A Ramanujan–Prime Spectral Framework for GUE Statistics and the Hilbert–Pólya Program

The remarkable agreement between the local statistics of the nontrivial zeros of the Riemann zeta function and the Gaussian Unitary Ensemble (GUE) suggests the existence of a hidden arithmetic spectral system. Motivated by Ramanujan–Fourier analysis, Montgomery’s pair-correlation conjecture, Odlyzko’s numerical experiments, and the Hilbert–Pólya philosophy, we investigate a conjectural framework in which Ramanujan spectral modes provide an arithmetic basis while primelogarithmic oscillations generate spectral fluctuations analogous to those of the Riemann zeros. A family of finite-dimensional self-adjoint operators is constructed from arithmetic kernels. The resulting framework naturally connects Ramanujan sums, prime-number oscillations, random matrix theory, quantum chaos, and spectral geometry. Although the construction does not constitute a proof of the Riemann Hypothesis, it provides a concrete mathematical setting within which arithmetic spectral realizations of the zeta zeros may be explored.