Branch-Aware Diagnostics for Generative Probability Paths

Score-based and flow-based generative models learn a single-valued field returning one velocity or score per state and time. When several branches stay compatible with an intermediate state, the target is multi-valued and the field can only return their average; the residual is an irreducible variance floor set by the path and coupling, not by model capacity. We introduce branch-variance decomposition (BVD), an exact identity (the law of total covariance) that splits this floor over a discrete branch variable into within-branch local uncertainty and between-branch semantic switching, and reads the between-branch share as a coupling diagnostic computable before any model is trained. The same identity holds on the score side, where the between-branch term equals the Fisher information of the branch posterior. In synthetic settings that separate geometry from meaning, a meaning-aware coupling drives branch disagreement from chance to near zero at only a few percent extra transport cost. On CIFAR-10, an unsupervised clustering recovers the between-branch signal that ground-truth labels give from binary to ten-way splits, once the neighborhood bandwidth scales with the branch count. On a trained Flow-Matching model, a posterior-weighted between-branch spread read from the conditional model predicts the velocity-MSE that per-branch conditioning recovers, matching its shape and its magnitude to within about ten percent; an independent estimate corroborates the shape.