Maxwell’s bound on electrostatic equilibria
Follow the latest progress by email.
Maxwell’s electrostatic-equilibrium conjecture proposes at most (n−1)² equilibria for n positive point charges in R³. The counterexample target uses five charges with at least twenty-four nondegenerate equilibria, exceeding sixteen.
Why it matters and what counts as progress
Why it matters: It asks whether a classical charge-count bound controls the number of isolated equilibrium points, with degeneracy excluded.
What counts as progress: A positive-charge configuration exceeding the proposed equilibrium bound with nondegenerate isolated equilibria, or a corrected upper bound.
How the problem developed
- 29 Jul 2026 · Counterexample [1]
Philip Arathoon, Gavin Ball and Matthew D. Kvalheim Constructed five point charges with at least 24 nondegenerate equilibria, exceeding Maxwell’s proposed (n−1)² bound.
AI contributions 2
From 12 to 6: Sharpening the Three-Charge Bound in Maxwell's Problem
For every α>0 and three positive point charges in R^n with n≥2, the potential has at most six nondegenerate equilibria. The new step proves a four-contact statement using separation at the unique saddle of a first integral.
The Maxwell Conjecture is False
Five positive point charges in R³ can have at least 24 nondegenerate electrostatic equilibria. A small perturbation makes all critical points nondegenerate and finite in number, exceeding Maxwell’s proposed (n−1)² bound of 16 when n=5.
Sources
Website timeline
Publication dates and editorial changes.
Claude helps reduce the three-charge equilibrium bound to six
GPT-suggested five-charge construction refutes Maxwell bound