Yesterday (Thursday), the car ran fine. Overnight, the battery went down, tested at 6 volts. I charged it up, and the auto parts store confirmed that the battery's fine. I reinstalled, and now:

a) the emergency lights are flashing, which the dealer says is the alarm system going off without the horn beeping (thank goodness for not beeping!), and
b) the car doesn't recognize either of the key fobs, offering the message, "key not found."

I've got a Monday appointment for the dealer to look at it, but does this sound like something I can correct myself? Can I convince the car to recognize the key fobs that it knew just yesterday?

Sounds like the immobilizer in action. Did you try locking/unlocking the door with the key blade (assuming that exists with this model)?
--
-- '96 850T at about 240K, 2005 XC70 at 100K, 1999 V70 at 200K; gone but not forgotten '97 850T5, '83 244 DL, '81 245 and '64 Amazon

I just went out and checked the battery, which held voltage fine overnight.

I then reconnected the battery. The car is no longer freaking out/flashing its lights. I then locked and unlocked the car a couple of times with the physical key (key blade), but, alas, the car still does not recognize the key fob.

Dang.

The last post in this thread is not very encouraging ...
https://www.volvoxc.com/forums/showthread.php?18047-Key-not-found-message
Seems like others have been able to solve the problem by disconnecting the battery for ten minutes, but you've already done that.
--
-- '96 850T at about 240K, 2005 XC70 at 100K, 1999 V70 at 200K; gone but not forgotten '97 850T5, '83 244 DL, '81 245 and '64 Amazon

Oh, THAT's comforting! We've never before had this problem, so I assumed it had to do with the battery being disconnected/reconnected. I hope it'll be a one time thing.

Oh well. Monday, someone from AAA gets the joy of coming off our narrow street, backing up our narrow and steep driveway (steep at first, changing to not so steep), loading up the car, and taking me to the dealer, where Serenity* the Service Angel will have a loaner car waiting for me.
--------
*Yes, that's the name her parents bestowed on her. It's a great name, and she's a lovely, competent person. In my limited contacts with her, she's really shown promise - I bet she'll be a service writer down the line.

Klaus pointed out that I hadn't updated my posts to report on the dealer's work on the key fob problem. Long story; short version here.

AAA refused to tow the car, because the driver didn't feel he could safely put the tow dolly under the front wheels. After a couple of days of frustrating exploration of ways to address that problem, Jen, of the marvelous couple who own the car repair place across the street, found the ritual necessary to prepare the car to acknowledge a key fob (something the dealer's service writer didn't know): use the manual key to unlock and lock the car; unlock and lock the car using the buttons on the fob/remote; unlock it again, get in, and start the car.

That worked for one fob (my wife's), but not for mine, so I drove it over to the dealer, where they spent a day trying to figure it out. The mechanic finally got it to recognize my key fob, and, a couple of hundred dollars later, I picked up the car. A day (or was it two?) later, I went to move the car, and it failed to recognize my fob again. It now recognizes my key fob for purposes of unlocking the car - sometimes, not always - and for adjusting the seat to my preferred settings, but not for starting the car.

At the time this occurred, we had so much going on in our lives that I just called the dealer and said, "You're gonna need to do this over, but I'll call later," and we went on with life. We were scheduled to go on our first ever cruise starting October 27, to celebrate our 50th anniversary (the illegal one - when we started living together); instead, we spent October 27 driving the RV to a town south of us, under a mandatory evacuation order due to the fire in our county (the Kincade Fire, for those following along at home). Maybe next week, I'll find the time to schedule the car for more attention at the dealer's; but I plan to ask the service writer to contact regional Volvo to see if they're aware of this kind of behavior.

I am NOT convinced that the convenience of the magic key fob outweighs the inconvenience when the software goes nuts. I certainly don't remember ever experiencing this kind of hassle with old-fashioned keys.

You might just have a dodgy ignition switch, not unknown on these cars and similar V/S70. It might just work with one and not the other due to wear. The wife's key might well be much less worn than yours. I'm showing how little I know about these cars if I ask if the car will start using the "manual" key? Does it switch on anything electrical?

No, the manual key is just for unlocking the driver's door. And we don't plug the fob into the dash slot, having learned that the plastic on the fob swells when exposed to petroleum distillates (including even the mild ones in hand lotions); we rely on the radio conversation between the fob and the car. Something's wrong in that radio back and forth.

That linked post was from 2011. Hopefully, there is a TSB regarding the problem. You didn't mention if the remote batteries have been changed. My daughter's and wife's backup key both had dead batteries after 5 years of sitting on the shelf.

Yes, I checked the batteries in the remotes. Second step, after determining that the car battery was healthy.