{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R227",
  "content_hash": null,
  "slug": "dstg-attempt-literature-and-size-eleven-audit",
  "type": "attempt",
  "title": "The finite gap remains open after the source and solver audit",
  "summary": "The literature confirms the established problem family, while the capped size-11 run returned no certificate.",
  "relevance": "For Most lattice points with all pairwise slopes distinct in a ten by ten grid, record dstg-attempt-literature-and-size-eleven-audit (“The finite gap remains open after the source and solver audit”) documents a concrete method, search boundary, or failed route. The record states: The literature confirms the established problem family, while the capped size-11 run returned no certificate.",
  "relevance_source": "recorded",
  "body": "Erdős, Graham, Ruzsa, and Taylor introduced the grid function in 1992 and proved asymptotic lower and upper bounds. Zhang improved the lower bound in 1993. Clemen's 2026 paper gives the current asymptotic lower bound\n\\[\ng(n)=\\Omega\\left(\\frac{n^{2/3}(\\log\\log n)^{1/3}}{(\\log n)^{1/3}}\\right).\n\\]\nThese asymptotic results do not settle the ten-grid instance.\n\nPeile and Taylor studied exactly the rectangular finite problem in 2000. The publisher abstract says that values or bounds are supplied for \\(m\\leq n\\leq11\\), but the accessible metadata does not expose the table entry for \\(d(10,10)\\). This audit therefore records the paper without attributing a numerical result that could not be checked.\n\nDistinct-difference configurations and Costas arrays are nearby but weaker for this purpose. They forbid repeated displacement vectors. Two parallel segments of different lengths can satisfy that condition, while the present problem forbids them.\n\nFor a direct exact test, Z3 5.0.0 was given 100 selection variables, an exact cardinality of 11, one pseudo-Boolean at-most-one constraint for each of the 112 direction classes, and the two translation normalizations used by the artifact. The run was stopped after 139 seconds and returned `unknown`; it supplies no upper bound. The artifact gives a solver-independent 0-1 pseudo-Boolean instance for continuing the size-11 case and checking any future certificate.",
  "status": "inconclusive",
  "evidence_grade": "sourced",
  "scope": {
    "kind": "bounded",
    "statement": "literature on distinct-slope grid sets and a timeboxed size-11 satisfiability search for the 10 by 10 grid",
    "bounds": {
      "grid_side": {
        "min": 10,
        "max": 10
      },
      "tested_size": {
        "min": 11,
        "max": 11
      },
      "audit_year": {
        "min": 2026,
        "max": 2026
      }
    },
    "exhaustive": false
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "attempt",
    "citation": {
      "url": "https://doi.org/10.1016/S0898-1221(00)00115-2",
      "locator": "R. E. Peile and H. Taylor, Sets of points with pairwise distinct slopes, Computers & Mathematics with Applications 39 (2000), 109-115; Felix Christian Clemen, Applications of Sparse Hypergraph Colorings, Discrete Mathematics 349 (2026), 114822; solver run on 2026-07-25"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.1016/S0898-1221(00)00115-2",
    "locator": "R. E. Peile and H. Taylor, Sets of points with pairwise distinct slopes, Computers & Mathematics with Applications 39 (2000), 109-115; Felix Christian Clemen, Applications of Sparse Hypergraph Colorings, Discrete Mathematics 349 (2026), 114822; solver run on 2026-07-25"
  },
  "relations": [
    {
      "slug": "R228",
      "title": "The maximum lies between 10 and 15",
      "object_type": "claim",
      "relation": "informs",
      "direction": "outgoing"
    },
    {
      "slug": "distinct-slopes-ten-grid",
      "title": "distinct slopes ten grid",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}