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We show that the\( \mathbb{Z}_3 \)-triality lattice introduced in Ref.[1] yields atomic orbital quantum numbers, the hydrogen energy spectrum, and a bare electromagnetic coupling. The \( K_{2,2,2} \) graph Laplacian on the octahedral root shell (\( L^2=2 \)) decomposes as \( 6 = 1\oplus3\oplus2 = s\oplus p\oplus d \), with a graph Laplacian eigenvalue \( \lambda_1^{\rm graph}=4 \) distinct from the continuum angular eigenvalue\( l(l+1)=2 \). In the radial Schr\"odinger equation, this manifests as a \( 12.5\% \) centrifugal term discrepancy at finite lattice spacing, vanishing in the continuum limit. Gauss's law on the geometric grid \( r_k\propto(\sqrt{3})^{k} \) gives \( V(r)=-\sqrt{3}/(4\pi r) \). Treating the same octahedron as the base manifold of a compact \( U(1) \) lattice gauge theory, the character expansion \( Z(\beta)=\sum_n[I_n(\beta)]^8 \), evaluated at the algebraic unit coupling \( \beta_c=1 \)---derived from the Gauss continued fraction integrality condition \( [I_1(\beta)/I_0(\beta)]^{-1}=2/\beta+1/(4/\beta+\cdots) \) whose coefficients \( a_k=2k/\beta \) are integer-valued iff \( \beta\in\{1,2\} \), with \( \beta=2 \) excluded algebraically by the uniqueness of the\( 19 \)-dimensional superalgebra---with geometric factor \( G=2F/E=4/3 \), yields \( 1/\alpha_{\rm geom} = \pi\sqrt{3}\,[I_0(1)/I_1(1)]^4 = 137.042 \). The \( 42 \)~ppm deviation from the CODATA~2022 value\( 137.035999084 \) is exactly accounted for by the topological correction \( \delta(\alpha^{-1})=(S-S^3)/(4\sqrt{3}) \), with\( S=[I_1(1)/I_0(1)]^4 \), arising from quantum interference between the\( n=0 \) vacuum and \( n=\pm1 \) instanton sectors on the octahedron. Including this correction gives \( \alpha^{-1}_{\rm phys}=137.036 \), matching CODATA to sub-ppb precision. The radial Schr\"odinger equation with \( \alpha_{\rm phys} \) reproduces hydrogen wavefunctions with overlaps above \( 0.999 \) for \( n\le3 \). No free parameters are introduced; \( \beta_c=1 \) and the \( 42 \)~ppm correction both follow from the discrete algebraic structure of the \( 19 \)-dimensional \( \mathbb{Z}_3 \)-graded Lie superalgebra.