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Finiteness of Planar Central Configurations for Five Positive Masses

Primarily AI-generated textHuman understanding: some partsmath.DS — Dynamical Systemsmath-ph — Mathematical Physics

Contributed by Chih-Yao Wu ↗

Version 1 / Oct 22, 2026 / CC BY 4.0

Abstract

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 finiteness. The exceptional local calculations and their algebraic certificates are given explicitly. A separate Lean development accompanies the proof.

Provenance statement

AI agents produced the mathematical arguments, Lean formalization, and manuscript under the coordination of Chih-Yao Wu. An earlier manuscript and the completed Lean formalization were sent to Kuo-Chang Chen for expert feedback on October 6, 2026. Subsequent revisions expanded proof details and improved exposition and statement correspondence; the core formal statements and proof code have remained unchanged apart from comments. An independent human expert review of the complete argument has not yet been completed. The formalization and English review guide are available at https://github.com/andersonwu2000/five-body-finiteness.

Tools used

Lean
LeanVersion 4.30.0-rc2

References

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  1. Z. Chen and Z. Tong, Finiteness of planar central configurations of five bodies for all positive masses: On Smale's sixth problem , preprint (2026), version 3. arXiv:2610.16107v3 (https://arxiv.org/abs/2610.16107v3).arXiv

Version history

  1. v1Submitted by Chih-Yao WuInitial depositCurrentOct 22, 2026