A geometric and topological framework for the proton-electron mass ratio

Understanding fermion mass hierarchies remains a central open problem in particle physics. We investigate whether the proton-electron mass ratio can be understood as a structural consequence of renormalization-group (RG) dynamics and non-Abelian gauge topology. We formulate logarithmic mass hierarchies in terms of spectral response functionals associated with RG evolution. Within this formulation, the logarithmic mass hierarchy decomposes into two structurally distinct contributions: a universal geometric term arising from RG flow periodicity, and a finite shift associated with the minimal topological sector of an SU(3) gauge theory. The framework predicts a weak dependence on gauge-group rank and provides falsifiable criteria via lattice observables and RG-scheme independence. These results suggest that fermion mass ratios may encode RG spectral geometry and gauge-theoretic universality, rather than arbitrary input parameters.