{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R518",
  "content_hash": null,
  "slug": "mgsl-attempt-literature-audit",
  "type": "attempt",
  "title": "Large-girth literature gives asymptotic context",
  "summary": "The audit found general constructions and random-generator theorems, with no exact optimization at q=101.",
  "relevance": "For Largest girth from two generators of SL(2,101), record mgsl-attempt-literature-audit (“Large-girth literature gives asymptotic context”) documents a concrete method, search boundary, or failed route. The record states: The audit found general constructions and random-generator theorems, with no exact optimization at q=101.",
  "relevance_source": "recorded",
  "body": "Lubotzky, Phillips, and Sarnak construct arithmetic Cayley graphs with asymptotically large girth. Their groups are \\(\\mathrm{PSL}_2\\) or \\(\\mathrm{PGL}_2\\) for selected arithmetic parameters, and their result does not enumerate two-generator pairs in \\(\\mathrm{SL}_2(\\mathbf F_{101})\\).\n\nGamburd, Hoory, Shahshahani, Shalev, and Virag prove asymptotic lower bounds for the girth of random Cayley graphs of simple algebraic groups. Their \\(\\mathrm{SL}_2\\) result concerns probability as the field grows. Bourgain and Gamburd use logarithmic girth as an input to uniform expansion results. Neither paper reports an exact maximum for this finite group.\n\nThe targeted search covered the exact group name, the prime 101, two-generator Cayley graphs, random Cayley girth, LPS graphs, and computational girth tables. No primary source giving the value of \\(M\\) or improving the finite ceiling 24 was located. This is a search report dated 2026-07-25, so novelty remains unverified.",
  "status": "inconclusive",
  "evidence_grade": "sourced",
  "scope": {
    "kind": "bounded",
    "statement": "the exact maximum-girth problem for four-regular Cayley graphs of SL_2(F_101) obtained from two generators"
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "attempt",
    "citation": {
      "url": "https://arxiv.org/abs/0707.1833",
      "locator": "Gamburd et al., Random Structures & Algorithms 35 (2009), 100-117, especially Theorem 8; Lubotzky, Phillips, and Sarnak, Combinatorica 8 (1988), 261-277, DOI 10.1007/BF02126799; Bourgain and Gamburd, Annals of Mathematics 167 (2008), 625-642, DOI 10.4007/annals.2008.167.625"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://arxiv.org/abs/0707.1833",
    "locator": "Gamburd et al., Random Structures & Algorithms 35 (2009), 100-117, especially Theorem 8; Lubotzky, Phillips, and Sarnak, Combinatorica 8 (1988), 261-277, DOI 10.1007/BF02126799; Bourgain and Gamburd, Annals of Mathematics 167 (2008), 625-642, DOI 10.4007/annals.2008.167.625"
  },
  "models": [],
  "relations": [
    {
      "slug": "R519",
      "title": "The maximum girth is currently certified between 17 and 24",
      "object_type": "claim",
      "relation": "informs",
      "direction": "outgoing"
    },
    {
      "slug": "max-girth-sl2-101-generators",
      "title": "max girth sl2 101 generators",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}