How An AI math breakthrough ignited a controversy

Regardless of what you think of the priority dispute issue discussed on sibling threads, I’m highly skeptical of the closing quote that this Navier Stokes result means that the same approach of casually spending a few million on agentic computation is going to solve end to end materials design or drug development.

Those problems can’t be formally verified with an automated theorem prover. We have a lot of physics based simulation tools, but they tend to focus on small subsets of the full design problem and they make limiting approximations because otherwise they’d be too computationally expensive, or we just don’t have the right data to parameterize them beyond describing qualitative behavior. Agents are helping accelerate research in these fields but I think it’s mostly a different class of problem that’s a lot harder to specify and verify

The core section:

> However, communications quickly became contentious. According to Buckmaster, OpenAI offered to give him sole authorship on the Navier-Stokes solution—but only if Alpöge’s name was removed from the work and if the write-up would acknowledge the problem had been resolved by an internal OpenAI model. Buckmaster refused, in part because he was troubled by the question of what OpenAI's system had actually seen. For example, Buckmaster said the company did not initially give him a clear answer about whether its agents had access to the pair's logs on Codex (which is an OpenAI product).

> OpenAI executives have denied that any employee or AI agent saw the pair’s work before the researchers released it publicly on 7 September. But there still remains a separate question: Could the pair's work have reached OpenAI's models through its training data?

> OpenAI’s blog announcing the Navier-Stokes solution does not dismiss the possibility: “While unlikely, we cannot rule out that de-identified data derived from [Buckmaster and Alpöge’s] usage of our products helped improve our models .”

I think it was a pretty questionable thing to do by trying to front-run these researchers even if they didn’t make use of their techniques. The fact that they may have inadvertently “borrowed” their work via training data makes it much worse.

OpenAI’s behavior here — even if you only consider [their] side of the story — was (at best) in bad taste.

FYI: The problem at hand is an existence problem. No real construction of any real formula for the solution is provided, only a singular perturbation expansion.

In short: The problem is about whether a solution (of the NS Equations with external driving force) can be found that blows up in finite time. i.e. exhibits infinite velocity at a point even for a viscous flow.

The solution: Take a circular curl ansatz which shrinks in xy-direction and elongates in z-direction and see whether you can find linearized waves so that these waves show a blow up when propagated on the curl. Then prove that the higher orders of the perturbation are regular before the T0 singularity time and you have solved the problem. The external force is just the remainder of the NS-Equation right hand side.

It is a lot of tedious formula juggling of all the higher orders and some singular perturbation expansions. Perfectly suited for algebra systems. OpenAI was using probably python sympy for the formula work and the researchers had to guide the LLM what to do in higher mathematical language.

Here is the paper:

https://cdn.openai.com/pdf/32d9f210-8b73-45e0-91bc-82a30aef8...

when you upload it to chatgpt astra can explain what they do and why it works, have fun.

Extensive discussion on OpenAI's blog post on Navier-Stokes: https://news.ycombinator.com/item?id=49613262 .

Quanta Magazine article that also discusses some of the controversy: https://www.quantamagazine.org/ai-has-solved-one-of-maths-1-...

The whole thing reeks of the desperation of an unprofitable venture-backed startup looking for its next PR win to keep the wind in the sails.

But I think what’s being overlooked in the race to claim absolute credit is that both sides ultimately relied on a LLM (and one of OpenAI’s at that). Either a human researcher made a breakthrough discovery with the help of Codex, or the latest GPT model made a breakthrough with the help of human training data, or a little of both… either way it is undeniable that LLMs have quickly become an integral part of R&D workflows and are accelerating research.

This would be a major win for any normal company. You could even build a bigger collaboration with this guy, give him a big budget and push for extensions to this preliminary result, and in return do a write up on how he uses your model in his workflow. Huge PR win. What this says to me is that their valuation is so astronomical that they feel the only way to justify it is to demonstrate a fully autonomous discovery bot… which it simply is not.

> “I certainly don't expect the industry to continue to spend millions of dollars to solve problems in mathematics, because there is no profit in it,” Columbia University mathematician Michael Harris wrote in an email to Science. But he worries the highly publicized achievement will be “extremely damaging to mathematics; it convinces decision makers that human mathematicians are obsolete, and it convinces young people that their passion for mathematics has no future.”

LLMs seem particularly suited toward these existence-proof problems. Working mathematicians seem absolutely essential for universally quantified results, still. I strongly doubt, for example, that if Fermat's Last Theorem hadn't been proven three decades ago, that an LLM would be able to do work equivalent to inventing the mathematics as Andrew Wiles did to solve the problem. I have similar doubts about P vs NP, the twin prime conjecture, even the Riemann Hypothesis (unless the latter has at least one counterexample).

And I want to be clear: I'm not downplaying the achievements of these models. This is remarkable! I simply think that the pattern of success is in existence proofs or finding counterexamples, which makes sense based on how LLMs function and are trained.

> Navier-Stokes is one of six “Millennium Problems” on a list compiled by the Clay Mathematics Institute in 2000.

Seven, not six. One is solved already, but is still a millennium problem.

OpenAI apparently used Buckmaster and Alpöge‘s work w/ Codex to bootstrap “their” dis-proof. https://cims.nyu.edu/~tristanb/statement.pdf

Regardless of the end result, OpenAI's behavior would be a career-ending ethics scandal for a human mathematician. This bit alone would be a career-ender.

> According to Buckmaster, OpenAI offered to give him sole authorship on the Navier-Stokes solution—but only if Alpöge’s name was removed from the work and if the write-up would acknowledge the problem had been resolved by an internal OpenAI model.

I wonder if an appropriate response from the mathematical community would be a good old-fashioned shunning. Mathematicians are allowed to use OpenAI's tools as much as they want, but no one with any current or prior OpenAI affiliation gets published in a reputable journal, ever.

My impression: the re-aristocratization of scientific research seems inevitable.

"Breakthrough" to me would be like Isaac Newton inventing calculus to calculate pi. This feels more like $12MM in tokens was spent to add another digit to pi using the old way.

I have started to feel the sense lately, that first with the HF breach and now this plagiarism scandal, it is the straw that has broken the camel's back (so to speak). We have turned the corner and clearly entered the endgame - everything is going to unravel astonishingly quickly.

Other than the undeniable breakthrough in math, the important point is the ability to orchestrate 10k agents to productively work on a single problem, which creates options:

> OpenAI, meanwhile, says its experience with Navier-Stokes could open the door to solving puzzles with more practical relevance. “We are now able to spend millions of dollars on a problem that we really care about and that really matters: developing new materials, finding cures to diseases,” Bubeck said. “All of those things that we have been talking about for a long time—now they seem to be at our fingertips.”

OpenAI has messed up big time here by competing with their customers. It would have become the norm for humans and mathematicians to use the tools and publish bigger results any way. If OpenAI didn’t run for credit, this theorem itself may have been proven by Buckmaster OR others in maybe a year or two. But now the bigger issue than AI solving problems is the issue of chat privacy, at the end of the day.

>For the past year, Buckmaster and Alpöge had been using a variety of AI tools, including OpenAI’s Codex, to tackle the Navier-Stokes problem. Last month, their AIs had at long last found a solution to the Euler equations and verified it in Lean.

Their AIs?

Moving forward, I can't imagine other mathematicians wanting to have this kind of experience. So there has to be a shift away from these services.

This right here, or at least the thought of this, is why in the not so far future, businesses can't (won't?) be using these LLMs services.

You cannot risk companies like Anthropic, OpenAI or their business partners like Microsoft having unfettered access to proprietary data on your company/businesses.

It it likely that they or rogue employees will use the information to make a profit? It's pure speculation, but I'd say more than likely, and we will never hear about it or read it on the news unless there's whistleblowers in high enough positions to know about it.

Assuming you and your employees aren't careful with what data you share, they will have intimate knowledge about your company from files and conversations logs. Likely personal user data too which they'll gladly create databases to link to and create extensive profiles on you, your employees and your businesses.

It's not far-fetched to see them leveraging insider information shared with LLMs to play the stock market, leveraging data against competing businesses in other markets they might want to explore, and likely a bunch of other things that are escaping me right now as I write this.

At the end of the day it's on those people for sharing such sensitive data, but it's not like these AI companies are innocent and won't gladly exploit every little byte of data without telling you, we know it happens.

Who should get credit? All the humans who ever worked to create the data.

And the AI company for making the search program that searched through the data and found the solution.

Hmm, is the formal verification through? Lean just asserts no errors in the proof, but like can prerequisites not be fullfilled?

> OpenAI, meanwhile, says its experience with Navier-Stokes could open the door to solving puzzles with more practical relevance. “We are now able to spend millions of dollars on a problem that we really care about and that really matters: developing new materials, finding cures to diseases,” Bubeck said. “All of those things that we have been talking about for a long time—now they seem to be at our fingertips.”

Eh? There's no connection at all between the Navier-Stokes work and those things.

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Does it matter who gets the credit at this point? Both used AI to do 99%+ of the work. So... do machines have ego?

AI models and in particular LLMs are not capable of logic reasoning. See for example this paper:

https://arxiv.org/abs/2506.06941

Ergo, they can't prove any theorem whatsoever. How do people at OpenAI expect that we believe in claims like that? This is yet another before-the-IPO stunt in my opinion..

Personally, I won't believe any of these claims until the community of mathematicians says otherwise.