Formalise the Maxwell conjecture on equilibria of point charges (arXiv:2607.27197)

## The conjecture

Let $q_1,\dots,q_n$ be non-zero charges at distinct points $a_1,\dots,a_n \in \mathbb{R}^3$, and let

$$V(x) \;=\; \sum_{j=1}^{n} \frac{q_j}{\lVert x - a_j\rVert}$$

be the resulting electrostatic potential on $\mathbb{R}^3 \setminus \{a_1,\dots,a_n\}$. The equilibria of the field are the critical points of $V$.

**Maxwell's conjecture** (as formulated by Gabrielov, Novikov and Shapiro from Maxwell's 1873 *Treatise on Electricity and Magnetism*, §113): if the critical points of $V$ are all non-degenerate, then there are at most $(n-1)^2$ of them.

**This is false.** Arathoon, Ball and Kvalheim (July 2026) exhibit five *positive* charges whose potential has at least $24$ non-degenerate critical points, against the conjectured bound of $(5-1)^2 = 16$: three unit charges at the vertices of an equilateral triangle inscribed in the unit circle of the plane $z = 0$, plus two charges of strength $q_\varepsilon = \tfrac34\varepsilon^3 - \tfrac{5}{32}\varepsilon^5$ at $(0,0,\pm\varepsilon)$, for all sufficiently small $\varepsilon > 0$. The three edge equilibria of the triangle persist while the central equilibrium bifurcates into 21.

## References

- [arXiv:2607.27197](https://arxiv.org/abs/2607.27197) — **The Maxwell Conjecture is False**, Philip Arathoon, Gavin Ball, Matthew D. Kvalheim (2026)
- [Mystery of point charges](https://doi.org/10.1112/plms/pdm012) — Gabrielov, Novikov, Shapiro, *Proc. London Math. Soc.* **95** (2007), 443–472 — where the conjecture is formulated, with the first general upper bound $2^{2n^2}(3n)^{2n}$
- [Counting equilibria of the electrostatic potential](https://arxiv.org/abs/2501.05315) — Edelsbrunner, Fillmore, Oliveira, *Proc. London Math. Soc.* **132** (2026) — best known upper bound $2^n(3n-2)^3$, and a record ratio of $25/7$ equilibria per unit charge
- [Maxwell's conjecture on three point charges with equal magnitudes](https://doi.org/10.1016/j.physd.2015.07.007) — Ya-Lun Tsai, *Physica D* **309** (2015), 86–98 — the $n = 3$ equal-magnitude case

## Proposed formalisation

`FormalConjectures/Arxiv/2607.27197/MaxwellConjecture.lean`, on `EuclideanSpace ℝ (Fin 3)`, with definitions for the Coulomb potential, its equilibria (via Mathlib's `gradient`, away from the charge locations), and non-degeneracy of the Hessian. Statements:

- the conjecture itself, as `answer(False) ↔ Conjecture` — `research solved`
- Theorem 1, for the explicit configuration above — `research solved`
- Theorem 1 part 2: a perturbation of the charge strengths makes the potential Morse, still with $\geq 24$ critical points — `research solved`
- Lemma 2: the limiting polynomial $\Phi_0 = \tfrac{5}{32}H_2 + \tfrac{15}{8}H_3 + \tfrac{3}{16}H_4$ has exactly 21 critical points, all non-degenerate — `research solved`
- Proposition 1: for every $m$, a configuration of $3 + 2m$ positive charges with $k \geq 4 + 20m$ non-degenerate equilibria — `research solved`
- the $n = 2$ bound — `textbook`
- Tsai's $n = 3$ equal-magnitude case — `research solved`
- the $n = 3$ case for arbitrary charges, still open — `research open`
- whether *any* polynomial bound in $n$ exists, still open: the known lower bound is now linear ($\sim 10n$) and every known upper bound is exponential — `research open`
- two proved `category test` lemmas pinning the counterexample coordinates

## Possible follow-ups

Gabrielov–Novikov–Shapiro also posed two conjectures about $V_p(x) = \sum_i \zeta_i^p / \lVert x - A_i\rVert^p$. Conjecture 1.8(a) was disproved by Edelsbrunner–Fillmore–Oliveira using the truncated octahedron. Conjecture 1.9 — that for all $p \geq 1$ the restriction of $V_p$ to any straight line in $\mathbb{R}^3$ has at most $2n-1$ equilibria — would be a natural addition; I haven't checked its current status.

I have a formalisation ready and will open a PR against this issue.

Start by reading FormalConjectures/Arxiv/2607.27197/MaxwellConjecture.lean and its definitions for the Coulomb potential, equilibria, and Hessian non-degeneracy. Review the listed conjecture, theorem, lemma, and proposition statements against the cited paper; done means the proposed formalisation is complete and the open or solved status is accurately represented.

Written by the indexing model from the issue text.