Boundary Interface Theory: Mathematical Rigor and Lagrangian Symmetry Analysis

This preprint is a follow-up study of the Boundary Interface Theory (BIT), focusing on its mathematical rigorization and Lagrangian symmetry analysis. Based on the 3-dimensional universal boundary interface as the fundamental spacetime, we redefine the boundary interface field, derive its 4D covariant form, prove energy-momentum conservation via Noether's theorem, obtain exact solutions of the field equation under static spherically symmetric sources, and clarify the mathematical relation and core differences between BIT and General Relativity. This work consolidates the theoretical foundation of BIT for future experimental verification and cosmological applications.