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REVIEW 1 major objections 2 minor 14 references

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

T0 review · 1 major / 2 minor · reviewed 2026-06-30 · grok-4.3

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Pith's one-line read The truncated Weil quadratic form at rising cutoffs matches the first Riemann zero to 168 decimal places.

desk verdict The paper gives the first public high-precision runs of the CvS Galerkin matrices up to c=100, with reported error drops of over 100 orders of magnitude on the first zero and eigenvector matches to 300+ digits, but the link to the CCM operator rests on applying a continuous lemma to finite-N matrices without a direct check. read the letter →

arxiv 2605.20224 v2 pith:N6BX4BWO submitted 2026-05-13 math.NT

Akiva Groskin This is my paper · ORCID
classification math.NT
keywords RiemannzerostruncatedWeilquadraticformGalerkinmatrixcriticallineeigenvaluespectrumprimecutoffnumericalapproximationevensector
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper constructs the Galerkin matrix for the truncated Weil quadratic form at cutoffs c from 13 to 67 and at c=100. It shows that the first positive zero extracted from the ground-state eigenvector converges to the first Riemann zero, with the absolute error dropping monotonically by 113 orders of magnitude as c increases. At c=100 the same eigenvector recovers the first ten Riemann zeros to 307-329 matching digits while the smallest even-sector eigenvalue reaches 10 to the minus 334. The results rely on an assumed unitary equivalence that lets the discrete spectrum be read as that of the underlying rank-one operator.

What carries the argument

The Galerkin matrix of the truncated Weil quadratic form indexed by cutoff c, whose ground-state eigenvector encodes the Fourier-Mellin zeros that approximate the Riemann zeros.

What would settle it

A run at cutoff 200 with basis size 300 in which the first-zero error stops decreasing or begins to increase would contradict the reported monotonic improvement.

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Extended reading notes

Core claim

The central claim is that the Fourier-Mellin zeros of the ground state of the truncated Weil quadratic form, obtained from its finite Galerkin matrix at cutoff c and basis size N, lie on the critical line and approach the Riemann zeros as c grows. Explicit computation yields a first-zero error of roughly 1.5 times 10 to the minus 168 at c=67 with N=100, and recovers the first ten zeros to hundreds of digits at c=100 with N=250. Aitken extrapolation of the smallest even-sector eigenvalue at the largest cutoff approaches the continuum heuristic value, while finite-cutoff artifacts in the spectrum are shown to vanish when the archimedean cutoff is enlarged.

Load-bearing premise

The unitary equivalence between the finite discretized matrix and the continuous truncated operator continues to hold at the cutoffs and basis sizes used.

Editorial extensions

If this is right

  • Raising the cutoff from 13 to 67 reduces the first-zero error by more than 100 orders of magnitude at fixed basis size.
  • The eigenvector at cutoff 100 recovers the first ten Riemann zeros to at least 307 matching digits.
  • Aitken extrapolation on the basis-size sweep at cutoff 100 predicts a limiting eigenvalue close to the continuum heuristic.
  • Negative eigenvalues present at finite archimedean cutoff disappear when that cutoff is enlarged, leaving the smallest positive eigenvalue as the genuine ground state.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the trend persists, the same discretization could compute additional Riemann zeros beyond currently tabulated lists by scaling c and N together.
  • The power-law fit observed up to c=67 is already falsified at c=100, suggesting the asymptotic rate with cutoff may differ from the moderate-cutoff regime.
  • Extending the archimedean cutoff further at c=100 would test whether positivity of the spectrum is preserved in the continuum limit.
  • The same matrix construction might be applied to related operators whose spectra are conjectured to encode other L-function zeros.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

1 major / 2 minor

Summary. The paper implements the Galerkin discretization of the Connes-van Suijlekom truncated Weil quadratic form at cutoffs c=13--67 (N=100) and c=100 (N=250), reporting that the smallest positive even-sector eigenvalue decreases monotonically from ~2×10^{-55} to ~1.5×10^{-168} for the first zero (113 orders of magnitude) and reaches ~10^{-334} at c=100, N=250, with the associated eigenvector recovering the first ten Riemann zeros γ_1 to γ_10 to 307--329 matching digits. These eigenvalues are interpreted, under unitary equivalence from CCM 2025 Lemma 5.1, as those of the CCM rank-one operator D_log^{(λ,N)}, with an Aitken-Δ² extrapolation at c=100 yielding log_{10}|λ_∞^{even}| ≈ -536.76 and -533.70 (approaching the Connes 2026 heuristic of ≈-530.38) and an empirical power-law fit on c≤67 that is falsified at c=100. Negative eigenvalues at c=100 are attributed to the finite archimedean cutoff T=800 and vanish upon increasing T; no proof is claimed.

Significance. If the numerical spectra and eigenvector recoveries are free of discretization artifacts, the results supply the first high-precision public evidence that the CvS truncated zeros approach the Riemann zeros, spanning 275 orders of magnitude in eigenvalue magnitude and recovering hundreds of digits. This directly tests the open convergence question in Connes 2026 and CCM 2025, supplies a falsifiable rate for finite-N behavior, and demonstrates that the method can produce machine-precision matches to known zeros without assuming the Riemann hypothesis.

major comments (1)
  1. The interpretation that the computed smallest-positive eigenvalues are those of the CCM rank-one operator (and hence genuine approximations to Riemann zeros) rests on the unitary equivalence of CCM 2025 Lemma 5.1. The lemma is formulated in the continuous setting; the manuscript applies it directly to the finite-N Galerkin matrices at N=100 and N=250 without an independent verification that the discretized spectrum coincides with the continuous spectrum to the reported precisions (hundreds of digits). The single caveat noted (negative eigenvalues at c=100) addresses only the archimedean cutoff T=800, not the N-discretization error. If the equivalence fails at finite N, the observed small positive eigenvalues could be discretization artifacts rather than approximations to the CCM operator spectrum. This is load-bearing for the central claim that the numerics approximate Riemann zeros via
minor comments (2)
  1. The manuscript should state explicitly how the even-sector restriction is imposed on the Galerkin matrix and how the archimedean cutoff T=800 is chosen and increased to remove the negative eigenvalues.
  2. Clarify whether the reported matching digits between eigenvector components and known Riemann zeros are obtained by direct comparison or via an auxiliary extraction step; include a brief description of the extraction procedure.

Simulated Author's Rebuttal

1 responses · 1 unresolved

We thank the referee for the careful reading and for noting the potential significance of the results. We respond below to the major comment on the application of the unitary equivalence.

read point-by-point responses

  1. Referee: The interpretation that the computed smallest-positive eigenvalues are those of the CCM rank-one operator (and hence genuine approximations to Riemann zeros) rests on the unitary equivalence of CCM 2025 Lemma 5.1. The lemma is formulated in the continuous setting; the manuscript applies it directly to the finite-N Galerkin matrices at N=100 and N=250 without an independent verification that the discretized spectrum coincides with the continuous spectrum to the reported precisions (hundreds of digits). The single caveat noted (negative eigenvalues at c=100) addresses only the archimedean cutoff T=800, not the N-discretization error. If the equivalence fails at finite N, the observed small positive eigenvalues could be discretization artifacts rather than approximations to the CCM operator spectrum. This is load-bearing for the central claim that the numerics approximate Riemann zeros via

    Authors: We agree that Lemma 5.1 is formulated in the continuous setting and that the manuscript does not contain an independent, rigorous verification that the finite-N Galerkin spectra coincide with the continuous spectra to hundreds of digits. Such a verification would require a complete a priori error analysis of the Galerkin method for this operator, which is not currently available. The existing caveat addresses only the archimedean cutoff. We will revise the manuscript to state explicitly that the lemma is applied heuristically to the discretized matrices and that the primary numerical claims (monotonic eigenvalue decrease, 113–275 orders of magnitude improvement, and 307–329 digit recovery of known zeros) stand on their own as empirical evidence. We will add a short discussion of why the observed stability under N-increase from 100 to 250 makes discretization artifacts unlikely for the reported small positive eigenvalues. revision: partial

standing simulated objections not resolved
  • Rigorous independent verification that the discretized spectrum coincides with the continuous spectrum to hundreds of digits

Circularity Check

0 steps flagged · score 0.0 of 10

Numerical results generated independently from direct matrix computations; no load-bearing reduction to self-citation or fitted inputs.

full rationale

The paper's core claims consist of explicit numerical outputs from implementing and diagonalizing the CvS Galerkin matrices at finite c and N (e.g., error shrinkage from 2e-55 to 1.5e-168 across c=13-67 at N=100, and eigenvector recovery to 307+ digits at c=100,N=250). These are obtained by direct computation and compared to tabulated Riemann zeros; the CCM 2025 Lemma 5.1 is invoked only post hoc for interpretive labeling of the eigenvalues as those of D_log^(λ,N), with an explicit caveat noted at c=100. No equation or claim reduces a reported quantity to a fitted parameter or prior self-result by construction. The empirical fit on c≤67 is explicitly labeled finite-N and shown falsified at higher values, so it is not presented as a derived prediction. The derivation chain is therefore self-contained against external benchmarks.

Assumptions & free parameters 3 free parameters · 2 assumptions · 0 invented entities

The paper relies on standard assumptions from the Connes framework and the cited lemma; no new entities are introduced. The cutoffs are explicit parameters varied in the study.

free parameters (3)
  • cutoff c
    Parameter controlling the inclusion of primes p ≤ c in the operator
  • matrix dimension N
    Size of the Galerkin discretization matrix
  • archimedean cutoff T = 800
    Finite cutoff used in the spectrum computation, noted as causing artifacts if too small
assumptions (2)
  • domain assumption Unitary equivalence with CCM 2025 Lemma 5.1
    Invoked to link the computed eigenvalues to the CCM operator
  • domain assumption The Galerkin method accurately discretizes the continuous operator for the chosen parameters
    Underlying assumption for the numerical results to approximate the true spectrum

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Cite this review

Pith. "Pith review of High-Precision Approximation of Riemann Zeros via the Truncated Weil Form." pith.science (2026). https://pith.science/paper/N6BX4BWO

@misc{pith2026260520224,
  author       = {Pith},
  title        = {Pith review of: High-Precision Approximation of Riemann Zeros via the Truncated Weil Form},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/N6BX4BWO}},
  note         = {Machine review of arXiv:2605.20224}
}
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 $|\gamma_1-\gamma_1^{\mathrm{Riemann}}|$ 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 $\gamma_1,\ldots,\gamma_{10}$ to 307-329 matching digits. 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 $D_{\log}^{(\lambda,N)}$. Aitken-$\Delta^2$ on the $c=100$ $N$-sweep gives $\log_{10}|\lambda_\infty^{\mathrm{even}}|\approx -536.76$ and $-533.70$, approaching the Connes 2026 Section 6.4 heuristic continuum prediction ($\approx -530.38$) monotonically in $N$. The empirical fit $|\log_{10}\lambda_{\min}|\approx 13.24\,c^{0.634}$ 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 $QW_\lambda$ is RH-equivalent and is not assumed at $\lambda=\sqrt{100}$). We make no claim of proof.

Figures

Figures reproduced from arXiv: 2605.20224 by the authors.

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-check (Sectio… view at source ↗
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 continuum p… view at source ↗
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 ↗
Figures from the paper (6 more)
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]
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. …
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 eigenvecto…
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 conver…
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:fig…
Figure 9
Figure 9. Figure 9: Nearest-neighbor spacing distribution of the 100 bulk eigenvalues ( [PITH_FULL_IMAGE:figures/full_fig_p030_9.png]

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