Flow Maps That Compose Exactly: When Consistency Costs Accuracy

Learned dynamics are often required to compose across time: a direct forecast should agree with a sequence of shorter forecasts. Deterministic dynamics on a sufficient state and Markov transition laws obey this identity, but conditional means, the targets of squared-error training, generally do not. We quantify the price of this mismatch. Under coverage and regularity assumptions, an exactly composing predictor incurs joint excess risk at least proportional to the squared composition defect of the true means. Matching examples attain the smoothness rate and small-noise risk order. A generator criterion identifies diffusions whose means compose, including nonlinear, noisy cases. Transformed linear diffusions provide compatible kernels with explicit representation limits. Fixed-model experiments isolate the forecasting consequences. On noisy Lorenz dynamics, propagating a residual distribution through the same fitted one-step map removes most of the long-horizon excess risk of its point rollout; a single intermediate collapse to the mean restores it. Without noise, autoregression is highly accurate. On motion capture, discarding a learned kernel's intermediate uncertainty degrades its forecasts, although a deterministic map remains the best point predictor. The guiding distinction is what should compose: under uncertainty, transition laws compose even when their means do not.