the extent of the source is off 0,0001 m. I have tried different ways of fixing this but none of them worked properly. Do any of you know the easiest way to fix this?
that is 0.00001 m shift or one tenth of a millimeter. Compared to your 4000 meter extent, I suspect that some floating point representation issue and the 'error' is virtual.
What are you doing that this value will be of concern?
What is your geodatabase XY resolution and tolerance?
The default is 1 mm (0.001m) and you are 2 orders of magnitude below this, and you are using a double sterographic projection which would not even have a accuracy remotely approaching this value