Atomic Shell Structure and Vacuum Crystal Field Splitting on the Z3 ⊗ Q3 Discrete Lattice

Standard quantum mechanics derives the atomic principal shell capacities 2n2 and the subshell splittings 2(2ℓ+1) from the continuous SO(3) rotational symmetry and the acciden- tal SO(4) degeneracy of the 1/r Coulomb potential. We demonstrate that these capacities and degeneracies emerge natively from the bound-state eigenspectrum of a purely discrete information-routing lattice. The vacuum substrate is specified as a simple cubic lattice Z3 partitioned into a Q3 internal space of three space-filling oblate square bipyramids per cubic unit cell. Modelling the lepton sector colour-singlet condition as a strong internal mixing penalty, and binding the resulting colourless walker by an effective Coulomb poten- tial motivated by the discrete Laplacian’s Green’s function, we find that continuous SO(3) spherical harmonics fracture into the exact direct sums of Oh (cubic) irreducible representa- tions required to recover the s,p,d,f subshells. Cumulative bound-state spatial capacities of 1,4,9,16 emerge without further input. The discrete origin of the lattice generates a falsifiable signature: a residual cubic-symmetry-breaking displacement of Eg d-orbital com- ponents into shorter radial extent than their T2g partners, persistent even in completely free atoms.