$ \newcommand{\R}{\ensuremath{\mathbb{R}}} \newcommand{\lat}{\mathcal{L}} \newcommand{\ensuremath}[1]{#1} $We show that for any lattice $\lat \subseteq \R^n$ and vectors $\vec{x}, \vec{y} \in \R^n$, \[ ρ(\lat + \vec{x})^2 ρ(\lat + \vec{y})^2 \leq ρ(\lat)^2 ρ(\lat + \vec{x} + \vec{y}) ρ(\lat + \vec{x} - \vec{y}) \; , \] where $ρ$ is the Gaussian measure $ρ(A) := \sum_{\vec{w} \in A} \exp(-π\| \vec{w} \|^2)$. We show a number of applications, including bounds on the moments of the discrete Gaussian distribution, various monotonicity properties of the heat kernel on flat tori, and a positive correlation inequality for Gaussian measures on lattices.