A Mathematical Theory of the Jacobian Lens

A paper submitted to arXiv on 22 October 2026 introduces a mathematical theory of the Jacobian lens (J-lens), the interpretability tool Anthropic researchers Gurnee et al. introduced in 2026 to find working-memory-like linear directions in frontier language model hidden states. The authors show the J-lens preferentially identifies coherent features that behave like leading, spatially coherent Lyapunov modes — perturbations along them amplify across layers, stay roughly constant across token positions, and hold across contexts — and validate the analysis on a Bayes filter and a toy transformer, each optimally predicting next tokens from a Hidden Markov Model. The paper also introduces a family of ω-J-lenses built to emphasize different temporal scales of information processing in LLMs.

Computer Science Machine Learning Submitted on 22 Oct 2026 Title:A Mathematical Theory of the Jacobian Lens View PDF https://arxiv.org/pdf/2610.31955 HTML experimental https://arxiv.org/html/2610.31955v1 Abstract:Recent work from Anthropic introduced a new interpretability tool, called the Jacobian lens J-lens, Gurnee et al., 2026 . This method identified linear directions in the hidden states of frontier language models that had properties reminiscent of working memory-like representations. However, it is unclear what mechanistic properties of such features make them detectable to the J-lens. To this end, we introduce a mathematical theory of the J-lens, and show that it preferentially identifies coherent features -- directions that act like leading, spatially coherent Lyapunov modes. Small perturbations along these directions amplify across layers, remain roughly constant across token positions, and are consistent across different contexts. While we are agnostic to whether these properties are necessarily associated with a Global Workspace'', the analytically derived preferences of the J-lens nevertheless explain prior empirical applied to both model classes. We validate our analysis on a Bayes filter and a toy transformer, each optimally predicting next tokens emitted from a Hidden Markov Model HMM . We show how modulating task parameters that change the coherence of features in the Bayes filter correspondingly and predictively modulates the response of the J-lens . We also introduce of a family of J-lenses built to emphasize different temporal scales of information processing in LLMs, which we call $\omega$-J-lenses. Additional Features Current browse context: cs.LG References & Citations Loading... Bibliographic and Citation Tools Bibliographic Explorer What is the Explorer? https://info.arxiv.org/labs/showcase.html arxiv-bibliographic-explorer Connected Papers What is Connected Papers? https://www.connectedpapers.com/about Litmaps What is Litmaps? https://www.litmaps.co/ scite Smart Citations What are Smart Citations? https://www.scite.ai/ Code, Data and Media Associated with this Article alphaXiv What is alphaXiv? https://alphaxiv.org/ CatalyzeX Code Finder for Papers What is CatalyzeX? https://www.catalyzex.com DagsHub What is DagsHub? https://dagshub.com/ Gotit.pub What is GotitPub? http://gotit.pub/faq Hugging Face What is Huggingface? https://huggingface.co/huggingface ScienceCast What is ScienceCast? https://sciencecast.org/welcome Demos Recommenders and Search Tools Influence Flower What are Influence Flowers? https://influencemap.cmlab.dev/ CORE Recommender What is CORE? https://core.ac.uk/services/recommender IArxiv Recommender What is IArxiv? https://iarxiv.org/about arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs https://info.arxiv.org/labs/index.html .