Branch-Aware Diagnostics for Generative Probability Paths

Score-based and flow-based generative models share a common structure: a single-valued vector field returning one velocity or score per state-time pair. When multiple semantic branches or transport paths remain compatible with the same intermediate state, the regression target is multi-valued and the model collapses it to a conditional mean. The residual mean-squared error is the conditional-velocity variance floor identified in recent variance-perspective work on Flow Matching, around which we develop a branch-level diagnostic framework. The law of total covariance over a discrete branch variable K decomposes the floor into within-branch local uncertainty plus between-branch semantic switching, with switching ratio R_switch = A_between / A_v as a coupling diagnostic. On the diffusion side, the responsibility-switching curvature in the mixture-score Jacobian equals the Fisher information of the branch posterior; the decomposition is refinement-monotone in the branch variable and L1-stable under proxy perturbation at a fixed branch set. Low-dimensional oracles support the framework: a binary Gaussian diffusion oracle reveals a finite middle-SNR commitment window; a crossing-path Flow Matching oracle matches the closed-form floor within 0.025%; small MLPs saturate it. Euclidean OT removes ambiguity when geometry and semantics align but fails when a weak semantic coordinate is dominated by high-dimensional nuisance geometry. On a 33-D conditional-mismatch toy over 10 seeds, a semantic-cost coupling whose λ → ∞ limit reduces to class-conditional minibatch OT-CFM cuts conditional semantic mismatch from 0.50 to 0.04 at ~5% raw transport overhead.