I don't think computability is the right framework to think about this question. Mostly this is because, for somewhat silly reasons, there exists a Turing machine that answers any fixed finite sequence of questions. It's just the Turing machine which prints out the answers to those questions. "But which Turing machine is that?" Dunno, this is an existence proof. But the point is that you can't just say "you can't build a device that solves the halting problem" because, for any fixed finite set of Turing machines, you can trivially build such a device, you just can't recognize it once you've built it...
It follows that in the framework of computability, to distinguish a "real halting oracle" from a rock with stuff scribbled on it, you need to ask infinitely many questions. And that's pretty hard to do.
I think Toby Ord actually has a good response to this objection:
It is worth responding, at this point, to a persistent argument that is made against the coherence of physical hypercomputation. It states that a machine must perform an infinite number of computations for it to count as hypercomputational. After all, for any function f on the integers, there is a Turing machine that computes f(n) for an arbitrarily large finite number of values of n. It is thus claimed either that we could not know that a prospective machine was a hypermachine after witnessing finitely many computations or that it simply would not be a hypermachine since its behaviour could be simulated by a Turing machine. Hypercomputation is thus claimed to be on shaky ground. However, it is easy to see that this argument cannot achieve what it hopes to since it does not just collapse hypercomputation to classical computation, but instead it collapses all computation on the integers down to that of finite state machines. This is because for any function f on the integers, there is also a finite state machine that computes f(n) on an arbitrarily large finite domain. Thus, as explained in Copeland [10], whatever force this argument is supposed to have with respect to hypercomputation, it must also have with respect to refuting the existence of classical (general recursive) computation. Since the latter is agreed to be on firm ground, it is difficult to see why this argument should count against the former.
It is thus claimed either that we could not know that a prospective machine was a hypermachine after witnessing finitely many computations or that it simply would not be a hypermachine since its behaviour could be simulated by a Turing machine. Hypercomputation is thus claimed to be on shaky ground.
The former suggestion seems like the more important point here. While true that the hypercomputer's behavior can be simulated on a Turing machine, this is only true of the single answer given and received, not of the abstractly defined problem being solved....
"Hypercomputation" is a term coined by two philosophers, Jack Copeland and Dianne Proudfoot, to refer to allegedly computational processes that do things Turing machines are in principle incapable of doing. I'm somewhat dubious of whether any of the proposals for "hypercomputation" are really accurately described as computation, but here, I'm more interested in another question: is there any chance it's possible to build a physical device that answers questions a Turing machine cannot answer?
I've read a number of Copeland and Proudfoot's articles promoting hypercomputation, and they claim this is an open question. I have, however, seen some indications that they're wrong about this, but my knowledge of physics and computability theory isn't enough to answer this question with confidence.
Some of the ways to convince yourself that "hypercomputation" might be physically possible seem like obvious confusions, for example if you convince yourself that some physical quality is allowed to be any real number, and then notice that because some reals are non-computable, you say to yourself that if only we could measure such a non-computable quantity then we could answer questions no Turing machine could answer. Of course, the idea of doing such a measurement is physically implausible even if you could find a non-computable physical quantity in the first place. And that mistake can be sexed up in various ways, for example by talking about "analog computers" and assuming "analog" means it has components that can take any real-numbered value.
Points similar to the one I've just made exist in the literature on hypercomputation (see here and here, for example). But the critiques of hypercomputation I've found tend to focus on specific proposals. It's less clear whether there are any good general arguments in the literature that hypercomputation is physically impossible, because it would require infinite-precision measurements or something equally unlikely. It seems like it might be possible to make such an argument; I've read that the laws of physics are consiered to be computable, but I don't have a good enough understanding of what that means to tell if it entails that hypercomputation is physically impossible.
Can anyone help me out here?