Universal cold spots of checkerboard lattices

We prove that the deep holes of the checkerboard lattice D_n, for every n at least four, are exactly the global minimizers of its translated Gaussian lattice sum at every positive Gaussian scale. The minimum equals one half of the nth power of a one-dimensional theta value. A sharp quartic theta inequality yields a stronger result with independent positive coordinate scales. In dimension four, the additional odd-integer minimizers occur exactly when all four scales coincide; in dimensions at least five, an explicit global quadratic lower bound controls the excess potential. We also determine the minimum of the heat kernel on the flat torus R^n/D_n at every positive time and classify the minimizers for positive Gaussian scale mixtures whose minimum is finite.