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

Abstract

The Connes-van Suijlekom truncated Weil quadratic form, indexed by a cutoff $c$ (controlling the primes $p\leq c$ in 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 $c=13$--$67$ at $N=100$, the first-zero error $|γ_1-γ_1^{\mathrm{Riemann}}|$ shrinks monotonically from $\sim 2\times 10^{-55}$ to $\sim 1.5\times 10^{-168}$ (113-OOM across fifteen cutoffs). The smallest-positive even-sector eigenvalue reaches $\sim 10^{-334}$ at $c=100$, $N=250$ (275-OOM span), whose eigenvector recovers $γ_1,\ldots,γ_{10}$ to 307-329 matching digits. Under the unitary equivalence with CCM 2025 Lemma 5.1, each $γ_k$ is (modulo a hypothesis-status caveat at $c=100$) an eigenvalue of the CCM rank-one operator $D_{\log}^{(λ,N)}$. Aitken-$Δ^2$ on the $c=100$ $N$-sweep gives $\log_{10}|λ_\infty^{\mathrm{even}}|\approx -536.76$ and $-533.70$, approaching the Connes 2026 Section 6.4 heuristic continuum prediction ($\approx -530.38$) monotonically in $N$. The empirical fit $|\log_{10}λ_{\min}|\approx 13.24\,c^{0.634}$ on $c\leq 67$, $N=100$ is a finite-$N$ rate, falsified at $c=100, N=200$ by 49 OOM. At $c=100$ the raw spectrum carries a small block of negative-sign eigenvalues at the finite archimedean cutoff $T=800$; these are an artifact of that cutoff and absent once $T$ is increased, so the smallest-positive even-sector eigenvalue is the genuine smallest one (continuum positivity of $QW_λ$ is RH-equivalent and is not assumed at $λ=\sqrt{100}$). We make no claim of proof.