We derive the complete quantum number hierarchy n, l, m, ms and the atomic shell capacities 2n²from the geometry of a Finsler-Randers substrate, without importing quantum mechanical postulates. The principal quantum number n arises from the Substrate Winding Number Theorem applied to guidance trajectories encircling a nuclear defect. The angular quantum numbers l and m arise from the representation theory of SO(3) on the 2-sphere surrounding the defect, enforced by the single-valuedness of the substrate phase and the rotational invariance of the substrate field equation. The constraint l < n is derived from the Maslov-corrected closure condition on guidance trajectories, without appeal to wavefunction normalizability. Spin- 1 2follows from the Hopf fibration structure of the minimal substrate loop. The Pauli exclusion principle is derived from the topological incompatibility of identical fermionic winding configurations withthe closure condition. The nuclear substrate charge is shown to be quantized as ρ0 = 2πZ, where Z is the first Chern number of the U(1) phase bundle over theenclosing 2-sphere, recovering the atomic number as a topological invariant and the Coulomb potential as the field of a topological defect. Spherical symmetry of the nuclear defect is derived as a theorem from the substrate field equation; no assumption remains unresolved. The shell capacities 2, 8, 18, 32 fall out from N (n) = 2n 2 with no free parameters and no quantum mechanical postulates. The fine structure constant α is identified as a fixed geometric ratio of the Randers substrate; its derivation from first principles is identified as the central open problem for future work.
David B Smith (Sun,) studied this question.