Learning goal: batchwise squared-cost assignment changes the endpoint coupling and can reduce transport cost and conflicting directions in this toy. Larger batches relax within-group restrictions, while small batches can already give substantial gains here. Reordering preserves the sampled endpoint sets; learned-model and numerical errors remain separate.

Green: paired reference lines; red: crossing pairs; thick: conditional-mean ODE paths; blue: sources; orange: targets.

Compare independent pairing with the full 192-point assignment. In this experiment, the crossing fraction changes by roughly 20% → 9% → 4% → 1.4% → 0.6% and one-step Euler error by 1.5 → 0.77 → 0.41 → 0.21 → 0.08. Here, batch 16 achieves a substantial fraction of the improvement, but this is not a universal sufficient batch size. Batchwise target reassignment is a permutation, preserving the empirical target marginal. It changes the joint coupling and intermediate field. High-dimensional cost concentration can affect gains, but does not universally make OT equivalent to random pairing.