A half-power vanishing-noise estimate for a quadratic kinetic mean-field game

We consider a variational kinetic mean-field game with quadratic acceleration cost, local quadratic congestion, and quadratic terminal congestion on $\T^d\times\R^d$. For a smooth, compactly supported probability datum, the density minimizers at velocity diffusivity $\nu$ and at zero diffusivity are at distance $O(\nu^{1/2})$ in spacetime $L^2$ and terminal $L^2$, in every dimension. The proof compares the two minimum values using kinetic Gaussian averaging and an entropy identity for the current flux. A positive convolution commuting with kinetic transport justifies this identity in the original weak admissible class, including the noncompact velocity domain and possible vacuum.