Post--FBHK Fibered Entropy--Transport Geometry with Petz Fibers and Type~III Persistence

This paper develops a Post–FBHK fibered entropy–transport (ET) geometry that couples classical Hellinger–Kantorovich / logarithmic entropy–transport (HK/Log–ET) dynamics on a Polish base space with HK-type transport on infinite-dimensional quantum state fibers endowed with Petz monotone Riemannian metrics. The configuration space is a product Z=X×Yadm Z = X Y₀₃₌Z=X×Yadm, where the base carries a canonical HK structure and the fiber is an admissible manifold of normal faithful states (including type III–compatible settings) equipped with a strong Petz metric. On the space of finite measures with mixed second moments, the paper introduces a product canonical HK point costC=cHKX+κ cHKYC = c₇₊^ X + \, c₇₊^ YC=cHKX+κcHKYand the associated Log–ET functional. The square root defines a complete geodesic distance dFETd₅₄ₓdFET, and an explicit Dirac reduction formula recovers the underlying HK kernel on Z ZZ. A central result is quarter rigidity: the Benamou–Brenier action with kinetic terms ∣vt∣2+κ∣wt∣2|vₜ|² + |wₜ|²∣vt∣2+κ∣wt∣2 matches dFET2d₅₄ₓ²dFET2 if and only if the reaction coefficient equals β=1/4 = 1/4β=1/4, fixing the canonical normalization uniquely through Dirac calibrations. The exposition is intentionally layered. The static core layer proves admissibility, completeness, geodesicity, Dirac reduction, and quarter rigidity under minimal Polish-space assumptions, without requiring PDE representations. The dynamic design layer records additional smoothness and convexity conditions under which one obtains a PDE/vector-field Benamou–Brenier formulation, Hamilton–Jacobi duality, EVI gradient flows for a grand-canonical entropy functional (classical KL plus fiberwise Araki entropy), and entropy-dissipative persistence/coarse-graining maps induced by classical Markov contractions and quantum channels. The framework generalizes and conceptually extends the earlier finite-dimensional Fibered Bures–HK Entropy–Transport (FBHK) program (DOI: 10. 5281/zenodo. 17512165) toward infinite-dimensional, noncommutative, and type-III-compatible regimes. Overall, the paper offers a mathematically explicit design theory for self-organizing systems with classical configurations and quantum-like internal states, highlighting how static HK/Log–ET geometry can be combined with admissible Petz fibers to support principled dynamics, entropy control, and persistence mechanisms in future AI, physics, and information-geometric applications.