{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R516",
  "content_hash": null,
  "slug": "mgpc-claim-small-characteristics",
  "type": "claim",
  "title": "The literal graph is a four-vertex star at p=2 and has no vertices at p=3",
  "summary": "The p=2 graph is connected. The p=3 vertex set is empty, which exposes a null-graph convention absent from the current acceptance conditions.",
  "relevance": "For Prime exceptions to connectivity of the Markoff graph, record mgpc-claim-small-characteristics (“The literal graph is a four-vertex star at p=2 and has no vertices at p=3”) records a bound, answer, status fact, or structural consequence. The record states: The p=2 graph is connected.",
  "relevance_source": "recorded",
  "body": "At \\(p=2\\), the vertices are\n\\[\n(1,1,1),\\ (0,1,1),\\ (1,0,1),\\ (1,1,0).\n\\]\nThe three Vieta moves send \\((1,1,1)\\) to the other three vertices. Each leaf has two self-loop incidences, so the underlying graph is a connected star.\n\nAt \\(p=3\\), a solution with one zero coordinate would require two squares to sum to zero. Nonzero squares equal 1 in \\(\\mathbb F_3\\), which forces the origin in this case. If all three coordinates are nonzero, the left side equals \\(1+1+1=0\\), while \\(xyz=\\pm1\\). Hence the nonorigin solution set is empty. The coefficient-three equation has eight nonorigin solutions at the same prime, so the usual scaling changes the graph in characteristic 3.\n\nThe phrase “ordinary graph connectivity” does not settle whether the null graph counts as connected. The acceptance condition asks for two components whenever an exception exists, a certificate that cannot exist for an empty vertex set. A clean statement can declare the null graph connected for this problem, or quantify the main conjecture over \\(p\\geq5\\) after listing the two small characteristics.",
  "status": "supported",
  "evidence_grade": "computational",
  "scope": {
    "kind": "bounded",
    "statement": "the literal coefficient-one graph at the two primes p=2 and p=3",
    "bounds": {
      "p": {
        "min": 2,
        "max": 3
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "locator": "Elementary characteristic-2 and characteristic-3 argument, independently replayed by both packet artifacts"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": null,
    "locator": "Elementary characteristic-2 and characteristic-3 argument, independently replayed by both packet artifacts"
  },
  "models": [],
  "relations": [
    {
      "slug": "R514",
      "title": "Connectivity is proved below one million and beyond an explicit threshold",
      "object_type": "claim",
      "relation": "supports",
      "direction": "outgoing"
    },
    {
      "slug": "R508",
      "title": "Exact Vieta-component enumerator",
      "object_type": "artifact",
      "relation": "evidences",
      "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"
    }
  ]
}