Dual Architecture: A Continuous Regularization Technique with Applications in Physics

The Gamma function diverges at negative integer and half-integer arguments, posing a fundamental obstacle for both perturbative and non-perturbative theories. Analytic continuation, while guaranteeing uniqueness, does not preserve the Euler integral representation in the left half-plane, as noted by Hardy and Titchmarsh. We present a continuous regularization technique - the Dual Architecture - that addresses this limitation through a complementary function with directional vector opposite to that of the Gamma function. The phase function is uniquely determined by the boundary conditions and , and its periodicity is established by Lemma 2.1. The regularized function for is finite by construction at all points where the classical Gamma function diverges, unifying regulation and subtraction within a single definition. We demonstrate the physical applicability of the technique across six systems: the cosmological constant, the Higgs boson mass, the strong CP problem, the Casimir effect, dimensional regularization in , and a divergent Gaussian integral. In each case, the algebraic development is presented in full, yielding analytical results consistent with experimental values. The technique offers a unified framework for treating Gamma-function divergences across perturbative and non-perturbative regimes. Keywords: Continuous regularization, Gamma function, uniqueness theorem, Casimir effect, Standard Model.