Spectral Dissipation Structures and Generator Regularity Hierarchies in Dissipative Covariance Flows: Directional Dissipative Resolution and Sobolev Extension

This paper investigates the structural recurrence of spectral irreversibility in dissipative covariance flows from three independent analytical layers: topological, measure-theoretic, and spectral. Starting from the KL divergence structure established in prior work 2, we verify that both the Fisher-Rao metric and the KL divergence fail to track dissipative directionality in the infinite-dimensional limit, and define a dissipative path functional Idir (t) = Tr ( (∂ₜ Cₜ) Cₜ⁻¹ (∂ₜ Cₜ) ) = 4 Σ n^4-2α e^-2n²t that tracks the spectral irreversibility of the flow. This functional is not an information metric on a statistical manifold, but a Lyapunov-type dissipative observable derived from the heat semigroup generator. The principal outcome is the emergence of a common spectral regularity hierarchy α > k + 1/2 (k = 0, 1, 2) shared across all three layers, corresponding precisely to the Sobolev spaces Hᵏ on ℓ². The convergence of Idir is guaranteed solely by t > 0, independently of the Gaussian measure existence condition, suggesting extensibility to broader dissipative systems. This paper is the third in a series: 1 established operational distinguishability collapse for dt k + 1/2 (k = 0, 1, 2) 를 공유한다는 구조적 발견이며, 이 계층은 ℓ² 위의 Sobolev 공간 Hᵏ와 정확히 대응한다. Idir의 수렴은 t > 0 조건 하나로 보장되며 상태공간에 무관하게 성립한다. 본 논문은 시리즈의 세 번째로, 1은 dt < 0 방향의 연산적 구별 가능성 붕괴를, 2는 이를 무한차원 측도론으로 재정식화하였다. Keywords: dissipative covariance flow, spectral regularity hierarchy, Sobolev spaces, Fisher-Rao metric, Feldman-Hajek theorem, heat semigroup, infinite-dimensional Gaussian measures, directional dissipative resolution