Order-Asymmetry in Diffusion Language Models: Discrete Curl and Conditional Compatibility

Diffusion language models decode tokens in flexible orders, but their denoisers provide only local conditionals for partially observed contexts, which need not be compatible with a single joint distribution over an unresolved block. As a result, different commitment orders can induce different order-specific pseudo-joints. We call this model-side failure conditional incompatibility, distinct from data-side dependence. For any two unresolved positions, we define curl as the log-ratio between the pseudo-joints induced by reversing the commitment order; it vanishes exactly when the two orders agree. We then prove that the order-dependent part of sequential decoding error is a data-weighted curl path integral, separate from conditional total correlation and marginal model error. Empirically, pairwise Fisher–Rao curl energy distinguishes list-operation regimes, recovers Sudoku $4\times 4$ constraint structure, and identifies a left-to-right preferred direction on GSM8K for both LLaDA-8B-Instruct and Dream-7B. Order sensitivity is therefore a measurable compatibility property of learned local conditionals.