The Maxwell Conjecture is False

Philip Arathoon; Gavin Ball; Matthew D. Kvalheim

The Maxwell Conjecture is False

Classical Physics 2026-07-29 v1 Mathematical Physics

Authors: Philip Arathoon , Gavin Ball , Matthew D. Kvalheim

Abstract

We exhibit a configuration of five point charges in Euclidean space whose electrostatic potential admits at least 24 critical points all of which are non-degenerate. Maxwell's conjecture that the field of $n$ point charges has at most $(n-1)^2$ critical points which are all non-degenerate is therefore false.

Cite

@article{arxiv.2607.27197,
  title  = {The Maxwell Conjecture is False},
  author = {Philip Arathoon and Gavin Ball and Matthew D. Kvalheim},
  journal= {arXiv preprint arXiv:2607.27197},
  year   = {2026}
}