Memory and Coherence in Linear and Delay Systems

This paper develops a theorem-oriented operator framework for memory and coherence in linear and delay dynamical systems. Memory is defined as nonlocal functional dependence in- duced by causal kernels, while coherence is formulated as phase-stable spectral correlation under deterministic transformation. The framework unifies convolution systems, Volterra operators, and delay-feedback architectures within a single structural perspective. Single-delay and multi-delay systems are analyzed analytically, establishing stability condi- tions and characterizing resonance density under rational independence of delays. Numerical simulations validate the operator-theoretic predictions by comparing memoryless, single-delay, and multi-delay systems under broadband excitation. Results indicate that increasing delay di- mensionality increases modal density and structural persistence without violating boundedness constraints. The framework situates structural memory and coherence entirely within standard systems theory and provides experimentally testable criteria for spectral persistence.