Language models exhibit strong reasoning capabilities, yet adapting them to structured domains remains challenging and can yield inconsistent outcomes. We identify representation compatibility, the extent to which a model effectively processes a representation for a structured task, as a key factor in adaptation. We study this in chess, which provides a controlled testbed with precise semantics, computable optimal actions, and multiple state representations, including a symbolic encoding (FEN) and a spatial format (ASCII). We find that models often process semantically equivalent inputs substantially differently, affecting both learning and generalization. Building on this observation, we propose representation-aligned auxiliary supervision, which uses environment-derived tasks expressed in compatible representations to improve adaptation to structured domains. Across models and representations, auxiliary supervision consistently improves optimal-move prediction relative to target-only training under identical target data. Tasks that expose environment dynamics provide larger and most consistent gains than surface-level or static supervision, while remaining competitive with substantially increasing the amount of target-task data. Moreover, ASCII-trained models transfer more effectively to FEN than FEN-trained models do to ASCII, even surpassing the FEN target-only baseline on FEN evaluation. The gains also extend beyond optimal-move prediction to open-ended, factually grounded commentary generation. Overall, our results show that auxiliary supervision in model-compatible representations can enable effective adaptation in structured domains.

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