Implementation--Ready Persistence--First Holographic Systems

This paper develops an implementation–ready companion theory for Persistence–First Holographic Systems (PFHS), focusing on how to compute PFHS dynamics on real CPU/GPU hardware without committing to a particular programming framework. The starting point is the PFHS framework of Aida & Takahashi, Persistence–First Holographic Systems (doi:10.5281/zenodo.17636648), which organises entropy–transport (ET) gradient flows and their bulk–boundary structure in a categorical way. On the analytic side, the work assumes standard EVI_λ theory for gradient flows in metric spaces and the convergence of Jordan–Kinderlehrer–Otto (JKO) type minimizing–movement schemes (Ambrosio–Gigli–Savaré, Jordan–Kinderlehrer–Otto, Liero–Mielke–Savaré, Maas, Mielke, Carlen–Maas). On top of this, the paper introduces an abstract implementation category Impl whose objects are finite–dimensional state representations (arrays, tensors, sparse vectors) and whose morphisms are compositions of CPU/GPU–friendly “primitive kernels” (e.g. sparse matvecs, convolutions, tensor contractions), each annotated with a Lipschitz bound and a cost in an ordered monoid. The central notion is a JKO–implementable ET gradient system: an analytic EVI–gradient flow equipped with a family of discretisations into Impl and a one–step JKO kernel whose iterates converge with rate O(τ + ε_h). Under mild functoriality assumptions, any PFHS whose bulk and boundary ET systems are JKO–implementable admits a PFHS implementation scheme: a pseudofunctor from the PFHS diagram into Impl that preserves persistence, interfaces, and autopoietic structure up to controlled error. The main theorem shows that every finite PFHS subdiagram can be realised numerically with a uniform global error bound of order O(τ + ε_h), while remaining compatible with coarse–graining and boundary observation. The framework is designed to be agnostic to the choice of coordinates (e.g. Wasserstein, Hellinger–Kantorovich, fibered Bures–HK geometries; see also Aida & Takahashi, doi:10.5281/zenodo.17483083; Takahashi, doi:10.5281/zenodo.17518572, 10.5281/zenodo.17576361, 10.5281/zenodo.17601860). It applies equally to finite Markov chains, measure spaces, and quantum Markov semigroups, as long as suitable JKO schemes and discretisations exist. The contribution is not a new numerical method or curvature inequality, but a universal categorical interface that separates (i) analytic ET structure, (ii) PFHS multiscale organisation, and (iii) hardware–level primitives. This separation is intended to support future PFHS implementations by other researchers or autonomous AI systems, including complexity–aware designs that exploit bulk–boundary dimension gaps and multiscale structure for compute–optimal simulation.