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arXiv · 2602.19109

Post-Routing Arithmetic in Llama-3: Last-Token Result Writing and Rotation-Structured Digit Directions

Yao Yan
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Abstract

We study three-digit addition in Meta-Llama-3-8B (base) under a one-token readout to characterize how arithmetic answers are finalized after cross-token routing becomes causally irrelevant. Causal residual patching and cumulative attention ablations localize a sharp boundary near layer~17: beyond it, the decoded sum is controlled almost entirely by the last input token and late-layer self-attention is largely dispensable. In this post-routing regime, digit(-sum) direction dictionaries vary with a next-higher-digit context but are well-related by an approximately orthogonal map inside a shared low-rank subspace (low-rank Procrustes alignment). Causal digit editing matches this geometry: naive cross-context transfer fails, while rotating directions through the learned map restores strict counterfactual edits; negative controls do not recover.

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Yao Yan. 2026-02-22. Post-Routing Arithmetic in Llama-3: Last-Token Result Writing and Rotation-Structured Digit Directions. https://arxiv.org/abs/2602.19109

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