Generalized Gauge Hierarchies

We generalize the analysis of coupling-constant renormalization in unified theories to allow for more than one scale of spontaneous symmetry breakdown between unification and SU(3) ⊗ SU(2) ⊗ U(1). We find a relation between $\alpha$, ${\alpha }_s$, ${{sin}^2\theta }_{\mathrm{W}}$, and the proton lifetime which is independent of the details of the gauge hierarchy and holds in a large class of unified theories.