The Bohr-Mollerup Theorem establishes the uniqueness of the Gamma function on the right half-plane, but makes no assertion regarding extensions to the left half-plane that preserve the integral form. This work proposes a Dual Architecture consisting of two integral operators with complementary domains: the Classical Gamma and a Symmetric Factorial, connected by an operator derived from the Hankel contour integral. When applied to the Riemann zeta function, this architecture replaces the classical functional equation—which exhibits indeterminate forms at integer points—with a formulation that is directly evaluable and preserves all values of the Dirichlet series. Analysis of the connection operator reveals that its real part vanishes exclusively on the critical line within the critical strip. Assuming a non-trivial zero off this line leads, via a closed cycle of the dual functional equation, to a contradiction involving the modulus of the Gamma function, as established independently through the Weierstrass representation and the maximum modulus principle. The result forces all non-trivial zeros of the Riemann zeta function to lie on the critical line.Keywords: Gamma function, Riemann zeta function, critical line, Hankel contour, Weierstrass representation.