Quantify the output-distribution gap for remasking and uniform-state samplers

Derive a quantitative lower bound on the divergence or other distributional discrepancy between the output laws of remasking and uniform-state diffusion samplers and the target training distribution, beyond the established per-step divergence results and support violations.

Background

The paper proves a per-step impossibility result: writing conditionally dependent positions independently incurs a divergence at least equal to their conditional total correlation. For absorbing-state samplers, Appendix A.5 shows that under prefix-measurable schedules these per-step errors add, yielding a complete argument about the output distribution.

For remasking and uniform-state samplers, positions can be written more than once and the relevant overwrite decisions may occur at states outside the support of the data distribution. The paper establishes that such decisions are not determined by the training distribution and that changing pinned positions either leaves the support or writes a dependent group, but it does not convert these facts into a quantitative gap between the final output law and the target distribution.