Electron Shell Structure and Exceptional Lie Algebras: Shell Capacities, the Division Algebra Ladder, and the Lanthanide Contraction as $G_2$ Rigidity

The quantum-mechanical $2(2\ell+1)$ rule for electron subshell capacities and the relativistic explanation of the lanthanide contraction are experimentally well-confirmed frameworks that are not challenged here.

This paper presents a speculative catalogue of structural coincidences between the numerical constants that appear in atomic electron shell structure and the geometry of the exceptional Lie algebras. It is a companion to https://doi.org/10.5281/zenodo.19960385, which established the same programme for nuclear magic numbers. The two papers together suggest that the associativity boundary at $G_2 = \text{Aut}(\mathbb{O})$ organises both nuclear and atomic structure — the nucleus on one side of the exceptional chain, the electron cloud on the other.

Three observations are developed. First, the electron subshell capacities $\{2, 6, 10, 14\}$ for $\ell = 0, 1, 2, 3$ coincide with the dimensions of specific objects in a sub-ladder of the division algebra tower: $\dim_\mathbb{R}(\mathbb{C}) = 2$, $|\Phi(A_2)| = 6$, $\dim(Sp(2)) = 10$, $\dim(G_2) = 14$. The progression $\mathbb{C} \to A_2 \to Sp(2) \to G_2$ terminates precisely at the first exceptional Lie algebra, where associativity fails.

Second, the cumulative electron shell capacities $\{2, 8, 18, 32\}$ — the period lengths of the periodic table — coincide with $2n^2$ for $n = 1, 2, 3, 4$, which is also the cumulative count obtained by filling the division algebra sub-ladder, equivalently mapping to the non-zero roots of the odd orthogonal Spin groups, $Spin(2N+1)$. The period structure of the periodic table terminates its clean $n^2$ pattern at $n=4$ (the f-block), corresponding to the $G_2$ boundary.

Third, the lanthanide contraction — the anomalous compression of ionic radii across the f-block — is quantitatively explained by relativistic kinematics. We observe that the f-block is precisely the regime where electron structure enters the 14-dimensional $G_2$ geometry, and we state a conjecture: that the relativistic contraction and the $G_2$ rigidity effect are two descriptions of the same phenomenon at different levels of abstraction.

This is a speculative working paper. No claim made here supersedes experimentally confirmed physical models. All proposed geometric interpretations are conjectural.