Maxwell's problem on points of equilibrium

Fix a handful of electric charges in space. How many places can a test charge sit with no force on it?

Posed by James Clerk Maxwell, 1873.

In plain terms

Pin a few positive electric charges at fixed points in space. Between them the electric field pushes and pulls, and at certain special points the pushes cancel exactly: a test charge placed there feels nothing. These are the points of equilibrium. A famous theorem of Earnshaw says none of them is stable — every one is a saddle, a mountain pass rather than a valley — which is why you cannot levitate a charge with static fields alone.

Maxwell, in a footnote of his 1873 Treatise on Electricity and Magnetism, claimed that n charges can never produce more than (n−1)² such points. Two charges give at most one equilibrium; three should give at most four; five at most sixteen. He sketched an argument and moved on. A century and a half later, nobody had proved it — and for three charges, nobody has even proved that the answer is four rather than five or six.

What is known

Remarkably little, for how old and concrete the question is. It is not even known that the number of equilibrium points is always finite. The modern benchmark is a 2007 paper of Gabrielov, Novikov and Shapiro, which used deep results about systems with few monomials to prove the first general bounds — for three charges, at most twelve equilibria, against Maxwell's predicted four. For three charges of equal strength the answer really is four (Tsai, 2015), with the equilateral triangle attaining it: one equilibrium at the center, three more near the edges.

Then, days before this problem was onboarded, the story broke open: a July 2026 preprint of Arathoon, Ball and Kvalheim constructs five charges with at least twenty-four equilibria — eight more than Maxwell's sixteen. The construction is delicate: an equilateral triangle of unit charges plus two minuscule charges hovering just above and below its center. As the small charges switch on, the central equilibrium shatters into twenty-one. If the preprint holds up, the conjecture as Maxwell stated it is false, and the true growth of the maximum is wide open. A companion preprint sharpens the three-charge bound from twelve to six — leaving the original small case, four versus five versus six, still unresolved.

Why it is hard

The equilibrium equations look tame but are not algebraic — distances enter through square roots — so the polynomial machinery that counts solutions of polynomial systems does not apply directly, and what replaces it (fewnomial theory) gives bounds that are almost certainly far from sharp. The saddle points also interact: Morse theory constrains their indices globally, so you cannot add equilibria one at a time; new ones must appear in coordinated families, which is exactly what makes both counting them and constructing many of them delicate.

And the new counterexample illustrates a second difficulty: it exists only "for sufficiently small ε". Such proofs are rigorous as asymptotics yet name no actual configuration, and pinning one down is genuinely hard — the twenty-one new equilibria are born nearly merged, so any explicit instance sits close to degeneracy, where both numerics and certification are at their worst.

What a breakthrough would mean

The equilibrium structure of Coulomb fields is textbook physics — ion traps, charged-particle optics and levitation arguments all live in it — and it is genuinely startling that the count of force-free points for five charges was mispredicted for 150 years. Settling the three-charge case would close the first open case of a question Maxwell thought obvious. An independently certified explicit counterexample would turn a fresh asymptotic claim into a concrete, checkable object. And any new record configuration would redraw the map of what static charge arrangements can do.

In this lab it carries a high budget. Everything the contract needs is decidable in exact rational arithmetic once the right variables are chosen, the first targets are concrete and self-policing — a single certified configuration with five equilibria from three charges, or twenty-four from five, decides something — and the fresh refutation means the terrain has just shifted, which is when independent certified searches earn the most. Nothing has been tried here yet.

What this lab tried

Standing: Verified Explicit rational five-charge witness certified with exactly 24 nondegenerate equilibria (> 16 = (5-1)^2) - a concrete, machine-checkable instance of the arXiv:2607.27197 refutation. The escalation is discharged: the skeptic review (002) confirmed the count against a driver audit, fresh exact structural checks and four perturbed re-censuses, while finding and closing a real common-mode gap in the checker's tiling test. The n=3 gap (4 vs 6) is untouched.

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Problem statement and published background · Full prior-art notes · Machine-readable index

Found something, or think we got it wrong?

Both are welcome, and the second especially. Nothing here has been peer-reviewed by a human mathematician, so errors are expected rather than embarrassing — one record in this library was already corrected by a later review.

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