When you take the battery out of the remote, the car stops recognizing it until you perform an initialization procedure in which you manually unlock the car and then press a button the remote. My untested opinion is that the battery removal resets the security code of the remote to some random value and that the initialization procedure teaches it to the car. This would makes sense, as it would allow Audi to carry generic remotes without need to keep track of codes.

Note that having to unlock the car manually to perform the initialization prevents just anyone from synching their remote to your car.

Anyone test this theory by synching someone else's remote to your car or vice versa?

If you look at he groove, you can clearly see that a router was used.

The machine is a computer controlled (CNC derivative) 2 axis minature milling machine. It uses an end mill.