{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R374",
  "content_hash": null,
  "slug": "heilbronn10-claim-exact-comellas-yebra-lower-bound",
  "type": "claim",
  "title": "The exact Comellas–Yebra construction has minimum area 0.0465374195825...",
  "summary": "Exact algebra checks all 120 triangles in the ten-point construction. Sixteen attain A = 5z²/8 - z³/2, and the other 104 have strictly larger area.",
  "relevance": "For Exact ten-point Heilbronn number in the unit square, record heilbronn10-claim-exact-comellas-yebra-lower-bound (“The exact Comellas–Yebra construction has minimum area 0.0465374195825...”) records a bound, answer, status fact, or structural consequence. The record states: Exact algebra checks all 120 triangles in the ten-point construction.",
  "relevance_source": "recorded",
  "body": "Let \\(c=(63+8\\sqrt{62})^{1/3}\\), \\(z=3/4-c/12-1/(12c)\\), \\(x=z/2\\), and \\(y=(1-z)(1-2z)\\). The ten points are\n\n\\[(x,0),(1-y,0),(0,x),(1,y),(1-z,z),(z,1-z),(0,1-y),(1,1-x),(y,1),(1-x,1).\\]\n\nThe parameter \\(z\\) is the unique real root of \\(12z^3-27z^2+20z-4\\), with \\(z\\approx0.3156111376086111\\). Exact determinant reduction modulo that cubic checks all \\(\\binom{10}{3}=120\\) triangle areas. Their minimum is\n\n\\[A=\\frac{5z^2}{8}-\\frac{z^3}{2}\\approx0.04653741958254177256.\\]\n\nSixteen triples attain \\(A\\). The remaining 104 are strictly larger. The next area is approximately 0.047806572540416697, attained by triples 134 and 568. This supplies the rigorous lower bound \\(\\Delta_{10}\\ge A\\).",
  "status": "supported",
  "evidence_grade": "computational",
  "scope": {
    "kind": "bounded",
    "statement": "the explicit ten-point Comellas-Yebra configuration and all 120 unordered triples of its points",
    "bounds": {
      "n": {
        "min": 10,
        "max": 10
      },
      "triangle_count": {
        "min": 120,
        "max": 120
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://arxiv.org/abs/2603.11107v2",
      "locator": "Exact reconstruction and 120-triangle certificate in heilbronn10-artifact-exact-certificate, executed 2026-07-28 UTC"
    },
    "runtime_seconds": 1.19,
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://arxiv.org/abs/2603.11107v2",
    "locator": "Exact reconstruction and 120-triangle certificate in heilbronn10-artifact-exact-certificate, executed 2026-07-28 UTC"
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R369",
      "title": "Exact 120-triangle certificate and symbolic family checks",
      "object_type": "artifact",
      "relation": "evidences",
      "direction": "incoming"
    },
    {
      "slug": "R375",
      "title": "The best-known ten-point area is 0.0465374195825..., with global optimality open",
      "object_type": "claim",
      "relation": "supports",
      "direction": "outgoing"
    },
    {
      "slug": "R376",
      "title": "The Comellas–Yebra three-parameter family has exact maximum 0.0465374195825...",
      "object_type": "claim",
      "relation": "supports",
      "direction": "outgoing"
    },
    {
      "slug": "R371",
      "title": "Audit the live record, primary sources, and current arXiv frontier",
      "object_type": "attempt",
      "relation": "informs",
      "direction": "incoming"
    },
    {
      "slug": "R373",
      "title": "Run the unrestricted Sudermann-Merx global model at n = 10",
      "object_type": "attempt",
      "relation": "informs",
      "direction": "outgoing"
    },
    {
      "slug": "heilbronn-square-ten",
      "title": "heilbronn square ten",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}