Sorry for being vague. My problem is experienced when the computer sleeps or is powered off. Ultimately I'd like my computer to sleep and remain asleep unless sent a magic packet or woken via mouse/keyboard (if possible), but the first step is getting it to stay off when powered off completely.
I turned the core and PC off overnight, and both remained off the next morning. This same test I performed with the core on and resulted in the PC being on in the morning, though I still haven't looked at the logs to figure out what time it woke (how long it stayed off for).