Self-Similar Scaling, Renormalization, and Universal Dust Morphology: A Detailed Analysis of Theorem 9 and Its Corollaries

Theorem 9 in Doucette (2025) advances a striking claim about the long-time behavior of dissipative Trojan dust clouds. Rather than treating the weakly dissipative circular restricted three-body problem as merely a perturbed Hamiltonian system whose details must be followed grain by grain, the theorem shows that, after orbit-averaging and suitable scaling, the system admits a renormalization-group interpretation. In that formulation, the asymptotic evolution is governed by a scale-invariant fixed point in action space and by a corresponding power-law density profile in physical space. The central result is that the action-space distribution approaches a universal inverse-action form, , and that the spatial density assumes a power-law decay , with  in the harmonic approximation. The associated corollaries deepen the theorem in ten distinct ways: they establish the uniqueness of the fixed point, identify the scaling symmetry of the transport equation, produce a self-similar renormalization orbit, show the existence of a scale-invariant measure, derive the spatial power law, recover the explicit harmonic-law exponent, organize systems into universality classes by mass ratio, verify compatibility with particle-number conservation, characterize exactly scale-symmetric initial data, and show the convergence of cloud morphology to a self-similar form. Taken together, the theorem and its corollaries do not merely supply a technical statement about one transport equation. They articulate a full asymptotic theory of how weak dissipation and resonant geometry combine to produce universal structure in Trojan dust clouds. This article presents that theory as an independent, coherent account, emphasizing its mathematical content, its physical meaning, its observational consequences, its possible empirical tests, its broader uses, and its future research directions.