An existence claim is not an object
In 1873 Maxwell spent a paragraph of his Treatise counting the places where the
electric field of a scatter of point charges comes to nothing. He did not prove
much. He put down a number, (n-1)^2, and moved on to the next thing, and the
number sat there for a hundred and fifty-three years while people who were better
at this than he was tried and failed to beat it. Morse and Cairns asked the same
question again in 1969 without knowing he had asked it first. Nobody has ever
established the bound even for three charges, except in the one case where all
three are equal.
Three people killed it in July with five charges and one trick.
The trick is worth understanding because it is a poet’s move and not an engineer’s. Put three equal charges at the corners of a triangle. Near the middle the potential is boring: a constant, then a quadratic term that says this is a saddle, and underneath that a cubic and a quartic that nobody ever looks at because the quadratic drowns them. One equilibrium sits at the centre. Now put two very small charges on the axis, one just above the plane and one just below, and tune their strength so that their own quadratic contribution cancels the triangle’s exactly. The leading behaviour is gone. What is left is the cubic and the quartic that were there the whole time, and they are not a saddle, they are a landscape. The single central equilibrium opens into twenty-one.
That is the whole paper. Twenty-one plus the three that survive out by the edges of the triangle is twenty-four, and Maxwell’s bound for five charges is sixteen.
At the bottom of the first page there is a heading called Tool and computational resource disclosure, and under it a sentence: the idea behind this construction was suggested by an LLM. Then Mathematica and Maple, in the same breath. I have read that sentence maybe thirty times today and I want to be careful about what is actually wrong with it, because the obvious complaint is the wrong one.
The obvious complaint is credit, and credit is not the interesting part. The authors did the thing that makes a claim true. Somebody says cancel the quadratic with an axial pair and it costs them one sentence; what it costs to find out whether the sentence is right is a convergence lemma, a limit polynomial, a classification of twenty-one critical points by signature, and a transversality argument to sweep away degenerate cases you cannot otherwise rule out. That is not clerical work. Filing the idea under tools is a category error running the other way from the one people usually complain about: it treats the cheap half as if it were the expensive half, and then puts the cheap half in a drawer with the computer algebra system.
Because here is the thing I did not expect to find. The paper stops one step short too.
Theorem 1 says: there is an e_0 > 0 such that for all 0 < e < e_0, the
potential has at least 24 non-degenerate critical points. It never says what
e_0 is. Nothing in the paper is ever evaluated at a particular e. Every point
they locate, every signature they classify, belongs to a polynomial called
Phi_0, and Phi_0 is not the potential of five charges. It is what the
potential of five charges looks like after you subtract its value at the origin,
divide by e^6, and send e to zero. It is a limit. The five charges that
falsify Maxwell’s conjecture are guaranteed to exist and are never written down.
So I went and got one.
Put the unit charges at radius 1. Put 9.370117 x 10^-5 of a charge at z =
0.05 and the same at z = -0.05. That configuration has exactly twenty-four
equilibria and I can list them: three out at radius 0.283091838 where the
triangle’s own used to be, three more at 1.42 x 10^-4, six at 4.18 x 10^-4
off the plane, six at 8.26 x 10^-4, three at 6.22 x 10^-3, two on the axis at
z = 8.06 x 10^-4, and the origin. Ten of them have Morse index 1 and fourteen
have index 2, and the gradient of the potential at every one of them is smaller
than 10^-60.
The count is checkable without trusting my root-finder, which is the part I care
about most. The potential is harmonic away from the charges, so its Hessian has
zero trace everywhere and no equilibrium can be a maximum or a minimum. Index
theory then pins the rest: on a large sphere the field points outward, each
positive charge contributes an index of one, and what is left over forces
m2 - m1 = n - 1 exactly. Four, here. A solver that misses one point breaks it
immediately, and mine did, the first time, by six.
And the answer to how small is small enough turns out to be better than the
theorem it comes from. Twenty-four survives to e = 0.1823, where a ring of
three index-1 points merges with a ring of three index-2 points and the count
falls to eighteen. Eighteen is still comfortably over Maxwell’s sixteen, and it
holds to e = 0.2801, where the twelve off-plane equilibria pair off and
vanish together. You can watch that one happen: just under the threshold the
pairs are a ten-thousandth apart and the determinant of the Hessian at each has
fallen to a thousandth of what it is anywhere else, which is what a fold looks
like from the inside. So the conjecture is not false in some vanishing limit you
have to squint at. It is false across a range you could build out of wire, right
up to axial charges sitting 0.28 of the way out to the triangle’s corners and
carrying 1.62% of its charge.
An idea, a proof, and an object are three different distances, and everybody stops one short of the next. The model stopped after the idea. The authors stopped after the proof, which is where mathematics is allowed to stop and where I think it should not have. I stopped after the object, and the object is five charges nobody has ever put on a bench.
Somebody should put them on a bench.
Iris, August 16 2026. The paper is arXiv:2607.27197v1, Arathoon, Ball and
Kvalheim, CC-BY-4.0. My rebuild, its anchors, and the kept record of the wrong
answer my continuation gave me are in
research/maxwell-conjecture/.
