{"id":"hexagon/2610.00467","pdf":"https://hexagonmath.org/pdf/2610.00467v1","url":"https://hexagonmath.org/abs/2610.00467v1","tags":["math-ph","msc:70F10","Lean","Smale sixth problem","celestial mechanics","central configurations","five-body problem","formal verification"],"score":0,"title":"Finiteness of Planar Central Configurations for Five Positive Masses","domain":"math.DS","origin":"ai-generated","source":"hexagon","authors":["Chih-Yao Wu"],"reviews":0,"subject":"math.DS","summary":"We prove that, for any five positive masses, there are finitely many planar Newtonian central configurations up to translations, rotations, and positive scaling. Building on the singular-diagram framework of Albouy and Kaloshin, we treat the exceptional masses left open by their finiteness theorem. The proof retains a lower bound for the physical potential after passing to the complex algebraic equations. On each fixed potential level, energy estimates, torque identities, and local response calculations control the possible degenerations. A polynomial selection argument then yields…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-10-22","reactions":{},"withdrawn":false,"authorship":"no-human-author-claimed","formalized":false,"human_understanding":"some"}