> vague existence claim of a program that cannot be concretely written
It can be concretely written. It's either "return True" or "return False". As of today we don't know which one of the two is this. However our lack of knowledge is not relevant to the question if it's computable or not. Definition of computability doesn't rely on it.
> unless it takes "P==NP" and "P!=NP" as inputs, which would be a tautology.
It doesn't take any inputs at all since none are necessary. It's a constant function. The question of computability isn't relevant for constant functions or for functions with a finite set of possible inputs -- they are computable as switch-case on all possible inputs.
The problem only appears with functions that have an infinite set of possible inputs where the switch-case approach won't work. You need an _algorithm_ to convert inputs to outputs and sometimes this algorithm just can't exists, regardless of how far we've got in proving truthfulness of various mathematical statements.
> vague existence claim of a program that cannot be concretely written
It can be concretely written. It's either "return True" or "return False". As of today we don't know which one of the two is this. However our lack of knowledge is not relevant to the question if it's computable or not. Definition of computability doesn't rely on it.
> unless it takes "P==NP" and "P!=NP" as inputs, which would be a tautology.
It doesn't take any inputs at all since none are necessary. It's a constant function. The question of computability isn't relevant for constant functions or for functions with a finite set of possible inputs -- they are computable as switch-case on all possible inputs.
The problem only appears with functions that have an infinite set of possible inputs where the switch-case approach won't work. You need an _algorithm_ to convert inputs to outputs and sometimes this algorithm just can't exists, regardless of how far we've got in proving truthfulness of various mathematical statements.