Spectral Analysis of a Fourier-Regularized Quantum Berry-Keating Hamiltonian and its Connection to the Riemann Hypothesis

This paper establishes a rigorous numerical and analytical framework for investigating the Hilbert-Pólya conjecture regarding the non-trivial zeros of the Riemann zeta function. We present a novel modification of the classical quantum Berry-Keating operator, explicitly introducing an exponential potential to balance phase-space dynamics. To resolve the long-standing issue of boundary discretization anomalies on finite domains, we implement a high-order Fourier spectral regularization method via the Fast Fourier Transform (FFT). Numerical diagonalization of the resulting Hermitian operator for N=1024 nodes yields highly stable discrete energy levels. Spectral unfolding and nearest-neighbor spacing distribution tests demonstrate an exact analytical match with the Gaussian Unitary Ensemble (GUE) Wigner surmise, confirming level repulsion and supporting the self-adjointness of the underlying Hamiltonian. This architecture provides a direct pathway for full logical formalization within interactive theorem provers such as Lean 4.