Second--Order Spectral Stability and the Yang--Mills Mass Gap

This paper develops an abstract second–order stability framework for families of self–adjoint operators arising in scale–dependent formulations of Yang–Mills theory. The analysis is based on refinement relations between effective Hamiltonians and on controlled second–order defects under scale doubling.Under strong resolvent convergence and mild summability assumptions on the refinement defect, the existence of a strictly positive spectral gap in the continuum limit is shown to follow from finite–scale positivity. The argument is entirely operator–theoretic and does not rely on explicit spectral computations, confinement scenarios, or perturbative expansions.The framework is independent of the choice of lattice, gauge fixing, or regularization scheme, and isolates the minimal structural mechanism responsible for the persistence of the mass gap across scales.