A Dual Architecture for the Gamma Function: Regularization of Feynman Integrals and the Yang-Mills Spectrum

This work demonstrates the existence of a mass gap in SU(N) Yang-Mills theory in four spacetime dimensions, based on the Dual Architecture of the Gamma function, a regularization scheme that uses exclusively Euler's convergent integral in both domains without analytic continuation. The framework defines a Regularized Gamma function that eliminates all poles at negative arguments while preserving the classical Gamma for positive arguments. The connecting factor between the two domains is shown to be unique via Sturm-Liouville theory as the only admissible eigenfunction satisfying physical boundary conditions and the Richel equation with negative curvature. The regularized Beta function emerges naturally from the gluon vacuum polarization Feynman integrals with a finite, uniquely determined value. Inserting this constant into the Yang-Mills beta function preserves asymptotic freedom while strengthening the infrared growth of the coupling, and the renormalization group equation produces a strictly positive mass gap with numerical values consistent with phenomenological QCD. The proof is supported by WKB validation at over seventeen hundred orders where the Dual Architecture eliminates the exponential deviation present in classical analytic continuation approaches, explicit two-loop verification in four-dimensional QCD, a general proof of scheme independence, all-order perturbative validity by structural induction, satisfaction of all six Wightman axioms with quantitative verification, and a non-perturbative proof via Dyson-Schwinger equations using the Schauder-Tychonoff fixed point theorem with exact asymptotic limits characterizing the gluon mass function completely. The framework preserves the Gribov horizon, the Faddeev-Popov determinant, and the theta-vacuum structure of the theory.