A Formal Symmetry Shell Model for the Structural Boundary of the Periodic Table up to Z=118: Axiomatic Derivation, Hilbert Space Representation, and Geodynamic Isomorphism

This paper presents a rigorous axiomatic mathematical framework for the Symmetry Shell Model (SSM), describing the natural termination of the periodic table at atomic number Z=118. Rather than competing with quantum mechanical orbital calculations, the SSM operates as a structural metatheory governing shell capacity, global mirror symmetry, and boundary conditions. We formulate the atomic state space within a finite-dimensional Hilbert space H_sym and introduce the Mirror Mantle Operator P_m. We prove that Z=118 emerges strictly as an axiomatic mathematical consequence of closing seven principal shells under mirror parity without presupposing the experimental limit. Furthermore, we provide a formal proof of non-existence for stable states at n >= 8 based on boundary discontinuity conditions and discrete calculus of the energy gap. Finally, we establish a conceptual and physical bridge termed Universal Structural Isomorphism, linking the atomic symmetry mantle to macroscopic boundary dynamics, specifically the Mohorovičić discontinuity (Moho mantle) in geophysics.