The Geometry of Rollout Error

Long-horizon rollout remains a central challenge in learned physical dynamics. Models with similar one-step accuracy can produce radically different trajectories, so local prediction error alone does not determine behavior under repeated composition. We analyze this separation through the finite-time transport of local residuals: errors introduced at different times and in different directions can have very different consequences depending on the subsequent learned dynamics through which they propagate. This perspective motivates separating three choices in autoregressive learned dynamics: temporal representation, which determines whether the model learns an instantaneous generator or the finite-time transition that is actually composed at deployment; forward exposure, which determines how many self-fed predicted states are encountered during training; and temporal credit, which determines how far learning signals propagate backward through that rollout. Across ten evaluation conditions spanning nine physical systems from The Well, DISCO-FLO, a direct finite-time-transition formulation of DISCO, achieves lower final-horizon median error than generator-based DISCO in all ten conditions despite lower one-step median error in only four. A matched-credit experiment shows that the representation effect persists when generator and transition models receive the same self-fed exposure and backward credit, while credit-depth ablations show that increasing temporal credit is not uniformly beneficial across systems. Together, these results suggest that long-horizon learned dynamics should be understood and optimized through the geometry of their deployed composition rather than inferred from local prediction accuracy alone.