Fireflies flashing, heart cells, a bridge full of pedestrians — independent oscillators, each nudged a little toward its neighbours. Below a critical coupling they stay a blur; above it they snap into one rhythm all at once. Slide the coupling and watch the swarm find a single beat.
Each oscillator’s phase obeys dθᵢ/dt = ωᵢ + (K/N)Σsin(θⱼ−θᵢ): every other phase pulls it a little. The order parameter r = |mean of e^{iθ}| is 0 for a uniform blur and 1 for perfect lockstep. The instrument integrates the real coupled system live; a fail-loud self-check runs it to steady state and throws unless r>0.95 at strong coupling (K=4) while staying low at K=0 — synchrony as an emergent phase transition, present in no single oscillator.
Identical natural frequencies are used so full lock is clean to see; with a spread of frequencies the transition is sharper-studied (the Kuramoto 1975 critical Kc). The integration step and count are illustrative; r’s definition and the K=0-vs-strong contrast are exact.