Geometric Standard Model: Part I. Emergence of $U(1)$ Gauge Dynamics and Low-Energy Electromagnetism

This study explores the geometric origin of macroscopic electromagnetism within the constrained SO(3,3) multi-time spacetime framework. We algebraically derive how the SO(2) geometric rotational symmetry of the transverse-time plane manifests as a local U(1) gauge field in 4D spacetime through the off-diagonal metric tensor of Riemannian geometry. By accounting for the massive energy barrier formed by the vacuum expectation value of the constraint field, we apply effective field theory (EFT) to integrate out heavy transverse-time fluctuations within the path integral. Consequently, we demonstrate that Maxwell's free-field Lagrangian and Coulomb's inverse-square law (1/r^2) emerge spontaneously from pure geometric constraints without empirical assumptions. Furthermore, we show that the Aharonov-Bohm phase shift fundamentally manifests as a natural geometric holonomy of transverse-time parallel transport, validating the physical reality of the gauge connection. While acknowledging the quantitative limitations of deriving exact Standard Model coupling constants from first principles without non-perturbative simulations, this paper provides a robust semi-classical proof of concept, illustrating that geometric topology can serve as the macroscopic foundation of electromagnetic forces.