Maxwell conjecture for point-charge equilibria

Maxwell’s question, discussed in 1873 and posed as an upper-bound problem by M. Morse and S. S. Cairns in 1969, asks whether $n$ positive point charges can have more than $(n-1)^2$ nondegenerate critical points. Gabrielov, Novikov, and Shapiro later formulated this as the Maxwell conjecture.

Known results

• Three equal-magnitude charges: at most $4$ nondegenerate equilibria; isolated-equilibrium counts are $0$, $2$, $3$, or $4$ (2015).
• Three positive charges in $\mathbb{R}^n$: at most $6$ nondegenerate equilibria, improving the prior general bound $12$ but not proving the conjectured bound $4$ (2026).
• General bounds were improved by V. Zolotov (2023) and H. Edelsbrunner, C. Fillmore, and G. Oliveira (2026).

July 2026 counterexample

Philip Arathoon, Gavin Ball, and Matthew D. Kvalheim construct five positive charges with at least $24$ nondegenerate critical points, exceeding $(5-1)^2=16$. Three original equilibria persist while a central one bifurcates into $21$; a small perturbation makes all critical points nondegenerate. The arXiv paper reports that GPT-5.6 Sol suggested the idea, while the authors verified and wrote the mathematics.

Current status (as of July 2026): The Maxwell conjecture is disproved by a published arXiv counterexample for $n=5$; sharper bounds for smaller cases, including whether $4$ is optimal for three unequal charges, remain open.