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Unpacking Softmax: How Temperature Drives Representation Collapse, Compression, and Generalization

T0 review · 4 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Softmax temperature—through the norm of the logits at initialization—biases deep classifiers toward low-rank, compressed representations, trading away out-of-distribution generalization while improving out-of-distribution detection.

desk verdict A broad, internally consistent empirical story about softmax temperature and representation rank, attached to a mechanism that is not actually derived for the architectures tested. read the letter →

arxiv 2506.01562 v1 pith:OAYRY7AP submitted 2025-06-02 cs.LG stat.ML

Wojciech Masarczyk , Mateusz Ostaszewski , Tin Sum Cheng , Tomasz Trzciński , Aurelien Lucchi , Razvan Pascanu This is my paper
classification cs.LGstat.ML MSC 68T0715A1815A42
keywords softmaxtemperaturerank-deficitbiasrepresentationcollapseneurallogitsnormsingularvaluealignmentout-of-distributiongeneralizationdetection
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

Softmax temperature is usually treated as a calibration or sampling knob, but this paper argues it silently steers what a deep classifier learns to represent. The authors claim that a low logits norm at initialization—produced by high softmax temperature or by architecture (narrow layers, small initialization, normalization placement)—systematically biases networks toward rank-deficit solutions whose internal representation has rank far below the number of classes, $\operatorname{rank}(M) \ll c-1$, in contrast with the full-rank structures predicted by neural-collapse theory. The reason this does not destroy accuracy is that softmax is a norm-amplifying map: growing the logits norm alone expands the post-softmax spectrum, so a low-rank pre-softmax matrix can still yield full-rank outputs. The paper documents a temperature-controlled trade-off: compressed representations generalize worse to out-of-distribution data but detect out-of-distribution inputs substantially better, giving practitioners a single dial that moves both properties at once. The authors themselves note the analysis is limited to supervised image classification, that the inner-product dynamics lack a formal treatment, and that only fixed-temperature training was examined (Section 7; Appendix N).

What carries the argument

The load-bearing object is the decomposition of the logits matrix $M = W A$ into the last layer's weights and the penultimate representations, analyzed through their singular value decompositions: $\|M\| = \|\Sigma_W V_W^\top U_A \Sigma_A\| \leq \|\Sigma_W \Sigma_A\|$. This inequality splits logits-norm growth into two mechanisms—singular-value scaling and singular-vector alignment—and the paper shows high-temperature training exploits the second: the top singular vectors of weights and activations align across layers, so the top singular value of the final logits becomes the product of per-layer top singular values, $\sigma_1^L = \prod_{i=1}^{L-1} \sigma_1^i \geq (\sigma_1^k)^L$, which grows exponentially in depth when the smallest top singular value exceeds $1$. That compounding produces representation collapse measured by numerical rank, and the bound $\operatorname{rank}(\partial L/\partial W^i) \leq \operatorname{rank}(A^{i-1})$ makes the collapse self-reinforcing through gradients. The counterbalancing piece is Proposition 3.1, a tight bound on the spectral gap of any column-stochastic matrix $S$, $0 \leq \sigma_1(S) - \sigma_n(S) \leq \sqrt{1+r} - \sqrt{\max\{1/n - r, 0\}}$ with $r$ the maximal sum of column inner products; this is the formal reason that raising $\|M\|$ alone raises the post-softmax rank of a fixed low-rank matrix. The supporting diagnostics are effective depth $\kappa$ (the first layer whose linear probe reaches $99\%$ of final accuracy, divided by depth), OOD generalization loss $\rho$ (the normalized gap between best and final-layer OOD probe accuracy), and solution rank (numerical rank of the pre-softmax logits).

What would settle it

Train a deep ReLU classifier at high temperature while recording, per layer, the cosine similarity between the top singular vectors of the weights and of the incoming activations; if numerical rank collapses before any alignment appears, or if the top singular value of the final logits does not track the product of per-layer top singular values, then the alignment-driven exponential-collapse mechanism is not the cause of rank-deficit bias.

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Extended reading notes

Core claim

The paper's central claim is that high-temperature training induces a rank-deficit bias: softmax-based deep networks converge to solutions with $\operatorname{rank}(M)$ much less than $c-1$, far below the rank predicted by Neural Collapse, and this bias is governed by the norm of the logits at initialization. Because softmax with temperature $T$ equals softmax with temperature $1$ applied to logits scaled by $1/T$, temperature is functionally a logits-norm control. High temperature flattens output distributions into a symmetric loss landscape, and the network escapes this symmetry by growing the logits norm; the growth path high-temperature models take is singular-vector alignment across layers, which compounds top singular values exponentially and collapses the numerical rank of representations. Collapsed representations then cap the rank of gradients, with $\operatorname{rank}(\partial L/\partial W^i) \leq \operatorname{rank}(A^{i-1})$, locking in the collapse. Softmax nevertheless restores output rank: Proposition 3.1 bounds the spectral gap of any softmaxed matrix by the mutual similarity of its columns, which the growing logits norm decreases, and a constructive proof (Proposition K.1) shows that pre-softmax rank $2$ suffices for full-rank post-softmax outputs with any number of classes. The empirical signature is uniform across MLP, VGG, ResNet, and ViT on CIFAR and ImageNet: comparable final accuracy, lower effective depth $\kappa$, higher OOD generalization loss $\rho$, and lower solution rank, together with better OOD detection under the NECO metric (for instance, a ResNet-34 on ImageNet-1k drops from solution rank 512 to 122 under high temperature).

Load-bearing premise

The cascade argument assumes that ReLU layers compound their top singular values the way perfectly aligned linear maps do, and the paper's own derivation (Section 3.1) sets the product formula $\sigma_1^L = \prod_i \sigma_1^i$ without accounting for the ReLU nonlinearity; if that alignment does not survive ReLU, the claimed exponential collapse and gradient-collapse feedback are not established by the presented math.

Editorial extensions

If this is right

  • Temperature becomes a deployment dial: training at high temperature (or with other low-logits-norm choices such as smaller initialization or narrower layers) yields models with comparable accuracy, compressed internal representations, and measurably better OOD detection scores on the NECO metric.
  • The compression–generalization exchange is systematic: lower solution rank and lower effective depth come with higher OOD generalization loss, so a model tuned for compactness should be expected to transfer worse to new distributions.
  • Rank-deficit bias is not a neural-collapse phase: it appears early in training, before the terminal phase, without weight decay or regularization, and it coexists with only partial satisfaction of neural-collapse conditions.
  • The effect is not specific to cross-entropy: training with MSE loss applied after softmax reproduces the same rank-deficit trends for ResNet-34 on CIFAR-100, and baseline VGG-19 is already collapsed because its default initialization produces low logits norms.
  • Architecture sets an implicit temperature: initialization scale, network width, and normalization placement all move the logits norm, so two networks trained identically can land on opposite sides of the trade-off.

Reading between the lines

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

  • Temperature scheduling—starting hot to compress and cooling late to restore OOD generalization—is the natural next experiment, and the paper explicitly leaves it open; the trade-off data suggest the two regimes' benefits could be combined in one run.
  • The same logits-norm logic should apply to the softmax inside transformer attention, where it may connect to attention entropy collapse and the need for register tokens; this is an extension, not a claim the paper makes.
  • Because the paper proves rank-2 logits can produce full-rank softmax outputs, classification capacity is not the reason models find high-rank solutions; training methods that bias toward near-rank-2 representations could compress models dramatically if the OOD-generalization cost is acceptable.
  • The observed negative correlation between OOD generalization and OOD detection may hold beyond high-temperature training, suggesting the trade-off is a property of softmax-trained classifiers rather than a temperature artifact.
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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 / 6 minor

Summary. The paper studies how the softmax temperature, and more generally the norm of pre-softmax logits, shapes the learned representations of deep classifiers. It introduces the notion of "rank-deficit bias": training with high softmax temperature or low initial logit norm yields solutions whose pre-softmax logits have numerical rank far below the Neural Collapse prediction c-1, together with compressed representations (lower effective depth) and worse out-of-distribution generalization but better out-of-distribution detection. The authors support this with experiments across MLPs, VGGs, ResNets, and ViT on CIFAR-10/100, ImageNet-100, and ImageNet-1k, and they propose a mechanism based on singular-vector alignment and logit-norm amplification. They also prove Proposition 3.1, a bound on the singular-value gap of column-stochastic matrices, and derive a theoretical minimal rank of 2 for full-rank post-softmax outputs.

Significance. If the core empirical relationship holds, this is a practically useful and fairly general finding: softmax temperature, or architectural choices that affect logit norms, can be used to trade representation compactness against OOD generalization and to improve OOD detection. The study is largely self-contained: the main correlation is tested by direct temperature and logit-norm interventions, and the experimental coverage across architectures and datasets is unusually broad. Proposition 3.1 is mathematically correct as stated, and the reported OOD generalization/detection tradeoff is a valuable empirical observation. However, the mechanistic derivation in Section 3.1 is not established for the ReLU networks tested, and several abstract-level claims overgeneralize beyond the data (notably CIFAR-10 shows no rank deficit). These issues are fixable within the scope of the manuscript, so the appropriate response is a major revision rather than rejection.

major comments (4)
  1. [Section 3.1] The collapse cascade relies on the formula sigma_1^L = product_{i=1}^{L-1} sigma_1^i for "maximally aligned networks." This identity is exact only for compositions of linear maps with aligned singular subspaces. For a ReLU network the layer map is A_i = ReLU(W_i A_{i-1}); ReLU is positively homogeneous and 1-Lipschitz but not linear, and the top singular value of the composed input-output map can be strictly smaller than the product of the layerwise weight singular values. The manuscript itself notes in Appendix D.2 that ReLU can restore full rank in deep unconstrained feature models, so the assumption that ReLU does not break the singular-vector chain is nontrivial and is not derived. As written, Sections 3.1-3.2 do not establish exponential collapse for the architectures tested; the cascade should either be labeled a heuristic or be supported by a proper analysis of ReLU compositions.
  2. [Section 3.1, Figures 3 and 10] The alignment reported in Figures 3 and 10 is the cosine similarity between singular vectors of W_i and A_i within the same layer, but the product formula used in the claimed collapse mechanism requires alignment between the dominant output direction of layer i and the dominant input direction of layer i+1. The plotted diagnostic therefore does not measure the quantity needed for multiplicative amplification, and it cannot by itself support the claim that alignment compounds across depth. The authors should either report a cross-layer alignment measure, such as alignment between the left singular vectors of W_i and the right singular vectors of W_{i+1}, or provide another direct test of the proposed cascade.
  3. [Section 3.2, Proposition 3.1 and Appendix J] Proposition 3.1 is a valid tight upper bound on sigma_1(S)-sigma_n(S), but it does not logically imply that increasing the logit norm boosts the numerical rank of the post-softmax matrix. As the column similarity r decreases, the bound tightens, which is consistent with rank increase, but a tighter upper bound alone does not force the spectral gap to shrink; the monotonic rank increase is an empirical observation from Figure 6. The text claims that the proposition shows the rank depends on column similarity and that norm amplification boosts rank; this should be rephrased so that the proposition's exact role is clear, or supplemented with a direct argument comparing ranks rather than only bounding the singular-value gap.
  4. [Table 1, Abstract, Section 7] The abstract and conclusions state that softmax networks converge to solutions of rank "much lower than the number of classes" and that this departs from Neural Collapse. However, for CIFAR-10 (10 classes) all reported SR values are 9, which equals the Neural Collapse value c-1 and is not much lower than c; Tables 7 and 8 confirm that temperature variation leaves SR at 9 on CIFAR-10. The rank-deficit phenomenon is clearly present for CIFAR-100 and ImageNet, but the universal formulation overstates the evidence. The claims should be restricted to settings with sufficiently many classes, or the authors should explain why the CIFAR-10 regime is not a counterexample to the stated definition of rank-deficit bias.
minor comments (6)
  1. [Section 2.1] The numerical-rank threshold gamma is a free parameter, and the statement "the default value of this hyperparameter proposed in PyTorch" does not define it precisely; please specify the exact tolerance and the matrix size used for each rank computation, since SR is a central quantity.
  2. [Equation (2)] The second equality in Equation (2) is attributed to singular vectors being orthonormal, but it actually follows from invariance of the Frobenius norm under orthogonal transformations; rephrasing would avoid confusion.
  3. [Abstract and throughout] There are several typographical issues, including "doessoftmax" in the Introduction, stray spaces around "softmax" in the abstract and conclusions, and inconsistent capitalization of "MSELoss" in Appendix F; a careful proofreading pass is recommended.
  4. [Table 1] The footnote "dagger shows when the rank is bounded by the size of the classifier" is not explained in the main text; for ImageNet-1k ResNet-34 the SR of 512 equals the penultimate width, so the comparison across architectures is affected by bottleneck dimensions and should be discussed.
  5. [Appendix F] The milder MSE results are attributed to "poorly designed parameters" without supporting evidence; either tune the hyperparameters or soften this explanation.
  6. [Appendix N] The limitation statement in Appendix N concedes that "a deeper theoretical analysis of the dynamics governing inner product evolution during training remains an important next step"; this is consistent with Major Comment 1 and should be reflected in the main text's characterization of the theoretical contribution.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the temperature-rank-OOD claims rest on direct empirical interventions, and the formal propositions are self-contained matrix inequalities and constructions.

full rationale

The central claims are empirical and self-contained: Table 1 and Figures 2-5, 11-16 directly vary softmax temperature or initialization logit norm and measure solution rank, effective depth, and OOD probe accuracy across several architectures; no fitted parameter is later relabeled as a prediction. The rank threshold is the PyTorch-default relative tolerance, so low logit norm does not automatically force low numerical rank by construction. The two formal results are independent: Proposition 3.1 (Appendix J) is a Gershgorin-based bound on the singular-value gap of any stochastic matrix, and Proposition K.1 explicitly constructs a rank-2 pre-softmax matrix whose scaled softmax is full-rank; neither assumes the rank-deficit phenomenon it is used to explain. The Section 3.1 product formula is a linear-algebra identity for aligned singular vectors, and Appendix N concedes that a deeper theoretical analysis of inner-product evolution is still needed, so the mechanism is admittedly underived rather than circularly assumed. Self-citations [12], [25], and [33] support background or auxiliary statements (temperature flattening, batch-norm rank preservation, intermediate-feature transfer) and are corroborated by the paper's own experiments (Figure 2, Appendix E), so they are not load-bearing. A footnote stating softmax_T(e)=softmax_1(e/(||e||T)) appears mathematically incorrect, but that is a correctness concern, not evidence that the conclusions reduce to their inputs.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The paper introduces no new physical or architectural entities. Its main conceptual contributions are the rank-deficit-bias label and the empirical temperature-control mechanism. The ledger above records the hand-chosen analysis thresholds and the two ad hoc theoretical assumptions that the mechanistic narrative depends on: linear propagation through ReLU layers and the Gram-matrix construction as a stand-in for classifier logits.

free parameters (2)
  • Numerical rank threshold gamma = PyTorch default tolerance
    Used to define solution rank SR in Section 2.1 via the number of singular values above gamma. The paper does not report the numeric value or study sensitivity to it, and all rank comparisons inherit this choice.
  • Effective-depth threshold = 99% of final-layer linear probe accuracy
    Used to define effective depth kappa in Section 2.1. The cutoff is arbitrary and not derived from theory; changing it would change the reported kappa values.
assumptions (5)
  • standard math Gershgorin circle theorem
    Used in the proof of Proposition 3.1 in Appendix J to bound the eigenvalues of S^T S.
  • ad hoc to paper Linearity of ReLU layers for the singular-value product formula
    The equation sigma_1^L = product_i sigma_1^i in Section 3.1 assumes maximal singular-vector alignment and ignores ReLU nonlinearities. This assumption is not stated or proven for the MLP, ResNet, VGG and ViT architectures used.
  • ad hoc to paper Row-normalized Gaussian Gram construction represents classifier logits
    Proposition K.1 shows that a rank-2 Gram matrix can become full-rank after row-wise softmax, but classifier logits have the form W A, not a symmetric Gram matrix. The paper uses this construction as the theoretical limit of rank-deficit bias without showing it applies to trained logits.
  • domain assumption Numerical rank of the raw activation matrix is the right proxy for representation capacity
    Appendix A.5 explicitly chooses to compute rank of the activation matrix rather than its covariance, arguing for interpretability. This choice affects all reported solution-rank values.
  • standard math Softmax temperature equivalence
    The identity softmax_T(e) = softmax_1(e/T) is used in Section 3.1 to map temperature changes onto logit-norm changes. This is mathematically trivial and does not introduce assumptions.

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Cite this review

Pith. "Pith review of Unpacking Softmax: How Temperature Drives Representation Collapse, Compression, and Generalization." pith.science (2026). https://pith.science/paper/OAYRY7AP

@misc{pith2026250601562,
  author       = {Pith},
  title        = {Pith review of: Unpacking Softmax: How Temperature Drives Representation Collapse, Compression, and Generalization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OAYRY7AP}},
  note         = {Machine review of arXiv:2506.01562}
}
read the original abstract

The softmax function is a fundamental building block of deep neural networks, commonly used to define output distributions in classification tasks or attention weights in transformer architectures. Despite its widespread use and proven effectiveness, its influence on learning dynamics and learned representations remains poorly understood, limiting our ability to optimize model behavior. In this paper, we study the pivotal role of the softmax function in shaping the model's representation. We introduce the concept of rank deficit bias - a phenomenon in which softmax-based deep networks find solutions of rank much lower than the number of classes. This bias depends on the softmax function's logits norm, which is implicitly influenced by hyperparameters or directly modified by softmax temperature. Furthermore, we demonstrate how to exploit the softmax dynamics to learn compressed representations or to enhance their performance on out-of-distribution data. We validate our findings across diverse architectures and real-world datasets, highlighting the broad applicability of temperature tuning in improving model performance. Our work provides new insights into the mechanisms of softmax, enabling better control over representation learning in deep neural networks.

Figures

Figures reproduced from arXiv: 2506.01562 by the authors.

Figure 1
Figure 1. Logits norm shapes the representations of neural networks leading to the best performance achieved at earlier layers for in distribution data when trained with low logits norm (right) compared to the baseline (left). These condensed representations trigger representation collapse, which harms generalization on out-of-distribution data. The plot presents the accuracy of linear probes attached to ResNet-34 trained on … view at source ↗
Figure 2
Figure 2. High-temperature training forces networks to increase logits norm to escape the loss symmetry induced by the softmax temperature compared to baseline. Experiment: ResNet-34 on CIFAR-100. Additional experiments in Appendix H. Breaking loss symmetry requires increasing logits norm To minimize the loss, the model must break symmetry by reducing softmax en￾tropy. This can be achieved either by decreas￾ing the temperatur… view at source ↗
Figure 3
Figure 3. (top) shows high-temperature models develop strong alignment early in the training, while baseline models exhibit negligible alignment throughout the whole training [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (22 more)
Figure 4
Figure 4. Figure 4: Rank of activations Ai bounds the rank of the gradients Gi+1, leading to low-rank gradients at deeper layers when trained with high temperature (right). Experiment: MLP trained on CIFAR-10. We now bridge our observations to Neural Collapse [13], a broader framework pre…
Figure 5
Figure 5. Figure 5: Evolution of logits rank before and after applying a softmax and logits norm before applying the softmax function. Experiment: ResNet-34 trained on CIFAR-100. In supervised classification, a neural network fθ(X) = softmaxT (M) must approximate target outputs Y ∈ R c×n.…
Figure 6
Figure 6. Figure 6: The evolution of post-softmax rank of random matrices M scaled by a con￾stant c (inverse of temperature). Each matrix M := AB⊤ ∈ R n×n , where A, B ∈ R n×k , and k = 1, . . . , 5, making M low-rank. The elements aij , bij ∼ U(−1, 1) [PITH_FULL_IMAGE:figures/full_fig_p…
Figure 7
Figure 7. Figure 7: Training with high temperature results in lower OrthoDev values compared to baseline model. Experiment: ResNet-34 CIFAR-100. Additional experiments in Appendix L. Studies have explored the relationship between Neu￾ral Collapse (NC) and a model’s ability to generalize t…
Figure 8
Figure 8. Figure 8: VGG-19 has orders of magnitude lower logit norm [PITH_FULL_IMAGE:figures/full_fig_p016_8.png]
Figure 9
Figure 9. Figure 9: VGG-19 trained with low temperature (right) reduces the effect of collapse, making deeper [PITH_FULL_IMAGE:figures/full_fig_p016_9.png]
Figure 10
Figure 10. Figure 10: reveals a striking difference in layer alignment dynamics between standard and high￾temperature training. The visualization tracks the evolution of inter-layer alignment (y-axis: training progress, x-axis: layer depth) through cosine similarity matrices of the top-15 …
Figure 11
Figure 11. Figure 11: Plot presents the impact of training with high temperature on learned representations and [PITH_FULL_IMAGE:figures/full_fig_p018_11.png]
Figure 12
Figure 12. Figure 12: Plot presents the impact of training with high temperature on learned representations and [PITH_FULL_IMAGE:figures/full_fig_p018_12.png]
Figure 13
Figure 13. Figure 13: Plot presents the impact of training with high temperature on learned representations and the model’s ability to generalize OOD. The higher the temperature, the lower the solutions’ rank found by the model. Experiment: ResNet-20 trained on CIFAR-10. The right bottom p…
Figure 14
Figure 14. Figure 14: Plot presents the impact of training with high temperature on learned representations and the model’s ability to generalize OOD. The higher the temperature, the lower the solutions’ rank found by the model. Experiment: ResNet-34 trained on CIFAR-100. 19 [PITH_FULL_IM…
Figure 15
Figure 15. Figure 15: Plot presents the impact of training with high temperature on learned representations and [PITH_FULL_IMAGE:figures/full_fig_p020_15.png]
Figure 16
Figure 16. Figure 16: Plot presents the impact of training with high temperature on learned representations and [PITH_FULL_IMAGE:figures/full_fig_p020_16.png]
Figure 17
Figure 17. Figure 17: (Top) Initialization effects on pre-training logits norm (x-axis) demonstrate a clear relationship between initialization scale and model behavior. Smaller σ values yield reduced logit norms, effectively equivalent to training with higher temperature. This relationshi…
Figure 18
Figure 18. Figure 18: Plot presents the evolution of logits norm and its impact on pre and post softmax rank leading to rank-deficit bias. When training with high temperatures, the post-softmax rank growth is triggered by the growth of the logits norm not the pre-softmax rank. Experiment: …
Figure 19
Figure 19. Figure 19: Plot presents the evolution of logits norm and its impact on pre and post softmax rank leading to rank-deficit bias. When training with high temperatures, the post-softmax rank growth is triggered by the growth of the logits norm not the pre-softmax rank. Experiment: …
Figure 20
Figure 20. Figure 20: Plot presents the evolution of logits norm and its impact on pre and post softmax rank leading to rank-deficit bias. When training with high temperatures, the post-softmax rank growth is triggered by the growth of the logits norm not the pre-softmax rank. Experiment: …
Figure 21
Figure 21. Figure 21: Applying softmax column-wise on rank-1 matrix A ∈ R 5×5 with various temperatures decreases the inner products between the columns with different indices of the highest element and increases the inner products between the columns with the same indices (bottom). The te…
Figure 22
Figure 22. Figure 22: Evolution of OrthoDev across layers for different temperatures. Consistently, networks trained with high temperature achieve low OrthoDev values. 31 [PITH_FULL_IMAGE:figures/full_fig_p031_22.png]
Figure 23
Figure 23. Figure 23: OOD generalization is negatively correlated with OOD detection. [PITH_FULL_IMAGE:figures/full_fig_p032_23.png]
Figure 24
Figure 24. Figure 24: The alignment (cosine similarity) between singular vectors of weights and represen￾tations of ResNet-18 trained on CIFAR-100 with high-temperature. 34 [PITH_FULL_IMAGE:figures/full_fig_p034_24.png]
Figure 25
Figure 25. Figure 25: The alignment (cosine similarity) between singular vectors of weights and represen￾tations of ResNet-18 trained on CIFAR-100 with temperature 1. 35 [PITH_FULL_IMAGE:figures/full_fig_p035_25.png]

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.