Gauge and holonomyThe arrow you carry around a loop — what a round trip can remember, and why “north” is a choice

Here is a question that sounds too simple to be deep. You carry something on a round trip — out and back to where you started — and you are careful never to twist it, turn it, or fiddle with it along the way. When you get home, is it the way it left? On a flat tabletop: obviously yes. On the surface of the Earth: no. It comes back rotated, by an angle you can predict exactly, even though you did everything right. That stubborn little rotation is called holonomy, and it is one of the most consequential facts in physics — it is what “curvature” actually does, it is how this site says loops remember, and the bookkeeping language built around it is what physicists mean by gauge theory. This module builds all of it from a walk, in three sections, with your hands on every figure.

It assumes the rotations primer and nothing else. By the end you own three words that unlock the back half of the mind essays: transport, gauge, holonomy.