04 — The formal core
Models, algorithms and the quiet structure underneath every system we claim to have invented.
4 essays
Two questions every articulated mechanism asks: given the joint angles, where is the tip? Given a tip pose, which angles get you there? The math is the same whether the arm is on a desk, a factory floor, or a simulation.
Twenty-seven methods, one question behind each: does this pattern mean something, or could it just be noise? Grouped by what they are for, with the reasoning and a worked example behind every one.
How a network turns a wrong guess into a better one — one derivative at a time. The chain rule, activation derivatives, the softmax–cross-entropy shortcut, and a worked example you can trace by hand.
One equation from 1961 predicts why your service falls over at 80% utilisation and not at 100%. It is the most useful mathematics I have ever applied to a production system.