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rebooted the machine this morning, and every time the folding begins, the cores crash.
Well...stop doing that!!! 
The most likely cause, IMHO, would be a corrupted WU. Probably one that didn't get written correctly at that first shutdown. There is a pretty active <TeamOSX forum site> that would get you more expert info, of course. If you aren't too far along with the WU (and even if you are and it's corrupted) you can delete it and get a new one. 
If you've been to those forums, you will notice the frustrating way you get from one place to another. The pages will open in a new tab/page but the one with the original link will also open at the new page leaving without the original list of topics.
That may be the way you use forums but I like to keep the list of topics available all the time. You can still do that at these forums by always using the 'back button' to return to the list. Several have complained about this behavior but no one seems to be able to do anything about it. Apparently their moderators/administrators are not as concerned about their users as some other sites I visit (like TS!)...
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Is this something you can do on your computer?
Absolutely! That's the whole point of these kinds of things. The technical name is "shared computational power" or "shared computing." By using thousands of individual computers, sometimes more can be accomplished than by using one huge, very expensive, dedicated computer. And it's almost free for the researchers as we volunteers do this for nothing but the pleasure of helping science/health research. And you are correct SETI is uses the same principles.
Come on in, the water's fine! You don't have to be online 24/7. Don't even have to have the computer on 24/7. You'll simply get work done quicker if you do that and may see your 'rankings ' climb faster.