The blueTooth remote use in some way a 2-way communications. When you sync the remote with the PS3 it's really the remote that receive the PS3 'IP' address. This is easy to prove. If you try the remote on a second system it wants work. So you sync again the remote with system 2. If you go back to system 1 you'll have to sync again with system 1. In conclusion, the remote learn the 'IP' address of the PS3 and not the reverse.

That said, I can't determine out of doubt if PS3 and remote are in beat. After you hit a key they are linked together and this can be observed easily: the current reach few milli-amps in this case. However after few minutes of inactivity (no key press) the remote will go down to full idle state: less than a milli-amp. So the BlueTooth link is not maintained in permanence. This feature is to preserve battery life.

So the answer is no, the remote can’t determine out of doubt that the PS3 is ‘on’ or ‘off’.