How exactly do Gaussian Processes (square dist kernel) enforce smoothness? (Aka how are they computed to do so?)
It's by definition, when you fit a guassian process you specify the mean function m(x) and the covariance function (or kernel) k(x,x'). Often the mean function is 0 and the covariance is the radial basis function or squared expential kernel which is smooth (in fact infinitly differentiable).