Clebsch Field Theory: Relativistic Continuum Kinematics and the Vacuum Maxwell Sector

This paper develops the Level A vacuum Abelian sector of Clebsch Field Theory (CFT). The theory assumes a relativistic Clebsch continuum as a foundational kinematic structure and shows that its vorticity two-form reproduces the homogeneous Maxwell equations as a geometric identity. A quadratic Lorentz-invariant action constructed from the vorticity leads, on regular local Clebsch charts, to the source-free Maxwell equations. The resulting stress–energy tensor, conformal structure, and transverse quantization are shown to coincide with those of standard free Maxwell electrodynamics up to an overall normalization constant. The formulation provides a local Clebsch realization of vacuum electromagnetism rather than a modification of Maxwell theory. Gauge invariance emerges from Clebsch representative freedom, and the physical degrees of freedom reduce to the two transverse photon polarizations in the standard way. This work establishes the kinematic and variational foundation of the CFT framework used in subsequent papers on Hopf solitons and anomaly structure.