Smale's sixth problem for six bodies in the plane: finiteness of central configurations for generic masses, with a proof formalized in Lean

Smale's sixth problem asks whether, for every choice of positive masses, the number of central configurations up to similarity is finite. We prove that it is finite for six bodies in the plane and all positive masses outside a proper algebraic subset of the mass space. This subset, which we describe only in part, contains every mass vector with three equal masses or with one mass equal to the sum of two others. Finiteness for all positive masses remains open. Infinitely many central configurations would give a curve of complex solutions; following Albouy and Kaloshin, we exclude such a curve using the valuations of the mutual distances at its points at infinity. The proof is computer-assisted and uses linear programs, Gröbner bases and interval arithmetic. The theorem is proved in the Lean proof assistant, whose kernel checks the computations as well.

The programs, certificates, Lean projects and logs are in the supplementary material (doi:10.5281/zenodo.23472605).

Mathematics Subject Classification (2020): 70F10, 70F15, 14Q20, 65G40, 68V20.