Anastrophic Theory: A Unified Framework for Functional Coherence and Information Geometry in Discrete Periodic Systems

We introduce Anastrophic Theory, a mathematical framework for the analysis of discrete periodic systems that unifies algebraic return structure, spectral geometry, and information-theoretic dynamics. Anastrophic elements are characterized by a finite structural return condition, capturing periodicity as an intrinsic invariant rather than a purely temporal phenomenon. This perspective extends naturally to functional settings, where constrained operators generate spectrally structured evolution. Spectral amplitudes evolve on the probability simplex, while phases reside on a compact toroidal manifold. Within this geometric formulation, we define Spectral Coherence as a global measure of phase alignment and introduce a Spectral Action principle governing optimal spectral trajectories under the Fisher–Rao information metric. The framework establishes a formal connection between algebraic periodicity, geometric coherence, and thermodynamic behavior. Applications include bounds on interference visibility in multi-mode quantum systems and a structural regularization scheme for continual learning, demonstrating how return invariants can stabilize dynamics and optimize information flow.