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arxiv: 2605.20224 · v1 · pith:N6BX4BWOnew · submitted 2026-05-13 · 🧮 math.NT

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

Akiva Groskin This is my paper

Pith reviewed 2026-05-21 07:59 UTC · model grok-4.3

classification 🧮 math.NT
keywords Riemann zerostruncated Weil formcritical linenumerical approximationoperator eigenvaluesconvergence rateGalerkin discretization
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The pith

The truncated Weil quadratic form with larger cutoffs approximates Riemann zeros with errors shrinking over 100 orders of magnitude.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper implements the truncated Weil quadratic form controlled by a cutoff parameter that includes more primes as it increases. Computations at cutoffs from 13 to 67 show the error in the first approximated zero decreasing from about 2 times 10 to the minus 55 to 1.5 times 10 to the minus 168. This 113-order-of-magnitude improvement across fifteen steps suggests that the zeros of the form's ground state converge to the actual Riemann zeros as the cutoff goes to infinity. If this convergence holds, it would link the spectral properties of the truncated operator directly to the distribution of prime-related zeros on the critical line.

Core claim

The ground state of the truncated Weil quadratic form has Fourier-Mellin zeros that lie on the critical line and converge to the Riemann zeros as the cutoff parameter increases, with the first-zero absolute error shrinking monotonically by over one hundred orders of magnitude between cutoffs 13 and 67 at fixed matrix size.

What carries the argument

The truncated Weil quadratic form indexed by cutoff c, whose ground-state Fourier-Mellin zeros are computed via Galerkin matrix discretization.

Load-bearing premise

The computed discrete zeros correspond to the eigenvalues of the underlying operator under the assumed unitary equivalence.

What would settle it

A computation at cutoff larger than 100 with matrix size exceeding 250 showing that the error in the first zero stops decreasing or begins to increase.

Figures

Figures reproduced from arXiv: 2605.20224 by Akiva Groskin.

Figure 1
Figure 1. Figure 1: First-zero absolute error |γ1 error| across fifteen cutoffs. The data spans 113 orders of magnitude. Dashed lines indicate the backward-error floors at dps = 80 and dps = 150. Remark 5.1 (Precision-floor warning for c = 43). At dps = 150, log10 |γ1 error| = −144.63 is only approximately 1 order of magnitude above the dps = 150 backward-error floor ε· ∥Q∥2 ≈ 6×10−150 (log10 ≈ −149.2). The dps = 200 spot-che… view at source ↗
Figure 2
Figure 2. Figure 2: Broken-axis N-sweep at c = 100, dps = 500 (smallest-positive even-sector eigenvalue). Top panel: the four measured data points log10 |λ even min | = −190.92, −247.19, −294.31, −333.68 at N = 100, 150, 200, 250, shown at their full ∼140-OOM range. Bottom panel (note the y-axis break and the change of scale): the two consecutive Aitken-∆2 extrapolations (−536.76, −533.70) and the Connes 2026 §6.4 heuristic c… view at source ↗
Figure 3
Figure 3. Figure 3: Matching-digit recovery of γ1, . . . , γ10 at c = 100 from three precision cells. N = 150 at dps = 500 (retight-tolerance baseline) gives ∼115–130 digits; N = 150 at dps = 1000 gives 219–242 digits (precision-doubling at fixed N); N = 250 at dps = 500 gives 307–329 digits (the headline cell of [PITH_FULL_IMAGE:figures/full_fig_p018_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The ratio |γ1 err|/λmin as a function of log c, showing slow monotone growth consistent with standard spectral approximation theory. The dashed line is the linear fit C(c) ≈ 6730 · log c − 11268. 22 [PITH_FULL_IMAGE:figures/full_fig_p022_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Sobolev regularity measurement at c = 23. The power-law fit through N = 40, 60, 80 yields s ≈ 46 with R2 > 0.9999. Saturation at N ≥ 100 (shaded) indicates the dps = 150 precision floor has been reached. A linear fit in log c gives s(c) ≈ 55 · log c − 128, R2 = 0.992. The empirical exponent s(c) increases from ≈ 9 at c = 13 to ≈ 75 at c = 43 — under the Babuˇska– Osborn identification, this would correspon… view at source ↗
Figure 6
Figure 6. Figure 6: Eigenvector overlap matrix |⟨ηc1 |ηc2 ⟩| across all fifteen cutoffs. All 105 pairwise overlaps are at least 0.9498 (the minimum, 0.94985, occurs at the maximally-separated pair (c = 13, c = 67); 104 of 105 pairs strictly exceed 0.950), indicating approximate eigenvector universality despite eigenvalues differing by up to 113 orders of magnitude. 25 [PITH_FULL_IMAGE:figures/full_fig_p025_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Eigenvector deviation 1 − |⟨ηc1 |ηc2 ⟩| versus cmin = min(c1, c2), grouped by prime-cutoff gap. At fixed gap, each series follows a clean power law ∼ c −α min with R2 > 0.999. The exponent ranges from α ≈ 2.7 (gap 2) to α ≈ 1.8 (gap 30). For fixed prime gap, the convergence rate follows a clean power law ( [PITH_FULL_IMAGE:figures/full_fig_p026_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Multi-zero convergence curves for γ1 through γ5 across all fifteen cutoffs (the curves for γ6 through γ10 are visually indistinguishable from these and are omitted for legibility; per-zero rate ratios for γ1 through γ5 and for γ10 are tabulated in [PITH_FULL_IMAGE:figures/full_fig_p027_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Nearest-neighbor spacing distribution of the 100 bulk eigenvalues ( [PITH_FULL_IMAGE:figures/full_fig_p030_9.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The paper presents the first public high-precision numerical implementation of the Connes-van Suijlekom truncated Weil Galerkin matrix at sixteen cutoffs (c=13 to 67 plus c=100). It reports that the absolute error |γ₁ − γ₁^Riemann| decreases monotonically from ∼2×10^{-55} to ∼1.5×10^{-168} (113 orders of magnitude) across c=13 to c=67 at fixed N=100, recovers the first ten Riemann zeros to 307–329 matching digits at c=100 and N=250, falsifies its own empirical rate fit at c=100, and obtains Aitken-Δ² extrapolations for the even-sector minimal eigenvalue that approach the Connes 2026 heuristic. The results are interpreted under unitary equivalence with CCM 2025 Lemma 5.1 (with an explicit hypothesis-status caveat at c=100), while noting that convergence as c→∞ remains open and that continuum positivity is not assumed.

Significance. If the reported computations are accurate, the work supplies the strongest numerical evidence to date that the CvS truncated Weil form at finite c approximates the Riemann zeros, with monotonic convergence spanning more than 100 orders of magnitude and recovery of multiple zeros to hundreds of digits. The explicit falsification of the finite-N empirical fit, the separation of the even-sector eigenvalue, and the independence of the matrix-eigenvalue data from any fitted parameters are notable strengths that increase the credibility of the numerical demonstration.

major comments (2)
  1. [Abstract / CCM-interpretation section] Abstract and the section discussing the CCM interpretation: the unitary equivalence with CCM 2025 Lemma 5.1 is invoked to identify the computed γ_k as eigenvalues of the rank-one operator D_log^(λ,N) at λ=√c, yet the finite-N Galerkin truncation and the hypothesis-status caveat at c=100 are not accompanied by an explicit verification or reference to the precise statement of the lemma that justifies the identification for the cutoff-dependent operator.
  2. [N-sweep / Aitken extrapolation paragraph] Section on the N-sweep at c=100: the Aitken-Δ² extrapolations yielding log₁₀|λ_∞^even| ≈ −536.76 and ≈ −533.70 are presented as approaching the Connes 2026 §6.4 heuristic (−530.38), but no error bounds or convergence diagnostics for the extrapolation itself are supplied, which is load-bearing for the claim that the finite-N trend is consistent with the continuum prediction.
minor comments (3)
  1. [Numerical results tables] The reference values for the Riemann zeros γ_k^Riemann used in the error tables should be stated with their own precision (e.g., source and number of digits) so that the reported matching digits can be independently assessed.
  2. [Implementation paragraph] The manuscript would benefit from a brief statement of the linear-algebra library and arbitrary-precision arithmetic settings (e.g., dps=500) employed for the Galerkin matrix diagonalizations, to facilitate reproducibility.
  3. [Notation throughout] Notation for the even-sector minimal eigenvalue λ_min^even and the cutoff-dependent operator should be introduced once and used consistently; occasional shifts between QW_λ and D_log^(λ,N) are mildly distracting.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive assessment and constructive comments. We address each major point below and will revise the manuscript accordingly where feasible.

read point-by-point responses

  1. Referee: [Abstract / CCM-interpretation section] Abstract and the section discussing the CCM interpretation: the unitary equivalence with CCM 2025 Lemma 5.1 is invoked to identify the computed γ_k as eigenvalues of the rank-one operator D_log^(λ,N) at λ=√c, yet the finite-N Galerkin truncation and the hypothesis-status caveat at c=100 are not accompanied by an explicit verification or reference to the precise statement of the lemma that justifies the identification for the cutoff-dependent operator.

    Authors: We agree that an explicit reference to the precise statement of CCM 2025 Lemma 5.1 is warranted to support the identification for the cutoff-dependent, finite-N operator. In the revision we will add a direct citation to the lemma together with a short paragraph clarifying how the unitary equivalence extends to the Galerkin truncation at finite c, while preserving the existing hypothesis-status caveat at c=100. This addition will not change any numerical claims. revision: yes

  2. Referee: [N-sweep / Aitken extrapolation paragraph] Section on the N-sweep at c=100: the Aitken-Δ² extrapolations yielding log₁₀|λ_∞^even| ≈ −536.76 and ≈ −533.70 are presented as approaching the Connes 2026 §6.4 heuristic (−530.38), but no error bounds or convergence diagnostics for the extrapolation itself are supplied, which is load-bearing for the claim that the finite-N trend is consistent with the continuum prediction.

    Authors: The referee is correct that formal error bounds or convergence diagnostics for the Aitken-Δ² procedure are not supplied. Because the underlying sequence is obtained from a non-standard operator truncation whose asymptotic behavior is not yet theoretically controlled, deriving rigorous a-priori error estimates lies outside the scope of the present numerical work. We will add a brief discussion of this limitation and emphasize that the reported values are heuristic indicators supported only by the observed monotonic approach; we do not claim quantitative accuracy for the extrapolated figures. revision: partial

Circularity Check

0 steps flagged

Numerical results are direct matrix computations independent of target values

full rationale

The paper implements the CvS truncated Weil Galerkin matrix at finite cutoffs c and fixed N, extracts eigenvalues and eigenvectors directly from the resulting matrix, and measures absolute deviation of the computed first zero from the independently known first Riemann zero. No parameters are fitted to the Riemann zeros; the reported monotonic error reduction (113 OOM) is an output of the eigenvalue solver. The empirical power-law fit on c≤67 is explicitly falsified at c=100, convergence as c→∞ is stated as open, and a hypothesis-status caveat is attached to the CCM interpretation. The central numerical demonstration therefore stands on its own matrix construction and does not reduce to any self-definition, fitted input, or load-bearing self-citation.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

The computations rest on the definition of the truncated Weil form, the choice of cutoff c and matrix dimension N as discretization parameters, and the unitary equivalence lemma from prior work; no new entities are postulated.

free parameters (2)
  • cutoff parameter c
    Determines which primes p ≤ c enter the operator; chosen as successive integers up to 100.
  • Galerkin matrix size N
    Finite dimension of the matrix approximation; values 100 and 250 are used.
axioms (1)
  • domain assumption Unitary equivalence with CCM 2025 Lemma 5.1
    Invoked to equate the computed zeros with eigenvalues of the CCM rank-one operator at λ = √c.

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Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages

  1. [1]

    A. C. Aitken. On Bernoulli’s numerical solution of algebraic equations.Proceedings of the Royal Society of Edinburgh, 46:289–305, 1926

    work page 1926

  2. [2]

    Ivo Babuˇ ska and John E. Osborn. Eigenvalue problems. In P. G. Ciarlet and J. L. Lions, editors, Handbook of Numerical Analysis, volume II, pages 641–787. North-Holland, Amsterdam, 1991

    work page 1991

  3. [3]

    Trace formula in noncommutative geometry and the zeros of the Riemann zeta function.Selecta Mathematica, New Series, 5:29–106, 1999

    Alain Connes. Trace formula in noncommutative geometry and the zeros of the Riemann zeta function.Selecta Mathematica, New Series, 5:29–106, 1999

    work page 1999

  4. [4]

    The Riemann Hypothesis: Past, present and a letter through time.arXiv preprint arXiv:2602.04022 [math.NT], 2026

    Alain Connes. The Riemann Hypothesis: Past, present and a letter through time.arXiv preprint arXiv:2602.04022 [math.NT], 2026

    work page arXiv 2026

  5. [5]

    Weil positivity and trace formula, the archimedean place

    Alain Connes and Caterina Consani. Weil positivity and trace formula, the archimedean place. arXiv preprint arXiv:2006.13771 [math.NT], 2020

    work page arXiv 2006

  6. [6]

    Spectral triples andζ-cycles.L’Enseignement Math´ ematique, 69:93–148, 2023

    Alain Connes and Caterina Consani. Spectral triples andζ-cycles.L’Enseignement Math´ ematique, 69:93–148, 2023. arXiv:2106.01715 [math.NT]

    work page arXiv 2023

  7. [7]

    Zeta zeros and prolate wave operators

    Alain Connes, Caterina Consani, and Henri Moscovici. Zeta zeros and prolate wave operators. arXiv preprint arXiv:2310.18423 [math.NT], 2023

    work page arXiv 2023

  8. [8]

    Zeta spectral triples.arXiv preprint arXiv:2511.22755 [math.NT], 2025

    Alain Connes, Caterina Consani, and Henri Moscovici. Zeta spectral triples.arXiv preprint arXiv:2511.22755 [math.NT], 2025

    work page arXiv 2025

  9. [9]

    van Suijlekom

    Alain Connes and Walter D. van Suijlekom. Quadratic forms, real zeros and echoes of the spectral action.arXiv preprint arXiv:2511.23257 [math.OA], 2025

    work page arXiv 2025

  10. [10]

    E. B. Davies and M. Plum. Spectral pollution.IMA Journal of Numerical Analysis, 24(3):417– 438, 2004

    work page 2004

  11. [11]

    Spectral pollution and second-order relative spectra for self-adjoint operators.IMA Journal of Numerical Analysis, 24(3):393–416, 2004

    Michael Levitin and Eugene Shargorodsky. Spectral pollution and second-order relative spectra for self-adjoint operators.IMA Journal of Numerical Analysis, 24(3):393–416, 2004

    work page 2004

  12. [12]

    Montgomery

    Hugh L. Montgomery. The pair correlation of zeros of the zeta function. In Harold G. Diamond, editor,Analytic Number Theory, volume 24 ofProceedings of Symposia in Pure Mathematics, pages 181–193. American Mathematical Society, 1973

    work page 1973

  13. [13]

    Parlett.The Symmetric Eigenvalue Problem, volume 20 ofClassics in Applied Mathematics

    Beresford N. Parlett.The Symmetric Eigenvalue Problem, volume 20 ofClassics in Applied Mathematics. SIAM, Philadelphia, 1998

    work page 1998

  14. [14]

    validates,

    Dominik ´Sliwi´ nski. Spectral analysis of thed(λ,N) log operators.arXiv preprint arXiv:2601.12133 [math.SP], 2026. 39 A Summary of Numerical Results For reference, we collect the final precision-resolved measurements in a single table. All rows use T= 800,N= 100, dps = 150 (c≤37) or dps = 200 (c≥41). Table 20: Complete 15-point results atN= 100,T= 800, d...

    work page arXiv 2026