Finiteness of planar central configurations of five bodies with positive masses

We prove that, for every choice of positive masses, the planar Newtonian five-body problem has finitely many central configurations up to similarity. After normalization and removal of the translation and rotation freedoms, every physical configuration is isolated in the complex solution variety. The proof combines the classification of singular sequences by Albouy and Kaloshin with a lower bound for the potential, asymptotic estimates, and local analytic reductions.