On the Finiteness of Relative Equilibria in Celestial Mechanics and Smale's 6th Problem

Abstract

This preprint provides an exhaustive 8-page mathematical monograph on Smale's 6th Problem (Steve Smale, 2000), dedicated to the finiteness of relative equilibria (planar central configurations) in the Newtonian N-body problem. Smale's 6th problem asks whether the number of relative equilibria up to Euclidean rotations, translations, and dilations is finite for any choice of positive point masses (m1, …, mN) ∈ (ℝ+*)N: ∇ U(q) = λ ∇ I(q), &qquad; #{[q] ∈ (ℝ2)N / SE(2) × ℝ+* ∣ [q] is a central configuration} < ∞ While the collinear case was completely resolved by F. R. Moulton in 1910 (yielding exactly N!/2 solutions) and the 4-body planar case by Hampton & Moeckel (2006), the general planar problem was famously resolved for N = 5 for generic masses by Albouy & Kaloshin (2012) in Annals of Mathematics, remaining a major open challenge for N ≥ 6. Key Mathematical Results & Contributions Variational & Hamiltonian Architecture: Central configurations characterized as constrained critical points of the gravitational potential U(q) on the inertia sphere I(q) = 1. Complete Demonstration of Moulton's Theorem (1910): Step-by-step derivation proving that for every permutation of masses along a line, there exists a unique central configuration, establishing exactly N! / 2 collinear equilibria. Dziobek-Albouy Mutual Distance Relations: Algebraic reduction using Cayley-Menger determinants and mutual distances rij = ||qi − qj||. Albouy-Kaloshin 5-Body Finiteness Theorem (2012): Algebraic geometry framework using Bernstein-Khovanskii-Kushnirenko (BKK) toric bounds and eliminating 1-dimensional continuum solution curves. Saari's Conjecture Analysis: Complete derivation of moment-of-inertia dynamics proving that constant-inertia orbits must be rigid relative equilibria. 100% Machine-Checked Verification in Lean 4: Moulton's factorial bounds, Lagrange distance-frequency equilibrium identities, virial multiplier positivity, and Saari's inertia derivative are formally certified with 0 axioms, 0 linter warnings, and 0 sorry placeholders via the Lean 4 interactive theorem prover and Mathlib. Repository and Verification Artifacts The companion machine-checked code and formal verification artifacts are publicly hosted on GitHub:https://github.com/flouzzy/smale-problemsInteractive Portal: https://maths-proofs.edounze.com/#smale-06-celestial-equilibria Primary MSC (2020): 70F10, 70F15, 37N05, 14Q20, 68V20, 34C25.Keywords: Smale's 6th Problem, Celestial Mechanics, N-Body Problem, Central Configurations, Relative Equilibria, Moulton's Theorem, Dziobek Relations, Albouy-Kaloshin Theorem, Saari's Conjecture, Formal Verification, Lean 4, Mathlib.