{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R513",
  "content_hash": null,
  "slug": "mgpc-claim-components-through-3001",
  "type": "claim",
  "title": "Exact Vieta enumeration connects every G_p for 5 <= p <= 3001",
  "summary": "A full breadth-first search visits all 1,169,185,980 nonorigin surface points across all 429 primes with 5 <= p <= 3001 and finds one component at every prime.",
  "relevance": "For Prime exceptions to connectivity of the Markoff graph, record mgpc-claim-components-through-3001 (“Exact Vieta enumeration connects every G_p for 5 <= p <= 3001”) records a bound, answer, status fact, or structural consequence. The record states: A full breadth-first search visits all 1,169,185,980 nonorigin surface points across all 429 primes with 5 <= p <= 3001 and finds one component at every prime.",
  "relevance_source": "recorded",
  "body": "For each prime \\(5\\leq p\\leq3001\\), the linked program constructs every point on\n\\[\nx^2+y^2+z^2=xyz\n\\]\nby solving the quadratic in \\(z\\), removes only \\((0,0,0)\\), and follows exactly the three stated Vieta involutions. It checks the point-count formula\n\\[\n|G_p|=p^2+3p\\left(\\frac{-1}{p}\\right)\n\\]\nand the zero-coordinate count, then starts breadth-first search at \\((3,3,3)\\). Every one of the 429 rows has one component whose size equals the checked point count.\n\nThe output also retains a shortest move word to a lexicographically selected farthest vertex. The largest recorded root eccentricity is 27, attained at \\(p=2711,2749,2909\\). At \\(p=2711\\), the word `121213123123213132131212312` takes \\((3,3,3)\\) to \\((1123,2036,2036)\\).",
  "status": "supported",
  "evidence_grade": "computational",
  "scope": {
    "kind": "bounded",
    "statement": "every prime p with 5 <= p <= 3001 for the coefficient-one Markoff surface and the three stated Vieta edges",
    "bounds": {
      "p": {
        "min": 5,
        "max": 3001
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://arxiv.org/abs/1812.07275",
      "locator": "Independent exact reproduction and extension through p=3001; compare the p<3000 connectivity computation and Proposition 2.1"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://arxiv.org/abs/1812.07275",
    "locator": "Independent exact reproduction and extension through p=3001; compare the p<3000 connectivity computation and Proposition 2.1"
  },
  "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": "markoff-graph-prime-connectivity-exceptions",
      "title": "markoff graph prime connectivity exceptions",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}