Quantum Theory of Entropic Time (QTET): Gauge-Theoretic Formulation of Operational Time

This work develops and calibrates the gauge-theoretic structure underlying the framework of entropic operational time introduced in our previous study. The focus is placed on the local equivalence relations implied by an operational definition of time and on their consistent realization within a covariant field-theoretic formulation. We show that local redefinitions of phase and of an entropic baseline necessarily give rise to a single complex Abelian gauge connection acting on the redundant description of quantum states. The resulting structure admits a minimal universal Lagrangian whose real sector reproduces standard electromagnetic dynamics, while the imaginary sector encodes operational rescalings of time associated with the entropic structure of the vacuum without introducing additional propagating degrees of freedom. The purpose of the present paper is primarily structural. We clarify the role of the complex gauge connection, fix normalization conventions, and establish correspondence limits required for consistency with known low-energy physics. Detailed phenomenological consequences are not addressed and will be discussed elsewhere.