Say, for the purposes of conjecture, that you have the necessary handwavium to uniformly cool it to any temperature you want, and decide to cool the water down to a temperature infinitely close to $0°$C.
What happens as $\lim_{T \to 0°C}$?
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Sign up to join this communitySay, for the purposes of conjecture, that you have the necessary handwavium to uniformly cool it to any temperature you want, and decide to cool the water down to a temperature infinitely close to $0°$C.
What happens as $\lim_{T \to 0°C}$?
Nothing would happen.
You could even cool to $-1 ^\circ$C without a phase change. Then, if nucleated maybe by a bump, a small fraction of the water would crystallise to a solid.
The molecular mechanics of particles are happening all the time as you go from one phase to another. Nothing about the processes changes as you lower the temperature, just the rate they occur.
Assuming you have some appropriate nucleation sites, then even at temperatures above freezing, some of the water molecules will bond to each other and join up. There's a chance that some other water molecule nearby will do the same. But at temperatures above freezing, the rate this happens is slower than the rate that one of the already-joined molecules leaves.
As you approach the freezing temperature, these two rates become similar. If you cool below freezing, then the only change is that the formation rate is now greater than the breakdown rate.