> [T]he group that produced the Navier–Stokes resolution involved on the order of 10,000 concurrent agents.
On the Navier–Stokes Millennium Prize Problem
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Re: On the Navier–Stokes Millennium Prize Problem
#12> Across all attempted problems, the agents sent 4.9 million messages and used about 300 billion output tokens Don't even try to do the math on how much that would cost at normal API prices. And we don't even know how much more expensive this internal-only model would be!
Napkin math if we assume gpt 6 astra on max is >$15 million (just for output tokens) for those wondering.
over 5 days, you couldn't achieve that level of testing and communication with humans on such a complex problem in that amount of time.
some might go so far as to call this a country of geniuses in a data center.
Re: On the Navier–Stokes Millennium Prize Problem
#13It seems like some other mathematicians (not affiliated with openAI) have also (or close to) done this. A statement was posted about the surrounding events by one of the them: https://cims.nyu.edu/%7Etristanb/statement.pdf
Also Terrence Tao's post: https://mathstodon.xyz/@tao/117233528517340774
Re: On the Navier–Stokes Millennium Prize Problem
#145 million messages, 300b output tokens, done in 5 days, and achieving something humans couldn't.
the first "Country of geniuses in a datacenter" moment.
Re: On the Navier–Stokes Millennium Prize Problem
#15"While unlikely, we cannot rule out that de-identified data derived from their usage of our products helped improve our models."
I think they should be able to unravel whether or not any sessions by Tristan or Levent went into the training data for this model.
Re: On the Navier–Stokes Millennium Prize Problem
#16> We’re sharing a solution to the Navier–Stokes existence and smoothness problem, one of the Millennium Prize Problems. This proof, produced by an internal OpenAI system, shows that the dynamics of the Navier-Stokes equations for fluid motion can develop a singularity in finite time. We’re sharing both a writeup of the proof and a formalization in Lean.
WOW?
Re: On the Navier–Stokes Millennium Prize Problem
#17From the methodology section:
> At all times we maintained the same strict safeguards that we apply to all our frontier model evaluations, including monitoring and isolation.
Looks like they're shifting away from the "unprecedented hacking ability" backroom-PR strategy into more benevolent messaging.
Re: On the Navier–Stokes Millennium Prize Problem
#18> Our goal in releasing this result is to report on the substantial progress of our AI models. We do not intend to claim the Millennium Prize for this result.
Does OpenAI have a policy of not claiming math prizes like this, or is this them trying to avoid any concerns (right or wrong, I'm sure we will hear more in the future) about how they got there?
Re: On the Navier–Stokes Millennium Prize Problem
#19the context here is super important, for those who haven't seen it yet. OAI maybe just trained on a real researchers solution and then celebrated having scored the goal unassisted save for the brief commentary at the bottom of this blog post. Here's the other side.
Re: On the Navier–Stokes Millennium Prize Problem
#20> Across all attempted problems, the agents sent 4.9 million messages and used about 300 billion output tokens Don't even try to do the math on how much that would cost at normal API prices. And we don't even know how much more expensive this internal-only model would be!
Napkin math if we assume gpt 6 astra on max is >$15 million (just for output tokens) for those wondering.
Yeah but they at least they got to steal $1 million from that nasty math prof who didn't want to remove his co-author.