AP=NP?

So assuming AP (all problems) can become NP (solvable problems where an answer can be verified), and then assuming you could have NP=AP within any given set of problems, there must be an algorithm and mechanism being used to solve them. This would create a logical paradox in the following way:

Could such an algorithm/mechanism conceive of a problem such that NP would not equal AP? If not, then the inquiry itself means that NP does not equal AP because hypothetically such a problem could be verified as NP does not equal AP if it were found (NP). If instead a problem could be found by the entity such that NP does not equal AP, than that solution (problem) would itself be the proof that NP does not equal AP. 

So then the question is, is it possible for NP to equal AP, like various perspectives suggest (God, Theory of Everything, advanced AI with time travel capabilities, etc). It is unclear to me how from the omniscient and omnipotent perspective how AP would not equal NP, given infinite continuous time.

ADDED NOTE 06/23/21

I am now realizing that if time were infinite then this would present a paradox. It seems the only solution that would not present a paradox is if the answer to this problem were toggling in the sense I suggest in this post: a new problem is found that makes total solved problems less than the sum of all problems, at which point it is solved by the algorithm, or alternatively no problem can be found that creates a new unsolved problem, at which point the algorithm must shift in order to find one that can be found, in order to solve its problem, which at that point is to create a problem. Either problem then returns the system to AP=NP (which seems to be the state of complete omniscience), and necessitates its return to identifying one of these two problems. One could see that that if an omniscient entity went back in time, carrying knowledge of the solution for the next x problems that would be encountered, that the state of AP=NP could be achieved for a time for the set of problems that originally existed there, but one's own perception of this problem set from their new (time traveling) perspective would necessarily alter the problem set, so their own personal timeline would be taken into consideration, for the perspective of overall amount of time passing when they currently are.

The only way for this toggling state to exist (disregarding time traveling concerns for now) is if time continued to evolve in some way. Since time would be an agreement based concept, I could possibly see a scenario where in the realm of authority omniscience existed (outside of time) but by adding the concept of agreement to an area (expanding time's bubble) this would create the potential for the examination of new types of problems, allowing for the AP=NP consideration to be run. This would only be possible though if there we actual permutations possible within the area of time that could not be completely predicted from outside of it (free will, two(+) rollers). In this way the expansion of the time bubble itself would act as a barrier to omniscience, but it seems like it would do so by adding possible things to know to a field that was already known (the realm of authority shifting to the realm of agreement), rather than frustrating any knowledge that was already gained.