Persistent entropy (PE) is an information-theoretic summary statistic of persistence barcodes that has been widely used to detect regime changes in complex systems. Despite its empirical success, a general theoretical understanding of when and why persistent entropy reliably detects phase transitions has remained limited, particularly in stochastic and data-driven settings. In this work, we establish a general, model-independent theorem providing sufficient conditions under which persistent entropy provably separates two phases. We show that persistent entropy exhibits an asymptotically non-vanishing gap across phases. The result relies only on continuity of persistent entropy along the convergent diagram sequence, or under mild regularization, and is therefore broadly applicable across data modalities, filtrations, and homological degrees. To connect asymptotic theory with finite-time computations, we introduce an operational framework based on topological stabilization, defining a topological transition time by stabilizing a chosen topological statistic over sliding windows, and a probability-based estimator of critical parameters within a finite observation horizon. We validate the framework on the Kuramoto synchronization transition, the Vicsek order-to-disorder transition in collective motion, and neural network training dynamics across multiple datasets and architectures. Across all experiments, stabilization of persistent entropy and collapse of variability across realizations provide robust numerical signatures consistent with the theoretical mechanism.

翻译：持久熵（PE）是持久条形码的一种信息论概要统计量，已广泛应用于检测复杂系统中的状态转变。尽管其经验上取得了成功，但对于持久熵何时以及为何能可靠检测相变的普遍理论理解仍然有限，特别是在随机和数据驱动的场景中。在本工作中，我们建立了一个通用的、模型无关的定理，提供了持久熵可证明区分两个相的充分条件。我们证明持久熵在相之间表现出渐近非消失的间隙。该结果仅依赖于持久熵沿收敛图序列的连续性，或在温和正则化条件下成立，因此广泛适用于各种数据模态、过滤方式和同调维度。为了将渐近理论与有限时间计算联系起来，我们引入了一个基于拓扑稳定化的操作框架，通过滑动窗口上稳定选定的拓扑统计量来定义拓扑转变时间，并在有限观测范围内构建基于概率的关键参数估计器。我们在Kuramoto同步转变、Vicsek集体运动中从有序到无序的转变，以及跨多个数据集和架构的神经网络训练动力学上验证了该框架。在所有实验中，持久熵的稳定化与不同实现间变异性的坍缩提供了一致的数值特征，与理论机制相吻合。