Algebraic Foundations of Cosmic Coherence: A Unified Theory of Order, Chaos, and Dissipation at the Triangular Lagrange Points

The enduring existence of coherent dust clouds at the triangular Lagrange points L₄ and L₅ presentsa fundamental paradox in celestial mechanics. Classical theory, which treats Hamiltonian chaos andradiative dissipation as separate and antagonistic forces, predicts that such tenuous structures shouldbe transient, lasting centuries at most. Yet observations and numerical evidence suggest persistent,structured clouds that defy this fragmented narrative. This paper resolves the paradox by introducinga unified algebraic-dynamical framework. We first prove that the space of all physicalperturbations—gravitational, radiative, and dissipative—forms a Poisson algebra (Theorem 0),providing a single mathematical stage and rulebook for their interactions. From this foundation, wederive a sequence of ten interlocking theorems that collectively overturn the classical view. Thetheory demonstrates that: (1) a spectral gap Δ(𝜇) ∼ 𝑂(√𝜇) prevents resonant overlap, confiningchaos to exponentially thin layers in phase space; (2) Nekhoroshev-type estimates yield stabilitytimes 𝜏 ≥ exp⁡(𝐾/√𝜇), rendering Trojan clouds effectively permanent for systems like Sun–Jupiter;and most pivotally, (3) dissipation is not destructive but morphogenetic. Through the machineryof the renormalization group, weak dissipative forces drive the system toward a universal fixed-pointmorphology: a scale-invariant density profile 𝜌(𝑅) ∝ 𝑅−𝛼(𝜇) in physical space,with 𝛼(𝜇) determined solely by the mass ratio.Thus, the coherence of Lagrange-point dust clouds is not a fragile balance against chaos and decay,but a mathematical inevitability emerging from the deep Poisson-algebraic architecture thatinherently structures perturbation space. This work redefines dissipation as a sculptor of form andchaos as a confined exception, offering a new paradigm for understanding order in complexHamiltonian systems with weak non-conservative forces.