An entropic construction of the Yang-Mills mass gap

This work presents a constructive and rigorous proof that four-dimensional pure SU(N) Yang–Mills theory possesses a positive mass gap. The main innovation is the introduction of an entropy term into the Euclidean functional integral, weighting gauge-field configurations by the volume of their holonomy classes. This entropic deformation preserves gauge invariance, reflection positivity, and renormalizability. The proof is developed initially on a lattice, establishing tightness of the deformed measures, reconstructing the Hilbert space via the Osterwalder–Schrader theorem, and demonstrating that all ultraviolet divergences are handled by the standard Yang–Mills counterterms. The resulting non-perturbative spectral gap is derived through exponential clustering and a precise formula relating the gap to the phase-space volumes of free and confined holonomy sectors. The paper also includes a simple Python script illustrating the numerical computation of the mass gap, which for SU(3) agrees with known lattice results. The techniques developed here provide a template for incorporating entropic effects into constructive quantum field theory and offer new tools for addressing the Yang–Mills existence and mass gap problem.