Unified Gravitational, Electromagnetic and Spin Dynamics: Four-Dimensional Kaluza–Klein Miracle from Boundary Geometrodynamics and S1 Clock Fiber

We prove a geometrodynamical description of electromagnetism based on metric-preserving Local Lorentz Transport, as a counterpart to the metric description of gravity. Motivated by wave mechanics, we implement intrinsic proper-time recurrence at the Compton period through covariant Periodic Boundary Conditions, thereby defining an effective \( S^1 \) clock fiber associated with a massive particle. Requiring consistency with the action principle restricts the admissible boundary geometrodynamics and induces an Abelian connection of curvature \( qF_{\mu\nu}=m\Omega^{EM}_{\mu\nu} \), where \( \Omega^{EM}_{\mu\nu} \) is the observer-adapted generator combining Fermi--Walker boosts and congruence-vorticity contributions. This also yields the Lorentz-force law as an induced Coriolis-like inertial effect and reproduces the ordinary minimally coupled electromagnetic matter action. The equivalent proper-time lifted formalism provides a four-dimensional interpretation of the Kaluza–Klein mechanism: the clock fiber formally plays the geometrical role of a cyclic extra dimension without introducing an additional physical spacetime dimension, while the lifted Ricci scalar yields the Maxwell kinetic term. The associated spin transport provides a non-trivial consistency check by reproducing the Bargmann–Michel–Telegdi structure, with admissible transverse rotations accounting for the anomalous magnetic-moment tensorial structure. Gravity and electromagnetism are described through complementary metric and boundary geometrodynamics, respectively.