Series: The Learn Arc — 50 posts through the Active Inference workbench. Previous: Part 40 — Session §8.1: Generalized coordinates Hero line. Eq 4.19 is Eq 4.13's continuous-time twin: a quadratic approximation to the variational free energy in generalized coordinates. Once you write it down, the rest of Chapter 8 is pattern-matching against neurons. From softmax to quadratic In discrete time, inference minimises a KL divergence, and when the posterior is categorical that KL is exactly a softmax update (Eq 4.13). In continuous time with gaussian beliefs, the KL simplifies into a quadratic form — and that is Eq 4.19. The quadratic is not a shortcut. It is the exact consequence of choosing gaussian variational densities in generalized coordinates. Five beats F is a sum of weighted squared errors. Prediction error at each order of generalized coordinate, weighted by its precision. Sensory error + dynamical error + prior error — squared and weighted. That is it. Gradient is linear in prediction error.…