Beyond the Map: Characterizing Representational Differences in Flow Maps

Consistency models (CMs) enable high-quality generation with few function evaluations, yet self-distillation objectives targeting the same flow map can behave differently. We study Lagrangian (LSD) and progressive self-distillation (PSD) in a controlled flow-map setting. Varying the flow-matching fraction reveals markedly different sensitivities: PSD benefits from stronger consistency training, whereas LSD relies more heavily on flow matching. Without flow-matching supervision, the two objectives empirically collapse to similar, seed-dependent degenerate solutions. Despite attaining comparable sampling quality, LSD- and PSD-trained models develop increasingly different hidden-representation geometries. These results show that equivalent target flow maps need not imply equivalent dependence on supervision or learned representations.