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REVIEW 4 major objections 7 minor 45 references

Understanding Human-like Solutions in Combinatorial Optimization via Learning and Search

T0 review · 4 major / 7 minor · reviewed 2026-07-30 · grok-4.5

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Pith's one-line read Human-like TSP tours emerge from optimal pretraining, RL fine-tuning, and Best-of-N search—not from copying optima or pure heuristics.

desk verdict Large human TSP corpus plus a clean SFT→RL→Best-of-N comparison; the ranking is real but the margin over pure optimal is thin and the cognitive-mechanism claim outruns the evidence. read the letter →

arxiv 2607.23854 v1 pith:URZ42X6K submitted 2026-07-26 cs.AI

Haijiang Yan , Jian-Qiao Zhu , Liqiang Huang , Ming Meng This is my paper
classification cs.AI
keywords CombinatorialOptimizationTravelingSalesmanProblemPointerNetworkHumanSolvingReinforcementLearningTest-timeSearchGeometricFeaturesCognitiveModeling
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

People solve Euclidean traveling salesman problems quickly and near-optimally even though the search space is combinatorial. This paper asks what geometric signature makes a tour human-like and what learning-plus-search recipe can produce it. On a new dataset of 150,000 instances and more than 20 million human tours, human solutions sit in a near-optimal geometric basin: they share many structural properties with true optima while keeping systematic, milder suboptimalities. Among models that never see human data, a Pointer Network pretrained on optimal tours, then fine-tuned by reinforcement learning on tour length, and decoded by Best-of-N sampling, matches human tours about as well as a network trained directly on humans. Classic geometric heuristics fall short of even a random human baseline. The claim is that human-like combinatorial solutions can arise from structured imitation of high-quality examples, reward-driven refinement, and modest test-time search under capacity limits—the same stack that powers many modern AI systems.

What carries the argument

Pointer Network policy distributions over valid tours, trained under four regimes (optimal supervised, pure RL, optimal-then-RL, human supervised) and decoded by greedy, beam, or Best-of-N sampling; human-likeness is scored by five tour-similarity metrics plus an eight-feature geometric basin (smoothness, turning entropy, edge variance, nearest-neighbor rank, convex-hull preservation, 2-opt, optimality gap, crossings).

What would settle it

On held-out TSP instances of similar size, show that another decoding or learning regime (for example pure RL with larger N, or a curriculum that never sees optimal labels) matches or exceeds the optimal+RL Best-of-N model on the same aggregate score and geometric-basin correlation, or that humans trained only on suboptimal feedback never acquire the near-optimal basin the model relies on.

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Extended reading notes

Core claim

Human Euclidean TSP tours occupy a near-optimal geometric basin rather than coinciding with optima. The computational account that best recovers those tours without ever training on human solutions is a Pointer Network pretrained by supervised learning on optimal tours, fine-tuned by RL on negative tour length, and decoded by Best-of-N sampling; that combination approaches the empirical upper bound set by a network trained directly on human tours.

Load-bearing premise

Matching the paper’s chosen similarity metrics and geometric features under this fixed architecture and Best-of-N decoding is taken as evidence that human TSP cognition itself implements the same supervised-plus-RL-plus-test-time-search stack.

Editorial extensions

If this is right

  • Human-like combinatorial solutions need not be hand-crafted heuristics; they can be the product of learned proposal distributions plus limited test-time search.
  • Supervised exposure to high-quality (even if not human) demonstrations followed by RL can place a policy in the same geometric region humans occupy.
  • Test-time compute (Best-of-N or similar) is a necessary modeling ingredient for human variability and quality, not an optional post-process.
  • Classic local geometric heuristics are insufficient models of human TSP once large-scale data and neural baselines are available.
  • The same three-stage recipe may transfer to other everyday combinatorial tasks that share route, schedule, or ordering structure.

Reading between the lines

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

  • If humans never see true optima at training scale, the pretrained policy must ultimately be explained by curriculum or meta-learning on simpler instances whose near-optima are discoverable—exactly the open direction the discussion flags.
  • Best-of-N at the first city offers a concrete process model for the long initial reaction times the behavioral data show, linking deliberation time to sample size.
  • The geometric-basin score itself could become a reusable diagnostic for whether other cognitive models of planning land in the human region rather than the pure optimum.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 7 minor

Summary. The manuscript introduces tsp150k, a large-scale behavioral dataset of human Euclidean TSP solutions (150,000 instances, >20M tours from 1,107 participants across five complexity levels), and uses it to evaluate a broad space of computational models: exact search (Concorde), classical heuristics (Nearest Neighbor, Convex Hull Cheapest Insertion, Largest Interior Angle), Elastic Net, and PointerNet policies trained under four objectives (supervised on optimal tours, RL, supervised+RL fine-tuning, supervised on human tours) each decoded three ways (greedy, beam, Best-of-N). Human tours are shown to be near-optimal (≤11% gap on average; best human tours within 0.3% of optimal) but systematically distinct, occupying a "near-optimal geometric basin" characterized by eight geometric features. The central modeling claim is that, among models never trained on human tours, the optimal-pretrained + RL-fine-tuned PointerNet decoded by Best-of-N (N=100) produces the most human-like tours, matching the human-trained upper bound on the aggregate tour-similarity score (0.600 vs 0.600, Table 2) and best reproducing the geometric feature profile (Fig. 3). The authors interpret this as evidence that human-like TSP solutions emerge from supervised structure learning + RL + test-time search.

Significance. If the results hold, the paper makes two real contributions. First, tsp150k is by far the largest human TSP corpus and is a genuinely valuable resource; the stylized facts in Appendix B (first-step deliberation RT, stable stepwise local optimality) are novel and interesting in their own right. Second, the modeling comparison is unusually broad and methodologically careful in several respects: a 50k-instance held-out test set, five independent tour-level similarity metrics plus eight geometric features, strong baselines including an exact solver, an explicitly labeled human-trained upper bound rather than an overclaimed mechanism, and fully specified training/decoding protocols in Appendices C–D. The qualitative finding — that a policy with broad support over near-optimal tours plus length-based test-time selection reproduces human-specific suboptimalities that the exact solver cannot — is a substantive and falsifiable claim about the structure of human solutions. That said, the headline quantitative claim ("best account," "matches the upper bound") rests on margins of ~0.008 aggregate points over parameter-free baselines, so its force depends on robustness analyses the current vers

major comments (4)
  1. [§3, Table 2 and note; Appendix D.1, Eq. (D14)] The central ranking claim ("the best account of human tours") hinges on an aggregate score that is an equal-weight mean of five min–max-normalized metrics, and the champion's margin is razor-thin: p_optimal+RL + Best-of-N(100) = 0.600 vs. Elastic Net 0.593 and Concorde 0.592 — a total spread of 0.008 across the top cluster. Moreover, on the most structural metric (EOR), Concorde (0.580) actually beats the champion (0.576), and LCS is tied (0.488); the champion's aggregate advantage comes entirely from LD, FD, and Kendall's τ. The note asserts standard deviations are "close to zero" but reports no per-instance paired comparisons, no confidence intervals, and no sensitivity analysis over the aggregation rule (e.g., rank-based aggregation, z-scoring, or alternative weightings). Because the min–max ranges are computed over the particular model set evaluated, the aggregate is also model-set-d
  2. [§3–4, Table 2] All reported results pool the five complexity levels, yet the metrics scale very differently with city count: LCS is divided by city count, LD grows with n, and FD is an absolute spatial quantity. The relative ranking of models plausibly varies across levels (e.g., at 10 cities humans are within 6.2% of optimal and optimal-tour overlap is high, so Concorde may dominate there, while at 24 cities human-specific deviations are largest). A per-complexity breakdown of Table 2 (or at least of the aggregate and EOR) is needed to establish that the champion's advantage is uniform rather than driven by a subset of levels, and to make the cognitive claim (which concerns human behavior at all difficulties) properly supported.
  3. [§4, Fig. 3; Appendix D.2, Eq. (D15)] The geometric-basin analysis uses beam-searched tours from p_human as the "human-like geometric benchmark" rather than the actual human tours in tsp150k. This introduces a circularity risk: the benchmark is itself the output of a fitted model (the same architecture family as the candidates being evaluated), so alignment with it may partly reflect shared PointerNet inductive biases rather than alignment with human geometry. The natural and available control is to compute the eight features directly on held-out human tours and re-run the Fig. 3 comparison against that empirical benchmark; if the champion's advantage (r=0.583, MAE=0.021) survives, the claim is much stronger. Relatedly, the Geometric Basin Score is min–max normalized across the evaluated models (Eq. D15), so the score changes if a model is added or removed; the normalization set should be stated and justified, or a model-ind
  4. [§4 (final paragraph) and §5 (Discussion)] The interpretive bridge from the winning pipeline to human cognition — supervised pretraining on optimal tours as an analogue of human learning — is the paper's weakest inferential step. Humans never observe optimal demonstrations at training scale, as the authors themselves acknowledge ("Humans are unlikely to directly observe optimal solutions... at the scale used in our supervised training"). The Discussion offers curriculum/meta-learning as future work, but the main text (e.g., "people may acquire broad geometric knowledge about good TSP solutions through exposure to high-quality or approximately optimal examples") states the analogy more strongly than the evidence supports. A concrete, in-scope test would materially strengthen the paper: train the supervised stage on data humans plausibly have — e.g., the best human tours per instance (which are within 0.3% of optimal per Appendix B
minor comments (7)
  1. [§3, §4] "PointNet" appears several times where "PointerNet" is meant (e.g., "the best PointNet model," "the PointNet policy of p_optimal+RL"). PointNet is a different, well-known architecture; this should be corrected throughout.
  2. [Table 2 note] The note says the gray row "indicates the best-performing PointerNet model," but p_human beam is best only on the aggregate and some metrics; the bold/gray conventions and the fact that the gray row is the human-trained upper bound could be stated more plainly. Reporting SDs or CIs in a supplementary table (even if small) would be preferable to omitting them entirely.
  3. [§3, metric definitions] Typo: "(e,g., two tours that share similar spatial trajectories...)". Also, the EOR/LCS/LD examples are helpful but the FD intuition ("corresponding points as the two tours are traversed") is compressed; a pointer to the formal definition in Table D4 would help readers.
  4. [Appendix C.5.3] Temperature-scaled sampling is introduced (Eq. C9) but the main results all use T=1; please state explicitly that T=1 was used for all reported Best-of-N results and whether any temperature exploration was done, since T interacts with the diversity argument made in §4.
  5. [References[7], [15], [17], [18]] Several entries have placeholder fields ("Princeton University Press, ???", "PMLR, ???", "MIT Press, ???"). Please complete the venue/publisher fields.
  6. [Fig. 3] The figure is dense: panel (b)'s 22-column numeric block is hard to parse in the rendered form, and the radar plots would benefit from a legend identifying which line corresponds to which N value. Consider moving the full numeric table to the appendix and showing only MAE/r summaries in the main figure.
  7. [Appendix A.4] The time limit (2× number of cities in seconds) is a potentially important design feature for interpreting human suboptimality and the deliberation account; it would be worth noting in the main text (§1 or Appendix B discussion) that human tours were produced under this constraint, since it bears on the interpretation of the first-step RT result.

Simulated Author's Rebuttal

4 responses · 0 unresolved

We thank the referee for a careful, constructive report that correctly identifies both the strengths of the work (the tsp150k resource and the breadth of the model comparison) and the four points where the current version overstates or under-supports its claims. We agree that the headline ranking claim rests on thin margins under one particular aggregation rule, that pooled results may mask heterogeneity across complexity levels, that the geometric benchmark introduces a potential circularity, and that the cognitive interpretation of supervised pretraining on optimal tours is the weakest inferential step. In the revision we will (i) add paired per-instance statistics, confidence intervals, and aggregation-rule sensitivity analyses; (ii) report a full per-complexity breakdown of Table 2; (iii) re-run the Fig. 3 analysis against features computed directly on held-out human tours; and (iv) train the suggested control model supervised on best-human tours, and soften the cognitive framing in the main text. We believe these revisions will either strengthen the conclusions substantially or, where the data do not support them, lead us to qualify them honestly.

read point-by-point responses

  1. Referee: Central ranking claim hinges on an equal-weight mean of min-max-normalized metrics with a razor-thin margin (0.600 vs 0.593/0.592); Concorde beats the champion on EOR and ties LCS; no per-instance paired comparisons, CIs, or aggregation-rule sensitivity; aggregate is model-set-dependent.

    Authors: The referee is right that the 0.008 aggregate spread cannot, by itself, carry the 'best account' claim, and that the min-max ranges being computed over the evaluated model set makes the aggregate non-portable. We also acknowledge that the note's 'standard deviations close to zero' refers to standard errors of the mean over 50k instances and does not speak to per-instance paired differences, which is the relevant comparison. In the revision we will: (1) report paired per-instance comparisons (champion vs. Concorde and vs. Elastic Net) for each metric, with bootstrap confidence intervals over instances on the difference in means, so the significance and effect size of each margin is explicit; (2) add a sensitivity analysis over aggregation rules: rank-based aggregation across models, z-scored aggregation, equal-weight mean as reported, and a leave-one-metric-out analysis showing whether the champion's top ranking survives any single metric's removal; (3) state explicitly that the aggregate is defined relative to the evaluated model set and refrain from treating its absolute value as meaningful. If the champion's advantage over Concorde/Elastic Net is not robust across aggregation rules, we will downgrade the claim to 'among the best accounts' and emphasize the geometric-feature analyses, where the champion's advantage (highest r on all eight features, lowest MAE on seven) is larger and arguably more diagnostic than the aggregate tour-similarity score. revision: yes

  2. Referee: All results pool five complexity levels, yet metrics scale differently with city count; ranking plausibly varies across levels. A per-complexity breakdown of Table 2 (at least aggregate and EOR) is needed.

    Authors: We agree; pooling was a presentational choice that obscures exactly the heterogeneity the referee describes, and the cognitive claim concerns behavior at all difficulty levels. The revision will include a full per-level version of Table 2 (five levels × all models, with the aggregate, EOR, and the other metrics), plus a per-level version of the paired comparisons requested in the previous point. We will report whether the champion's advantage is uniform, and where it is not — e.g., we share the referee's expectation that at 10 cities, where optimal-tour overlap is high, Concorde will be highly competitive — we will say so plainly and interpret the result accordingly (the near-optimal-basin account predicts the champion's distinctiveness should grow with complexity, which is itself a testable and interesting pattern). revision: yes

  3. Referee: Geometric-basin analysis uses beam-searched p_human tours as benchmark, risking circularity (shared PointerNet inductive bias). Natural control: compute eight features on held-out human tours directly. Also, Geometric Basin Score normalization is model-set-dependent.

    Authors: This is a fair and important criticism, and the proposed control is entirely feasible with our data — the eight features are computed directly from tours, so nothing prevents applying them to held-out human tours. In the revision we will recompute the benchmark from the actual human tours in the held-out test set (averaging features across the ~139 tours per instance, as already done for Fig. 1b) and re-run the full Fig. 3 comparison against this empirical benchmark. The p_human beam-search benchmark will be retained only as a secondary, clearly-labeled model-based reference. We will also state the normalization set for Eq. (D15) explicitly and add a robustness check in which the basin score is recomputed with fixed normalization ranges (e.g., ranges from the human-tour distribution) so that adding or removing a candidate model cannot change existing models' scores. We note one partial mitigation already in the paper: the champion is not the architecture most similar to p_human in training objective (p_optimal and p_RL share the architecture too), so shared architecture alone does not explain the champion's advantage; but we agree only the empirical benchmark fully answers the circularity concern. If the r=0.583/MAE=0.021 advantage does not survive against real human tours, we will revise the geometric-basin claims accordingly. revision: yes

  4. Referee: Interpretive bridge from supervised pretraining on optimal tours to human learning is the weakest inferential step; humans never observe optimal demonstrations at scale. Suggested test: train the supervised stage on data humans plausibly have, e.g., best human tours per instance (within 0.3% of optimal).

    Authors: We agree this is the paper's weakest step, and we are grateful for the concrete suggestion, which is directly implementable: tsp150k contains, for each instance, the best human tour, which is within 0.3% of optimal. In the revision we will train an additional PointerNet supervised on best-human tours per training instance (optionally the best-k human tours to match demonstration counts), optionally with RL fine-tuning, and evaluate it in the same framework. This model is supervised on data humans demonstrably generate — not data they observe, but a far more psychologically plausible stand-in than Concorde solutions, effectively operationalizing 'learning from one's own best past solutions.' If this pipeline matches or exceeds p_optimal+RL, the cognitive interpretation becomes substantially better supported; if it underperforms, that is itself informative about what the optimal-pretraining stage contributes. Independently of the outcome, we will soften the main-text language: the sentence about acquiring knowledge 'through exposure to high-quality or approximately optimal examples' will be reframed explicitly as an analogy with stated limitations, and the Discussion's curriculum/meta-learning caveat will be moved earlier and given more prominence. revision: yes

Circularity Check

1 steps flagged · score 1.0 of 10

Empirical model comparison against held-out human tours; only a mild benchmark-proxy step in the geometric analysis, not a by-construction reduction of the main claim.

  1. other [Section 4 (Characterizing Geometric Features); Fig. 3 caption]
    "Here, the geometric properties of beam-searched tours from p_human(π|x) served as the empirical benchmark for human-likeness. We then compared the corresponding geometric features of tours generated by the two most human-like search algorithms (i.e., Concorde and Elastic Net) and the PointerNet policies decoded using Best-of-N sampling against this benchmark"

    The geometric ‘human-like’ target is defined as the geometry of a PointerNet trained on human tours and beam-decoded, then other models are scored by MAE/r against that proxy. This slightly folds the human-trained model into the definition of the geometric benchmark rather than using only raw human-tour features. It is not a by-construction reduction of the main claim: Table 2 still compares models directly to human tours, p_human is labeled an upper bound, and different policies/N still separate cleanly on the eight features.

full rationale

The paper’s load-bearing claim is comparative and empirical, not a first-principles derivation: among policies never trained on human tours, p_optimal+RL decoded by Best-of-N(100) best matches held-out human solutions on tour-level metrics (Table 2) and on eight geometric features (Fig. 3). Those metrics are computed between model tours and actual human tours on a held-out test set; p_human is explicitly an upper bound, not the claimed mechanism. Training objectives (supervised optimal, RL, optimal+RL) and decoding procedures do not embed human tour geometry by definition, so the champion’s alignment is not forced by construction. The only mild circularity-adjacent move is using beam-searched p_human tours as the ‘human-like geometric benchmark’ in §4 rather than raw human-tour geometry alone; that is a proxy choice for the geometric panel, not a self-definitional reduction of Table 2 or of the main claim. No fitted parameter is renamed as a prediction, no uniqueness theorem is imported from overlapping authors, and self-citations are ordinary background. Score 1 reflects that single non-load-bearing proxy step; the central result remains independently evaluated.

Assumptions & free parameters 5 free parameters · 6 assumptions · 3 invented entities

The work is empirical modeling, not a derivation from axioms. Load-bearing commitments are standard TSP geometry, the adequacy of PointerNet policies plus three decoders as the hypothesis class, the chosen similarity/geometry feature set as the definition of human-likeness, and the analogy from model stack to human learning despite humans lacking optimal demonstration oracles.

free parameters (5)
  • PointerNet width (embed/hidden dim 128, single-layer LSTMs) = 128
    Architecture capacity chosen by authors; affects how much probability mass spreads over near-optimal tours and thus Best-of-N gains.
  • Best-of-N sample size N and beam width B = N in {1,10,100}; B=100
    Test-time compute knobs that directly move models toward the human/optimal basin; N=100 and B=100 are selected reporting points.
  • PPO and supervised training hyperparameters = as in Appendix C.5.2
    Learning rate 1e-4, batch/rollout sizes, clip ε=0.2, entropy coef 0.01, 50k SL steps / 200k or 10k RL iterations, grad clip 1.0—chosen to train policies whose samples are later compared to humans.
  • Min–max normalization ranges for aggregate and geometric basin scores = per-metric min/max over evaluated models
    Composite scores depend on min/max over the included model set, so absolute score levels are relative to that menu of models.
  • Attention temperature T at decode = T=1 (training and Best-of-N)
    Controls exploration of the tour distribution; training uses T=1, Best-of-N reported at T=1.
assumptions (6)
  • standard math Euclidean TSP tour quality is total closed path length under L2 distance; valid tours are permutations with return to start.
    Formalized in Appendix C.1; used for rewards, Concorde optima, and Best-of-N selection.
  • domain assumption Concorde returns globally optimal tours for n≤24 instances used here.
    Appendix C.2; all optimality gaps and supervised optimal targets depend on this.
  • domain assumption A PointerNet autoregressive pointing policy is an adequate hypothesis class for human tour distributions.
    §2 and Appendix C.5; alternative cognitive process models are not given equal learning+search treatment.
  • ad hoc to paper Tour-level metrics (EOR, LCS, LD, Fréchet, Kendall τ) plus eight geometric features define human-likeness for model selection.
    §3–4 and Appendix D; different feature menus could reorder models.
  • ad hoc to paper Best-of-N sampling is a reasonable computational proxy for human test-time deliberation (motivated by long first-step RT).
    Appendix B.e and §5; not identified from process-tracing beyond RT pattern.
  • domain assumption Participants’ online WeChat game tours under time caps are representative human TSP solutions for cognitive modeling.
    Appendix A; convenience sample, Chinese WeChat users, paid/gamified setting.
invented entities (3)
  • Near-optimal geometric basin (Geometric Basin Score)
    purpose: Summarize human-like tour geometry as a region in an eight-feature space and rank models by proximity to human benchmark tours.
    Constructed in §4 and Appendix D.2 by normalizing and averaging eight features; useful descriptive aggregate but not an independently measured latent in the brain or in prior TSP theory.
  • Eight geometric feature suite (G_sm, G_tae, G_elv, G_nnr, G_ch, G_og, G_2opt, G_cr)
    purpose: Decompose why human tours differ from optimal/heuristic/PointerNet tours.
    Features draw on known TSP geometry ideas but the specific battery and instance-dependent vs independent split are paper-defined evaluation tools.
  • tsp150k human TSP corpus independent evidence
    purpose: Provide large-scale behavioral ground truth for model comparison.
    New dataset from the authors’ experiment; existence is empirical, not a theoretical postulate, but access is not evidenced as public in the manuscript text.

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Pith. "Pith review of Understanding Human-like Solutions in Combinatorial Optimization via Learning and Search." pith.science (2026). https://pith.science/paper/URZ42X6K

@misc{pith2026260723854,
  author       = {Pith},
  title        = {Pith review of: Understanding Human-like Solutions in Combinatorial Optimization via Learning and Search},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/URZ42X6K}},
  note         = {Machine review of arXiv:2607.23854}
}
abstract

Humans often find good solutions to combinatorial optimization problems that are computationally hard even for advanced computer algorithms. In the Euclidean traveling salesman problems (TSP), people rapidly produce tours that are near-optimal, despite severe limits on time and computation. What makes a tour human-like, and how might such solutions be learned? Here we address these questions through a large-scale behavioral and computational investigation of human performance in Euclidean TSP. We sampled a broad space of TSP instances, collected human solutions, and compared them with neural policies based on Pointer Networks, which are recurrent neural networks with an attention-based pointing mechanism that define probability distributions over valid tours. We trained these networks under multiple objectives, including reinforcement learning (RL), supervised learning from optimal tours, supervised learning from human tours, and RL fine-tuning after optimal-supervised pretraining. Human tours were not identical to optimal tours, but occupied a near-optimal geometric basin: they shared many structural properties with optimal solutions while preserving systematic human-specific deviations. The best account of human tours was not direct imitation of optimal tours, but a model pretrained on optimal tours, fine-tuned by RL, and decoded through $\text{Best-of-}N$ sampling. These findings suggest that human-like solutions may emerge from a combination of structured supervised learning, RL, and test-time search, echoing computational principles underlying many modern artificial intelligence systems.

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Pith tools

Reviewed July 30, 2026 · model on record in the stance chip above.