PFUSRC_48 Group Isomorphism and Structural Invariance: A Rigorous Mathematical Proof of the PFUSRC Renormalized Unified Cosmic System

This paper takes Group Isomorphism and Structural Invariance as core mathematical tools to provide a complete, rigorous, and self-consistent formal proof of the PFUSRC_48 Renormalized Double-Cone Frustum Unified Cosmic System. The paper first identifies six fundamental crises in modern physics and cosmology. Based on the primitive axiomatic system of PFUSRC_48, the 45° coaxial double-cone frustum renormalized geometry, the rigid dimensional evolution path 1→5→11, the global dynamic invariant operator β₁, and the intrinsic scale-invariant constant π₁=12/11, all crises are systematically resolved. Using the three conditions of group isomorphism—bijectivity, operation preservation, and structural invariance—the paper rigorously proves that PFUSRC_48 satisfies geometric uniqueness, dimensional rigidity, high-dimensional homomorphic projection, gravity-quantum unification, and falsifiability. Finally, key objections are addressed to complete logical closure. This paper proves that PFUSRC_48 is a mathematically rigorous, geometrically unique, physically necessary, observationally falsifiable, and logically self-closed unified paradigm. Group isomorphism and discrete structure constitute the ultimate ontological essence of the universe.