There is a point where the force becomes net zero and it isn't immediately after compression. I acknowledge that. It is pushing back initially but not for the entirety of contact.
Jaden
Well, that just has to do with the resiliency of the tip. Different materials will yield different results. I have no idea how a leather tip springs back. There's no fundamental physical reason why it would have to be so, though, other than at the very tippy end of the decompression...because if not, the tip would fly apart, so clearly it's slowing down on it's own. How much, starting where, etc is a physical property of the tip and nothing else, really.
But if it's in contact, it really must still be pushing, and I can give a bit of a nuanced argument why if you're interested.
For the tip to be in contact with the cue ball, yet impart no force, it must be expanding at precisely the same rate that the cue ball is traveling. Let's neglect friction, because we know that all this happens in a very short amount of time, and slowing down due to friction is negligible (and will actually only slant the problem more in my favor).
We already know that the rate the tip expands MUST go back down to zero very quickly, or else the tip flies apart. So think about this: for the tip to be in contact and not push, it must be traveling at constant velocity to just barely keep up with the cue ball. That would be an incredible coincidence. What really happens is that any time the tip is in contact, it IS pushing. The only reason they're still in contact is that the tip is pushing on the cue ball, and the cue ball's not getting out of the way fast enough. There's a instant in there where the tip's velocity (due to the being on the cue + it's expansion) finally slows up enough that the cue ball skitters away, but it's overwhelmingly likely that any time the tip and cue ball are in contact they're pushing on each other because it would require theoretical perfection for it to happen any other way.
I hope that makes sense. I know it's not easy to follow and is kind of a hand wavy sort of argument, not a proof.