Uniform Modular Coercivity and a Nonperturbative Proof of the Yang–Mills Mass Gap

We prove the existence of a mass gap for four-dimensional SU(2) Yang–Mills theory by developing a new analytic framework combining modular Dirichlet forms, gauge-adapted op- timal transport, and multiscale logarithmic Sobolev inequalities. At the mesoscopic scale, we establish a uniform MLSI for Yang–Mills block measures by exploiting gauge-averaged con- vexity of the Wilson action. A gauge-adapted Wasserstein distance is introduced to control inter-block fluctuations, yielding a Talagrand-type transport inequality on the quotient by local gauge transformations. These two ingredients allow us to construct a multiscale entropy iteration that produces a global MLSI with a strictly positive coercivity constant independent of lattice size. Through the Cipriani–SauvageotnoncommutativeDiracformalismandOsterwalder–Schraderreconstruction, this coercivity yields a positive spectral gap for the continuum Yang–Mills Hamiltonian. Our approach is entirely nonperturbative and does not rely on lattice approximation limits beyond Mosco convergence of Dirichlet form