16GB 4S Shuts down randomly

Question

I've got an ongoing saga with my new-ish iPhone 4S.
Seemingly at random, my phone will shut down as though the battery has been completely drained. This can occur with anywhere from 17% battery to 45% battery, and it always happens the same way. The phone displays the Apple logo, vibrates twice, then displays a white circling 'please wait' animation, then goes black. When I plug it back into a power source, it shows a red, totally depleted battery and begins to charge. A second or so later, it boots all the way back on and shows that it actually has much more battery -- 17% or 45% or whatever it was when it shut down.
I have tried reseting, restoring from a backup, restoring from cloud, and restoring as a new phone. I have also had two replacement phones from my numerous trips to the Genius Bar. My most recent move was to delete all of my local apps and uninstall and reinstall iTunes. I've been keeping my phone at default factory settings (not even a new ring tone) to see if it happens again.
It happened again this morning. My phone was restored two days ago from a known-good copy at the local Apple Store. Since then I have never plugged it into iTunes at home and have not downloaded any apps, any music, or even used the internet browser for anything. I think I've maybe made one or two phone calls.
Anyone have any clue what's going on? I'm about to go back to the Apple Store for the seventh time in three weeks, and it's becoming increasingly clear that the folks there are just guessing at this point.

Answer 1

The serial numbers match and were never mistyped, so Apple gave up and sent me to Verizon. I spent an hour with Verizon last night. From them I learned that Vodafone actually owns Verizon, so all Verizon iPhones have CTNs belonging to Vodafone. This doesn't exactly jive with what two Apple Geniuses and a store manager told me, which is that they've never seen Vodafone listed on a Verizon number. Verizon eventually decided that they had no idea, and their only advice was to give me a replacement and offered to hook me up with a less-than-new (but thoroughly inspected!) replacement phone. Yeah, thanks, no.
The only thing left to do is go back to Apple and get another (actually brand new) replacement phone. In the incredibly remote chance that I had a bad app that somehow corrupted my hardware (by magic, or something), I'm hoping that getting a new phone that has never ever had any contact with my old iTunes, those apps, or the old corrupted profiles or previous phones, will finally solve the issue. If this doesn't work, I will assume that my phone has basically half the battery that it is supposed to, never allow it to drop below 50% capacity, and eventually end up going back to Droid phones the next time I'm up for an upgrade.
Or those Nokia Windows phones! Those look nice, right?