So, I called support. "I'm sorry, we can't help you with this, because it's over 90 days. This call will cost $50 if you want to get it diagnosed."
Wtf? It's *FOUR MONTHS OLD*. So, I'm going to have to pay probably the entire worth of the bloody thing to get it both diagnosed, and then fixed. And ... I didn't do anything to it!
edit later: I did order the "Fix me" box from Apple, via the website referenced in many comments because of ralphmelton's help. But by the time I finally got to that part, I had to run out the door to go swimming, and didn't get to edit this entry... I am now! Just a bit late.
So at this point, I guess I get to wait for the thing to be fixed. Sigh.