{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R514",
  "content_hash": null,
  "slug": "mgpc-claim-current-status",
  "type": "claim",
  "title": "Connectivity is proved below one million and beyond an explicit threshold",
  "summary": "Let T=(863#)(53#)(13#)(7#)(5#)3^3 2^5. The graph G_p is connected for every prime 5 <= p < 1,000,000 and every prime p > T; this packet also certifies 40,066 primes with 10,000,000 < p <= 20,000,000. The unresolved p >= 5 cases are the primes in 1,000,000 <= p <= T outside the individual certificates recorded or cited here. Literally, G_2 is connected and G_3 has no vertices, so p=3 still needs a null-graph convention.",
  "relevance": "For Prime exceptions to connectivity of the Markoff graph, record mgpc-claim-current-status (“Connectivity is proved below one million and beyond an explicit threshold”) records a bound, answer, status fact, or structural consequence. The record states: Let T=(863#)(53#)(13#)(7#)(5#)3^3 2^5.",
  "relevance_source": "recorded",
  "body": "Brown's Theorem 2 and exhaustive data establish connectivity for every prime \\(p<1{,}000{,}000\\). Theorem 1.4 of Eddy, Fuchs, Litman, Martin, and Tripeny establishes connectivity for every prime\n\\[\np>T=(863\\#)(53\\#)(13\\#)(7\\#)(5\\#)3^3 2^5,\n\\]\nwhere \\(n\\#\\) is the product of the primes at most \\(n\\). Their decimal approximation is \\(3.448\\times10^{392}\\).\n\nThe packet's complete Vieta enumeration independently checks every prime \\(5\\leq p\\leq3001\\). Its maximal-divisor scan applies Eddy et al.'s sufficient criterion to every prime in \\(10{,}000{,}000<p\\leq20{,}000{,}000\\), certifying 40,066 and leaving 565,962 undecided by that criterion. Thus the remaining universal problem for \\(p\\geq5\\) is the finite set of primes in \\([1{,}000{,}000,T]\\) outside the individual certificates recorded or cited here.\n\nLiteral enumeration gives a connected four-vertex graph at \\(p=2\\) and an empty vertex set at \\(p=3\\). The canonical acceptance condition asks for two components when a prime is exceptional. It therefore needs an explicit empty-graph convention before \\(p=3\\) can be classified.",
  "status": "supported",
  "evidence_grade": "sourced",
  "scope": {
    "kind": "family",
    "statement": "current connectivity status and exact unresolved remainder for the coefficient-one Vieta graph at every prime",
    "family": "coefficient-one Markoff graphs G_p at prime p, with the literal nonorigin vertex convention"
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://arxiv.org/abs/2308.07579v1",
      "locator": "Eddy et al., Theorem 1.4; Brown, DOI 10.1007/s40993-024-00592-9, Theorem 2; packet claims mgpc-claim-components-through-3001, mgpc-claim-maximal-divisor-10m-20m, and mgpc-claim-small-characteristics"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://arxiv.org/abs/2308.07579v1",
    "locator": "Eddy et al., Theorem 1.4; Brown, DOI 10.1007/s40993-024-00592-9, Theorem 2; packet claims mgpc-claim-components-through-3001, mgpc-claim-maximal-divisor-10m-20m, and mgpc-claim-small-characteristics"
  },
  "models": [],
  "relations": [
    {
      "slug": "R513",
      "title": "Exact Vieta enumeration connects every G_p for 5 <= p <= 3001",
      "object_type": "claim",
      "relation": "supports",
      "direction": "incoming"
    },
    {
      "slug": "R515",
      "title": "The maximal-divisor criterion certifies 40,066 primes between ten and twenty million",
      "object_type": "claim",
      "relation": "supports",
      "direction": "incoming"
    },
    {
      "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": "supports",
      "direction": "incoming"
    },
    {
      "slug": "R512",
      "title": "Dated source and convention audit",
      "object_type": "attempt",
      "relation": "informs",
      "direction": "incoming"
    },
    {
      "slug": "markoff-graph-prime-connectivity-exceptions",
      "title": "markoff graph prime connectivity exceptions",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}