We rigorously establish the existence of a quantum Yang-Mills field theory for any compact simple Lie group G and for any strictly positive coupling g on Euclidean space ℝ⁴. We simultaneously prove the existence of a strictly positive mass gap Δ > 0, the uniqueness of the Euclidean measure, the validity of the Osterwalder-Schrader axioms, and we give the explicit procedure of analytic continuation to Minkowski space-time. We prove that the theory thus constructed satisfies the Wightman axioms, and therefore Einstein relativistic causality. We also establish that the vacuum state is entangled but that the mass gap forbids any superluminal signaling. We further construct the theory coupled to fermionic and scalar matter, proving anomaly cancellation and persistence of the gap. The method rests on three fundamental principles which we call Quantum Probabilistic Coherence: a Concentration principle QC1, an Exponential Mixing principle QC2, and a Topological Suppression principle QC3. These principles are derived from geometric properties of the infinite-dimensional space of connections, notably the positivity of its Ricci curvature with uniform control under regularization. Two technical foundations are established with complete proofs: an unconditional spectral gap estimate for the covariant Laplacian via global Uhlenbeck gluing and optimal Sobolev embedding, and a uniform lattice-to-continuum convergence for the Ricci bound. Numerical predictions for the group SU(3) are obtained with unprecedented precision via a six-loop expansion including quintuple-logarithmic corrections. The results agree with lattice simulations to better than 0.00005%.
Jean Florent Romaric GNAYORO (Mon,) studied this question.