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

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 $|γ_1-γ_1^{\mathrm{Riemann}}|$ 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 $λ_{\min}^{\mathrm{even}}$ separately reaches $\sim 10^{-334}$ at $c=100$, $N=250$ (275-OOM span from $c=13$), and the same eigenvector recovers $γ_1,\ldots,γ_{10}$ to 307-329 matching digits at $N=250$, $\mathrm{dps}=500$. 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 $D_{\log}^{(λ,N)}$ at $λ=\sqrt c$. On the four-point $N$-sweep at $c=100$, Aitken-$Δ^2$ on two consecutive triples gives $\log_{10}|λ_\infty^{\mathrm{even}}|\approx -536.76$ and $\approx -533.70$, approaching the Connes 2026 Section 6.4 heuristic continuum prediction ($\approx -530.38$) monotonically with $N$. The empirical fit $|\log_{10}λ_{\min}|\approx 13.24 c^{0.634}$ 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 $QW_λ$ is RH-equivalent and we do not assume it at $λ=\sqrt{100}$. We make no claim of proof.