Conditional Mean Dependence Governs One-Step Endpoint Regression

One-step generative policies may match iterative samplers or collapse to a conditional mean. We characterize this outcome for $L^2$ endpoint regression. Let $\rho$ denote the ratio of target within-condition variance reproduced by a one-step map. For the optimal map, $\rho ={\rho }^{\ast }$, the squared correlation ratio between source noise and target given context. Collapse occurs exactly when the target's conditional mean is independent of the source noise; full statistical independence is sufficient but unnecessary. Moreover, ${\rho }^{\ast }$ depends on the training pairing, not on the target marginal alone. A fixed-marginal coupling family makes ${\rho }^{\ast }$ span $[0,1]$, giving every target-marginal-only estimator worst-case absolute error at least $1/2$; paired training samples permit estimation under regularity assumptions. Synthetic experiments recover the predicted curve. A cost-matched construction further shows that quadratic transport cost does not identify endpoint retention. An area-downsampled full-image MNIST check recovers the predicted ordering but exposes the failure of a generic high-dimensional estimator. An exploratory OGBench test recovers the map-spread ordering yet refutes the behavioral prediction: the collapsed policy succeeds more often. Thus the criterion diagnoses endpoint-map spread, not task utility.