Gauge structure of two-mode eikonal interference in $(2+1)$-dimensional space-time

We formulate an intrinsic $U(1)$ geometric structure in $(2+1)$-dimensional space-time arising from two-mode eikonal interference in geometrical optics. The complex ratio of the two modes defines a space-time phase field whose gradient generates a natural gauge connection. Phase singularities occur at amplitude zeros and form topological defects whose circulation is quantized. This framework provides a simple geometric basis for topological optical configurations and for formulations related to Abelian Chern--Simons theory.