Shared circuits predict whether LLMs generalize across formats in arithmetic reasoning

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

cs.CLSep 3, 2026

Shared circuits predict whether LLMs generalize across formats in arithmetic reasoning

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

Organizations: Massachusetts Institute of Technology

Abstract

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 más cinco'), and Italian ('due più 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.

Figures & tables

Panel of Figure 1 — Figure 1: Analysis pipeline. We first identify the target circuit as the top-1% of units in the numeric domain (left); then, we use attribution scores on the target circuit in the verbal domain to predict correctness.

Figure 2: Per-format accuracy. Accuracy is high in numeric format and drops in English, Spanish, and Italian.

Figure 3: Circuit engagement predicts item-level accuracy. A Rows: verbal formats. The paired violins indicate the loading onto the numeric circuit for incorrect (left) and correctly solved items (right) with point-biserial $r$ below; non-significant pairs ( $p\geq 0.05$ ) are in light gray. B Across models, accuracy is near zero at low loading and increases with it. C The uncertainty (entropy) of the outcome also increases with numeric loading.

Figure 4: Circuit loading predicts item-level accuracy beyond trained probes and confidence measures. Share of the variance in item-level correctness attributed to each predictor by an $R^{2}$ decomposition. Hatched bars indicate predictors that are not significant ( $p\geq 0.05$ ).

Appendix figures & tables3 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 5: The numeric circuit causally drives item-level accuracy. A Point-biserial correlation between the fraction of preference restored by activation patching and item-level accuracy for random units (gray) and the numeric circuit (colored). Each faint line is one model, squares are means, and brackets are paired Wilcoxon tests ( $p<0.01$ **, $p<0.001$ ***). B Across models and formats, the causal correlation ( y ) correlates with the attribution-based correlation of the main analysis ( x ).

verbal	$S^{\text{all}}$	$S^{\text{correct}}$	$S^{\text{incorrect}}$
English	$0.38$	$0.38$	$-0.05$
Spanish	$0.30$	$0.30$	$0.11$
Italian	$0.38$	$0.38$	$0.10$

Appendix

Table 1: Item-level point-biserial correlation between loading on the numeric circuit and accuracy, under three definitions of the numeric circuit.

n=2{,}000

per format. Results from Llama-3.1-8B.

Figure 6: Every format’s circuit predicts item-level accuracy in every other format. Point-biserial correlation between loading on the circuit identified in the anchor format (rows) and item-level correctness in the target format (columns), computed on the balanced samples ( $n=2{,}000$ per format, 80% correct). Asterisks mark significance at $p<0.05$ (Holm-corrected). Results from Llama-3.1-8B.

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Are Arithmetic Heuristic Neurons Form-Invariant? A Mechanistic Analysis of Symbols, Text, and Code in LLMs

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Large language models often succeed on one formulation of a problem while failing on an equivalent formulation. Whether these failures arise from distinct internal circuits or different activation states of a shared circuit remains unknown. Recent mechanistic interpretability studies suggest that arithmetic in LLMs emerges from a "bag of heuristics," encoded by a sparse set of MLP neurons that represent distinct arithmetic strategies. We investigate whether arithmetic heuristic neurons are form-invariant across symbolic arithmetic, natural language word problems, and Python code in three Llama-3 models. In each format, we identify arithmetic heuristic neurons using a two-stage pipeline combining attribution patching and activation patching. A compact set of neurons is shared across all three formats, and targeted interventions show this shared circuit is both necessary and sufficient for late-layer arithmetic computation. Transferring the shared neurons' activations from a successful execution in one format to a failed execution in another recovers most incorrect predictions, exceeding 97% for addition and subtraction, indicating that cross-format failures arise from activation states rather than distinct circuits. Moreover, shared neurons consistently belong to the same heuristic families across formats, demonstrating that arithmetic computation in LLMs is largely form-invariant at the neuron level.

Abstract

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