Eventual Square Pyramid Optimality for Five-Point Riesz Energy
Manuscript 14 October 2026 · Online 14 October 2026
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.
Record
- Affiliation
- Mercury Software GmbH
- Contact
- [email protected] · GitHub
- Result
- Complete scoped proof
- Categories
- math.MG · math.CA
- Manuscript
- 14 October 2026
- Online release
- 14 October 2026
- Version
- 1.0
- License
- Creative Commons Attribution 4.0 International
Files and verification
The PDF is the canonical reading copy. The source archive contains the LaTeX manuscript, bibliography, reproducibility material, and audit documents without build artefacts.
Citation
Alper Ferudun, “Eventual Square Pyramid Optimality for Five-Point Riesz Energy,” EulerSolve Research Papers, AIM-GEOMETRY-0095, 2026. https://doi.org/10.5281/zenodo.23365473.
BibTeX
@misc{ferudun2026fivepointriesz,
author = {Ferudun, Alper},
title = {Eventual Square Pyramid Optimality for Five-Point Riesz Energy},
year = {2026},
howpublished = {EulerSolve Research Papers},
url = {https://eulersolve.org/papers/aim-geometry-0095/},
doi = {10.5281/zenodo.23365473},
note = {AIM-GEOMETRY-0095; unrefereed preprint}
}