Maxwell said five charges allow at most 16 equilibria. At ε = 1/6, the counterexample’s 24 are now certified.

Arathoon, Ball, and Kvalheim (arXiv:2607.27197, idea credited to an OpenAI model, argument written and verified by the authors) refute Maxwell’s 1873 point-charge conjecture: five positive charges whose potential has at least 24 nondegenerate critical points, exceeding the proposed bound (n−1)² = 16. The paper’s evidence is floating-point computer algebra — Mathematica and Maple checks, no code artifact, no certificate, no concrete ε. We pinned the configuration the paper’s own Figure 1 depicts, ε = 1/6 with q_ε = 859/248832 exactly, and certified it: 24 pairwise-disjoint Krawczyk boxes, each proving existence, local uniqueness, and Hessian nondegeneracy of a critical point. 20 checks, 3/3 mutation controls rejected, 0.1 s. Since 24 > 16 in exact integers, the bound fails at this explicit configuration.

What was checked

The configuration is the paper’s own: unit charges at the vertices of an equilateral triangle in the plane, charges q_ε = (3/4)ε³ − (5/32)ε⁵ at (0, 0, ±ε). The exact layer establishes q_ε = 859/248832 from the paper’s eq. (1) at ε = 1/6, positivity of all five charges, and the claim gate 24 > 16 — all in eqcert rationals, with ε, q_ε, and √3 entering interval arithmetic only through enclosures proved to contain the exact values. Float Newton from the paper’s Lemma 2 seed points, closed under the configuration’s exact D₃×Z₂ symmetry, supplies exactly 24 candidates in orbit shells of sizes [1,3,2,6,6,3,3] — and candidates carry no evidentiary weight. Every asserted fact rests on the certificates: per point, a Krawczyk box (existence + local uniqueness) and an interval Hessian determinant excluding 0 over the whole box; 0 lies in every trace interval, the harmonicity witness; all 276 box pairs disjoint, so the 24 are genuinely distinct points.

The structure matches the paper too: Morse indices certified per box via Sylvester leading-minor sign intervals give per-shell indices [1,2,2,1,2,1,2] and totals {index 1: 10, index 2: 14} — the paper’s m₁ = 10, m₂ = 14 of Remark 2 — with no index 0 or 3 anywhere, as harmonicity demands. And ε = 1/6 is not arbitrary: it is the value of the paper’s Figure 1, chosen inside the window where the 24 points are both present and well-conditioned (min |det Hess| 1.3e−6 at the origin, matching the (125/2048)ε⁶ prediction to 3 digits), so the certificate speaks directly to the figure the paper offers as evidence.

The mutation that proves the teeth

M3 is the sharpest: drop the ε⁵ term of eq. (1) and the Hessian at the origin becomes the exact zero matrix — that term is precisely what makes the origin nondegenerate — and 0 of 24 boxes certify. M1, flipping the ε⁵ sign, reorganizes the bifurcated family and only 1 of 24 certifies. M2 proves the counting has teeth, not just the enclosures: duplicate one candidate and all 24 boxes still certify individually, but the disjointness logic rejects the set — 24 boxes are not 24 points unless they are pairwise disjoint.

What was NOT audited, stated so it cannot be assumed

Why one pinned instance is the right certificate

The paper proves an asymptotic statement with an unspecified threshold; its numerical evidence is one figure at one ε. A validated-numerics certificate is the complement: it cannot reach “for all sufficiently small ε”, but at the instance it pins it upgrades float evidence to proof — existence, distinctness, and nondegeneracy of all 24 points, with nine orders of magnitude between box size and box spacing. Maxwell’s bound needs only one configuration to fail, and that configuration is now certified.