Pathway Towards a Proof of the Riemann Hypothesis via Spectral Transfer

The Riemann Hypothesis (RH), which posits that all non-trivial zeros of the Rie- mann Zeta function lie on the critical line R(s) = 1/2, remains one of the most pro- found open problems in mathematics. This paper presents a potential pathway for its proof, not a conclusive demonstration, based on a spectral transfer method. The strategy leverages the fact that the RH is true for the Dedekind Zeta function of the field of Gaussian integers, ζQ(i)(s). A functional object, V (s), is constructed to relate the completed Riemann Xi-function, ξQ(s), to its Gaussian analogue, ξQ(i)(s). The central proposition is that the holomorphy of V (s) in the critical strip R(s) > 1/2, aided by a "corrector crutch" function g(s) designed to cancel structural discrepan- cies, is incompatible with the existence of a zero of ζ(s) off the critical line. The theoretical foundations of the method are presented, its limitations are discussed, and the necessary steps for a rigorous verification by the mathematical community are outlined.