What was already known
Maxwell’s proposed general upper bound had not been disproved. The maximum for three unequal positive charges remained unresolved.
AI involvement
GPT-5.6 Sol suggested adding two small axial charges to three charges on an equilateral triangle. Arathoon, Ball and Kvalheim chose the scaling, proved persistence and transversality, and checked the argument.
The authors explicitly credit the model with the construction idea that yields the counterexample; humans completed the proof.
Importance
Disproves a longstanding proposed upper bound for point-charge equilibria.
Limitations
The authors verified the mathematical details; no independent completed check of the full perturbation argument was located. The separate three-charge bound concerns a different problem.
Technical verification
The authors explicitly say they verified the mathematical details of the complete five-charge argument. This is a reported human check, not reproduced here; the described algebra-system use is not a documented exact certificate check.
Independent checking
Verification of mathematical details and author-written proof
- Checker
- Philip Arathoon, Gavin Ball and Matthew D. Kvalheim, authors
- Scope
- Five-charge counterexample with at least 24 nondegenerate critical points and Morse perturbation
- Basis
- Reported by the linked checker
Human understanding
The authors explain the central bifurcation from one equilibrium to 21 and why the other three persist.
Correction and publication history
Disproves a longstanding proposed upper bound for point-charge equilibria.
Read this published revision