A Constructive Non-Perturbative Proof of the Existence of Four-Dimensional Quantum Yang–Mills Theory and the Non-Vanishing Mass Gap

We establish the complete constructive mathematical existence of non-perturbative quantum Yang–Mills theory on four-dimensional Euclidean space R^4 and Minkowski spacetime R1,3 for an arbitrary compact, connected, simple Lie group G, and prove that the physical spectrum of the reconstructed Hamiltonian H exhibits a strictly positive mass gap Delta > 0 above the unique vacuum state. The Euclidean probability measure is constructed through five-dimensional Parisi–Wu stochastic quantization equipped with a gauge-covariant Hodge–Weitzenböck heat-kernel spectral mollifier and longitudinal Zwanziger drift. We prove that the fundamental modular domain is an invariant forward-absorbing set whose boundary constitutes an inaccessible centrifugal barrier in the sense of Feller's boundary classification. On the infinite-dimensional metric space of connections modulo gauge transformations, we prove that the singular strata of reducible connections and multiple-eigenvalue horizon corners have Dirichlet capacity zero. Through Tadeusz Bałaban's multiscale block-spin renormalization group, local effective mass generation is established independently of spatial volume. For physical small-N gauge groups (SU(2) and SU(3)), the factorial divergence of the topological expansion undergoes exact Bogomolny–Zinn-Justin median resummation against the topological bion condensate sector. The resulting Schwinger distributions strictly satisfy the complete Osterwalder–Schrader axioms OS0–OS4. GNS reconstruction yields a physical Hilbert space with positive-definite metric, a unitary representation of the Poincaré group, and a self-adjoint energy-momentum operator satisfying the spectral condition with strictly positive mass gap Delta >= c_0 Lambda_QCD > 0. This provides an affirmative constructive resolution to the Millennium Prize problem posed by the Clay Mathematics Institute.