A Nonperturbative Framework for the Yang–Mills Mass Gap Functional Integral Inequalities, Confinement, and Spectral Positivity in ( \mathbb{R}^4 )

Abstract
We present a nonperturbative framework for the emergence of a mass gap in
four-dimensional pure Yang–Mills theory with compact gauge group (SU(N)). The
construction integrates renormalization group flow, dimensional transmutation, confinement
via Wilson loop area law, and functional integral inequalities derived from reflection
positivity. We establish that exponential decay of gauge-invariant correlation functions
follows from confinement and yields a strictly positive lower bound in the spectrum of the
Hamiltonian. The mass gap is thus identified as a spectral consequence of a finite correlation
length induced by a dynamically generated infrared scale. The formulation is compatible with
the Osterwalder–Schrader axioms and provides a structurally consistent route toward a
rigorous construction of Yang–Mills theory on (\mathbb{R}^4).