When the blur is slower than the flow: clipped and corrected samplers on the critically damped Langevin flow

Critically damped Langevin diffusion (CLD) generates data in position–velocity phase space. The standard CLD specifies its noise through a stochastic differential equation and never meets the threshold studied here. We combine the CLD flow instead with an isotropic noise schedule specified by its marginals, σ(t) = σ_min e^{rt}. A reverse-time SDE with state-independent noise then exists only for r ≥ κ, the largest stretching rate of the flow; below κ the standard clipped sampler is biased, while a corrected sampler and the probability-flow ODE stay exact. On a four-dimensional Gaussian-mixture test, the measured shrinkage places the threshold at κ_fit/κ = 0.992 ± 0.025. An exact error equation shows that the damage is made while the blur is close to σ_min, does not depend on σ_max, and exceeds 10% only when σ_min is comparable to the data scale or the CLD mass is far from its default (up to 18.6% in mixtures). With learned scores the clipped sampler breaks as with the exact score, and the corrected sampler fixes it without retraining. In typical settings, with σ_min below the data scale, the threshold effect is a few per cent, smaller than the error of a small learned score. For a single Gaussian mode in the noise-dominated regime, the probability-flow ODE never damps a score error, it only rotates it, whereas the corrected SDE damps it. Code, data and verification: https://github.com/neginyan/cld-corrected-sampler (doi:10.5281/zenodo.23689208).