You now know what a vortex is and that only annihilation can remove one. The question that decides everything: will a warm lattice actually contain free vortices? If yes, order is doomed — free whirlpools wander and scramble every long-distance agreement. If they can only exist as tight bound pairs, order survives. The answer turns out to be a competition between two quantities that both grow with the size of the system, both logarithmically — so the winner is decided purely by their coefficients. One coefficient is πK. The other is the number 2. The wall is where they tie: K = 2/π. This section builds that argument with both logarithms measured, not asserted.
First, the accounting concept the argument runs on — stated plainly, because the essays use it constantly and no page has stopped to define it. Energy is what a configuration costs: here, misaligned neighbors (the chord from primer 00 — bonds pay for disagreement). Entropy is how many ways a configuration can happen: a vortex pinned to one spot is one way; a vortex allowed anywhere in the box is millions of ways. Temperature is the exchange rate between the two — how much cost the ambient jiggling will happily pay to buy options. The ledger nature actually settles is their difference, the free energy:
F = E − T·S (cost minus options, at the going rate)
A thing spontaneously appears when its free energy is negative — when the options it opens outweigh the cost of building it. Not because anything wants disorder; simply because there are so many more ways to be disordered than ordered that, past a price point, disorder is what you get by counting.
Here is the miracle that makes the transition sharp. The cost: a lone vortex torques the field all the way out — arrows at distance r from the core must still tilt by about 1/r, so the energy collected out to radius R grows as πK·ln(R/a) (a is the core size, the lattice spacing). Not a guess: the figure below builds a vortex, adds up real bond-by-bond misalignment cost ring by ring, and plots it against ln R — a straight line whose measured slope you can compare to πK. The payoff: the entropy of placing that free vortex anywhere in a box of radius R counts (R/a)² sites, so T·S = 2T·ln(R/a) — the same logarithm, coefficient 2 (in the site's units, T = 1). Two straight lines on a ln R axis. Whichever is steeper wins at every scale at once.
The ledger is an argument about one ideal vortex. Below is a live thermal lattice making the decision for real: 28×28 coupled oscillators at temperature 1, each nudged by its neighbors (coupling K) and kicked by noise, with every vortex the field currently contains marked on top — orange cores winding +1, purple −1. Slide the coupling and watch the population change regime: weak coupling, a gas of free whirlpools; strong coupling, a nearly combed lawn where vortices appear rarely, in tight pairs, and vanish again.