A Sharp Gaussian Log-Partition Bound for Mutual Information

Let X have an arbitrary Borel probability law mu on R^n, and let N be an independent standard Gaussian. We prove that I(X;X+N) is at most the Gaussian expectation of log integral exp(<z,x>) mu(dx). The constant one is optimal; no moment, centering, or log-concavity assumption is required. Caffarelli's contraction theorem, applied from the Gaussian convolution to standard Gaussian measure, gives a coupling whose correlation is at least twice the channel mutual information. The conditional Gibbs inequality proves the bound, and conditional truncation handles arbitrary input laws. This answers the first numbered question in Jingbo Liu's contribution to Oberwolfach Report 14/2022, page 695. The separate generalized dual Sudakov question is not addressed. This is an unrefereed preprint; no absolute priority or formal-verification claim is made.