I'm in my final year of my undergraduate studies doing work on modelling the n-body problem numerically and I also have some interest in theoretical guarantees. Now, I've been looking for a theorem that guarantees the following:

$$\text{KAM stable} \implies \text{smooth trajectories} \tag{*}$$

I haven't come across such a theorem so far but I'm certain it must be true.

Note 1: I haven't read all the literature on KAM theory but I've read a few introductions on the subject(ex. Jacques Fejoz).

Note 2: Smoothness would rule out collision or non-collision singularities but it doesn't imply KAM stability as there are smooth trajectories which aren't KAM stable. Ex: all solutions to the three body problem besides the figure eight solution. So the converse is obviously false.