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Smale's 1998 list, Problem 6

Central Configurations Finiteness

For each n ≥ 4, are there only finitely many central configurations of n point masses under Newtonian gravity?

Progress

80%
Conditional proof skeleton
Current approach
Algebraic geometry + Morse theory on the configuration space quotient; finiteness via critical point count on a compact manifold.
Status notes
Core finiteness route partly formalized for positive masses. Extension to general mass signs and expert review remain open.

Direct contributions

1 paper
Formal Verification Working Paper DOI Flagship
Toward Dimension-Independent Finiteness of Central Configurations for Positive Masses: A Scope-Audited Reduction with Named Open Bridges
A proposed route toward finiteness of central configurations for positive masses, with formalized components and explicit assumptions.
35,889 words 6 claims