The full answer is: they cannot return if there are only finitely many balls, but they *can* if there are infinitely many.

Let’s first assume that there are finitely many balls. As Thomas pointed out, we can assume that the center of mass is fixed. Let’s consider R defined to be the distance from the center of mass to the furthest ball and call that furthest ball B (which ball that is might change over time). R might be decreasing at the start—we might start with B going towards the center of mass. But if R decreased forever then we would know that they never return to their starting location (since R would be different)! So at some point it must become at least as large as it was at the start. At that point either the derivative of R is 0 or it is positive. In either case, R must increase forever onwards—which again shows it can’t return to its original starting point. Why is it always increasing from that point onwards? Well, the only way for the ball B to turn around and start heading back towards the center is if there is another ball further away than it to collide with it. But that can’t be, since B is the furthest out ball! (Edit: I see now that this is essentially equivalent to cousin_it’s argument.)

For infinitely many balls, you *can* construct a situation where they return to their original position! We’re going to put a bunch of balls on a line (you don’t even need the whole plane). In the interval [0,1], there’ll be two balls with initial velocity heading in towards each other at unit speed, with one ball at the left edge of the interval and one ball at the right. Then do the same thing for each interval [k,k+1]. When you let them go, each pair in each interval will collide and then be heading outwards with unit interval. Then they’ll collide at the boundary with the next interval with the ball from the next interval. That sets them back at the starting position. I.e. all balls collide first with their neighbor on one side, then their neighbor on the other side, setting them back to their starting position.

The key question is your second paragraph, which I still don’t really buy. Taking an action like “attempt to break out of the box” is penalized if it is done during training (and doesn’t work), so the very optimization process will be to find systems that do

notdo that. It might know that it could, but why would it? Doing so in no way helps it, in the same way that outputting anything unrelated to the prompt would be selected against.