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REVIEW 3 major objections 5 minor 24 references

Calibrated Physics-Informed Uncertainty Quantification

T0 review · 3 major / 5 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read A physics-informed conformal prediction framework converts a neural PDE surrogate's deviation from the governing equations into calibrated error bars, giving marginal and joint coverage guarantees without labelled data.

desk verdict Sound application of split conformal to PDE residuals with a genuinely useful data-free calibration scheme, but the abstract and contributions claim OOD coverage and relaxed exchangeability that the theorem does not support. read the letter →

arxiv 2502.04406 v2 pith:HVBSJF63 submitted 2025-02-06 cs.LG cs.AIphysics.comp-ph

Vignesh Gopakumar , Ander Gray , Lorenzo Zanisi , Timothy Nunn , Daniel Giles , Matt J. Kusner , Stanislas Pamela , Marc Peter Deisenroth This is my paper · ORCID
classification cs.LGcs.AIphysics.comp-ph MSC 62G1568T07
keywords conformalpredictionuncertaintyquantificationphysics-informedmachinelearningneuralPDEsurrogatesresidualsfinite-differencestencilsmarginalcoveragejoint
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 tries to establish that a neural network solving partial differential equations can be given statistically valid error bars using only the physics it is supposed to obey, with no ground-truth simulation outputs required. The proposal is to feed the model's predictions through the PDE residual operator and use the resulting residual magnitudes $|D(\hat f(X))|$ as conformal nonconformity scores, calibrating a quantile that bounds physics violation at a user-chosen coverage level. This matters because neural PDE solvers are fast but can be confidently wrong; a calibrated residual band makes it possible to flag or reject predictions that violate conservation laws and reroute them to expensive numerical solvers. The paper also shows the band can be computed efficiently by expressing finite-difference stencils as convolutional kernels, and demonstrates marginal and joint coverage on wave, fluid, magnetohydrodynamic, and fusion-relevant benchmarks.

What carries the argument

The machinery is the Physics Residual Error (PRE) used as a conformal nonconformity score, evaluated with finite-difference stencils implemented as convolutional kernels. For a differential operator $D$, the PRE is $|D(\hat f(X))|$, computed by convolving prediction tensors with stencils such as the 2D Laplacian kernel, which is memory-efficient and model-agnostic. Conformal calibration takes the $\lceil (n+1)(1-\alpha)\rceil/n$ quantile of calibration PREs, giving input-independent prediction sets; marginal CP uses cell-wise scores while joint CP uses the supremum of normalized scores times their standard deviation. Theorem A.1 proves coverage under the exchangeability of residuals.

What would settle it

Run a neural PDE surrogate and its CP-PRE band on a distribution shift deliberately chosen to break exchangeability, such as initial-condition parameters sampled outside the calibration range or a changed PDE coefficient, and count the fraction of new residuals $|D(\hat f(X_{n+1}))|$ inside the calibrated band. A repeated rate clearly below $1-\alpha$ at a fixed $\alpha$ would show the coverage claim does not survive outside the theoretical regime.

Watch

Extended reading notes

Core claim

The paper's central claim is that a neural PDE surrogate's failure to obey its governing equations can be measured, calibrated, and turned into a statistically valid error bar without ever seeing ground-truth solutions. Because an exact solution of a well-posed PDE has residual zero, the residual space has a fixed 'true answer' regardless of input, so the usual conformal nonconformity score $|\hat f(X) - Y|$ can be replaced by $|D(\hat f(X))|$, the norm of the PDE residual. Calibrating the quantile $\hat q_\alpha$ of these residual norms gives a symmetric band $[-\hat q_\alpha, \hat q_\alpha]$ that contains a new prediction's physics residual with probability at least $1-\alpha$ whenever calibration and deployment residuals are exchangeable; marginal CP applies per cell and joint CP gives a domain-wide band. The paper reports coverage at the requested level for wave, Navier-Stokes, and magnetohydrodynamics surrogates, and for two tokamak fusion surrogates (plasma blob evolution and Grad-Shafranov equilibrium).

Load-bearing premise

The entire coverage guarantee rests on the physics residuals of calibration and deployment being exchangeable; if deployment shifts the residual distribution—new initial conditions, changed coefficients, or a different grid—the guarantee has no theoretical basis and must be verified empirically.

Editorial extensions

If this is right

  • No ground-truth simulation data is needed for calibration: only sampled model outputs and the differential operator are required, which is why the reported evaluation time drops to tens of seconds versus thousands for data-driven CP-AER.
  • Marginal CP supplies per-cell error bars that localise spatio-temporal regions where the surrogate violates the physics.
  • Joint CP supplies a domain-wide band that can accept or reject whole predictions, enabling a fallback pipeline that routes rejected cases to numerical solvers.
  • Guaranteed marginal and joint coverage is reported for wave, Navier-Stokes, magnetohydrodynamic, and two tokamak fusion surrogates (plasma blob evolution and Grad-Shafranov equilibrium).
  • The framework extends to any forward model written in the canonical form $Ax - b = 0$, including ODEs and algebraic equations, not just PDEs.

Reading between the lines

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

  • Beyond the paper: the guarantee lives on the residual, so a prediction can satisfy the band and still be wrong in solution space if the added error is annihilated by the differential operator (for example, adding a homogeneous solution of a linear PDE); relying on the band thus assumes the residual is a faithful proxy for solution error.
  • Beyond the paper: the theorem's exchangeability assumption means the strong out-of-distribution coverage seen in the experiments should be expected to degrade somewhere; a practical safeguard is to monitor coverage online or re-calibrate on a domain that brackets the deployment distribution.
  • Beyond the paper: the joint reject/accept criterion is a natural active-learning signal—retrain the surrogate on the initial conditions whose residuals exceed the band, and the quantile should tighten as the model learns the hard physics cases.
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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

3 major / 5 minor

Summary. The paper proposes CP-PRE, a conformal prediction framework for neural PDE solvers in which the nonconformity score is the norm of the PDE residual of the surrogate prediction. The framework is post-hoc, model-agnostic, and does not require target labels for calibration: it computes a quantile qhat of calibration residual scores and forms the prediction set [-qhat, qhat] for the residual at a new input. The manuscript presents a theorem claiming marginal coverage under exchangeability, a joint-CP variant, a convolutional finite-difference implementation of residual evaluation, and experiments on the wave, Navier-Stokes, and MHD equations plus two tokamak applications. The paper's headline claims include 'guaranteed coverage across prediction domains' and 'relaxing exchangeability restrictions.'

Significance. If the in-distribution guarantee is correctly stated, CP-PRE is a practically valuable and very cheap UQ scheme for neural PDE surrogates: it turns a physics residual into a calibrated, label-free error bound, and the convolutional stencil implementation is a nice engineering contribution with released code. The in-distribution result follows from standard split conformal prediction applied to a new score, so the core theoretical claim is credible. However, the manuscript substantially overclaims the theoretical scope by presenting out-of-distribution coverage as a derived guarantee and by asserting that the method relaxes exchangeability restrictions; these claims are not consequences of the theorem. The marginal-CP per-cell coverage interpretation also needs clarification, because the validation reports averaged empirical coverage, which is weaker than pointwise per-cell guarantees.

major comments (3)
  1. [Section 5, Tables 3-5, Theorem A.1] The claim of 'guaranteed coverage for both in and out-of-distribution testing' (Section 5, text above Table 3, and repeated in the abstract as 'coverage guarantees across prediction domains') is not supported by Theorem A.1, which assumes that calibration residuals and the new residual are exchangeable. Under a shifted deployment distribution there is no guarantee that the coverage remains at 1-alpha; the OOD results in Tables 3-5 are empirical observations on unspecified shifts. Moreover, the appendix descriptions for the 1D and 2D experiments state that 'the same bounded domain ... is used for calibration and validation' (Sections G.3, H.3, J.3, K.3, L.2), which makes the nature of the 'out-of-distribution' columns unclear. Please either remove the OOD guarantee language, explicitly describe the actual distribution shift and reframe the results as empirical, or provide a theoretical extension for the specific shift considered.
  2. [Section 4.1, Figure 3] Section 4.1 states that marginal-CP 'provides independent error bars for each cell' and that the resulting intervals are 'compliant with Equation (5) for each cell.' The implementation appears to compute a single scalar qhat from the pooled cell-wise residuals and then applies this same value to every cell; the empirical validation in Figure 3 reports 'average empirical coverage,' which is consistent with coverage over the pooled distribution rather than per-cell coverage. If the quantile is global, the 'per cell' guarantee is not valid and the language should be revised to state that coverage holds on average over cells. Please clarify the calibration protocol and, if per-cell quantiles are used, specify how they are estimated.
  3. [Contributions, Section 7] The bullet point in the contributions stating that the method provides 'input-independent prediction sets while relaxing exchangeability restrictions' is incorrect. Theorem A.1 requires exchangeability of the residuals (or of the inputs when the model is fixed); the fact that the prediction set [-qhat, qhat] does not depend on the new input X_{n+1} does not remove or relax this requirement. The related sentence in the Discussion that the framework 'loosens exchangeability restrictions as we can modify the calibration domain by reformulating the PRE' is also unsupported. Please correct these statements to avoid claiming a property that the theory does not provide.
minor comments (5)
  1. [Appendix A, proof of Theorem A.1] The proof states P(pi(R_{n+1}) <= ceil((n+1)(1-alpha))) = ceil((n+1)(1-alpha))/n. Under exchangeability the rank is uniform on {1, ..., n+1}, so the probability is ceil((n+1)(1-alpha))/(n+1), not over n. The final inequality still holds because ceil((n+1)(1-alpha))/(n+1) >= 1-alpha, but the displayed equality is mathematically incorrect and should be fixed.
  2. [Section 5.1 and Appendix J] The 'out-of-distribution' columns in Tables 3 and 4 are not accompanied by a definition of the distribution shift. Please state explicitly how the out-of-distribution test initial conditions were sampled (e.g., wider parameter ranges, different PDE coefficients, or a different grid), since the current appendix descriptions suggest that validation uses the same bounded domain as calibration.
  3. [Table 2] The 'Data-Free' label for CP-PRE could be misleading: the method still requires sampling initial conditions and running the surrogate to generate calibration residuals, although no target labels are needed. Consider using 'Label-Free' or 'Target-Free' to describe the property more precisely.
  4. [Section B] The sentence 'the true output Y for PRE, irrespective of the PDE is always 0' is confusing because the residual operator D acts on the model output and the score is the norm of that residual field. Please rephrase to clarify that the score measures the deviation from satisfying the PDE, i.e., |D(fhat(X)) - 0|, rather than claiming the output Y itself is zero.
  5. [Section D.1] Typo: 'control in teh temporal discretisation' should read 'control in the temporal discretisation.' Please proofread throughout.

Circularity Check

0 steps flagged · score 2.0 of 10

The central coverage theorem is a standard split-conformal lemma applied to a new score; the paper's OOD coverage claims exceed the theorem but do not render the derivation circular.

full rationale

Theorem A.1 proves P(|D(f_hat(X_{n+1}))| <= q_hat) >= 1-alpha from exchangeability of the scalar residuals; this is the textbook split-conformal argument and is self-contained in the appendix. The PRE score is a legitimate nonconformity measure: it maps each input to a scalar physics residual, and the calibrated quantile q_hat is not a parameter fitted to the validation targets. The in-distribution coverage experiments are a standard empirical check of this theorem rather than a circular prediction. The abstract and Tables 3-5 additionally claim coverage "across prediction domains" and "out-of-distribution"; these statements are not consequences of Theorem A.1 and would need a separate distribution-shift argument, but an unsupported scope claim is a correctness/validity concern, not a circular reduction. Self-citations to Gopakumar et al. (2024a) are used to justify not re-testing multiple architectures, but the model-agnostic property follows from the CP theorem independently, so no load-bearing argument reduces to the self-citation.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The central claim rests on the standard conformal prediction quantile, on exchangeability of residuals, and on interpreting the PDE residual as a meaningful score. The method introduces no new physical entity. The main free quantity is the calibration quantile q_hat, which is the fitted output of the procedure.

free parameters (1)
  • residual quantile q_hat = empirical (ceil((n+1)(1-alpha))/n)-quantile of calibration PRE scores
    The prediction set is [-q_hat, q_hat]; q_hat is estimated from model outputs on the calibration set, so the coverage check is a check on this fitted quantile.
assumptions (5)
  • domain assumption Exchangeability of residuals {|D(f_hat(X_i))|} between calibration and deployment.
    Theorem A.1 states this as the premise for the coverage guarantee; Section 3.2 and Section A rely on it.
  • standard math The true solution satisfies the PDE, so its residual is zero.
    PRE is justified by the PDE residual being zero for the exact solution; used in Section 4 and Appendix B.
  • domain assumption Finite-difference stencils computed as convolutions approximate the PDE residual up to truncation error.
    Section 4.3 defines PRE with FD stencils; Appendix D.1 acknowledges truncation error affects band width.
  • domain assumption Boundary conditions are either periodic or disregarded in the main coverage experiments.
    Section 4.3 states boundary conditions are disregarded; Appendix E shows boundary error bars separately. Main PRE scores are interior residuals.
  • domain assumption The surrogate f_hat is fixed during calibration.
    CP-PRE is a post-hoc method; the residual distribution is over the fixed model's outputs, not over retraining.

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Cite this review

Pith. "Pith review of Calibrated Physics-Informed Uncertainty Quantification." pith.science (2026). https://pith.science/paper/HVBSJF63

@misc{pith2026250204406,
  author       = {Pith},
  title        = {Pith review of: Calibrated Physics-Informed Uncertainty Quantification},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HVBSJF63}},
  note         = {Machine review of arXiv:2502.04406}
}
read the original abstract

Simulating complex physical systems is crucial for understanding and predicting phenomena across diverse fields, such as fluid dynamics and heat transfer, as well as plasma physics and structural mechanics. Traditional approaches rely on solving partial differential equations (PDEs) using numerical methods, which are computationally expensive and often prohibitively slow for real-time applications or large-scale simulations. Neural PDEs have emerged as efficient alternatives to these costly numerical solvers, offering significant computational speed-ups. However, their lack of robust uncertainty quantification (UQ) limits deployment in critical applications. We introduce a model-agnostic, physics-informed conformal prediction (CP) framework that provides guaranteed uncertainty estimates without requiring labelled data. By utilising a physics-based approach, we can quantify and calibrate the model's inconsistencies with the physics rather than the uncertainty arising from the data. Our approach utilises convolutional layers as finite-difference stencils and leverages physics residual errors as nonconformity scores, enabling data-free UQ with marginal and joint coverage guarantees across prediction domains for a range of complex PDEs. We further validate the efficacy of our method on neural PDE models for plasma modelling and shot design in fusion reactors.

Figures

Figures reproduced from arXiv: 2502.04406 by the authors.

Figure 1
Figure 1. Neural PDE framework: Neural PDE solvers use data from traditional numerical solvers to quickly approximate PDEs across various conditions (shown by black arrows). To ensure reli￾ability, these models incorporate uncertainty quantification (UQ) methods. If the predicted error exceeds a coverage threshold ϵ, the numerical solver is utilised, further adding to the training data; otherwise, predictions are used as outp… view at source ↗
Figure 2
Figure 2. Schematic of physics-informed uncertainty quantifica￾tion workflow. Initial conditions generate neural PDE predictions autoregressively, over which physics residual errors are estimated. Calibration via marginal and joint conformal prediction yields er￾ror bars - pointwise for marginal-CP and domain-wide for joint-CP. The usage of PDE residuals under the guise of Physics￾Informed Machine Learning (PIML) (Karniadakis… view at source ↗
Figure 3
Figure 3. Validation plots demonstrating coverage guarantee detailed in Equation (5) obtained by performing CP using PRE across experiments. The average empirical coverage obtained experimentally is given on the y-axis (ranging from 0 to 1, with 1 representing 100% coverage), while the theoretical coverage is represented on the x-axis. We obtain guaranteed coverage while using marginal-CP formulation and near-to-ideal coverag… view at source ↗
Figures from the paper (28 more)
Figure 4
Figure 4. Figure 4: Wave: (From left to right) neural PDE (FNO) prediction at the last time instance, physics residual error of the prediction, Upper error bars obtained by performing marginal-CP and joint-CP respectively (90% coverage). For brevity, we have only shown the upper error bar…
Figure 5
Figure 5. Figure 5: Reduced MHD: CP-PRE using the Temperature equa￾tion (Eqn. 3 in (Gopakumar et al., 2024b)) of reduced-MHD to bound the plasma surrogate models. The PRE captures the model error relatively well, allowing us to provide lower and upper error bars corresponding to our requi…
Figure 6
Figure 6. Figure 6: Grad-Shafranov: The PRE for a specific poloidal field coil configuration is indicated on the left, and the lower and upper bars for 50% are displayed adjacent to it. Aside from guaranteeing coverage, the CP-PRE framework allows us to discard physically inconsistent equ…
Figure 8
Figure 8. Figure 8: CP-PRE provides guaranteed coverage irrespective of the discretisation associated with the model outputs., however, the width of the obtained coverage bounds indicates the discretisation error associated with the gradient estimation. Coverage taken for α = 0.1 ∼ 90% co…
Figure 9
Figure 9. Figure 9: Error bars obtained over the boundary conditions over the right wall of domain of the Navier-Stokes Equation using Marginal and Joint CP. The empirical coverage obtained using the boundary condition as the PRE nonconformity score is also given. 21 [PITH_FULL_IMAGE:fig…
Figure 11
Figure 11. Figure 11: Burgers’ Equation: (Left) Comparing the neural PDE (FNO) performance with that of the physics-based numerical solver at the last time instance. (Middle) Upper and lower bounds for 90% coverage obtained by performing marginal-CP. (Right) Upper and lower bounds for 90% …
Figure 12
Figure 12. Figure 12: Advection Equation: Marginal-CP with α = 0.5 G.3. Calibration and Validation To perform the calibration as outlined in Section 5, model predictions are obtained using initial conditions sampled from the domain given in [PITH_FULL_IMAGE:figures/full_fig_p023_12.png]
Figure 13
Figure 13. Figure 13: Advection Equation: joint-CP with α = 0.5 23 [PITH_FULL_IMAGE:figures/full_fig_p023_13.png]
Figure 14
Figure 14. Figure 14: Burgers Equation: Marginal-CP with α = 0.75 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 x 0.3 0.2 0.1 0.0 0.1 0.2 0.3 u t = 5 Residual Lower Upper 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 x 0.2 0.1 0.0 0.1 0.2 u t = 10 Residual Lower Upper 0.00 0.25 0.50 0.75 1.0…
Figure 15
Figure 15. Figure 15: Burgers Equation: joint-CP with α = 0.75 25 [PITH_FULL_IMAGE:figures/full_fig_p025_15.png]
Figure 16
Figure 16. Figure 16: CP-PRE provides guaranteed coverage irrespective of the model performance, however, the width of the obtained cov￾erage bounds indicates the accuracy of the model in obeying the underlying physics. Coverage taken for α = 0.1 ∼ 90% coverage. 26 [PITH_FULL_IMAGE:figure…
Figure 17
Figure 17. Figure 17: Wave Equation: Temporal evolution of field associated with the wave equation modelled using the numerical spectral solver (top of the figure) and that of the FNO (bottom of the figure). The spatial domain is given in Cartesian geometry. J.3. Calibration and Validation…
Figure 18
Figure 18. Figure 18: Analysing the PRE over the ground truth and the prediction. Though the neural PDE solver is capable of learning seemingly indistinguishable emulation of the physics while exploring the PRE over each tells a different story. As opposed to the smooth Laplacian of the PR…
Figure 19
Figure 19. Figure 19: Navier-Stokes Equations: Temporal evolution of velocity and pressure modelled using the numerical spectral solver (top of the figure) and that of the FNO (bottom of the figure) 30 [PITH_FULL_IMAGE:figures/full_fig_p030_19.png]
Figure 20
Figure 20. Figure 20: Navier-Stokes: CP using the Momentum Equation (13) as the PRE for a neural PDE surrogate model trained to model fluid dynamics [PITH_FULL_IMAGE:figures/full_fig_p031_20.png]
Figure 21
Figure 21. Figure 21: Navier-Stokes: CP using the Continuity Equation (12) as the PRE for a neural PDE surrogate model trained to model fluid dynamics [PITH_FULL_IMAGE:figures/full_fig_p031_21.png]
Figure 22
Figure 22. Figure 22: MHD Equations: Temporal evolution of velocity and pressure modelled using the numerical solver (top of the figure) and that of the FNO (bottom of the figure). (Continued on next page) 33 [PITH_FULL_IMAGE:figures/full_fig_p033_22.png]
Figure 23
Figure 23. Figure 23: MHD Equations: Temporal evolution of velocity and pressure modelled using the numerical solver (top of the figure) and that of the FNO (bottom of the figure). (Continued from previous page) 34 [PITH_FULL_IMAGE:figures/full_fig_p034_23.png]
Figure 24
Figure 24. Figure 24: MHD: Slice plots along the x-axis (sliced at y = 0.5m) indicating the marginal and joint coverage (90%) obtained over the neural PDE modelling the MHD equations using the induction equation Equation (17) (on the left) and the energy equation Equation (16) (on the righ…
Figure 25
Figure 25. Figure 25: MHD: CP using the Induction Equation (17) as the PRE for a neural PDE surrogate model solving the Ideal MHD equations. The last time instance of the prediction is shown. x y PRE: Denergy( , v, P, B) 5 0 5 ×10 2 (a) PRE of the Energy Equation Equa￾tion (16) over the FN…
Figure 26
Figure 26. Figure 26: MHD: CP using the Energy Equation (16) as the PRE for a neural PDE surrogate model solving the Ideal MHD equations. The last time instance of the prediction is shown. Error bars obtained using joint CP are an order of magnitude higher than that obtained by marginal CP…
Figure 27
Figure 27. Figure 27: MHD: CP using the Continuity Equation (14) as the PRE for a neural PDE surrogate model solving the Ideal MHD equations. x y PRE: DGauss(B) 5 0 5 ×10 2 (a) PRE of the Divergence Equation Equa￾tion (18) over the FNO prediction x y Marginal CP (+q) 2 4 6 8 ×10 2 (b) Uppe…
Figure 28
Figure 28. Figure 28: MHD: CP using the Gauss’s law for magnetism Equation (18) as the PRE for a neural PDE surrogate model solving the Ideal MHD equations. 36 [PITH_FULL_IMAGE:figures/full_fig_p036_28.png]
Figure 29
Figure 29. Figure 29: Empirical coverage obtained by performing CP-PRE over JOREK-FNO predictions 6 orders of magnitude faster than the numerical solver. For a complete description of the MHD system and additional experimental details, we refer readers to (Gopaku￾mar et al., 2023b). M.4. C…
Figure 30
Figure 30. Figure 30: Sample Equilibrium plot showcasing the magnetic equi￾librium as the contour plots observed for a given poloidal field (PF) configuration (PF coil locations are indicated in blue blocks). The poloidal cross-section of the tokamak is shown here, with the structural boun…
Figure 31
Figure 31. Figure 31: A simple schematic diagram of a generic tokamak with all of the main magnetic components and fields shown (Li et al., 2014). The poloidal field coil magnets (grey) are that which this work aims to optimise. R,Z Mesh PF Coil Locations Equilbirum Encoder Decoder [PITH_…
Figure 32
Figure 32. Figure 32: Conditional auto-encoder developed as a surrogate model for mapping the poloidal field coil locations to the cor￾responding magnetic equilibrium under constant coil currents. equilibria. This allows us to explore the design space quickly while adding trustworthiness t…
Figure 34
Figure 34. Figure 34: Marginal and joint empirical coverage obtained by per￾forming CP-PRE over the Grad-Shafranov equation 39 [PITH_FULL_IMAGE:figures/full_fig_p039_34.png]

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Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.