In a previous question that I asked here it turned out that for every instance of the halting problem, being the matter whether a certain computer program halts or runs forever, there exists a consistent theory that decides and proves this instance. It also turns out there exists a sound theory that decides and proves this instance. For example, consider the program that checks the Goldbach conjecture and halts when it finds a counterexample. If this program never halts, then a proof for it exists in the theory that consists of the Peano Arithmetic axioms plus the axiom "The Goldbach checking program never halts". This theory trivially proves the non-halting of the Goldbach checking instance, but the problem is that mathematicians would not have trust in the soundness of this theory, even though the theory may very well be sound and consistent. For this reason I will make a new attempt to formalize my question:
Is there a proof and a theory for every instance of the halting problem, in such a way that the proofs and the theories that arise would be accepted by arbitrarily large and complex theorem provers as being correct proofs in a sound theory?