Kalman Folding 3: Derivations

In Kalman Folding, Part 1, we present basic, static Kalman filtering as a functional fold, highlighting the unique advantages of this form for deploying test-hardened code verbatim in harsh, mission-critical environments. The examples in that paper are all static, meaning that the states of the model do not depend on the independent variable, often physical time. Here, we present mathematical derivations of the basic, static filter. These are semi-formal sketches that leave many details to the reader, but highlight all important points that must be rigorously proved. These derivations have several novel arguments and we strive for much higher clarity and simplicity than is found in most treatments of the topic.