whatA field of arrows, each easing toward its neighbours' direction (XY coupling) while temperature rattles it -- seeded Langevin dynamics, deterministic to the last flip of noise. Around every lattice square the piece sums the winding: a full +2pi turn is a vortex, marked with a plus core; -2pi an antivortex, marked minus.

whyThe claim is the 1973 result that won a Nobel: defects here are not free. Below the Kosterlitz-Thouless temperature a vortex and its antivortex are BOUND -- they come in pairs, orbit tightly, annihilate; the census stays near zero and every plus has a minus close at hand (the HUD measures the pairing distance). Cross T_KT ~ 0.89 and the pairs UNBIND: defects proliferate and stride apart, melting the field's stiffness without ever magnetizing -- a phase transition with no order parameter, only topology. Click injects a fresh pair: cold, it collapses on itself; hot, the crowd swallows it.

howOverdamped Langevin on the lattice with a seeded Gaussian noise stream; defect detection by plaquette winding; pairing measured as mean distance to the nearest opposite charge. The auto-story holds cold, heats past T_KT, and cools again. One integer seeds every arrow and every rattle.