Let R denote the class of all computable, causal functionals that are rate-independent in the classical sense (invariant under monotone time reparametrizations), and let Pi_n be the Preisach extremum stack of an input sequence u_{0:n}. We prove a characterization theorem establishing that every F in R satisfies Fu = f(Pi_n) for a computable f, and derive two information-theoretic results. First, under any probability measure on u_{0:n}, the equality I(u_{0:n}; Fu) = I(Pi_n; Fu) holds for every F in R and is an immediate corollary of the characterization theorem. Second, the main result: Pi_n is a Shannon-minimal sufficient statistic in the sense that I(u_{0:n}; Pi_n) <= I(u_{0:n}; S) for every random variable S from which all R-queries are computable. The proof uses the finite indicator family of [Frydrych, 2026] to reconstruct Pi_n from any sufficient S. As a corollary, online maintenance of Pi_n suffices for rate-independent estimation: the NNLS estimator of the Preisach measure mu can be assembled from the incremental stack process (Pi_t)_{t=0}^n in O(k * L^2) memory per step, where k = |Pi_t| and L is the grid resolution.

翻译：令R表示所有经典意义下率无关（在单调时间重参数化下保持不变）的可计算、因果泛函类，并令Pi_n为输入序列u_{0:n}的普赖萨赫极值堆。我们证明了一个刻画定理，确立每个F∈R均满足Fu = f(Pi_n)（其中f为可计算函数），并导出两个信息论结果。首先，在u_{0:n}的任意概率测度下，对每个F∈R均有等式I(u_{0:n}; Fu) = I(Pi_n; Fu)成立，该等式是刻画定理的直接推论。其次，主要结果：Pi_n是香农最小充分统计量，即对每个足以计算所有R查询的随机变量S，均有I(u_{0:n}; Pi_n) ≤ I(u_{0:n}; S)。证明过程利用[Frydrych, 2026]的有限示性族从任意充分S重构Pi_n。作为推论，在线维护Pi_n足以实现率无关估计：普赖萨赫测度μ的NNLS估计器可通过增量堆过程(Pi_t)_{t=0}^n以每步O(k * L^2)内存进行组装，其中k = |Pi_t|且L为网格分辨率。