Sharp Variance Certificates for Log-Partition Gaps

When does the variance of a log density ratio certify, rather than merely approximate, the error of a Gaussian approximation? For a standard Gaussian reference $Q$ and a smooth log ratio $g$ that is square-integrable with its gradient, we prove that ${\nabla }^{2}g⪯(1-\kappa )I$, $0<\kappa \le 1$, implies $\mathrm{KL}(Q\Vert P)=\log {\mathbb{E}}_Q{e}^g-{\mathbb{E}}_{Q}g\le {\mathrm{Var}}_Q(g)/(2\kappa )$ for the tilted target $P\propto e^{g}Q$. The coefficient is optimal for every $\kappa$, and the bound extends the known concave case to nonconcave log ratios without bounded-gradient or small-error assumptions. A matrix refinement resolves direction and Hermite degree. It never exceeds the population matrix Fisher (log-Sobolev) certificate, is exact for Gaussian targets when the curvature matrix is their relative precision, and shows that weak curvature is paid only through the degree-one and degree-two components. These are the stationarity residuals of Gaussian variational inference and are known to vanish at its exact optimum. Consequently, at an exact full-covariance VI stationary point of a smooth log-concave target, $\mathrm{KL}(Q\Vert P)\le 3{\mathrm{Var}}_Q(g)/2$ under mild growth conditions, with no curvature margin. Explicit examples separate the variance, degree-resolved and Fisher certificates at nonvanishing error, and the bound extends to non-Gaussian references with two-sided curvature control. Gaussian polynomial moments and residual bounds make the certificates computable without the target normalizer. On Bayesian logistic posteriors, the scalar certificate is 0.29–0.67 times matrix Fisher across 48 historical fits, and under strong regularization it certifies KL tolerances that the Fisher certificate cannot reach. Under a fixed prior, the degree-resolved certificate decreases as data accumulate, and independent estimates place it within a factor of two of the estimated KL at moderate priors.