Why does my Mac Studio never start the Screen Saver and wakes up randomly after sleep?

This seems like it ought to be straight forward, however my Mac Studio NEVER starts the screen saver, and only rarely shuts the display. What is more, if I use the apple menu to sleep the system, it often wakes randomly, sometimes even before I can leave the room.

Wake for network access is off. No streaming sites are open. Mouse has new batteries and should not be sending spurious signals.

Can someone tell me what I am doing wrong?

[Re-Titled by Moderator]