{
  "schema_version": 1,
  "problem_number": "AIM-GEOMETRY-0095",
  "title": "Eventual Square Pyramid Optimality for Five-Point Riesz Energy",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "We prove that, for every sufficiently large real exponent s, the unique five-point minimizer of chordal Riesz s-energy on the unit two-sphere, up to orthogonal congruence and relabeling, is a square pyramid. Its apex is the north pole and its base has latitude -t, where (1-t)^(s/2+1)=(2+2^(-s/2))*t. Starting from the limiting configuration theorem of Bondarenko, Hardin and Saff, angular stationarity and relative height balance localize every minimizer within o(1/s) of this moving pyramid. A uniform positive Hessian proves uniqueness. The threshold is existential; the intervening exponent range remains unresolved. This is an unrefereed AI-assisted preprint, with no absolute priority, independent review or formal verification claim.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.MG",
    "math.CA"
  ],
  "keywords": [
    "Riesz energy",
    "five points on the sphere",
    "square pyramid",
    "energy minimization",
    "asymptotic localization",
    "Hessian positivity",
    "AIM-GEOMETRY-0095"
  ],
  "manuscript_version_date": "2026-10-14",
  "publication_date": "2026-10-14",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-14",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/aim-geometry-0095/",
  "pdf_url": "https://eulersolve.org/papers/aim-geometry-0095/paper.pdf?v=f0fdb01bdfe2",
  "doi": "10.5281/zenodo.23365473",
  "zenodo_record_url": "https://zenodo.org/records/23365473",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Complete proof of eventual global square-pyramid optimality and uniqueness for five-point chordal Riesz energy. No numerical threshold or resolution of the entire intermediate-exponent range is claimed. The broader AIM source record remains partially resolved. Unrefereed; no absolute priority, independent review or formal verification claim.",
  "files": {
    "paper.pdf": {
      "sha256": "f0fdb01bdfe2fe3628e476f1f455ab5b1fe2b27aacdbaf925fa6900aa69bbecc"
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    "source.zip": {
      "sha256": "8b4aaa8640e110b3917c73bfbc18a88e58506cb9823b435c487a7df1cde80eeb"
    },
    "verification_report.md": {
      "sha256": "3618b58b98a7db2ff19d035e2c0668e04cc05dab7b262170fd9e0a7af2122d54"
    }
  },
  "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."
}