Divide and Conquer: A Distributed Approach to Five Point Energy Minimization
Richard Evan Schwartz
TL;DR
This work confirms a long-standing conjecture about a phase transition in the 5-point energy minimization problem on the sphere under power-law potentials. The author introduces a framework of avatars, hybrid potentials, and domain restrictions to reduce the problem to manageable, verifiable components, combining interpolation, local convexity, and symmetrization with a substantial computer-assisted calculation. The main result identifies a transition interval around $S=15+\frac{24}{512}$ to $15+\frac{25}{512}$ and a transition exponent $\shin\in(15,15_{+})$, where TBP minimizes for small exponents, a FP minimizes for large exponents, and TBP ties with a FP at the transition; at $s=1$ the result recovers Thomson’s classical 5-electron problem. The work combines rigorous analytic bounds with exhaustive, verifiable computational procedures to certify global energy minima across the specified parameter ranges, advancing the understanding of how symmetry and geometry govern energy minimization on spheres. The techniques, including the Averaging System and hierarchical division with interval arithmetic, offer a blueprint for tackling similar discrete-energy problems in higher dimensions or with more particles.
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.