Why Can't Reverse Hashing Prove $P \neq NP$? Lets assume a hashing algorithm that runs in polynomial time and produces a hash that is mathematically impossible to reverse. As far as I know, the $P=NP$ problem states that if they are indeed equal, then if an algorithm exists to verify a solution in polynomial time, then an algorithm exists to find a solution in polynomial time. In this case, we can verify a solution by just running it through the hashing algorithm and comparing the final results. However, there cannot exist an algorithm to find a solution, since the hash is impossible to reverse.
Give a polynomial time hash function that takes exponential time to compute the inverse of.


That’s an open problem. It can’t currently be used to resolve $P$ vs $NP$ because no one knows if there is an example of such a function. In CS, we often assume that $P\neq NP$ and that there are cryptographically secure hashes, but that’s an (experienced based) assumption, not a probable fact.