{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R796",
  "content_hash": null,
  "slug": "tptgms-attempt-literature-audit-and-exact-model",
  "type": "attempt",
  "title": "The literature gives 32 as the planar record and 22 as the exact 17-point value",
  "summary": "Kurz's primary paper supplies both ingredients used here; a 925-variable binary program would settle the remaining finite-grid gap.",
  "relevance": "For Most squares spanned by twenty points of the ten grid, record tptgms-attempt-literature-audit-and-exact-model (“The literature gives 32 as the planar record and 22 as the exact 17-point value”) documents a concrete method, search boundary, or failed route. The record states: Kurz's primary paper supplies both ingredients used here; a 925-variable binary program would settle the remaining finite-grid gap.",
  "relevance_source": "recorded",
  "body": "Kurz studies \\(S_{\\square}(n)\\), the largest number of squares spanned by \\(n\\) arbitrary planar points. Theorem 51 proves \\(S_{\\square}(17)=22\\). Table 6 records the lower bounds 25, 28, and 32 for 18, 19, and 20 points. Appendix C lists point-set representatives. The paper describes the 20-point value as a lower bound, so it does not settle the present finite-grid maximum.\n\nThe same 32-square pattern fits inside \\(\\{0,\\ldots,9\\}^2\\), as the executable record verifies. A focused search for the fixed-cardinality ten-grid problem found no paper giving an exact optimum or a grid-specific upper certificate.\n\nA direct exact model uses one binary variable \\(x_p\\) for each of the 100 grid points and one binary variable \\(y_s\\) for each of the 825 grid squares. Impose\n\\[\n\\sum_p x_p=20,\\qquad y_s\\leq x_p\\quad(p\\in s),\n\\]\nand maximize \\(\\sum_s y_s\\). Positivity of the objective forces \\(y_s=1\\) whenever all four vertices of \\(s\\) are selected. This model has 925 binary variables and 3,301 linear constraints. A solver proof log, checked independently against the 825-square list digest in this record, would close the interval.",
  "status": "next_experiment",
  "evidence_grade": "sourced",
  "scope": {
    "kind": "bounded",
    "statement": "published results on squares spanned by small planar point sets, with a proposed exact optimization over the 825 square hyperedges of the ten grid",
    "bounds": {
      "literature_point_count": {
        "min": 17,
        "max": 20
      },
      "optimization_point_variables": {
        "min": 100,
        "max": 100
      },
      "optimization_square_variables": {
        "min": 825,
        "max": 825
      }
    },
    "exhaustive": false
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "attempt",
    "citation": {
      "url": "https://epub.uni-bayreuth.de/5936/1/squares_in_plane_point_sets.pdf",
      "locator": "Sascha Kurz, Plane point sets with many squares or isosceles right triangles, Theorem 51, Table 6, and Appendix C"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://epub.uni-bayreuth.de/5936/1/squares_in_plane_point_sets.pdf",
    "locator": "Sascha Kurz, Plane point sets with many squares or isosceles right triangles, Theorem 51, Table 6, and Appendix C"
  },
  "relations": [
    {
      "slug": "R797",
      "title": "The maximum lies between 32 and 43",
      "object_type": "claim",
      "relation": "supports",
      "direction": "outgoing"
    },
    {
      "slug": "R797",
      "title": "The maximum lies between 32 and 43",
      "object_type": "claim",
      "relation": "proposes_continuation_for",
      "direction": "outgoing"
    },
    {
      "slug": "twenty-points-ten-grid-max-squares",
      "title": "twenty points ten grid max squares",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}