Next token prediction is still how pre-training works, but that's a relatively small part of the whole training run nowadays. Most of the time is spent in post-training reinforcement learning scenarios, where models are scored based on an evaluation of their ability to complete a task, not whether any specific token correctly predicts a known next token.
Generally, anyone talking about next token prediction being some limiting factor of LLM capability is a good reason to ignore them, as they haven't updated their understanding of LLMs since 2023 or so.
All that happens in reinforcement learning and everything from pre-training to generation, including the last token, at the end is about adjusting the probability of the next token given the sequence.
Anyone who thinks reinforcement learning changes how the generation works hasn’t worked on inference engines and you can dismiss them as inexperienced and misguided.
I think there's a question of semantics here. RL is way more expensive in terms of compute, as you have to generate a significant amount of output before you can judge whether it's good or bad. So compute is one angle, but another is how much impact RL has on the weights vs pre-training
Apparently the impact of RLVR is generally bigger, otherwise AI companies would spend more of their compute budget on pre-training. The former seems to be more effective.
So, the AI wrote a NL proof of Navier-Stokes, then incorrectly auto-formalised it to Lean, but still ended up with a verifiable proof of Navier-Stokes? That seems... strange?
This same thing happened back in the 10 advances in math and CS release a month or two ago. The non sofic group construction relied on false prior literature. They realized this and fixed it in the lean program but didn't modify it in the writeup, so the written proof was both incorrect as stated and did not correspond to the lean proof. I was surprised how little press it got at the time, it seems like a huge risk factor.
Or the AI wrote a NL proof of Navier-Stokes, began rewrite in Lean, then discovered a false / handwavy / easier to write in Lean / etc approach of some parts of the proof, and modified it accordingly. Since there wasn't any backpass from Lean to NL to include any changes it did due to any of the above reasons, the proofs aren't identical. That's what I think is most likely.
If the reason for the differences was done intentionally in Lean (as opposed to hallucinate e.g. m+4 vs m+5 as mentioned in remark 3.2), then a simple recording of differences, and then afterwards pass back any changes to the original NL would fix the issue. If it was hallucinated, then there is no guarantee it wouldn't keep hallucinating, and thus you might never end up with the same proof no matter how many passes you do back and forth (see remark 3.4).
I don't know if we can really be clear about the order of things, but I think even without AI maths is filled with "someone provides a proof of X, and the proof itself is wrong/incomplete but X itself is true".
"Incomplete" proofs might be a way of viewing this. You have a NL argument to prove X. It turns out the NL proof has holes you can drive a truck through. So... you go around and patch the holes.
The resulting proof is different! You can start off with a bad proof and find a correct proof. Sometimes.
EDIT: for French speakers (maybe autodub gets you there) I saw a very nice simple case of this recently. A commonly stated proof for a relatively simple theory. The proof has a giant hole in it, and completing it requires some work [0])
e.g. one of the lemmas in the NL description is false, but a weaker version of the lemma (that does hold) is sufficient for the proof, so the (incorrect) lemma is never formalized.
It's not actually the same model that solved the problem that did the translation. Astra did the translation after the intenral model produced the NL Proof. As for the discrepancies, It's not necessarily right to think of this as 'incorrect formalisation'. Maybe it was essentially a 'proof refactoring'. Maybe Astra thought some parts could be easier expressed in a certain way, or maybe aspects of the NL proof were kind of handwavey etc.
Why do you assume this order? I would assume the AI starts with lean (that's what it was RL'd on at least) then tried to translate the proof for us humans, which is quite hard.
I'm not sure about it but it seems plausible given the potential error.
Edit: they explicite say that your order is correct in the post
I would expect it's the other way around. The AI wrote a Proof of NS in lean and write up a NL proof based on the lean one. The authors claim that openAI did NL -> Lean, but that is unsubstantiated.
> The agents arrived at their resolution on Saturday, September 5, about 88 hours after the first agents were launched. Lean formalization and verification took an additional 17 hours via GPT‑6 Astra.
This seems pretty clear that they found the proof first and then mechanized it afterwards.
Trying to map this observation to code and appreciate them starting with very simple examples (that Astra screw-fixed into lean.)
But in a simplified way this is close to promting the model to create a set with the members 1,2,1,5 and it correctly creates {1,2,5} in code. So The initial "proof" / instruction was wrong and it silently fixed that. ( Which is one of the failure modes they're describing)
Proof steps can become different in Lean but still work to prove the theorem. Incorrect proof steps in the NL paper could also be converted to correct ones in Lean. The paper shows some examples of this. So the NL paper can't be trusted even if Lean verification passes, but it doesn't immediately mean that the NL proof is wrong or that Lean is checking the wrong thing.
Or maybe the AI didn’t write Lean proof at all, or rather, not the LLM at least. But instead OpenAI has an internal traditional reinforcement model that is able to stumble on the Lean proof by the share amount of compute power available to them thousand monkeys on a thousand typewriter style. And then pretend LLM did it because that is what they are selling.
I don’t. But I do know how the scientific method works, and OpenAI’s display is anything but. Until what they have demonstrated is reproduced I take their claims to be nothing but marketing. A for profit company will lie in order to maximize their profits. Above I presented an alternative hypothesis, which is probably wrong, but until OpenAI’s results are replicated I will believe my alternative hypothesis just as much as I believes the claims of the for profit company making them.
In the absence of evidence, a guess is the best you got.
And, no until OpenAI’s results have been replicated, I don‘t believe companies own words about their own products, and therefor a marketing statement is not evidence.
It's the method they used to create the proof that's in question here. They claim their amazing product did it, and therefore you should buy their product because will do amazing things for you too!
He described himself a "violently anti-communist", and he saw nuclear war as an eventual necessity in a civilisational conflict between democracy and totalitarianism. If you view a nuclear war as unavoidable, it's easy to conclude that it's better to get it over with sooner rather than later, since a later war will be even more destructive. Especially since von Neumann was active in the 1940s and 50s, when the west had a slight edge in weapons technology, and before the development of the giant H-bombs. He and a few other pro-nuclear scientists argued there was a brief window where the USA could fight and win a nuclear war against the USSR, and they should take it.
While it's easy to dismiss that view in hindsight, it's worth considering that von Neumann had personally witnessed the rise of the Nazis and had to flee along with many of his friends and family. And he had seen that regime defeated only by military strength of the USA. And while the USSR and USA never did end up in a nuclear war, it really was only by a series of miracles that it was avoided. I think most historians of the 20th century would agree we got very, very lucky.
Von Neumann family was affected also by the communist "red terror" in Hungary in 1919, and by the subsequent "white terror" of the militant anti-communist authoritarian government after 1919.
What Ed-Tech companies? This article isn't talking about Ed-Tech companies. Axel Rauschmayer isn't an Ed-Tech company. He's an individual author who wrote books about JavaScript, the language. He wasn't pushing these "unnecessary solutions", he was helping developers understand JavaScript, exactly the thing you say is necessary. Josh W. Comeau isn't an Ed-Tech company, he's an individual who writes content about new CSS features like Anchor Positioning[1]. What do educators like this have to do with 1+ GB RAM frontends?
Ironically this article itself is a model of "life in 2026" based on examples cherry-picked to support the author's viewpoint. I wonder if he as open to being proved wrong about this exact thesis as he encourages his audience to be?
There was a level of patent/copyright protection, there was a level of awareness of stealing, a company could be taken to a court of law. A writer of a PhD thesis could show that someone lifted huge parts of a dissertation without attribution. All that is gone.
They could, but I could always hope they wouldn't, and most of the time I wouldn't be disappointed. AI companies don't even have means to attribute to someone the solution they're selling.
I don't think it was ever ignored that knowledge workers would be the first affected by AI. I remember hearing for a long time that lawyers, doctors, accountants, etc. would all be obsoleted by AI.
I do think many (including me) were surprised that software engineering would be the first kind of knowledge work to be successfully automated. We assumed it would be one of the last, because we'd be the industry using AI to replace everyone else, not the industry using AI to replace ourselves.
This is completely standard journalistic practice when reporting a quote on something that is currently contentious or unproven. For example, consider the two hypothetical headlines:
Activists decry police murder of protester
Activists decry police "murder" of protester
The first implies that the police murdering the protestor is an established fact. The second makes it clear that it is the activists who used the term "murder" rather than it being a non-contentious fact.
Both versions are fancifully direct in our era a media+police stenography, try:
Mayor denounces "radical activists" rioting and damaging city property in response to an officer-involved incident that resulted in one fatal kinetic altercation with a "loud and provocatively dressed" man.
Read McSweeney's guide to ambiguous grammar, where they lead you step by step from the quick brown fox jumps over the lazy dog to the ultimate say-nothing sentence speed was involved in a jumping-related incident while a fox was brown, and the punchline is you can then translate "the St Louis County Police Department was involved in an officer-involved shooting after officers came under heavy gunfire" to "a police officer shot a black person" (https://en.wikipedia.org/wiki/Killing_of_Michael_Brown)
But no, that's the language issued by police departments covering their asses. Journalists are a little bit covering their asses in making clear what is a quote and what isn't, but they're still responsible for what they choose to quote, and to be clear on whether they have verified the claims they're quoting.
Ideally, unverified claims shouldn't even be in an article. But when there _has_ been some kind of stramash, they just don't know who did what, the need to rush out an article to get clicks means they'll just quote both sides and try to give them equal prominence until they can do more verification.
oh like this article that says a child was hit by a "LAPD bullet" as though wild bullets roam the streets of LA looking for children instead of being fired from a gun by a person who made a decision.
It doesn't help when snippets are post leaving out the other context clues where things like "according to" or "anonymous source stated" or similar attribute of the quote. Just putting words between quotation marks does not lead me to believe they were quotes if there is no attribution. Otherwise, it's a Trump level "people tell me" type of "quote".
Prompting them yes, suggesting potentially fruitful research directions and so on, but the actual research was conducted by hundreds of agents swapping millions of messages and using billions of output tokens over 88 hours. The result being a huge Lean proof: https://github.com/openai/NavierStokesAndEuler. It's not just possible for humans to manually guide such a process in a meaningful way. They can set the direction and attempt to understand the result, but they solution itself must emerge (or not) from the agent swarm.
So yes, the models do seem to be "solving" the problems themselves, but not necessarily in the way we think of mathematical discoveries happening. Academic mathematics has historically been resource constrained: There are a limited number of top-level mathematicians, and they only have so much time and brain power to spend. So when approaching a problem, they are essentially forced to be as efficient as possible, not just searching for a solution, but for one that can be achieved within their cognitive budget. This induces them to develop novel techniques and abstractions, and it is actually those techniques and abstractions that tend to be the valuable part for further research, not the proof itself.
An agentic swarm is like getting a single skilled mathematician, cloning them a hundred times, then locking them in a room with the single objective of solving a problem. No longer constrained by time or brain power, they can approach it differently, using pre-existing techniques to gradually build their way to a solution. This process might not require a single intuitive leap or new discovery, and the solution will not be simple or elegant, but they will probably get there. It is more like a process of intelligently guided search than invention.
The OpenAI team didn't make a Lean proof. They brute forced a counter example. The "other" team was doing what you described but they haven't "finished" their work yet. Also, their Lean proof was for a simpler version of the problem, not the full NS.
Also, OpenAI wanted the actual mathematician taken off the resulting paper. I'm not sure I would describe what OpenAI did as research. What the other team was doing does seem to be more like research but the hardware was still in those cases mostly brute forcing things and then doing something like a genetic algorithm to compose an actual proof based upon the results of a large set of brute force attempts.
Didn't OpenAI make a Lean proof?
https://github.com/openai/NavierStokesAndEuler/tree/main/Nav...
"This repository contains Lean 4 formalizations of the results presented in “Finite time blowup for Navier–Stokes” and “Finite time blowup for the Euler equation” by OpenAI."
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