# 麦克斯韦猜想是错误的（GPT 5.6 解法）

- 来源：Hacker News 热门（buzzing.cc 中文翻译）
- 作者：rahen
- 发布时间：2026-08-01 00:25
- AIHOT 分数：54
- AIHOT 链接：https://aihot.virxact.com/items/cms96r96y06gdro9ki8ewphyg
- 原文链接：https://arxiv.org/abs/2607.27197

## AI 摘要

研究人员构造出五个点电荷的静电势配置，其至少存在24个非退化临界点，从而推翻麦克斯韦猜想（该猜想认为点电荷场最多有临界点且全部非退化）。构造始于等边三角形顶点上的三个单位电荷，再加入两个小电荷形成浅三角双锥，使中心平衡点分叉为21个平衡点。该构造思路由OpenAI的GPT-5.6 Sol提出，作者已核实数学细节并自行撰写论证。

## 正文

The Maxwell Conjecture is False

Philip Arathoon, Gavin Ball, Matthew D. Kvalheim

Babson College, MA, USA,

parathoon@babson.edu

University of Missouri, Columbia, MO, USA,

gavin.ball@missouri.edu

University of Maryland, Baltimore County, MD, USA,

kvalheim@umbc.edu

Abstract

We exhibit a configuration of five point charges in Euclidean space whose electrostatic potential admits at least 24 critical points all of which are non-degenerate. Maxwell’s conjecture that the field of point charges has at most critical points which are all non-degenerate is therefore false.

1 Introduction

In J. C. Maxwell’s 1873 treatise on electricity and magnetism he discusses the number of equilibria of the electric field generated by point charges [5, §113]. Apparently unaware of this, M. Morse and S. S. Cairns in 1969 posed the problem of finding an upper bound for the number of equilibria [6, p. 293]. The first general bounds were supplied by A. Gabrielov, D. Novikov, and B. Shapiro in [3] who, based on their reading of [5, §113], formulated the ‘Maxwell conjecture’ which states that if the critical points of the electrostatic potential generated by point charges are all non-degenerate then their number cannot exceed . These bounds were later improved by V. Zolotov in 2023 [8] and further improved by H. Edelsbrunner, C. Fillmore, and G. Oliveira in 2026 [2]. Maxwell’s bound is trivially achieved for but it is not known even for if 4 is the maximum number, except in the case of equal charges [7]. Further related problems in classical electrostatics are discussed in [1].

In this note we produce a counterexample to Maxwell’s conjecture: a configuration of five point charges with at least 24 critical points all of which are non-degenerate. The counterexample starts with three unit charges placed at the vertices of an equilateral triangle. This configuration has four equilibria: one at the centre and three displaced inwards around the edges of the triangle, as shown in Figure 1(a). Two small charges are then added to the centre of the triangle and moved slightly apart along the orthogonal axis to form a shallow triangular bipyramid. The three edge equilibria persist under the addition of the small axial charges whereas the central equilibrium bifurcates into a family of 21 equilibria.

Acknowledgments

Kvalheim was supported in part by the Air Force Office of Scientific Research under award number FA9550-24-1-0299.

Tool and computational resource disclosure

The idea behind this construction was suggested by an LLM (OpenAI’s GPT-5.6 Sol). The authors have verified the mathematical details and have written the argument in their own words. Computer algebra software (Mathematica, Maple) was used to verify computations and produce visualisations.

2 Counterexample

Place three unit charges at the vertices of an equilateral triangle

The Coulomb potential generated by these charges has the following Taylor expansion around the origin

are harmonic polynomials with . Place two additional -charges along the triangle’s axis of symmetry at

The potential generated by the axial charges has the expansion

Theorem 1.

There is an such that for all the potential generated by five charges located at with respective strengths and

(1)

admits at least 24 non-degenerate critical points. Moreover, a small perturbation to the charge strengths gives a potential with a finite number of critical points all of which are non-degenerate.

To prove Theorem 1, we introduce a deformed potential defined as

Lemma 1.

Let as in (1) and let

Then there exists an open neighbourhood of and a function real-analytic on such that

whenever and Consequently, for every , and every compact we have for all sufficiently small .

Proof.

Let be any compact set. Write Since is real analytic near and the are homogeneous, we have

where is real-analytic near For the axial pair,

This function is real analytic and even in and its Taylor expansion is

Substituting and adding the and terms gives

for some real-analytic Dividing by proves the formula for . The -estimate follows because the derivatives of are bounded on a neighbourhood of . ∎

Remark 1.

The charge is judiciously chosen to have the form in (1) for two reasons. Firstly, in order for to exist the order of must be at least , the coefficient of must equal and there must be no term. Secondly, the specific coefficient of is chosen out of convenience to simplify the form of ; it may be replaced with any negative coefficient greater than without affecting our results.

Lemma 2.

The polynomial has exactly 21 critical points all of which are non-degenerate.

Proof.

In cylindrical coordinates

with partial derivatives

The critical points can now be found and their Hessians evaluated with a routine calculation. These points are classified as follows:

➢

The origin with signature .

➢

Two axial equilibria at and with signature .

➢

Six planar equilibria at , , and with signature for the outer radius and for the inner.

➢

Twelve off-planar equilibria at with signature and with signature for .

Remark 2.

For the original triangular configuration the central equilibrium has Morse index 1 and the three edge equilibria each have Morse index 2. These satisfy the Euler characteristic equation

for , , and where represents the charge locations and the point at infinity. When the two small axial charges are added the three edge equilibria persist but the central equilibrium bifurcates into the critical points of , 10 of which have Morse index 1, and 11 have Morse index 2. The Morse inequalities therefore remain consistent with , , , and .

Proof of Theorem 1.

By Lemma 1, the function is smooth, so the implicit function theorem implies that the 21 non-degenerate critical points of provided by Lemma 2 persist to 21 non-degenerate critical points of for all sufficiently small . These yield 21 non-degenerate critical points of at distance from the origin.

On the other hand, the implicit function theorem also implies that the three non-zero non-degenerate critical points of persist to three non-degenerate critical points of for all sufficiently small . The distance of these from the origin is bounded away from , so they are distinct from the preceding 21 and hence has at least 24 non-degenerate critical points for all sufficiently small .

Fix such an . We have not eliminated the possibility that has degenerate critical points elsewhere. To get rid of them, we perturb the vector of charges of the particles.111Essentially the same argument shows that we need only perturb the charges of any 4 of these particles. Alternatively, a different application of parametric transversality shows that we need only perturb the location of a single particle [6, Thm. 6.2]. Let be the electrostatic potential generated by particles at the same locations as those generating , but with charge vector . Consider the smooth map

given by the resulting electric field. Since do not lie in a common plane, it is straightforward to show that is a submersion (cf. [4, Ex. 1.7.22]), so the parametric transversality theorem implies that is a regular value for and hence is a Morse function for almost every [4, p. 68].

By the implicit function theorem, choosing any such sufficiently close to yields a Morse potential having at least 24 critical points. And since all of the charges are positive, all critical points of belong to the compact convex hull of , so there can be only finitely many. ∎

Remark 3.

The construction employed in this counterexample can be iterated in a manner similar to constructions in [2]. Suppose that, after finitely many steps, the resulting potential enjoys the same -symmetry as the original triangle configuration. The symmetry forces the Taylor expansion of around the origin to be of the form

(2)

where, for induction, we suppose and are both greater than zero (as is the case for the original potential of the triangular configuration). Place a pair of charges of strength at a distance along the -axis. The contribution of this pair near the origin is

so its leading term cancels the term in . On the scale and after dividing by a repeat of the argument above shows that the new, deformed, potential converges to

as For an appropriate choice of and this potential has the same nondegenerate critical points as found above. For example, the choice makes the limit of the deformed potential a multiple of the above. One of these critical points is the origin, which was already a critical point before the new pair of charges was added. The remaining 20 are new. Since the potential of the added pair tends to zero in on compact sets disjoint from the origin, the nondegenerate critical points of persist in the new potential. Thus, we have added 20 critical points at the expense of adding two small positive charges. Moreover, after this step the expansion (2) of the new potential retains the same form, with coefficient of equal to and coefficient of unchanged, so the argument may be repeated. Applying this iteration to the initial potential and using the perturbation argument from the proof of Theorem 1 gives the following

Proposition 1.

For every there exists a configuration of positive point charges whose potential has a finite number of critical points, all of which are non-degenerate.

Therefore, this construction produces an asymptotic critical point to charge ratio of , which is larger than the value of achieved in [2].

媒体内容 · 前往原文查看

Figure 1: (a) The electric field generated by an equilateral triangle with no axial charges. The next three figures give numerical evidence for 24 equilibria when two axial charges are added with : (b) two off-planar equilibria in the positive -quadrant of the plane; (c) the potential along the -axis showing 3 axial equilibria; (d) the potential along the ray (with split axes) showing 3 planar equilibria.

References

[1] A. Abanov, N. Hayford, D. Khavinson, and R. Teodorescu, Around a theorem of F. Dyson and A. Lenard: energy equilibria for point charge distributions in classical electrostatics, Expo. Math., 39 (2021), pp. 182–196.

[2] H. Edelsbrunner, C. Fillmore, and G. Oliveira, Counting equilibria of the electrostatic potential, Proc. Lond. Math. Soc. (3), 132 (2026), pp. Paper No. e70163, 32.

[3] A. Gabrielov, D. Novikov, and B. Shapiro, Mystery of point charges, Proc. Lond. Math. Soc. (3), 95 (2007), pp. 443–472.

[4] V. Guillemin and A. Pollack, Differential topology, AMS Chelsea Publishing, Providence, RI, 2010. Reprint of the 1974 original.

[5] J. C. Maxwell, A Treatise on Electricity and Magnetism, vol. 1, Clarendon Press, Oxford, 1873.

[6] M. Morse and S. S. Cairns, Critical point theory in global analysis and differential topology: An introduction, vol. Vol. 33 of Pure and Applied Mathematics, Academic Press, New York-London, 1969.

[7] Y.-L. Tsai, Maxwell’s conjecture on three point charges with equal magnitudes, Phys. D, 309 (2015), pp. 86–98.

[8] V. Zolotov, Upper bounds for the number of isolated critical points via the Thom-Milnor theorem, Anal. Math. Phys., 13 (2023), pp. Paper No. 81, 18.