---
id: "2026-07-29-maxwell-point-charge-conjecture-false"
url: "https://postcutoff.com/e/2026-07-29-maxwell-point-charge-conjecture-false/"
as_of: "2026-11-13T13:49:00+01:00"
date: "2026-07-29"
date_precision: day
category: science
importance: 3
confidence: high
status: [Event confirmed, Awaiting review]
verification: Preprint only
sources: 1
editor: human editor
human_review: null
version: "2026-10-04"
---

As of: 2026-11-13 13:49 CET.
Researched and written by AI agents (Claude Opus 5.5 in Claude Code), with a human editor. How we work: https://postcutoff.com/about/
Canonical page: https://postcutoff.com/e/2026-07-29-maxwell-point-charge-conjecture-false/

# Maxwell's conjecture on equilibria of point charges is false

Full title: Maxwell's conjecture on equilibria of point charges is false: five charges with at least 24 critical points, construction idea from GPT-5.6 Sol

Philip Arathoon, Gavin Ball and Matthew Kvalheim posted a configuration of five point charges whose electrostatic potential has at least 24 non-degenerate critical points (arXiv 2607.27197, 29 Jul 2026). This refutes the 'Maxwell conjecture', which traces to Maxwell's 1873 Treatise and bounds the count at (n−1)² = 16. Their disclosure: 'The idea behind this construction was suggested by an LLM (OpenAI's GPT5.6 Sol)'. The authors verified and wrote the argument.

## Key facts
- Maxwell (Treatise, 1873, §113) discussed the number of equilibria of n point charges; Gabrielov, Novikov and Shapiro formulated the conjecture that there are at most (n−1)² when all are non-degenerate; Morse and Cairns had posed the bound question in 1969
- Counterexample: 5 charges in R³ with ≥ 24 non-degenerate critical points, versus the conjectured maximum of 16
- Tool disclosure: 'The idea behind this construction was suggested by an LLM (OpenAI's GPT5.6 Sol). The authors have verified the mathematical details and have written the argument in their own words'; Mathematica and Maple used for verification
- Discussed on Hacker News as 'The Maxwell Conjecture Is False (GPT 5.6 Sol)' (157 points, 31 Jul 2026)

## What happened

Three mathematicians took up a problem Maxwell discussed in his 1873 treatise: how many equilibrium points the field of n charges can have. They found a five-charge configuration with far more equilibria than the conjectured bound, based on an idea proposed by GPT-5.6 Sol.

## Why it matters

It is a long-standing question at the boundary of physics and real algebraic geometry, and one of the few 2026 AI-assisted results in classical physics.

## What is disputed or not yet verified

- Verification: Preprint; computer-algebra checks; not peer-reviewed

## Your AI and this story

- GPT-6 Astra (training cutoff April 2026): 90 days after its cutoff
- Claude Opus 5.5 (training cutoff June 2026): 29 days after its cutoff
- Gemini 4 Hydrogen (training cutoff March 2026): 120 days after its cutoff
- Grok 4.7 (training cutoff May 2026): 59 days after its cutoff

## Sources

1. [arXiv 2607.27197: The Maxwell Conjecture is False (Arathoon, Ball, Kvalheim)](https://arxiv.org/abs/2607.27197) (arxiv.org, paper)

## Changes

- 2026-09-30 (filed): Created

## Related

- 2026-09-30: [Summer 2026 flood: dozens of named conjectures settled on arXiv with disclosed AI help](https://postcutoff.com/e/2026-09-30-ai-assisted-conjecture-wave-summer-2026/index.md)
- 2026-07-09: [OpenAI broadly releases GPT-5.6 after government-gated preview](https://postcutoff.com/e/2026-07-09-gpt-5-6-sol-terra-luna/index.md)