Polynomial Memory Scaling in Marginally Stable Non-Mixing Dynamical Systems

This paper studies how additive noise limits the effective memory of stochastic dynamical systems. We introduce an operational definition of memory depth based on the persistence of initial condition influence under noise and analyze its scaling behavior across different dynamical regimes. We show that uniformly chaotic systems exhibit logarithmic noise–memory scaling, while marginally stable, non-mixing systems display polynomial scaling. This distinction provides a quantitative criterion for separating chaotic and marginal dynamics and yields a falsifiable prediction that can be tested numerically in concrete systems. The results identify a structural boundary between chaotic and marginally stable dynamics based solely on measurable scaling behavior.