{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R207",
  "content_hash": null,
  "slug": "ddg9-claim-eight-point-witness",
  "type": "claim",
  "title": "Eight points satisfy both restrictions",
  "summary": "The candidate witness has 28 distinct squared distances and all 56 of its triples have nonzero signed area.",
  "relevance": "This records a result, observation, or conjecture used to assess the target problem.",
  "relevance_source": "derived",
  "body": "Take\n\\[\nS=\\{(6,0),(8,8),(1,1),(1,3),(8,7),(5,7),(5,1),(6,5)\\}.\n\\]\nIts sorted squared-distance list is\n\\[\n[1,2,4,5,8,9,10,13,16,17,20,25,26,29,32,34,36,41,45,50,52,53,58,65,68,74,85,98].\n\\]\nThe list has \\(\\binom82=28\\) entries and no repetition. For each of the \\(\\binom83=56\\) triples, the verifier evaluates the integer cross product\n\\[\n(x_2-x_1)(y_3-y_1)-(y_2-y_1)(x_3-x_1).\n\\]\nEvery value is nonzero. Hence \\(S\\) has distinct mutual distances and no three collinear points.",
  "status": "established",
  "evidence_grade": "computational",
  "scope": {
    "kind": "bounded",
    "statement": "the stated eight-point subset of {0,...,8}^2",
    "bounds": {
      "grid_side": {
        "min": 9,
        "max": 9
      },
      "set_size": {
        "min": 8,
        "max": 8
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://doi.org/10.1007/BF01299744",
      "locator": "Direct integer verification in ddg9-artifact-exhaustive-nine-search"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.1007/BF01299744",
    "locator": "Direct integer verification in ddg9-artifact-exhaustive-nine-search"
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R208",
      "title": "The exact value is G_9=8",
      "object_type": "claim",
      "relation": "supports",
      "direction": "outgoing"
    },
    {
      "slug": "distinct-distance-grid-9",
      "title": "distinct distance grid 9",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}