The Rigidity of Frequencies: A Signal Processing and Dual-Manifold Formulation of the Riemann 1859 paper

This article formalizes a geometric and signal-processing framework to contextualize Bernhard Riemann’s 1859 peper. Rather than searching for non-trivial zeros directly on the asymmetric landscape of the complex plane, we map the problem onto two perpendicular, interacting manifolds: the Input Signal Manifold ($M_s$) and the Spectral Frequency Manifold ($M_t$). By establishing a strict, one-to-one Fourier duality between the unalterable, history-dependent sequence of prime numbers and Riemann's completed $\xi(t)$ function, we demonstrate that any deviation from a purely real frequency ($t_{\text{im}} \neq 0$) mathematically requires a structural deformation of the prime sequence itself. Backed by Hans von Mangoldt’s exact verification of the prime staircase error terms, this closed-loop information system demonstrates that the rigidity of arithmetic forces all resonant frequencies to lie strictly on the real axis of $M_t$.