The Dual Architecture of the Gamma Function: From Vacuum Energy in Odd Dimensions to the Cosmological Constant and Neutrino Mass

The Gamma function, defined by the Euler integral, converges only in a restricted half-plane. Outside this domain, the integral diverges and one resorts to analytic continuation, which does not preserve the original integral form. This work proposes the Dual Architecture of the Gamma Function, a formulation based exclusively on integral representations and their natural domains of convergence. The architecture is built from two complementary objects: the Classical Gamma, convergent for positive real parts, and the Symmetric Gamma, convergent for negative real parts. Their connection is established through a multiplicative inverse relationship that yields unity, providing a parameter-free mechanism to regularize all divergences across the complex plane. A real trigonometric operator encodes a Dirichlet boundary condition with unit reflection coefficient, geometrically corresponding to a specific phase rotation that incorporates wave backscattering at impenetrable boundaries. We investigate the physical consequences of this dual structure in three key areas. First, the vacuum energy density in odd dimensions is evaluated, where the dual architecture produces a sign alternation that agrees with reference values in all analyzed cases. Second, the high-order WKB expansion for the quartic potential is examined, where the dual evaluation completely absorbs an exponential correction of previously unexplained physical origin, confirming backscattering as its source. Third, the cosmological constant problem is addressed by summing the regularized vacuum energy over the dimensional spectrum; the series converges naturally to a fundamental constant that predicts the neutrino mass and places the seesaw scale at the Grand Unified Theory regime. The formulation preserves the core principles of quantum field theory, positioning the dual architecture as a complementary regularization scheme for problems involving boundaries, odd dimensions, and the global structure of the vacuum.