My point is indeed that "you can't compare [arbitrary sized numbers] in constant time" is a statement about some model of computation, and not all of them.
So again, the issue is not that you can't compare numbers of arbitrary magnitude in constant time -- I never said that. The issue is that it is a contradiction to say that it takes linear time to inspect all the digits of a number AND that it only takes constant time to compare O(n) digits of two numbers.
You're essentially saying that you can axiomatize your mathematical universe with nonsense axioms, which yes, you could do, but that's not useful to point out.
Your statement was "If the numbers can be arbitrary in size, then you can't compare them in constant time". I was merely pointing out that this is not a true statement.
Again, for the third time: you cannot "assume" that reading the digits of a number takes O(n), but reading the digits of two numbers to compare them takes O(1). That is absolutely, obviously true. Your response, that you can "assume" the latter is true as part of the model of computation is just wrong, plain and simple. There's nothing else to say about it.
The original claim directly implies this, and if you can't address that point, then just save us both the time and don't respond.