Navier–Stokes Lost in Translation (arxiv.org)
190 points by nill0 7 hours ago
ComplexSystems 5 hours ago
Aside from the usual squabbling about AI, it seems the bombshell claim is this:
"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.
mkarrmann 5 hours ago
No, they're not claiming that.
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.
nayroclade 4 hours ago
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?
famouswaffles an hour ago
latent-person 3 hours ago
runarberg 4 hours ago
nyeah 3 hours ago
>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.
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.
measurablefunc 4 hours ago
If it doesn't correspond to the original proof then you don't know what it is actually formalizing. It could be a buggy proof of ⊥.
aureianimus 4 hours ago
auggierose 4 hours ago
omnicognate 5 hours ago
If I understand the abstract correctly (big caveat), they aren't saying they didn't prove it. They're saying they gave two proofs, one in natural language and one in Lean, that are not equivalent to each other. I assume the main significance is that the Lean proof is not a formal verification of the natural language one and the natural language proof is not a readable explanation of the Lean one. Both of those things can be desirable, so to complete the set we'd get 4 proofs.
TeMPOraL 5 hours ago
But just to clarify: is either of them actually addressing the real Navier-Stokes, or will it turn out we'll end up with two pairs of proofs about something irrelevant to the actual problem?
fasterik 4 hours ago
kzrdude 5 hours ago
From computer science perspective the conclusion is obvious: untenable to have two representations without an exact translation or machine checked correspondence between then. All we have is a vibe translation using the LLM. The methodology should obviously be improved.
dgacmu 4 hours ago
lovasoa 4 hours ago
If I understand well, they mean that the thing they proved in lean is not Navier Stokes. And they don't make any statement about whether the natural language proof is correct or not.
nicf 5 hours ago
I read them as making a much weaker claim than this: not that the Lean proof isn't valid, just that it is not actually a formalization of the natural-language proof in the PDF they provided alongside it. I haven't heard any PDE people claim that the Lean proof is invalid, and I have heard things from a lot of them that imply that they think it is valid. (I'm a former research mathematician, but this is very far from my specialty, so I'm not really equipped to evaluate this claim myself.)
fasterik 5 hours ago
The claim is about the equivalence between two proofs and says nothing about the correctness of either proof. This seems to be confusing a lot of people.
pohl 5 hours ago
> has not formalized the original "natural language" idea of Navier-Stokes incorrectly
Did you mean “not…correctly”?
ActorNightly 26 minutes ago
Nobody has proven or disproven NS equations.
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.
stared 6 hours ago
For a refreshment of what is Navier-Stokes in a few words: https://p.migdal.pl/equations-explained-colorfully/#navier-s...
sleet_spotter 5 hours ago
This is so lovely. I desperately wish I could color code all math!!
stared 5 hours ago
You can. Not only the code is there, but also an interactive editor.
neutronicus 4 hours ago
Hmm.
I don’t think that actually explains the idea of a momentum density transport equation well at all.
buzzy_hacker 6 hours ago
If I'm understanding correctly, this is questioning the equivalence between the natural language proof and the lean proof, but not the correctness of the lean proof?
caughtinthought 6 hours ago
If the lean proof doesn't match the natural language one (which is the one the AI generated to solve the problem), it sounds like the lean proof isn't verifying the intended claim?
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."
hyperpape 5 hours ago
The material is interesting, but unless the statement that is proved in lean is not blowup for Navier-Stokes, then it's still proven.
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?
ammar2 5 hours ago
That assumes the natural language paper came first and then was formalized in lean. I haven't looked too deeply into how these labs solve these problems (or if they even specify this publicly) but you could also start with lean and then write the natural language proof based on it.
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.
dcre 4 hours ago
Not quite — the formal statement of the problem in Lean may be correct, and therefore the Lean proof gives quite a lot of confidence that the statement is true. It's just that the proof given in natural language doesn't necessarily match up with the Lean proof, so the natural language proof might be unsound even though the statement it's proving is true.
icedrift 2 hours ago
empath75 5 hours ago
> If the lean proof doesn't match the natural language one (which is the one the AI generated to solve the problem), it sounds like the lean proof isn't verifying the intended claim?
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.
latent-person 5 hours ago
caughtinthought 5 hours ago
OrderlyTiamat 5 hours ago
The lean proof being correct is easy to verify, whether it proves the thing we care about is much harder.
If your code compiles, are you sure it's bug free?
ndriscoll 5 hours ago
I'm pretty sure Mathlib has had enough human authored definitions to formalize the basic calculus necessary to state Navier-Stokes for quite some time? Some other problems admittedly need quite a bit of machinery built up to even try to say what the question is, but every undergrad learns multiple approaches to formally define everything necessary to write down a PDE.
lanstin 2 hours ago
nyeah 5 hours ago
jansport123 5 hours ago
syntax vs semantics
jrflo 5 hours ago
It doesn't look like they've found an error in the NL proof either, just that they are different?
kccqzy 5 hours ago
Indeed. The natural language proof is incorrect but the Lean proof is correct.
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.
kurtis_reed 5 hours ago
How do you know the natural language proof is incorrect?
zmgsabst 5 hours ago
Yes — because there are many non-equivalent statements that are easier to prove.
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.
empath75 5 hours ago
Yes, exactly. There's no real pressure on AI to get the natural language version of the proof correct, and no way to really judge it automatically.
kurtis_reed 5 hours ago
Yes however, whether a natural language proof and a formal proof "correspond" is subjective.
infogulch 4 hours ago
The paper shows that the Lean proof and the prose (pdf) proof do not match exactly. But if the Lean theorem Lean accepted is equivalent to original problem statement published by the Clay Institute, this mismatch is of no consequence to the validity of the proof itself. That's not a trivial if: stating the problem precisely is often as hard as the proof. Validation efforts should concentrate on whether the Lean theorem is equivalent to the one published by the Clay Institute.
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.
latent-person 2 hours ago
> That's not a trivial if: stating the problem precisely is often as hard as the proof.
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...
vanyle 28 minutes ago
This paper is a large amount of nothing. First, natural language is not as precise as lean, so you have multiple ways to translate a NL argument to Lean. As shown in Fig 1, the LLM did a decent job at translating the argument about roots in a succint way.
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.
sigbottle 5 hours ago
Will we ever run into a theory of meaning crisis?
_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.)
dooglius 5 hours ago
Given the high-level description of the examples, I think it's less of a "mis-translation" as it is the LLM tweaking the proof as it formalized it. Going between m+4 and m+5 is a pretty different thing than the sort of ambiguities that generally arise in parsing natural-language mathematical statements.
Sniffnoy 5 hours ago
Hm, looking through here, I don't see where they state what it is that OpenAI actually proved instead of Navier-Stokes blowup with forcing. I see where they do this for some other particular statements used along the way, but not for the headline result.
notrealyme123 5 hours ago
I get the feeling a lot of people propose that we can write a verifier for every proof in lean.
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
skywalqer 5 hours ago
Well, I believe the incompleteness theorems speak about provability, not about how the proofs themselves are expressed.
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.
jcranmer 5 hours ago
The incompleteness theorems state that every sufficiently complicated logic lets you construct a statement that is effectively "this statement has no proof," so either there exists true statements that lack proofs (incompleteness) or there exists false statements with proofs (incorrectness).
ezwoodland 5 hours ago
Just all the useful proofs. You can get arbitrarily more complicated and uninteresting theorem statements by making meta statements about the system you are doing proofs in. At some level the system can't answer questions about itself.
hypersoar 5 hours ago
The incompleteness theorem says that there are statements which can be neither proven true nor false in a given axiomatic system. If there is a proof to write in lean, then the statement is already outside the bounds of incompleteness.
arbirk 5 hours ago
It was a piston in a non-compressible fluid so to speak (ie. storm in a glass of water)
empath75 6 hours ago
I recently spent 3 weeks with claude formalizing a CS paper about a borrow checker in lean, for a personal project.
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.
ted_dunning 5 hours ago
This is the common experience in replicating a published paper by hand ... it is common to find "obvious" aspects that are anything but.
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.
ndriscoll 5 hours ago
I have hopes that this is primarily a matter of needing more engineering work on ergonomic formal languages and better building a language that "looks like math." e.g. when doing linear algebra stuff, a linear combination might be defined as a finitely supported function from an index set to your space, which is fine, but ugly and maybe conceptually overwhelming on first meeting, so I did some toying with little macros and eventually a small python Lean -> HTML renderer to do some basic transformations to make it look more like typical math notation with like \Sigma_{i \in I} a_i, or with a_0+...+a_n, etc. (to... not fantastic success, but I think there's still something to the idea).
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.
hgoel 5 hours ago
I enjoy running into those details when implementing papers, since it usually leads to improved understanding of the subject and an ability to approach the matter with more rigor in some way that I had not noticed before. It does also involve a lot of work and lost sleep though.
We should be very careful about relinquishing sorting through such details to AI.
empath75 3 hours ago
Claude could not fix them without a lot of help, so i did not relinquish sorting through those details in general. Just the drudgery of grinding through proof obligations.
dekhn 5 hours ago
As a second rate scientist, nothing makes me happier than finding a "hot" paper in my field, reading it, converting it to code, and demonstrating the authors made systematic errors that mean the paper is more likely false than true.
I've been criticized for doing this, but to me it emphasizes how much attention goes to the hot, wrong papers.
j2kun 5 hours ago
I think this highlights that, at the very least, coverage of AI-generated proofs should describe them as "claims" to solve problems, until, like all other works, the community has had time to review and digest them.
The idea that an AI company is beyond peer review is harmful.
fasterik 5 hours ago
As far as I understand it, nobody is disputing the correctness of the Lean proof, or that it proves the conjecture it actually claims to prove. That's sufficient to consider the problem "solved". The natural language proof is a "nice to have".
abstrakraft 5 hours ago
The claim in TFA is that the formalization(in Lean) of the problem does not correspond to the natural language statement of the problem, such that the statement proven is not the conjecture for which proof is required for the problem to be considered "solved".
sigmar 4 hours ago
fasterik 5 hours ago
Arodex 5 hours ago
[flagged]
dang 2 hours ago
fasterik 5 hours ago
j2kun 5 hours ago
kurtis_reed 5 hours ago
john_strinlai 5 hours ago
>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?
abdullahkhalids 5 hours ago
OpenAI has expressed this sentiment by not submitting to or saying they will submit their results to peer reviewed journals.
fasterik 5 hours ago
john_strinlai 5 hours ago
TeMPOraL 4 hours ago
yieldcrv 5 hours ago
setgree 5 hours ago
swiftcoder 5 hours ago
I've seen a lot of breathless reporting about various mathematical things being "proven" on the basis of the LLM-generated Lean formulation compiling. We probably wouldn't declare that for a human-written proof until peers had checked the proof for errors
fatcatsbestcats 5 hours ago
john_strinlai 5 hours ago
j2kun 5 hours ago
DoctorOetker 3 hours ago
Anyone who doesn't understand peer review (its intended workings, its negative effects by implementation flaws, etc.) automatically assumes expression is beyond academic peer review, so thats potentially a lot of people...
jrflo 5 hours ago
So my guess is that they have the AI system attempt to prove the theorem in natural language, then try to generate a Lean proof for it, and in that process they end up with a slightly different solution as the autoformalizer is essentially rewriting the NL proof to make it formalizable? Do we just need a "reverse pass" to re-align the NL proof with the Lean code?
Also, it doesn't seem that they are questioning the truthfulness of either proof, just that they are different?
ted_dunning 5 hours ago
Generating the lean proof first is a viable approach as well followed by an explanatory pass.
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.
palisade an hour ago
ok
essai57 an hour ago
I don't think that's an accurate summary of what this paper or its abstract actually claim.
palisade 24 minutes ago
Okay, but if they write another open letter or give another keynote proclaiming the end of the world then you owe me a beer.
129983-asf 5 hours ago
Two leading experts on Navier Stokes still do not know whether their methods were used:
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.
FrustratedMonky 5 hours ago
Not a mathematician. Why not just always use LEAN? Why use natural language at all?
ted_dunning 5 hours ago
Because it is really hard to read and the level of detail is so high that even lemmas that you can read may have such enormous levels of detail that makes real understanding difficult given that humans have limited working memory.
matusp 5 hours ago
Why not always write machine code? Why use programming languages at all?
FrustratedMonky 3 hours ago
If a programming language compiler isn't guaranteed to be re-producible, then yeah, you'd have to revert to machine code.
Jtarii 5 hours ago
Lean is a write only programming language.
jansport123 5 hours ago
Same reason humans write code not only for a compiler to translate into machine code but also so other humans can understand what we write, learn from it, modify it etc...
Jaxan 5 hours ago
Not only that, we also have code comments and standalone documentation.
FrustratedMonky 3 hours ago
A programming language, when compiled, is a guaranteed reproducible result. If you recompile a program, you get the same thing each time.
The point of the article is that natural language is not these things.
binlog 5 hours ago
Because people need to understand what is being proven.
caughtinthought 5 hours ago
The example in Figure 1 should help understand why... the NL version is much more approachable for humans.
FrustratedMonky 3 hours ago
If its ambiguous or wrong, then what are you understanding ?
upboundspiral 3 hours ago
cs702 26 minutes ago
TL;DR:
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 with irony.
le-mark 6 hours ago
> In particular, we highlight that the problem of resolving ambiguities in mathematical NL text, which is necessary in order to provide semantically faithful translation
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
ted_dunning 5 hours ago
Natural language is ambiguous, but the Lean formalization is very well defined and unambiguous.
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.
hyperpape 5 hours ago
> the downvotes will show many disagree
> gotcha bitch!
You may have misdiagnosed the problem.
ballmerpoint 5 hours ago
This shouldn’t be a surprising result. We’ve known almost since LLMs became a thing that they can “prefer” modifying the terms or context of a problem when they can’t solve it directly (what one might call “cheating” if there were any volition involved). Often that happens in a way that isn’t immediately obvious to the user.
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.
sebzim4500 5 hours ago
No one is disputing the correctness of the lean proof, the problem is that they did a bad job converting it to natural language.
_flux 5 hours ago
Actually, as an earlier commenter noticed, it seems that the proof was done in natural language, and only then translated to Lean, as https://openai.com/index/navier-stokes-solution/ says:
> 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.
ballmerpoint 4 hours ago
I am also not disputing the correctness of the Lean proof. I even emphasized this in my comment: “correct but irrelevant”.
Oh, well. I suppose I should avoid getting involved in these AI threads, but now it’s about half the forum.
ForHackernews 5 hours ago
Indeed. I've never used AI to translate between natural language and Lean but I have gone from English to Golang, Python, Typescript and SQL and its interpretations can be... creative, let's say.