The point is that you can’t use the result that there exists a hard function, since all you know is that the function is hard, not whether it’s in NP, which is a basic problem for your scheme.
Your counting argument for Turing Machines also likely have this problem, and even if not, I see no reason why I couldn’t relativize the results, which is a no-go for P vs NP proof attempts.
The point is that you can’t use the result that there exists a hard function, since all you know is that the function is hard, not whether it’s in NP, which is a basic problem for your scheme.
Your counting argument for Turing Machines also likely have this problem, and even if not, I see no reason why I couldn’t relativize the results, which is a no-go for P vs NP proof attempts.