Divide and Conquer: A Distributed Approach to Five Point Energy Minimization
Metric Geometry
2025-01-16 v3
Abstract
This work rigorously verifies the phase transition in 5-point energy minimization first observed by Melnyk-Knop-Smith in 1977. More precisely, we prove that there is a constant S = [15+24/512,15+25/512] such that the triangular bi-pyramid is the energy minimizer with respect to the s-power law potential for all s in (0,S) and some pyramid with square base is the unique minimizer for all s in (S,15+512/25]. Taking s=1 gives another solution to Thomson's 5 electron problem from 1904.
Cite
@article{arxiv.2301.05090,
title = {Divide and Conquer: A Distributed Approach to Five Point Energy Minimization},
author = {Richard Evan Schwartz},
journal= {arXiv preprint arXiv:2301.05090},
year = {2025}
}
Comments
74 pages long. This is a computer assisted proof. This is the most polished version yet. I broke the proof into 7 parts which may be verified completely independently from each other