Shared proof readers

Read the mathematics first: precise hypotheses, displayed equations and a step-by-step proof. Lean, reused library facts and source evidence are optional. These shared results do not certify an entire textbook chapter.

1. Pairing a relative score with transport displacement

   The analytic estimate behind the Cauchy–Schwarz step in Chewi’s proximal-sampling argument; first variation remains a separate theorem.

2. The exact law of one random-coordinate update

   Turn the instruction “resample a uniformly selected coordinate conditionally” into an explicit transition probability.

3. Why random-scan heat bath is reversible

   Follow the proof from symmetric atomic flux to Mathlib’s full set-integral definition; zero target atoms are handled explicitly.

Complete declaration-by-declaration teaching coverage