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A Measurement of the Largest-Scale CMB E-mode Polarization with CLASS

T0 review · 4 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read A ground-based telescope has measured the CMB's largest-scale E-mode polarization and detected cosmic reionization at 99.4% significance.

desk verdict The pixel-space transfer-matrix extension of xQML is the real contribution here, and the tau measurement is an honest but fragile first step from the ground that deserves a serious referee. read the letter →

arxiv 2501.11904 v2 pith:YBKZGMKC submitted 2025-01-21 astro-ph.CO

Yunyang Li , Joseph Eimer , John Appel , Charles Bennett , Michael Brewer , Sarah Marie Bruno , Ricardo Bustos , Carol Chan
show 29 more authors
David Chuss Joseph Cleary Sumit Dahal Rahul Datta Jullianna Denes Couto Kevin Denis Rolando Dunner Thomas Essinger-Hileman Kathleen Harrington Kyle Helson Johannes Hubmayr Jeffrey Iuliano John Karakla Tobias Marriage Nathan Miller Carolina Morales Perez Lucas Parker Matthew Petroff Rodrigo Reeves Karwan Rostem Caleigh Ryan Rui Shi Koji Shukawa Deniz Valle Duncan Watts J. Weiland Edward Wollack Zhilei Xu Lingzhen Zeng
This is my paper · ORCID
classification astro-ph.CO
keywords cosmicmicrowavebackgroundE-modepolarizationreionizationopticaldepthground-basedCMBobservationpixel-spacetransfermatrixquadraticmaximum-likelihoodestimatorCLASSlargeangularscale
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

This paper reports the first ground-based measurement of the reionization optical depth, $\tau$, from the largest-angular-scale E-mode polarization of the cosmic microwave background. Using 115 detector-years of CLASS 90 GHz observations cross-correlated with Planck maps, the authors detect the reionization signal at 99.4% significance and measure $\tau = 0.053^{+0.018}_{-0.019}$, consistent with Planck-only results. The paper's central methodological claim is that time-domain filtering in maximum-likelihood map-making can be precisely characterized and corrected with a pixel-space transfer matrix, allowing recovery of the largest angular scale polarization modes that were previously thought to be lost. If this approach is correct, ground-based experiments can participate in low-multipole reionization measurements that were previously the exclusive domain of space missions, and the path to a cosmic-variance-limited $\tau$ measurement from CLASS becomes clear.

What carries the argument

The paper's key object is the pixel-space transfer matrix $\mathbf{F}_{16,N_F}$, a linear operator constructed by reobserving and downgrading single-pixel input maps with the map-making pipeline's reobservation operator $\mathbf{R}$, which fixes the noise model to a realization-independent form. This matrix forward-models the effect of time-domain filtering on sky signals at the map level, and is paired with a modified quadratic maximum-likelihood cross-spectrum estimator (xQML) that incorporates the transfer matrix into the signal covariance, allowing unbiased power spectrum recovery down to $\ell=2$. The transfer matrix is evaluated at $N_\mathrm{side}=32$ for signal maps and $N_\mathrm{side}=16$ for noise covariance matrices, and it is also used to filter foreground templates so that cleaning is performed consistently with the filtered data.

What would settle it

An end-to-end simulation that applies the measured map-making nonlinearity correction at $\ell=4$ specifically, recomputing the likelihood, would settle whether the $\tau$ central value shifts by more than the quoted uncertainty; alternatively, an independent ground-based or balloon-borne low-$\ell$ E-mode measurement with comparable sky coverage returning $\tau$ outside $0.053 \pm 0.019$ would contradict the detection.

Watch

Extended reading notes

Core claim

The central discovery is that the large-scale E-mode signal, which encodes the reionization history and is enhanced roughly as $\tau^2$ on the largest angular scales, is recoverable from a ground-based instrument after the time-domain filtering bias is forward-modeled and corrected. The authors demonstrate this by cross-correlating the CLASS 90 GHz map with foreground-reduced Planck 100/143 GHz maps, obtaining a detection of reionization at 99.4% confidence and $\tau = 0.053^{+0.018}_{-0.019}$. They further show that the only uncorrected bias, arising from the nonlinearity of the maximum-likelihood map-maker's noise weighting, is less than 3% at $\ell=3$ and is small enough not to be applied in the spectra. The cross-spectrum is validated by null tests, consistency with Planck at intermediate angular scales ($\ell > 30$), and robustness to masks, multipole ranges, and external data sets, with the caveat that the significance weakens when multipoles below $\ell=5$ are dropped.

Load-bearing premise

The measurement assumes the map-making noise model is linear and independent of the sky signal; the residual nonlinearity, uncorrected and measured at under 3% at $\ell=3$, could be larger at $\ell=4$, the multipole that drives the $\tau$ result.

Editorial extensions

If this is right

  • Ground-based CMB polarization experiments can recover the largest-scale E-modes after a transfer-matrix filtering correction, opening low-$\ell$ reionization measurements to the ground.
  • The measured $\tau = 0.053^{+0.018}_{-0.019}$ is consistent with Planck HFI-based values, supporting the lower range of $\tau$ estimates that reduce the $\sigma_8$ tension via $A_s e^{-2\tau}$.
  • The xQML estimator with transfer-matrix correction is unbiased up to $\ell=40$ in validation tests, providing a reusable tool for other experiments facing filtering-induced signal loss.
  • Forecasts in the paper indicate that modest filtering optimization (a 20% improvement) could bring CLASS's $\tau$ precision close to the cosmic variance limit of about 0.003 for its sky patch.

Reading between the lines

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

  • Editorial extension: if the transfer-matrix correction remains valid as newer modulators (reflective half-wave plate) and a second 90 GHz telescope come online, the CLASS dataset could provide a $\tau$ measurement with precision comparable to Planck's, and possibly an independent check on the lensing-anomaly inflating $A_s$ values.
  • Editorial extension: the same pixel-space transfer matrix machinery is directly transferable to other ground-based and balloon-borne CMB experiments (e.g., those with heavier time-domain filtering), including for B-mode analyses on large scales where filtering is even more aggressive.
  • Editorial extension: the paper's finding that the $\tau$ constraint is driven mainly by the single multipole $\ell=4$ implies that closer attention to the map-making nonlinearity at exactly that scale should be a priority for future low-$\ell$ analyses; a targeted end-to-end simulation at $\ell=4$ with more realizations could sharpen the result.
  • Editorial extension: the discarded closeout-period data, which represents nearly half the survey, might be recoverable if the low-$\ell$ noise modeling improves, potentially more than doubling the effective sensitivity without new observations.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The paper presents the CLASS 90 GHz polarization analysis with the goal of recovering the largest-angular-scale CMB E-mode signal. The authors introduce a pixel-space transfer-matrix formalism to correct for time-domain filtering in maximum-likelihood map-making, implement it in a quadratic cross-spectrum estimator (xQML), and validate the pipeline with 500 end-to-end simulations, internal null tests, and cross-checks against Planck. The central scientific claim is a measurement of the reionization optical depth from a CLASS×Planck cross-correlation, tau = 0.053 (+0.018, -0.019), corresponding to a 99.4% rejection of the no-reionization hypothesis, which the paper labels as the first ground-based attempt at such a measurement. At intermediate multipoles (ell>30) the results are consistent with Planck.

Significance. If the central claim holds, this is a substantial methodological and observational milestone: it demonstrates that ground-based data, after careful filtering correction, can contribute to low-ell E-mode polarization measurements and to constraints on the reionization optical depth. The paper is unusually transparent: it provides extensive validation with 500 simulations, a full PTE table for null tests, a public implementation of the modified xQML estimator, and explicit statements of residual unmodeled biases. These strengths are real and should be credited. However, the headline detection significance and the tau value are carried by a single multipole, ell=4, and the per-multipole accuracy of the transfer-matrix correction, foreground cleaning, and noise covariance at that exact multipole is not separately demonstrated. The significance of the result is therefore conditional on the fidelity of one band power and on the accuracy of its total variance, including sample-variance contributions.

major comments (4)
  1. [§5.4, Figure 16] The headline claim of a 99.4% detection of cosmic reionization is not robust to the omission of a single multipole: the paper states that dropping ell=4 reduces the detection significance below threshold, and Figure 16 shows that no detection is claimed when multipoles below ell=5 are excluded. Since the entire reionization detection and the tau estimate rest on the ell=4 band power, the manuscript should provide per-multipole validation of the transfer-matrix-corrected xQML estimator at ell=4, including foreground-cleaning residuals and noise covariance accuracy at that multipole. The current validation (e.g., Figure 19 and Table 1) is aggregate over ell ranges and does not establish that the ell=4 band power and its variance are unbiased at the level needed for the 99.4% significance claim.
  2. [§3.4.4, Figure 7] The map-making non-linearity bias is characterized as less than 3% at ell=3 and is left uncorrected, but no estimate is given at ell=4, which is the multipole that drives the tau result. Because the likelihood at ell=4 depends both on the mean shift in C_ell and on the sample-variance term in Equation (15), a systematic error in the ell=4 band power or in its variance at the level of even a moderate fraction of the quoted error would alter both tau and the detection significance. The authors should report the non-linearity bias at ell=4 (and at each ell used in the likelihood) and, if it is not negligible, include it as a systematic uncertainty or correct it.
  3. [§4.1, Appendix B] The low-ell EE null test for the VPM sync split fails with PTE below 2e-4, driven by ell=8, and the paper attributes this to inadequate noise modeling when detector-pair cancellation is compromised. The authors argue that this failure does not affect the final cross-correlation because the ell=8 outlier is not present in the CLASS×Planck spectrum. However, the failure demonstrates that the noise model can be substantially inaccurate for some detector configurations, and the final analysis uses the same noise model. The manuscript should provide a direct test that the final (unsplit) CLASS noise covariance and the ell=4 band-power variance are unaffected, for example by comparing a null test constructed with the final noise model or by showing that the VPM sync split null spectrum is consistent with simulations after the proposed noise-model improvement.
  4. [§4.2, §5.4] The tau measurement is not an independent ground-based determination: the CMB channel is the Planck HFI coadded map, the absolute polarization calibration is fitted to Planck high-ell spectra (Section 4.2), and the likelihood in Section 5.4 imposes a Gaussian prior on A_s e^{-2tau} derived from Planck. The paper acknowledges that calibration does not compromise tau independence only to the extent that A_s e^{-2tau} is well constrained, but the abstract and conclusion should state more explicitly that the result is a cross-correlation measurement whose parameter interpretation is tied to Planck-based calibration and priors. The 99.4% significance is the significance of the cross-spectrum being nonzero, not of CLASS independently detecting reionization; this distinction should be made explicit in the title or abstract claims.
minor comments (5)
  1. [Abstract vs. §3.3] The abstract quotes a polarization sensitivity of 78 muK arcmin, while the body text (Section 3.3) reports 82 muK arcmin; these values should be reconciled.
  2. [Figure 9] The labels 'az-2ε' and 'az-4ε' appear to be typographical errors for 'az-2π' and 'az-4π' used in Figure 8 and the text.
  3. [§3.4.4] The sentence 'The maximum of this bias is found to be less than 3% at ell=3' is ambiguous: it should clarify whether 3% is the maximum over all ell and occurs at ell=3, or whether ell=3 is the only multipole at which the bias is estimated.
  4. [§4.2, Eq. (7)] The use of the square root of the auto-spectrum transfer function for cross-spectra is an approximation that the paper states is negligible; a sentence quantifying the difference or citing a validation test would improve clarity.
  5. [§5.4] The description of the likelihood step 6 should explicitly state that the uniform prior on tau is over 0 to 0.3 and that the Gaussian prior on A_s e^{-2tau} is a choice imported from Planck; these choices are central to the reported error bars and should be restated in the results paragraph.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the tau measurement is a cross-spectrum result with disclosed external calibration and prior, not a self-referential derivation.

full rationale

The derivation of tau is not circular. The central quantity is the CLASS x Planck xQML EE cross-spectrum (Section 5.3), and the filtering correction is validated against reobservation simulations independent of the data (Appendix A, Figure 19), so the transfer-matrix step does not impose the reported tau. The absolute calibration eta=1.053 is fitted to ell>30 spectra; the paper explicitly states that this 'does not compromise the experiment's ability to measure tau independently of the calibrator at large angular scales (ell<30), to the extent that A_s e^{-2tau} is considered a well-constrained parameter' (Section 4.2) - an external degeneracy-breaking assumption, not a fitted parameter renamed as a prediction. The Gaussian prior on 10^9 A_s e^{-2tau} is a standard Planck-based constraint; it does not fix tau itself, and the data's tau posterior (0.053+0.018-0.019) is obtained from the low-ell cross-spectrum. The 99.4% significance is computed by replacing CLASS data with tau=0 simulations while keeping the same pipeline, so it is a frequentist null test of the CLASS map's correlated signal, not a self-fulfilling construction. The paper's own robustness checks (dropping ell=4 removes significance; <3% uncorrected map-making nonlinearity; tau-dependent sample variance at ell=4) are limitations and statistical-fragility concerns, not cases where an output equals an input by definition. Self-citations to L23 and other CLASS papers supply methodology that is re-derived and validated here, so they are not load-bearing circular premises.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The central tau estimate does not require new physical entities; it rests on a small set of fitted calibration and foreground parameters and on the linearity assumptions of the transfer-matrix correction. The most important is the realization-independent noise model assumption, whose violation is bounded at less than 3% by simulations but is not corrected in the reported spectra.

free parameters (4)
  • Absolute polarization calibration factor eta = 1.053 +/- 0.014
    Fitted by minimizing chi2 between CLASS x SEVEM cross-spectra and Planck fiducial CMB spectra at 31 <= ell < 301 (Section 4.2). It scales all CLASS polarization amplitudes, so any error propagates linearly into the low-ell cross-spectrum and the tau estimate.
  • Synchrotron template coefficient alpha (CLASS-K baseline) = 1.24e-2 +/- 0.08e-2
    Fitted in foreground cleaning over the f070 mask (Section 5.2, Table 1). It sets how much WMAP K-band synchrotron emission is subtracted from CLASS 90 GHz maps and affects the cleaned low-ell spectrum.
  • Dust template coefficient beta (CLASS-K baseline) = 1.39e-2 +/- 0.01e-2
    Fitted in foreground cleaning using the SRoll2 353 GHz dust template (Section 5.2, Table 1). Residual dust mis-subtraction would bias the low-ell EE spectrum.
  • Time-domain filter truncation order = 12 harmonic modes each for 8pi and 2pi scan-synchronous filters
    Chosen by hand in Section 3.2 to remove scan-correlated systematics. The transfer matrix correction depends on this choice, and a different truncation would change the recovered low-ell power and tau.
assumptions (6)
  • domain assumption The maximum-likelihood map-maker can be approximated as a linear, realization-independent operator (reobservation R) with a noise model fixed to the final template iteration.
    Used to construct the pixel-space transfer matrix F in Section 3.4.2. The residual non-linearity is bounded at less than 3% by simulations (Section 3.4.4) but not corrected.
  • domain assumption The absolute polarization calibration derived from high-ell (ell>30) CMB spectra, which constrain A_s e^{-2tau}, remains valid at low ell and does not bias tau.
    Section 4.2 explicitly relies on A_s e^{-2tau} being a well-constrained parameter. This is an external input from Planck, not derived in this paper.
  • domain assumption Foreground emission at 90 GHz is well traced by linearly scaled WMAP K-band and Planck 30/353 GHz templates with no significant spatial decorrelation.
    Used for foreground cleaning in Section 5.2. The fitted coefficients and residuals are checked, but template mis-tracing is a known limitation of template cleaning.
  • standard math CMB realizations are Gaussian and isotropic, so synfast realizations with CAMB spectra and the simulation-based likelihood describe the data distribution.
    Used throughout Section 5.4 to build the tau likelihood and PTE values. Standard in CMB analysis.
  • domain assumption The fiducial Planck 2018 TTTEEE+lowE cosmology and the Gaussian prior on A_s e^{-2tau} are valid inputs.
    Section 5.4 fixes other cosmological parameters and draws tau uniformly with A_s chosen to satisfy A_s e^{-2tau}=1.884e-9; the final posterior uses a Gaussian prior from Planck. This imports external constraints into the measurement.
  • ad hoc to paper Scan-synchronous systematics are adequately described by the first 12 harmonic modes of 8pi and 2pi azimuth patterns.
    Section 3.2: the filter basis is chosen to remove velocity-induced and ground-fixed signals. The choice is motivated by observed systematics but is not derived from a first-principles model.

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Cite this review

Pith. "Pith review of A Measurement of the Largest-Scale CMB E-mode Polarization with CLASS." pith.science (2026). https://pith.science/paper/YBKZGMKC

@misc{pith2026250111904,
  author       = {Pith},
  title        = {Pith review of: A Measurement of the Largest-Scale CMB E-mode Polarization with CLASS},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YBKZGMKC}},
  note         = {Machine review of arXiv:2501.11904}
}
abstract

We present measurements of large-scale cosmic microwave background (CMB) E-mode polarization from the Cosmology Large Angular Scale Surveyor (CLASS) 90 GHz data. Using 115 det-yr of observations collected through 2024 with a variable-delay polarization modulator, we achieved a polarization sensitivity of $78\,\mathrm{\mu K\,arcmin}$, comparable to Planck at similar frequencies (100 and 143 GHz). The analysis demonstrates effective mitigation of systematic errors and addresses challenges to large-angular-scale power recovery posed by time-domain filtering in maximum-likelihood map-making. A novel implementation of the pixel-space transfer matrix is introduced, which enables efficient filtering simulations and bias correction in the power spectrum using the quadratic cross-spectrum estimator. Overall, we achieved an unbiased time-domain filtering correction to recover the largest angular scale polarization, with the only power deficit, arising from map-making non-linearity, being characterized as less than $3\%$. Through cross-correlation with Planck, we detected the cosmic reionization at $99.4\%$ significance and measured the reionization optical depth $\tau=0.053^{+0.018}_{-0.019}$, marking the first ground-based attempt at such a measurement. At intermediate angular scales ($\ell>30$), our results, both independently and in cross-correlation with Planck, remain fully consistent with Planck's measurements.

Figures

Figures reproduced from arXiv: 2501.11904 by the authors.

Figure 1
Figure 1. CLASS daily observation efficiencies (see definition in text) of the 90 GHz telescope since the initial deployment in 2018 through the recent modulator replacement in June 2024. The gray region represents the total detector uptime when the modulator (VPM) was also operational. The yellow region indicates the data volume initially selected at the DataPkg level while the green region shows the fraction retained after … view at source ↗
Figure 2
Figure 2. Galactic avoidance masks used for this analysis (first 5 panels) and the 90 GHz hits map (last panel) in celestial coordi￾nates. The first two rows show the low-resolution Galactic avoid￾ance masks at Nside = 16, with retained sky fraction ranging from 70% to 40%. The region not surveyed by CLASS but used for Planck analysis is painted in lighter blue. We use the “f0XX” no￾tation to denote the retained sky fraction … view at source ↗
Figure 3
Figure 3. CLASS 90 GHz linear polarization maps shown in Celestial coordinates under Mollweide projection. Gray regions indicate portions of the sky not surveyed by CLASS. These maps have been smoothed to 2◦ FWHM for visualization. likelihood map-making algorithm (Dünner et al. 2013; Romero et al. 2020; L23). The core of this method involves constructing noise covariance matrices for data segments that capture most of the tem… view at source ↗
Figures from the paper (17 more)
Figure 4
Figure 4. Figure 4: EE power spectrum transfer functions. The blue curve shows the beam window function (b 2 ℓ ), and the orange curve repre￾sents the effect of the anti-aliasing low-pass filtering before down￾sampling the demodulated data (˜b 2 ℓ ). The green curve is the isotropic mappi…
Figure 5
Figure 5. Figure 5: Example of the transfer matrix row elements (at native Nside = 256 before the downgrade operation) corresponding to the filtering of a single Nside = 16 pixel in the Stokes Q map. The peak value in the filtered map (among the central red pixels) is 0.94. The “cross” sh…
Figure 6
Figure 6. Figure 6: Comparison of reobservation and the transfer matrix approximation. The first column shows the Q/U maps from reobserving the Planck 100 GHz maps and downgrading them to Nside = 16. The second and third columns show the same maps but processed with the transfer matrix ev…
Figure 7
Figure 7. Figure 7: Low-ℓ bias from the maximum-likelihood map-making noise weighting. The error bars are derived from Monte Carlo sim￾ulations and represent the uncertainty on the mean. The gray shade is the theoretical calculation of the sample variance on the mean, which is larger than…
Figure 8
Figure 8. Figure 8: Definition of the scan-related data splits. The bot￾tom panel shows the typical 2degs−1 azimuth scan pattern of the CLASS telescope over two back-and-forth scanning periods (four sweeps). The four az-related splits divide every scan period into two halves and are displ…
Figure 9
Figure 9. Figure 9: Probability-to-exceed (PTE) values for the low-ℓ and mid-ℓ null tests. The PTE values are tabulated for every null test split and each of the three linear polarization spectra for the two ℓ￾ranges. The cell color indicates the PTE value—red (purple) for PTE values less…
Figure 10
Figure 10. Figure 10: E-mode power spectra from the CLASS 90 GHz and Planck data. The black data points are cross-spectra between CLASS 90 GHz and Planck PR4 SEVEM CMB maps for EE and T E cross-correlations. The T E spectrum on the right panel uses the CLASS E-mode map and the Planck tempe…
Figure 11
Figure 11. Figure 11: Low-resolution CLASS 90 GHz linear polarization maps before (top) and after (bottom) foreground reduction. Small residuals appear mostly around the Galactic center and the anticen￾ter (Tau A) where the data are not used for determining the template coefficients. These…
Figure 13
Figure 13. Figure 13: shows the per-multipole uncertainty from the noise (i.e., not including sample variance, black) and the to￾tal error budget (blue, including the sample variance from the fiducial model) [PITH_FULL_IMAGE:figures/full_fig_p015_13.png]
Figure 12
Figure 12. Figure 12: Transfer matrix-corrected xQML EE cross spectra between CLASS 90 GHz and the Planck 100/143 GHz coaddition map. The spectra are taken with a baseline f050 Galactic mask. The Planck 100×143 GHz result from P20 is shown as brown data points for comparison. The error bar…
Figure 14
Figure 14. Figure 14: Transfer matrix-corrected xQML spectra correlation matrix (2 ≤ ℓ ≤ 10) for CLASS × Planck over the f050 mask. The matrix is built from 500 simulations combining the foreground￾cleaned simulation suite with signals drawn from the fiducial ΛCDM model. The correlation am…
Figure 15
Figure 15. Figure 15: PTE values for the Null × Planck and CLASS × Planck cross spectra for each multipole ℓ and every polarization spectrum. In the former case, the null spectra are compared to the null model, and in the latter case, the comparison is made against the best-fit model with …
Figure 16
Figure 16. Figure 16: Marginalized posterior distribution of τ from CLASS × Planck spectra over f050 mask starting at ℓmin = 2 (baseline) and results from alternative analysis choices including masking, polar￾ization spectra used for likelihood, data set (using WMAP instead of Planck), and…
Figure 17
Figure 17. Figure 17: Top: τ likelihood for a single multipole, ℓ = 4, of the EE power spectrum evaluated at the measurement CˆEE ℓ=4. Gray dots are the likelihood values computed for 600 (τ ,As) parameter pairs. Blue and orange dots highlight two examples with τ = 0.05 and τ = 0.03, respe…
Figure 18
Figure 18. Figure 18: Forecast on the τ constraint precision, στ , from CLASS data alone as a function of improved map depth (decreasing map noise) assuming filtering optimizations. The black curve represents the forecast for the baseline analysis without filtering optimization; each color…
Figure 19
Figure 19. Figure 19: Validation of the filtering correction in the xQML im￾plementation. Top: EE power spectra of 500 signal-only fiducial ΛCDM reobservation simulations computed using xQML with (in black) and without (in gray) the filtering correction. The error bars reflect the uncertai…
Figure 21
Figure 21. Figure 21: The low-ℓ EE null spectra for the VPM sync split (top) and the radial split (bottom) for comparison. The shaded regions represent the 1/2/3-σ range from noise-only simulations. Nevertheless, the “VPM sync” detector split is designed to partially check for this. Given …

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.