{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R228",
  "content_hash": null,
  "slug": "dstg-claim-certified-interval-10-15",
  "type": "claim",
  "title": "The maximum lies between 10 and 15",
  "summary": "A new 10-point witness improves the lower bound, while the complete direction count gives 15.",
  "relevance": "For Most lattice points with all pairwise slopes distinct in a ten by ten grid, record dstg-claim-certified-interval-10-15 (“The maximum lies between 10 and 15”) records a bound, answer, status fact, or structural consequence. The record states: A new 10-point witness improves the lower bound, while the complete direction count gives 15.",
  "relevance_source": "recorded",
  "body": "Let \\(M_{10}\\) denote the maximum. The point set\n\\[\nA=\\{(0,7),(1,2),(2,0),(4,6),(5,3),(6,9),(7,1),(8,1),(9,5),(9,8)\\}\n\\]\nhas ten points. Reducing every difference vector by the greatest common divisor of its coordinates, with a fixed sign convention, produces 45 distinct primitive directions. Thus \\(M_{10}\\geq10\\).\n\nFor the upper bound, every segment direction has a unique representative \\((a,b)\\) with \\(0\\leq a\\leq9\\), \\(-9\\leq b\\leq9\\), \\(\\gcd(a,|b|)=1\\), and \\(a>0\\), apart from the vertical representative \\((0,1)\\). Exact enumeration gives 112 supported undirected directions. A \\(k\\)-point set in the problem uses a different direction for each of its \\(\\binom{k}{2}\\) pairs, so\n\\[\n\\binom{k}{2}\\leq112.\n\\]\nSince \\(\\binom{15}{2}=105\\) and \\(\\binom{16}{2}=120\\), this proves \\(M_{10}\\leq15\\). The certified interval is therefore\n\\[\n\\boxed{10\\leq M_{10}\\leq15}.\n\\]\nThe exact value remains unresolved in this entry.",
  "status": "established",
  "evidence_grade": "reproduced",
  "scope": {
    "kind": "bounded",
    "statement": "all subsets of the 10 by 10 lattice grid with pairwise distinct undirected slopes",
    "bounds": {
      "grid_side": {
        "min": 10,
        "max": 10
      },
      "certified_lower_bound": {
        "min": 10,
        "max": 10
      },
      "certified_upper_bound": {
        "min": 15,
        "max": 15
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://doi.org/10.1016/j.disc.2025.114822",
      "locator": "Exact construction and direction-count replay in dstg-artifact-witness-direction-check"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.1016/j.disc.2025.114822",
    "locator": "Exact construction and direction-count replay in dstg-artifact-witness-direction-check"
  },
  "relations": [
    {
      "slug": "R226",
      "title": "Exact witness verifier and size-11 continuation instance",
      "object_type": "artifact",
      "relation": "evidences",
      "direction": "incoming"
    },
    {
      "slug": "R227",
      "title": "The finite gap remains open after the source and solver audit",
      "object_type": "attempt",
      "relation": "informs",
      "direction": "incoming"
    },
    {
      "slug": "distinct-slopes-ten-grid",
      "title": "distinct slopes ten grid",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}