Covariant Multi-Scale Negative Coupling on Dynamic Riemannian Manifolds: A Geometric Framework for Topological Persistence in Infinite-Dimensional Systems

Abstract

Dimensional reduction is a generic consequence of dissipation in nonlinear evolution equations, often leading to attractor collapse and the loss of dynamical richness. To counteract this, we introduce a geometric framework for Covariant Multi-Scale Negative Coupling Systems (C-MNCS), formulated intrinsically on smooth Riemannian manifolds for a broad class of semilinear dissipative PDEs. The proposed coupling redistributes energy across dynamically separated spectral bands, inducing a scale-balanced feedback that prevents finite-dimensional degeneration. We establish the short-time well-posedness of the coupled state-metric evolution system in Sobolev spaces and derive a priori estimates for phase-space contraction rates. Furthermore, under a global boundedness hypothesis, we prove that the global attractor possesses a strictly finite Hausdorff and Kaplan-Yorke dimension. To bridge abstract topological bounds with physical realizability, we isolate the core Adaptive Spectral Negative Coupling (ASNC) mechanism for numerical validation. High-resolution experiments, utilizing a fully coupled ETDRK4 scheme and continuous QR-based Lyapunov exponent computation on a conformally flat 2D dynamic scalar manifold, corroborate the theoretical predictions. These computations explicitly demonstrate the stabilization of high-dimensional attractors against severe macroscopic dissipation. This geometrically consistent mechanism offers a new paradigm for maintaining structural complexity and multiscale control in infinite-dimensional dynamical systems.