The Dual Architecture of the Gamma Function and the Regularization of Asymptotic and Logarithmic Divergences

The Gamma function, defined by the Euler integral , converges only for . Outside this domain, the integral diverges and one resorts to analytic continuation. However, as observed by Titchmarsh, this extension does not preserve the original integral form, and the Bohr-Mollerup Theorem guarantees directional uniqueness throughout the complex plane, but does not ensure univocity for opposite directions in certain domains, leaving the negative axis without an equivalent unique characterization. The present proposal organizes this domain through a dual architecture with four complementary functional objects — the Classical Gamma and the Symmetric Gamma — connected by the operator , which encodes a Dirichlet condition with reflection coefficient and introduces wave backscattering. For negative half-integer arguments, the dual architecture and analytic continuation produce distinct structures. This difference is investigated in two contexts: the vacuum energy density in odd dimensions ( to ), where the sign coincides with reference values in 100% of cases, and the WKB expansion of Noreen & Olaussen for the potential up to order 1704, where a correction without apparent physical origin had been identified. The dual architecture reveals that replacing the traditional Beta functions with in the coefficients completely eliminates the deviation , with the ratio absorbing the correction (-1 and confirming backscattering as the physical origin. Keywords: Dual architecture, Gamma function, negative half-integer arguments, vacuum energy density, operator , backscattering, Noreen & Olaussen.