# Commutator Memory: Sparse, Path-Local Reading and Steering in Language Models

arXiv: https://arxiv.org/abs/2609.34348 · HTML: https://mlsanity.com/paper/2609.34348/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 34 of its 48 references that are in the map.
- First posted on arXiv 28 Sep 2026.

## Builds on (most cited first, 30 of 34)

- [Proximal Policy Optimization Algorithms](https://mlsanity.com/paper/1707.06347/) (2017)
- [The Llama 3 Herd of Models](https://mlsanity.com/paper/2407.21783/) (2024)
- [DeepSeekMath: Pushing the Limits of Mathematical Reasoning in Open Language Models](https://mlsanity.com/paper/2402.03300/) (2024)
- [Qwen3 Technical Report](https://mlsanity.com/paper/2505.09388/) (2025)
- [Direct Preference Optimization: Your Language Model is Secretly a Reward Model](https://mlsanity.com/paper/2305.18290/) (2023)
- [Qwen2.5 Technical Report](https://mlsanity.com/paper/2412.15115/) (2024)
- [Overcoming catastrophic forgetting in neural networks](https://mlsanity.com/paper/1612.00796/) (2016)
- [Locating and Editing Factual Associations in GPT](https://mlsanity.com/paper/2202.05262/) (2022)
- [The Pile: An 800GB Dataset of Diverse Text for Language Modeling](https://mlsanity.com/paper/2101.00027/) (2020)
- [Membership Inference Attacks against Machine Learning Models](https://mlsanity.com/paper/1610.05820/) (2016)
- [Gradient Episodic Memory for Continual Learning](https://mlsanity.com/paper/1706.08840/) (2017)
- [Understanding Black-box Predictions via Influence Functions](https://mlsanity.com/paper/1703.04730/) (2017)
- [Gradient Surgery for Multi-Task Learning](https://mlsanity.com/paper/2001.06782/) (2020)
- [Interpretability in the Wild: a Circuit for Indirect Object Identification in GPT-2 small](https://mlsanity.com/paper/2211.00593/) (2022)
- [Editing Models with Task Arithmetic](https://mlsanity.com/paper/2212.04089/) (2022)
- [Mass-Editing Memory in a Transformer](https://mlsanity.com/paper/2210.07229/) (2022)
- [Membership Inference Attacks From First Principles](https://mlsanity.com/paper/2112.03570/) (2021)
- [Optimizing Neural Networks with Kronecker-factored Approximate Curvature](https://mlsanity.com/paper/1503.05671/) (2015)
- [A Watermark for Large Language Models](https://mlsanity.com/paper/2301.10226/) (2023)
- [Towards Automated Circuit Discovery for Mechanistic Interpretability](https://mlsanity.com/paper/2304.14997/) (2023)
- [Eliciting Latent Predictions from Transformers with the Tuned Lens](https://mlsanity.com/paper/2303.08112/) (2023)
- [UltraFeedback: Boosting Language Models with Scaled AI Feedback](https://mlsanity.com/paper/2310.01377/) (2023)
- [Fine-Tuning Pretrained Language Models: Weight Initializations, Data Orders, and Early Stopping](https://mlsanity.com/paper/2002.06305/) (2020)
- [Studying Large Language Model Generalization with Influence Functions](https://mlsanity.com/paper/2308.03296/) (2023)
- [Provable Robust Watermarking for AI-Generated Text](https://mlsanity.com/paper/2306.17439/) (2023)
- [TRAK: Attributing Model Behavior at Scale](https://mlsanity.com/paper/2303.14186/) (2023)
- [Scaling Monosemanticity: Extracting Interpretable Features from Claude 3 Sonnet](https://mlsanity.com/paper/2605.29358/) (2026)
- [SeedPrints: Fingerprints Can Even Tell Which Seed Your Large Language Model Was Trained From](https://mlsanity.com/paper/2509.26404/) (2025)
- [Mechanistic Data Attribution: Tracing the Training Origins of Interpretable LLM Units](https://mlsanity.com/paper/2601.21996/) (2026)
- [Commute Your Domains: Trajectory Optimality Criterion for Multi-Domain Learning](https://mlsanity.com/paper/2501.15556/) (2025)

## Abstract (arXiv)

Gradient updates on different data generally do not commute: training a language model on two data sources in opposite orders gives different weights, even with the same data and total exposure. Loss or benchmark deltas show that the models differ, not where. We ask whether this path dependence leaves a parametric training-history memory: a weight component that flips sign when the two sources are swapped, is localized in output space, changes the held-out loss gap between the two orders under targeted interventions, and reveals which trained model came from which order. For one small SGD step of size η on each of sources A and B, the weight difference θAB-θBA is, to leading order, η2 bAB, where bAB=HBgA-HAgB is the Lie bracket of the two gradient fields at the base model. We define commutator memory by projecting the bracket through the logits into one score per vocabulary token; the scores sum to the bracket's prediction of the gap. The scores are localized: on three models, the same readout of the measured θAB-θBA, or of a bracket from disjoint batches, shares 82-99% of the original top-20 tokens, versus 35-49% for norm-matched random directions. They are causally actionable: in Qwen-3-4B SFT, downweighting the ten tokens with the largest predicted share of the gap closes a median 32% of the measured gap, while frequency-matched tokens with near-zero scores have almost no effect. The weights themselves carry the component: projecting the difference between the two trained models onto bAB identifies which came from which order in 92% of cases across four LLMs (chance 50%). Controlled tests also cover matched-batch DPO, a frozen-rollout GRPO-style objective, and an AdamW endpoint check. The memory is defined per source pair, not per example, and its projection on bAB decays with further training.