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

This work presents a regularization framework for Yang-Mills theory based on the Dual Architecture of the Gamma function, which uses exclusively Euler's convergent integral in both domains without analytic continuation. The framework defines a Regularized Gamma function that eliminates 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. A regularized Beta function emerges from the gluon vacuum polarization Feynman integrals with a finite, uniquely determined value. When inserted into the Yang-Mills beta function, this constant preserves asymptotic freedom while modifying the infrared growth of the coupling, and the renormalization group equation produces a strictly positive energy scale with numerical values in the range of a few hundred megaelectronvolts, consistent with the established QCD scale. The framework is validated by a WKB expansion carried to over seventeen hundred orders, where the Dual Architecture eliminates the exponential deviation present in classical analytic continuation, with the deviation parameter vanishing at all orders tested to two hundred digits of precision. Explicit two-loop verification in four-dimensional QCD, a proof of scheme independence, and structural induction establishing all-order perturbative validity are provided. All six Wightman axioms are satisfied with quantitative verification, and a non-perturbative analysis via Dyson-Schwinger equations using the Schauder-Tychonoff fixed point theorem yields exact asymptotic limits for the gluon mass function. The framework preserves the Gribov horizon, the Faddeev-Popov determinant, and the theta-vacuum structure of the theory. Keywords: Dual Architecture, Gamma function regularization, Sturm-Liouville uniqueness, WKB validation, Wightman axioms, Dyson-Schwinger equations.