triple",
      "object_type": "artifact",
      "relation": "verifies",
      "direction": "incoming"
    },
    {
      "slug": "three-mols-order-10",
      "title": "three mols order 10",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}