The nuclear spin-orbit coupling has been a fitted parameter since Goeppert Mayer and Jensen's shell model in 1949. This paper derives it algebraically: D₄ triality in the 600-cell provides the vector–spinor exchange that mixes orbital and intrinsic angular momentum.
The 600-cell sits on S³ = SU(2). Hyperspherical harmonics on S³ have degeneracy (l+1)²; the 600-cell is the unique regular 4D polytope whose adjacency spectrum preserves all-perfect-square multiplicities (1²–6²). It contains embedded 24-cell substructures forming the D₄ root system — the unique Lie algebra with triality, the outer automorphism exchanging vector and spinor representations. That exchange is the algebraic mechanism for spin-orbit coupling.