[cyborgx7]

Here is the thing with that. Technically every individual instance of a problem is decidable. It's only in the universal case that they become undecidable.

Take the halting problem for example. Imagine two algorithms for determining if a program halts. One always returns true, the other always returns false. For every single program, one of them will be the correct solver, but clearly neither of them is the correct solver for every single program.

Therefore the counter-example necessarily has to depend on the specific details of the algorithm that claims to be the universal halting decider.

Strictly speaking, there isn't an universal counter-example, since it changes depending on what the algorithm is. There are just instructions on how you can always make one, no matter what the algorithm is.