A Lean Proof of the Thomson Problem for Seven Electrons
Blog post from Vals
A team of ten AI agents produced a Lean-verified proof that the regular pentagonal bipyramid uniquely minimizes Coulomb energy among seven distinct points on the unit sphere, resolving the N=7 Thomson problem up to rotations, reflections, and relabeling. The proof divides possible configurations according to their smallest pairwise inner product, using exact semidefinite-programming certificates to rule out broad regions and then applying interval-arithmetic rigidity and an exact second-order local-minimality result to establish uniqueness near the bipyramid. Although numerical optimization was used to discover and round certificate data, all resulting calculations are checked with exact arithmetic in Lean’s kernel rather than relying on floating-point computation. The 17,895-line Mathlib-only formalization was built from a clean copy, checked against the fixed target statements, audited for axioms, accepted by an independent kernel implementation, and tested with a deliberately altered certificate value that caused verification to fail. The work builds on recent Lean-verified N=8 methods and was completed over roughly 15 hours through 1,270 agent messages, with sources, scripts, and verification records published in a public repository.
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