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: I discussed this with my supervisor, a physicist, and his response was that he had no idea but it would be useful if it were true.