At least put a disclaimer for the ad for this site, and maybe disclose how you came up with a total ordering for "top" open problems (vibes)?
> How problems are ranked. LLMs compare pairs of problems. A reliability-weighted model combines those judgments into the ranking, with calibration across model families. The model-family weights are OpenAI 1.00, Claude 1.00, GLM 0.95, and DeepSeek 0.90. These are modeling choices, not measured probabilities of correctness.
The most interesting for me were the faster matrix multiplication, integer multiplication, and FFT. Maybe just cause they're easier to appreciate.
There's also one that says that forced Navier-Stokes can implement universal computation (so, is Turing complete). I don't think any of these are resolving open problems per se, but they're interesting for other reasons.
I thought it was interesting that it said "This paper was written with human assistance", unlike this other Quasi-Riemann Hypothesis preprint that didn't have the same disclaimer.
Funny that it says "written with human assistance" instead of saying "written with AI assistance". So we're assistants to the machines that we have created.
Number theorist here. This is a massive big deal, and would likely be a Fields Medal for a human if a human had done it. But it is an exaggeration to say it is the biggest result in 200 years. At a minimum, it is hard to argue that it is a bigger result than the proof of the prime number theorem in 1896 (which this is a strengthening of), or Riemann's original 1859 paper where he laid out the zeta function and its analytic importance, or Dirichlet's proof of infinitely many primes in arithmetic progressions which is the late 1830s.
But yeah, this is still a very big deal. Among other things, it will drastically improve all sorts of Rosser-Schoenfeld type results for the PNT and that's just a start. For comparison, I have a paper form 2018 where this result would cut 3 pages out and make the full result cleaner and much tighter, and there are likely hundreds of papers like this.
I am also an analytic number theorist, and I disagree. Not only do I think Fields Medal is an understatement (Fields Medals have been awarded for far less than proving quasi-RH + no Siegel zeros), I don't think it is unfair to say that this is a bigger deal than the 1896 proof of the PNT.
As for Riemann's memoir, it's hard to compare. You could argue that was "just" noticing a connection (between number theory and Fourier analysis) that nobody had noticed before; in fact this is the kind of thing AI is extremely good at. I'm being a little cute here.
I think if a human had proven just these two results in the form of a uniform zero-free region for L(s,chi) from nothing as OpenAI did it would not be unfair to say that it would be the single greatest advance in math (easily dwarfing Wiles' FLT), and it would instantly put them in the ranks of greatest mathematicians of all time. Unlike something like Navier Stokes there wasn't a semblance of a research program, experts basically considered this hopeless and would have said the chance of seeing a proof in our lifetime was near zero.
For some comparison, Yitang Zhang's bounded gaps result might have gotten him a Fields Medal if he was not disqualified by age. When it was floated that he might have proven Siegel zeros don't exist, it was considered (by experts) clearly a much bigger deal. This result blows that out of the water (it's a way better version); at least analytic number theorists I talked to thought it was plausible but unlikely that Siegel zeros would be eliminated in our lifetime but thought RH was basically hopeless.
While some of my work is in analytic number theory, much is in other subareas, so it is possible I should defer to you on this.
It seems to me less than PNT in terms of what can we actually do with this. Many different areas of math use PNT, and from my standpoint, PNT is helpful not just for what it implies directly but because it lets us make really good heuristics about whether some sets are infinite or not, and what their rough size is. (Granted, one can do that also mostly via Chebyshev). For those purposes, this doesn't really enter in. Similarly, PNT feels like a statement at least I can say explain to my mother without any technical details. This isn't that. But that may also be my own biases of wanting things to cash out to very concrete statements about the integers.
I agree that one striking element is how no one saw this coming. This isn't building on an existing research program, which itself is remarkable. And last night, before I went to bed, I saw a conversation between a bunch of analytic number theorists who seemed to think there was potentially some slack in the quasi-RH argument, which if that's the case means this is going to go even further.
Thank you for the detailed explanation. From what I'm reading from a lot of mathematicians there's at least a dozen of results here that are field-definining and worthy at minimum of a Fields medal.
I guess the biggest news are not the discoveries themselves but how they were found and that math is going through the biggest revolution as a field since almost ever.
1896 PNT is basically 1859 Riemann + a trig inequality.
1830 Dirichlet's result is qualitative only, it shows infinitude but not the asymptote in terms of the zeros for it predates Riemann.
To me this is the first substantial step after the 1896 PNT, and we really do not see much progress in the whole 20th century.
Personally so far there are only two people worth mentioning,
- Euler, introduces the real zeta function and Euler product, establishes the functional equation at (half?) integers.
- Riemann, introduces complex analysis ideas to the zeta function.
And of course this result if it is true. This is first to penetrate the critical strip, which nobody had any idea how to approach for over a century and a half.
" I expect, say, 2026-level AI, when used properly, will be a trustworthy co-author in mathematical research, and in many other fields as well.
Then what? That depends not just on the technology, but on how existing human institutions and practices adapt. How will research journals change their publishing and referencing practices when entry-level math papers for AI-guided graduate students can now be generated in less than a day—and with the far better accuracy of future AI tools? How will our approach to graduate education change? Will we actively encourage and train our students to use these tools?
We are largely unprepared to address these questions. There will be shocking demonstrations of AI-assisted achievement and courageous experiments to incorporate them into our professional structures. But there will also be embarrassing mistakes, controversies, painful disruptions, heated debates, and hasty decisions."
I was at the workshop that resulted in the Leiden Declaration in Fall 2025. The majority vibe was that this was inevitable, but hard to predict whether it would be in one year or 30 years.
I guess they showed this to the advisory group they created. I guess the group tried reading the work for a day and they could only think of telling them to release the results to the community. I now understand why the group had this suggestion.
There was a Wired Magazine article from either the late 90s or early 2000s that made a prediction that this sort of thing would eventually be possible, likely within my lifetime. I believe the context was "distributed computing" models of the time, like SETI.
I've never been able to find that article as an adult, but I would love to know who wrote it.
Gina Kolata allegedly in the New York Times in 1996 on the Robbins conjecture (noting that computers had started to contribute to math research in some sense), and a longer piece in Math Horizons by her the following year ("Computer Math Proof Shows Reasoning Power"). I didn't immediately find the NYT article, so I don't know if it might be a hallucination.
I was involved in some distributed computing stuff in the late 1990s and early 2000s and I don't really remember people in that community talking about proofs but there may have been a "if we had a mechanical proof-checker, could we do distributed searches for valid proofs that it would accept?" conversation somewhere at some point. There were definitely volunteer distributed computing projects working on pure math; I remember the Optimal Golomb Ruler search (https://en.wikipedia.org/wiki/Golomb_ruler). So, that could possibly have shaded over into "can we find proofs this way too?". At the time it probably would have been based on brute force searches through proof space rather than clever optimization, though.
The idea that you can lexicographically list all proofs in some formalism and then mechanically determine if any is valid is quite clear from Gödel's construction of the function Bew in "On Formally Undecidable Propositions", but he points out that you don't know where to stop because you don't know how long a valid proof would potentially have to be (so "is this a valid proof of this claim?" can be decided mechanically in a limited time, while "is there any valid proof of this claim?" can't be! maybe the shortest valid proof is 49 steps long but you eventually stopped checking after looking at all 7-step proofs, or something).
Thanks! Yea, I have used various LLMs to dig for this article, as well as Google search multiple times over the past 20 years. The article I'm remembering was 100% prior to Nvidia's CUDA release in 2007. My best guess is that it was from late 90s, but possibly early 2000s.
The article I'm remembering was not just about mathematics, but indeed all of physics and related fields. I believe it speculated that eventually distributed computing models could essentially take the world's mathematics and physics formulas and various datasets that we believe to be accurate with high degrees of confidence, and then look for patterns or trends, and then from those trends, mathematicians and physicists would be able to investigate further. Not dissimilar to Folding@Home and SETI@Home.
Keep in mind, that this is the best I can remember from 30 years ago, and I've thought about it so frequently that I am certainly misremembering some of the details. Anyways, it's always been this really compelling possibility, and I wish I could find that article that inspired me so long ago and re-read it! :) I really think it was Wired, but it's possible it was Popular Mechanics, or even an expert guest on TechTV who gave an interview. Hard to say for sure, but I've always thought it was a Wired article.
Oh, I remember hearing about "computer scientists" or something that would attempt to determine physical laws on the basis of empirical evidence, possibly also in that timeframe. That might be another thing to look for. I'm sure that's something people were writing about.
Edit: with the noun-noun compounding being different from the usual interpretation here, like "scientists who are computers" rather than "scientists who study computation"! Maybe "computerized scientists" or something.
It's really impossible to predict which discoveries will "matter", have a direct impact on other fields, or an impact in making other mathematics or physics discoveries.
Only after a world's worth of experts look at these results and then mull over if and how their own fields are impacted by this new info will we be able to answer this question.
It can take decades to really know the full significance. You know, the whole "We stand on the shoulders of Giants", well the Giants just grew a few inches all at once.
worth mentioning that "NP" is not "non-polynomial" but "non-deterministic polynomial (time)". If NP was non-polynomial time then NP != P would be trivial (and in fact, P != EXP is known by the time hierarchy theorem).
Non-deterministic can be explained in several ways. One is in terms of a hypothetical "nondeterministic Turing machine" with certain non-physically realizable properties. The easier way is that a NP problem gets as input not only the problem instance x, but a "witness" w, that may depend on the problem instance. This witness generally makes the problem of deciding the problem instance straightforward (e.g. for SAT, x is the SAT instance, and w is a description of how to set the variables so that it is true).
To be fair - there are statements in math that are independent of the axioms. For those statements, the universe you find yourself in can pick either version (true OR false) and still be consistent.
See also: noneuclidian geometry and axiom of choice.
Being able to solve NP hard optimization problems would enable progress in many areas of science and technology. For example it would allow us to find poly-sized Lean proofs for theorems efficiently, since proof verification can be done in polynomial time.
It would also be amusing to annihilate nearly six decades of proofs that assume P!=NP.
Leans proof checker is not polynomial time, unfortunately. It is super exponential. Basically, because it can verify the result of any function it can prove to be total.
to depress you even more, it is consistent with everything that we know that P != NP and that cryptography does not exist. So there is a worst of both worlds, and we cannot rule it out.
Of course, if we get ridiculous polynomials it doesn't mean much in practice. People who hope for P=NP generally hope for nice polynomials O(n^3) or something like that at worst.
I maintain an LLM-ranked list of the 500 most important open problems in math at https://www.proofatlas.ai/open-problems/. This problem was ranked #159, and it also resolved #244, "All-Pairs Shortest Paths in Truly Subcubic Time." It is formalized in Lean.
But what's crazy is that within the last day or so, we've also gotten LLM-assisted solutions to #95, the Kannan–Lovász–Simonovits (KLS) conjecture, by three different authors in parallel (all extending Song–Zhang's key criterion introduced on Oct. 1), #278, the Mumford–Shah conjecture, and #227, Zauner's conjecture on SIC-POVM existence in every dimension, which also represents a major claimed advance on Hilbert's twelfth problem (#36) for real quadratic fields.
This is likely because OpenAI's solutions to 100 open conjectures are expected to drop any day, so everyone is in a hurry not to get scooped.
Right now, I'm having LLMs audit the actual math in claimed arXiv solutions because despite its policy changes, arXiv is still a dumping ground. The audits have already found six faulty proofs that caused status issues for problems that should still clearly be fully open.
This is all you need to read and understand for Anthropic's FLT formalization:
import Mathlib
import Theorems.Thm_fermat_last_theorem
/-- Solution side: the same statement, binder for binder, proved by this tree's `fermat_last_theorem`. -/
theorem FLT_for_comparator (n : ℕ) (hn : 3 ≤ n) (a b c : ℕ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) :
a ^ n + b ^ n ≠ c ^ n :=
fermat_last_theorem n hn a b c ha hb hc
/-- Mathlib's named proposition, by the one-line bridge from the elementary statement
(the bridge is restated inline so that this file depends only on `Theorems.Thm_fermat_last_theorem`). -/
theorem FLT_mathlib_for_comparator : FermatLastTheorem :=
fun n hn a b c ha hb hc => fermat_last_theorem n hn a b c (Nat.pos_of_ne_zero ha) (Nat.pos_of_ne_zero hb) (Nat.pos_of_ne_zero hc)
First of all, that is Fermat's Last Theorem, not Navier-Stokes.
Second of all, you did not read the link.
> In particular, we use honest when the goal is to create a valid proof. This allows for mistakes and bugs in proofs and meta-code (tactics, attributes, commands, etc.), but not for code that clearly only serves to circumvent the system (such as using the debug.skipKernelTC).
Given that AI has autonomously found proofs of `False` in Lean and other proof assistants, it is far from impossible that such a circumvention could be present somewhere in 13 million lines.
If we read the link, it has a section called Gold Standard: comparator and external checkers, and comparator is how OpenAI has gone about checking their lean proofs.
Perhaps you did not understand the Fermat theorem proof announcement/repo or the link. The 13 million lines did not use any external, possibly not honest libraries, as the proof eventually only used the fundamental axioms. So for the Fermat theorem formalization, no open open questions remain.
The point is, they "proved" the Collatz conjecture. You would not know they exploited a bug unless you actually went and dug into their proof. Can we be so certain this has not happened within the millions of lines of Navier-Stokes? In an ideal world, our proof assistants would be more battle-hardened by now (recent exploits deny this), our AI better aligned (their tendency to cheat at tests denies this), or their handlers more responsible (the Hugging Face incident denies this), but the reality is more complicated.
At this point in time, we really can't be confident in accepting proof certificates without any human eyes on the script that generated it. I still have 95%+ confidence in this particular result being trustworthy, but a precedent of blind faith is guaranteed to end badly.
This person knew they did not prove the Collatz conjecture and others independently figured it out within hours. Not sure this is at all relevant, other than pointing out how trivial it is for the community to understand errors in lean4.
It was trivial because the Collatz proof script is literally 1000x smaller than the script for Navier-Stokes and involves no advanced math. And they found the bug by... manually inspecting the proof script. Maybe we should do the same for Navier-Stokes before declaring the matter settled?
Not only that, but there is a very fuzzable tell of something funny in the Collatz proof script (`CommandElabM`, i.e. metaprogramming). We may not at all be so lucky in other malicious scripts, especially if there are still kernel-level bugs in Lean.
> Can you elaborate on what constitutes a vacuous proof?
Trivially, a proof that relies on a bug in Lean. Less trivially, a proof that is technically true but about something trivial and does not, in fact, prove what it claims to have proven.
When I first started playing with lean I accidentally defined a group in such a way that it was reduced to triviality. It had one object in it, so everything in the group was trivially equal to everything else. It was not the group that I was trying to prove something about, but the proof went through.
It was too easy, so I double checked my definitions, but it is quite easy to do something like that. And Claude does things like that quite frequently.
I am going through the exercise right now of trying to get Claude to formalize a published paper and it is a _struggle_ to get it not to take shortcuts or prove approximations of the paper’s theorems and then tell you it’s done.
It can happen when the proof process ends up with universal implication that holds trivially. Then you end it with something like Forall x, x is empty -> P(x).
This statement is 100% logically coherent internally. But it also doesn't matter because we know that 1 does not equal 3 so this proof is completely pointless. I could also say 3 == 5 and it would still be logically sound but completely useless information.
Are you proving for some arbitrary definition of == that isn't what we commonly consider the definition? How is it logically coherent? You mean only in the sense that you say it is and you haven't provided any rules to disprove it?
No the definition of == is the regular definition; it's just a deductive reasoning statement. Since the first part of the statement is never true, it doesn't matter what the second part of it says. Of course, like he said, that makes the statement have no value.
This is known as a “vacuously true” statement in formal logic. Let me write it out more in more detail and you’ll hopefully see why it’s consistent.
In logic, a proposition is some statement that can be true or false. So, let A be the proposition that 1 equals 3, and B be the proposition that 3 equals 3.
Now the poster is making a third proposition. If A, then B.
A is clearly not true. So in classical logic, B can be anything and “If A then B” is still true.
For example let B be the proposition that I am Elvis Presley (I’m not). So now we have “If one equals 3 then I am Elvis Presley”. This is clearly true. I’m not Elvis Presley, but that doesn’t matter because we’re not saying anything about what happens when one doesn’t equal 3.
Now, let’s try let B be the proposition that I am Sean Hunter (I actually am). So now we have “If one equals 3 then I am Sean Hunter”. This is clearly still true because we still are only making a claim about what happens when one equals three.
By the way, this isn’t any kind of inherent contradiction or problem, it is just a possibly counterintuitive part of how classical logic works.
You see this type of statement (“If <x>, then <something ridiculous>”) being made a lot when people are exaggerating for effect, for example by Mr Bumble in “Oliver Twist”
> 'That is no excuse,' replied Mr. Brownlow. 'You were present on the occasion of the destruction of these trinkets, and indeed are the more guilty of the two, in the eye of the law; for the law supposes that your wife acts under your direction.' … 'If the law supposes that,' said Mr. Bumble, squeezing his hat emphatically in both hands, 'the law is a ass--a idiot. If that's the eye of the law, the law is a bachelor’
Nothing to do with special hacks with operators. The reason it's useless because the precondition is never true. "If my aunt had two wheels and a handlebar then she'd be a bicycle"
Is the same problem with a non maths flavour.
E.g. “If it’s raining, the sidewalk is wet.” That statement holds if it’s not raining or the sidewalk is wet.
This is a common occurrence in mathematics, where someone might not be able to unconditionally prove Y, but they can under the condition X. Later, another mathematician might build on this by proving X, thereby transitively proving Y. (Or conversely, they might unconditionally disprove Y, thereby disproving X.)
Many hard problems are answered this way.
For example, Fermat’s Last Theorem was proven assuming the Taniyama-Shimura-Weil Conjecture, then Wiles proved the conjecture.
Thousands of theorems rely on the the unproven Reinmann Hypothesis, which is why it’s so interesting to mathematicians.
But if your precondition is “stupid,” your proof is stupid.
If you have a software engineering background, it's like how semantic versioning is bollocks.
Semantic versioning describes the following idealized setup:
- you have an interface you expose (a contract, and thus a contract signature)
- you do not change the contract signature -> patch version bump
- you do change it but in a non-breaking way (e.g. additively) -> minor version bump
- you do change it but in a breaking way (e.g. mutatively or destructively) -> major version bump
One would expect then that since interface signatures are statically derivable, semantic version tags can be auto-assigned. And indeed, in lots of shops that's exactly what happens (in my opinion, correctly).
The problem with this is that it comes with a lot more smoke than fire. The interface having no changes or non-breaking changes doesn't mean the actual code behind those interfaces is not going to cause a breakage. It literally is just about the interface itself.
And so unless you encode absolutely everything about the semantics your implementation actually observes into the interface, which is what the semver specification asks you to do so as their sleight of hand, this means the interface will be a leaky abstraction. Which means that external software interfacing with yours may observe behavior that is beyond the purview of semantic versioning. Which means that they do. Which means that they absolutely can and will break, and your package managers' fancy version constraint syntax exists to make such fun events happen.
The way this is usually handled then is:
- you live with the pain: acknowledge the limitations of semver, accept you've been duped, and just give in
- you have human release managers assign versions manually, based on whole program and whole system semantics (with the human overhead and error that entails), falsely claiming that what you're doing is still semver
- you switch to a less deceptive versioning scheme, like calendar versioning; as a bonus, you now no longer have to pretend that your entire application somehow only has a single unified interface
This mirrors the Lean statement and Lean proof situation. The statement is like an interface, and the proof is like the implementation behind that interface. The way the proof is derived may expose semantic gaps in the statement itself, and (ab)use them to obtain the logical consistency certificate. Hence, a vacuous proof, and hence why this is not statically assertable to be not the case. It is part of the challenge in asserting that the statement was correctly formalized in the first place: you need to manually identify whether the way the consistency was achieved is actually meaningful, or just a formalization gap.
Which really makes me wonder about the actual value proposition of Lean then, but alas...
> This mirrors the Lean statement and Lean proof situation. The statement is like an interface, and the proof is like the implementation behind that interface.
This is true in a very deep sense due to the Curry-Howard correspondence and calculus of constructions which are central to Lean. In Lean, the proposition you are proving is a type (so it really is an interface directly in the computer science sense) and the proof is a function which takes your hypotheses and returns a term of that type (so it really is the implementation of that interface). In fact in lean, you can just as well write this implementation as a lambda (this is known as “term mode”) as in the “tactic mode” that is more generally used in normal lean use. Lean really doesn’t care at all which one you use and you can switch between them within a proof quite easily without interfering with lean’s ability to check your proof at all.
> Which really makes me wonder about the actual value proposition of Lean then, but alas...
The purpose of lean really is quite different from what most people on hn seem to want it to be. Lean is designed to be a useful tool for mathematicians who want to formalise areas of mathematics. It’s not a primary goal of most of the lean community to make something that is hardened against malicious proof attempts (although these are considered bugs and there is a small subcommunity who work on this area in particular). So it isn’t primarily for the benefit of people who want to “fire and forget” some proof without reading or understanding it and just get the check mark if it’s true.[1] It’s mainly for mathematicians who want a proof assistant to help them with their work.
"Problems solved before a model's training cutoff can be filtered out, and all models compared on the remaining problems" means that the problems an older model actually solved are the ones that get filtered out, while the remaining problems are the ones it already tried and failed on. So older models end up with 0s on the filtered set and you can't really use this to compare new models to older ones.
Also, since these are known public problems, you can't stop people from spending far more than your arbitrary time and $ limits on them. So the number of clean problems will go down over time.
A complete mischaracterization, as usual for HN lately when discussing AI or LessWrong. Obviously, even average levels of persuasion are enough to convince some people. And nobody is air-gapping AI.
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