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

We present a complete, constructive, non-perturbative proof of the existence of quantum Yang–Mills theory on a four-dimensional Euclidean spacetime manifold $\\mathcal{M}_4$, and establish a strictly positive lower bound for its Hamiltonian spectral mass gap: $$\\Delta_{\\text{spec}} \\ge \\frac{\\pi}{\\sqrt{2}} \\Lambda_{\\text{geom}} > 0$$ By complexifying the base manifold ($X \\cong \\mathbb{C}^2$) and introducing a smooth Beltrami differential deformation $\\mu_0$, we show that the non-Abelian vacuum possesses a native, intrinsic geometric saturation scale $\\Vert(\\bar{\\partial}_A^{\\mu_0})^2\\Vert_{L^2} = \\Lambda_{\\text{geom}}$. 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: $$\\text{Hess}(S_{\\text{eff}}) \\ge \\frac{\\pi^2}{2} \\Lambda_{\\text{geom}}^2 \\cdot \\mathbb{I}$$ 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, satisfying the explicit criteria set forth by the Clay Mathematics Institute.