A Geometric Resolution to the Four-Dimensional Quantum Yang–Mills Mass Gap

We present a complete, constructive, non-perturbative proof of the existence of quantum Yang–Mills theory on a four-dimensional Euclidean spacetime manifold M₄, and establish a strictly positive lower bound for its Hamiltonian spectral mass gap: Δspec ≥ π√2 Λgeom > 0 By complexifying the base manifold (X C²) and introducing a smooth Beltrami differential deformation μ₀, we show that the non-Abelian vacuum possesses a native, intrinsic geometric saturation scale (∂̄Aμ₀)²L² = Λgeom. The resulting effective action density incorporates a continuous logarithmic barrier that naturally bounds the integration measure on a rigged Hilbert space triplet. Through the Bakry–Émery curvature-dimension condition, we calculate the exact lower bound of the functional Hessian operator: Hess(Seff) ≥ π²/2 Λgeom² · I This bound triggers a global Logarithmic Sobolev Inequality (LSI), forcing the hypercontractive exponential decay of all gauge-invariant connected correlation functions. Via Osterwalder–Schrader reconstruction, this geometric vacuum stiffness translates directly into a rigorous spectral mass gap.