A Mathematical Theory of the Jacobian Lens
arxiv.org
arxiv.org
It answers one of the conceptual questions that came to mind as to why it's even a well-defined concept:
> When the Jacobian is studied in an LLM, it provides directions in the residual stream that are “verbalizable” (Gurnee et al., 2026), which is to say that it identifies how perturbations change the linguistic output of the model. However, this property holds only in the local neighborhood of the point on which the Jacobian is defined, and the J-lens is defined by averaging over prompts, tokens, and future positions in the output layer. An average Jacobian has no reason to behave like any of its elements.