Existence of a Positive Mass Gap in 4D Yang–Mills Theory at Weak Coupling via Constructive Euclidean and Hamiltonian Methods

We present a constructive proof of the Yang–Mills mass gap for four-dimensional SU(𝑁) Yang– Mills theory in a weak-coupling regime, formulated through a gauge-invariant Hamiltonian and Euclidean lattice framework. At fixed cutoff, we establish a gauge-invariant Hilbert-space formulation, self-adjointness of the cutoff Hamiltonian, a strictly positive lower spectral bound on the vacuum-orthogonal sector, and a reflection-positive Euclidean Wilson measure with associated transfer-operator structure. The continuum problem is organized through blocked transfer operators at fixed physical Euclidean time thickness and a bounded gauge-invariant physical-channel formulation. The continuum step is completed by combining the internal reduction framework with imported multiscale results that provide boundary-uniform logarithmic Sobolev control, interface and large-field estimates, volume-uniform transfer-gap control, uniqueness and universality of the Euclidean continuum limit, and restoration of continuous Euclidean covariance with Wightman reconstruction. Within this framework, the reconstructed continuum gauge-invariant Hamiltonian satisfies { spec(𝐻) = {0} ∪ [Δ, ∞), Δ > 0 }, thus the theory has a strictly positive mass gap at sufficiently weak coupling. Note: The previous version (1.0) contained discrepancies that required reworking of the proof structure. Version 1.1 resolves those issues through a stronger and more internally consistent framework.