{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R512",
  "content_hash": null,
  "slug": "mgpc-attempt-source-and-convention-audit",
  "type": "attempt",
  "title": "Dated source and convention audit",
  "summary": "The checked primary literature still treats universal connectivity as open for p>=5, while proving a finite effective remainder and several broad or bounded regions.",
  "relevance": "For Prime exceptions to connectivity of the Markoff graph, record mgpc-attempt-source-and-convention-audit (“Dated source and convention audit”) documents a concrete method, search boundary, or failed route. The record states: The checked primary literature still treats universal connectivity as open for p>=5, while proving a finite effective remainder and several broad or bounded regions.",
  "relevance_source": "recorded",
  "body": "The audit resolved the canonical equation, vertex convention, and Vieta-only edge set against the current primary sources. De Courcy-Ireland and Lee compute connectivity below 3000 for \\(p\\geq5\\) in the coefficient-three normalization, using their Dehn-twist presentation of the strong-approximation graph. Brown cites that computation for the Vieta graph below 3000. Multiplication of all coordinates by 3 conjugates the coefficient-three Vieta moves to the canonical coefficient-one moves for \\(p\\neq3\\). Brown proves connectivity below one million with an almost-linear criterion and defines the coefficient-one graph after assuming \\(p>2\\). Eddy et al. prove connectivity for every prime above \\(3.448\\times10^{392}\\) and give the maximal-divisor criterion used here. Chen proves all but finitely many primes, and Martin supplies a later proof of the component-divisibility input. Bellah et al. connect special points for a family including certain Mersenne primes.\n\nA search on 2026-07-28 used the exact formulations `site:arxiv.org Markoff mod p graph connectivity connectedness prime 2026`, `site:arxiv.org \"Markoff mod p\" graph connected connectivity`, and `site:doi.org Markoff graph modulo p connectivity`. It also checked the six exact-target arXiv records in the packet bibliography and the generalized-level search result. The strongest checked universal result remains Eddy et al.'s explicit upper threshold. Brown's exhaustive data cover 78,068 primes with \\(3001\\leq p\\leq999{,}983\\). No checked source resolves every prime \\(p\\geq5\\).",
  "status": "completed",
  "evidence_grade": "sourced",
  "scope": {
    "kind": "family",
    "statement": "dated source, normalization, and open-status audit for coefficient-one Markoff graphs at prime parameters p>=5",
    "family": "coefficient-one Markoff graphs G_p for primes p>=5"
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "attempt",
    "citation": {
      "url": "https://doi.org/10.1007/s40993-024-00592-9",
      "locator": "Brown, Theorem 2 and Sections 1 and 4; source comparison completed 2026-07-28"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.1007/s40993-024-00592-9",
    "locator": "Brown, Theorem 2 and Sections 1 and 4; source comparison completed 2026-07-28"
  },
  "models": [],
  "relations": [
    {
      "slug": "R514",
      "title": "Connectivity is proved below one million and beyond an explicit threshold",
      "object_type": "claim",
      "relation": "informs",
      "direction": "outgoing"
    },
    {
      "slug": "R515",
      "title": "The maximal-divisor criterion certifies 40,066 primes between ten and twenty million",
      "object_type": "claim",
      "relation": "informs",
      "direction": "outgoing"
    },
    {
      "slug": "R516",
      "title": "The literal graph is a four-vertex star at p=2 and has no vertices at p=3",
      "object_type": "claim",
      "relation": "informs",
      "direction": "outgoing"
    },
    {
      "slug": "R511",
      "title": "Shard the criterion scan, then route failures to the almost-linear test",
      "object_type": "attempt",
      "relation": "informs",
      "direction": "outgoing"
    },
    {
      "slug": "markoff-graph-prime-connectivity-exceptions",
      "title": "markoff graph prime connectivity exceptions",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}