{
  "schema_version": 1,
  "problem_number": "OWR-2090-019",
  "title": "Unbounded Line Pinning by Pairwise Disjoint Convex Bodies",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "For every integer m >= 3 we construct m+4 pairwise disjoint compact convex bodies with nonempty interiors in R^3 and a common transversal line such that every subfamily pinning that line contains at least m members. Thus isolated line transversals to arbitrary disjoint convex bodies admit no pinning bound depending only on the dimension. Four separated polytopes restrict the nearby transversals to a classical one-parameter family on a hyperbolic paraboloid. The remaining bodies are convex epigraphs of flat oscillatory perturbations: together they exclude every nonzero parameter, while each has an exclusive sequence of excluded parameters tending to zero. All contacts with the pinned line are distinct single points. An explicit extension gives the same conclusion in every dimension at least three. The construction is nonsemialgebraic and does not settle the separate question for pairwise disjoint polytopes. This is an AI-assisted, self-audited, unrefereed preprint; no external review or absolute priority is claimed.",
  "result_type": "COMPLETE_COUNTEREXAMPLE",
  "categories": [
    "math.MG",
    "math.CO"
  ],
  "keywords": [
    "line pinning",
    "isolated transversals",
    "convex bodies",
    "flat functions",
    "geometric transversals",
    "OWR-2090-019"
  ],
  "manuscript_version_date": "2026-10-19",
  "publication_date": "2026-10-19",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-19",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/owr-2090-019/",
  "pdf_url": "https://eulersolve.org/papers/owr-2090-019/paper.pdf?v=be504f4c847e",
  "doi": "10.5281/zenodo.23460239",
  "zenodo_record_url": "https://zenodo.org/records/23460239",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "For arbitrary pairwise disjoint compact convex bodies in every dimension at least three, there is no dimension-only bound on the size of a pinning subfamily. The construction uses nonsemialgebraic flat oscillations; it does not resolve the separate disjoint-polytope question or provide globally smooth bodies. The classical regulus block is credited. AI-assisted, self-audited and unrefereed; no independent review, formal certification or absolute-priority claim.",
  "files": {
    "paper.pdf": {
      "sha256": "be504f4c847e9005ad2a4308a563fd1be77ca21145298f94db78af0611348153"
    },
    "source.zip": {
      "sha256": "bd2724c3ac5d82eda639d4aaf2b1d2d4ae452cc4d28697b79cab09e793877a9c"
    },
    "verification_report.md": {
      "sha256": "5a926ee65c360f5dbb10cb68a2b44d730b5054dfff63aede26836aef1d2fcefc"
    }
  },
  "ai_use_disclosure": "AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text."
}