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

arXiv: https://arxiv.org/abs/2602.19109 · HTML: https://mlsanity.com/paper/2602.19109/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 10 of its 16 references that are in the map.
- First posted on arXiv 22 Feb 2026.

## Builds on (most cited first, 10 of 10)

- [The Llama 3 Herd of Models](https://mlsanity.com/paper/2407.21783/) (2024)
- [Locating and Editing Factual Associations in GPT](https://mlsanity.com/paper/2202.05262/) (2022)
- [Refusal in Language Models Is Mediated by a Single Direction](https://mlsanity.com/paper/2406.11717/) (2024)
- [Probing Classifiers: Promises, Shortcomings, and Advances](https://mlsanity.com/paper/2102.12452/) (2021)
- [Towards Best Practices of Activation Patching in Language Models: Metrics and Methods](https://mlsanity.com/paper/2309.16042/) (2023)
- [How to use and interpret activation patching](https://mlsanity.com/paper/2404.15255/) (2024)
- [Arithmetic Without Algorithms: Language Models Solve Math With a Bag of Heuristics](https://mlsanity.com/paper/2410.21272/) (2024)
- [Pre-trained Large Language Models Use Fourier Features to Compute Addition](https://mlsanity.com/paper/2406.03445/) (2024)
- [Understanding Addition in Transformers](https://mlsanity.com/paper/2310.13121/) (2023)
- [All for One: LLMs Solve Mental Math at the Last Token With Information Transferred From Other Tokens](https://mlsanity.com/paper/2509.09650/) (2025)

## Abstract (arXiv)

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.