Verifiable Mathematical Reasoning as a Probe of Emergent Global Workspaces

Probing the internal mechanisms of reasoning models is hard when correctness is ambiguous and each problem admits a single answer. Verifiable, multi-solution mathematical problems are unusually powerful probes: exact verification handles behavior, and many valid solutions let one intervention separate \emph{whether} a model produces a valid solution from \emph{which} one it produces---a distinction accuracy alone conflates. On multi-solution $n$-queens completion, we fit Jacobian lenses to the task logits of a recursive latent reasoner, which refines a latent state rather than extending tokens, and an autoregressive LLM. In the recursive reasoning model, a per-example subspace amounting to a fraction of a percent of the latent state is causally necessary across most of the trajectory; conditioned on sampled branches, ablating it destroys \emph{which} solution is returned while sparing \emph{whether} one is. The autoregressive model instead concentrates constraint enforcement in a narrow band of late decoder layers. In both, the workspace is a functional \emph{role}, not a module---a thin channel in the shared state---yet the two architectures differ sharply in where and how it emerges, suggesting why recursive refinement may reason more effectively than autoregressive decoding, and a concrete target for reasoning-model design. Moreover, a global workspace that concentrates reasoning into a compact, broadcastable channel is inherently domain-general: identifying it opens a path to steering it and transferring reasoning across domains and applications.