Beyond Population Loss: Statistical Coupling Design for Flow Matching

Changing a flow-matching coupling changes both the regression problem and the statistical cost of learning it. We show that these effects can be separated exactly: for Gaussian endpoints and linear spatial models, cross-covariance fixes the entire population objective, but higher-order dependence still changes finite-sample generation risk. Smooth non-Gaussian couplings strictly outperform the entire jointly Gaussian family at first order. The separation holds for every fixed Euler budget in a constant-field model and, for every scale s > 1, in a time-dependent class with a realizable Gaussian reference and exact integration. The latter has an explicit relative improvement exceeding 27/256. On the constructive side, covariance-constrained quartic transport characterizes the global first-order coupling oracle, with O(n^-2) regret under stabilized least squares. Finite cell-transport programs retain the continuous Gaussian marginals exactly, converge to the oracle, and admit smooth full-support implementations. Executed fresh-training experiments connect the optimized designs to terminal-risk improvements; at 64 cells, the smooth designs lower first-order coefficients by 44.9% and 65.5% in the two controlled protocols. Matrix reachability certificates and fixed-feature residual scores provide complementary structural results.