Which is why I said it was still a “never say never” and not an already solved problem.

The halting problem is impossible for Turing machines, but if hypercomputation ends up possible, it isn’t impossible.

For example, an oracle machine as proposed by Turing, or a ‘real’ computer using actual real values.

The latter in particular may even end up a thing in the not too distant future assuming neural networks continue to move into photonics in such a way that networks run while internals are never directly measured. In that case the issue would be verifying the result - the very topic of the paper in question.

Effectively, while it is proven that we can never be able to directly measure a solution to the halting problem, I wouldn’t take a bet that within my lifetime we won’t have ended up being able to indirectly measure a solution to the problem and directly validate the result.