{
  "schema_version": 1,
  "problem_number": "OWR-11136-022",
  "title": "Favourite Distances in Three-Space: Exact Suspension Counts and a Three-Arc Gap",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Assign a positive radius to each point of a finite set in Euclidean three-space and count directed pairs at the assigned distance. Put B(n)=ceil(n^2/4+5n/2)+1. We prove that the exact maximum among n-point suspensions is B(n) for every n>=13, using Swanepoel's construction for sharpness. A maximum principle on the finitely generated group of circle angles supplies a rational increment, and a rational sine-product obstruction excludes equality in the earlier upper bound. We then prove that a suspension core with at least eight circle points and one genuine exceptional point has at most B(n) arcs; with two exceptions it has at most B(n)+3. Combining this with Swanepoel's structural theorem gives B(n)<=f_3(n)<=B(n)+3 for all sufficiently large n. Two exceptions in a large extremizer must be reflections across the circle plane. The exact unrestricted maximum and the conjecture that every sufficiently large extremizer is a suspension remain unresolved. AI-assisted, self-audited and unrefereed; no independent review, formal verification or absolute priority is claimed.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.CO",
    "math.MG"
  ],
  "keywords": [
    "OWR-11136-022",
    "math.CO",
    "math.MG"
  ],
  "manuscript_version_date": "2026-10-22",
  "publication_date": "2026-10-23",
  "publication_date_kind": "first public online release",
  "version": "1.1",
  "date_modified": "2026-10-23",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.1",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/owr-11136-022/",
  "pdf_url": "https://eulersolve.org/papers/owr-11136-022/paper.pdf?v=ef50d16e91d5",
  "doi": "10.5281/zenodo.23587294",
  "zenodo_record_url": "https://zenodo.org/records/23587294",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Every suspension with c>0 has at most lc+l+4c-1 arcs and exact n-point maximum B(n)=ceil(n^2/4+5n/2)+1 for n>=13. For cores with c>=8, one genuine exception has at most B(n), and two have at most B(n)+3. Swanepoel's credited structural theorem and construction imply B(n)<=f3(n)<=B(n)+3 for all sufficiently large n. A non-reflected pair cannot be extremal when n>=23. This is a self-audited, AI-assisted, unrefereed preprint. No independent certification, formal proof verification or absolute-priority claim is made.",
  "files": {
    "paper.pdf": {
      "sha256": "ef50d16e91d57778fa997256ad0902e0b30fd6e6f5d6fb8431e546ad1d63598e"
    },
    "source.zip": {
      "sha256": "8c98460ebb15cd0173fb747546b4c9c82ff5b1bc6fb8f3d696f170885381cba3"
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    "verification_report.md": {
      "sha256": "4e77e963facb4df059357f87c96bbc3fd12ed5d6197035499a214dac24580c1d"
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  },
  "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."
}