It goes something like this. Modern ECUs have multiple maps which they work with to get you what you finally get in terms of drivability/power of an engine. At the very basic level there are two maps, a base map and a correction value map. These maps exist for all variables that an engine has (fuel/timing/boost) at the very least. Once the ECU power is cut, the values the ECU has stored when taking the base map, applied the correction values map and gotten the final values the engine seems to like, this information is erased. It gets rebuilt as you drive the car and can take a few miles to get back to proper values as you'll have to hit multiple load points on each of these maps to get it all perfect. There are literally hundreds if not thousands of possible combinations once you hit all these load points.
With that part understood, the next variable is what the tuner does in terms of modifying the tune to get additional gains. Are they modifying the base maps or the correction value maps or possibly both? If the tuner modifies the base maps, then the difference in output should be felt immediately with only a minor improvement over the adaptation period. If the tuner modifies the correction values map, then there will be no felt difference initially and only after adaptation will you feel an improvement.
This is how I have seen ECUs from other platforms work in the past and I'm sure that while a bit more complex, the current Audi ones work the same way at the core concept.
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