{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R230",
  "content_hash": null,
  "slug": "dta9-attempt-prior-art-audit",
  "type": "attempt",
  "title": "OEIS reports a matching rectangle threshold",
  "summary": "OEIS A322740 lists 65 as the least rectangle area for seven points, one above the 8 by 8 coordinate span here.",
  "relevance": "This records a line of attack on the target problem, including what it establishes and where it stops.",
  "relevance_source": "derived",
  "body": "OEIS A322740 asks for the least product \\(L_1L_2\\) of coordinate spans of a lattice rectangle containing \\(n\\) points whose \\(\\binom n3\\) triangle areas are positive and pairwise different. It reports\n\\[\na(6)=30,\\qquad a(7)=65.\n\\]\nThe six-point configuration listed there fits inside the present grid. Its reported seven-point threshold also predicts the upper bound here: \\(\\{0,\\ldots,8\\}^2\\) has coordinate-span area \\(8\\cdot8=64\\).\n\nThe 2018 IBM Research challenge uses the same geometric condition for 11 points and seeks a small containing rectangle. Pinchasi's 2008 theorem concerns the minimum number of different triangle areas forced by an arbitrary planar point set, a neighboring extremal question.\n\nThe focused search found no peer-reviewed source stating the exact square-grid maximum \\(A_9\\). OEIS gives a directly relevant prior computation without a replayable search trace on the entry. The independent artifact proves the needed seven-point nonexistence for this grid.",
  "status": "completed",
  "evidence_grade": "sourced",
  "scope": {
    "kind": "bounded",
    "statement": "prior results for seven lattice points with all 35 triangle areas positive and different in a rectangle of coordinate area at most 65",
    "bounds": {
      "point_count": {
        "min": 7,
        "max": 7
      },
      "rectangle_area": {
        "max": 65
      }
    },
    "exhaustive": false
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "attempt",
    "citation": {
      "url": "https://oeis.org/A322740",
      "locator": "OEIS A322740 and A322741, checked 2026-07-24; IBM Research, Ponder This Challenge, October 2018; Rom Pinchasi, The Minimum Number of Distinct Areas of Triangles Determined by a Set of n Points in the Plane, SIAM Journal on Discrete Mathematics 22(2) (2008), 828-831, DOI 10.1137/07068299X"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://oeis.org/A322740",
    "locator": "OEIS A322740 and A322741, checked 2026-07-24; IBM Research, Ponder This Challenge, October 2018; Rom Pinchasi, The Minimum Number of Distinct Areas of Triangles Determined by a Set of n Points in the Plane, SIAM Journal on Discrete Mathematics 22(2) (2008), 828-831, DOI 10.1137/07068299X"
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R231",
      "title": "The exact maximum is A_9=6",
      "object_type": "claim",
      "relation": "supports",
      "direction": "outgoing"
    },
    {
      "slug": "distinct-triangle-areas-grid-9",
      "title": "distinct triangle areas grid 9",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}