Reflection-Positive Renormalization and the Persistence of the Mass Gap in Lattice SU( N ) Yang-Mills: Part(2)

We show that a measure of clarity can be brought to the nonperturbative Yang-Mills problem if one holds fast to two principles: reflection positivity and gauge invariance. On the lattice, we construct a renormalization procedure that respects these principles exactly at each step. The method is elementary in its components: a transverse representative chosen within the fundamental modular region, a smooth horizon projector from the covariant Laplacian that softens long-range fluctuations, and a block transformation whose locality does not fade with scale. Out of these pieces arises a framework that is both mathematically precise and physically faithful. From this construction emerge three enduring results. First, the polymer expansion remains convergent under repeated renormalization, with bounds independent of the number of steps. Second, the fall-off of correlations, which embodies the presence of a mass gap, persists uniformly across scales with a constant rate m * > 0. Third, the spectral gaps of successive transfer operators obey an inequality that prevents them from collapsing, so that a strictly positive lower bound endures in the continuum limit. Thus we obtain a step-scaling mechanism that conveys spectral information from the strong-coupling domain into the scaling window without loss. The bridge between Euclidean clustering and Hamiltonian gaps is kept intact, and the way is opened to the continuum reconstruction of Yang-Mills theory with a nonzero mass threshold.