What was already known
Gabrielov, Novikov and Shapiro had proved an upper bound of 12. Maxwell’s proposed bound of four remains open for three unequal charges.
AI involvement
Claude assisted with parts of the new saddle-separation argument. Gabrielov, D. Novikov, T. Novikov and Shapiro developed the final proof and report independently verifying all results.
The disclosed Claude work includes the paper's main new saddle-separation idea, within a human-led proof.
Importance
Halves the previous general three-charge upper bound from 12 to six.
Limitations
This does not reach or refute the bound of four for three unequal charges. Claude’s exact share of the saddle-separation lemma is not specified; symbolic and numerical checks are not documented as exact certificates.
Technical verification
The four authors explicitly report independent verification of all results. Symbolic and numerical checks are also mentioned, but their scope and certificate character are not documented sufficiently to count as machine checking.
Independent checking
Independent author verification of all results
- Checker
- Gabrielov, D. Novikov, T. Novikov and Shapiro, authors
- Scope
- Theorem 1.1, at most six nondegenerate equilibria for three positive charges and alpha>0
- Basis
- Reported by the linked checker
Human understanding
The authors explain the separated-variable saddle, four oval contacts and how the earlier Rolle count gives six.
Correction and publication history
Halves the previous general three-charge upper bound from 12 to six.
Read this published revision