The Preisach Extremum Stack is a Shannon-Minimal Sufficient Statistic for Rate-Independent Functionals

Let R denote the class of all computable, causal functionals that are rate-independent in the classical sense (invariant under monotone time reparametrizations 1), and let Πn be the Preisach extremum stack of an input sequence u0:n.We prove a characterisation theorem establishing that every F ∈ R satisfies Fu(n) = f (Πn) for a computable f , anduse this to derive two information-theoretic results.First, under any probability measure on u0:n (the sequence itself is the random object, not a parameter), the equalityI(u0:n; Fu(n)) = I(Πn; Fu(n))holds for every F ∈ R and is an immediate corollary of the characterisation theorem.Second, the main result: the stack Πn is a Shannon-minimal sufficient statistic in the sense that I(u0:n; Πn) ≤ I(u0:n; S ) for every random variable S from which all R-queries are computable. The proof uses the finite indicator family of 2 to reconstruct Πn from any sufficient S .As a corollary, online maintenance of Πn suffices for rate-independent estimation: the NNLS estimator of the Preisach measure μ can be assembled from the incremental stack process (Πt)nt =0 in O(k · L2) memory per step, where k = |Πt| and L is the grid resolution.