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.
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.