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: Gamma_R(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) = int_0^infty 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: Gamma_R⁽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.