{
  "schema_version": 1,
  "problem_number": "OWR-12012-001",
  "title": "The Minimum Distance of Unshifted Spherical Fibonacci Lattices",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "For every Fibonacci number N=F_n>=2, we prove that the unshifted rational spherical Fibonacci lattice has least Euclidean distance2/sqrt(N), uniquely attained by the north pole and the next point. This answers the separation question in Brauchart's contribution to Oberwolfach Report40/2012,p2438 (UnsolvedMath OWR-12012-001). The proof combines a global Lambert-map distance inequality with Zaremba's classical formula for the Fibonacci lattice figure of merit. More generally, a rational N-point Lambert lift with Zaremba index rho>=N/4 has exactly this minimum; rho>N/4 also gives uniqueness. The construction includes a pole and uses the exact rational Fibonacci angle; the claim does not concern the different shifted irrational Fibonacci spiral, discrepancy, or optimal packing among all point sets. The four-page written proof is accompanied by symbolic and integer regression checks and separately labelled floating-point diagnostics. This is a self-audited, unrefereed preprint prepared with AI assistance, without independent review, formal proof-assistant verification, or an absolute-priority claim.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.MG",
    "math.NT",
    "math.NA"
  ],
  "keywords": [
    "OWR-12012-001",
    "spherical Fibonacci lattice",
    "minimum distance",
    "Lambert equal-area map",
    "Zaremba index"
  ],
  "manuscript_version_date": "2026-10-27",
  "publication_date": "2026-10-27",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-27",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/owr-12012-001/",
  "pdf_url": "https://eulersolve.org/papers/owr-12012-001/paper.pdf?v=bc25740976ad",
  "doi": "10.5281/zenodo.23681420",
  "zenodo_record_url": "https://zenodo.org/records/23681420",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "For every n>=3, the unshifted rational spherical Fibonacci lattice with N=F_n points and angle increment2pi F_(n-1)/F_n has least Euclidean distance2/sqrt(N), uniquely attained by its north pole and next point. More generally the rational Lambert lift has squared separation at least min(4/N,16rho(N,a)/N^2); rho>=N/4 gives exact pole attainment and rho>N/4 gives uniqueness. The exact result concerns the unshifted rational Fibonacci construction in the source question. The distinct shifted irrational Fibonacci spiral is outside its scope. Zaremba's classical index formula is credited; this is a self-audited, unrefereed proof. This preprint is AI-assisted, self-audited and unrefereed; no independent or formal verification or certified novelty or absolute-priority claim is made.",
  "files": {
    "paper.pdf": {
      "sha256": "bc25740976ad1c6e51843cae9df0050ce20f2fd845bb622e5d0b49cb307b9e3c"
    },
    "source.zip": {
      "sha256": "bbae5e4ae8b78800d0a99909100e65b13f4ff1cf096de93cd16a1247209cb658"
    },
    "verification_report.md": {
      "sha256": "c78ebd163699c317e3e63566e3fe1d552a0981c9451d24374b2f154ec7c2315a"
    }
  },
  "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."
}