Contextual Cross-Jacobians: Query-Specific Transport in Diffusion Language Models

Vocabulary lenses make intermediate language-model states readable. Most either decode a state directly or replace the downstream network with one map averaged over many prompts and positions. The latter is useful for asking what a state is generally poised to verbalize, but it obscures a different question: what does this source contribute to this recipient in this execution? The distinction is especially visible in diffusion language models, where one visible token can influence several unresolved positions. We introduce the Contextual Cross-Jacobian (CCJ) lens, which linearizes the remaining computation from one source to one recipient in the current forward pass, transports the source state through that local map, and decodes the result with the model's own unembedding. On Dream and LLaDA, CCJ improves target-blind multihop recovery over globally averaged J- and R-fields when every method searches every eligible source layer. Holding the forward pass fixed while changing only the recipient changes the decoded vocabulary, and following the same route through denoising reveals latent concepts before they are highly ranked in the current output distribution. Finally, finite interventions follow CCJ-J's predicted scale, and layers selected without answer labels support activation-only two-way binding swaps. A matched autoregressive comparison finds the same advantage; diffusion provides the additional ability to trace unresolved recipients over time. These results establish CCJ as a training-free mechanistic lens for execution-specific source-to-recipient computation that reusable global maps average away.