{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R519",
  "content_hash": null,
  "slug": "mgsl-claim-certified-window",
  "type": "claim",
  "title": "The maximum girth is currently certified between 17 and 24",
  "summary": "An explicit generating pair attains girth 17, while the degree-four Moore bound rules out girth 25 and above.",
  "relevance": "For Largest girth from two generators of SL(2,101), record mgsl-claim-certified-window (“The maximum girth is currently certified between 17 and 24”) records a bound, answer, status fact, or structural consequence. The record states: An explicit generating pair attains girth 17, while the degree-four Moore bound rules out girth 25 and above.",
  "relevance_source": "recorded",
  "body": "Write \\(M\\) for the maximum in the problem. The pair\n\\[\nA=\\begin{pmatrix}69&82\\\\70&51\\end{pmatrix},\\qquad\nB=\\begin{pmatrix}41&48\\\\47&92\\end{pmatrix}\n\\quad(\\bmod 101)\n\\]\ngenerates all 1,030,200 elements of \\(\\mathrm{SL}_2(\\mathbf F_{101})\\), and its undirected Cayley graph has girth exactly 17. This proves \\(M\\geq17\\).\n\nEvery graph under consideration is 4-regular on\n\\[\n101(101^2-1)=1{,}030{,}200\n\\]\nvertices. A 4-regular graph of girth at least 25 would contain a tree ball with\n\\[\n1+4\\sum_{i=0}^{11}3^i=1{,}062{,}881\n\\]\ndistinct vertices. This exceeds the group order, so \\(M\\leq24\\). Together,\n\\[\n17\\leq M\\leq24.\n\\]\nThe upper bound ranges over every eligible pair. The lower-bound computation certifies one pair and does not enumerate simultaneous-conjugacy classes of all pairs.",
  "status": "supported",
  "evidence_grade": "supported",
  "scope": {
    "kind": "bounded",
    "statement": "all ordered pairs (A,B) in SL_2(F_101)^2 that generate SL_2(F_101) and have four distinct elements A, A^{-1}, B, B^{-1}",
    "bounds": {
      "group_order": {
        "min": 1030200,
        "max": 1030200
      },
      "cayley_degree": {
        "min": 4,
        "max": 4
      }
    },
    "exhaustive": false
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://arxiv.org/abs/0707.1833",
      "locator": "Exact lower-bound certificate in mgsl-artifact-word-and-group-verifier; Moore counting argument included here"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://arxiv.org/abs/0707.1833",
    "locator": "Exact lower-bound certificate in mgsl-artifact-word-and-group-verifier; Moore counting argument included here"
  },
  "models": [],
  "relations": [
    {
      "slug": "R521",
      "title": "The displayed generating pair has girth exactly 17",
      "object_type": "claim",
      "relation": "supports",
      "direction": "incoming"
    },
    {
      "slug": "R520",
      "title": "The Moore bound gives a universal ceiling of 24",
      "object_type": "claim",
      "relation": "bounds",
      "direction": "incoming"
    },
    {
      "slug": "R518",
      "title": "Large-girth literature gives asymptotic context",
      "object_type": "attempt",
      "relation": "informs",
      "direction": "incoming"
    },
    {
      "slug": "max-girth-sl2-101-generators",
      "title": "max girth sl2 101 generators",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}