Addition in Four Movements: Mapping Layer-wise Information Trajectories in LLMs

Multi-digit addition is a clear probe of the computational power of large language models. To dissect the internal arithmetic processes in LLaMA-3-8B-Instruct, we combine linear probing with logit-lens inspection. Inspired by the step-by-step manner in which humans perform addition, we propose and analyze a coherent four-stage trajectory in the forward pass:Formula-structure representations become linearly decodable first, while the answer token is still far down the candidatethis http URLcomputational features then emergethis http URLdeeper activation layers, numerical abstractions of the result become clearer, enabling near-perfect detection and decoding of the individual digits in thethis http URLthe output, the model organizes and generates the final content, with the correct token reliably occupying the topthis http URLtrajectory suggests a hierarchical process that favors internal computation over rote memorization. We release our code and data to facilitate reproducibility.

@article{yan2025_2506.07824,
  title={ Addition in Four Movements: Mapping Layer-wise Information Trajectories in LLMs },
  author={ Yao Yan },
  journal={arXiv preprint arXiv:2506.07824},
  year={ 2025 }
}