The Covariant Transverse Slicing Operator: Gauge Interactions and Charge Quantization (Part III.A of the ODD Series)

We extend the Operator-Derived Dimension (ODD) framework to include gauge interactions by promoting the Transverse Slicing Operator to its covariant form. By introducing the 5D gauge field AM , we derive the exact interaction Hamiltonian for charged particles localized on the 4D domain wall.We distinguish between the transverse gauge component Aw, which generates scalar mass perturbations via the modified P¨oschl-Teller potential, and the longitudinal component Aμ, which projects to the 4D observable photon. We demonstrate that geometric charge quantization emerges directly from the topological winding of the U (1) fiber, and prove that the Supersymmetric (SUSY) algebra of the domain wall is robustly preserved under local gauge transformations.