Reflection-Positive Continuum Reconstruction of SU ( N ) Yang-Mills Theory with a Nonzero Mass Gap: Part(3)

We seek order in a hard matter by trusting two guides that do not mislead: positivity and invariance. From a reflection-positive lattice formulation of SU(N) Yang-Mills, choosing on each time slice a definite transverse form and gently damping distant modes, we proceed by steps that preserve these guides. From such simple means we pass to the continuum: Euclidean correlation functions satisfy the Osterwalder-Schrader axioms, and the Wightman theory with a unique vacuum and positive energy follows. The decisive point is the scale: time-correlations are completely monotone and decay uniformly, so the spectral measure cannot touch zero; and along the same flow the vacuum-orthogonal evolution decays at a fixed rate, which survives in the limit. Thus a strictly positive, volume-independent spectral gap is obtained for the continuum Hamiltonian, and the bridge from Euclidean positivity to Hilbert-space dynamics remains intact.