High-Precision Approximation of Riemann Zeros via the Truncated Weil Form

TL;DR

This paper presents a high-precision numerical implementation of the Connes-van Suijlekom truncated Weil form, achieving unprecedented accuracy in approximating Riemann zeros and eigenvalues related to the Riemann Hypothesis.

Contribution

First public implementation of the CvS Galerkin matrix at multiple cutoffs, demonstrating exponential convergence in approximating Riemann zeros and eigenvalues with high precision.

Findings

Monotonically decreasing error in first zero approximation across cutoffs

Eigenvalues reach magnitudes around 10^{-334} at c=100, N=250

Eigenvector recovery of multiple zeros to over 300 digits of accuracy

Abstract

The Connes-van Suijlekom truncated Weil quadratic form, indexed by a cutoff parameter $c$ that controls the primes $p\leq c$ entering the operator, has a ground state whose Fourier-Mellin zeros provably lie on the critical line; whether they converge to the Riemann zeros as $c\to\infty$ is open (Connes 2026; Connes-Consani-Moscovici 2025). We present, to our knowledge, the first public implementation of the CvS Galerkin matrix at sixteen cutoffs ($c=13$ through $67$, plus $c=100$). Across $c=13$ through $c=67$ at $N=100$, the first-zero absolute error $|\gamma_1-\gamma_1^{\mathrm{Riemann}}|$ shrinks monotonically from $\sim 2\times 10^{-55}$ to $\sim 1.5\times 10^{-168}$ -- a 113-OOM convergence across fifteen cutoffs. The smallest-positive even-sector eigenvalue $\lambda_{\min}^{\mathrm{even}}$ separately reaches $\sim 10^{-334}$ at $c=100$, $N=250$ (275-OOM span from $c=13$), and the same eigenvector recovers $\gamma_1,\ldots,\gamma_{10}$ to 307-329 matching digits at $N=250$, $\mathrm{dps}=500$. Under the unitary equivalence with CCM 2025 Lemma 5.1, each $\gamma_k$ is (modulo a hypothesis-status caveat at $c=100$) an eigenvalue of the CCM rank-one operator $D_{\log}^{(\lambda,N)}$ at $\lambda=\sqrt c$. On the four-point $N$-sweep at $c=100$, Aitken-$\Delta^2$ on two consecutive triples gives $\log_{10}|\lambda_\infty^{\mathrm{even}}|\approx -536.76$ and $\approx -533.70$, approaching the Connes 2026 Section 6.4 heuristic continuum prediction ($\approx -530.38$) monotonically with $N$. The empirical fit $|\log_{10}\lambda_{\min}|\approx 13.24 c^{0.634}$ on $c\leq 67$, $N=100$ is shown to be a finite-$N$ rate, falsified at $c=100, N=200$ by 49 OOM. The raw spectrum at $c=100$ carries 3, 5, 8, 11 negative-sign eigenvalues for $N=100,150,200,250$; continuum positivity of $QW_\lambda$ is RH-equivalent and we do not assume it at $\lambda=\sqrt{100}$. We make no claim of proof.