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On the Navier–Stokes Millennium Prize Problem

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Re: On the Navier–Stokes Millennium Prize Problem

#921

Earlier quoted context omitted.

The implication from their last couple of published articles[1][2] is that they think they’ve achieved “recursive self improvement”. [1] https://openai.com/index/research-acceleration-view-inside-o... [2] https://openai.com/index/an-alien-mind/

Recursive self improvement of their upcoming IPO value maybe. They are fluffy PR pieces otherwise.

This comment was applicable 2 years ago. It isn't any longer.

Re: On the Navier–Stokes Millennium Prize Problem

#922

Buried under the drama is the fact that OpenAI is claiming that an internal model they’ve been training for less than two weeks is more than twice as capable in mathematics as Astra, which was only made public a week ago. Even if this improvement is limited to mathematics, that is an astounding feat.

Agent systems become most credible when they produce artifacts that can be independently checked, not when they merely produce persuasive explanations.

Re: On the Navier–Stokes Millennium Prize Problem

#923
post #413

Earlier quoted context omitted.

The point of lean proofs (as it stands) is simply one bit of information: that a given mathematical statement is indeed true. It's a way to be absolutely certain (modulo bugs in the lean kernel) that a proof you came up for a statement is indeed correct. It is really not meant to be analyzed, much less now that they are fully llm written.

Well, how do we know there aren't errors in their construction within the lean code? Does it just "not compile" or something, or is it deeper / more fundemental than that.

that's essentially it, if the proof is incorrect it does not compile which signifies a problem in some step.

Re: On the Navier–Stokes Millennium Prize Problem

#924

Earlier quoted context omitted.

Recursive self improvement of their upcoming IPO value maybe. They are fluffy PR pieces otherwise.

How can you possible say this sort of thing in context of what looks like a millenium prize being solved. I swear there's nobody blinder than those who won't see.

I don't think we should assume a millenium puzzle has been solved, yet. Astra showed impressive capacity for cheating when it was faced with impossible cybersecurity challenges. It seems equally plausible at this stage that it's found a bug in Lean.

Re: On the Navier–Stokes Millennium Prize Problem

#925
"Across all attempted problems, the agents sent 4.9 million messages and used about 300 billion output tokens."

At a conservative estimate of GPT 6 Astra pricing, this would have cost upwards of 15 Million dollars for anyone using the OpenAI API!

To me this is the one silver lining. Yes, they can solve millennium prize problems, but it still costs a fortune.

Re: On the Navier–Stokes Millennium Prize Problem

#927
post #127

This is utterly shocking. Even the AI optimists did not expect this to happen in 2026. Wow. Millennium Prize Problems were used as examples of something the current approach to AI just wasn't capable of, discussions that would result in "we'll need a totally new architecture".

> This is utterly shocking. Even the AI optimists did not expect this to happen in 2026. Wow. Wrong.

Right: https://x.com/dioscuri/status/2097418466571485272

Re: On the Navier–Stokes Millennium Prize Problem

#928
post #631

Earlier quoted context omitted.

Most experimental physics and other natural sciences are strongly driven by their theoretical siblings, i.e. in particle research nothing gets built without a solid theoretical foundation of what you expect to find (or where you expect existing theories to break down), the same is true in other areas, no one is doing an experiment in quantum physics before they have a solid theoretical understanding of the effects th…

Most of high energy theoretical physics is very non-rigorous or even hand-wavy. I think AI isn’t there yet for such problems.

Please make a benchmark for it, that'd be super interesting! My guess is we'd see models climbing it quickly, but maybe not

Re: On the Navier–Stokes Millennium Prize Problem

#929

Earlier quoted context omitted.

Why is it unlikely?

Because the models are trained on hundreds of billions of user conversations, across more than a billion different humans. The conversations are anonymized and not easily traceable back to a specific user. It's unknowable and not possible to prove if any one specific conversation contained the insights for solving Navier–Stokes. We also don't know if the authors unintentionally provided data to OpenAI through alterna…

Anonymized doesn't mean there's no way to know whether it is in there. My ballot is anonymized, but it's known to be in the box because a checkmark was put next to my name when my ID was verified. OpenAI can trivially check their account settings to know what happened to their chats. The fact that they are being vague about this likely indicates that they have already done so and discovered that the data did go into the training set.

Further, given that this is all in the open now, they can search the training data. No way somebody is using some specific unique cutting edge mathematical approach to solve a fluid dynamic problem 99.9% of people have never heard of and it's not locatable. Considering they spent $15,000,000 already on this, they could afford to grep around to be able to state that their hands are clean.

Re: On the Navier–Stokes Millennium Prize Problem

#930

"we cannot rule out that de-identified data derived from their usage of our products helped improve our models ." What a landmine sentence to bury in this report, you can't rule out your models were spying on other researchers?

I've been saying it for a while now, but no one gives a fuck. Let me repeat it again. THE BIG LABS CLEAN ROOM YOUR DATA (CREATE SYNTHETIC DATASETS ON IT), EVEN IF YOU OPT OUT, SO THEY CAN BYPASS COPYRIGHT LAWS AND THEIR OWN LOOSELY WORDED TERMS OF SERVICE. "TOS: We don't train on your data" -> Correct. They train on the synthetic version of your data. I guess we're just going to ignore this forever though. Who cares…

For sure. Even if it wasn't a measure to avoid copyright, you pre-process LLM training data to remove errors, characters that can't be tokenized, etc etc. Doing so with another LLM has been standard for a while.
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