A Vanishing Theorem for the Exceptional Fiber

We develop a geometric reinterpretation of the Riemann hypothesis by applying Hodge--de Rham complex to the Salem integral criterion. By constructing an appropriate manifold equipped with a Fisher information metric derived from the Fermi--Dirac kernel, we reformulate the analytic condition of Salem's theorem as the vanishing of certain cohomology classes. We propose a proof strategy utilizing the Bochner--Weitzenb"ock formula, spectral rigidity from $E_8$ symmetry, and a Kodaira-type vanishing argument to establish the triviality of the Salem operator's kernel in the critical strip.