Learned scores on the critically damped Langevin flow: a corrected sampler in 281 steps, and a metric-dependent fragility of the probability flow

We study learned scores for the critically damped Langevin (CLD) flow with an isotropic noise schedule specified by its marginals, added in a rescaled metric Y = PZ. The standard CLD, with velocity-only noise given by its SDE, does not show the failures studied here. On a four-dimensional Gaussian mixture, from 28 networks and pre-registered tests, we report two results and a partial mechanism. First, a Heun-type integrator of the corrected reverse SDE, with a learned score and no retraining, reaches a median |L| ≤ 0.083 at 281 steps in all four metrics tested, where Euler–Maruyama gives 0.42–0.55; at 140 steps it no longer suffices. Second, the learned probability-flow ODE broke for 10 of 12 networks with P = I (κ_P = 2 > r = 1) and for 3 of 16 with metrics of κ_P ≤ 0; at the network level only one pre-registered comparison was significant. Broken runs contain at most 0.12% outliers: a few tail trajectories, which start in the tail of the prior and pick up most of their deviation while σ is large, fly far at the end. Thirteen of nineteen pre-registered predictions failed; we report all of them. This paper continues "When the blur is slower than the flow" (paper G, doi:10.5281/zenodo.23689912). Code, the 28 trained networks, stored results, pre-registration files and verification: https://github.com/neginyan/cld-learned-score-samplers (doi:10.5281/zenodo.23898962).