The von Mises module gave the site's central number a birth certificate: R0(KBKT) ≈ 0.303 is the mean resultant of a circular cloud at the critical coupling. But it left the deepest question standing: why is there a critical coupling at all, and why does it sit at exactly K = 2/π — a clean ratio of small integers and π, with no dial anyone tuned? This module is the answer. It is a story about defects: little whirlpools in a field of phases, the energy it costs to have one, the freedom you gain by letting it roam, and the knife-edge where those two logarithms exactly cancel. That knife-edge is the wall the whole library keeps returning to.
This is an advanced module. It assumes phase, winding numbers (you will meet old friends — the loops you drew there come back as whirlpools), and the resultant. Everything else is built here, including the site's missing plain-words account of free energy.