Numerical Verification of Low-Lying Riemann Zeta Zeros, Gap Statistics, Pair Correlations, and a Toy Hilbert–Pólya Determinant Test

This upload contains a short research note/paper presenting numerical verification results for the Riemann zeta function non-trivial zeros, including a toy Hilbert–Pólya style operator test. The document reports computed low-lying zeros, basic gap statistics, determinant scans, and consistency checks. The goal is to provide reproducible numerical evidence and a structured exploration toward a Hilbert–Pólya formulation. Keywords: Riemann Hypothesis, Riemann zeta function, Xi function, Hilbert–Pólya, spectral operator, numerical verification, non-trivial zeros.