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)
Science result
- Field
- physics / classical electrostatics / real algebraic geometry
- Problem
- Maxwell's conjecture on the number of equilibria of n point charges (open since 1873)
- Result
- A five-charge configuration with at least 24 non-degenerate critical points, exceeding the conjectured bound (n−1)² = 16.
- AI system
- GPT-5.6 Sol
- Human role
- AI suggested the construction idea; humans verified it with computer algebra and wrote the paper
- Verification
- Preprint; computer-algebra checks; not peer-reviewed
- Status
- pending
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.
Changelog
- 2026-09-30: created
Related events
- OpenAI broadly releases GPT-5.6 (Sol, Terra, Luna) after government-gated preview ★★★★
- Summer 2026 flood: dozens of named conjectures settled on arXiv with disclosed AI help (July–September catalogue) ★★★★
Sources (1)
id: 2026-07-29-maxwell-point-charge-conjecture-false · updated 2026-09-30 · open in the interactive timeline