Short Horizons and Sparse Concepts: a Mathematical View of the Readout in the J-lens

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.

Lens methods inspect model computation by mapping intermediate activations to vocabulary tokens. Yet the concepts humans need to read out often span multiple tokens---entities, phrases, intermediate objects---making token-level readouts incomplete. Reliable multi-token readout with little model-specific preparation remains challenging. We introduce Concept Lens: token-level lens clues guide candidate concept search, then the model derives a representation for each candidate and scores it against the original activation. Across 2,400 multi-hop clozes on five LLMs (8B--70B), Concept Lens instantiated with J-lens and R-lens achieves average Rank@10 scores of 36.6% and 54.5%, respectively, compared with 21.7% for Template Lens. Concept-swap interventions on derived concept representations shift model answers toward those associated with the replacement concepts. Further experiments show that Concept Lens can also reveal what a model recognizes along the way, beyond what appears in its final answer. Our code is available at https://github.com/XijieGo/c-lens

A language model's prediction of its next token develops across layers, and lens methods track this process by decoding intermediate hidden states into tokens. But a lens reading reflects both the hidden state and the readout (the unembedding matrix) used to decode it. Many lenses are fit on a corpus, and we show that two lenses differing only in their fitting corpus can report different tokens for the same hidden states. We call this dependence corpus conditionality. To examine readout structure independently of the fitting corpus, we introduce Sparse Readout Prism (SRP), which decomposes the readout using only its weights and expresses any token logit or logit difference as a sum of contributions from sparse readout features. This reveals readout features as a new unit of analysis for lens readings, exposing structure that token identities can obscure and enabling comparisons across tokens, contexts, layers, and lenses. Replacing the original readout with SRP's sparse approximation reconstructs 8.9-17.3 percentage points more of the tested logit differences than the strongest of six baselines built on geometric relations among readout rows. Ablating features shifts logit differences in proportion to their SRP contributions. Although token readings vary with the fitting corpus, the dominant readout feature remains stable. Because SRP uses no corpus in its construction, it provides a control independent of the fitting corpus for lens analyses.