Dual Architecture of the Gamma Function: Complementary Extension to the Left Half-Plane and Regularization of Logarithmic Divergences

This work proposes a complementary extension of the factorial function to the real line, based on the separation of the bilateral integral of e^-Ax into two disjoint domains. For A > 0, the classical Gamma function Gamma (z) emerges in the right half-plane; for A < 0, we introduce the Symmetric Gamma Function bar (Gamma) (z) = e^-i pi (z+1) Gamma (-z), defined in the left half-plane, with poles at the positive integers. The duality between the integral kernels underlies a regularization mechanism for logarithmic divergences: GammaR (z) = 1/bar (Gamma) (z-1) and bar (Gamma) R (z) = 1/Gamma (z+1), replacing each pole by an exact finite value. An alternative real representation B (x) = int₀ⁱnfty e^-u u^-x du (convergent for x < 1) is constructed, together with a trigonometric factor C (x) = 2 sin (pi x) + cos (pi x), defining F (x) = C (x) B (x) for x <= 0, yielding F (-n) = (-1) ⁿ n!, thereby unifying the complex branch, real regularization, and Laurent expansion formalisms. The relation F (-n) * Res (Gamma, -n) = 1 establishes the fundamental duality. The method is validated in arithmetic progressions, QED, and QCD. The extension to k-loops is systematic via exponentiation: GammaR^ (k) (-n) = (-1) ⁿ/n!ᵏ. Distinct physical prediction: In d = 5 dimensions, the vacuum energy density changes sign — a repulsive force where the MS-bar predicts attraction.