{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R231",
  "content_hash": null,
  "slug": "dta9-claim-exact-maximum-six",
  "type": "claim",
  "title": "The exact maximum is A_9=6",
  "summary": "An explicit six-point set has 20 different positive doubled areas, and exhaustive search excludes every seven-point subset.",
  "relevance": "This records a result, observation, or conjecture used to assess the target problem.",
  "relevance_source": "derived",
  "body": "The six points\n\\[\n\\{(8,7),(2,7),(3,6),(0,2),(7,1),(1,0)\\}\n\\]\ngive the 20 doubled areas\n\\[\n1,6,7,8,9,10,13,17,21,28,29,30,31,34,35,36,37,41,42,43.\n\\]\nThey are positive and pairwise distinct, proving \\(A_9\\geq6\\).\n\nEvery doubled triangle area in the grid is an integer between 1 and 64. The alphabet count \\(\\binom{k}{3}\\leq64\\) gives the preliminary bound \\(k\\leq8\\). The exhaustive certificate then enumerates every increasing seven-element subset of the 81 grid points. When a candidate point is appended, it forms one new triangle with each pair of selected points. The branch continues exactly when these new doubled areas are positive, mutually different, and absent from the prior area mask.\n\nThe search reaches no depth-seven leaf. Since the condition passes to subsets, any valid set of at least seven points would contain a valid seven-point subset. Thus \\(A_9\\leq6\\), and\n\\[\nA_9=6.\n\\]",
  "status": "established",
  "evidence_grade": "computational",
  "scope": {
    "kind": "bounded",
    "statement": "all subsets of the 81 lattice points {0,...,8}^2 whose triangle doubled areas are positive and pairwise distinct",
    "bounds": {
      "coordinate_minimum": {
        "min": 0,
        "max": 0
      },
      "coordinate_maximum": {
        "min": 8,
        "max": 8
      },
      "grid_points": {
        "min": 81,
        "max": 81
      },
      "set_size": {
        "min": 6,
        "max": 7
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://oeis.org/A322740",
      "locator": "Independent exhaustive certificate dta9-artifact-complete-seven-search, executed on 2026-07-24"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://oeis.org/A322740",
    "locator": "Independent exhaustive certificate dta9-artifact-complete-seven-search, executed on 2026-07-24"
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R229",
      "title": "Complete seven-point branch certificate",
      "object_type": "artifact",
      "relation": "evidences",
      "direction": "incoming"
    },
    {
      "slug": "R230",
      "title": "OEIS reports a matching rectangle threshold",
      "object_type": "attempt",
      "relation": "supports",
      "direction": "incoming"
    },
    {
      "slug": "distinct-triangle-areas-grid-9",
      "title": "distinct triangle areas grid 9",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}