Persistence–First Holographic Systems (PFHS) is a structural framework for organising gradient-flow dynamics in entropy–transport (ET) geometries across multiple scales and interfaces. The paper assumes the standard existence, uniqueness, and contractivity theory for EVI_ gradient flows in classical, discrete, and quantum settings (Ambrosio–Gigli–Savaré, Liero–Mielke–Savaré, Maas, Erbar–Maas, Mielke, Carlen–Maas) and asks a different question: how can many such ET gradient systems, living on a multiscale “world + boundary” architecture, be packaged so that persistence, self-like structure, and curvature control become transparent? To this end, the work introduces persistence fields, persistence–first holographic systems (PFHS), autopoietic interfaces, and contractive ET gradient representations. A PFHS consists of a world category, a boundary category, a persistence monoid acting by coarse-graining functors, and an interface functor that is equivariant with respect to this action. Within this setting, a reflective “self-like” subcategory and its boundary orbit play the role of an autopoietic layer. A contractive ET gradient representation assigns to each world and boundary object an ET gradient system together with Lipschitz, entropy-decreasing maps that intertwine the associated semigroups. The main result is an “autopoietic invariance package”: under explicit structural assumptions, the self-like world layer and its boundary orbit form closed families of ET gradient systems whose curvature parameters are controlled by simple image–EVI bounds. The paper illustrates the framework on finite Markov chains with multiscale coarse-graining and sketches how similar ideas extend to classical–quantum ET geometries based on fibrewise Bures–HK metrics (Takahashi, 2025). The contribution is conceptual rather than analytic: it provides a unifying categorical language for multiscale dissipative systems, complementary to compositional frameworks such as decorated cospans, open Markov processes, and Markov categories.
M. et al. (Tue,) studied this question.