One Transition Score, Two Graphs: Untangling Dynamics and Synchronous Innovation Dependence

Finite-interval sampling folds dynamical propagation and synchronous innovation dependence into the same transition, making the two mechanisms difficult to distinguish. We study how to jointly recover the dynamics and synchronous graphs from irregularly sampled observations. For additive-noise diffusions with correlated innovations, differentiating the conditional transition score with respect to the terminal increment and the starting state yields a curvature block and a cross block, respectively. The curvature block characterizes synchronous dependence and demixes the cross block to reveal dynamical sensitivity, allowing the two blocks to jointly identify both graphs. We construct systems with identical cross blocks but different two-layer structures, establishing that the cross block alone cannot identify the two graphs. We also derive support-recovery conditions that account for finite observation gaps and score-estimation error. CAST (Curvature and Sensitivity from Transitions) learns a terminal score conditioned on the actual observation gap directly from irregular transition pairs, without estimating trajectory derivatives or imputing onto a regular grid. Experiments on synthetic systems and external benchmarks validate the two-block readout and gap-conditioned learning. CAST outperforms published baselines for dynamical structure learning on three of four benchmark datasets.