Structure and Limits of Composable Coarse-to-Fine Generative Models

We study coarse-to-fine generative models that combine deterministic coarse evolution with stochastic reconstruction. We characterize the temporal laws of Markov rollouts with transition kernels $Q_t(u)=G({\varphi }_t(Ru))$ that compose exactly at all positive times. Under Lipschitz regularity of the restriction, quadratic-Wasserstein Lipschitz regularity of the decoder, and controlled coarse-flow regularity, every positive-time joint law is a mixture of independent emissions along a deterministic flow of persistent classes. This representation does not require exact consistency between reconstructed samples and their prescribed coarse states. It implies that mean-square continuity forces the visited class laws to be point masses. Within the same exactly composable setting, we bound the path-law distance to independent reconstruction under approximate fiber consistency, with dependence on decoder sensitivity and fiber error. Explicit constructions show that weaker decoder regularity permits stochastic class evolution and that increasing decoder sensitivity can preserve temporal dependence as the fiber error vanishes. Controlled temporal experiments separate identical marginals from path statistics, and a small learned reconstruction study shows why improved temporal correlation alone is insufficient.