I'm not a mathematician or any other kind of scientist. Just a toiler in the software mines, and an onlooker.

> It goes entirely against the culture of research mathematics to hoard secrets. If you've done something interesting, you want to share it, and it's in your best interest to share it.

You're describing an enviable human culture with a reward system suited to people. People publish papers etc. to enrich the human domain and to get recognition. AIs don't care about things - and neither, I'm asserting, will their owners. That culture will become a vestige.

Mathematics as the underpinning of science and technology that can be exploited for inter-state competition, including cryptography but not limited to that (image analysis, targeting algorithms, conflict modelling, whatever) won't see the light of day. The AIs that discover the mathematics will be paid for by the warfighting budgets of states, and by the corporations that feed on them.