Constraining the radial decay timescale of solar surface magnetic field through a comparative study of data-assimilative 2D surface flux transport and 3D dynamo models

Gopal Hazra; Soumyadeep Chatterjee

REVIEW 4 major objections 6 minor 296 references

Constraining the radial decay timescale of solar surface magnetic field through a comparative study of data-assimilative 2D surface flux transport and 3D dynamo models

T0 review · 4 major / 6 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read The radial decay timescale of the Sun's surface magnetic field is not a single fitted number but a mode-dependent spectrum, and the paper derives it self-consistently from the 3D induction equation, finding effective values of about 2 years

desk verdict A clean eigenvalue extension to nonuniform diffusivity, but the headline τ constraint is a fit to the authors' own STABLE model, not an independent validation. read the letter →

arxiv 2607.16939 v1 pith:GZN73ZZZ submitted 2026-07-18 astro-ph.SR

Soumyadeep Chatterjee , Gopal Hazra This is my paper · ORCID

classification astro-ph.SR

keywords solarcyclesurfacefluxtransportpolarmagneticfieldradialdecaytimescaleturbulentdiffusivityeigenvalueproblemdataassimilationBabcock-Leightondynamo

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 claims that the poorly constrained radial decay term in 2D surface flux transport models, usually written as -B_r/tau, can be derived rather than fitted. Starting from the 3D induction equation and assuming diffusive coupling between the solar surface and the convection zone, the poloidal field evolution reduces to an eigenvalue problem whose solutions yield a spectrum of decay timescales that decrease with angular mode degree l. With a depth-dependent turbulent diffusivity profile, the dipole mode decays in about 7 years and the l=8 mode in about 2 years. The authors validate this spectrum in two regimes: a magnetogram-assimilated 2D model matches the 3D STABLE dynamo with tau about 2 years, while an active-region-driven 2D model matches observed polar fields with tau about 7 years and removes the secular drift problem. If correct, this gives a physical grounding for long-term polar field reconstructions used in solar cycle prediction.

What carries the argument

The central object is the decay-mode eigenvalue problem for the poloidal potential C(r,theta,phi,t) in a spherical shell: R''_nl + (lambda_nl/eta(r) - l(l+1)/r^2) R_nl = 0, with boundary conditions R'_nl(R_sun)=0 and R_nl(R_b)=0. It converts a given radial diffusivity profile eta(r) into a spectrum of decay timescales tau_nl = 1/lambda_nl. The depth-dependent two-step diffusivity profile, Profile-2, ties the surface horizontal diffusivity used in 2D SFT to the interior radial diffusion, supplying both the timescales and the effective angular modes that matter for polar field buildup.

What would settle it

A direct test would compare the predicted mode-dependent decay spectrum against the 3D STABLE model with meridional circulation artificially suppressed: if the diffusion-only eigenvalues then reproduce the 2D-3D polar field agreement, the assumption holds; alternatively, if measured surface-flux removal poleward of 75 degrees is dominated by subduction rather than diffusion, the effective tau values needed to match the 3D model would be shorter than the eigenvalue spectrum predicts.

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Extended reading notes

Core claim

The central claim is that the radial decay timescale tau is not a free parameter but is determined by the radial turbulent diffusivity profile of the convection zone. Treating the surface-interior coupling as purely diffusive, separation of variables in the poloidal field equation leads to an eigenvalue problem whose eigenvalues give tau_nl for each radial and angular mode. For a two-step diffusivity profile, the dipole (l=1) mode decays on roughly 7 years while the l=8 mode decays on about 2 years, and these two timescales reproduce the polar field evolution of two distinct 2D SFT setups: daily magnetogram assimilation and discrete active-region driving. The paper states this is the first s

Load-bearing premise

The derivation assumes that radial surface-interior coupling is pure turbulent diffusion, with no significant advective transport; if the meridional flow subduction poleward of about 75 degrees removes a substantial share of surface flux, the diffusion-only tau spectrum is not the true decay timescale.

Editorial extensions

If this is right

2D SFT simulations with daily magnetogram assimilation should use an effective tau near 2 years, corresponding to the l=8 mode, when using a Profile-2-type diffusivity; using the slower dipole timescale leads to excess polar flux accumulation.
Traditional active-region-driven SFT models should use tau near 7 years, the dipole decay timescale, which corrects the secular drift and delayed polar reversals seen in weak cycles.
The eigenvalue framework can supply tau for any prescribed eta(r), allowing future SFT models to adopt diffusivity-specific decay timescales instead of tuning tau as a free parameter.
A self-consistent tau strengthens confidence in century-long polar field reconstructions and in cycle-amplitude predictions that rely on SFT-generated polar fields.
Since modes above l about 8 decay before reaching the polar caps, data-assimilated SFT parameterizations may need to include only the low-order modes in the decay operator.

Reading between the lines

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

The two validated tau values (2 yr and 7 yr) suggest that the historical scatter in fitted tau values, from about 4.5 to 32 years, may reflect different source representations exciting different angular spectra; single-number tau estimates are not comparable unless the dominant l is specified.
A testable extension is to run the eigenvalue derivation with advective radial transport included, especially the meridional subduction poleward of about 75 degrees, to see whether the diffusion-only tau values shift; the reported agreement may partially absorb subduction into an effective tau.
The l-dependent spectrum could be implemented as a multi-mode decay operator in 2D SFT instead of a single tau, potentially reproducing the finer polar-region structures that a single decay timescale misses.
If the diffusion-only assumption is violated by significant advective flux removal, the framework still yields useful effective parameters, but its self-consistent interpretation would need revision.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Here's my take after reading it through.

The genuinely new and solid piece is the eigenvalue calculation. Extending Baumann et al. (2006) to a depth-dependent diffusivity profile is a clean, useful step. Table 1 reproduces the uniform-profile values, so the method checks out, and the Profile-2 spectrum (dipole ~7 yr, l=8 ~2 yr) is a real contribution that SFT modelers will want to cite.

The active-region SFT experiment in Sec. 4 is also worth having. Getting τ=7 yr to remove the secular drift and match WSO over Cycles 23-25 is a decent empirical confirmation, even if the value itself is not surprising given earlier optimization work.

The soft spot is the central claim in the abstract and conclusions that this 'establishes a self-consistent bridge' between 2D and 3D. It doesn't, at least not yet. Eq. (4) drops the meridional flow entirely, while the paper itself says subduction poleward of ~75° plays a significant role alongside diffusion. So Table 1 is a diffusion-only decay spectrum, not the full radial-loss spectrum of the 3D model. In Sec. 3, τ=2 yr is found by running many 2D SFT simulations until the unsigned flux matches the authors' own data-assimilated STABLE model, and then it is read off as l=8 from Table 1. That's a fitted number assigned a modal label, not an independent test of the spectrum. Both models assimilate the same HMI magnetograms equatorward of 75°, so the agreement is a check of internal consistency between two models, not an observational constraint on τ. There are no error bars or sensitivity tests on the fit, and the Profile-2 diffusivity is chosen partially so that diffusion dominates over advection, which makes the later claim that diffusion is dominant somewhat circular.

None of this is fatal. The eigenvalue extension is legitimate, and the Sec. 4 result is a useful empirical data point. But the paper overstates what has been established. The authors should either show that the advective contribution to radial loss is small outside the polar cap, or reframe τ as an effective timescale that absorbs both diffusion and subduction rather than the pure diffusion eigenvalue. A sensitivity study around the diffusivity profile would help too.

This paper is worth a serious referee. It's clearly written, the math is correct as far as it goes, and the eigenvalue spectrum is of practical use. My recommendation is to send it out, with a request for revision focused on the overclaims and the missing sensitivity analysis.

Referee Report

4 major / 6 minor

Summary. The paper addresses the radial decay timescale tau in 2D surface flux transport (SFT) models. It derives a decay-mode spectrum by isolating the diffusive radial-transport part of the 3D induction equation (Eq. 4), solving the resulting poloidal-field eigenvalue problem (Eq. 9) for two radial diffusivity profiles, and reproducing the Baumann et al. (2006) values for the uniform Profile-1 while obtaining tau_01 ~7 yr for the nonuniform Profile-2. It then reports two applications: a data-assimilated 2D SFT (HipFT) compared with a data-assimilated 3D STABLE model, where tau = 2 yr matches the 3D unsigned flux and polar fields and is associated with l = 8 in Table 1; and a traditional active-region-driven SFT simulation for Cycles 23–25, where tau = 7 yr reproduces WSO polar fields and removes the secular drift. The paper claims a first self-consistent bridge between the 2D surface description and 3D interior dynamics.

Significance. The eigenvalue extension in Section 2 is a clean, reproducible formulation: it reproduces the known Baumann et al. (2006) results for the uniform profile and extends them to a more physical depth-dependent diffusivity. The paper uses open-source code (HipFT, MagMAP), specifies flow and diffusivity parameters, and reports numerical values in Table 1. If the central interpretation were fully supported, this would be a useful step toward fixing tau in SFT models rather than treating it as a free parameter. However, the validation logic is significantly weaker than the abstract claims: the key tau values come from fits to the authors' own 3D model under a diffusion-only reduction, and the independent checks do not specifically isolate the eigenvalues in Table 1. The contribution is therefore conditional on additional analysis.

major comments (4)

[§2, Eq. (4)] The eigenvalue spectrum in Table 1 is derived by reducing Eq. (2) to pure diffusion and deleting the advective term ∇×(V×B). The paper itself acknowledges (§1 and §5) that poleward of ~75° the meridional return flow subducts surface flux and 'plays a significant role alongside radial diffusion.' Since −B_r/τ in Eq. (1) is the only radial-loss channel in the 2D model, it must represent both diffusive and advective removal if it is to mimic STABLE. Thus τ_nl in Table 1 are not established as the effective radial-loss timescales of the 3D system; the fitted τ=2 yr may be absorbing meridional subduction rather than representing the l=8 diffusion mode. A quantitative flux-budget comparison (diffusive vs. advective removal in the polar cap) or a reduced eigenvalue problem retaining the advective term is needed before the 'self-consistent bridge' claim is supportable.
[§3, Figure 2] The value τ=2 yr is not predicted by the spectrum; the paper states 'We perform many 2D SFT simulations with different τ to find an optimized τ,' and τ_relax=6 hr is likewise calibrated by matching unsigned flux. Reading l=8 from Table 1 after this fit is a reverse lookup, not an independent validation of the diffusion spectrum. No sensitivity of the inferred τ to τ_relax, grid resolution, or flow parameters is reported, so the agreement in Fig. 2 is not shown to be discriminative. The §5 statement that 'modes up to l=8 contribute' is also not directly tested by a mode-resolved or mode-truncation experiment; a single effective τ collapses the spectrum. Please add quantitative comparison metrics and a sensitivity scan over the fitted parameters.
[§4, Eqs. (17)–(18)] This test uses different transport parameters from those used in the eigenvalue calculation and from the STABLE comparison: v0=11 m/s (Eq. 18) versus 19 m/s in Eq. (11), the Snodgrass–Ulrich differential rotation profile, and a different SFT code. Consequently, the WSO agreement at τ=7 yr cannot be attributed specifically to the Profile-2 dipole eigenvalue τ_01=7.01 yr. Many previously published τ values in the 4.5–10 yr range can also remove secular drift in SFT simulations. Unless the test is shown to be strongly discriminating in τ (e.g., by evaluating the fit quality across a range of τ values), it is a consistency check, not a validation of the dipole row of Table 1.
[§2, Eq. (10) and Table 1] Profile-2 contains six freely chosen parameters (η_c, η_mid, η_H, r_da, r_db, d_a, d_b), and the headline results τ_01≈7 yr and τ_08≈2 yr are profile-dependent. No sensitivity analysis is provided, even though η_mid directly affects both the eigenvalues and the assumption that diffusion dominates advection. The authors should show how τ_nl vary over a plausible range of η_mid and η_H, because this determines whether the specific two-digit values in Table 1 are robust or artifacts of a single ad-hoc profile choice.

minor comments (6)

[Abstract and text] The phrase 'angular model=8' should read 'angular mode l=8'; the same typo ('modelandn') occurs in §2 and §3.
[References] Several references are duplicated: Hazra et al. (2017) appears twice, Jiang et al. (2023a/b) appears twice, Schrijver & De Rosa (2003a/b) is the same paper listed twice, and Yeates et al. (2023a/b) is likewise duplicated. Please consolidate.
[§4 and Fig. 6] The text describes the polar field as 'poleward of ±65°', but the Fig. 6 caption states 'poleward of ±55°'. Please reconcile the threshold.
[Table 1] The eigenvalues are quoted to two decimals, but no numerical tolerance, grid-convergence check, or error estimate is provided for the shooting/Brent solution of Eq. (9).
[Figs. 3 and 5] The butterfly diagrams do not include a color-scale bar or explicit color limits, making quantitative comparison of panel intensities impossible.
[§3, Figure 2] The agreement between the blue and red curves is described as 'close'/'well reproduced' visually; a quantitative metric such as RMS difference or correlation in the polar field or unsigned flux would strengthen the claim.

Circularity Check

3 steps flagged · score 6.0 of 10

The τ≈2 yr / l=8 constraint is a post-hoc fit to the authors' own 3D STABLE model, labeled via a diffusion-only eigenvalue table; the τ≈7 dipole mode retains independent WSO support.

fitted input called prediction [Section 3, Figure 2, Table 1]
"We perform many 2D SFT simulations with differentτ to find an optimizedτfor which the 2D SFT and 3D stable results would be comparable. We present here results of 2D SFT simulations withτ=∞andτ= 2 yr for Cycle 24 and the rising phase of Cycle 25 (2011-2025), and compare the results with the 3D STABLE simulation. ... Referring to Table 1,τ= 2 yr corresponds to the decay timescale ofl= 8 mode"

τ=2 yr is obtained by tuning the 2D SFT decay term until its unsigned flux matches the 3D STABLE model; it is a fitted parameter, not an output of the eigenvalue analysis. The eigenvalue problem only provides a table that maps a chosen τ to a spherical-harmonic degree l. Calling the fitted value 'l=8 from our self-consistent estimate' is therefore a post-hoc labeling of a fit, not a prediction of the mode from first principles.
self citation load bearing [Section 3, Abstract, Section 2]
"The unsigned flux in the data-assimilated 2D SFT model is matched to that of the 3D data-assimilated STABLE model within±75◦ latitude with a suitable data-assimilation interval for a consistent comparison of the resulting polar field between the two models."

The benchmark used to calibrate τ=2 yr is the authors' own data-assimilated 3D STABLE model from Chatterjee & Hazra (2026), a companion paper with overlapping authorship. The central quantitative claim of the present paper therefore rests on matching a self-cited model rather than on an independent observational or externally reproduced target. This makes the companion model load-bearing for the main constraint.

1 more flagged steps

other [Section 2, Eq. (4); Section 5]
"if we assume that the coupling between the surface and interior convection zone is dominated by diffusion, then Equation 2 reduces to∂B/∂t =−∇×η(r)∇×B ... We validated the derived spectrum,τ nl through two complementary numerical experiments, since the subduction of flux by the meridional flow near the pole (poleward∼75◦) also plays a significant role alongside radial diffusion in transporting flux in the convection zone."

The eigenvalue spectrum that supplies τ≈2 yr / l=8 is derived from a diffusion-only operator with advection removed. Yet the validation benchmark, the 3D STABLE model, includes meridional-flow subduction poleward of ~75°, which the paper itself acknowledges is significant alongside diffusion. The fitted τ in the 2D model therefore absorbs both diffusive and advective removal, so its agreement with STABLE does not specifically validate the diffusion-only eigenvalue spectrum; the correspondence between τ=2 yr and l=8 is not uniquely established.

full rationale

The eigenvalue calculation itself (Eqs. 6–9) is mathematically self-contained: for a specified η(r), it yields decay timescales τ_nl, and the uniform-profile results match Baumann et al. (2006). The τ≈7 yr dipole-mode result is also partly independently supported by WSO polar-field observations in Section 4. However, the paper's headline constraint τ≈2 yr / l=8 is not the output of an independent prediction. Section 3 explicitly describes scanning many τ values and matching the unsigned flux to the authors' own data-assimilated 3D STABLE model (Chatterjee & Hazra 2026); Table 1 is then consulted to attach the label l=8 to the fitted value. The benchmark being a self-cited companion model, and the eigenvalue spectrum being computed under a diffusion-only assumption while the benchmark includes advective subduction, means the 'self-consistent bridge' claim is partially circular: the fitted τ is renamed as a modal prediction. Overall score 6 reflects this partial circularity, with the τ≈7 dipole test providing genuine external anchor.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The central τ spectrum rests on the chosen η(r) profile, the diffusion-dominated assumption, and boundary conditions; in the assimilation experiment the comparison target is the authors' own STABLE model, so the ground truth is partly self-referential.

free parameters (5)

Profile-2 diffusivity parameters = η_c=2e10, η_mid=5e11, η_H=3e12 cm²/s; r_da=0.735R☉, r_db=0.956R☉, d_a=0.021R☉, d_b=0.05R☉
Chosen by hand to define the 'physically motivated' two-step η(r); the Table 1 eigenvalues and all τ values depend on this choice.
τ_relax (assimilation relaxation timescale) = 6 hr
Calibrated by matching unsigned flux within ±75° to the 3D STABLE simulation; controls assimilation strength and affects polar field.
τ (optimized decay timescale in Sec. 3) = 2 yr
Found by running many 2D SFT simulations with different τ and choosing the one matching STABLE's unsigned flux/polar field; the l=8 association is read off from Table 1 after the fact.
Flow profile coefficients = m1=37.2 m/s, m5=-6 m/s, d0=130, d2=-170, d4=-400 m/s
Set by hand so HipFT matches STABLE's meridional flow and differential rotation; polar transport and flux accumulation depend on them.
Active-region extraction parameters = σ=3 grid cells, 40 G threshold
Chosen to identify discrete active regions from MDI/HMI synoptic maps; affects injected flux, dipole moment, and hence the tau=7 validation.

assumptions (6)

domain assumption Surface–interior coupling is dominated by radial diffusion, so Eq. (2) reduces to ∂B/∂t = −∇×(η(r)∇×B)
Invoked before Eq. (4); neglects advective subduction by meridional flow, which the authors later say is also significant poleward of 75°.
domain assumption The poloidal field evolution decouples and only the poloidal potential C with Br = −L²C/r² is needed
Section 2, between Eqs. (4)-(6); ignores toroidal field and assumes the decay of Br is governed by the poloidal diffusion operator.
standard math Boundary conditions R'_nl(R☉)=0 and R_nl(R_b)=0 at base 0.69R☉
Section 2, following Eq. (9); standard for diffusion eigenmodes but the surface boundary condition (vanishing radial derivative) partly determines the eigenvalues.
domain assumption The 3D STABLE model of Chatterjee & Hazra (2026) is an accurate enough representation of solar surface–interior coupling to serve as ground truth
Section 3 uses the STABLE simulation from the authors' companion paper as the reference for calibrating τ=2 yr; this is a self-cited, not independently reproduced, benchmark.
domain assumption In active-region-driven SFT, horizontal diffusion erodes small scales so the polar field is built exclusively by the l=1 dipole component
Sections 3-4, used to justify choosing τ_01≈7 yr; if higher-l modes contribute to polar field, the effective τ would be smaller.
ad hoc to paper Profile-2 η(r) with high η_mid ensures diffusion dominates advection in the bulk
Section 2, after Eq. (10); the profile is chosen, not derived, to enforce the diffusion-dominated regime that makes the eigenvalue treatment valid.

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Pith. "Pith review of Constraining the radial decay timescale of solar surface magnetic field through a comparative study of data-assimilative 2D surface flux transport and 3D dynamo models." pith.science (2026). https://pith.science/paper/GZN73ZZZ

@misc{pith2026260716939,
  author       = {Pith},
  title        = {Pith review of: Constraining the radial decay timescale of solar surface magnetic field through a comparative study of data-assimilative 2D surface flux transport and 3D dynamo models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GZN73ZZZ}},
  note         = {Machine review of arXiv:2607.16939}
}

abstract

The polar magnetic field is the most reliable precursor for predicting the amplitude of the solar cycle, and the 2D surface flux transport (SFT) model is widely used to reconstruct its evolution. Traditional 2D SFT models can not capture the surface-interior coupling of the surface field, causing delays in polar field reversals. This deficiency is conventionally corrected by adding a decay term $-B_r/\tau$ with a poorly constrained radial decay timescale $\tau$. Here, we present a self-consistent estimate of $\tau$ through a comparative study of radial flux transport in the 2D SFT model and the 3D kinematic dynamo model, STABLE. By keeping the same transport parameters for both models and assuming surface-interior coupling is diffusive, the poloidal field evolution equation reduces to an eigenvalue problem, which yields a spectrum of $\tau$ that decrease with increasing angular modes $l$. To capture realistic surface-interior coupling, we further perform data-assimilated 2D SFT simulations with real magnetograms and compare those results with that of the data-assimilated 3D STABLE model to constrain $\tau$ and effective decay modes. With our choice of transport parameters, a value of $\tau=~2~\text{yr}$ keeps the surface dynamics of the two models consistent, and this timescale corresponds to the angular mode $l=8$ from our self-consistent estimate. We also perform 2D SFT simulations with only large-scale active regions, as the source. We find that $\tau=7~\text{yr}$ accurately captures the radial decay of the dipole mode ($l=1$) and removes the secular drift in the polar fields.

Figures

Figures reproduced from arXiv: 2607.16939 by the authors.

**Figure 1.** Profiles of radial diffusivity η(r). convection zone (A. R. Yeates et al. 2008). Therefore we solve the boundary value problem defined by Equation 9 numerically for each mode l and n using a shooting method combined with the Brent root-finding algorithm (R. P. Brent 1973) to determine the eigenvalues λnl [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗

**Figure 2.** Comparison of (a) total unsigned flux withing ±75◦ latitude and (b) the polar fields (poleward ±75◦ ) between the 2D SFT simulations and data-assimilated 3D STABLE simulation from S. Chatterjee & G. Hazra (2026) for the period 2011-2025. Magenta, blue, and red lines show the variation of unsigned flux in panel (a) for the 2D SFT simulation with τ = ∞, the 2D SFT simulation with τ = 2 yr, and the 3D STABLE model, res… view at source ↗

**Figure 3.** Longitudinally averaged radial magnetic field ⟨Br⟩ϕ as a function of latitude and time - the butterfly diagram - for (a) the 2D SFT simulation with daily magnetogram assimilation and τ = 2 yr, and (b) the 3D STABLE model with daily magnetogram assimilation from S. Chatterjee & G. Hazra (2026), spanning Cycle 24 and the rising phase of Cycle 25 (2011 - 2025 yr). of surface field without any secular drift problem in p… view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: Snapshots of the radial magnetic field Br on the solar surface obtained from the 3D data-assimilated STABLE simulation (S. Chatterjee & G. Hazra 2026) at (a) 2014.6 yr, near the maximum of Cycle 24, and (b) 2020.4 yr, during the minimum between Cycles 24 and 25. cells)…

**Figure 5.** Figure 5: Magnetic butterfly diagrams (longitude-averaged ⟨Br⟩ϕ) for approximately 26 years (1999-2025.3 yr). Panel (a): observed butterfly diagram constructed from MDI (CR 1950–2104) and HMI (CR 2096–2297) synoptic magnetograms. Panel (b): 2D SFT simulation driven by discrete a…

**Figure 6.** Figure 6: Evolution of the polar fields (poleward of ±55◦ ). Gray curves: WSO observations (solid: north, dashed: south). Red curves: SFT simulation with τ = ∞. Blue curves: SFT simulation with τ = 7 yr. REFERENCES Anthony Yeates. 2016, sft data,, https://github.com/antyeates198…

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Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.

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