Three Equivalent Mathematical Forms of the Core Constraint in Time Field Theory and Their Unified Physical Ontology

This paper systematically demonstrates three equivalent mathematical formulations of the core constraint of Time Field Theory (TFT) — the flux conservation equation, also known as the spacetime periodicity law: ∇·(γ∇γ)=0. The three forms are the three-dimensional complex scalar modulus harmonic form, the two-dimensional holomorphic function form, and the phase integral variational form. All forms are strictly derived from the perfect circle postulate and the principle of metric self-consistency, with no additional physical assumptions introduced. They are not independent mathematical tools, but necessary manifestations of the same principle of periodic self-consistency from three perspectives: differential field, complex geometry, and path integral. This work completes the paradigm shift of the TFT core equation from algebraic constraint to geometric ontology. It serves as the mathematical supplement and geometric manifestation of TFT Series No.33 (Necessity of Flux Conservation), and constitutes the core mathematical bridge between Series No.30 (General Outline of Metretike) and Series No.34 (Endogenous Derivation of Riemannian Geometry).