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A Fourier Space Perspective on Diffusion Models

T0 review · 3 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read The paper claims that the standard diffusion noising schedule corrupts high-frequency details so fast it breaks the reverse step's Gaussian assumption, degrading generated fine detail; an equal-SNR schedule fixes this at no FID cost.

desk verdict Clear paper on frequency-aware forward processes, but the headline causal claim is confounded by changing loss weighting alongside forward noise, so the high-frequency gains cannot yet be attributed to equal-rate noising. read the letter →

arxiv 2505.11278 v1 pith:N6N2FHNE submitted 2025-05-16 stat.ML cs.CVcs.LGstat.ME

Fabian Falck , Teodora Pandeva , Kiarash Zahirnia , Rachel Lawrence , Richard Turner , Edward Meeds , Javier Zazo , Sushrut Karmalkar This is my paper · ORCID
classification stat.MLcs.CVcs.LGstat.ME
keywords diffusionmodelsFourierdomainsignal-to-noiserationoisescheduleshigh-frequencygenerationnormalityassumptionfrequencyhierarchygenerativemodelling
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 argues that the standard DDPM noising schedule does not corrupt all image frequencies at the same rate: because natural signals have exponentially less energy at high frequencies, white noise destroys fine detail first and fastest. It claims this fast noising breaks the approximation that the reverse denoising step is Gaussian, and that the resulting errors accumulate into visibly wrong high-frequency statistics in generated images. As a remedy it introduces EqualSNR, a forward process that keeps the signal-to-noise ratio identical across all frequencies at every timestep, removing the low-to-high frequency generation hierarchy of DDPM. Empirically, EqualSNR matches or beats DDPM's FID on standard image benchmarks while producing high-frequency content that a simple spectral detector can no longer tell apart from real data. If correct, the paper shows that the frequency hierarchy is an artifact of the noising schedule, not a necessary ingredient of diffusion models.

What carries the argument

The load-bearing object is the per-frequency Signal-to-Noise Ratio in the Fourier domain, $\mathrm{SNR}_t(i) = \frac{\alpha_t C_i}{(1-\alpha_t)\Sigma_{ii}}$, where $C_i$ is the variance of the $i$-th Fourier coefficient of the data and $\Sigma_{ii}$ the variance of the added noise at that coordinate. Under DDPM the noise variance is flat, so for power-law data the high-frequency SNR is orders of magnitude below the low-frequency SNR at every timestep; 'faster noising' then means a larger SNR drop per time increment, and 'earlier' means a smaller SNR at every fixed time. EqualSNR re-weights the noise so that $\Sigma_{ii} \propto C_i$, making the SNR equal across all frequencies at each timestep and removing the hierarchy. The framework also carries the explanation: the reverse kernel $q(y_{t-1} \mid y_t)$ is Gaussian only when the forward noise is small relative to the signal variance, and the SNR is exactly the quantity that measures that ratio, so it pinpoints where the Gaussian assumption breaks and defines a calibration procedure for comparing schedules fairly.

What would settle it

Construct (or find) a natural-image-like dataset whose high-frequency Fourier marginals are exactly Gaussian while still following the power law in variance, and compare DDPM against EqualSNR on it: if the DDPM high-frequency deficit persists, the normality violation is not the main driver; if it vanishes, the proposed mechanism is confirmed.

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Extended reading notes

Core claim

The central claim is that the DDPM forward process, acting on data whose Fourier variance decays as a power law, noises high-frequency components faster and earlier than low-frequency ones, and that this unequal treatment breaks the Gaussianity assumption of the reverse process for exactly those components. In a finite-time model each reverse transition $q(y_{t-1} \mid y_t)$ is approximated by a Gaussian; the paper proves that when the forward noise is large relative to the signal variance, the true posterior can stay a constant total-variation distance away from any Gaussian, and that this is precisely the situation at high frequencies under DDPM. On CIFAR-10 the KL divergence of the reverse posterior from its Gaussian approximation is $1.8\times10^{-2}$ at a high frequency versus $3.1\times10^{-4}$ at a low frequency, and the paper ties this to the observable failure of DDPM: a logistic regression on two summary statistics of the spectral magnitudes separates DDPM-generated images from real ones almost 100% of the time. The proposed alternative, EqualSNR, sets the per-frequency SNR equal across all coordinates at every timestep by making the noise variance proportional to the data variance in Fourier space. With this schedule the normality violations shrink to match low-frequency levels, the same detector drops to near-chance accuracy, FID on CIFAR-10, CelebA, and LSUN Church is on par with DDPM (better on the highest-resolution dataset), and a synthetic sparse-dots task where high frequencies dominate shows a marked quality gain. The paper therefore claims that the low-to-high frequency generation hierarchy is not essential to diffusion models.

Load-bearing premise

The argument depends on high-frequency detail patterns in real images being far from bell-shaped, as in the paper's two-peak toy example; if they were effectively bell-shaped, the reverse step's Gaussian approximation would hold and fast noising would not produce the degradation the paper blames on it.

Editorial extensions

If this is right

  • The low-to-high frequency generation hierarchy is a side effect of the DDPM noise schedule, not an ingredient diffusion models need: a hierarchy-free schedule trains and samples stably with equal or better fidelity.
  • Any modality with a power-law spectrum, such as images, audio, video, and protein density maps, inherits the same fast-noising distortion, so equal-SNR forward processes become a natural default there.
  • EqualSNR reaches its final FID with fewer timesteps than DDPM, so the schedule also offers a sampling-cost saving at fixed quality.
  • The per-frequency SNR defines a direct design space for task-tuned schedules, such as spending more steps on the frequencies that matter for a downstream application, without changing the network architecture.
  • Spectral summary statistics exposed a quality gap that FID missed, so frequency-aware evaluation should accompany FID when fine detail matters.

Reading between the lines

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

  • The paper's own Figure 1 shows the same power-law spectrum in audio, video, and protein density maps, yet its experiments stop at natural images; the direct next test is EqualSNR on those modalities, where the same high-frequency gain is predicted.
  • An alternative reading of the results is plainer than the normality story: giving every frequency an equal SNR also gives fine details as many refinement steps as coarse structure, so per-frequency quality should improve under any account of discretisation error; measuring per-frequency reconstruction error per timestep would separate the resource-allocation effect from the Gaussianity mechanism.
  • If high-frequency fidelity is what defeats spectral detectors, equal-rate schedules also lower the cost of realistic forgeries in detail-rich domains; the paper flags the DeepFake implication in its impact statement but does not quantify how much harder detection becomes at full resolution.
  • All experiments use deterministic DDIM sampling while the normality argument is made for the stochastic reverse kernel; repeating the comparison with true ancestral sampling would show whether the equal-rate benefit belongs to the forward process or to the sampler.
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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 studies the forward noising process of DDPM in Fourier space. It observes that for data with a power-law spectral variance, standard white-noise corruption makes high-frequency components reach low SNR faster and earlier than low-frequency components. The authors argue that this fast noising violates the Gaussianity assumption for the reverse-process conditional at finite discretization, leading to degraded high-frequency sample quality. They propose EqualSNR, a Fourier-space forward process whose noise covariance is matched to the data covariance so that all frequencies have equal SNR at each timestep, and provide an ELBO-based training loss and a DDIM-style sampler. Experiments on CIFAR10, CelebA, LSUN Church, and a synthetic dots dataset report FID comparable to DDPM and improved high-frequency statistics under a Fourier-space discriminator.

Significance. If the central claim holds, the paper would provide a crisp explanation of a widely observed phenomenon and show that the low-to-high frequency generation hierarchy is not essential to diffusion models. The paper's empirical contribution is substantial: EqualSNR is competitive on standard benchmarks and improves high-frequency fidelity, with careful calibration of average SNR. The ELBO connection (Proposition 2) and the covariance-matching construction (Definition 2) are clean. However, the causal mechanism asserted in the abstract is not yet isolated: the comparison changes both the forward process and the loss weighting, and the theoretical results are toy examples rather than a general theorem. The paper also ships many auxiliary analyses (SNR heatmaps, per-frequency variance trajectories) that are useful for the community.

major comments (3)
  1. [§4, Algorithm 1, Table 1] The experimental comparison does not isolate the forward process from the loss objective. EqualSNR is trained with L_t = ||C^{-1/2}(y0 - \hat y0)||^2, while the DDPM baseline is trained with unweighted pixel-space MSE; in Fourier space this is an unweighted loss. Because high-frequency entries of C are orders of magnitude smaller than low-frequency ones, the C^{-1/2} weighting massively increases the penalty on high-frequency errors, so a U-Net with limited capacity will devote more capacity to high frequencies even if the forward SNR profile is unchanged. Table 1 compares EqualSNR against DDPM under different calibrations but never varies the loss weighting while holding the forward process fixed (or vice versa). The high-frequency gains in Table 2 and Figs. 5-6, and the dots result in Fig. 6, could therefore be due to the objective rather than to equal-rate noising. The manuscript should include an ablation where, for example, DDPM is trained with the C^{-1/2}-weighted loss or EqualSNR is trained with an unweighted loss, to identify the cause of the improvements.
  2. [§3.2 / §3.1] The theoretical justification of the Gaussian-violation mechanism is developed for the stochastic reverse process q(yt-1|yt), but all experiments (Algorithm 2 and the standard DDIM sampler) use deterministic DDIM sampling. The statement in §3.2 that 'a similar property holds here' for DDIM is asserted without derivation or direct verification. Since the empirical claims in §4 are obtained with DDIM, the paper should either prove or empirically test the analogous normality-violation statement for the deterministic DDIM update, or at least discuss why the stochastic analysis transfers.
  3. [§3.1, Proposition 3] Proposition 3 is a counterexample with a two-component Gaussian-mixture prior D0 = 1/2 N(-1, delta^2) + 1/2 N(1, delta^2) and forward noise variance 4. It shows that such a prior can produce a non-Gaussian reverse conditional. However, the paper does not establish that actual high-frequency Fourier coefficients of natural images satisfy the premise; these coefficients are approximately sums over many pixels and are expected to be near-Gaussian by the central limit theorem. The empirical KDE/KL evidence in Fig. 3 uses one timestep and shows KL values of order 10^-2, and the claim that such small violations accumulate into the visible high-frequency degradation in §4 is not supported by a quantitative error-propagation argument. The abstract's causal statement 'faster noising ... results in violations ... leads to degraded generation quality' is therefore stronger than what the theoretical results and experiments currently establish.
minor comments (5)
  1. [§2, Eq. (4)] The calculation of the noise covariance in Fourier space assumes that the discrete Fourier transform is unitary (up to a normalization constant); this should be stated explicitly so the reader can follow the covariance computation.
  2. [Definition 2] The phrase 'universal constant' for c should be replaced by 'constant independent of frequency' to avoid confusion with the standard meaning of universal constants in probability theory.
  3. [§3.1 and §A.6] The numbering of propositions is inconsistent: the informal Proposition 1 in §3.1 is formalized as Proposition 3 in §A.6; please align the numbering.
  4. [§A.3] The derivation of the calibrated alpha_eq^t should spell out the averaging measure (equal weights over frequencies) and state that the calibration is performed on the training set, so that the values in Table 1 are reproducible.
  5. [§B.1, Eq. (13)] Equation (13) uses y0 on the right-hand side while the surrounding text says the model predicts a clean estimate; the notation should be corrected to \hat y0.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the EqualSNR equal-SNR property is definitional, not a fitted prediction, and the main claims rest on independent theoretical and empirical evidence.

full rationale

The paper's derivation chain is self-contained. The claim that DDPM noises high frequencies faster and earlier follows directly from the SNR definition s_DDPM_t(i)=α_t C_i/(1−α_t) together with the empirically documented Fourier power law; it is an implication, not an assumed conclusion. EqualSNR is defined by choosing noise covariance Σ_ii=cC_i, so equal SNR across frequencies holds by construction, but the paper does not present that definition as a derived prediction; the substantive contributions are the normality-violation analysis, the alternate forward process, and the empirical comparisons. Proposition 3 is a standalone counterexample: a two-component mixture prior with added noise yields a non-Gaussian reverse conditional. It does not assume the high-frequency generation result, and the bridge to real data is supported by the Monte Carlo and KDE evidence in Fig. 3 and the KL numbers in §3.1, not by the proposition alone. The experiments are benchmarked against external data and metrics: the Dzanic-style classifier is trained on real versus generated Fourier magnitudes, and FID is reported against standard imaging datasets. No parameter is fitted to the outcome and then renamed a prediction; the calibration in §A.3 merely matches average SNR across forward processes as a normalization. The self-citations in §5 (Williams et al. 2023, Falck et al. 2022) are related-work references on U-Net spectral properties and are not load-bearing for the central claim. Two caveats are worth flagging, but they are not circularity in the formal sense. First, the EqualSNR comparison in Section 4 changes both the forward noise covariance and the training loss L_t=||C^{-1/2}(y0−ŷ0)||^2, so the observed high-frequency improvements are not isolated to the noising rate; this is an experimental confound, not a reduction by construction. Second, the statement in §3.2 that a similar normality-violation property holds for deterministic DDIM is asserted without derivation even though DDIM is used in experiments; this is an omitted justification, not a circular step. Appendix B's stated limitations about resolution, FlippedSNR, and modality coverage also indicate an honest, non-circular presentation. Overall, the paper's main comparison is against external benchmarks and its theoretical results do not reduce to their inputs by definition.

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

The core additions that the reader must accept without independent proof are: the power-law data assumption, the transfer of the toy counterexample to real high-frequency coefficients, the unproved extension of the normality analysis to DDIM sampling, and the claim that U-Net inductive bias is preserved under the Fourier-space loss. No new physical entities are introduced; EqualSNR and FlippedSNR are forward-process designs, not entities. The covariance C is estimated from data as part of the method.

free parameters (1)
  • Fourier signal variance matrix C = Diag(Cov(y0)) estimated from training data
    The EqualSNR forward process (Definition 2, Algorithm 1) scales noise variance per frequency by C_i. It is estimated from the dataset, not optimized, so it is a data-dependent schedule input rather than a fitted free parameter in the usual sense.
assumptions (5)
  • domain assumption Target data modalities (images, audio, proteins, materials) exhibit exponentially decaying variance and magnitude in the Fourier domain (Fourier power law).
    Invoked in Section 2 and Fig. 1 to argue that DDPM noises high frequencies faster and earlier than low frequencies. The paper shows this property on several datasets, but assumes it holds for the modalities it claims to cover.
  • ad hoc to paper The toy two-component mixture in Proposition 3 transfers to real high-frequency Fourier coefficients of natural images.
    Section 3.1 states that for CIFAR10 high frequencies the noise variance is much larger than the data variance, so the same phenomenon occurs. No proof shows that real high-frequency coefficients have a bimodal marginal; the paper relies on KDE/KL empirics instead.
  • ad hoc to paper The reverse-process normality violation observed for stochastic DDPM also holds for deterministic DDIM sampling.
    Section 3.2 asserts this without derivation, yet all experiments use DDIM or Algorithm 2 for sampling. This assumption is needed to connect the theory to the experimental results.
  • domain assumption A U-Net retains its inductive bias when operating on pixel-space inputs while the training loss is computed in Fourier space.
    Algorithm 1 trains f_theta in Euclidean/pixel space with an MSE loss on Fourier coefficients. The paper states this is to maintain the U-Net inductive bias, but provides no analysis of how the Fourier-space loss interacts with the architecture.
  • standard math In the limit T to infinity, the reverse transition q(y_{t-1}|y_t) converges to a Gaussian (Feller 1954).
    Section 3.1 uses this standard result as the baseline for the finite-T Gaussian assumption that the paper interrogates.

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Cite this review

Pith. "Pith review of A Fourier Space Perspective on Diffusion Models." pith.science (2026). https://pith.science/paper/N6N2FHNE

@misc{pith2026250511278,
  author       = {Pith},
  title        = {Pith review of: A Fourier Space Perspective on Diffusion Models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/N6N2FHNE}},
  note         = {Machine review of arXiv:2505.11278}
}
read the original abstract

Diffusion models are state-of-the-art generative models on data modalities such as images, audio, proteins and materials. These modalities share the property of exponentially decaying variance and magnitude in the Fourier domain. Under the standard Denoising Diffusion Probabilistic Models (DDPM) forward process of additive white noise, this property results in high-frequency components being corrupted faster and earlier in terms of their Signal-to-Noise Ratio (SNR) than low-frequency ones. The reverse process then generates low-frequency information before high-frequency details. In this work, we study the inductive bias of the forward process of diffusion models in Fourier space. We theoretically analyse and empirically demonstrate that the faster noising of high-frequency components in DDPM results in violations of the normality assumption in the reverse process. Our experiments show that this leads to degraded generation quality of high-frequency components. We then study an alternate forward process in Fourier space which corrupts all frequencies at the same rate, removing the typical frequency hierarchy during generation, and demonstrate marked performance improvements on datasets where high frequencies are primary, while performing on par with DDPM on standard imaging benchmarks.

Figures

Figures reproduced from arXiv: 2505.11278 by the authors.

Figure 1
Figure 1. [Left] The Fourier power law observed in (top-left) images (Krizhevsky et al., 2009), (top-right) videos (Kay et al., 2017), (bottom-left) audio (Tzanetakis, 1999), and (bottom-right) Cryo-EM derived protein density maps (wwPDB Consortium, 2023). [Center] A DDPM forward process on these modalities noises high-frequency components substantially faster (SNR changes more per time increment), and earlier than low-freque… view at source ↗
Figure 2
Figure 2. Comparing the SNR (dB scale) for DDPM and the alternate EqualSNR in Fourier space. In the forward process, SNR is computed as a Monte Carlo estimate of Eq. (5) on CIFAR10 (referring to § B for details on the reverse process). low frequencies. As the generative process of diffusion mod￾els reverses the forward process, low-frequencies are gen￾erated earlier than high-frequencies, which can be viewed as being generate… view at source ↗
Figure 3
Figure 3. Fast noising of high frequencies leads to violations of normality in the DDPM reverse process. We plot Monte [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (26 more)
Figure 4
Figure 4. Figure 4: The forward process controls when frequencies are generated in the reverse process. We visualise the forward and backward process of [left] DDPM and [right] EqualSNR in pixel space (rows 1,3,4) and Fourier space (magnitudes; row 2). Rows 3 and 4 are low- and high-pass …
Figure 6
Figure 6. Figure 6: EqualSNR outperforms DDPM on data where high-frequency information is dominant. Pixel intensity dis￾tribution (sorted descendingly) of 1000 generated samples for DDPM and EqualSNR, and two examples. ples). In [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]
Figure 7
Figure 7. Figure 7: Visualisation of the [left] low- and [right] high-pass filter masks. [PITH_FULL_IMAGE:figures/full_fig_p020_7.png]
Figure 8
Figure 8. Figure 8: Dots Examples in the DDPM and EqualSNR case. Yellow corresponds to ‘white pixels’ (highest intensity) on this [PITH_FULL_IMAGE:figures/full_fig_p021_8.png]
Figure 9
Figure 9. Figure 9: Analysis of violations of the Gaussian assumption in DDPM (1 of 3). [PITH_FULL_IMAGE:figures/full_fig_p023_9.png]
Figure 10
Figure 10. Figure 10: Analysis of violations of the Gaussian assumption in DDPM (2 of 3). [PITH_FULL_IMAGE:figures/full_fig_p024_10.png]
Figure 11
Figure 11. Figure 11: Analysis of violations of the Gaussian assumption in DDPM (3 of 3). [PITH_FULL_IMAGE:figures/full_fig_p025_11.png]
Figure 12
Figure 12. Figure 12: Analysis of violations of the Gaussian assumption in EqualSNR (1 of 3). [PITH_FULL_IMAGE:figures/full_fig_p026_12.png]
Figure 13
Figure 13. Figure 13: Analysis of violations of the Gaussian assumption in EqualSNR (2 of 3). [PITH_FULL_IMAGE:figures/full_fig_p027_13.png]
Figure 14
Figure 14. Figure 14: Analysis of violations of the Gaussian assumption in EqualSNR (3 of 3). [PITH_FULL_IMAGE:figures/full_fig_p028_14.png]
Figure 15
Figure 15. Figure 15: presents the C matrix for the three standard imaging benchmarks used in § 4 [PITH_FULL_IMAGE:figures/full_fig_p030_15.png]
Figure 16
Figure 16. Figure 16: The alternate FlippedSNR forward process noises [PITH_FULL_IMAGE:figures/full_fig_p030_16.png]
Figure 17
Figure 17. Figure 17: The reverse process learns to mirror the forward process. We show the variances at time t of the [top] forward and [bottom] backward process in Fourier space, for [left] DDPM, [centre] ESNR, and [right] FlippedSNR. In [PITH_FULL_IMAGE:figures/full_fig_p031_17.png]
Figure 18
Figure 18. Figure 18: DDPM on images normalised to [−1, 1] is not variance-preserving, i.e. the variance changes throughout the forward process. To see this, we plot the coordinate-wise signal variances diag(Cov(x0)) in pixel space (in contrast to [PITH_FULL_IMAGE:figures/full_fig_p031_18.png]
Figure 19
Figure 19. Figure 19: SNR of the [left] DDPM and [right] EqualSNR forward processes on CelebA ( [PITH_FULL_IMAGE:figures/full_fig_p032_19.png]
Figure 20
Figure 20. Figure 20: SNR of the [left] DDPM and [right] EqualSNR forward processes on LSUN Churches ( [PITH_FULL_IMAGE:figures/full_fig_p032_20.png]
Figure 21
Figure 21. Figure 21: Flipped SNR forward process on CIFAR10 ( [PITH_FULL_IMAGE:figures/full_fig_p032_21.png]
Figure 22
Figure 22. Figure 22: Forward process of DDPM, at uniformly-spaced timsteps between [PITH_FULL_IMAGE:figures/full_fig_p033_22.png]
Figure 23
Figure 23. Figure 23: Forward process of EqualSNR, at uniformly-spaced timsteps between [PITH_FULL_IMAGE:figures/full_fig_p034_23.png]
Figure 24
Figure 24. Figure 24: Forward process of FlippedSNR, at uniformly-spaced timsteps between [PITH_FULL_IMAGE:figures/full_fig_p035_24.png]
Figure 25
Figure 25. Figure 25: Reverse process of DDPM, at uniformly-spaced timsteps between [PITH_FULL_IMAGE:figures/full_fig_p036_25.png]
Figure 26
Figure 26. Figure 26: Reverse process of EqualSNR, at uniformly-spaced timsteps between [PITH_FULL_IMAGE:figures/full_fig_p037_26.png]
Figure 27
Figure 27. Figure 27: Reverse process of FlippedSNR, at uniformly-spaced timsteps between [PITH_FULL_IMAGE:figures/full_fig_p038_27.png]
Figure 28
Figure 28. Figure 28: Samples of a diffusion model trained on CIFAR10 with a [left] DDPM and [right] EqualSNR (calibrated) forward process, using T = 200 steps at inference time. 39 [PITH_FULL_IMAGE:figures/full_fig_p039_28.png]
Figure 29
Figure 29. Figure 29: Samples of a diffusion model trained on CelebA with a [left] DDPM and [right] EqualSNR (calibrated) forward process, using T = 200 steps at inference time. 40 [PITH_FULL_IMAGE:figures/full_fig_p040_29.png]
Figure 30
Figure 30. Figure 30: Samples of a diffusion model trained on LSUN Church with a [left] DDPM and [right] EqualSNR (calibrated) forward process, using T = 200 steps at inference time. 41 [PITH_FULL_IMAGE:figures/full_fig_p041_30.png]

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.