Spectral Geometry of the Riemann Zero Log-Gas: A Weighted Graph Laplacian, Four Theorems, and a Precise Open Problem

We establish four theorems about the spectral geometry of the Riemann zero log-gas and its connections to the de Bruijn–Newman flow. The central observation, which appears to be new, is that the Hessian of the log-gas potential W = Σj<k log|tj −tk| is exactly the negative of a weighted graph Laplacian L; the proof is one line. This immediately gives (Theorem 2) a strictly positive spectral gap λ2(L) > 0 for any finite collection of distinct real numbers, with a two-line proof from graph connectivity. Theorem 3 gives explicit two-sided bounds cj nn ≤ cj ≤ cj nn + (4/dmin 2)(π2/6−1) on the local stiffness, with the constant π2/6−1 ≈ 0.6449 arising from the Basel problem. Theorem 4 uses the Rayleigh quotient to prove that consecutive entries of the Fiedler vector satisfy |vj −vj+1|2 ≤ λ2·d2/2, establishing that the most susceptible perturbation to the zero configuration is a smooth global deformation rather than a local collision, for every finite N. All four results hold unconditionally for any finite set of distinct real numbers; applied to the Riemann zeros they yield structural observations about the de Bruijn–Newman flow. The paper closes with a precisely stated open problem (Open Problem 7.1) about whether the infinite sequence of spectral gaps λ2(LN) has a positive infimum, which we identify as the natural next question for researchers in log-gas theory, random matrices, and the de Bruijn–Newman constant.