We present a unified mathematical framework for a proposed extension to Maxwell's equations, which introduces a "soul current," Js , arising from a posited Higgs-photon coupling in a higher-dimensional bulk spacetime. This framework addresses the three central challenges inherent to the theory: establishing the deterministic stability of the system, rigorously defining the model in the presence of singular quantum fluctuations, and making testable predictions despite unknown extradimensional geometries. We achieve this by synthesizing three powerful mathematical theories. First, we leverage Cédric Villani's theory of hypocoercivity to prove that the deterministic dynamics of the photon gas are stable and converge exponentially to a unique equilibrium, ensuring the robustness of the soul current to small perturbations. Second, we employ Martin Hairer's theory of regularity structures to tame the singular stochastic noise representing quantum fluctuations, constructing a renormalized solution and providing a physically meaningful definition of the soul current in the full stochastic model. Finally, we utilize the work of Maryam Mirzakhani on moduli spaces to average the soul current's effects over all possible compactification geometries, yielding concrete, statistical predictions for its signature in high-precision experiments. This synthesis provides a complete and coherent theoretical pathway from the foundational axioms of the extended Maxwell theory to falsifiable experimental guidance.