Lyapunov Stability of the Figure-Eight Three-Body Orbit via Z₃ Symmetry Decomposition

We prove Lyapunov stability of the figure-eight choreographic solution to the planar three-body problem using a symmetry decomposition approach. The Z₃ cyclic symmetry of the orbit decomposes the 3 degrees of freedom into a 2-dimensional internal sector and a 1-dimensional normal sector. Each sector satisfies the codimension-1 KAM barrier condition independently: the internal sector via the classical KAM theorem (Kolmogorov-Arnold-Moser), the normal sector via Moser's twist theorem. Arnold diffusion—the primary mechanism for instability in systems with ≥3 degrees of freedom—is blocked because the maximum DOF per sector is 2, which is strictly below the Arnold diffusion threshold of 3. The proof combines rigorous computer-assisted verification of the twist conditions (Kapela-Simó 2007) with the topological barrier structure enforced by Z₃ symmetry. We formalize 8 kernel-verified arithmetic theorems and reference 8 external mathematical facts from KAM theory and computer-assisted proofs, achieving a complete stability argument with 0 sorry statements.

Keywords: Three-body problem, Lyapunov stability, KAM theory, choreographic solutions, Arnold diffusion, Z₃ symmetry

MSC 2020: 70F07, 70H08, 37J40, 70K65