A Mathematical Theory of the Jacobian Lens
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 ω-J-lenses.
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BibTeX
@misc{ostrow2026mathematical,
title = {{A Mathematical Theory of the Jacobian Lens}},
author = {Mitchell Ostrow and Daoyuan Qian and Toni Liu and Ila Fiete},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.31955},
eprint = {2610.31955},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.31955},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Mathematical Theory of the Jacobian Lens
AU - Mitchell Ostrow
AU - Daoyuan Qian
AU - Toni Liu
AU - Ila Fiete
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.31955
UR - https://arxiv.org/abs/2610.31955
AB - 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 ω-J-lenses.
ER -