Kolmogorov Complexity Characterisation of Rate-Independent Functionals via Extremum Stack

A recent mathematical demonstration confirms that the extremum stack of a discrete sequence serves as a minimal sufficient statistic for all computable, causal, and rate-independent functionals, evaluated through Kolmogorov complexity. The findings indicate that the shortest program addressing every inquiry within the class R fulfills K_R(u_{0:n}) = K(Pi_n) ± O(1), with the additional terms being independent of both the sequence length n and stack depth k. The classical wiping property of the Preisach hysteresis operator underpins sufficiency, while minimality is established via a finite indicator family with confirmed rate-independence. Consequently, any compression of a hysteresis-driven stream that maintains the entire class R must preserve at least K(Pi_n) - O(1) bits. The theorem suggests a stack-based compression algorithm that ensures Kolmogorov optimality, a feature absent in conventional time-series methods. This research is available on arXiv with the identifier 2605.18885.