Uniform vanishing-viscosity convergence for a local quadratic mean field game

We prove uniform vanishing-viscosity convergence for the local mean field game system with quadratic Hamiltonian and linear density coupling on a flat torus in dimensions one, two and three. For every smooth nonnegative probability initial density, smooth fixed terminal cost and positive time horizon, the entire family of value functions converges uniformly on each interior time slab to a single continuous function. Initial vacuum is allowed. Averaging finite-energy perturbations of optimal trajectories turns a density-gradient estimate into a modulus of continuity independent of viscosity. A convex variational limit and path calibration then identify the occupied-region value, while homogeneous Hamilton--Jacobi comparison determines its vacuum extension.