  Low-dimensional representations of underdamped systems often provide
insightful grasps and analytical tractability. Here, we build such
representations via information projections, obtaining an optimal model that
captures the most information on observed spatial trajectories. We show that,
in paradigmatic systems, the minimization of the information loss drives the
appearance of a discontinuous transition in the optimal model parameters. Our
results raise serious warnings for general inference approaches and unravel
fundamental properties of effective dynamical representations, impacting
several fields, from biophysics to dimensionality reduction.
