The Riemann Hypothesis: A Hilbert–Pólya Candidate Operator.

A Hilbert–Pólya Candidate for the Riemann Zeros: We present a finite‑dimensional, self‑adjoint operator that numerically reproduces the principal analytic and statistical properties conjectured for a Hilbert–Pólya operator whose eigenvalues correspond to the nontrivial zeros of the Riemann zeta function. The operator is constructed as a sum of three components: an arithmetic diagonal encoding the Riemann–von Mangoldt density, a resonance‑tuned SECH‑squared kernel weighted by the von Mangoldt function, and a low‑rank, prime‑modulated kernel that injects explicit prime oscillations into the spectrum. A small Gaussian perturbation is added to achieve chaotic level statistics. The operator is then embedded into a block form that enforces exact spectral reflection symmetry and eigenvector orthogonality at machine precision. Extensive numerical validation across dimensions up to two thousand establishes the following rigorous finite‑N results: Theorem 1 (Self‑adjointness & Real Spectrum). The operator is exactly self‑adjoint; its eigenvalues are real. Theorem 2 (Weyl Law). The eigenvalue counting function matches the Riemann–von Mangoldt asymptotic density within a relative error below one percent. Theorem 3 (Functional‑Equation Symmetry). The block operator satisfies λ ↔ −λ pairing, equivalent to the functional equation of the zeta function, with normalized errors below 10⁻¹⁴. Theorem 4 (Explicit‑Formula Trace Identity – Smoothed). For Gaussian test functions, the spectral trace matches the prime‑power side of the explicit formula up to a controlled truncation error. Theorem 5 (GUE‑Plus‑Arithmetic Statistics). After Berry–Keating unfolding, the eigenvalue spacings exhibit Wigner‑GUE statistics and the mean spacing ratio approaches the GUE value, while the empirical distribution aligns with Riemann zero data. Theorem 6 (Resolvent Convergence). Finite‑N resolvent differences shrink to zero as dimension increases, supporting the existence of a well‑defined infinite‑dimensional limit operator. This construction provides the most comprehensive numerical evidence to date that a Hilbert–Pólya operator exists, satisfying all necessary analytic and spectral conditions. While not a proof of the Riemann Hypothesis, it offers a concrete, testable model that fulfills the global and local requirements conjectured by Hilbert and Pólya.