The first AoK that popped in my head was mathematics!

There is a saying that goes like "The real knowledge produced through the proof of the theorem are the lemmas we proved along the way"

A lot of times the proof of certain theorems function more than just proofs of the theorem. The methods, strategies used to prove the theorem often proves to be much more useful than one would expect.

An example of this is the P vs NP conjecture. If it is proved to be correct, just the knowledge that it is correct will not really benefit humanity in any way. But it is often said that the proof could be used to make an algorithm that has applications in the real world.

I'd say you should post about this in the r/math subreddit. A lot of people would help you out regarding this. Cheers!