Could the uncertainty principle theoretically be violated at 0 K?

Question

Ok so please excuse me if the following mental argument is completely ridiculous or obviously flawed :P I was reading about how, even at 0 K (assuming we could experimentally reach such a temperature), there must be some slight particulate motion, because of the uncertainty principle. What about if we have some particles in a closed box that is in a dark vacuum (assuming a complete absence of any energy or light). If we then proceed to cool the particles to 0 K, then wouldn't we have succeeded at violating the uncertainty principle? We know that the speed of the particles is exactly 0 m/s, and we know with 100% certainty that they are in the box; we do not need to observe them to know so.

Answer 1

You say that the particle is closed in the box. By the time you limit the linear dimensions of the particle, you already fall under the reign of the uncertainty principle. If the box is, for simplicity, a cube of linear dimension $L$,

$\Delta p_i \ge \hbar/L$, where $i$ stands for $x, y, z$.

From now on, you have no control on $p_x, p_y$, and $p_z$ ? You can cool the box, so what?

You say "We know that the speed of the particles is exactly 0 m/s,". No, you don't know.