{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R521",
  "content_hash": null,
  "slug": "mgsl-claim-witness-girth-seventeen",
  "type": "claim",
  "title": "The displayed generating pair has girth exactly 17",
  "summary": "All reduced words through length eight have distinct values, and a reduced relation of length 17 closes.",
  "relevance": "For Largest girth from two generators of SL(2,101), record mgsl-claim-witness-girth-seventeen (“The displayed generating pair has girth exactly 17”) records a bound, answer, status fact, or structural consequence. The record states: All reduced words through length eight have distinct values, and a reduced relation of length 17 closes.",
  "relevance_source": "recorded",
  "body": "Use lowercase letters for inverses. Exact multiplication modulo 101 gives\n\\[\nA^4BA B^{-1}A^{-2}B^{-2}A^{-1}B^5=I.\n\\]\nIn letter form, one such relation is `AAAABAbaabbaBBBBB`, of length 17.\n\nFor the lower bound, enumerate every freely reduced word of length at most eight. Their matrix values are all distinct. The layer sizes are\n\\[\n1,4,12,36,108,324,972,2916,8748,\n\\]\nwhose sum is 13,121, exactly the radius-eight ball in the 4-regular tree. Any relation of length at most 16 can be split into two words of length at most eight with the same matrix value. The injective word ball rules this out. An edge between two vertices in the outer layer supplies the displayed length-17 relation, so the girth is 17.\n\nA separate full BFS reaches 1,030,200 determinant-one matrices. Since\n\\(|\\mathrm{SL}_2(\\mathbf F_{101})|=101(101^2-1)=1{,}030{,}200\\), the matrices generate the stated group. Their inverses and one another are pairwise distinct, so the Cayley graph has degree four.",
  "status": "established",
  "evidence_grade": "computational",
  "scope": {
    "kind": "bounded",
    "statement": "the single Cayley graph of SL_2(F_101) generated by the displayed matrices A and B",
    "bounds": {
      "word_length_checked": {
        "min": 0,
        "max": 17
      },
      "radius_eight_ball_vertices": {
        "min": 13121,
        "max": 13121
      },
      "group_vertices_reached": {
        "min": 1030200,
        "max": 1030200
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://arxiv.org/abs/0707.1833",
      "locator": "Reproduced by mgsl-artifact-word-and-group-verifier on 2026-07-25"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://arxiv.org/abs/0707.1833",
    "locator": "Reproduced by mgsl-artifact-word-and-group-verifier on 2026-07-25"
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R519",
      "title": "The maximum girth is currently certified between 17 and 24",
      "object_type": "claim",
      "relation": "supports",
      "direction": "outgoing"
    },
    {
      "slug": "R517",
      "title": "Exact word-ball and full-group verifier",
      "object_type": "artifact",
      "relation": "evidences",
      "direction": "incoming"
    },
    {
      "slug": "max-girth-sl2-101-generators",
      "title": "max girth sl2 101 generators",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}