A Nonperturbative Framework for the Yang–Mills Mass Gap Functional Integral Inequalities, Confinement, and Spectral Positivity in (R⁴)

AbstractWe present a nonperturbative framework for the emergence of a mass gap infour-dimensional pure Yang–Mills theory with compact gauge group (SU (N) ). Theconstruction integrates renormalization group flow, dimensional transmutation, confinementvia Wilson loop area law, and functional integral inequalities derived from reflectionpositivity. We establish that exponential decay of gauge-invariant correlation functionsfollows from confinement and yields a strictly positive lower bound in the spectrum of theHamiltonian. The mass gap is thus identified as a spectral consequence of a finite correlationlength induced by a dynamically generated infrared scale. The formulation is compatible withthe Osterwalder–Schrader axioms and provides a structurally consistent route toward arigorous construction of Yang–Mills theory on (R⁴).