Modern physics is confronted with the fundamental incompatibility be tween General Relativity (GR) and Quantum Field Theory (QFT), as well as core puzzles including cosmic singularities, the nature of dark matter and dark energy, and the black hole information paradox. Starting from an in depth geometric analysis of relativistic spacetime, this paper systematically elaborates the background, core axioms and rigorous mathematical deriva tions of the Unified Membrane Grand Unified Theory (UMGUT). Based on this theory, we improve the cosmic evolution model and the Standard Model of particle physics, eliminating the inherent contradictions of traditional the ories. Ultimately, UMGUT is applied to uniformly resolve 12 core puzzles in modern physics. With a 2-dimensional bilaterally covariant unified mem brane as its core carrier and no additional higher-dimensional assumptions, UMGUT achieves a seamless unification of gravity and quantum field the ory, providing a self-consistent and verifiable unified framework for modern physics.