Construction of Finite Hilbert--P\'olya Matrices from Weil's Explicit Formula

Abstract

Starting from the Riemann--$\Xi$ specialization of Weil's explicit formula, we construct finite real-symmetric Prime--Weil matrices $(\mathbf S)$ from pole, archimedean, and finite prime-power data. This construction is a finite-dimensional arithmetic model within the Hilbert--P\'olya program, which seeks a self-adjoint spectral realization of the nontrivial zeta-zero parameters. Their off-diagonal entries form a Loewner-type divided-difference matrix with a rank-two displacement identity. We formulate the spectral quotient as a Hermitian definite generalized eigenproblem on the fixed zero-mean contrast space. This realization is invariant under positive affine rescaling, avoids ground-vector normalization and an ill-conditioned oblique projector, and preserves the finite quotient spectrum. For a dimension-matched zero-side matrix built from $N$ distinct positive ordinates $\gamma_k$, rational interpolation gives the exact contrast-pencil spectrum $\{\pm\gamma_1,\ldots,\pm\gamma_N\}$ and positive-parity square spectrum $\{\gamma_1^2,\ldots,\gamma_N^2\}$. Since the ordinates are inputs, this is a reconstruction theorem. Assuming RH, the same interpolation vector proves $\lambda_{\min}(\mathbf S)\to0$. At $N=L=13$, Lemke's ground-state quotient and the contrast pencil agree numerically and reproduce the first three zeta ordinates to the reported precision. The remaining problem is a relative prime-to-zero perturbation theorem with uniform control of the compressed metric. No proof of RH is claimed.