{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R797",
  "content_hash": null,
  "slug": "tptgms-claim-certified-interval-32-43",
  "type": "claim",
  "title": "The maximum lies between 32 and 43",
  "summary": "A 20-point set in a 5 by 5 window spans 32 squares, while deletion averaging from the exact 17-point planar theorem gives a universal upper bound of 43.",
  "relevance": "For Most squares spanned by twenty points of the ten grid, record tptgms-claim-certified-interval-32-43 (“The maximum lies between 32 and 43”) records a bound, answer, status fact, or structural consequence. The record states: A 20-point set in a 5 by 5 window spans 32 squares, while deletion averaging from the exact 17-point planar theorem gives a universal upper bound of 43.",
  "relevance_source": "recorded",
  "body": "Write \\(M\\) for the requested maximum. The following points lie in the ten grid:\n\\[\n\\begin{aligned}\nP=\\{&(2,2),(2,3),(2,4),\\\\\n&(3,2),(3,3),(3,4),(3,5),\\\\\n&(4,1),(4,2),(4,3),(4,4),(4,5),\\\\\n&(5,1),(5,2),(5,3),(5,4),(5,5),\\\\\n&(6,2),(6,3),(6,4)\\}.\n\\end{aligned}\n\\]\nExact enumeration finds 32 squares in \\(P\\). Their squared side lengths have multiplicities\n\\[\n1:11,\\quad 2:8,\\quad 4:4,\\quad 5:7,\\quad 8:1,\\quad 10:1.\n\\]\nHence \\(M\\geq32\\).\n\nFor the upper bound, let \\(U_n\\) bound the number of squares in every \\(n\\)-point set in the plane. If an \\(n\\)-point set spans \\(q\\) squares, sum the square counts after deleting each point. Every square survives exactly \\(n-4\\) deletions, so\n\\[\n(n-4)q\\leq nU_{n-1}.\n\\]\nKurz proved \\(U_{17}=22\\). Iterating the displayed inequality and taking integer parts gives\n\\[\nU_{18}\\leq\\left\\lfloor\\frac{18\\cdot22}{14}\\right\\rfloor=28,\n\\quad U_{19}\\leq\\left\\lfloor\\frac{19\\cdot28}{15}\\right\\rfloor=35,\n\\quad U_{20}\\leq\\left\\lfloor\\frac{20\\cdot35}{16}\\right\\rfloor=43.\n\\]\nThis applies to every planar 20-point set, including subsets of the ten grid. Therefore\n\\[\n\\boxed{32\\leq M\\leq43}.\n\\]\nThe exact value remains open in this record.",
  "status": "partial",
  "evidence_grade": "reproduced",
  "scope": {
    "kind": "bounded",
    "statement": "all 20-element subsets of the 100 lattice points {0,1,...,9}^2, counting every nondegenerate Euclidean square once",
    "bounds": {
      "grid_width_in_points": {
        "min": 10,
        "max": 10
      },
      "grid_height_in_points": {
        "min": 10,
        "max": 10
      },
      "chosen_points": {
        "min": 20,
        "max": 20
      },
      "certified_lower_bound": {
        "min": 32,
        "max": 32
      },
      "certified_upper_bound": {
        "min": 43,
        "max": 43
      }
    },
    "exhaustive": false
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://arxiv.org/abs/2112.12716",
      "locator": "Sascha Kurz, Plane point sets with many squares or isosceles right triangles, Theorem 51 for S_square(17)=22 and Table 6 for the 20-point lower bound 32; finite-grid witness replay in tptgms-artifact-exact-square-verifier"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://arxiv.org/abs/2112.12716",
    "locator": "Sascha Kurz, Plane point sets with many squares or isosceles right triangles, Theorem 51 for S_square(17)=22 and Table 6 for the 20-point lower bound 32; finite-grid witness replay in tptgms-artifact-exact-square-verifier"
  },
  "models": [],
  "relations": [
    {
      "slug": "R795",
      "title": "Exact four-subset square verifier",
      "object_type": "artifact",
      "relation": "supports",
      "direction": "incoming"
    },
    {
      "slug": "R796",
      "title": "The literature gives 32 as the planar record and 22 as the exact 17-point value",
      "object_type": "attempt",
      "relation": "supports",
      "direction": "incoming"
    },
    {
      "slug": "R796",
      "title": "The literature gives 32 as the planar record and 22 as the exact 17-point value",
      "object_type": "attempt",
      "relation": "proposes_continuation_for",
      "direction": "incoming"
    },
    {
      "slug": "twenty-points-ten-grid-max-squares",
      "title": "twenty points ten grid max squares",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}