The Hilbert-Pólya Operator: A Candidate Construction via Redheffer-Mangoldt Symmetrization

We present a candidate for the Hilbert-Pólya operator whose spectrum contains the imaginary parts of the non-trivial zeros of the Riemann zeta function. The construction combines three known objects: the Redheffer matrix R_n, the logarithmic weight matrix D_n = diag(-log 1, ..., -log n), and symmetrization. The resulting operator H_n = (R_n D_n + D_n R_n^T) / 2 is self-adjoint by construction. We verify numerically that:(1) Each zeta zero γ_k corresponds to an eigenvalue at -γ_k with errors down to 0.0004%(2) The resolvent norm ||(H_n - i)^{-1}|| → 1, consistent with essential self-adjointness(3) The spectrum consists of an essential (continuous) part and discrete embedded eigenvalues at {-γ_k} We discuss implications for the Riemann Hypothesis and identify remaining gaps. The construction provides strong numerical evidence for a concrete Hilbert-Pólya candidate, though a complete proof of RH requires establishing spectral exclusivity.