The Extremum Stack is a Minimal Sucient Statistic for Rate-Independent Functionals: A Kolmogorov Complexity Characterisation
Piotr Frydrych · Zenodo (CERN European Organization for Nuclear Research) · 2026
We prove that the extremum stack Π_n of a discrete sequence u_{0:n} ∈ G_L^{n+1} is a minimal sufficient statistic for the class R of all computable, causal, rate-independent functionals, in the sense of Kolmogorov complexity. Specifically, we establish: K(Π_n) − O(1) ≤ K_R(u_{0:n}) ≤ K(Π_n) + O(1), where K_R(u_{0:n}) is the length of the shortest program answering every query in R, and the O(1) overhead is independent of both the sequence length n and the stack depth k. Sufficiency follows from the classical wiping property of the Preisach hysteresis operator. Minimality is established via a finite indicator family whose rate-independence is verified explicitly. Any compression of a hysteresis-driven stream that preserves the full class R must therefore retain at least K(Π_n) − O(1) bits; the stack-based compression algorithm implied by the result carries a Kolmogorov optimality guarantee absent from standard time-series compression methods. Submitted to Information Processing Letters (Elsevier).