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 ≤ c entering the operator, produces a ground state whose Fourier–Mellin zeros provably lie on the critical line; whether they converge to the Riemann zeros as c → ∞ is open (Connes 2026; Connes–Consani–Moscovici 2025). We present, to our knowledge, the first independent public implementation of the Connes–van Suijlekom Galerkin matrix at sixteen cutoffs (c = 13 through 67, plus c = 100). Across the in-sample window c = 13 through c = 67 at N = 100, the first-zero absolute error |γ1 − γ1Riemann| shrinks monotonically from ∼2×10−55 to ∼1.5×10−168, a 113-OOM convergence across fifteen cutoffs. The smallest-positive even-sector eigenvalue λmineven separately reaches ∼10−334 at c = 100, N = 250 (275-OOM span from c = 13). Out-of-sample test at c = 100. On the four-point N-sweep N ∈ {100, 150, 200, 250} at dps = 500, consecutive first-difference ratios 0.837 and 0.836 match to two decimal places. Aitken-Δ2 on the two overlapping triples yields log10|λ∞even| ≈ −536.8 and ≈ −533.7, approaching the Connes 2026 §6.4 heuristic prediction (≈ −530.4) monotonically with N (6.4 and 3.3 OOM gaps out of |x∞| ∼ 530). The same eigenvector recovers γ1, …, γ10 to 307–329 matching digits at N = 250, dps = 500. Under the unitary equivalence with Connes–Consani–Moscovici Lemma 5.1, this is the deepest such Galerkin-truncation recovery in the public Connes–van Suijlekom / Connes–Consani–Moscovici literature, subject to a hypothesis-status caveat. The raw finite-N matrix carries a small block of negative-sign eigenvalues at the finite archimedean cutoff T = 800; these are an artifact of that cutoff and are absent once T is increased, so the smallest-positive even-sector eigenvalue is the genuine smallest one (continuum positivity of QWλ is RH-equivalent and is not assumed at λ = √100). The fit |log10 λmin| ≈ 13.24 c0.634 on c ≤ 67 at N = 100 is shown to be a finite-N rate, falsified at c = 100, N = 200 by 49 OOM in the direction of faster decay. Structural observations include approximate eigenvector c-invariance (overlap ≥ 0.9498 on all 105 cutoff pairs despite eigenvalues differing by 113 OOM), multi-zero convergence universality (all ten detectable zeros within 3.8% of each other), an empirical Galerkin-convergence exponent s(c) ≈ 55 log c − 128, un-rescaled Galerkin bulk-spectrum Poisson statistics (β < 0.05; this is a structural diagnostic of the truncated operator, not a test of Montgomery's conjecture, which applies to locally-rescaled zero spacings), and tight bulk invariants log|det Qc| ≈ −65.6 c + 542 (R2 = 0.997). We make no claim of proof; the contribution is reproducible numerical data and its careful interpretation under the existing CvS / CCM framework. All code, data, and ancillary files are publicly available. Versions and corrections. This is Version 3.7, which revises the manuscript text; every numerical-data file in this deposit is identical to the corresponding file in Version 3.6. It folds in the correction recorded on 2026-08-14: the manuscript's statement that raising the archimedean cutoff to T = 1200 removes the small block of negative-sign eigenvalues at c = 100 was wrong. That block rearranges with the cutoff rather than clearing at any cutoff we tested, and the conclusion rests instead on a cutoff-free evaluation of the archimedean entries, certified by a rigorous interval factorization in Arb, which leaves the even sector with no negative eigenvalues. This revision also records that the odd sector, which the computations here do not probe, has since been probed independently by M. Osman, whose experiments we have reproduced. Five corrections to the paper have been recorded since the first version, none of which changes a measured value, and all are incorporated in the manuscript text. They are listed in full in ERRATA.md, deposited here as a file, with the version history in CHANGELOG.md beside it. Earlier versions remain available, and the concept DOI always resolves to the current one.