Finsler Spacetime, Entropy–Transport Gradient Flows, and Fibered Bures–HK Geometry: From Relativistic Kinetic Gases to Persistence–First Holographic Universes

This record collects three companion papers that develop a unified entropy–transport (ET) perspective on Finsler spacetime gravity, relativistic kinetic theory, and quantum field degrees of freedom: Finsler–Entropy–Transport Gradient Systems for Relativistic Kinetic Gases (Paper A)This paper constructs Finsler–entropy–transport (FET) gradient systems for relativistic kinetic gases on Lorentz–Finsler spacetimes. Using Hellinger–Kantorovich (HK) / Wasserstein–Fisher–Rao type distances, it realises Finsler–Friedmann cosmological expansion as an EVI (Evolution Variational Inequality) gradient flow of a free-energy functional on a Finsler–HK configuration space. The work links Finsler cosmology, relativistic kinetic theory, and optimal transport–based gradient flows, and identifies a positive “world curvature parameter” that controls large-scale stability and exponential expansion without postulating a dark-energy sector by hand. Persistence–First Holographic Finsler Universes (Paper B)Building on Paper A, this work embeds Finsler–ET universes into the Persistence–First Holographic Systems (PFHS) framework. It organises “world”, “self-like subsystem”, and “observation boundary” dynamics as a hierarchy of ET gradient systems connected by entropy–transport morphisms. Using image–EVI curvature transfer, the paper shows how positive world-level curvature induces quantitative persistence margins for self-like and boundary subsystems in Finsler cosmology, providing a holographically organised notion of stability for observers and interfaces. Fibered Bures–HK over Finsler Spacetime: Towards a Unified Entropy–Transport Picture of Gravity and the Standard Model (Paper C)The third paper introduces a fibered Bures–HK geometry over a Finsler–ET universe, combining HK/ET transport on the classical spacetime–kinetic base with Bures/Petz-type monotone metrics on quantum state fibers. Under standard assumptions on quantum Markov semigroups and the Fibered Bures–HK Entropy–Transport (FBHK) template, it constructs a single ET gradient system whose total entropy functional couples (i) Finsler–Friedmann kinetic entropy, (ii) quantum relative entropy for local Standard Model degrees of freedom, and (iii) coarse-grained gauge/Yukawa interaction terms. The result is a grand unified structure—not a predictive GUT—where gravity and Standard Model fields are jointly realised as components of one EVI gradient flow, compatible with PFHS and quantum information geometry. Together, these three works provide a coherent theoretical stack: from Finsler–HK cosmological dynamics, through persistence-first holographic organisation of observers, to a fibered Bures–HK framework that integrates Finsler spacetime, entropy–transport geometry, and coarse-grained Standard Model quantum fields in a single metric gradient-flow picture.