Oops! Dropped copper injector seal ring into valve compartment
You are correct; most likely that copper ring is sitting down at the bottom of the valve-cover right beneath that injector 'tube', well, at least for now that is.
Pretty much impossible to recover unless you pull the valve-cover since copper is non-magnetic, and there's just no clearance to get down in there really, although, if you have one of those flexible 'grabber' tools...you know, ithas the handle with metal cable attached, and when you push button or pull lever, then out comes a set of four prongs that can grab small stuff like nuts/bolts/o-rings, etc. They market those to mechanics as a recovery tool to be able to get into tight/deep places, etc. Definitely worth a shot, IMHO. I've seen them in auto parts stores and lots of other places too.
I dropped a steel nut from my injector #6 tensioner down into 'the abyss', so I went fishing too, but since it was steeI, I made a magnet 'search tool' from a zip tie and tiny fridge magnet that fell out of something long ago, and some electrical tape both to hold the magnet and dull it down a bit, and to attach to the zip tie firmly, etc, given it was way down on the very end, and I heard it bounce when if fell in, so didn't know exactly where it ended up. So I had to check all around the injector seat region. It took some patience and trying different approaches for sure, but I got it out. Best fish I ever caught!
Good Fishing!

I'm really curious as to whether the OP managed to find it!
So, unless you got data that points directly at the one cylinder misfiring/ injector being the fault, I'd be looking around for other faults like air leaks, airflow sensors, and the things that control the feedback loop for fueling. If you installed/coded a new injector with proper seals and the seat was clean, and torqued the retainer properly (on Gen 2 TDI motor the injector bolts are single-use only) I'd leave that alone. If you reused the old injector, then sure, the injector itself can develop internal problems, but do the little stuff like cleaning the harness connector to the injector and endure positive/locked connections, etc.
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So, unless you got data that points directly at the one cylinder misfiring/ injector being the fault, I'd be looking around for other faults like air leaks, airflow sensors, and the things that control the feedback loop for fueling. If you installed/coded a new injector with proper seals and the seat was clean, and torqued the retainer properly (on Gen 2 TDI motor the injector bolts are single-use only) I'd leave that alone. If you reused the old injector, then sure, the injector itself can develop internal problems, but do the little stuff like cleaning the harness connector to the injector and endure positive/locked connections, etc.
in the end replacing the bolt is an obvious need to do
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in the end replacing the bolt is an obvious need to do
The OE is just a standard, albeit high-pressure, multi-spray, Bosch fuel injector labeled with Audi part#; Bosch is the OEM and sells it under their brand for less, and also can be found as remanufactured for additional cost savings, but requires you pay a core fee/return the old injectors . It would still cost quite a lot per injector regardless, so if you got an unreasonably great deal on it then that'd certainly be suspect. The Bosch, in-house, OEM part # is not the same as the Audi labeled OE part#, but you may see the cross reference info shown on listings. It is however, the exact same part. The thing about eBay is you have to pay strict attention to whether it's OE, OEM, new, like new, remanufactured, or a recycled/used part. A used fuel injector could mean trouble.
Not aware of any delays in ECU 'learning' the new injector past the initial start/run-in drive cycle after the injector was coded/adapted to the car. I'd be looking at the fueling balance across all injectors and for other common sources of 'stumble' or roughness, which on a TDI basically comes down to the air metering and fueling. Don't forget about vacuum leaks though...very common for the junction at brake booster vac line to engine vac line to crack/leak, and that can also create similar idle and running roughness, especially when brake pedal is in use.










