Operational Depth Implies Marginal Stability in Recursive Dynamical Systems

Many physical, biological, and computational systems appear to operate near marginal stability, with maximal Lyapunov exponent close to zero. Existing explanations often invoke fine-tuning, self-organized criticality, or optimization principles.In this work, we provide a structural explanation independent of optimization. We define operational depth as an information-theoretic property of recursive dynamical systems requiring persistent influence of perturbations across unbounded iteration depth, formalized through power-law decay of conditional mutual information.We prove that systems satisfying operational depth cannot be strongly stable (contractile) nor mixing-chaotic, and must therefore have zero maximal Lyapunov exponent. This result partitions recursive systems into operationally deep and shallow classes and establishes marginal stability as a necessary condition for persistent information propagation in deterministic discrete-time dynamics.The framework is illustrated through canonical examples including the logistic map, irrational circle rotations, and quasiperiodic motion on the torus. The results are testable, falsifiable, and impose structural constraints on systems exhibiting long memory and deep information processing.