You can't come up with a counterexample for P != NP because there isn't a formula to disprove. For P = NP you would propose a general algorithm to convert all NP problems into P in P time, and an AI could then find a counterexample which would disprove that particular method. To demonstrate P != NP you need to prove that no possible algorithm can convert any NP into P which is much harder than providing a counterexample.

AI has just gotten to the intelligence that it can make clever counterexamples to mathematical conjectures, but the frontier isn't quite smart enough that it can make novel contributions to mathematics. We are really close though. Only a matter of months away.

The paper does not resolve P versus NP, but it does make an important advance in a closely related area. To prove that P ≠ NP, it would be enough to show that every algorithm for an NP-complete problem requires superpolynomial time. We cannot prove anything remotely that strong. For explicit NP-complete problems in unrestricted computational models, we cannot even prove superlinear lower bounds. There is therefore an enormous gap between the lower bounds we can prove and the superpolynomial bounds we would need.

VP and VNP are closely related algebraic analogues of P and NP. Here the paper proves new lower bounds for computing the permanent, a VNP-complete polynomial, in particular models of arithmetic computation: roughly (n^2\log\log n) arithmetic gates for unrestricted division-free circuits, and (n^4/\log n) size for the more restrictive formula model. These are still polynomial bounds, so they do not separate VP from VNP. But lower bounds on the resources needed to compute explicit functions are exactly what would ultimately be required for such a separation, and meaningful lower bounds of this kind are exceptionally rare.