A Fourier–Hilbert Approach to the Riemann Hypothesis

We develop a Fourier–Hilbert decomposition of the spectral increment associated with the nontrivial zeros of the Riemann zeta function. By introducing a series expansion for the increments Delta-S-n in terms of Fourier modes with factorial suppression, we establish uniform bounds showing that amount Delta-S-n < 0.5 holds if and only if the nontrivial zeros lie on the critical line. We prove that oscillatory cancellations and factorial decay ensure this inequality uniformly, thereby yielding a new approach to the Riemann Hypothesis. The method is entirely analytic and relies on a precise bounding of Fourier integrals, but it also admits a natural interpretation as an entropy-stability condition in a Hilbert–diagram framework developed for physics.