Note that the halting problem isn’t very relevant here. You can take a much simpler problem, like computing the sum of two integers. By the same argument, it’s just as impossible to fully test a black box that claims to be an adding machine, but outputs garbage for inputs greater than some N. Moreover, you can’t always determine whether a box is an adding machine even if you’re allowed to look inside the box and inspect its algorithm, by Rice’s theorem.
Yeah, that’s what I kind of figured, too...
Note that the halting problem isn’t very relevant here. You can take a much simpler problem, like computing the sum of two integers. By the same argument, it’s just as impossible to fully test a black box that claims to be an adding machine, but outputs garbage for inputs greater than some N. Moreover, you can’t always determine whether a box is an adding machine even if you’re allowed to look inside the box and inspect its algorithm, by Rice’s theorem.