GapFlow: Cross-Variable Priors for Flow-Matching Time-Series Imputation on Contiguous Gaps

Flow matching is a strong framework for multivariate time-series imputation. Replacing the Gaussian source with a structured prior that already interpolates the observed entries shortens the transport the network must learn, and yields large gains over diffusion and attention baselines. We show that this design has a systematic failure mode. The prior is fitted *per variable*, so it carries nothing across channels; when missingness comes in a contiguous block rather than scattered points, a variable's own history is severed and the prior degenerates to a straight line through the gap. On three of six benchmarks the resulting imputer is beaten by a plain transformer by up to $46.8\%$: in that regime the structured prior does active damage. We introduce GapFlow, which conditions the source prior on the *observed channels* through a ridge-regularised Gaussian conditional, drives the probability path by a two-sided distance to the nearest observation, and refines the sample by self-conditioning. GapFlow attains the lowest MAE against nine baselines in $72$ of $72$ settings, spanning six datasets, four missingness mechanisms, two ratios and four window lengths. Its gains are mechanism-dependent: $8$–$46\%$ under block missingness against $\approx 0\%$ under MCAR, with no exception across six datasets.