Current CMB data show a 2-parameter compression of nearly-Gaussian fluctuations, but novel primordial non-Gaussianities may arise not strongly CMB-constrained so far, and that may be more easily detected in Large Scale Structure Surveys. We have a good story, even with evidence, that all structure in the universe evolved from the diffusion of quantum-fluctuations that broke the adiabaticity of coarse-grained kinematic field trajectories during inflation.  I will describe the use of linear dynamical systems theory and highly-nonlinear pseudo-spectral lattice simulations to describe the non-G possibilities.  Kolmogorov-Sinai entropy is used to measure linear instabilities and coarse-grained Shannon entropy (the entropy we observe here and now) is used for classical-fluctuation generation during heating. This kinematical entropic picture also applies to short-lived instabilities while inflation is ongoing, and I describe in-out state lattice simulations with non-G arising from through multi-field potential-surface features.