# Interpreting the Latent Structure of Operator Precedence in Language Models

arXiv: https://arxiv.org/abs/2510.13908 · HTML: https://mlsanity.com/paper/2510.13908/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Placed in Mathematical reasoning, one of 17 directions in LLM reasoning.
- Builds on 11 of its 15 references that are in the map.
- First posted on arXiv 14 Oct 2025.

## Placed in

- LLM reasoning / [Mathematical reasoning](https://mlsanity.com/lr/mathematical-reasoning/)

## Builds on (most cited first, 11 of 11)

- [UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction](https://mlsanity.com/paper/1802.03426/) (2018)
- [Sparks of Artificial General Intelligence: Early experiments with GPT-4](https://mlsanity.com/paper/2303.12712/) (2023)
- [Emergent Abilities of Large Language Models](https://mlsanity.com/paper/2206.07682/) (2022)
- [Understanding intermediate layers using linear classifier probes](https://mlsanity.com/paper/1610.01644/) (2016)
- [GSM-Symbolic: Understanding the Limitations of Mathematical Reasoning in Large Language Models](https://mlsanity.com/paper/2410.05229/) (2024)
- [A Mechanistic Interpretation of Arithmetic Reasoning in Language Models using Causal Mediation Analysis](https://mlsanity.com/paper/2305.15054/) (2023)
- [Interpreting and Improving Large Language Models in Arithmetic Calculation](https://mlsanity.com/paper/2409.01659/) (2024)
- [Towards Uncovering How Large Language Model Works: An Explainability Perspective](https://mlsanity.com/paper/2402.10688/) (2024)
- [Large Language Models and Mathematical Reasoning Failures](https://mlsanity.com/paper/2502.11574/) (2025)
- [Adaptive Circuit Behavior and Generalization in Mechanistic Interpretability](https://mlsanity.com/paper/2411.16105/) (2024)
- [Integrating Arithmetic Learning Improves Mathematical Reasoning in Smaller Models](https://mlsanity.com/paper/2502.12855/) (2025)

## Built on by (most cited first, 2 of 2)

- [Locate, Steer, and Improve: A Practical Survey of Actionable Mechanistic Interpretability in Large Language Models](https://mlsanity.com/paper/2601.14004/) (2026)
- [Towards Understanding the Robustness of Sparse Autoencoders](https://mlsanity.com/paper/2604.18756/) (2026)

## Abstract (arXiv)

Large Language Models (LLMs) have demonstrated impressive reasoning capabilities but continue to struggle with arithmetic tasks. Prior works largely focus on outputs or prompting strategies, leaving the open question of the internal structure through which models do arithmetic computation. In this work, we investigate whether LLMs encode operator precedence in their internal representations via the open-source instruction-tuned LLaMA 3.2-3B model. We constructed a dataset of arithmetic expressions with three operands and two operators, varying the order and placement of parentheses. Using this dataset, we trace whether intermediate results appear in the residual stream of the instruction-tuned LLaMA 3.2-3B model. We apply interpretability techniques such as logit lens, linear classification probes, and UMAP geometric visualization. Our results show that intermediate computations are present in the residual stream, particularly after MLP blocks. We also find that the model linearly encodes precedence in each operator's embeddings post attention layer. We introduce partial embedding swap, a technique that modifies operator precedence by exchanging high-impact embedding dimensions between operators.