Posterior Means Underestimate Diffusion: Auditable Identification of Position-Only Stochastic Dynamics

Learning stochastic dynamics from noisy positions requires estimating forces and diffusion without observing velocity. We show that the standard smooth-then-regress approach systematically underestimates diffusion in our benchmarks because posterior-mean reconstruction discards uncertainty in velocity increments. We introduce OMWG-C, an endpoint-corrected weak-moment estimator that restores this missing variance from smoother cross-time covariances. Drift and diffusion are unknown functions represented in supplied candidate dictionaries; the second-order kinematic structure is assumed. In a frozen synthetic confirmation, OMWG-C lowers diffusion error in every unit, with a substantial median reduction relative to the posterior-mean baseline, while development results show little change in one-step prediction. For data without ground truth, we introduce a pre-set cross-sampling-rate audit that refits the dynamics after downsampling. Endpoint-corrected fits fail this audit in all cell-migration cohorts, despite better prediction in most cohorts, whereas ocean-drifter point estimates pass despite worse prediction. Passing tests rate stability, not physical correctness. These results establish predictive fit, parameter recovery, and sampling-rate stability as distinct dimensions of position-only equation learning.