Here's the verdict:
Pulled battery.
Pressed buttons on dead fob to make sure it was dead.
With dead fob in hand pressed button on door: nothing
Used key to get in
Ugly computer displays "No Key"
With dead fob in hand tried ignition: nothing
Stuck my finger in fobgina
while pressing ignition and holding the fob NEAR the fobgina :nothing
Interestingly, when I stuck my finger in the fobgina the computer displayed the "Clutch" screen but that was as far as it went.
I would bet that there is some sort of RFID in the fob.