An Operator-Theoretic Formulation of the Symmetry Shell Model: Structural Metatheory and Periodic Table Closure at Z=118

This paper presents a rigorous operator formulation of the Symmetry Shell Model (SSM), establishing a structural metatheory for the architecture of the periodic table and electronic subshell closure up to Z=118. By formalizing a mirror symmetry operator P̂_m as a unitary involution on a direct sum Hilbert space H_sym = ⊕_{n=1}^7 H_n, we demonstrate that the structural completion of electronic shells arises from global geometric boundary conditions rather than microscopic relativistic parameter fitting. We prove that three core structural postulates—the degeneracy rule C(l) = 2(2l+1), the upper angular momentum bound l_max = 3, and mirror reflection symmetry around n_c = 4—non-circularly yield the maximum atomic boundary Z_limit = 118. Furthermore, we demarcate the epistemological scope of SSM from microscopic Dirac-Fock calculations and decouple electronic shell closure from nuclear instability mechanisms. The model provides a unified algebraic foundation for periodic classification and shell capacity bounds.