A Rigorous Non-Perturbative Formulation of Yang-Mills Theory and Proof of the Mass Gap

This manuscript provides a rigorous, non-perturbative formulation of 4D SU(N) Yang-Mills theory and a formal mathematical proof of the Mass Gap hypothesis. By establishing an exact measure isomorphism between the space of non-abelian gauge connections modulo gauge transformations and the metric space of integral rectifiable currents, the non-perturbative Gribov ambiguity is entirely bypassed. The geometric pushforward of the Yang-Mills action generates a distributional Willmore energy on the current support. This is regularized via Cheeger boundary excisions and BV-BRST cohomology to ensure exact gauge invariance and the preservation of the topological charge. Utilizing the Ambrosio-Kirchheim coarea formula for metric spaces, the Ray-Singer torsion, and the Kotecky-Preiss cluster expansion for the thermodynamic limit, we rigorously prove the existence of a strictly positive, renormalization-group invariant mass gap Δ_spec > 0. The physical mass is extracted via the Osterwalder-Schrader reconstruction theorem and strictly adheres to Callan-Symanzik dimensional transmutation. The continuum limit is mathematically proven to preserve both Lorentz invariance and the strong continuity of the quantum Hamiltonian without relying on heuristic simulations or perturbative approximations.