Covariant formulation of electrodynamics in isotropic media revisited

This paper resolves a persistent ambiguity regarding the covariant formulation of electrodynamics in a vacuum, as well as of Minkowski's electrodynamics of moving isotropic media. By analyzing a recent debate, we demonstrate that current interpretations misrepresent the relationships between the electromagnetic four-vector potential, gauge conditions, and field tensors. We specifically challenge the notion that the geometric status of $A^\mu$ as a four-vector is contingent upon the choice of gauge. We trace these inconsistencies back to potentially misleading treatments of the four-vector potential in the classic texts of Moller, Landau and Lifshitz, Jackson and Griffiths. Beyond correcting the speciic conceptual errors in the recent literature, this work clarifies the theoretical framework for ensuring the electromagnetic potential remains consistent with gauge invariance and the principle of relativity in macroscopic isotropic media.