Structural Emergence of Gauge sector from Entropy-Hessian dynamics

This work develops the structural origin of gauge interactions within the entropy(https://doi.org/10.5281/zenodo.18773391)–probability(https://doi.org/10.5281/zenodo.18779307)–connection(https://doi.org/10.5281/zenodo.18780579) framework introduced in the earlier studies on scalar-generated geometry and statistical spacetime structure. We also tried to use this work in non-equilibrium and investigated it's properties(https://doi.org/10.5281/zenodo.18832964). In that program, spacetime geometry emerges from an entropy functional in equilibrium and its compatibility with probability flow, while gravitational dynamics arise from the geometric structure induced by these statistical principles. Building on this foundation, the present paper investigates how internal gauge fields arise from the same underlying primitives. Starting from entropy, probability, and affine connection as the fundamental variables, we show that gauge interactions emerge from the compatibility conditions governing probability transport on the emergent geometric manifold. Internal gauge fields appear naturally as components of the connection acting on the probability bundle, and the Yang–Mills kinetic structure is generated through the heat-kernel expansion of the entropy functional. The resulting construction demonstrates that gauge dynamics need not be postulated independently but can arise from the same statistical–geometric framework that gives rise to spacetime and gravitational structure.