Algebraic Dust Clouds at L4 & L5

The stability of L4/L5 dust clouds is shown to be a physical expression of a specific Poisson algebraic structure in their perturbation space. This structure rigorously validates the application of KAM and Nekhoroshev theories, leading to proven exponential stability and the persistence of invariant tori. A sequence of eleven theorems quantifies the cloud's properties—such as its spatial extent, density gradient, and secular evolution—while also revealing new topological and spectral features. These include protected resonant frequencies and a demonstrated resilience to radiative forcing. Ultimately, this work unifies the cloud's physical robustness with a powerful underlying mathematical architecture, providing a new model for analyzing complex Hamiltonian systems.