Analytic Resolution of the Yang–Mills Existence and Mass Gap Conjecture via Stochastic Quantization, BPS Topological Rigidity, and Logarithmic Sobolev Inequalities

The Yang–Mills existence and mass gap conjecture requires a rigorous constructive demonstration that a pure non-Abelian quantum gauge theory exists on four-dimensional spacetime and exhibits a strictly positive spectral mass gap consistent with axiomatic quantum field theory. Traditional static path-integral formulations restricted to bounded modular gauge regions encounter a fundamental analytical obstruction, because bounded domains in infinitedimensional functional spaces lack topological compactness and therefore fail to guarantee a discrete energy spectrum. In this treatise, we resolve the conjecture analytically by replacing static domain compactness with dynamic diferential measure convexity through the framework of covariant stochastic quantization. By evolving the non-Abelian gauge connection along an endogenous stochastic relaxation flow governed by a master difusion operator with non-linear osmotic regularization and constrained by self-dual topological action rigidity, we construct a unique stationary gauge-invariant probability measure without global gauge-fixing singularities. Applying infinite-dimensional curvature-dimension calculus on the quotient space of gauge orbits, we prove that the efective Ricci curvature tensor of the stochastic generator is strictly bounded from below by a positive uniform constant. This uniform positive curvature enforces a dimension-independent logarithmic Sobolev inequality on the invariant quantum measure. A perturbative Taylor linearization of the relative entropy functional around the vacuum state directly yields a spectral gap inequality, establishing a strictly positive lower bound for the mass of the lightest physical excitation.Furthermore, the equivalence between the logarithmic Sobolev inequality and semigroup hypercontractivity guarantees the exponential clustering of gauge-invariant correlation functions at macroscopic distances, while the exacl algebraic cancellation of the intrinsic gluonic yield potential isolates a parameter-free physical confinement scale anchored by the pure pseudoscalar glueball resonance.