N identical oscillators start at random phases. Each instant, a controller may push every oscillator's phase at whatever rate it likes, as long as the total power spent right now stays under a fixed budget. The greedy law spends that budget, every instant, on whatever push grows the population's order parameter r fastest right now. It's exactly optimal for two oscillators — and provably not, for more.
The order parameter is R = (1/N)Σ e^{iφⱼ}, a single complex number: its length r is how synchronized the population is (1 = perfectly in phase), its angle is the population's average phase. Greedy control at power budget P sets each oscillator's push to uⱼ = −c·sin(φⱼ−ψ), where ψ is R's angle and c is scaled every instant to spend exactly P. That's the direction of steepest instantaneous ascent of r — the obvious thing to do, and it's genuinely optimal when N=2.
The right-hand plot is the actual point of this page: it traces R(t) as a path through the complex plane, for the same random starting phases, at three different power budgets. They're the same curve. More power only travels that curve faster — it never bends it. That's the geometric shape of greedy's failure: rescaling the budget reparametrizes time along one fixed path, so no amount of power ever lets greedy reach synchrony by a shorter route. A smarter law would have to bend the curve itself, not just travel it quicker — and greedy structurally can't.
each dot is one oscillator's phase · the arrow is R, the order parameter
three power budgets, same starting phases — watch them trace one curve
"Sim time" is arbitrary — natural frequencies are all zero, so nothing moves without the controller, and only the ratio of power to time means anything. Doubling the power budget roughly halves the time to sync for the same starting phases; watch the three reference curves confirm it while sharing one path. Synchrony is called at r≥0.999, not exactly 1 — r only reaches 1 in the limit, never in finite time, once oscillators get close enough that the greedy push itself shrinks toward zero.