Hilbert–Pólya via de Branges and the Weyl m-Function: Vertical Convolution and the Riemann Hypothesis

Abstract

We present a deterministic analytic–spectral route to the Riemann Hypothesis. The argument begins with a Gaussian spectral regularization of the zeta function around a variationally selected center and encodes the functional symmetry at the level of a completed companion that is entire and exactly symmetric. A vertical convolution identity with a Gaussian kernel, together with uniform analytic estimates, produces locally uniform convergence from the regularized completion to the classical one.

On the operator side, we construct, for each regularization scale, explicit self-adjoint Schrödinger operators with exponential confinement whose Weyl–Titchmarsh data are calibrated to the completed arithmetic model; equality of spectral measures follows. This calibration forces a Hermite–Biehler structure for the completed functions, which in turn places all zeros on the critical line at every fixed scale. Passing to the limit, the zero set is transferred and the Riemann Hypothesis is established.

Conceptually, the work realizes the Hilbert–Pólya philosophy at each scale and, in the limit, furnishes a self-adjoint model at the level of Weyl data for the classical completion. Methodologically, the proof uses only standard tools from complex analysis, de Branges spaces, and Weyl–Titchmarsh theory, and it is organized so that each step is transparent and verifiable on its own. We also record stability results for zero counts on crossing rectangles and robustness under changes of smoothing and compact perturbations of the confining potential.