I've already run the diagnostics that came both with the system and with the Apple Care package, which has its own diagnostics tests, and everything passed. I also reloaded OSX.
I suspect the fault is in hardware, and it might be something simple like a loosened connection somewhere. I think that because the machine seems to boot up more often when it's warmer, if that makes any sense. (When I turn it on for the first time in the day, I get a gray screen, with nothing on it. If I immediately turn it off and try again, I usuallly get a gray screen, but sometimes the Apple logo, but then no little time spiral, whatever you call it. After I've gone through this for a few iterations, it'll boot, and then everything runs fine. If I skip all the turn-ons, and just turn it on and let it run for a couple of minutes with the gray screen, then turn it off, it'll often boot all the way on only the second try. The only reason I can think that it'd do this is that the heat from the machine is actually firming up a connection somewhere; but then, I really don't know what I'm talking about.)
I'm taking it to Apple repair tomorrow. I really don't care about the money as much as the principle of the thing. It's as if you bought a health insurance policy, paid up in full, and then found out you couldn't get coverage when you need it because you lost the original receipt, written years before, even though you still have the policy itself in your hand. What I really need to do is get my ass kicked (which I'm sure the repair bill will do for me) for taking the word of a sales clerk who was probably hustling t-shirts the week before, and gym shoes the week after.
I'm still seriously annoyed with Apple because, after all, it IS their policy, and if they have people out selling it, those people should know about the key components of the deal.