{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R208",
  "content_hash": null,
  "slug": "ddg9-claim-exact-value-eight",
  "type": "claim",
  "title": "The exact value is G_9=8",
  "summary": "An explicit eight-point set satisfies both restrictions, and exhaustive backtracking rules out every nine-point set.",
  "relevance": "This records a result, observation, or conjecture used to assess the target problem.",
  "relevance_source": "derived",
  "body": "The exact extremal value is\n\\[\n\\boxed{G_9=8}.\n\\]\nThe set in `ddg9-claim-eight-point-witness` proves the lower bound. It has 28 distinct squared distances and no collinear triple.\n\nFor the upper bound, `ddg9-artifact-exhaustive-nine-search` examines increasing subsets of the 81 grid points. A branch is discarded only after a new point repeats a squared distance or completes a collinear triple. Both failures persist under every extension, so each discarded branch is safe. The search reaches no depth-nine node. Therefore no valid nine-point subset exists. Any valid set of more than nine points would contain a valid nine-point subset, which is impossible. The two bounds give the claimed equality.",
  "status": "established",
  "evidence_grade": "computational",
  "scope": {
    "kind": "bounded",
    "statement": "subsets of the 81 lattice points {0,...,8}^2 with distinct squared distances and no collinear triple",
    "bounds": {
      "grid_side": {
        "min": 9,
        "max": 9
      },
      "grid_points": {
        "min": 81,
        "max": 81
      },
      "cardinality_decision": {
        "min": 8,
        "max": 9
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://doi.org/10.1007/BF01299744",
      "locator": "Independent witness and exhaustive certificate in ddg9-claim-eight-point-witness and ddg9-artifact-exhaustive-nine-search"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.1007/BF01299744",
    "locator": "Independent witness and exhaustive certificate in ddg9-claim-eight-point-witness and ddg9-artifact-exhaustive-nine-search"
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R207",
      "title": "Eight points satisfy both restrictions",
      "object_type": "claim",
      "relation": "supports",
      "direction": "incoming"
    },
    {
      "slug": "R205",
      "title": "Exhaustive nonexistence certificate for nine points",
      "object_type": "artifact",
      "relation": "evidences",
      "direction": "incoming"
    },
    {
      "slug": "R206",
      "title": "Prior work covers the distance-only problem",
      "object_type": "attempt",
      "relation": "corroborates",
      "direction": "incoming"
    },
    {
      "slug": "distinct-distance-grid-9",
      "title": "distinct distance grid 9",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}