"Lukas, I don't understand your objection at all. How does disrupting the physical adding machine mid-process prove that it isn't doing addition? One can also disrupt electronic computers mid-process..."

I didn't say it doesn't do addition; I said that it doesn't do the same addition that the 'theoretical' adder is doing. That's what Elizier called 'artificial addition'.

All you can say is that the physical adder will most of the time do a 'physical addition' that corresponds to the 'theoretical addition'; but you need to make a lot of assumptions about the environment of the physical adder (it doesn't melt, it doesn't explode etc.), and those assumptions don't need to hold.

You don't need to make assumptions for the theoretical adder: You define it to do addition.