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
Event confirmedAwaiting review
The takeaway
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).
Status
- Claim
Event confirmedAwaiting review
- Our reporting
- High confidence
- Verification
- Preprint only
- Importance
- 3 of 5
- Last verified
- 30 September 2026
Your AI and this story
- GPT-6 Astra90 days after its cutoff
- Claude Opus 5.529 days after its cutoff
- Gemini 4 Hydrogen120 days after its cutoff
- Grok 4.759 days after its cutoff
None of these four assistants can know about it. The closest, Claude Opus 5.5, stops 29 days before it.
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| Verification | Preprint; computer-algebra checks; not peer-reviewed |
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Sources
1 source from 1 site. Numbers match the chips in the text.
1 source: 1 primary
Primary
Changes
- Filed