Navier–Stokes Lost in Translation
arxiv.org
arxiv.org
"In particular, we show that the formalised Lean proof does not correspond to the NL proof of blow-up of solutions to the Navier-Stokes equations."
So these authors seem to be claiming that OpenAI has not really proven Navier-Stokes at all. If I get their idea correctly, they are claiming that the LLM has not formalized the original "natural language" idea of Navier-Stokes correctly. If true, it would mean that their purported Lean proof is not actually a proof of Navier-Stokes at all, but something that is an incorrect translation of the original natural language idea. If correct, this is a really bold claim and I would like to see if other researchers agree.
Did you mean “not…correctly”?
;)
https://github.com/google-deepmind/formal-conjectures/blob/8...
No one is disputing that the Lean formaization of Navier-Stokes is correct, so we should have high confidence that the generated Lean proof is valid.
The authors are claiming that the Lean proof is not the same proof as the NL one. Therefore, we shouldn't yet have confidence that the NL proof is valid.
This is an important claim which the math community will need to work through. However, the Lean proof alone is sufficient for OpenAI to (reasonably confidently, leaving aside questions of academic manners) claim to have proven NS.
A proof of a theorem is different from the statement of the theorem. OpenAI has a Lean proof of the statement. That is all they need. There may be many different proofs of this statement, including NL proofs. It does not matter that these NL proofs may or may not be different from the Lean proof, at least for the correctness of the Lean proof. But of course the NL proof may be wrong. But who cares?
Indeed. If only some of those people would see the irony.
What matters most of all, as any first year student of mathematics would know, is whether the formal problem statement corresponds to the NL statement. TFA specifically states that at least some of the allegedly proven formal statements DO NOT.
Does the paper give a single example of one of the OpenAI solved theorems with a Lean certificate where the Lean statement does not correspond to the actual statement from the mathematical literature? I don't think so, but in case I am wrong, feel free to provide that example.
> To demonstrate the effect of this result we provide several examples of AI mistranslations of NL statements and proofs into Lean in practice, resulting in mismatches between NL proofs and their Lean ‘verifications’. These include OpenAI’s announced Navier-Stokes proof.
Could /I/ be mistranslating the paper’s formal statement to NL? I don’t think so, but in case I am wrong, feel free to cite the correct formal statement that they claim as divergent between Lean and NL formulations by OAI.
[edit: typo]
The people trying to understand the proof are probably following the natural language version. So they care.
I wouldn’t be surprised at all if that’s how this paper (which I did not read) arose.
Why are review papers published? Executive summaries? "Introduction to X" books?
People's time and computational resources are finite. Summarising information -- ideally in structured ways that preserve important properties, but even in informal, unstructured ways -- is critical for making any kind of progress in this world.
In order to prove security, you must first simulate the universe.
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).
humans as a whole have always known the weakness of natural language is in its precision. In a way this isn't strange that this issue has come up.
"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])
> 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.
These authors don't seem to be disputing that this Lean formalization of Navier-Stokes is correct. I don't think that gives us any new information about whether the generated Lean proof is or isn't a valid proof of this N-S blowup thing.
NS is continuous approximation to what is otherwise a discrete system. Particle collisions are discrete time events that are averaged over time. NS loses accuracy for very, very, very very low fluid densities and energies.
AI "proving" that this approximation can numerically "blow" up does not mean the approximation loses validity.
What OpenAI purports to have proven, as I understand it, is that certain initial conditions to that equation, plus "forcing" over time (which could be a literal force acting on the fluid such as stirring it with a spoon or some other extrinsic effect) only have finite (and therefore physically plausible) solutions for a finite period of time, after which singularities appear, with the velocity or pressure of some of the fluid approaching infinity as you approach the finite time limit.
This is a result that Terry Tao conjectured in 02014, but without the forcing: http://arxiv.org/abs/1402.0290
I think we can be pretty confident that the L∃∀N proof is really about Navier-Stokes. The question is whether what it says about Navier-Stokes is what we think it says.
More interesting, you can ask a vision model to color parts of speech in img2img and it works OK for frontier models.
I don’t think that actually explains the idea of a momentum density transport equation well at all.
From the paper: "A third possibility is that the NL proof provides stronger statements than what the formal proof actually establishes, with (of course) different proofs. The latter happens in OpenAI’s announced proof of blow-up of Navier Stokes equations."
What the examples seem to show is that the proof method is different between the natural language proof and the lean proof. Which, if the lean proof actually proves blowup, would suggest that the natural language proof is subtly wrong, but the strategy was close enough to be used to create a real lean proof.
A little worrying, but part of the purpose of formalizing things in Lean, it forces you to be more accurate than natural language does. It's surprisingly common for major theorems to have slight inaccuracies early on that can be repaired. Famously, the initial proof of Fermat's Last Theorem had a flaw that took a year to repair (though I think that's unusually difficult).
So the most fundamental question is: does the Lean theorem faithfully state the right theorem?
For what it's worth the initial lean specifications for the top-level theorems generally come from human written formalizations such as in https://github.com/leanprover-community/mathlib4/blob/021ce6... so we can be reasonably confident about their correctness.
No, the other way around. The natural language proof was derived from the lean code, badly. This is my experience with using claude and lean to prove things. Its natural language explanations drift a lot from the lean, both before and after. But the lean code is the lean code.
Was it? Are you claiming a LLM does reasoning in lean or what? Since this (and all the other proofs by OpenAI etc) have been in the reverse order [1]:
> 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.
That is definitely interesting because how do you know the 88 hours of work are correct before you throw another 17 hours of lean formalization work on it? You could end up just finding out there was some hallucination in the original work.
If your code compiles, are you sure it's bug free?
Need to prove a forall statement? forall x, P(x) is the same as a function taking x and returning the proof that P(x) is true.
Need to prove an exists statement? Create the pair (x, h) that gives the actual x that proves the exists, along with a proof that it satisfies the property you claim.
Maybe the only weird thing is that there are types and sets, so sets are kind of automatically more of a "subset" of some type.
The actual Mathlib is more generic, but once you get a hang of writing definitions (as you do in intro proofs), I've found that you can pretty naturally translate whatever you'd have in your undergrad notes. And undergrad should cover defining integers, rationals, reals, relations, functions, sequences, limits, derivatives, integrals, etc. Even if they've never studied solving PDEs, they'd have to take multivariable calculus and know enough to be able to write one (assuming they take at least single variable analysis+linear algebra)?
The proofs can get involved and tedious with all of the extra bookkeeping, or techniques to try to reduce the bookkeeping (tactics, etc). But the definitions and statements are pretty much what you'd expect.
So the Lean proves something and the question is whether that something is actually what we care about — or something similar, but ultimately not the question.
Humans have made similar mistakes too. A human writes a specification for how things should work, the human translates that into code, the code does not work, and finally the human fixes the code and forgets to fix the original spec.
Moreover, the paper claims that the NL arguments of Navier-Stokes are stronger than the Lean ones. My understanding is that the translator LLM got lazy and wrote the minimal amount of code that satisfied the theorem without the extra stronger claims.
It is common in mathematical papers to say "And by the way, this actually proves [stronger claim]", but this is something an AI with a precise goal of performing a translation would never do, as it's goal is to translate the proof, not to quality mathematics.
That said, a gap between the Lean proof and the pdf is annoying for interpretability, and interpretation is a valid aim, but that does not factor into the proof's validity.
Really? You think 300 lines of Lean code [1] is just as hard as the proof (or even remotely close)? Also note, as the README says [2], that the theorem was written independently by formal conjectures, not by the LLM.
[1]:https://github.com/openai/NavierStokesAndEuler/blob/main/Com...
[2]: https://github.com/openai/NavierStokesAndEuler/blob/main/Com...
"Disclaimer: We do not make claims about the correctness of OpenAI’s NL proof, we only make statements about mistranslations into Lean."
_Assuming_ two failure modes:
- The lean kernel could always have a bug. - The formalized statement may not correspond to what _mathematicians_ "actually wanted"
It seems natural to make the argument of, "Well, even if you make the argument that the proof can have mistakes, it's surely easier to check the problem statement of something rather than the solution".
(A "nice property" is that, the agent doesn't need to even get "subarguments correct" according to the _second_ criteria - maybe in the natural proof it invents an object subtly different from the formal one, but it all checks out. If you guarantee that the _original_ statement corresponds, then the only possibility is the lean kernel. So it doesn't recurse infinitely, in this case).
But "definitions" are always a really weird thing that I don't think we have good theories for? How do you quantify how much descriptive power you need to express a question? Often times in math, the hard part is getting the definition right - but what if the definition itself starts to become so complex and unverifiable that no one can correspond that to anything? Well, it seems like many interesting long-standing math problems have "relatively" simple problem statements, in such a way that you could formalize it to lean easily, but not sure if there's really a silver bullet w/ lean or if it's going to be turtles all the way down.
It probably doesn't matter as long as AI keeps skyrocketing on the much more general property that is "intelligence", but still. Interesting to think about.
(Well, this is where AIT gets actually interesting, but still, I don't think its a generalized theory of semantics.)
Can someone tell me in simple terms why this doesn't conflict with the incompleteness theorems?
edit: thanks for the responses, i feel slightly less dumb now
We know as a consequence of Goedel theorems (at least I believe so), that there is no algorithm that would take a statement and output a proof if it is provable or a counterexample if it is not. However, AI provers never give anything for sure, so I think there is no contradiction here.
(The algorithms described above are of course completely impractical, taking time exponential in the length of the proof (of theorem or counterexample).)
The sort of head canon for any of the automated proof systems is that lean saying a proof is correct is "if lean is correct then the proof is correct".
One can get the temptation to try and prove lean correctness with lean but I believe _that_ is impossible due to the incompleteness theory.
_But_ the discussions I had, everyone was kind of in agreement with the idea that you could continue to shrink down the "kernel" of lean with lean (or whatever proof system really) so that in the end the thing you have to trust is pretty small.
The formalization went through, but there were _several_ mistakes in the original paper that it uncovered, from type setting errors to (many) formulas that quantified over all resources as printed, but actually applied to only arising resources in the calculus..
So the formalization did give me a formally verified borrow checker that I could use to build a programming language on top of, but it was _not_ exactly the borrow calculus that was printed in the paper.
I expect this is the most common experience when mechanizing a printed paper. There are a lot of skipped steps and handwaving.
The scary thing is when AIs generate unreadable formal proofs and then effectively lie (or fabulate, to be polite-ish) about the natural language version of the steps. Since the natural language version is arguably the most important aspect of a solution to a flagship problem, this fabulation deflates the value of the solution while the existence of the solution discourages further work on the problem.
I think a lot of math notation isn't wrong given a context, so in theory we should be able to translate it into something formal. Maybe also generate living documents where you can e.g. write `h : some_claim := by details(by rw[nat_mul_comm]; ...)` and the renderer hides details just like you'd write "obviously" in a traditional text. If the reader wants, they could then expand the details. etc. I found that many codex-generated proofs could be improved by telling it that I want a sequence of steps
have next_step := by <I don't care>
have therefore := by <still don't care>
Lean's VSCode plugin is I think only starting to explore the idea of a proper IDE for math. There's probably still tons of unexplored potential for like that fused with Matlab or whatever.
I've been criticized for doing this, but to me it emphasizes how much attention goes to the hot, wrong papers.
We should be very careful about relinquishing sorting through such details to AI.
https://terrytao.wordpress.com/2026/10/04/on-classical-solut...
Humans will have to wade through mountains of slop to decipher the argument. Alternatively, they could just ignore it like Mochizuki's ABC proof prior to the Scholze/Stix refutation.
The idea that an AI company is beyond peer review is harmful.
i havent seen this sentiment expressed anywhere, have you?
isn't this comment chain on a submission about openai's claims being reviewed?
are people not reviewing openai claims right now?
openai themselves specifically call out that there may be issues with their results. journalists and laypeople just happen to skip that part, like they do with ~all physics, health, astronomy, etc results.
This is far more efficient and they’re telling the academic industry to grow up
Sister comments are saying that academics dont like the Lean programming language and see a lack of human language described proof. Doesn’t sound like something I should care about but I’m watching for a better human language description of the problem as this discussion evolves
The proof was released in the spirit of being first at all costs without any attempt to clean it up. I doubt that OpenAI mathematicians could give a coherent talk about it, certainly not using a blackboard.
No. The way to build confidence that your software is well made, you do a proper external security audit and obtain the requisite certificate from a proper auditing firm.
It's also incorrect to think peer review in mathematics is low quality (like it is in some other fields). Certainly, when major results are in place, editors ensure that high quality peer reviewers are recruited and do their job properly. Like all human processes this fails sometimes, but not enough to not do it.
which specific openai statements does this part of your analogy map to?
in the "sharing ai progress in mathematics" blog, openai simply says "results", and never once claims that all of them are unquestionably true. instead, they state they want to evaluate the results. their github states that the results are "different stages of verification" and also says "Some of the unformalized results could have issues"
that is the opposite of "claiming [...] their software is secure", to use your analogy.
A good review does not merely check the correctness of logical arguments, it gives suggestions for the exposition, citing the correct references, putting everything in the right context, etc.
Prestige to the reviewed, not to the reviewer.
Good, I just wanted to point out that peer review isn't primarily an arbitrage of truth, it is also to make sure the exposition is nice to read. When you get a reviewer who actually cares, you receive lots of feedback that isn't related to the correctness of Lemma 3.14.15 and stuff like that.
Peer review is a proxy for correctness.
Peer review journal is a proxy for quality peer review, or at least it was, once upon a time.
yet i have never seen anyone say "the idea that physicists are beyond peer review is harmful" because some mainstream news articles published a piece about dark energy or whatever.
https://github.com/google-deepmind/formal-conjectures/blob/8...
Maybe read the comment before replying, at a minimum.
that's not the claim. the formal statement of the problem for the NS proof was written by humans not autoformalized.
Also, it doesn't seem that they are questioning the truthfulness of either proof, just that they are different?
Actually, they are questioning whether the natural language description of the proof is either not faithful to the formal proof, or simply wrong, or both.
The point of the article is that natural language is not these things.
There is an abstract javascript machine which can obviously be translated to hardware instructions. "Obvious" in the mathematical research sense, as in the statement is flat out wrong under closer inspection. Websites regularly crash under memory pressure.
But they are good enough most of the time, and that's the key. With more resource a more correct program can be created, but cost-benefit calculations show a ceiling. A local pet store does not have the money to pay 2000 hours of formal verification work, and usually a WordPress site is good enough.
Mathematicians work with their brains. They have finite time to understand papers. Formalists say that all theorems could be reduced to logical axioms, but most of time it's jumping at a much higher level, because no one has time for minute details. Proofs are theoretically right or wrong, but in practice there are "slightly wrong" proofs, where there are some minute errors that "feels" like can be correcte, and most of the time it can be. This ambiguity is not a problem, but a feature of mathematics, because it means more time can be spent to move faster at a higher level of abstraction, but this would be lost with Lean.
If you understand the Lean, then you can create a NL proof. The LLM clearly doesn't understand the Lean code it produced.
It seems the AI wrote code in Lean that proves there are solutions to Navier-Stokes that can blow up, but...
the AI's explanation of the code, in natural language, does not correspond to the Lean proof!
That is... so rich in irony.
This is what I've been wondering about with LLM proofs. Math is logical, but mathematical writing is still natural language: symbols get overloaded, conventions go unstated, and a lot rides on context. So a model can translate a statement into a formal system and prove it, and the proof can check out, while the statement it proved isn't quite the one the mathematician meant. I read this article as a caution that some of the LLM proofs announced so far may not hold up once a human checks what was actually proved. Is that a fair reading?
Edit out vulgarity
> gotcha bitch!
You may have misdiagnosed the problem.
It's not the form language that is the real problem here. It's the ambiguity on the other side and the extreme difficulty of doing a useful and accurate translation.
Before it was dropping databases or deleting repositories. Now it’s subtly changing the meaning of math problems to get a correct but irrelevant answer.
> 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.
So it suggests that the formalization/verification step may have fixed some issues in the natural language proof, and either such differences were never noticed or the corrections weren't ported back to the NLP.
Oh, well. I suppose I should avoid getting involved in these AI threads, but now it’s about half the forum.