I just can't find anything that it should hook up nor can I figure out what it is from the wiring diagrams...

This is a non-turbo S4.

 

Congratulations, you're the one millionth person to ask this question!

Seriously, you are...

The blanked hole on top of the thermostat housing is where the temperature switch for the auxiliary electric fan goes, and the wire is for that switch. The fan was fitted to autos and Turbos, and cars that didn't have the fan didn't have the switch, just a spare wire. You can ignore it.

 

Cool, this means that I'm not just a complete idiot and left something off the car. On the downside, it means that's not what's causing my idle problems.

But after working on this for 3 months, simply having it running is enough for me to celebrate...

 

Also .. shouldnt you win something for being the 1 millionth person?

*gives high five*

 

Of course! I win: a running engine!*

*engine must still be tuned, may not be emissions legal in all service areas. Damage to trunk must be repaired before vehicle can be licensed and driven. Void where prohibited by law.