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Shared circuits predict whether LLMs generalize across formats in arithmetic reasoning

Andrea Gregor de Varda, Sana Pandey, Pengrui Han, Jacob Andreas, Evelina Fedorenko

arXiv:2609.04463v1cs.CLcs.AI

TL;DR

The paper asks whether internal mechanisms can predict LLM generalization across arithmetic formats, since behavioral evaluations cannot cover all equivalent variations. Using attribution patching, it finds that circuit overlap tracks verbal-format difficulty and item correctness, matching supervised probes without labeled data.

  • Problem

    The paper asks whether models generalize across input formats, since behavioral evaluation cannot test every equivalent variation or identify the internal mechanism behind a correct answer.

    Source: §1 · p. 1

  • Method

    Attribution patching compares each model’s own answers to arithmetic prompts and sign-flipped counterparts to localize units favoring the original prompt.

    Source: §2.3 · p. 2

  • Results

    Circuit overlap tracks verbal-format difficulty and item correctness, matching supervised probes despite requiring no labeled data.

    Source: §1 · p. 2, §4 · p. 5

  • Takeaways & Limitations

    Generalization of behavior is grounded in circuit generalization: without numeric-circuit engagement models almost always fail, but engagement does not guarantee success.

    Source: §4 · p. 4

    Decision-point entropy explains as much or more variance than circuit loading, especially in lower-resource formats, though circuit loading contributes unique variance in all three.

    Source: Limitations · p. 5

Abstract

from arXiv · show

In many forms of reasoning, including arithmetic reasoning, generalizing across superficial changes in input format is effortless for humans: anyone who can solve 2+5 can also solve 'two plus five'. In contrast, LLMs are more brittle to surface variations of the prompts: for example, they solve numeric arithmetic problems almost perfectly but are substantially less accurate on verbal renditions of the same problems. Here, we ask whether generalization across formats can be predicted from the models' internals. Using attribution patching, we first independently localize the circuit that each model recruits to solve numeric arithmetic problems (2+5) vs. verbal ones, in three languages: English ('two plus five'), Spanish ('dos m\'as cinco'), and Italian ('due pi\`u cinque'); then, we test whether overlap with the model's own numeric circuit predicts its generalization to the verbal formats. Indeed, we find support for this idea at three levels: circuit overlap accounts for the relative difficulty of the three verbal formats, for which models generalize best, and for which items are solved correctly, rivaling supervised probes while requiring no labeled data.

1 Introduction

The paper asks whether internal mechanisms can predict LLM generalization across arithmetic formats, since behavioral evaluation cannot cover every equivalent variation. It finds that overlap with a numeric circuit tracks verbal-format difficulty and item correctness, matching supervised probes without labeled data.

1 Introduction

  • Behavioral performance alone cannot establish the mechanism behind an answer or cover every equivalent variation of a task.The paper therefore asks whether internal mechanisms can help predict generalization to new formats.Source: §1 · p. 1
  • The study tests whether reliance on the same internal circuit predicts generalization from numeric to verbal arithmetic formats.It localizes units for numeric, English, Spanish, and Italian arithmetic with attribution patching across 13 models from six families.Source: §1 · p. 1
  • Circuit overlap accounts for verbal-format difficulty and item correctness, matching supervised probes despite requiring no labeled data.The analysis uses the numeric circuit as a reference for predicting behavior in verbal formats.Source: §1 · p. 2, Figure 1 · p. 2

2 Methods

The methods build a balanced arithmetic dataset, evaluate 13 language models, and use attribution patching to compare format-specific circuits with each model’s numeric circuit.

2.1 Dataset

  • Dataset items are positive-integer addition or subtraction problems, balanced across operand length, term count, and carry requirement.Each item has a sign-flipped counterpart and is rendered in numeric, English, Spanish, and Italian formats; analyses use 2,000 items.Source: §2.1 · p. 2

2.2 Models

  • The study evaluates 13 base language models spanning six families and 0.6B–32B parameters, with one-shot prompts and greedy decoding.Each model receives an in-context exemplar in the target format.Source: §2.2 · p. 2

2.3 Attribution patching

  • Attribution patching compares each model’s own answers to original and sign-flipped prompts to localize units favoring the original answer.This avoids relying on gold answers and remains defined when a model is usually wrong; items with unequal token lengths or identical outputs are excluded.Source: §2.3 · pp. 2–3
  • The numeric circuit is each model’s top 1% of units by numeric-format attribution; verbal-format overlap and item loading are measured against it.Overlap uses top-1% Jaccard, while item loading sums attribution scores on units in the numeric circuit.Source: §2.3 · p. 3

3 Results

Across arithmetic formats, accuracy declines from numeric to verbal problems, and overlap with the numeric circuit tracks format-level and item-level performance. Circuit loading also predicts item correctness beyond trained probes and confidence measures.

3.1 Accuracy varies dramatically with format

  • Median accuracy fell from 87.4% numeric to 36.4% English, 17.9% Spanish, and 6.8% Italian; Numeric > English ≥ Spanish ≥ Italian held in every model.Source: §3.1 · p. 3

3.2 Circuit overlap predicts format- and model-level accuracy

  • Mean top-1% Jaccard overlap declined from English (0.145) to Spanish (0.078) to Italian (0.069).Overlap predicted accuracy across models in English (r = 0.75, p = 0.003) and Spanish (r = 0.58, p = 0.04), but the Italian trend was not significant (r = 0.47, p = 0.10).Source: §3.2 · p. 3

3.3 Circuit loading predicts item accuracy

  • Correct items had higher numeric-circuit loading in adequately sampled model-format combinations; without loading, accuracy was near zero, but loading did not ensure success.Activation patching confirmed causal involvement; other source formats also predicted targets when item-set accuracy was balanced, although verbal sources were impractical because their accuracy was generally low.Source: §3.3 · pp. 3–4

3.4 Circuit loading predicts accuracy beyond probes and confidence

  • 11.0% vs. 11.9% (English loading vs. entropy); loading remained significant in every format and matched or exceeded both probes.In Spanish and Italian, entropy accounted for more variance than circuit loading.Source: §3.4 · p. 4, Figure 4 · p. 4

4 Discussion

The discussion frames cross-format generalization as grounded in circuit generalization, while emphasizing that circuit engagement predicts failure more confidently than success. It relates this approach to prior work and notes its potential use beyond arithmetic.

4 Discussion

  • If a model does not engage the numeric circuit, it almost always fails on verbal formats; engagement makes success more likely but does not guarantee it.Source: §4 · p. 4
  • This extends Sun (2025) from circuit stability across arithmetic subtasks to surface-format changes and individual-problem predictions.Source: §4 · p. 4
  • A high-performing format can provide a reference circuit for weaker formats in other domains, including programming languages and cross-language question answering (Ahuja et al., 2023).Source: §4 · pp. 4–5
  • The method requires no supervision or ground truth; knowing which format is strong is sufficient, and its metric predicts on par or better than probes trained on thousands of examples.Source: §4 · p. 5

Limitations

The authors report that they lack a theory explaining why circuit overlap predicts generalization or when that relationship holds. Their circuits identify engaged units, not the algorithms those units implement, and circuit loading is not always the strongest correctness predictor.

Limitations

  • The authors cannot yet explain why circuit overlap predicts generalization or identify the conditions under which that relationship holds or breaks.They present the empirical finding as a basis for developing a theory grounded in internal computations.Source: Limitations · p. 5
  • Because the circuits identify engaged units rather than the computations they support, the analysis offers statistical prediction rather than formal guarantees of correctness.The authors say linking implementation-level components to algorithmic-level procedures would be necessary for formal guarantees.Source: Limitations · p. 5
  • Decision-point entropy explains as much or more variance than circuit loading, especially in lower-resource formats, but loading contributes significant unique variance in all three formats.Entropy indicates uncertainty, whereas circuit loading links that uncertainty to internal structure.Source: Limitations · p. 5

Ethical Considerations

The authors characterize the work as fundamental interpretability research on arithmetic reasoning using synthetic data and publicly available models.

Ethical Considerations

  • The work studies arithmetic reasoning with synthetic data and publicly available models, and the authors foresee no risks beyond those of the underlying models.They report using Claude Opus 4.5–4.8 for coding and writing assistance, while attributing scientific content, design decisions, and claims to the authors.Source: Ethical Considerations · p. 5

A Supplementary related work

Prior work describes circuit reuse across tasks and languages and characterizes arithmetic circuits and shared representations. The authors distinguish their study by predicting cross-format generalization from an independently localized reference circuit at the item level.

Circuits are reused across tasks and languages

Arithmetic circuits

Circuits as predictors

  • Unlike Sun (2025), which predicts subtask-level accuracy within one numeric format, this study uses an independently localized reference circuit for cross-format, item-level prediction.The authors state that, to their knowledge, prior work has not made these cross-format predictions at the level of individual problems.Source: A Supplementary related work · p. 7

B Attribution patching

Attribution patching scores units by approximating how replacing their activations changes a model-output preference metric. The metric is built from the model’s own responses to an original prompt and its sign-flipped alternative, without using gold answers.

B Attribution patching

  • The preference metric compares how strongly the model favors its greedy answer to the sign-flipped prompt’s greedy answer when reading an input.It is rescaled to equal 1 on the original prompt and 0 on the sign-flipped prompt, and uses no gold answer.Source: B Attribution patching · p. 7
  • Attribution patching linearly approximates the effect of replacing each unit’s activation with its value from the original prompt, avoiding an exhaustive forward pass per unit.The method assigns an attribution score to each unit based on the approximate replacement effect on the preference metric.Source: B Attribution patching · p. 8
  • The method scores MLP units at the prompt’s last position using activation differences and the metric gradient, with two forward passes and one backward pass per item.The activations come from the original and alternative prompts, and the gradient is evaluated on the alternative prompt.Source: B Attribution patching · p. 8

C Causal validation of the item-level prediction

The authors test whether attribution-based item-level predictions reflect a causal role for the numeric circuit. Activation patching shows that stronger restoration of the original answer preference is associated with correct answers.

C Causal validation of the item-level prediction

  • The intervention swaps in activations from the original prompt at the last prompt token while the model reads a sign-flipped alternative.The measure is the fraction of the preference difference restored; random units matched in number provide a control.Source: C Causal validation of the item-level… · p. 8
  • Across 38 model–format combinations, the point-biserial correlation between activation-patching restored fraction and item-level accuracy was positive in 35 and significant in 32; it exceeded random-unit correlations in every format.Source: C Causal validation of the item-level… · p. 8
  • Causal and attribution-based item-level correlations covary across models and formats (r = 0.74).Paired Wilcoxon tests found larger numeric-circuit than random-unit correlations in English and Spanish (p < 0.001) and Italian (p < 0.01).Source: C Causal validation of the item-level… · p. 8, Figure 5 · p. 8

D Probe and confidence-measure details

The section describes comparison predictors and confidence measures, then tests whether numeric-circuit accuracy prediction depends on identifying the circuit from mostly correct examples. Correct-only and all-item circuits predict verbal accuracy similarly, unlike an incorrect-only circuit.

D Probe and confidence-measure details

  • Two logistic-regression probes predict correctness from the last prompt token’s residual stream or MLP activations, transferring from numeric to verbal arithmetic.Both selected layer 25 and achieved crossvalidated AUCs of 0.93 for residual stream and 0.92 for MLP activations on numeric problems.Source: D Probe and confidence-measure details · pp. 8–9
  • Mean log-probability averages teacher-forced probabilities over the model’s answer tokens, while entropy measures the next-token distribution at the prompt’s final token.Source: D Probe and confidence-measure details · p. 9
  • The LMG method decomposes linear-probability-model R2 by averaging each predictor’s contribution across all predictor-entry orders.Source: D Probe and confidence-measure details · p. 9

E Disentangling correctness and format

  • The test addresses whether numeric-circuit prediction reflects the circuit’s high accuracy on its identification sample rather than a numeric-format-specific advantage.The study uses Llama-3.1-8B and independently samples 1,600 correct and 400 incorrect items per format.Source: E Disentangling correctness and format · p. 9
  • For each verbal format, loadings on Sall and Scorrect predict verbal-format accuracy at essentially the same magnitude, whereas loadings on Sincorrect do not predict accuracy across formats.The variants were built on the matched numeric sample: Sall used all 2,000 items, Scorrect 1,600 correct items, and Sincorrect 400 incorrect items.Source: E Disentangling correctness and format · p. 9

F Cross-format prediction for all format pairs

The authors test cross-format item-level prediction using balanced samples and circuits identified separately in each format. Every format’s circuit predicts accuracy in every other format, with directional differences in prediction strength.

F Cross-format prediction for all format pairs

  • Every anchor circuit predicts every target format (rpb = 0.26 to 0.53, all Holm-corrected p < 0.05); verbal anchors predict verbal targets more strongly than numeric anchors do.Verbal-to-verbal correlations are 0.30 to 0.53, numeric-to-verbal 0.30 to 0.38, and verbal-to-numeric 0.26 to 0.28.Source: Figure 6 · p. 9, F Cross-format prediction for all… · p. 10
  • To control for accuracy differences, each format uses 2,000 balanced items at 80% accuracy; the top 1% of units in each anchor circuit is tested against target-item correctness.The numeric format is the main analysis anchor, and its median accuracy is 87.4%.Source: F Cross-format prediction for all… · p. 9
  • Cross-format prediction is not specific to the numeric circuit: any format’s circuit predicts accuracy in every other format, consistent with formats tapping a shared arithmetic circuit to different degrees.Source: F Cross-format prediction for all… · p. 10
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