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 F(z) with directional vector opposite to that of the Gamma function. The phase function C(z)=cosu2061(πz)-sinu2061(πz) is uniquely determined by the boundary conditions F(0)=1 and F(1/2)=-√π, and its periodicity is established by Lemma 2.1. The regularized function RΓ(z)=1/[C(z)Γ(1-z)] for R(z)≤0 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 d=3, 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.