This work develops a structural, not analytic, layer on top of the reversible Persistence–First Holographic Systems (rPFHS) framework. Starting from an EVI gradient flow of a total entropy functional on a fibered Bures–HK entropy–transport space, represented by a reversible Markov kernel with a unique invariant “origin law”, the paper introduces value-anchored natural-law fields that are intrinsically tied to the same dynamics. At the microscopic level, a value anchor is a nonnegative scalar field on state space, inducing a law-level value functional. The paper formulates origin-anchoring and value-reproduction axioms and shows, via a corrected Poisson equation, that such value anchors exist in finite-state reversible FBHK systems with a persistence functional. On networks of value-anchored rPFHS cores, the node-level evolution of expected values is closed—by explicit modelling assumption—into a discrete-time reaction–diffusion system on a quasi-lattice graph, with Fisher–KPP type reaction terms. Under these closure hypotheses, standard results for Fisher–KPP equations and branching random walks on graphs imply a strictly positive minimal asymptotic front speed and a strictly positive interior value plateau for any compactly supported excitation. The paper does not claim new PDE or branching-process theorems; instead, it packages existing front-speed results into a value-anchored, natural-law blueprint. All assumptions (FBHK geometry, reversible core, value axioms, closure parameters) are kept explicit and modular, so that they can be calibrated, derived, or replaced in concrete physical or computational models.
Takahashi, K. (Tue,) studied this question.