An Algebraic-Quantum Perspective on Atomic Shell Structure: The m-Squared Identity and Chemical Periodicity

Atomic electrons exist under the central Coulomb potential of the nucleus, a constraint that mandates their wave functions be described by spherical harmonics. While standard quantum mechanics treats these functions as static geometric solutions, the empirical structure of the periodic table (2, 8, 8, 18, 18, ...) reveals a distinct "doubling" behavior that standard hydrogenic models fail to predict. This work proposes that atomic shells are governed by a specific algebraic symmetry where the spherical harmonic manifolds themselves exhibit an effective doubling in the filling sequence. We derive the shell capacity formula directly from the arithmetic series identity applied to the degeneracy of angular momentum states, demonstrating that this "doubling" is a necessary geometric response to multi-electron correlations and centrifugal barriers. We introduce the Algebraic-Quantum Shell Evolution Model (AQSEM), which perfectly reproduces the noble gas atomic numbers (Z = 2, 10, 18, 36, 54, 86, 118). This model integrates Unsold's Theorem, Lin's Information Theory, Alexa's variational flows, and Pearson's maximum hardness to establish two key principles: the Symmetry-Driven Energy Minimization, which explains the prevalence of high-symmetry partial configurations (e.g., Carbon sp3 hybridization) by minimizing structural information content, and the Enhanced Stability Principle, which posits that molecules achieve global minimum energy when each constituent atom satisfies local Unsold spherical symmetry.The model reveals that covalent bonding requires a threshold of effective nuclear strength: newly opened shells exhibit insufficient nuclear attraction to sustain localized orbital overlaps, mostly favoring delocalization (metallic conductivity) or electron loss (ionic bonding). Stable covalency occurs where nuclear attraction sufficiently stabilizes localized spherical charge distributions, identifying Hydrogen as algebraically distinct from alkali metals and necessitating its covalent behavior. This framework provides quantitative insight into the octet rule and hybridization, and establishes a predictive model for bonding behavior and molecular orbital formation. The model's "doubling" index aligns with Density Functional Theory, manifesting as a derivative discontinuity in the chemical potential (Janak's Theorem), and functions as a topological invariant for the atomic vacuum state, thereby linking the periodic table to topological phases of matter.