Why Knowing Both Hops Is Not Enough: Understanding Two-Hop Generalization in Language Models

Heng Wang; Herun Wan; Minnan Luo; Yilin Wang; Zili Zhang

REVIEW 3 major objections 6 minor 14 references

Why Knowing Both Hops Is Not Enough: Understanding Two-Hop Generalization in Language Models

T0 review · 3 major / 6 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read Two-hop reasoning fails when upper layers learn to map facts instead of reasoning over them.

desk verdict A well-designed symbolic study with a plausible layer-mismatch story and a looped-weights fix that works, but the 'complete explanation' is stronger than the causal evidence. read the letter →

arxiv 2608.07261 v1 pith:ZOGCZMTD submitted 2026-08-07 cs.CL

Zili Zhang , Yilin Wang , Heng Wang , Herun Wan , Minnan Luo This is my paper · ORCID

classification cs.CL

keywords two-hopreasoningmechanisticinterpretabilityimplicitrepresentationalignmentloopedtransformerout-of-distributiongeneralizationlogitlenstraining-inferencemismatch

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

This paper asks why a transformer can store two atomic facts perfectly yet fail to combine them into a two-hop answer. In a controlled symbolic environment, the authors find a sharp asymmetry: generalization succeeds when the second hop is in-distribution and fails when it is out-of-distribution, even though the model demonstrably knows the out-of-distribution fact. They trace the failure to a layer-wise division of labor — lower layers build a consistent internal representation of the bridge entity, but upper layers were trained on atomic facts only to map an already-formed representation to the output, not to reason over an intermediate one. The claimed consequence is that the failure is a training-inference mismatch rather than missing knowledge, and that aligning representations with lower-layer inputs, for example by sharing weights between the bottom and top layers in a looped architecture, restores out-of-distribution two-hop generalization.

What carries the argument

The load-bearing object is the bridge entity representation $\mathcal{R} = h^5_{r_1}$, the hidden state at the first-relation position of layer 5, identified by logit lens as the point where the model has resolved the intermediate entity $e_2$ before starting the second hop. Three instruments carry the argument: entity patching, which swaps $\mathcal{R}$ for a same-entity representation from another context and checks whether the answer survives; a linearity probe, which fits a linear map from intermediate to final-layer states and shows near-perfect linearity for single-hop facts but a nonlinear gap for two-hop queries; and a sliding-window attention mask, which shows upper layers barely attend to the bridge position on atomic facts yet fail catastrophically when blocked from it on two-hop queries. The proposed fix is a looped architecture in which the top four layers share parameters with the bottom four, which the authors show forces entity and relation hidden states to become compatible with input embeddings so the same reasoning circuit works on surface tokens and on intermediate representations.

What would settle it

Repeat the entity-patching and linearity-probe experiments with the anchor shifted to layers 4 and 6: if the patching success rates and the single-hop-versus-two-hop linearity gap do not follow the same pattern at adjacent layers, the layer-5 bridge representation is a correlate rather than the mechanism. A second check is to find any standard (non-looped, non-representation-trained) transformer that reaches above-chance Test-OO accuracy, which would contradict the claim that the upper layers' mapping habit is what blocks second-hop out-of-distribution generalization.

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Extended reading notes

Core claim

The paper's central claim is that two-hop generalization in a transformer is governed by where computation happens, not by whether the knowledge is stored. Using logit lens, the authors identify layer 5 as the transition where the bridge entity $e_2$ is resolved at the first-relation position, and they designate the hidden state there as the bridge representation $\mathcal{R}$. Generalization on Test-II and Test-OI tracks the alignment of this representation across contexts: substituting $\mathcal{R}$ with the same entity's representation from another context preserves the answer exactly when the model generalizes. Failure on Test-IO and Test-OO is attributed to a functional mismatch: atomic-fact training teaches the upper layers a near-linear map from a fully formed entity representation to the output space, whereas a two-hop query requires nonlinear aggregation of the bridge representation with the second relation. Linear-probing and attention-masking experiments support the distinction, and both representation-based continual training of the upper layers and a looped architecture that ties the bottom and top four layers' parameters substantially recover the failing splits, with the looped strategy carrying over to a natural-language two-hop dataset.

Load-bearing premise

The explanation rests on the logit-lens identification of layer 5's hidden state at the first-relation position as the bridge representation the upper layers actually consume; the paper never varies that anchor layer to test whether the same patching and linearity results hold elsewhere.

Editorial extensions

If this is right

The failure on Test-IO and Test-OO is not a knowledge deficit: the model answers single-hop questions about the OOD facts while systematically failing to chain them, so any effective intervention must change how layers use representations, not add facts.
Continual training of only the upper layers (6–8) on OOD atomic facts presented as hidden states lifts Test-IO/Test-OO accuracy from near zero to substantial levels, confirming that the missing capability is reasoning over intermediate representations.
A looped architecture with shared lower/upper parameters generalizes across all four splits without extra supervision, and the loop-end and bridge-layer cosine-similarity metrics rise sharply under it, evidence that representation-input alignment is the operating mechanism.
The looped results transfer to a Llama-style backbone and to a natural-language two-hop dataset extracted from Wikipedia-derived data, so the paper argues the mismatch is an architectural phenomenon rather than an artifact of the symbolic setup.
Deeper looped models generalize better and faster on the previously failing splits, indicating that the alignment strategy benefits from added capacity.

Reading between the lines

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

If the mapping-versus-reasoning diagnosis is right, the same asymmetry should appear in other compositional tasks, such as arithmetic chains or transitive relations, wherever training rewards upper layers for reading off an already-resolved operand; a cross-task replication would test the mechanism's generality.
The anchor-layer choice is the argument's most exposed point: a perturbation study that patches at layers 4 and 6 and reproduces the same success rates and linearity gap would confirm the mechanism is specific to layer 5 rather than a correlate of it.
Looped weight-sharing can be read as a soft prior that intermediate representations should resemble input embeddings; a testable extension is whether a regularization loss pulling $h^5_{r_1}$ toward the embedding of $e_2$ achieves comparable gains without tying parameters, isolating the alignment effect from other consequences of sharing.
The appendix's comparison with explicit chain-of-thought training shows explicit supervision reaches near-perfect Test-OO accuracy while looped training does not; whether multiple loop passes or combining the two closes that residual gap is a question the paper leaves open.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Short version: this is a solid mechanistic study with a plausible central explanation and an intervention that works, but the 'complete explanation' claim outruns the evidence. The split design is clean, the controlled symbolic setting is appropriate, and the convergent probes (entity patching, linearity, attention masking) all point in the same direction. The looped architecture result is the most useful single contribution: sharing bottom and top weights recovers most of the failing Test-IO/Test-OO accuracy without extra supervision. That is a real finding worth building on.

What is new: prior work documented the compositionality gap; this paper gives a layer-level account—upper layers learn to map already-formed representations to outputs on atomic facts, so they do not reason over OOD bridge representations—and a mechanism-grounded fix. The representation-consistency result for the successful splits is also well executed.

Soft spots, in proportion. The layer-5 anchor is selected via logit lens on the same model; the stress-test note is right that this is not a causal identification. The patching experiment runs on Train-II, where the model already succeeds, so it shows association between representation alignment and generalization, not that misalignment at layer 5 causes the OOD failure. The linearity divergence in Section 4.3 is a decodability result: high linear fit on single-hop facts can also mean the answer direction is already linearly readable after lower layers do the lookup, even if upper layers are doing nonlinear work. So the mapping-versus-reasoning dichotomy is plausible but not established. The attention-masking result is consistent with the story but not decisive—any two-hop mechanism needs the upper layers to look at the bridge position. The representation-based training in Section 5.1 is the strongest support, but it shows sufficiency of an adapted upper stack, not that the standard failure is located at layer 5. The bottleneck could be upstream, in the readability of the bridge representation in OOD first-hop contexts, or in attention routing. The real-data Table 2 has no standard-training baseline, so part of the gain might come from the dense-subgraph construction rather than the loop. Results are single-seed, and no code or data are shipped.

These are addressable, not fatal. The central phenomenon—asymmetric two-hop generalization—is robust, and the looped fix is genuinely interesting. The paper deserves a serious referee. I would engage, asking for an anchor-layer perturbation test, a standard baseline on the real-data looped experiments, multiple seeds, and code.

Referee Report

3 major / 6 minor

Summary. This paper studies why a from-scratch transformer trained on symbolic atomic facts (e1,r,e2) and in-distribution two-hop compositions fails to generalize to two-hop queries whose second hop is out-of-distribution. In a controlled knowledge-graph environment (2,000 entities, 200 relations, 40,000 atomic facts, ID/OOD split of 95/5, Train-II size proportional to the ID atomic count with the reported ratio φ=7.2), an 8-layer GPT-2-style model reaches near-perfect accuracy on Test-II and Test-OI but stays at chance on Test-IO and Test-OO. The authors propose a mechanistic explanation: successful generalization is driven by context-invariant bridge-entity representations that emerge at layer 5 (supported by logit-lens and entity-patching experiments), while the failures arise from a training-inference mismatch in which the upper layers, trained on atomic facts, learn to linearly map already-resolved representations to outputs rather than to reason nonlinearly over the bridge representation at the r1 position (supported by a linear-probing divergence and an attention-masking control). Two interventions, representation-based continual training of the upper layers and a looped weight-sharing architecture that aligns intermediate representations with lower-layer inputs, recover most of the lost Test-IO/Test-OO accuracy. Extension experiments cover a Llama-style backbone and a Wikidata-derived real-world two-hop dataset.

Significance. If the explanation holds, the paper makes a useful contribution: it converts a puzzling asymmetry (failure on two-hop despite full atomic knowledge) into a concrete claim about layer-wise functional specialization, and it offers a mechanism-grounded remedy with a measurable alignment signature. Strengths to credit include: held-out evaluation splits that were not used to fit the model or the measurements; entity-patching with noise-injection and position-swap controls; the correlation between patching success and Test-II/Test-OI accuracy across training (Figure 5); two independent interventions that both point to missing representation-based reasoning over OOD atomic facts (Section 5.1 and Section 5.2); and the Appendix H ablations that attempt to reproduce the looped-training effect with alternative training strategies. The main risk is that the causal chain from 'upper layers are linear mappers' to 'Test-IO/OO failures' has gaps: the linearity measurement is a decodability result, and the failing splits are never directly probed for bridge-representation quality. The interventions, not the linearity measurement, carry the weight of the diagnosis.

major comments (3)

[§4.3, Fig. 6] The linear-probing experiment establishes that the transformation h^L_r ≈ W h^ℓ_r is well fit for the r-position of single-hop atomic facts and poorly fit for the r2-position of two-hop queries, but this contrast is a decodability statement rather than a causal characterization of what the upper layers compute. For an atomic fact (e2, r2, e3), the lower layers have already performed the lookup so that the residual stream at r2 is answer-directed; a high linear fit then only says the answer direction is linearly readable, which is equally consistent with the upper layers implementing nonlinear operations. Conversely, for a two-hop query the correct answer is not locatable at r2 from the input by construction, so low linearity at r2 is expected of any successful second-hop computation, including one performed by an OOD-capable model. The reported divergence therefore does not by itself support the statement in §4.3 that 'the upper layers are trained on atomic facts to perform linear mapping, but multi-hop reasoning requires them to execute non-linear information aggregation.' The causal load is carried instead by the Appendix E masking experiment and the Section 5.1/5.2 interventions, which do support the mismatch story. To make Figure 6 load-bearing, the authors should (i) state which two-hop split is used for the two-hop curve (Train-II versus Test-IO/OO), and (ii) measure the same linearity metric on the failing Test-IO/OO splits and on the looped or representation-trained models, where the mismatch is allegedly repaired. The predicted signature is that the linearity divergence at the bridge layer shrinks for OOD second hops after the intervention.
[§4.1–§4.2, Fig. 5] The entire analysis is anchored on layer 5, which is selected as 'the bridge layer' by logit lens on the very models used for all subsequent measurements, and this anchor choice is never perturbed. Appendix D only varies the training stage; it does not vary the anchor layer in the patching, linearity, or attention-masking analyses. In addition, the entity-patching experiment (Figure 5) is run on Train-II two-hop queries, i.e., on the successful in-distribution cases; it never tests whether h5(r1) is a well-formed, causally required bridge representation in the failing Test-IO/Test-OO examples. Since the paper's central claim is about why Test-IO/OO fail, the bridge-quality measurement on those splits and a sensitivity analysis across anchor layers (e.g., repeating the patching protocol with representations taken from layers 4, 6, or 7) are needed to rule out the possibility that the patching-success and linearity metrics track a correlate rather than the actual reasoning pathway. Without such a check, the statement in §4.1 that h5(r1) 'serves as an anchor for subsequent analysis' is an assumption, not a validated fact.
[Appendix A, Table 4] The reported dataset counts are arithmetically inconsistent with the construction description. The construction yields 2,000 entities × 20 relations = 40,000 atomic facts, partitioned 95/5 into 38,000 ID and 2,000 OOD facts, but Table 4 lists 38,000 training ID plus 3,000 test ID atomic facts and 2,000 training plus 2,000 test OOD atomic facts, totaling 45,000 atomic facts. In addition, §2 states that the ID/OOD partition uses 'proportion ϕ,' whereas Appendix A defines ϕ=7.2 as the ratio of Train-II facts to ID atomic facts. These numbers must be reconciled and the definition of ϕ made consistent for the experiments to be reproduced exactly, since the claimed training dynamics and the asymmetry pattern depend on the split sizes and on the ϕ=7.2 choice.

minor comments (6)

[§5.3 and Appendix B, Table 5] Two cross-references are inconsistent: §5.3 says the alignment results explain 'the generalization observed in Section 5.3,' while the accuracy results for the looped model appear in §5.2, and Table 5 in Appendix B also attributes the looped training to 'Section 5.3' rather than §5.2.
[Table 1] The text in §5.3 claims that looped training produces 'strong alignment with lower-layer inputs' for both e2 and r2, but the Loop-End Alignment value for r2 actually decreases from 0.4121 (standard) to 0.3141 (looped); the alignment claim is only supported by the Bridge-Layer Alignment columns (0.8132 and 0.8487) and should be qualified accordingly.
[§5.1 vs. Appendix F] Section 5.1 states that the continual training is applied 'only to the upper layers (layer 5 and above),' whereas Appendix F specifies that the representation pairs (h_e2, h_r2) are fed into Layer 6 and that Layers 6–8 are trained; the layer-5-versus-layer-6 discrepancy should be resolved, since layer 5 is the layer whose representations are being consumed.
[Appendix E, Fig. 13] The rendered axis tick labels in Figure 13 appear as garbled glyph sequences, which obscures the masking-window center values; the figure should be regenerated with readable labels.
[Throughout] There are several small typos and reference-format issues: §1 contains 'Tset-II' and 'Dirven' instead of 'Test-II' and 'Driven'; several reference entries list 'and 1 others' (Wei et al., Elhage et al., Ye et al., Zhu et al., Zhao et al. 2025a); and the Radford et al. entry lacks a year and venue.
[§4.3] The claim that 'for Test-II and Test-OI, all their second hops in the form of intermediate representation have appeared during training' is an overstatement, since Train-II (273,600 examples) covers only a subset of the possible ID two-hop chains; the authors should either quantify the coverage of representational-form second hops or soften 'all' to reflect partial coverage.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main claims are supported by held-out evaluations and interventions, not by construction.

full rationale

The paper's central diagnosis—that Test-IO/OO failures arise from a training-inference mismatch in which upper layers learn linear mapping rather than representation-based reasoning—is not defined into existence. The bridge-layer anchor h5(r1) is selected by logit-lens observation on the same model, which is a localization assumption rather than a circular reduction: the paper does not define the bridge representation in terms of the failure it is meant to explain. The linear-probing experiment fits a linear map W on atomic-fact hidden states and reports high cosine similarity for single-hop and low for two-hop; this is an empirical measurement, not a fitted parameter renamed as a prediction. The entity-patching experiments are causal interventions, and the attention-masking experiment probes held-out evaluation queries; these provide independent evidence for the layer-wise story. The representation-based training intervention in §5.1 supervises upper layers on OOD atomic facts in representational form and then shows Test-IO/OO improvement; although this is a constructive test that directly supplies the hypothesized missing capability, it does not leak two-hop supervision, and it is corroborated by the looped-architecture experiment in §5.2, which improves Test-IO/OO without any such representational supervision. No load-bearing self-citation chain or uniqueness-imported-from-authors pattern is present. The weakest point is that layer 5 is chosen as the bridge layer via logit lens on the same models and then used as the anchor for subsequent measurements; varying the anchor layer would strengthen the causal claim, but this is a robustness/correctness concern, not circularity.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The central claim is driven by design choices (phi and the ID/OOD split) and by interpretability assumptions about logit lens, causal patching, and linear probes. No new physical or theoretical entities are postulated; the looped architecture is a parameter-sharing scheme rather than an invented entity. The paper's conclusions rest on these assumptions being valid for the controlled setting and transferable to real language data.

free parameters (2)

phi (Train-II ratio) = 7.2
Ratio of two-hop training samples to ID atomic facts, taken from Wang et al. (2024). It controls whether the model groks Test-II and therefore shapes the entire phenomenon; it is not swept in this paper.
ID/OOD atomic split proportion = 95% ID / 5% OOD
The assignment of atomic facts to in-distribution and out-of-distribution subsets is chosen by hand and determines the composition of all two-hop splits.

assumptions (4)

domain assumption Logit lens projections through the unembedding matrix are a valid way to localize where the model represents entities and relations.
Used to select layer 5 as the bridge entity layer in Section 4.1; if the method is unreliable, the anchor for all subsequent analysis is invalid.
domain assumption The symbolic training environment is representative of the multi-hop reasoning behavior of larger language models.
The paper extrapolates the mechanism to real-world corpora; the real-data experiments use small models on a filtered subgraph, so this assumption is only partially tested.
domain assumption Causal patching of hidden states is an appropriate intervention for testing whether a representation is used in downstream computation.
The entity patching experiment in Section 4.2 relies on this; if residual streams carry other channels, patching alone may not isolate the representation's causal role.
domain assumption Fitting a linear map h_L approximately W h_l measures whether upper layers compute linear maps, and near-linear maps imply mapping rather than reasoning.
Linear probes are a standard tool, but the interpretation that linearity implies absence of reasoning is an interpretive leap.

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Pith. "Pith review of Why Knowing Both Hops Is Not Enough: Understanding Two-Hop Generalization in Language Models." pith.science (2026). https://pith.science/paper/ZOGCZMTD

@misc{pith2026260807261,
  author       = {Pith},
  title        = {Pith review of: Why Knowing Both Hops Is Not Enough: Understanding Two-Hop Generalization in Language Models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZOGCZMTD}},
  note         = {Machine review of arXiv:2608.07261}
}

read the original abstract

Large language models (LLMs) can solve complex multi-hop problems yet exhibit puzzling failures on simple two-hop queries: although a model may correctly store each individual hop, it often fails to combine them. To understand the internal mechanisms of this phenomenon, we train transformers from scratch in a controlled symbolic environment. Our experiments reveal a pattern in two-hop generalization: models generalize reliably when the second hop follows the training distribution, but always fail when it deviates. Through mechanistic analysis, we provide a complete explanation for these distinct generalization behaviors: in settings where models generalize successfully, performance is driven by the emergence of consistent intermediate representations for the same entities across contexts, whereas failures on settings where the second hop is out-of-distribution arise from a mismatch across layers: lower layers correctly construct these intermediate representations, but upper layers, while trained on corresponding atomic facts, primarily learn to map them to outputs rather than to reason over them. Driven by this insight, we propose a recurrent-style training strategy, which enables transformers to reuse their reasoning circuitry across input forms and substantially improves generalization on out-of-distribution two-hop queries.

Figures

Figures reproduced from arXiv: 2608.07261 by the authors.

**Figure 1.** The pipeline of data construction. Training Configuration: Our training split consists of all atomic facts (ID and OOD) and Train-II. All other two-hop splits (Test-II, Test-IO, Test-OI, and Test-OO) are used for evaluation to comprehensively assess the generalization ability of the model. We train a GPT2-style decoder-only transformer (Radford et al.) with 8 layers and a hidden dimension of 768. The model is opt… view at source ↗

**Figure 2.** Training dynamics of a transformer trained on all atomic facts and in-distribution two-hop compositions. [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** Logit lens reveals a progressive reasoning [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

Figures from the paper (16 more)

**Figure 4.** Figure 4: (Left) A two-hop query composed of two single-hop facts, where the bridge entity e2 connects (e4, r4) and (e2, r3) to yield the target entity e3. (Right) The entity patching procedure: the hidden representation of the bridge entity e2 at layer 5 is replaced with an alt…

**Figure 5.** Figure 5: ) ensures that patching success stems from representation alignment rather than the model ignoring the position. 2To rule out e3 being pre-encoded at r3 in early training, we replace r3 with that from another two-hop query whose answer is not e3, so predicting e3 must…

**Figure 6.** Figure 6: High cosine similarity and low MSE from layer 5 onward in single-hop queries indicate that upper layers act as linear mappers, whereas the lower linearity in two-hop queries suggests nonlinear information aggregation at the bridge layer. 4.3 Why Second-Hop OOD Fails: …

**Figure 7.** Figure 7: (Left) In a two-hop query (e1, r1, r2)→e3, lower layers first form the bridge entity representation e2 at r1. Upper layers must then combine Re2 with r2 to infer e3. (Right) For a single-hop fact (e2, r2, e3), the representation of e3 is already formed at r2 in lower l…

**Figure 8.** Figure 8: (a) Continual training of upper layers on atomic facts presented in a representational form. We only depict the process of continual training; the standard training process is the same as in [PITH_FULL_IMAGE:figures/full_fig_p007_8.png]

**Figure 9.** Figure 9: Histograms of the set size distributions for the [PITH_FULL_IMAGE:figures/full_fig_p013_9.png]

**Figure 10.** Figure 10: Illustration of the local graph structure cen [PITH_FULL_IMAGE:figures/full_fig_p013_10.png]

**Figure 11.** Figure 11: Relationship between pairwise set overlap and representation similarity for the four attribute sets [PITH_FULL_IMAGE:figures/full_fig_p014_11.png]

**Figure 12.** Figure 12: Logit lens results at different stages of the standard training process. (a) Memorization stage (checkpoint [PITH_FULL_IMAGE:figures/full_fig_p015_12.png]

**Figure 13.** Figure 13: Accuracy of predicting the target entity un [PITH_FULL_IMAGE:figures/full_fig_p016_13.png]

**Figure 15.** Figure 15: Logit lens of the final model trained under the [PITH_FULL_IMAGE:figures/full_fig_p017_15.png]

**Figure 16.** Figure 16: Comparison of two training strategies described in Appendix [PITH_FULL_IMAGE:figures/full_fig_p018_16.png]

**Figure 17.** Figure 17: Training results for the two training strategies described in Appendix [PITH_FULL_IMAGE:figures/full_fig_p019_17.png]

**Figure 18.** Figure 18: Test accuracy during training on looped transformers with varying model scale, controlled by the number [PITH_FULL_IMAGE:figures/full_fig_p020_18.png]

**Figure 19.** Figure 19: Visualization of how a model trained with a looped architecture transfers embedding-based atomic fact [PITH_FULL_IMAGE:figures/full_fig_p020_19.png]

**Figure 20.** Figure 20: Layerwise cosine similarity heatmaps comparing models trained under a recurrent architecture (left [PITH_FULL_IMAGE:figures/full_fig_p024_20.png]

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Reference graph

Works this paper leans on

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Pith tools

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