The insane part is placing the burden on AI labs to advance mathematical understanding. Why should this be their responsibility?

If they want to simply generate theorems and proofs and post them on the internet, why not? Does doing that really inflict "substantial negative externalities ... on the mathematical community"?

Also, people would get mad at them (rightly) if they sat on a proof of a major unsolved problem for months while bringing in human mathematicians to convert it into a more human-friendly form

(They definitely should spend SOME time working with human mathematicians before putting out a press release, but that's exactly what they've done every time)

If math is a "burden" on AI labs, why do they willingly spend tens of millions of dollars for a purely theoretical and unphysical result? What's the ROI for that when the frontier labs are already notoriously cash-strapped? Their motives pervade the entire discussion and cannot be ignored here.

Presumably some combination of marketing existing models and training/improving future models.

My point was about what is and isn't the responsibility of AI labs. Why should they be required to invest more (or fewer) resources into math than they otherwise want to?

(If you ask me, safety and security are what they need to invest more into.)

Simple. Because they are benefiting from output of the math community, in fact they were required for creating these models in the first place.

...Could you not ask this of anyone who works on pure math problems? What on Earth are you talking about??

"Academics are paid paltry salaries, university departments are cash-strapped, why are they studying these abstruse math problems instead of building rockets?"

Successfully advancing fields of math and science is a good thing. Why would they not do it. Why would they not want to prove their increasingly intelligent computers are able to solve unsolved, complex math problems humans have failed to solve for centuries and then release the solutions to the public. What do you want them to do, not try to solve them?

They are not advancing the field. At least they didn't in the case of the N-S debacle. The model most likely used the conversation text of a mathematician which had discovered and was pursuing a new avenue towards the goal, then pumped millions upon millions to hurry up ahead to an opaque solution which leaves no-one the wiser, beyond just settling a theoretical question. The only impact on science is to deflate the potential minds who might have been motivated to join the effort.

Is it not more democratic, if AI companies announce new theorems and proofs, and the discipline of mathematics furthers "our understanding" - instead of that the wealthiest universities in existence (all of them in the US) use their advantageous access to AI to produce them, along with their exclusive furthering of our understanding?

In a world, where expensive AI models running thousands of agents produce the most remarkable results, how should the discipline of mathematics look like, when money and access to compute plays such a pivotal role?

With the Navier-Stokes solutions being out in the wild, any mathematician at an impoverished university can make a stab at it and help our understanding. Assume the solutions being held back by those who produced them at Harvard & Co. (until they have their nice articles ready), what would the role of mathematicians be at less affluent universities?

I do not assume AI companies will be ready to provide equal access to their AIs to worldwide mathematics.

Isn't easier to understand something backawards? It seems much simpler to understand something once you already have the solution.