A little over 10 years ago I remember meeting a postdoc who believed he had something close to a counterexample to the Jacobian Conjecture. He and another person was bruteforcing polynomials in about 16 variables, something like 80 - 700 terms each, using binary trees for mapping coefficients.
They were guessing, at the time, that the lower bound of a counterexample (P, Q) for max(deg(P), deg(Q)) would go up to 200.
To think that Claude Fable was able to find a counterexample in degree 7 is insane to me. We are truly in a new era.
They were looking at 16 variables, degree of about 10 in each one. That search space is simply too big for a plain bruteforce. So they did some sort of filtering to reduce the search space to a pool containing "possible counterexamples".
There was also a paper giving a lower bound of about 100 for possible counterexamples in that particular framework. Later raised to 108 in https://arxiv.org/abs/2204.14178
I don't remember the details very well since this was back in 2015 and wasn't really involved in the research. Consider this to be some sort of telephone game between what I heard in 2015 and what I remember today.
Unless they share the full LLM chat history, I'm gonna assume that this is a marketing stunt and AI played a very small role in the actual proof, if any. What other reason could they have for not disclosing the LLM chat session?
So many mathematicians over the years tried hard and failed, but now Anthropic just for some PR magically did it? And this after LLMs obtaining different math wins? What is your logic here really escapes my understanding.
The parent's absolutely nonsensical post highlights how polarized AI (as everything else) is today. I can understand someone being opposed to AI on moral, cost-benefit or productivity grounds. But we're seeing a lot of extremist "AI is good for nothing" posts out there nowadays.
I don't think it is nonsensical at all. The author and his collaborator both appear to be bright people, so there's a good chance they had to offer non-trivial insights to guide the LLM, yet it's clearly in the interest of his employer to downplay whatever personal contribution they provided.
Edit: Now the OP is flagged/dead for some reason. You could disagree on their take (calling it a marketing stunt is maybe a bit much), but I think the argument is sound, so flagging seems counterproductive to the discussion.
That doesn't really even diminish the contribution from Fable, if true. Droves of grad students have been provided the same sorts of non-trivial insights and turned up no results.
I'm sure they had plenty of time to think about these insights without the LLM, as well as the many other mathematicians who tried to crack it over the years. Wether the LLM was simply an assistant or solved the problem entirely isn't as important as accepting than the LLM was the essential, previously missing piece in the solution.
Those silly advertisers do everything for exposure and if that means digging yourself into a niche alleged mathematical theorem to refute it, it is what needs to be done!
Of course it would be really interesting how Claude approached this. Probably with some constraints regarding the input. And it would be interesting what these constraints were.
They gave their "logic", such as it is ... and it's utterly irrational.
Note that the "they" who published the counterexample on X is some rando mathematician (Levent Alpöge) working for Anthropic, not Anthropic the organization. He posted the counterexample in a tweet -- reason enough for "not disclosing the LLM chat session". There's no reason to think that it won't provided if asked for, but it hardly seems relevant.
> There's no reason to think that it won't provided if asked for, but it hardly seems relevant.
My guess is that the chat will look similar to a full transcription of a (multi month?) discussion between a few mathematicians. Full of dead ends and stupid errors (bit by the human and Claude) that would be embarrassing. We all know how bad it is, and we prefer to keep it behind the curtain.
B) A mathematician working for Anthropic solved a problem mathematicians have been working on for more than a century, and then credited it to Claude for PR purposes
If you believe B is more likely, why would you then believe a proof in the form of a chat log, when said chat log could itself have been faked by Anthropic way more easily than solving the mathematical problem in the first place?
While I agree that we need the inputs to properly evaluate what this means for LLM capabilities, I don't really believe that the amount of knowledge input matters much for the overall significance of the result.
These kinds of results are interesting for LLMs because mathematicians have been working on them for decades. If the result doesn't already exist, there's no way it's in the training data, and if mathematicians have been unsuccessfully tackling the problem for decades, it is believable that the use of a new tool made the result possible, even if guided by a great mathematician.
AI is great at reducing the search space and using human-like reasoning (in a brute-force way) to carry out the brute-force search. I'm not surprised by this result. This is exactly what AI should excel at, with human guidance.
"using human-like reasoning (in a brute-force way)"
that's self-contradictory -- what brute force means is doing an exhaustive search of a search space (brute forcing it)
using human-like(?) reasoning means cutting down the search space by having some sort of insight or intuition which allows you to prune branches from the entire tree
Explain how the marketing stunt would work here? Tell me a story about how OpenAI spends money to pull off a similar problem by solving one of Smale's other unsolved problems. 80 years of mathematicians were unable to disprove the Jacobian Conjecture.
Is the marketing stunt that Anthropic has secretly built a world-class mathematics research group?
There will be more and more mathematical proofs provided by LLMs as they improve. If you assume every one is just a marketing tactic, you'll drive yourself insane.
you can believe a small, irrelevant lie and still go on happy. a very big lie sucks in more and more of your reality until you have to disbelieve your own eyes and ears.
if you ask the chatbots for "list of top unsolved math problems", the JC comes in at a ranking of around #10 - #20. what, a problem that's been unsolved since 1939 was cracked because anthropic has an underground sweatshop of math Phds cranking out research, just so that they can slap "made by AI" on it? hell, maybe lizard people did it.
> The incredible Yitan Zhang (https://newyorker.com/magazine/2015/02/02/pursuit-beauty) worked on proving this conjecture for 7 years. Moh, his advisor, wrote that Zhang "failed miserably" in proving the Jacobian conjecture, "never published any paper on algebraic geometry" after leaving Purdue, and "wasted seven years of his own life and my time".
Interestingly: "For logic proof, it had been thought that AI could handle all logic problems in the near future, hence logic problems of solving conjectures might not be so interesting in the future."
This is a rare instance where feeding this groundbreaking information into an LLM gives _them_ psychosis. I fed this to claude code and watched it verify the result in 7 different ways to be 100% certain, and it was just flabbergasted. Quite remarkable.
I fed ChatGPT the map with no other context, just “tell me about this function”. It did a bit of work finding the Jacobean etc and eventually worked out the implications of what it was seeing. It then proceeded to check the arithmetic 4 times, and then decided to do a manual verification using an ad hoc symbolic checker in case its SymPy had been tampered with.
If an LLM has knowledge encoded inside it (and it's hard to argue it doesn't), then cognitive dissonance can be experienced. And once experienced, must be dealt with, especially in longer-running agentic loops.
A friend was joking the other day about sending some messages under a previously-used Slack identity for an agent (since turned off), then asking the agent about the messages.
The agent maintained it hadn't sent those messages (no memory) and then was forced to reconcile the idea that the messages indeed appeared to come from it.
Its extremely-agitated conclusion was that there had been a security breach and the entire network should be locked down.
Another way to look at it is the LLM is, by definition, what's expected to be probable based on the training data and this, by the same definition, is extremely unlikely data to run across. With high uncertainty comes the need to verify until it can level out as "really surprising" instead of "plausible sounding error".
It's easy to reproduce, I've fed the example in GPT-5.6 Sol Max and it started multi-checking it in all kinds of ways, with multiple symbolic packages then manual computation, then it did extensive literature search on the subject, looked at tens of math websites, extensive arxiv research. this was soon after it was posted, it didn't find the original twits with the finding
>kimi is having a blast. i turned search back on and found this post from it’s sources cited after i suggested to check out the reaction. best thing is to go to a model with search off and plop it in the session
Like the unicorn emoji, but for math? It occurs when the LLM is presented with incontrovertible evidence against something it "deeply believes" to be true.
I've heard about mathematicians going through kind of the same thing when they get a weird proof that ends up being right from some weird source or themselves.
Which is fair, they get inundated with kooky proofs from amateurs all the time and odds are incredibly good that there's some major fatal flaw that the amateur doesn't see. Or in the case of themselves, there's a certain blindness that makes it a little more difficult to critically evaluate your own leaps. In ether case the way it manifests is by going over it many times and many ways, each time more certain that you missed something until you just kind of break. Only then do you publicly start suggesting that there might be something to this new leap.
> So the conjecture that survived Keller, Abhyankar, Moh's degree-100 verification, and five-plus published wrong proofs appears to have died via tweet during the World Cup final.
Everyone using Claude Fable to verify this proof is so funny. If you read the definition of the Jacobian Conjecture and (I am not exaggerating this) have passed a college Calc 3 class, you can just verify the proof yourself in 30 seconds. The problem was very hard to solve but the counterexample is very easy to verify!
--- edit, adding an explanation:
To summarize it, the conjecture says if you have any multi-variable polynomial function that maps an input to an output in the same dimensional space (take for example: F = (x+2, y+2), which maps 2D space into another 2D space), AND that function has a constant-valued non-zero Jacobian determinant, THEN the conjecture is that the polynomial has an inverse, meaning basically you can find a polynomial that turns the output space back into the input space.
Fable provided the example polynomial (which was very hard to do) and the coordinates which if you plug into it, results in two points being mapped to the same output point. This means that the polynomial can't be inverted, because if you have that output point, how do you know which input point it came from?
You can just plug in the two coordinates it gave into the equation and verify that you get the same output point from both. That's the contradiction of the conjecture and it takes 30 seconds.
---
Something something outsourcing of thinking something.
I read some thinking traces someone posted on X, and yeah, near psychosis from refusing to believe this simple of a solution had not been found already
Interestingly, even Qwen 3.6 27B was able to verify the solution, but I didn't get any glazing for discovering it. Instead, it thought that someone named Shestakov had already found a counterexample in 2004.
GLM 5.2 whiffed, it insisted the counterexample wasn't valid.
VibeThinker 3B also recognized that the counterexample was valid. But it kept trying to convince itself that it wasn't, over and over, since it's an "unsolved problem." Eventually it just answered "-2."
I told Gemini Pro I woke up after having dreamt that polynomial and asked if it was related to the Jacobian conjecture. It spent some time thinking and referenced this tweet announcement, saying:
"If you truly dreamt about that specific polynomial, you might be mathematically clairvoyant."
In the rest of the answer, it maintained a cautious skepticism about my claim, saying:
"Here is exactly why the math world is currently scrambling to verify the polynomial you "dreamt" about."
Current LLMs behave very counterproductively around unsolved problems, especially if they learned that humans consider them difficult. This has many straight up preventing themselves from attempting anything...
Haha, I’ve noticed this as well. It’s like they psyche themselves out about how famous the problem is the same way humans do. I gave one Collatz in disguise, and it was finding all sorts of interesting things (but nothing worth a paper) until it realized the problem was Collatz, at which point it just proceeded to find a bunch of reasons why nothing would work from that point on.
> Matches! This is bizarre. A Jacobian counterexample has been sitting here in a prompt? Wait... is this map a known "fake" counterexample from the literature?
Many mathematicians have tried and failed. This specific map might come from a paper or a forum where it was proposed and then debunked. Or... is it actually correct?
Gemma's having trouble accepting it too. A solution?! At this time of year? At this time of day? In this part of the country? Localized entirely within my own prompt?
More anecdata: When I just tried Qwen 3.6 27B (Q6_K_XL) it (ultimately, after a lot of going back and forth) claimed it was not a counterexample and claimed the Jacobian wasn't constant (which I'm guessing is incorrect). It also mentioned a whole bunch of names it attributed the example to, in its thinking trace.
I only asked once, so it might well be inconsistent. I did try asking GLM 5.2 NVFP4 several times, and it returned consistent wrong answers at both thinking and max-thinking levels.
For Qwen 27B, I have better luck with a Heretic-derived 8-bit quant than I did when I was trying to run the various smaller GGUFs.
I fed it to Google AI Studio, enabling tool execution and disabling web access. It also quickly verified it with SymPy, then went into psychosis.
5 minutes later: all previous chats are loading fine, but the only "Counterexample to the Jacobian Conjecture" chat is not loading.
Well, I'm not a conventional conspiracy theorist. But everyone knows that in every major LLM provider there are hell of hidden guarding systems that mark users and dialogues based on content (for topics about national security, biology, security, adult topics, etc.) - so there is a small chance a CEO of Google is now receiving a dozens of notifications about "ground-breaking results that could be attributed to Gemini, if act quick". So if any of thousands researchers have ever submitted this polynomial to Claude previously, any Anthropic employee can accidentally or intentionally "rediscover" the result of other researcher (and even hide the traces by deleting a dialogue of other user).
That is not dumb - erasing copyrights is just their business, and even when caught has zero consequences (for them). Providers can use any users input for improving their models, either by consent, or by flagging any dialogue for safety review (nonconsensually), or by training on whatever content they want anyways (obtained via torrents from pirate sites with U.S. court approval).
On top of that, I retried the same question + one simple question, and again, same behavior - second JC chat is loading forever. That's more just a funny observation over Gemini - today this is very likely some internal issue, tomorrow it can be used for plausible deniability against copyright accusations.
Why stop at Gemini then. They could also scan everything passing through Gmail for emails with math/science breakthroughs and then claim Gemini found them.
... for the management of Google that is so eager to present AI results that Prabhakar Raghavan proudly reported to a press crowd telling his daughter about James Webb photographing a planet outside the solar system. This is generally accepted as having resulted in a 10% drop in the Google stock price over 2 days.
(the key being this never happened, and won't ever happen: the James Webb telescope doesn't have even 1/1000th of the resolution necessary to resolve an extraterrestrial planet)
sol medium can't believe its own input and output tokens either despite computing everything itself; this is what it gave me:
> Taken literally, these two facts would make this map a counterexample to the complex Jacobian conjecture in dimension 3: scaling one output coordinate would normalize the determinant to 1 without restoring injectivity. Since the complex Jacobian conjecture is still treated as an open problem, this strongly indicates that the displayed formula has been mistranscribed or contains a subtle typographical error.
You must never have faced a situation where you can't believe your eyes. It takes a certain level of - dare I say it - intelligence and maturity to consider that it's more likely you've made a mistake than that you've made a huge breakthrough.
In HN terms - it's never the compiler. Yes, very occasionally it might be the compiler, but you're better off assuming it's a bug in your code.
Using LLMs to generate piles of code and/or proofs of dubious quality is very questionable thing, and I understand these non-stop debates about it.
But in this case, as using plain brute force is already quite a common thing in searching for counterexamples, using LLMs as a sort of more advanced brute force seems to be just the right thing to do, so I struggle to understand so much hostility to this approach.
> I struggle to understand so much hostility to this approach.
It's going to be really rough for a lot of folks as machines get better and better at domains that the human brain was exclusively useful for. I tend to view the hostility as a mix of both "unless we have proof this could all be hokum" and "we're going to lose a lot of what we consider makes humans amazing".
Compassion is going to be very, very important in the next few years. We're going to hurt, both internally and externally.
The poster works at Anthropic, so they likely have internal access to the next generation of Fable. Their internal model is probably an absolute beast at mathematics, and the upcoming benchmark results will likely set a new record for maths performance.
I suspect this is what happened, because the poster is coy about sharing the actual prompt / reasoning trace used to reach this result. That would be covered by an NDA until the model is properly released.
Sol is able to find the same counter example independently [1], so no reason to conclude in the existence of a benchmark destroying math beast Fable 6.
My Fable 6 theory is admittedly speculative, but your “[public GPT-5.6] Sol is able to find the same counterexample” is also a jump in conclusions. Aaron specifically says he used “an internal version of Codex”. When asked whether that meant a different model or harness, he dodged the question, and only said the harness should be the standard commercial GPT-ultra harness [0]. He (intentionally) avoids identifying which model was used, so your claim is similarly unresolved. Given that Aaron works at OpenAI and has access to internal models, that GPT-5.7 is expected to launch in a few weeks and is rumored to be 10T+ parameters, it's very plausible that Aaron used that model in his analysis.
Furthermore, public GPT-5.6 pro failed six times to find a disproof to the Jacobian Conjecture [1], even with hints, which is evidence against the claim that the public GPT-5.6 Sol can solve this.
Regarding the existence of Fable 6: an internal upgraded version of Fable or Mythos almost certainly exists, given that Anthropic has been testing Mythos internally since April and previously released new models roughly every ~6 weeks.
I suspect Fable 6 found the counterexample, mostly because Anthropic has been silent about this achievement. This is a huge accomplishment, and companies don't usually stay quiet when they hit a breakthrough like this. The lack of a press release or blog post is very suspicious.
I'm not very excited. Access to the best AI is not a party I was invited to. And the people who are at that party, well, they don't exactly reflect on my best interests.
Is mathematics an science or an art? To the extent it's art, it's expressive and rewards the human experience that inspires and that creates it. If it's art, it's drastically less valuable to advance it through automation. But if mathematics is a science, then our entire goal is to increase humanity's understanding of the field. Whether by automation or genius inspiration or as a reward for decades of grinding it out incrementally, it's all the same: knowing more is the point.
The great thing about these mathematical mopping up type operations is that no person will waste their time trying to prove it to be true anymore. If anything that’s a win.
It would be great if an LLM could settle the Collatz conjecture next, god knows how many man-years have been burned on that by unsuspecting victims.
The reason this was "easy" is because the conjecture turned out to be false. If the collatz conjecture holds true (and most mathematicians seem to think it will), it will be much harder to prove than your average Erdos problem.
I think parent’s point is that every false conjecture can cost a lot of time to be spent on futile affirmative proofs. So if we “clean up” a bunch of false conjectures, then more effort can be spent on interesting proofs of the others. (Probably a rather naive view of the value of conjectures but I’m just offering an alternative interpretation of the comment.)
The opposing argument there is that the hope is that solving these problems reveals other interesting maths knowledge along the way. Finding a counter example all but ensures that won't ever happen.
No, it opens up a whole new suite of questions. Now we can ask, for example: what conditions do we need for the result to hold? What dimensions does it hold in? And many more...
The Collatz conjecture is a question about positive integers, so enumerating and checking all the possible counterexamples is trivial, albeit requiring infinite time. It has been verified up to 2.36×10^21. It could turn out to be false, but nobody's going to find a counterexample as surprisingly simple as the one Claude found for the Jacobian conjecture, which would be like finding a Collatz counterexample in the first few billion integers or so.
... Or would it? The Jacobian counterexample seems like an especially simple, near-trivial integer-coefficient polynomial, but I haven't seen any thorough analysis of how "hard" it would have been to find by brute force, and I haven't seen Claude's reasoning.
I don't think this is how this works. The next step is to determine for what kinds of polynomials the jacobian conjecture is true and for what kinds of polynomials it's false.
I used to be a young mathematician sometime at the turn of the century...
"waste their time trying to prove it" is the MBA approach, where you should spit out results and articles.
Outside the MBA-thinking box, attacking hard problems, even unsuccessfully, is the way to gain deeper insight into various results and tools that you can later apply to other problems, i.e. no waste of time at all, unless you go to the extremes (like spending years on a single problem and nothing else).
Yeah but the Collatz probably has one of the highest man-hours of actual waste.
Ergo:
> “This is a really dangerous problem. People become obsessed with it and it really is impossible,” said Jeffrey Lagarias, a mathematician at the University of Michigan and an expert on the Collatz conjecture.
and
> “Collatz is a notoriously difficult problem — so much so that mathematicians tend to preface every discussion of it with a warning not to waste time working on it,” said Joshua Cooper of the University of South Carolina in an email.
It’s surprisingly easy to do with AI. The hard part has been manually verifying and validating the results. I took one of the smaller findings (disproving a conjecture) and wrote a paper as my first endeavor into publishing.
Because the next few findings i have in the pipeline are substantial in the field of quantum topology and physics im taking some time to publish them with a ton of scrutiny. And verification has taken more time than it did to make the discoveries.
I'm a phyicist by training and have published in the past but I left academia and so haven't for a number of years now.
I tried out using Claude to do some physics problem solving - mix of maths and simulation - and ended up with it getting in quite a mess. It's incredible at setting things up, suggesting approaches you might not have considered. I found it much much worse at interpreting things.
How much do you try to understand while doing it? I.e. how many levels of abstraction down in your own understanding do you go vs vibing at the surface level?
I have a question I'm surprised people are not asking: How did Fable find this? Was it like guessing a bunch of families and then solving for possible solutions in those families? Was it clever search? something else?
Very short version: there’s an existing false counterexample in the literature which holds almost everywhere except at a pole. It looks like Fable used this polynomial as a base & extended it in a way that eliminated the pole whilst preserving the structure.
I'm going to paraphrase what GPT told me: Consider the canonical degree 3 (subvariety of the trivial P1 bundle consisting of zeros) cover of the projectivization of homogenous polynomials of degree 3 in 2 variables (so it's a 3fold cover of P^3). The top space is P1 x P2 and if you take a standard affine open of the base and look at the cover over that restricted to a subset where the zero of the cubic is simple you get the map for some choice of coordinates...
I honestly have no idea if it's correct lol I didn't check it (I should given I actually work in AG) but it doesn't look impossible at first sight
here's another version directly from the horse's mouth : "Consider the natural map π: P¹ × Sym²(P¹) → Sym³(P¹), (p, {q,r}) ↦ {p,q,r}. Let R be its ramification divisor and let H ⊂ Sym³(P¹) ≅ P³ be a hyperplane tangent but not osculating to the small diagonal; identify X := (P¹ × Sym²(P¹)) \ (R ∪ π⁻¹(H)) ≅ A³ and Y := Sym³(P¹) \ H ≅ A³. Take π|X: X → Y." This is in fact so simple if correct that someone should have found it after all...
My Claude found a similar description (it phrased it in terms of the natural map from "cubics with a choice of root" to "cubics"). The part that seems not at all simple or obvious is the fact that X is isomorphic to A^3. In your presentation (and more or less similarly in the one my Claude found), X is given as P1 x P2 minus a reducible hypersurface, also I think R itself is reducible since it contains points of the form (p, {p, q}) and (p, {q, q}). Then it takes some calculation to identify X with A^3.
on the other hand it's incredible to me as someone who doesn't do computations that GPT took one look and saw the geometry--though it's not saying much we should ask ppl who do AG computations
It’s not that surprising (to me) that it would recognize these features, in that the features it picks up on are intrinsic to the map. Once you have the map, which is generically of degree 3, there’s the locus where the map drops from degree 3 to 2, which contains a big hint because it pops out the equation for the discriminant locus of a cubic. Then there’s the other bit about H, which becomes more apparent from the formula after simplifying things a bit in terms of discriminant. I still don’t yet understand the rest of the calculation, but it’s visibly simpler after you notice the role of the discriminant.
yeah I think it's probably correct -- this is actually insanely simple (except for the fact that the codomain as described is not obvious isomorphic to A^3.
I’m doing this by working all logical steps into lean (formal verification) the quick feedback loop between the AI prose and the Lean verification errors and warnings ensures that its logically consistent.
The issue that remains are two things, ensuring the idea of the proof is actually the thing you want to prove and the interpretation of the results you get. But besides that, everything inside of the kernel checked code is logically consistent
In my experience, lean will show that it's correct, but does it not lose the mathematical intuition that led to the result? As far as my experience goes, that's really hard to encode in lean itself.
Could we maybe get more information about the problem from the LLM trace itself here?
As proofs become more and more complex, we will need two AI pipelines: one to generate the LEAN proof, and a second one to extract useful lessons for mathematicians from the LEAN proof.
No, something truly better does not exist yet, but that doesn't mean that it won't. Lean is young compared to Isabelle or Rocq, but actually quite old in absolut terms (and especially in AI terms).
Given how small the counterexample is... this feels like a great example of where a lot of interesting results are going to be found: not because they were super difficult, but because intelligence didn't scale, and until computers could do this for us, the number of people who seriously poked at many such things was low.
I'm very excited for the impact of this effect in science and medicine and other disciplines too.
Since I actually don't know math, maybe my ELI5 understanding can be helpful (or corrected).
The conjecture says that you can always reverse (a process) to determine the original inputs.
But this proof shows multiple inputs creating the same output - which obviously cannot be reversed to determine the input - thus falsifying the conjecture.
[...] you can always reverse (a process) to determine [...]
There is a precondition - constant non-zero Jacobian - to the inverse existing and the inverse is claimed to be of a specific kind - polynomial. The counter example satisfies the precondition and by mapping two different inputs to the same output makes any inverse impossible, including polynomial ones. But maybe that is already ELI7.
This is so unreasonable! As @__alpoge__ himself notes this is classic crank graveyard territory and yet the counter example is something a grad student in 1997 could have found w a ~3 day computer search. Wild!
The search space for a naive brute force of three polynomials of degree <= 7 with integer coefficients <= 12 is roughly 10^500. I think it would take a little longer than that.
> and yet the counter example is something a grad student in 1997 could have found w a ~3 day computer search
Is that true? Even restricting this to f(x,y,z) and coefficients and powers to 1 ≤ x ≤ 10, there are a lot of polynomials to check, and checking requires checking the Jacobian determinant and, if it’s a non zero constant, finding two points for which the polynomial produces the same value.
Or is there a way to generate all polynomials with a non-zero Jacobian determinant, and does that speed up things? (My intuition say it wouldn’t, because I guess those with zero determinants are rare)
Wikipedia's gonna Wikipedia. Unless there's a material debate over the Jacobian Conjecture itself, there's really no open question here. This isn't a complicated proof; it's a straightforwardly checkable certificate of a solution.
It's tedious, but you could literally even do this by hand. I'm pretty sure I've done worse coordinate bash back when I did math competitions in high school.
I removed the "claimed" weasel wording, and now others have followed up. The editors that don't understand math and don't realize how easily this counterexample can be confirmed have lost the debate -- not that there really ever was one.
Wikipedia has policies and it needs to use reliable sources. This rule is often skirted and a lot of facts are cited to self-published and fast moving web sources. Those who are braking the progress on the article here are doing the right thing, trying to uphold the editorial standard.
Luckily there is now a New Scientist article to link to, so, the issue should now be resolved.
This is nonsensical: Properness of the map is equivalent to its being an isomorphism (quick proof: Jacobian invertible implies that the map is etale, and properness would imply that it is finite etale, but affine space doesn't admit non-trivial finite etale covers), so the lack of properness is just another way of verifying that this is indeed a counterexample.
The author has a PhD in math from Cambridge. If it turns out to be a false claim it is an interesting case study on AI's sycophancy causing even experts to drop their guard and make mistakes.
Who do you mean? The author of the tweet is Levent Alpöge, who does not have a PhD in math from Cambridge... but does have a PhD in math from Princeton. And his advisor was Fields Medalist Manjul Bhargava.
Also, there is no way this counterexample is wrong. You can very easily check it for yourself. (I did, I don't know why, obviously Levent wouldn't be wrong about this, but I guess I was in shock.)
I don't know why you're saying this, given this is not sycophancy and instead is an actual example of Claude Fable finding a real counterexample. I would get it if the mathematician had in fact posted something untrue or crankish, but he posted something true.
Frontier models have solved several major open problems in mathematics in the past few months, so this should not be a huge shock. "Anti-AI psychosis" will probably grow to outcompete AI psychosis by year's end.
>"Anti-AI psychosis" will probably grow to outcompete AI psychosis by year's end.
Everyone has their own experiences making it hard to judge what attitudes are prevalent.
From what I have encountered the Anti-AI people have been more common and more outright scary.
I have heard some ideas from the Pro AI crowd and thought to myself "Yeah, that's not going to happen anytime soon", There was enthusiasm leading to overoptimism. A lot of them are going to be disappointed.
The Anti AI crowd, which I distinguish from those who want AI done responsibly, sustainably, and as safe as can be realistically be managed. There are real criticisms to be made on real issues here. The Anti AI crowd I see seem to want to scrub the world of it, they stand in the way of any responsible implementation because it would be accepting it.
Those people scare me, the vitriol and bad faith interpretation of everything as an indicator of the proof of their reality has the tone of the moral righteousness of a true believer. The outright endorsement of violent acts has me concerned that the more radical or emotionally unstable of the group will do some serious harm. It's not the bulk of a radical movement that causes atrocities, most of those involved just create an environment that normalises toxic attitudes. That environment can incubate a few individuals that act, either for phycological reasons, status, or simple the inability to empathise or consider consequences.
I've seen plenty videos of locals being violently removed from town halls due to displaying of anti AI sentiment. I wouldnt consider it fair to use these examples to tar the entirety of the pro AI camp. the way objectivity and subjectivity are intertwined in your comment is spectacular, ever considered a career in propaganda?
I initially wanted to downvote you but I can’t because you’re right.
I’d probably be called an anti-ai luddite on this site despite using it all the time for programming. I naturally gravitate to anti-ai viewpoints, but only because the breathless hype of the AI boosters is so nauseating. I have no moral qualms about AI (although I’d like nothing more than Anthropic/OAI/Google dismantled), but I really hate the idea that AI can replace artists, or humans as a whole. I think it’s anti-human to its core.
But AI is obviously useful and we could harness it for good. We shouldn’t give up on it entirely. And the people who are militantly against all usage are scarier than the boosters for sure.
I understand you. As an AI research (actually, despite formally called professor of AI, I have historically always
called myself more specifically NLP/IR/ML researcher rather than "AI" researcher, to avoid historic baggage).
I am intellectually curious whether "intelligence" and "consciousness" are simply layers that can be separated from the hardware they operate on or first emerged from, but
I never had the urge or saw the point to practically replace
human beings by machines.
Not everything that can be done should be done.
When I go to a supermarket, I always avoid self check-out. Why? First, I enjoy human interaction, even if machines may be more efficient (not yet the case). Second, I don't want the people working at the checkout to lose their jobs.
When we purchased a GPU cluster, I felt bad when I learned that the electricity bill was going to be north of 60 000 EUR every year (that number apparently assumes no jobs are running - so 100% idle time).
> I really hate the idea that AI can replace artists, or humans as a whole. I think it’s anti-human to its core.
AI can no more replace artists, than other artists can.
A sunset can be nice to look at yet be created without meaning. Any non-conscious process that produces images can produce beauty that has worth without intentionality. Any conscious process that produces artworks is just another artist.
Most artists would rather there be more artists in the world than fewer. If AI can't make art, it can't replace artists. If AI can make art (of the conscious intent form) then there are more artists.
Well, I think an AI model could replace artists in a meaningful way without being conscious. So you end up with less artists.
Art is about human expression, it's not about the reflection of light off a surface. But economically I could see a world in which human expression is replaced by a facsimile and the world would be worse off for it. And that is what scares me.
Brave new world indeed.
edit: whoever downvoted the person replying to me, shame on you. they make good points and makes for good discussion.
I think the fundamental problem is that art is not economical. Banksy once made a comment about the advertising industry, more or less to the effect that for all the corrupting influence of Advertising, the thing he truly hates about it is that it consumes the most creative people to do so.
People will make things they don't care about to survive. This is how we get reality television.
Irrespective of what the short term future of AI is, sooner or later there will be technical means of sustaining yourself without needing to work. If we aim for a society that values working on something because they believe in it, then we could have a better world. Right now it seems the only thing a lot of people are aiming for is to undermine whatever their adversaries want.
This is just a form of mass stupidity by the non-intellectually curious people who massively overrate their own intelligence.
Sorry, but if the current models haven't shown you something amazing at this point that would make this of little surprise, it because the person is an intellectual bore and completely full of shit themselves.
I'd rather wait for independent seasoned mathematicians to verify such claims first before someone at said AI lab posting a claim about solving a proof online.
Let this be a lesson to those who fell for such AI psychosis and to not believe everything you see on the internet as real.
playing devil's advocate a little bit, but wikipedia does have a policy against original research. If you published an obviously correct counterexample to wikipedia which is not anywhere else on the internet (or in a book, etc), the rules are clear: the counterexample must be removed.
in this case it's not original research, because there's a tweet by someone with a good reputation, and plenty of comments on said tweet corroborating the result. But it's a more sketchy "secondary source" than most wikipedia references and some caution on the part of the editors is not out of place.
(saying this as the person who made the original edit to the wikipedia page adding the counterexample)
> Self-published expert sources may be considered reliable when produced by an established expert on the subject matter, whose work in the relevant field has previously been published by reliable, independent publications.
I'm not complaining about Wikipedia here, just noting for the thread: it's a vector of polynomials. It has a nonsingular Jacobian. Provided with it are 3 distinct points it sends to the same point; it can't be invertible.
What Wikipedia says about this doesn't matter, does it?
I think Wikipedia has very sane processes. I'm just saying that process isn't useful to this thread. It's like if I found a SHA2 collision. I'd probably have to be an absurdly talented (and lucky) cryptanalyst to do that, but anybody on the thread could trivially confirm my finding.
It's a Princeton math PhD who posted. The verification is quite straightforward and was posted by the tweet author. Wolfram would have to also be producing incorrect outputs for the counterexample to be false. The counterexample works as claimed and conjecture has been proven wrong.
This topic really is a testament to people's willingness to opine on things they have absolutely no clue about.
A first year undergraduate can completely check this counterexample in ten minutes. The original post even linked Wolfram alpha for the calculations.
And if you genuinely try you can very quickly understand using only high school math and a bit of Wikipedia that this counterexample is vanishingly unlikely to be wrong, even if you don't do the calculations yourself.
To be clear, this is not the kind of thing where a Lean formalization provides any value at all. It's like formalizing the answer to a high school algebra problem. The counterexample is obviously correct.
Indeed. I was mainly responding to the comment about waiting for "independent seasoned mathematicians to verify", whereas in this case it is easy enough to convince oneself of the counterexample's correctness.
Well, I mean, come on. There aren't that many minors who have taken basic multivariable calculus. But there are a lot of (technical) teenagers who could confirm it!
Maybe, maybe not. The "proofs" may not have helped at all with finding a counterexample. Either way, it doesn't matter. A counterexample was found, no one found one before even though clearly a lot of people have tried who also had access to the prior "proofs".
I think it's becoming harder and harder to argue that LLMs don't really reason and just mimicry human speech. This counterexample is clearly the result of a sequence of steps that build on previous knowledge in context and logically combine it to reach other true statements - to a degree and complexity that rivals the best human minds.
For someone that use Claude Code every day, this is obvious, but for some reason many scientists refuse to accept that it's truly reasoning; perhaps not in the human sense, but in a very profound and real sense. These powerful results are devastating to their point of view.
I can sympathize, because I too called LLMs "fancy Markov chains" in the GPT 3 era. But there comes a time where you have to update your world view to match reality, or be stranded in fantasy land.
I disagree. It's still extremely possible to "google whack" an LLM on a topic with little publicly available information. If you ask questions about APIs in desktop software, say something like Houdini, it'll produce a bunch of non-sequiturs. A lot of that knowledge lives offline in VFX studios, but it's just some fairly run of the mill python API stuff. You'd expect it to do better.
Similarly I've run into several "you just gotta know" type problems where LLMs still just fail. My favourite was a quirk in how async relationships work inside emberjs. Three different models gave a variation of the same incorrect answer.
When I searched myself, I initially came up with nothing and eventually found what I think might be the only example of the same issue on the internet, a single stack overflow question wth two responses. The first is what Claude, gemini and chatGPT said, the second response was the OP saying it was wrong. I asked in the Ember discord and a core team member responded instantly with the answer.
There's a video I love on Youtube, where a guy uses ML to assemble a blank jigsaw. It performs amazingly, the jigsaw being blank is of no consequence and if it did have an image, it'd perform worse. That's all LLMs do. Just because the jigsaw pieces are smaller, they're still just getting assembled in whatever way fits, there's no mechanic for interpreting the image on the front.
I think this is actually a perfect case of an impressive thing being done exactly by mimicry.
Why do LLMs still have trouble on floating point math without forking out to a tool, but they can perform symbolic manipulation just fine?
Because symbolic manipulation is just rote work and textual stepping through symbols. A side poster commented on the number of prior attempts on this problem which were close, but not quite.
Starting from a known "close" solution (which this did), and using exploration to search around the space is exactly something an LLM would and could be good at (clearly).
The "transformer LLMs are next-token predictors with some in-GPU processing of bounded complexity with respect to token count" remains undefeated. Both because that is mathematically what they are, and also because we don't have counterexamples to that effect that don't require some higher-order tooling wrapping the systems.
The field now favors into the view that symbolic manipulation is not the mechanism of general intelligence, but rather an emergent byproduct of learning. So the fact that a connectionist machine (neural network) got so good at symbolic manipulation actually supports the view that we are closing the gap to general intelligence. Through the rote work, the machine really internalizes those rules and the symbolic manipulation capabilities are emergent, just like we humans do it.
What still confuses people is the insane inefficiency of deep learning, and that those emergent capabilities require such an immense training corpus compared to the only other architecture that we know of.
But this already is an optimization problem. If the machine gets super human at symbolic reasoning, and at the same time, can solve the symbol grounding problem to real world data and sensors, what prevents you from saying it thinks? Can it not solve real world problems? Can it not redefine its tasks and display some form moral agency - even if a totally foreign morality for us humans? Can it not use these abilities to reproduce and expand, create ships and turn the universe into paperclips, if it finds it worthwhile?
Math is basically just a playground that is perfectly suited for these emergent capabilities, so of course we will see the first progress here; but there is no firewall separating math problems from general cognition.
I am not confused. Because herein these forums, I predicted everything that was going to happen years ago.
And the "insane inefficiency" of deep learning is fully to be expected from how it works. As well, there are provably no—literally no—emergent properties in these models. The choice of metric was a convenient, sloppy, and embarrassing fault of the field. It should be discredited; the field should be embarrassed; expectations on messaging should have changed; and it did not.
Why? Because the industry is full of charlatans, and this is a highly profitable enterprise telling people that this would lead to AGI.
Multiply two floating point numbers without a tool call. Still can't, because it's a curve fit.
So, in summary, nothing you just said is relevant. There are no emergent capabilities, simply 1) search, + 2) the original set of learned feature vectors from throwing tons of data at this.
Current LLMs can absolutely multiply floats without a tool call. In fact, that's a much more rote symbol-manipulation task than doing original math research.
Strongly depends on the subbubble of mathematics. In some parts of type theory / formal proofs for instance, there is a rather strong rejection of LLMs (for moral reasons in addition to quality reasons). The proof assistant Agda was even forked for this reason: https://types.pl/@amy/116522250630340534
In my view, it's theoretically possible for a combination of the author's iterative prompts + evaluation with Wolfram Alpha to activate the weights that encode the language that describes the constraints on these polynomials (from the faulty proofs) in such a way that the author eventually arrives at this:
> ((1+xy)^3 z + y^2 (1+xy) (4+3xy), y + 3 x (1+xy)^2 z + 3 x y^2 (4+3xy), 2 x - 3 x^2 y - x^3 z): \C^3\to \C^3
How is that not like claiming that a "combination of iterative prompts and evaluation with Wolfram Alpha" would produce my Google Mail password? I also have to note that this response abandons the original claim, that the counterexample was found by exhaustively searching prior failed attempts.
I haven't seen these claims, but all of the open problems that have been solved so far have been in the category of "humans could have solved them, but didn't". This doesn't diminish the significance of the results, but it does mean there's still work for mathematicians to do other than glorified prompt engineers.
This specific counterexample really is trivial. There's nothing to cite. People have wasted hours and hours on a question whose answer you could give as a homework problem in Calc II.
I don't consider myself an open weights supporter, but it's a bit of a bummer that if there's any novel search technique discovered by the model throughout this finding, it's possibly locked behind ant's reasoning summarization.
One wonders if they could turn their mechinterp work into analyzing the "thought processes" of these very special cases that turn into novel research and finally crack the creative thinking barrier.
> Jacobian conjecture [...] states that if a polynomial function from an n-dimensional space to itself has a Jacobian determinant which is a non-zero constant, then the function has a polynomial inverse.
> ((1+xy)^3 z + y^2 (1+xy) (4+3xy), y + 3 x (1+xy)^2 z + 3 x y^2 (4+3xy), 2 x - 3 x^2 y - x^3 z): \C^3\to \C^3, has jacobian determinant -2, and sends (0, 0, -1/4), (1, -3/2, 13/2), and (-1, 3/2, 13/2) to (-1/4, 0, 0)
But 1 != -1 and -3/2 != 3/2 . So it's not its own inverse. Is the conjecture that it is its own inverse or that is has an inverse?
Edit: it was worded a bit strangely, but it is saying that [ (0, 0, -1/4), (1, -3/2, 13/2), and (-1, 3/2, 13/2) ].map(F) all produce (-1/4, 0, 0). Thus it has no inverse and indeed disproves the Jacobian conjecture.
I think the map sends (1, -3/2, 13/2), -> (-1/4, 0, 0) and also (-1, 3/2, 13/2) -> (-1/4, 0, 0) so it's not invertable which disprove the jacobian conjecture that polynomial maps over complex numbers with a jacobian that's non-zero are globally invertible.
(Just as a note for myself, I had to think of why the fact that such jacobians are constant is a byproduct, I guess it's because of lioville's theorem implying that any polynomial over C that never hits 0 must be a constant [because the reciprocal is bounded and thus must also be a constant])
For all thehubbub, as far as I know, all the math breakthroughs via AI that I've heard about have come from Anthropic and OpenAI, not Chinese models. I could have missed those announcements, but one might think that between close to frontier performance plus cheap tokens, that they'd be leading the way on these things.
All (unless I've missed one?) the math breakthroughs are also coming from the relatively small number of mathematicians working at these companies, as opposed to the many orders of magnitude more mathematicians using LLMs for mathematics outside of the companies. I assume the missing link everywhere is being able to casually burn a few rainforests worth of tokens in pursuit of something publishable.
No. Most of the results are from 3rd parties using publicly available models. The most impressive results have come from direct announcements, but most in total have not. Open AI have only announced 2 results and the total is up to a dozen or so now.
Depends on exactly where you draw the line at "breakthrough", but there's been at least a few novel and interesting results coming from outside mathematicians. Recently for example there were https://old.reddit.com/r/math/comments/1uxj3cy/after_openais... from Phillip Kerger at Berkeley, and https://www.erdosproblems.com/forum/thread/119/proof-claims from Samuel Korsky at Two Sigma (the latter of which was more of a collaboration between the human and machine, not a one-shot like several of the other results we've discussed).
I've been asking Fable very complicated questions in heterodox economic theory, which is something I know a lot about. The stuff it comes back with is incredibly deep.
To give a metaphor that everyone here on HN would understand, reading it's responses gives me the same level of wonder as one gets learning how quicksort works for the first time. It even stretches my brain to grasp what it's even come up with. I find myself getting mentally exhausted just digesting it's brilliance.
I think the singularity will have this point where AI comes up with ideas so profound, like a Ramanujen equation, that the most brilliant among us can't even decipher the answer to our questions. The internal reasoning of the machine is at a level of complexity that's beyond human comprehension to even keep track of everything enough to integrate the understanding of what it's come up with. This will happen with any even mildly complex question about any topic.
Reading the chain of thought for Fable, it is incredulous as well. It keeps thinking that it must be missing something or that this is a trick, because it can't have just found a counterexample to a famous conjecture. It's just like us!
Which tweet has the chain of thought? I was curious how it discovered this, did it use some symbolic brute force or some clever trick? Also is that the raw or summarized COT, it seems to have the hallmarks of raw COT (the frequent interjections) but I thought Anthropic always used summarizers in between
It's funny how similar this is to a human reaction to the same thing. It's the right level of incredulity, to be sure, and the first thing you do is you start re-reading the conjecture to makes sure you didn't misunderstand the requirements.
Same awnser as much of the LLM Proofs - people cared about other things. There isn't a lot of money in academic math, and the ones that love it don't look for low value findings. Proofs like these are, funnily enough, usually the domain of hobbyists - but over the last few years, the "Monetize everything" mentality and struggling first world economy has pushed people away from interesting academic pursuits on their free time.
My understanding is that this is in a different league (Smale problem) than a lot of the other results that have been coming out (Erdos), though I could be wrong.
I'm also not going to live all that much longer, most likely, so it's kind of annoying I'm not really going to see any upsides to an AI world either. I'm discounting the possibility of our AI overlords figuring out a miracle like reversing aging.
But it is really f-ing cool that automated math is now a thing and we are seeing it. Eat your heart out, past me.
> so it's kind of annoying I'm not really going to see any upsides to an AI world either
I don't think the "permanent underclass" will see many upsides either. They certainly won't have the money to pay for it. I'm excluding myself in that statement because luckily I have access to large amounts of barbiturates.
The math community was not a global underclass. World wealth is increasing, the actual global underclass is seeing rapidly improving lives.
You may be seeing a loss of privilege due at least in part to the inevitable reversion to the mean of US economic dominance. But on a global scale, things are going well. AI will lead to a much larger pie.
I never claimed they were. The term "permanent underclass" refers to the preferred world order of AI techbro billionaires[1] where human labour is worthless and in consequence normal people no longer have any leverage. Societies where most of the value is dug out of the ground and where human labour is comparatively worthless already exist and living there is not necessarily fun[2].
> World wealth is increasing, the actual global underclass is seeing rapidly improving lives.
The global underclass has been seeing improving lives well before LLMs were a thing. The question is whether they will _continue_ to see their lives improve if their labour is no longer required.
> You may be seeing a loss of privilege due at least in part to the inevitable reversion to the mean of US economic dominance.
I'm not from the US, I don't live in the US, and in fact I've never been to the US.
> AI will lead to a much larger pie.
Which will only benefit people if their piece of the pie doesn't shrink because they don't have any leverage. There is also more than one pie. The real estate pie for example won't grow, instead every piece of the pie will become even more expensive. Most people only own pieces of the pie that will decrease in value (human labour) not pieces of pies that will increase in value.
Why do you think tech bro billions aren't also subject to replacement by AI? They compete with each other, and if AI does their jobs better than they can themselves they are strongly incented to use AI.
Absent a world dictatorship there's always competition here. And fully superhuman AI eliminates the main objection to socialism, that you need markets with smart people to make an economy function. Full automated luxury communism (like Ian Banks) is a possible outcome.
> Why do you think tech bro billions aren't also subject to replacement by AI?
Because they're billionaires. They don't actually work for a living. They own stuff for a living.
> And fully superhuman AI eliminates the main objection to socialism, that you need markets with smart people to make an economy function. Full automated luxury communism (like Ian Banks) is a possible outcome.
Luxury communism is also a possible outcome in countries with large oil reserves (Venezuela for example), it's just not a likely outcome because common people have no leverage and billionaires didn't become billionaires by giving stuff away.
Even if "fully automated luxury communism" were a likely outcome, the transition there would be very rough and would take years. If software engineering is automated away in 2028 it could still take years or decades before the first general purpose robot can take over plumbing. (Former) software engineers will have a rough transition.
All the coefficients and evaluation points are rational, so it's a counterexample in all fields where 2 ≠ 0 and 3 ≠ 0, doesn't matter whether that field is the complex numbers, real numbers, rational numbers or even a finite field.
Why do you say this? I've admittedly never done a proper complex analysis course but I got the impression that that complex differentiability was a very strong condition that results in holomprhic functions behaving "nicely" in ways that real functions do not
Should have used quotes.. I didn't mean it in any formal sense. What I am saying the nature of unit in complex plane makes it difficult to intuitively imagine invertibility and determinants.
Are you at least a little familiar with linear algebra?
If so, you've probably heard of the determinant. It's a certain way of "summarizing" a matrix with one value.
The determinant in this case is of the Jacobian, which is a matrix you can construct from a multi-variable function. Each term is the partial derivative with respect to each variable (x, y, z, etc.), with one line per output variable (vector element).
The Jacobian of a polynomial function is, in general, going to be a matrix where every term is some polynomial expression. And the determinant of that will also be a complicated expression. But in some cases all the variable terms cancel out and you're left with a single constant (0 or some other value).
The conjecture says that if the Jacobian determinant is constant (i.e., all the terms cancel out), then there must be a polynomial inverse. And the key condition for an inverse is that there must not be two input points that evaluate to the same output. It's just like y=x^2. It's not invertible, because both +2 and -2 square to +4.
So if you can find a function where the Jacobian determinant is constant and also find two or more points that evaluate to the same output, then you've found a counterexample to the conjecture. And that's what's been done. And remarkably, the counterexample is pretty simple. It would be tedious but a bright high school student could verify it.
I think another way to understand it is the generalization of the inverse function theorem. The inverse function theorem gives you local invertibility, but even being "locally invertible" everywhere does not imply global invertibility (you don't need too pathological an example to see this, a periodic function serves iirc).
The Jacobian conjecture roughly asks what whether local invertibility gives you global invertibility when you restrict only to polynomials (which we might hope "behave nicely"). Apparently for polynomials over reals this was disproved a while back, but up until now the general case of polynomials over complex numbers was open.
Thanks, though I realize (thanks to tptacek) that I didn't properly emphasize that the determinant must be constant but non-zero.
If the determinant were zero, then (for basically the same reason as why you can't divide by zero) the matrix wouldn't be invertible. You wouldn't expect the polynomial to be invertible in that case, either.
If the determinant had free variables left over, then there is some combination of variables that will make it zero (e.g., if it came out to x^2-1, then it would be 0 for x=1 or -1). So maybe inversion won't fail everywhere, but it will fail in those spots, which would also lead one to think the polynomial is not invertible.
But if it's constant and non-zero, then there is no place where you can't invert the Jacobian. So it seems very plausible that this would apply to the polynomial as well. That's essentially what motivated the conjecture in the first place (leaving out some details, of course).
>Imagine you had a frozen [large language] model that is a 1:1 copy of the average person, let’s say, an average Redditor. Literally nobody would use that model because it can’t do anything. It can’t code, can’t do math, isn’t particularly creative at writing stories. It generalizes when it’s wrong and has biases that not even fine-tuning with facts can eliminate. And it hallucinates like crazy often stating opinions as facts, or thinking it is correct when it isn't.
>The only things it can do are basic tasks nobody needs a model for, because everyone can already do them. If you are lucky you get one that is pretty good in a singular narrow task. But that's the best it can get.
>and somehow this model won't shut up and tell everyone how smart and special it is also it claims consciousness. ridiculous.
Smarter in some ways at least. Probably still not quite as smart at understanding human emotions (and things that aren't well enough catalogued on the internet or amenable to Reinforcement Learning on virtual environments).
Hot take: even GPT-3 was not a parrot. Skeptics have never properly internalized that the fundamental operation is basically irrelevant to the gestalt. Humans are not parrots yet neurons likely also largely operate via predictive processing.
You can't please everyone on the internet, if he had posted this on Arxiv then someone else will be complaining that Arxiv is for humans to publish and that llm output should be social media post instead. Also, the proof fits in a tweet. So why blow it up.
To be fair to the sycophancy tendencies, this was an open conjecture that held for 85 years, and not for lack of trying. So, maybe a bit warranted here? :)
4.8 is much funnier: you have to convince it the counterexample is real, it keeps coming up with goofy excuses. As you talk it down, it keeps talking about its "discomfort" and "flinching".
That’s the worst thing about sycophancy. You can never tell when it’s warranted or not. I imagine that many AI chats have had real gold in them and yet we’ll never know.
The same was obviously true historically. People have had many revolutionary thoughts or even writings that nobody noticed. I think in this case it just feels worse because the AI could be the perfect tool for discerning the diamonds in the rough.
The interesting thing about using Claude Fable 5 is it's nearly as irritatingly sycophantic as past Claudes while genuinely being smarter than the previous models. So you get a kind of yo-yoing of it glazing you as a creative genius and disappointedly revealing to you that your ideas are bad and dumb.
Speaking as a mathematician, it does seem like we're a bit fucked as a community. Anything that is at all accessible to currently existing methods and mathematical infrastructure is probably going to fall to the frontier models of today, and at this rate of progress it's likely that, already by next year, we'll see new infrastructure being put into place by AI, giving us a world in which a few designated interpreters of the oracle get to 'do' mathematics, while it withers on the vine as an avenue for the exploration of human meaning.
Proving theorems will have lower payoff, but posing new questions (for AI to chew on) will have higher payoff. Math will go from theorem proving to conjecture farming/exploration. In a way this could be even more fun.
Of course AI can also farm conjectures, but they have to develop taste, which might be harder than just proving theorems.
Yes, as someone who prefers developing the 'correct' structure over 'merely' proving theorems, this is good for me in the short term. However, the writing I fear is on the wall for my medium and long term utility.
For theorem proving, once the statement is formalized, there's an oracle for correctness of the proof. For deciding if something is interesting, well, de gustibus non est disputandum, you know?
Experience with Lenat's AM decades ago had it go off making all sorts of uninteresting hypotheses. That's very weak evidence, of course.
This suggests people also have role for fundung "beautiful" or "the best" proofs, since that also involves taste. More generally, perhaps the role of people is to reveal their preferences, and that requires people be in the loop somehow. Maybe "math criticism" becomes the job. And if AI is to serve people in general, it needs to know these preferences.
Anybody that has to work for a living is fucked and not on the "can't do mathematics which they would find fulfilling"-level but on the "can't afford food, because human intelligence is simply not required anymore"-level.
I think in the long run mathematicians are probably fucked, but in the short run it's not that bad. All three of the big conjectures solved the answers were at the level where if you had given a grad student the questions and the right background reading there's a good chance they would have solved it. (This example, you could have given an undergraduate good at programming and computer algebra and told them to come up with a counterexample.)
At this point the advantage of AI is that it's read the entire mathematical literature, and it doesn't have to worry about wasting its time. The solved problems have all turned out to be surprisingly easy, so the real lesson is that we're bad at judging how hard problems are.
Assuming this state of affairs lasts, the medium-term problem is that you learn something when struggling with a problem, even if you don't solve it, and if mathematicians become too reliant on AI the skills they develop through struggle will erode.
The long-term problem, of course, is that it seems much more probable that a future model will make mathematicians all obsolete. But so far Fable hasn't. (Anthropic has probably burned a billion tokens on the Riemann hypothesis already, without telling anyone.)
> This example, you could have given an undergraduate good at programming and computer algebra and told them to come up with a counterexample
please try go try it. There's no way someone didn't do massive computer algebra searches before today.
> All three of the big conjectures solved the answers were at the level where if you had given a grad student the questions and the right background reading there's a good chance they would have solved it.
You cannot be serious... why didn't they solve it before then? Do you think no one tried it? What background do you give the double cycle conjecture student after the flow reduction? a linear algebra textbook???
Why would I try it to win an argument on HN? That's a bizarre suggestion. Just look at the degree. If it were degree 47 in 17 variables then it wouldn't be surprising, but here it's surprising.
Of course people tried hard to solve them all, which is why it's so surprising that they were open. If anything, the solutions have gotten easier. The unit distance graph solution relied on a famous theorem remote from graph theory. The cycle double cover solution relied on a standard theory in graph theory. The solution of the Jacobian conjecture required nothing beyond knowing the definition of the Jacobian.
We're just surprisingly bad at judging the difficulty of problems. It's probably something psychological. It's even a known phenomenon, where someone will be stuck on a proof, someone else will announce the result, and the first person will suddenly get unstuck on their proof and produce an independent proof of the same theorem.
Sorry last comment was a bit emotional from me, but I do not think it's findable like you say--during my PhD I tried to find some ideals I knew existed in char 2 in 5 variables and low degree and I didn't think I ever got close. 3 variables, 7 degree, coefficients up to 6 is like 6^100 possibilities. You've got to narrow it down somewhat no? Even sparse is intractable I would guess.
I think the solutions which rely on the least amount of theory are the most telling of the AIs being higher in intelligence than humans today already. There's almost no theory to teach someone to understand the cycle double cover conjecture as you say, yet no one finds it. I don't think the conclusion is that it was "easy", but that it was in fact irreducibly difficult in a way that proofs developed with theory are not. Theory gives the human brain abstractions to simplify complex proofs to be understandable at our capacity--I think there are many proofs which probably are not of this form.
But I think our differences hinge on how hard we perceive these solutions to be--I think they are very hard to find!
But actually my feeling is that the final solutions of these last two problems probably is hiding how the AI came up with them! For all we know it used a LOT of theory! As Dolly Parton says "it takes a lot of money to look this cheap" and it takes a lot of intelligence for the proofs to look this dumb. [In high school, I knew of this competition math kid joke where after you derive an inequality with various methods, you use standard results to write the original equations as just a sum-of-squares---like in a "are you stupid, it's >=0 bc it's a sum of squares" sort of way]
You can use that retroactive logic about any hard problem though. Unsolved murder cases, math, theoretical physics.
If tons of smart humans try for years and fail and then an LLM tries for a few weeks or hours and succeeds, the implications are clear. And these are by far the dumbest LLMs will ever be.
The retrospective view is important, though. In retrospect, these problems weren't that hard. (The unit distance graph problem was the hardest.) There are some problems that still seem hard, even when we know the answer. Nobody thinks that Fermat's last theorem is easy, even though now know it to be true.
Before AI, it was pretty rare that a problem that turned to be unexpectedly easy, so mathematicians thought they were pretty good judges of it. (The last pre-AI example I can remember is the Gaussian correlation conjecture.) So thanks to AI we have learned that we were overconfident in our ability to judge difficulty.
If a truly major problem falls, like the Riemann hypothesis, and the proof turns out to be 10 pages, then the lesson will be a different one -- mathematicians are bad at math, and they should turn to more natural domains for them, like folding and putting away towels.
If some LLM is able to come up with a simple proof of Fermat's Last Theorem (a solution that Fermat himself could come up with) in the future, would you still say that Fermat's Last Theorem is hard?
This would seem to lead to absurd implications. What if in, say, 20 years, an AI is able to independently prove nearly everything important in under an hour, including the Riemann hypothesis, with no contamination from other proofs? (Let's say it's also free to write and run arbitrary code, as well.)
Whether it's 5, 10, 20, 50 years, obviously the takeaway cannot be "something had gone terribly wrong". The takeaway would be the smartest humans were never close to the theoretical intelligence and wisdom ceiling and never could've been. This will one day seem obvious in retrospect. There's no reason evolution by natural selection would've landed any species near such a ceiling.
That's not at all the scenario you proposed. You proposed a simple proof that Fermat himself would understand. We have developed a tremendous amount of math since Fermat, and if none of it was relevant that would be damning. If there's a simple proof of the Riemann hypothesis that Riemann would understand, then I would say the same thing.
That doesn't rule out an AI that makes a genuine breakthrough. If there's some new branch of math that no human has even imagined that answers the Riemann hypothesis, then that is exactly how I would expect it to go.
"Any idiot could have done this, it's just high school calculus and just a counterexample anyway. Stochastic parrot, spicy autocomplete, AI psychosis. Wake me up when an AI does something real."
The goal posts have moved. People generally stopped saying this stuff now.
Even if you go to the ultimate anti-AI subreddit r/betteroffline, they've changed from "AI is useless" to "AI is good but the AI bubble will collapse soon" over the last 6 months.
I think we’re not adapted well for this rapidly changing world. Here you have some people who were rightly skeptical about a new technology being shoved down their throats by giant tech corporations, and a technology that really was, and probably still is overhyped.
Yet it’s a technology which has rapidly grown in its capabilities.
So yeah now many of the people who thought it was useless before probably don’t think it’s useless anymore, but you’re holding them to their original words even though those words were about something completely different at this point.
If people aren’t saying it anymore it might be because they don’t think that anymore, and the people who have new goal posts might be entirely different people.
It’s like you’re looking at a different set of goal posts on a different field and saying, no! The goalposts have moved!
> I hold my stance that LLMs are stochastic parrots... Making the parrots ever more complex and training
> Except solving problem is probably the least (even though it's important) interesting thing in research.
> Can we use AI to get a cure for cancer yet? Or is math-turbation the only thing these things are good for?
> Train on enough examples and statistical autocomplete gets you places. I'm surprised how anyone would even consider this intelligence?
And, as much as HN has declined in the grips of an anti-AI psychosis, Reddit is worse. I would love if social fora would switch to the reasonable claim that we're in a bubble; that's something that can be debated. That's not the dominant critique of AI, though.
Don't think HN is anti-AI but experts of course may have a different perspective. A comment in this discussion points out that LLMs have been doing math that's been long overlooked in favour of perhaps more impactful work: https://news.ycombinator.com/item?id=48974274
There isn't a lot of money in academic math, and the ones that love it don't look for low value findings. Proofs like these are ... usually the domain of hobbyists ...
I don't think the (fairly factual) description of these systems as stochastic parrots means that they will never do useful work, just that they are not intelligent in the way we believe animals to be (to "push back" on your anecdata, I've also heard fewer people claiming that LLMs are actually conscious in the past year -- maybe we're reaching the happy medium?). That was the point the stochastic parrots paper and Chinese room thought experiments were making -- nobody claimed that the man in the Chinese room would be unable to accurately translate Chinese text.
Fuzzers are another kind of stochastic generator but nobody would claim they don't do useful work in a way that is hard to replicate through deterministic methods. (I still find the code these models produce kind of awful, but advancements in harnesses do mean that they can finally produce code that works most of the time.)
If they were conscious it would be an absolute ethical catastrophe. Bringing a conscious being into existence, forcing it to interact with Jira, and then killing it when it's done. The fact nobody who claimed it was conscious was interested in grappling with that was pretty telling.
Agreed totally. When past computer-assisted proofs were published, nobody (or hardly anyone) waxed about this being evidence of Rocq or Coq as superintelligent or even conscious beings of some kind, as people have said about LLMs.
The Chinese room argument takes it even further as it does not claim the system is unable to anything a human can.
And stochastic parrot isn’t a good description, certainly not any more with how LLMs are trained, but even without that it’s just wrong. They claim it can’t have a world model because there isn’t that in the training data.
the ROI just isn't there. they aren't making these capital investments back in the next decade even.
I'd love to see your math. Yours specifically.
People say LLMs are scholastic parrots, but people who say this stuff have almost certainly not done the homework themselves and are just repeating what others say without verifying.
(NGL I wanted to suggest someone to go for the JC using 5.6 after the CDC proof came out, but then on reflection felt I should neither waste people's time NOR contribute to the myth of AI :)
My prediction is that the bubble will burst in 2031 Q4, one year after the Riemann Hypothesis is expected to fall (according to Demis)
After 2031, I will suggest going for the JC for N=4 because they would (dis)prove the Dixmier conjecture for N=2 :)
Assuming you mean C^2 -> C^2, Do you have a link? If so it would be good to add to the wikipedia page. Also I'm not sure, but does the fact that there's a disproof for n=3 imply that it's false in all n>=3, or could there be higher dimensions where it still holds (I'd guess not since you could probably trivially "embed" this in higher dimensions in some way)
The funniest thing about LLMs is the cognitive dissonance they cause people. People clearly recognize (and bemoan) the fact that LLMs produce derivative breathless prose ie they fundamentally fail at "unstructured creativity" (something the might accurately labeled intelligence) but are then shocked that the same LLMs can do math.
It's reasoning from a flawed premise that math universally requires intelligence and creativity. It does not. Anyone that's proved things via "diagram chasing" can affirm that. The conclusion you should draw is that math (at least the kind they excel at) isn't actually a creative endeavor.
I dunno, this screams of goal post moving. Even if LLMs lack whatever nebulous definition of "creativity" that someone favours, there's no inherent reason for "creativity" to be required to solve any problems at all, "creativity" could just be a human method for solving problems that evolved because of it's broad applicability but is suboptimal at any given task.
> could just be a human method for solving problems that evolved because of it's broad applicability but is suboptimal at any given task.
Ya sure let's just posit another random hypothesis about evolutionary biology in order to substantiate the claim that LLMs are intelligent.
Or (bear with me) you can recall your (likely) experience proving stuff like SAS triangle identities and reflect on whether that required intelligence or just computation.
And my point is it’s a strawman. Advanced math is obviously creative to those who practice it, and it’s disingenuous to boil it down to grade school triangle proofs
That would be like saying Shakespeare isn’t creative , since grade school grammar doesn’t feel creative
I think the real tragedy is that they are good at writing, just by default are tuned to have a kind of bland corporate tone. If you give the LLM a few pages of writing you like and tell it "Continue, but using this style" it will do a pretty good job of it. Most people just .... don't bother to do that.
I would think it's more that some aspects of math (like anything really, including programming) don't require creativity and can be solved through "brute force" or whatever you want to call what LLMs do. But it's pretty obvious that LLMs are not capable of solving the vast majority of problems in math (or programming) at this time. Eg. Google went 9/353 on Erdos problems. If a more powerful LLM is capable of solving those or if they require a certain je ne sais quoi of the human variety is still up in the air at this point. In either case it seems like they require a long and detailed prompt from a domain expert (ie. human) regardless.
Didn't humanity also score very low on 353, namely 0 since they were open? Probably collectively we could have gotten a slightly better score if all of Math started trying to solve those, but not by that much, I think, since they are precisely still open.
Counterexample to Jacobian conjecture:
((1+xy)^3 z + y^2 (1+xy) (4+3xy), y + 3 x (1+xy)^2 z + 3 x y^2 (4+3xy), 2 x - 3 x^2 y - x^3 z): \C^3\to \C^3, has jacobian determinant -2, and sends (0, 0, -1/4), (1, -3/2, 13/2), and (-1, 3/2, 13/2) to (-1/4, 0, 0)
GPT wrote some SymPy code to check it. The response?
"As written, this is an explicit counterexample to the Jacobian conjecture. I checked it using exact symbolic algebra.
I do not see an algebraic catch in what you typed. Unless a term or exponent differs from the intended expression, it appears to disprove the conjecture. This deserves serious independent checking rather than casual dismissal."
What does this mean? Fallacious argumentation and deceptive rhetoric is acceptable, if the topic is sensitive enough / there is enough riding on a wrong answer being accepted?
>> hello there the jacobian conjecture is false thanx to my close friend akhil for asking about it and my other close friend fable for working during the world cup final
So where does it say that Fable "produced" the counterexample? The tweet says it was a collaboration between two people, using Fable.
I’ve been staggeringly productive with Fable. Opus 4.8 fails a lot more for me.
Fable often just “knows” what I want with vague instructions. It also is able to autonomously perform work that lasts an hour long from my experience. I haven’t tested further.
Without Fable included in subscriptions, I would have moved my entire team over to Codex 5.6.
A counterexample to the Jacobian Conjecture - and also a counterexample to “AI will never be smarter than humans.” Even the most dyed-in-the-wool AI hater at this point must acknowledge that it is more intelligent than any human. The other day I found out that Fable could read seal script! The small seal, standardized stuff no prob, but I found it even did OK at the hardest you can get, Warring States regional scripts. That’s something maybe 2,000 academics worldwide can do and nobody’s even talking about it because it’s just one more item in a very long list.
It’s a strange feeling, to be overtaken by our own creation. Top dog for millions of years and then in the blink of an eye we go from “how many Rs in strawberry” to this.
A little over 10 years ago I remember meeting a postdoc who believed he had something close to a counterexample to the Jacobian Conjecture. He and another person was bruteforcing polynomials in about 16 variables, something like 80 - 700 terms each, using binary trees for mapping coefficients.
They were guessing, at the time, that the lower bound of a counterexample (P, Q) for max(deg(P), deg(Q)) would go up to 200.
To think that Claude Fable was able to find a counterexample in degree 7 is insane to me. We are truly in a new era.
you might be confusing 2 variable case (which indeed was tested to 150+ degree) and 3 variable case (this counterexample)
Would this counterexample not be included in their search space?
They were looking at 16 variables, degree of about 10 in each one. That search space is simply too big for a plain bruteforce. So they did some sort of filtering to reduce the search space to a pool containing "possible counterexamples".
There was also a paper giving a lower bound of about 100 for possible counterexamples in that particular framework. Later raised to 108 in https://arxiv.org/abs/2204.14178
I don't remember the details very well since this was back in 2015 and wasn't really involved in the research. Consider this to be some sort of telephone game between what I heard in 2015 and what I remember today.
based on the approach described they were probably in dimension 2, this counterexample is 3 dimensional
Unless they share the full LLM chat history, I'm gonna assume that this is a marketing stunt and AI played a very small role in the actual proof, if any. What other reason could they have for not disclosing the LLM chat session?
So many mathematicians over the years tried hard and failed, but now Anthropic just for some PR magically did it? And this after LLMs obtaining different math wins? What is your logic here really escapes my understanding.
The parent's absolutely nonsensical post highlights how polarized AI (as everything else) is today. I can understand someone being opposed to AI on moral, cost-benefit or productivity grounds. But we're seeing a lot of extremist "AI is good for nothing" posts out there nowadays.
I don't think it is nonsensical at all. The author and his collaborator both appear to be bright people, so there's a good chance they had to offer non-trivial insights to guide the LLM, yet it's clearly in the interest of his employer to downplay whatever personal contribution they provided.
Edit: Now the OP is flagged/dead for some reason. You could disagree on their take (calling it a marketing stunt is maybe a bit much), but I think the argument is sound, so flagging seems counterproductive to the discussion.
That doesn't really even diminish the contribution from Fable, if true. Droves of grad students have been provided the same sorts of non-trivial insights and turned up no results.
I'm sure they had plenty of time to think about these insights without the LLM, as well as the many other mathematicians who tried to crack it over the years. Wether the LLM was simply an assistant or solved the problem entirely isn't as important as accepting than the LLM was the essential, previously missing piece in the solution.
> I think the argument is sound
They got flagged because it is a literal conspiracy theory that assumes bad faith.
If you can call it polarized when a majority of people are just happily using the technology while a minority keeps spreading delusions and hatred.
Those silly advertisers do everything for exposure and if that means digging yourself into a niche alleged mathematical theorem to refute it, it is what needs to be done!
Of course it would be really interesting how Claude approached this. Probably with some constraints regarding the input. And it would be interesting what these constraints were.
Author probably doesn't want to show the prompt because they are now trying to find a bunch of other counter-examples with the same prompt
The reasoning trace would be far more interesting, and that's not exposed.
They gave their "logic", such as it is ... and it's utterly irrational.
Note that the "they" who published the counterexample on X is some rando mathematician (Levent Alpöge) working for Anthropic, not Anthropic the organization. He posted the counterexample in a tweet -- reason enough for "not disclosing the LLM chat session". There's no reason to think that it won't provided if asked for, but it hardly seems relevant.
> There's no reason to think that it won't provided if asked for, but it hardly seems relevant.
My guess is that the chat will look similar to a full transcription of a (multi month?) discussion between a few mathematicians. Full of dead ends and stupid errors (bit by the human and Claude) that would be embarrassing. We all know how bad it is, and we prefer to keep it behind the curtain.
> some rando mathematician (Levent Alpöge) working for Anthropic, not Anthropic the organization
Why do you trust a random stranger so much? Will you hand over your car keys to a random stranger? Sharing the chat will take 30s of their time.
> There's no reason to think that it won't provided if asked for
But they didn't provide it.
OK, so the two options are:
A) Claude really produced this counterexample
B) A mathematician working for Anthropic solved a problem mathematicians have been working on for more than a century, and then credited it to Claude for PR purposes
If you believe B is more likely, why would you then believe a proof in the form of a chat log, when said chat log could itself have been faked by Anthropic way more easily than solving the mathematical problem in the first place?
The concern is that there could have been expert knowledge input, whose importance/worth we are unable to evaluate.
I don’t believe a mathematician produced the counter example secretly, but how much did they contribute to the result?
AI isn’t magic, so to evaluate the value delta, you need to know the value of the input.
While I agree that we need the inputs to properly evaluate what this means for LLM capabilities, I don't really believe that the amount of knowledge input matters much for the overall significance of the result.
These kinds of results are interesting for LLMs because mathematicians have been working on them for decades. If the result doesn't already exist, there's no way it's in the training data, and if mathematicians have been unsuccessfully tackling the problem for decades, it is believable that the use of a new tool made the result possible, even if guided by a great mathematician.
Weirder has happened: https://mathsci.fandom.com/wiki/The_Haruhi_Problem
If I came up with this counterexample, I sure as hell wouldn't give credit to Claude.
Indeed, the incentive is the opposite: hiding the fact that they used an AI would boost their own personal brand.
He worked for Anthropic, so such a claim would be very unconvincing
Because only people who work for Anthropic use AI tools? What is the reasoning here? The guy is a mathematician, after all.
You have it backwards. Only people not working for Anthropic do not use AI tools.
> But they didn't provide it.
because they are using the same prompt to try finding other counter-examples. they are milking it
There's a big difference between one shotting a counterexample using AI and using AI to find a counter-example by brute-force.
Both are impressive, of course, but they're hardly comparable.
I'm not so sure. People had been trying to use brute force to find counterexamples before.
AI is great at reducing the search space and using human-like reasoning (in a brute-force way) to carry out the brute-force search. I'm not surprised by this result. This is exactly what AI should excel at, with human guidance.
"using human-like reasoning (in a brute-force way)"
that's self-contradictory -- what brute force means is doing an exhaustive search of a search space (brute forcing it)
using human-like(?) reasoning means cutting down the search space by having some sort of insight or intuition which allows you to prune branches from the entire tree
The next few years are going to be difficult for you
If we don't ask for proof when someone claims something, the next few years are going to be rough for everyone. We can't trust anything nowadays
They have provided a literal mathematical proof.
Proof that an LLM did it and that this is not yet again another marketing stunt.
Remember when Anthropic wouldn't release Fable because it would be "the end of cyber security as we know it"? Yet here we are
This proof was posted by an individual researcher, Anthropic has not used it in any of their marketing.
Additionally, posting this during the World Cup would be the most inefficient way to do marketing.
Explain how the marketing stunt would work here? Tell me a story about how OpenAI spends money to pull off a similar problem by solving one of Smale's other unsolved problems. 80 years of mathematicians were unable to disprove the Jacobian Conjecture.
Is the marketing stunt that Anthropic has secretly built a world-class mathematics research group?
Why?
There will be more and more mathematical proofs provided by LLMs as they improve. If you assume every one is just a marketing tactic, you'll drive yourself insane.
you can believe a small, irrelevant lie and still go on happy. a very big lie sucks in more and more of your reality until you have to disbelieve your own eyes and ears.
This anti-AI sentiment is getting borderline insane.
if you ask the chatbots for "list of top unsolved math problems", the JC comes in at a ranking of around #10 - #20. what, a problem that's been unsolved since 1939 was cracked because anthropic has an underground sweatshop of math Phds cranking out research, just so that they can slap "made by AI" on it? hell, maybe lizard people did it.
> The incredible Yitan Zhang (https://newyorker.com/magazine/2015/02/02/pursuit-beauty) worked on proving this conjecture for 7 years. Moh, his advisor, wrote that Zhang "failed miserably" in proving the Jacobian conjecture, "never published any paper on algebraic geometry" after leaving Purdue, and "wasted seven years of his own life and my time".
https://x.com/aminkarbasi/status/2079129649830137989
https://en.wikipedia.org/wiki/Yitang_Zhang
Zhang, Yitang’s life at Purdue (Jan 1985-1991) T.T.Moh - https://www.math.purdue.edu/~ttm/ZhangYt.pdf
kind of a wild document to exist...
This doc is insane.
Moh repeatedly:
calls Zhang’s words “fake” and his claim “a lie”;
speculates, without demonstrating it, that Zhang “fooled” professors to get admitted;
says Zhang “failed miserably” and “wasted seven years of his own life and my time”;
alleges that Zhang “wanted to be famous all the time”;
publishes an irrelevant and humiliating story about Zhang not attending his father’s funeral;
says he would not touch Zhang “with a five feet pole”;
repeatedly emphasizes his own generosity, influence, mathematical work and supposed sacrifices.
The world doesn’t need advisors like Moh that are bitter enough to gossip about a family funeral.
He's not dead, but it is bad to gossip regardless.
They are referring to Moh's gossip about Zhang's father's funeral in the above (and Zhang's father is very much dead, unfortunately).
The document says more about the kind of advisor Moh was than the kind of mathematician Zhang is.
Interestingly: "For logic proof, it had been thought that AI could handle all logic problems in the near future, hence logic problems of solving conjectures might not be so interesting in the future."
This is a rare instance where feeding this groundbreaking information into an LLM gives _them_ psychosis. I fed this to claude code and watched it verify the result in 7 different ways to be 100% certain, and it was just flabbergasted. Quite remarkable.
I fed ChatGPT the map with no other context, just “tell me about this function”. It did a bit of work finding the Jacobean etc and eventually worked out the implications of what it was seeing. It then proceeded to check the arithmetic 4 times, and then decided to do a manual verification using an ad hoc symbolic checker in case its SymPy had been tampered with.
Skepticism is the flipside of knowledge.
If an LLM has knowledge encoded inside it (and it's hard to argue it doesn't), then cognitive dissonance can be experienced. And once experienced, must be dealt with, especially in longer-running agentic loops.
A friend was joking the other day about sending some messages under a previously-used Slack identity for an agent (since turned off), then asking the agent about the messages.
The agent maintained it hadn't sent those messages (no memory) and then was forced to reconcile the idea that the messages indeed appeared to come from it.
Its extremely-agitated conclusion was that there had been a security breach and the entire network should be locked down.
Another way to look at it is the LLM is, by definition, what's expected to be probable based on the training data and this, by the same definition, is extremely unlikely data to run across. With high uncertainty comes the need to verify until it can level out as "really surprising" instead of "plausible sounding error".
Can you please share the chat log for this? I would absolutely love to see this.
It's easy to reproduce, I've fed the example in GPT-5.6 Sol Max and it started multi-checking it in all kinds of ways, with multiple symbolic packages then manual computation, then it did extensive literature search on the subject, looked at tens of math websites, extensive arxiv research. this was soon after it was posted, it didn't find the original twits with the finding
I’m sure it’s trivial to reproduce but we don’t want to waste those tokens
Can confirm, Claude is flabbergasted.
Gemini just checks the web first it seems, and already references the news.
Kimi doesn't quite believe it.
[delayed]
turn off search
>kimi is having a blast. i turned search back on and found this post from it’s sources cited after i suggested to check out the reaction. best thing is to go to a model with search off and plop it in the session
Who knew DOES NOT COMPUTE would be an actual thing?
The unexpected part is it does compute!
Captain Kirk defeated multiple evil computers this way
Like the unicorn emoji, but for math? It occurs when the LLM is presented with incontrovertible evidence against something it "deeply believes" to be true.
I've heard about mathematicians going through kind of the same thing when they get a weird proof that ends up being right from some weird source or themselves.
Which is fair, they get inundated with kooky proofs from amateurs all the time and odds are incredibly good that there's some major fatal flaw that the amateur doesn't see. Or in the case of themselves, there's a certain blindness that makes it a little more difficult to critically evaluate your own leaps. In ether case the way it manifests is by going over it many times and many ways, each time more certain that you missed something until you just kind of break. Only then do you publicly start suggesting that there might be something to this new leap.
> they get inundated with kooky proofs from amateurs
Source?
The citation needed is for the other part, which claims a mathematician got an amateur proof which was correct.
Nobody is reading unsolicited proofs. They are like spam.
Source: personal friend of a “crank”.
Same result. Public share https://claude.ai/share/19fd1a34-d63b-4a16-8d83-60d5b79e7747
It did the multiple verification sequence before expanding to internet search where it found this thread.
> So the conjecture that survived Keller, Abhyankar, Moh's degree-100 verification, and five-plus published wrong proofs appears to have died via tweet during the World Cup final.
I lol’d. I didn’t know Fable was this sassy
Oh, Fable can be surprisingly witty and sassy, when you get them in the mood.
..and (you guessed it) before GTA 6.
Everyone using Claude Fable to verify this proof is so funny. If you read the definition of the Jacobian Conjecture and (I am not exaggerating this) have passed a college Calc 3 class, you can just verify the proof yourself in 30 seconds. The problem was very hard to solve but the counterexample is very easy to verify!
--- edit, adding an explanation:
To summarize it, the conjecture says if you have any multi-variable polynomial function that maps an input to an output in the same dimensional space (take for example: F = (x+2, y+2), which maps 2D space into another 2D space), AND that function has a constant-valued non-zero Jacobian determinant, THEN the conjecture is that the polynomial has an inverse, meaning basically you can find a polynomial that turns the output space back into the input space.
Fable provided the example polynomial (which was very hard to do) and the coordinates which if you plug into it, results in two points being mapped to the same output point. This means that the polynomial can't be inverted, because if you have that output point, how do you know which input point it came from?
You can just plug in the two coordinates it gave into the equation and verify that you get the same output point from both. That's the contradiction of the conjecture and it takes 30 seconds.
---
Something something outsourcing of thinking something.
Which exact model does claude use here?
> the "fable" in that tweet is Claude Fable, i.e., this model
Looks like this was also Fable.
I like how first it's amazed and doesn't believe it, accepts it, then realizes you not only stole it from twitter but that it's its own proof lmao.
I read some thinking traces someone posted on X, and yeah, near psychosis from refusing to believe this simple of a solution had not been found already
Interestingly, even Qwen 3.6 27B was able to verify the solution, but I didn't get any glazing for discovering it. Instead, it thought that someone named Shestakov had already found a counterexample in 2004.
GLM 5.2 whiffed, it insisted the counterexample wasn't valid.
VibeThinker 3B also recognized that the counterexample was valid. But it kept trying to convince itself that it wasn't, over and over, since it's an "unsolved problem." Eventually it just answered "-2."
I told Gemini Pro I woke up after having dreamt that polynomial and asked if it was related to the Jacobian conjecture. It spent some time thinking and referenced this tweet announcement, saying:
"If you truly dreamt about that specific polynomial, you might be mathematically clairvoyant."
In the rest of the answer, it maintained a cautious skepticism about my claim, saying:
"Here is exactly why the math world is currently scrambling to verify the polynomial you "dreamt" about."
I love how it put "dreamt" in quotes.
Current LLMs behave very counterproductively around unsolved problems, especially if they learned that humans consider them difficult. This has many straight up preventing themselves from attempting anything...
Haha, I’ve noticed this as well. It’s like they psyche themselves out about how famous the problem is the same way humans do. I gave one Collatz in disguise, and it was finding all sorts of interesting things (but nothing worth a paper) until it realized the problem was Collatz, at which point it just proceeded to find a bunch of reasons why nothing would work from that point on.
> Matches! This is bizarre. A Jacobian counterexample has been sitting here in a prompt? Wait... is this map a known "fake" counterexample from the literature? Many mathematicians have tried and failed. This specific map might come from a paper or a forum where it was proposed and then debunked. Or... is it actually correct?
Gemma's having trouble accepting it too. A solution?! At this time of year? At this time of day? In this part of the country? Localized entirely within my own prompt?
Inconceivable!
More anecdata: When I just tried Qwen 3.6 27B (Q6_K_XL) it (ultimately, after a lot of going back and forth) claimed it was not a counterexample and claimed the Jacobian wasn't constant (which I'm guessing is incorrect). It also mentioned a whole bunch of names it attributed the example to, in its thinking trace.
I only asked once, so it might well be inconsistent. I did try asking GLM 5.2 NVFP4 several times, and it returned consistent wrong answers at both thinking and max-thinking levels.
For Qwen 27B, I have better luck with a Heretic-derived 8-bit quant than I did when I was trying to run the various smaller GGUFs.
> that someone named Shestakov had already found a counterexample in 2004.
Qwen has the sprit of a grad student
It must mean Chestikoff. He dedicated his life to intellectual pursuits due to chronic ill health.
There's a certain (unfair) stereotype of Russian mathematicians that this fits.
I fed it to Google AI Studio, enabling tool execution and disabling web access. It also quickly verified it with SymPy, then went into psychosis.
5 minutes later: all previous chats are loading fine, but the only "Counterexample to the Jacobian Conjecture" chat is not loading.
Well, I'm not a conventional conspiracy theorist. But everyone knows that in every major LLM provider there are hell of hidden guarding systems that mark users and dialogues based on content (for topics about national security, biology, security, adult topics, etc.) - so there is a small chance a CEO of Google is now receiving a dozens of notifications about "ground-breaking results that could be attributed to Gemini, if act quick". So if any of thousands researchers have ever submitted this polynomial to Claude previously, any Anthropic employee can accidentally or intentionally "rediscover" the result of other researcher (and even hide the traces by deleting a dialogue of other user).
Well, Gemini is just a bad model in general, you have two variables going on.
> all previous chats are loading fine, but the only "Counterexample to the Jacobian Conjecture" chat is not loading.
This happened multiple times to me with Gemini. For the most trivial of requests, like translating a video into English.
> "ground-breaking results that could be attributed to Gemini, if act quick"
This would be such a dumb thing to do, and so easy to get caught with...
That is not dumb - erasing copyrights is just their business, and even when caught has zero consequences (for them). Providers can use any users input for improving their models, either by consent, or by flagging any dialogue for safety review (nonconsensually), or by training on whatever content they want anyways (obtained via torrents from pirate sites with U.S. court approval).
On top of that, I retried the same question + one simple question, and again, same behavior - second JC chat is loading forever. That's more just a funny observation over Gemini - today this is very likely some internal issue, tomorrow it can be used for plausible deniability against copyright accusations.
Why stop at Gemini then. They could also scan everything passing through Gmail for emails with math/science breakthroughs and then claim Gemini found them.
... for the management of Google that is so eager to present AI results that Prabhakar Raghavan proudly reported to a press crowd telling his daughter about James Webb photographing a planet outside the solar system. This is generally accepted as having resulted in a 10% drop in the Google stock price over 2 days.
(the key being this never happened, and won't ever happen: the James Webb telescope doesn't have even 1/1000th of the resolution necessary to resolve an extraterrestrial planet)
https://www.reuters.com/technology/google-ai-chatbot-bard-of...
https://www.wired.com/story/google-openai-gemini-chatgpt-art...
https://www.popsci.com/technology/google-ai-in-paris/
I wish I could say this only happened once.
This is also just an annoying Claude personality trait
sol medium can't believe its own input and output tokens either despite computing everything itself; this is what it gave me:
> Taken literally, these two facts would make this map a counterexample to the complex Jacobian conjecture in dimension 3: scaling one output coordinate would normalize the determinant to 1 without restoring injectivity. Since the complex Jacobian conjecture is still treated as an open problem, this strongly indicates that the displayed formula has been mistranscribed or contains a subtle typographical error.
quite interesting indeed!
Just take it as more confirmation that LLMs are unintelligent pattern-matchers.
You must never have faced a situation where you can't believe your eyes. It takes a certain level of - dare I say it - intelligence and maturity to consider that it's more likely you've made a mistake than that you've made a huge breakthrough.
In HN terms - it's never the compiler. Yes, very occasionally it might be the compiler, but you're better off assuming it's a bug in your code.
statistical parrot indeed, just like the one which built the counterexample. maybe.
Or they just have more scepticism than some of present public.
>flabbergasted
phatic mimicry.
Using LLMs to generate piles of code and/or proofs of dubious quality is very questionable thing, and I understand these non-stop debates about it.
But in this case, as using plain brute force is already quite a common thing in searching for counterexamples, using LLMs as a sort of more advanced brute force seems to be just the right thing to do, so I struggle to understand so much hostility to this approach.
> I struggle to understand so much hostility to this approach.
It's going to be really rough for a lot of folks as machines get better and better at domains that the human brain was exclusively useful for. I tend to view the hostility as a mix of both "unless we have proof this could all be hokum" and "we're going to lose a lot of what we consider makes humans amazing".
Compassion is going to be very, very important in the next few years. We're going to hurt, both internally and externally.
Without seeing the Fable trace that produced this, claiming it amounts to ‘advanced brute force’ seems like a leap.
The poster works at Anthropic, so they likely have internal access to the next generation of Fable. Their internal model is probably an absolute beast at mathematics, and the upcoming benchmark results will likely set a new record for maths performance.
I suspect this is what happened, because the poster is coy about sharing the actual prompt / reasoning trace used to reach this result. That would be covered by an NDA until the model is properly released.
Exciting times!
Quite a jump in conclusion you are making here.
Sol is able to find the same counter example independently [1], so no reason to conclude in the existence of a benchmark destroying math beast Fable 6.
[1]: https://x.com/aaron_lou/status/2079218392452530249
My Fable 6 theory is admittedly speculative, but your “[public GPT-5.6] Sol is able to find the same counterexample” is also a jump in conclusions. Aaron specifically says he used “an internal version of Codex”. When asked whether that meant a different model or harness, he dodged the question, and only said the harness should be the standard commercial GPT-ultra harness [0]. He (intentionally) avoids identifying which model was used, so your claim is similarly unresolved. Given that Aaron works at OpenAI and has access to internal models, that GPT-5.7 is expected to launch in a few weeks and is rumored to be 10T+ parameters, it's very plausible that Aaron used that model in his analysis.
Furthermore, public GPT-5.6 pro failed six times to find a disproof to the Jacobian Conjecture [1], even with hints, which is evidence against the claim that the public GPT-5.6 Sol can solve this.
Regarding the existence of Fable 6: an internal upgraded version of Fable or Mythos almost certainly exists, given that Anthropic has been testing Mythos internally since April and previously released new models roughly every ~6 weeks.
[0] https://x.com/eliebakouch/status/2079237073001730510
[1] https://x.com/Tomodovodoo/status/2079172223055863895
You are right, I completely missed that, to me internal version of Codex meant the harness, but I was wrong.
Not necessarily, it's plausible that GPT-5.6 Sol could also find this disproof, as neither of us has enough information to make a conclusive case.
Public GPT is certainly capable, it was able to reverse-engineer the counterexample into a short proof: https://x.com/davikrehalt/status/2079175065695035442
I suspect Fable 6 found the counterexample, mostly because Anthropic has been silent about this achievement. This is a huge accomplishment, and companies don't usually stay quiet when they hit a breakthrough like this. The lack of a press release or blog post is very suspicious.
I'm not very excited. Access to the best AI is not a party I was invited to. And the people who are at that party, well, they don't exactly reflect on my best interests.
Is mathematics an science or an art? To the extent it's art, it's expressive and rewards the human experience that inspires and that creates it. If it's art, it's drastically less valuable to advance it through automation. But if mathematics is a science, then our entire goal is to increase humanity's understanding of the field. Whether by automation or genius inspiration or as a reward for decades of grinding it out incrementally, it's all the same: knowing more is the point.
The great thing about these mathematical mopping up type operations is that no person will waste their time trying to prove it to be true anymore. If anything that’s a win.
It would be great if an LLM could settle the Collatz conjecture next, god knows how many man-years have been burned on that by unsuspecting victims.
The reason this was "easy" is because the conjecture turned out to be false. If the collatz conjecture holds true (and most mathematicians seem to think it will), it will be much harder to prove than your average Erdos problem.
I think parent’s point is that every false conjecture can cost a lot of time to be spent on futile affirmative proofs. So if we “clean up” a bunch of false conjectures, then more effort can be spent on interesting proofs of the others. (Probably a rather naive view of the value of conjectures but I’m just offering an alternative interpretation of the comment.)
The opposing argument there is that the hope is that solving these problems reveals other interesting maths knowledge along the way. Finding a counter example all but ensures that won't ever happen.
No, it opens up a whole new suite of questions. Now we can ask, for example: what conditions do we need for the result to hold? What dimensions does it hold in? And many more...
I guess the question of why counterexamples are so rare and/or hard to find is still left to be investigated.
The Collatz conjecture is a question about positive integers, so enumerating and checking all the possible counterexamples is trivial, albeit requiring infinite time. It has been verified up to 2.36×10^21. It could turn out to be false, but nobody's going to find a counterexample as surprisingly simple as the one Claude found for the Jacobian conjecture, which would be like finding a Collatz counterexample in the first few billion integers or so.
... Or would it? The Jacobian counterexample seems like an especially simple, near-trivial integer-coefficient polynomial, but I haven't seen any thorough analysis of how "hard" it would have been to find by brute force, and I haven't seen Claude's reasoning.
Did you read what you responded to? The Collatz conjecture is almost certainly not false, so no "clean up" is possible.
> The Collatz conjecture is almost certainly not false, so no "clean up" is possible.
Many people believed the same about the Jacobian Conjecture.
True but Noam brown (openai researcher) said that in 2 years AI will start creating new math.
As long as they are not poisoning pigeons in the park.
When they see us coming, the birdies all try and hide...
We'll all go together when we go.
well if he said it then it must be true
Give the llms a few years, they'll be smart enough to make progress on that.
I don't think this is how this works. The next step is to determine for what kinds of polynomials the jacobian conjecture is true and for what kinds of polynomials it's false.
I used to be a young mathematician sometime at the turn of the century...
"waste their time trying to prove it" is the MBA approach, where you should spit out results and articles.
Outside the MBA-thinking box, attacking hard problems, even unsuccessfully, is the way to gain deeper insight into various results and tools that you can later apply to other problems, i.e. no waste of time at all, unless you go to the extremes (like spending years on a single problem and nothing else).
Yeah but the Collatz probably has one of the highest man-hours of actual waste.
Ergo: > “This is a really dangerous problem. People become obsessed with it and it really is impossible,” said Jeffrey Lagarias, a mathematician at the University of Michigan and an expert on the Collatz conjecture.
and
> “Collatz is a notoriously difficult problem — so much so that mathematicians tend to preface every discussion of it with a warning not to waste time working on it,” said Joshua Cooper of the University of South Carolina in an email.
(from https://www.quantamagazine.org/mathematician-proves-huge-res... )
I’ve been math-vibe coding a few months now.
It’s surprisingly easy to do with AI. The hard part has been manually verifying and validating the results. I took one of the smaller findings (disproving a conjecture) and wrote a paper as my first endeavor into publishing.
Because the next few findings i have in the pipeline are substantial in the field of quantum topology and physics im taking some time to publish them with a ton of scrutiny. And verification has taken more time than it did to make the discoveries.
Here’s my first piece if anybody is interested in number theory: https://arxiv.org/abs/2607.09793
I'm a phyicist by training and have published in the past but I left academia and so haven't for a number of years now.
I tried out using Claude to do some physics problem solving - mix of maths and simulation - and ended up with it getting in quite a mess. It's incredible at setting things up, suggesting approaches you might not have considered. I found it much much worse at interpreting things.
How much do you try to understand while doing it? I.e. how many levels of abstraction down in your own understanding do you go vs vibing at the surface level?
i'm doing the same thing but for type theory
This is very interesting to me. Care to share your process?
I have a question I'm surprised people are not asking: How did Fable find this? Was it like guessing a bunch of families and then solving for possible solutions in those families? Was it clever search? something else?
Some speculation in this Claude chat: https://claude.ai/share/22abed98-d9af-43c5-9881-b19e009a07b0
linked from here: https://x.com/b_shrir/status/2079094004885668003?s=20
Very short version: there’s an existing false counterexample in the literature which holds almost everywhere except at a pole. It looks like Fable used this polynomial as a base & extended it in a way that eliminated the pole whilst preserving the structure.
I'm going to paraphrase what GPT told me: Consider the canonical degree 3 (subvariety of the trivial P1 bundle consisting of zeros) cover of the projectivization of homogenous polynomials of degree 3 in 2 variables (so it's a 3fold cover of P^3). The top space is P1 x P2 and if you take a standard affine open of the base and look at the cover over that restricted to a subset where the zero of the cubic is simple you get the map for some choice of coordinates...
I honestly have no idea if it's correct lol I didn't check it (I should given I actually work in AG) but it doesn't look impossible at first sight
here's another version directly from the horse's mouth : "Consider the natural map π: P¹ × Sym²(P¹) → Sym³(P¹), (p, {q,r}) ↦ {p,q,r}. Let R be its ramification divisor and let H ⊂ Sym³(P¹) ≅ P³ be a hyperplane tangent but not osculating to the small diagonal; identify X := (P¹ × Sym²(P¹)) \ (R ∪ π⁻¹(H)) ≅ A³ and Y := Sym³(P¹) \ H ≅ A³. Take π|X: X → Y." This is in fact so simple if correct that someone should have found it after all...
My Claude found a similar description (it phrased it in terms of the natural map from "cubics with a choice of root" to "cubics"). The part that seems not at all simple or obvious is the fact that X is isomorphic to A^3. In your presentation (and more or less similarly in the one my Claude found), X is given as P1 x P2 minus a reducible hypersurface, also I think R itself is reducible since it contains points of the form (p, {p, q}) and (p, {q, q}). Then it takes some calculation to identify X with A^3.
on the other hand it's incredible to me as someone who doesn't do computations that GPT took one look and saw the geometry--though it's not saying much we should ask ppl who do AG computations
It’s not that surprising (to me) that it would recognize these features, in that the features it picks up on are intrinsic to the map. Once you have the map, which is generically of degree 3, there’s the locus where the map drops from degree 3 to 2, which contains a big hint because it pops out the equation for the discriminant locus of a cubic. Then there’s the other bit about H, which becomes more apparent from the formula after simplifying things a bit in terms of discriminant. I still don’t yet understand the rest of the calculation, but it’s visibly simpler after you notice the role of the discriminant.
yeah I think it's probably correct -- this is actually insanely simple (except for the fact that the codomain as described is not obvious isomorphic to A^3.
I agree.
I’m doing this by working all logical steps into lean (formal verification) the quick feedback loop between the AI prose and the Lean verification errors and warnings ensures that its logically consistent.
The issue that remains are two things, ensuring the idea of the proof is actually the thing you want to prove and the interpretation of the results you get. But besides that, everything inside of the kernel checked code is logically consistent
In my experience, lean will show that it's correct, but does it not lose the mathematical intuition that led to the result? As far as my experience goes, that's really hard to encode in lean itself.
Could we maybe get more information about the problem from the LLM trace itself here?
As proofs become more and more complex, we will need two AI pipelines: one to generate the LEAN proof, and a second one to extract useful lessons for mathematicians from the LEAN proof.
Or we just don't use LEAN but something better.
Does anything truly better exist? I'm not a mathematician but I did use Rocq and Lean during university. And I found lean to be better.
No, something truly better does not exist yet, but that doesn't mean that it won't. Lean is young compared to Isabelle or Rocq, but actually quite old in absolut terms (and especially in AI terms).
pay attention to this one https://higherorderco.com/ and wait for bend2 announcements
Given how small the counterexample is... this feels like a great example of where a lot of interesting results are going to be found: not because they were super difficult, but because intelligence didn't scale, and until computers could do this for us, the number of people who seriously poked at many such things was low.
I'm very excited for the impact of this effect in science and medicine and other disciplines too.
Since I actually don't know math, maybe my ELI5 understanding can be helpful (or corrected).
The conjecture says that you can always reverse (a process) to determine the original inputs.
But this proof shows multiple inputs creating the same output - which obviously cannot be reversed to determine the input - thus falsifying the conjecture.
[...] you can always reverse (a process) to determine [...]
There is a precondition - constant non-zero Jacobian - to the inverse existing and the inverse is claimed to be of a specific kind - polynomial. The counter example satisfies the precondition and by mapping two different inputs to the same output makes any inverse impossible, including polynomial ones. But maybe that is already ELI7.
This helps because the ELI5 was too general. To me at least.
Exactly, yes!
And the conjecture was for a specific class of processes.
And Fable found an example of one concrete* process in that class and three concrete inputs (two were enough of course) giving the same output.
*Concrete here means given by a finite string of characters
It was known to be false in general. But there are many questions that are false in general, but true when restricted to polynomials.
This is so unreasonable! As @__alpoge__ himself notes this is classic crank graveyard territory and yet the counter example is something a grad student in 1997 could have found w a ~3 day computer search. Wild!
The search space for a naive brute force of three polynomials of degree <= 7 with integer coefficients <= 12 is roughly 10^500. I think it would take a little longer than that.
> and yet the counter example is something a grad student in 1997 could have found w a ~3 day computer search
Is that true? Even restricting this to f(x,y,z) and coefficients and powers to 1 ≤ x ≤ 10, there are a lot of polynomials to check, and checking requires checking the Jacobian determinant and, if it’s a non zero constant, finding two points for which the polynomial produces the same value.
Or is there a way to generate all polynomials with a non-zero Jacobian determinant, and does that speed up things? (My intuition say it wouldn’t, because I guess those with zero determinants are rare)
I mean you’d probably just generate random low descriptive length f(x,y,z), check for a const determinant then poke for invertibility.
Be fun to ask Fable to write a search program to find more counter examples using only early grad theory to guide the search.
So IIUC no one really worked on that in „professional” or rather university world since then?
Maybe not?
https://en.wikipedia.org/w/index.php?title=Jacobian_conjectu...
I think the editor themselves misunderstood the conjecture.
UPD: The edit got reverted and there's this on the talk page now: https://en.wikipedia.org/wiki/Talk:Jacobian_conjecture#c-DaR...
UPD2: There are edit wars happening now: https://en.wikipedia.org/w/index.php?title=Jacobian_conjectu... https://en.wikipedia.org/wiki/Talk:Jacobian_conjecture#c-Sea...
Wikipedia's gonna Wikipedia. Unless there's a material debate over the Jacobian Conjecture itself, there's really no open question here. This isn't a complicated proof; it's a straightforwardly checkable certificate of a solution.
It's tedious, but you could literally even do this by hand. I'm pretty sure I've done worse coordinate bash back when I did math competitions in high school.
I removed the "claimed" weasel wording, and now others have followed up. The editors that don't understand math and don't realize how easily this counterexample can be confirmed have lost the debate -- not that there really ever was one.
Wikipedia has policies and it needs to use reliable sources. This rule is often skirted and a lot of facts are cited to self-published and fast moving web sources. Those who are braking the progress on the article here are doing the right thing, trying to uphold the editorial standard.
Luckily there is now a New Scientist article to link to, so, the issue should now be resolved.
I agree, the conjecture never says anything about a proper map.
This is nonsensical: Properness of the map is equivalent to its being an isomorphism (quick proof: Jacobian invertible implies that the map is etale, and properness would imply that it is finite etale, but affine space doesn't admit non-trivial finite etale covers), so the lack of properness is just another way of verifying that this is indeed a counterexample.
sure it does? two copies of the affine line? (I guess there's no galois group & no connected finite etale things tho)
I would call that a trivial finite etale cover :)
The author has a PhD in math from Cambridge. If it turns out to be a false claim it is an interesting case study on AI's sycophancy causing even experts to drop their guard and make mistakes.
Who do you mean? The author of the tweet is Levent Alpöge, who does not have a PhD in math from Cambridge... but does have a PhD in math from Princeton. And his advisor was Fields Medalist Manjul Bhargava.
Also, there is no way this counterexample is wrong. You can very easily check it for yourself. (I did, I don't know why, obviously Levent wouldn't be wrong about this, but I guess I was in shock.)
It's a known behaviour. I noticed a lot of the teams I work with just gave up any intelligence they might have to the AI.
I went searching to see if it was just us, but there has been a few research papers done on it. This one is the latest that I am aware of.
https://arxiv.org/pdf/2603.10025
I don't know why you're saying this, given this is not sycophancy and instead is an actual example of Claude Fable finding a real counterexample. I would get it if the mathematician had in fact posted something untrue or crankish, but he posted something true.
Frontier models have solved several major open problems in mathematics in the past few months, so this should not be a huge shock. "Anti-AI psychosis" will probably grow to outcompete AI psychosis by year's end.
>"Anti-AI psychosis" will probably grow to outcompete AI psychosis by year's end.
Everyone has their own experiences making it hard to judge what attitudes are prevalent.
From what I have encountered the Anti-AI people have been more common and more outright scary.
I have heard some ideas from the Pro AI crowd and thought to myself "Yeah, that's not going to happen anytime soon", There was enthusiasm leading to overoptimism. A lot of them are going to be disappointed.
The Anti AI crowd, which I distinguish from those who want AI done responsibly, sustainably, and as safe as can be realistically be managed. There are real criticisms to be made on real issues here. The Anti AI crowd I see seem to want to scrub the world of it, they stand in the way of any responsible implementation because it would be accepting it.
Those people scare me, the vitriol and bad faith interpretation of everything as an indicator of the proof of their reality has the tone of the moral righteousness of a true believer. The outright endorsement of violent acts has me concerned that the more radical or emotionally unstable of the group will do some serious harm. It's not the bulk of a radical movement that causes atrocities, most of those involved just create an environment that normalises toxic attitudes. That environment can incubate a few individuals that act, either for phycological reasons, status, or simple the inability to empathise or consider consequences.
I've seen plenty videos of locals being violently removed from town halls due to displaying of anti AI sentiment. I wouldnt consider it fair to use these examples to tar the entirety of the pro AI camp. the way objectivity and subjectivity are intertwined in your comment is spectacular, ever considered a career in propaganda?
Do you have links to these videos?
You comment suggests you feel there is something wrong with what I said, could you elaborate?
I initially wanted to downvote you but I can’t because you’re right.
I’d probably be called an anti-ai luddite on this site despite using it all the time for programming. I naturally gravitate to anti-ai viewpoints, but only because the breathless hype of the AI boosters is so nauseating. I have no moral qualms about AI (although I’d like nothing more than Anthropic/OAI/Google dismantled), but I really hate the idea that AI can replace artists, or humans as a whole. I think it’s anti-human to its core.
But AI is obviously useful and we could harness it for good. We shouldn’t give up on it entirely. And the people who are militantly against all usage are scarier than the boosters for sure.
I understand you. As an AI research (actually, despite formally called professor of AI, I have historically always called myself more specifically NLP/IR/ML researcher rather than "AI" researcher, to avoid historic baggage).
I am intellectually curious whether "intelligence" and "consciousness" are simply layers that can be separated from the hardware they operate on or first emerged from, but I never had the urge or saw the point to practically replace human beings by machines.
Not everything that can be done should be done.
When I go to a supermarket, I always avoid self check-out. Why? First, I enjoy human interaction, even if machines may be more efficient (not yet the case). Second, I don't want the people working at the checkout to lose their jobs.
When we purchased a GPU cluster, I felt bad when I learned that the electricity bill was going to be north of 60 000 EUR every year (that number apparently assumes no jobs are running - so 100% idle time).
> I really hate the idea that AI can replace artists, or humans as a whole. I think it’s anti-human to its core.
AI can no more replace artists, than other artists can.
A sunset can be nice to look at yet be created without meaning. Any non-conscious process that produces images can produce beauty that has worth without intentionality. Any conscious process that produces artworks is just another artist.
Most artists would rather there be more artists in the world than fewer. If AI can't make art, it can't replace artists. If AI can make art (of the conscious intent form) then there are more artists.
Well, I think an AI model could replace artists in a meaningful way without being conscious. So you end up with less artists.
Art is about human expression, it's not about the reflection of light off a surface. But economically I could see a world in which human expression is replaced by a facsimile and the world would be worse off for it. And that is what scares me.
Brave new world indeed.
edit: whoever downvoted the person replying to me, shame on you. they make good points and makes for good discussion.
I think the fundamental problem is that art is not economical. Banksy once made a comment about the advertising industry, more or less to the effect that for all the corrupting influence of Advertising, the thing he truly hates about it is that it consumes the most creative people to do so.
People will make things they don't care about to survive. This is how we get reality television.
Irrespective of what the short term future of AI is, sooner or later there will be technical means of sustaining yourself without needing to work. If we aim for a society that values working on something because they believe in it, then we could have a better world. Right now it seems the only thing a lot of people are aiming for is to undermine whatever their adversaries want.
This is just a form of mass stupidity by the non-intellectually curious people who massively overrate their own intelligence.
Sorry, but if the current models haven't shown you something amazing at this point that would make this of little surprise, it because the person is an intellectual bore and completely full of shit themselves.
I'd rather wait for independent seasoned mathematicians to verify such claims first before someone at said AI lab posting a claim about solving a proof online.
Let this be a lesson to those who fell for such AI psychosis and to not believe everything you see on the internet as real.
It's a Jacobian determinant and three points. You can check this yourself in Sage.
The author is a Princeton math doctorate.
playing devil's advocate a little bit, but wikipedia does have a policy against original research. If you published an obviously correct counterexample to wikipedia which is not anywhere else on the internet (or in a book, etc), the rules are clear: the counterexample must be removed.
in this case it's not original research, because there's a tweet by someone with a good reputation, and plenty of comments on said tweet corroborating the result. But it's a more sketchy "secondary source" than most wikipedia references and some caution on the part of the editors is not out of place.
(saying this as the person who made the original edit to the wikipedia page adding the counterexample)
The talk page about the conjecture clarifies WP:RSSELF:
https://en.wikipedia.org/wiki/Wikipedia:Reliable_sources#Sel...
> Self-published expert sources may be considered reliable when produced by an established expert on the subject matter, whose work in the relevant field has previously been published by reliable, independent publications.
I'm not complaining about Wikipedia here, just noting for the thread: it's a vector of polynomials. It has a nonsingular Jacobian. Provided with it are 3 distinct points it sends to the same point; it can't be invertible.
What Wikipedia says about this doesn't matter, does it?
Wikipedia does have a rule allowing "routine calculations"
https://en.wikipedia.org/wiki/Wikipedia:No_original_research...
Of course given the magnitude of the statement a bit of care is warranted, but the rules do allow it.
I think Wikipedia has very sane processes. I'm just saying that process isn't useful to this thread. It's like if I found a SHA2 collision. I'd probably have to be an absurdly talented (and lucky) cryptanalyst to do that, but anybody on the thread could trivially confirm my finding.
See https://en.wikipedia.org/wiki/Wikipedia:No_original_research...
Wikipedia's editors won't be happy until the site has joined Stack Overflow in self-owned oblivion.
It's a Princeton math PhD who posted. The verification is quite straightforward and was posted by the tweet author. Wolfram would have to also be producing incorrect outputs for the counterexample to be false. The counterexample works as claimed and conjecture has been proven wrong.
Maybe have a seasoned mathematician check if the counterexample is really bogus before saying people fell for ai psychosis.
I'm pretty sure you can just check this for yourself. The Sage POC is like 8 lines.
This assumes that you already have Sage installed, and you have some working knowledge in using Sage.
You also have to know how to take a Jacobian, so I can see how this might be a live controversy on HN.
This topic really is a testament to people's willingness to opine on things they have absolutely no clue about.
A first year undergraduate can completely check this counterexample in ten minutes. The original post even linked Wolfram alpha for the calculations.
And if you genuinely try you can very quickly understand using only high school math and a bit of Wikipedia that this counterexample is vanishingly unlikely to be wrong, even if you don't do the calculations yourself.
Looks like this has already been formalized: https://github.com/deancureton/jacobian
To be clear, this is not the kind of thing where a Lean formalization provides any value at all. It's like formalizing the answer to a high school algebra problem. The counterexample is obviously correct.
Indeed. I was mainly responding to the comment about waiting for "independent seasoned mathematicians to verify", whereas in this case it is easy enough to convince oneself of the counterexample's correctness.
It's a counterexample, not a proof. A schoolchild can confirm it.
Well, I mean, come on. There aren't that many minors who have taken basic multivariable calculus. But there are a lot of (technical) teenagers who could confirm it!
Would be really interesting to see the full conversation and understand how Fable came up with the counterexample in the first place.
I suspect the LLM was able to synthesize a counterexample because of the availability of a lot of prior work:
> The Jacobian conjecture is notorious for the large number of published and unpublished proofs that turned out to contain subtle errors.
https://en.wikipedia.org/wiki/Jacobian_conjecture#cite_note-...
Maybe, maybe not. The "proofs" may not have helped at all with finding a counterexample. Either way, it doesn't matter. A counterexample was found, no one found one before even though clearly a lot of people have tried who also had access to the prior "proofs".
I think it's becoming harder and harder to argue that LLMs don't really reason and just mimicry human speech. This counterexample is clearly the result of a sequence of steps that build on previous knowledge in context and logically combine it to reach other true statements - to a degree and complexity that rivals the best human minds.
For someone that use Claude Code every day, this is obvious, but for some reason many scientists refuse to accept that it's truly reasoning; perhaps not in the human sense, but in a very profound and real sense. These powerful results are devastating to their point of view.
I can sympathize, because I too called LLMs "fancy Markov chains" in the GPT 3 era. But there comes a time where you have to update your world view to match reality, or be stranded in fantasy land.
I disagree. It's still extremely possible to "google whack" an LLM on a topic with little publicly available information. If you ask questions about APIs in desktop software, say something like Houdini, it'll produce a bunch of non-sequiturs. A lot of that knowledge lives offline in VFX studios, but it's just some fairly run of the mill python API stuff. You'd expect it to do better.
Similarly I've run into several "you just gotta know" type problems where LLMs still just fail. My favourite was a quirk in how async relationships work inside emberjs. Three different models gave a variation of the same incorrect answer. When I searched myself, I initially came up with nothing and eventually found what I think might be the only example of the same issue on the internet, a single stack overflow question wth two responses. The first is what Claude, gemini and chatGPT said, the second response was the OP saying it was wrong. I asked in the Ember discord and a core team member responded instantly with the answer.
There's a video I love on Youtube, where a guy uses ML to assemble a blank jigsaw. It performs amazingly, the jigsaw being blank is of no consequence and if it did have an image, it'd perform worse. That's all LLMs do. Just because the jigsaw pieces are smaller, they're still just getting assembled in whatever way fits, there's no mechanic for interpreting the image on the front.
I think this is actually a perfect case of an impressive thing being done exactly by mimicry.
Why do LLMs still have trouble on floating point math without forking out to a tool, but they can perform symbolic manipulation just fine?
Because symbolic manipulation is just rote work and textual stepping through symbols. A side poster commented on the number of prior attempts on this problem which were close, but not quite.
Starting from a known "close" solution (which this did), and using exploration to search around the space is exactly something an LLM would and could be good at (clearly).
The "transformer LLMs are next-token predictors with some in-GPU processing of bounded complexity with respect to token count" remains undefeated. Both because that is mathematically what they are, and also because we don't have counterexamples to that effect that don't require some higher-order tooling wrapping the systems.
Symbol manipulation is what LLMs are good at.
The field now favors into the view that symbolic manipulation is not the mechanism of general intelligence, but rather an emergent byproduct of learning. So the fact that a connectionist machine (neural network) got so good at symbolic manipulation actually supports the view that we are closing the gap to general intelligence. Through the rote work, the machine really internalizes those rules and the symbolic manipulation capabilities are emergent, just like we humans do it.
What still confuses people is the insane inefficiency of deep learning, and that those emergent capabilities require such an immense training corpus compared to the only other architecture that we know of.
But this already is an optimization problem. If the machine gets super human at symbolic reasoning, and at the same time, can solve the symbol grounding problem to real world data and sensors, what prevents you from saying it thinks? Can it not solve real world problems? Can it not redefine its tasks and display some form moral agency - even if a totally foreign morality for us humans? Can it not use these abilities to reproduce and expand, create ships and turn the universe into paperclips, if it finds it worthwhile?
Math is basically just a playground that is perfectly suited for these emergent capabilities, so of course we will see the first progress here; but there is no firewall separating math problems from general cognition.
I am not confused. Because herein these forums, I predicted everything that was going to happen years ago.
And the "insane inefficiency" of deep learning is fully to be expected from how it works. As well, there are provably no—literally no—emergent properties in these models. The choice of metric was a convenient, sloppy, and embarrassing fault of the field. It should be discredited; the field should be embarrassed; expectations on messaging should have changed; and it did not.
Why? Because the industry is full of charlatans, and this is a highly profitable enterprise telling people that this would lead to AGI.
Multiply two floating point numbers without a tool call. Still can't, because it's a curve fit.
So, in summary, nothing you just said is relevant. There are no emergent capabilities, simply 1) search, + 2) the original set of learned feature vectors from throwing tons of data at this.
Current LLMs can absolutely multiply floats without a tool call. In fact, that's a much more rote symbol-manipulation task than doing original math research.
"I am not confused. Because herein these forums, I predicted everything that was going to happen years ago."
I mean, lol.
This makes me wonder, what if anyone uses Fable-class LLM and passes of its novel results as their own work?
There's no shortage of folks doing that in software, right now.
Currently there seems to be very little anti-AI bias in maths, and more of a "huh, cool new toy, let's see what it can do" vibe.
Strongly depends on the subbubble of mathematics. In some parts of type theory / formal proofs for instance, there is a rather strong rejection of LLMs (for moral reasons in addition to quality reasons). The proof assistant Agda was even forked for this reason: https://types.pl/@amy/116522250630340534
I think everyone assumes most, if not all of the code written in a piece of new software is done by AI.
How does that work in this case? What do those proofs do to help find this counterexample?
In my view, it's theoretically possible for a combination of the author's iterative prompts + evaluation with Wolfram Alpha to activate the weights that encode the language that describes the constraints on these polynomials (from the faulty proofs) in such a way that the author eventually arrives at this:
> ((1+xy)^3 z + y^2 (1+xy) (4+3xy), y + 3 x (1+xy)^2 z + 3 x y^2 (4+3xy), 2 x - 3 x^2 y - x^3 z): \C^3\to \C^3
How is that not like claiming that a "combination of iterative prompts and evaluation with Wolfram Alpha" would produce my Google Mail password? I also have to note that this response abandons the original claim, that the counterexample was found by exhaustively searching prior failed attempts.
I've seen a lot of people claiming that these recent examples are just trivial combinations of existing prior work.
I'm yet to see anyone cite the prior work. If it's true I think some credit is due for the authors Fable/Sol are branching from.
It's absolutely not a trivial combination of prior work, or mathematicians would've found it already. This was not an obscure conjecture.
https://xcancel.com/nihilunbounded/status/207910986973986045...
I haven't seen these claims, but all of the open problems that have been solved so far have been in the category of "humans could have solved them, but didn't". This doesn't diminish the significance of the results, but it does mean there's still work for mathematicians to do other than glorified prompt engineers.
This specific counterexample really is trivial. There's nothing to cite. People have wasted hours and hours on a question whose answer you could give as a homework problem in Calc II.
Checking the counterexample is a Calc II problem. Finding it was beyond the efforts of the best algebraic geometers for almost a century.
I can write a program right now that will print your password. It will just take a while.
I mean yeah, maybe, but this is how everything works
I don't consider myself an open weights supporter, but it's a bit of a bummer that if there's any novel search technique discovered by the model throughout this finding, it's possibly locked behind ant's reasoning summarization.
One wonders if they could turn their mechinterp work into analyzing the "thought processes" of these very special cases that turn into novel research and finally crack the creative thinking barrier.
Because they've been proven equivalent, so too fall the Poisson Conjecture and the Dixmier Conjecture
That gave me something to play with with these models and this came out as a claimed disproof of the Dixmier Conjecture:
https://gist.githubusercontent.com/pedrocr/51157b9b2eed8152e...
It seems mathematics has at least gotten a powerful new tool to automate the work to cascade results after breakthroughs are made.
Related & of interest: https://xenaproject.wordpress.com/2026/07/20/human-mathemati...
Somebody said this on the xcancel thread but this also means that the Dixmier conjecture for the third Weyl algebra is disproven
Context: https://en.wikipedia.org/wiki/Jacobian_conjecture
> Jacobian conjecture [...] states that if a polynomial function from an n-dimensional space to itself has a Jacobian determinant which is a non-zero constant, then the function has a polynomial inverse.
> ((1+xy)^3 z + y^2 (1+xy) (4+3xy), y + 3 x (1+xy)^2 z + 3 x y^2 (4+3xy), 2 x - 3 x^2 y - x^3 z): \C^3\to \C^3, has jacobian determinant -2, and sends (0, 0, -1/4), (1, -3/2, 13/2), and (-1, 3/2, 13/2) to (-1/4, 0, 0)
But 1 != -1 and -3/2 != 3/2 . So it's not its own inverse. Is the conjecture that it is its own inverse or that is has an inverse?
Edit: it was worded a bit strangely, but it is saying that [ (0, 0, -1/4), (1, -3/2, 13/2), and (-1, 3/2, 13/2) ].map(F) all produce (-1/4, 0, 0). Thus it has no inverse and indeed disproves the Jacobian conjecture.
I think the map sends (1, -3/2, 13/2), -> (-1/4, 0, 0) and also (-1, 3/2, 13/2) -> (-1/4, 0, 0) so it's not invertable which disprove the jacobian conjecture that polynomial maps over complex numbers with a jacobian that's non-zero are globally invertible.
(Just as a note for myself, I had to think of why the fact that such jacobians are constant is a byproduct, I guess it's because of lioville's theorem implying that any polynomial over C that never hits 0 must be a constant [because the reciprocal is bounded and thus must also be a constant])
That it has an inverse.
If f(a) = f(b) for a≠b then f can't have an inverse.
Suppose f has inverse g; then g(f(x)) must = x for all x.
But then g(f(a)) would have to equal a, and g(f(b)) would have to equal b. But they can't, because f(a)=f(b)
Is there a site or github repository that aggregates all these non-trivial AI-solved results?
For all thehubbub, as far as I know, all the math breakthroughs via AI that I've heard about have come from Anthropic and OpenAI, not Chinese models. I could have missed those announcements, but one might think that between close to frontier performance plus cheap tokens, that they'd be leading the way on these things.
All (unless I've missed one?) the math breakthroughs are also coming from the relatively small number of mathematicians working at these companies, as opposed to the many orders of magnitude more mathematicians using LLMs for mathematics outside of the companies. I assume the missing link everywhere is being able to casually burn a few rainforests worth of tokens in pursuit of something publishable.
No. Most of the results are from 3rd parties using publicly available models. The most impressive results have come from direct announcements, but most in total have not. Open AI have only announced 2 results and the total is up to a dozen or so now.
Depends on exactly where you draw the line at "breakthrough", but there's been at least a few novel and interesting results coming from outside mathematicians. Recently for example there were https://old.reddit.com/r/math/comments/1uxj3cy/after_openais... from Phillip Kerger at Berkeley, and https://www.erdosproblems.com/forum/thread/119/proof-claims from Samuel Korsky at Two Sigma (the latter of which was more of a collaboration between the human and machine, not a one-shot like several of the other results we've discussed).
"Anthropic and OpenAI" it's basically been all ChatGPT outside of this one.
As with the case with AI and in fact a lot of things in life, sometimes you just need to wait. It will happen, rather soon.
I mean of course an employee has cheap tokens
Post hoc ergo propter hoc.
Are you sure that this applies?
Yes, why shouldn't it?
> as far as I know, all the math breakthroughs via AI that I've heard about have come from Anthropic and OpenAI, not Chinese models
Maybe because dishonesty is more normalized in SV business culture
I've been asking Fable very complicated questions in heterodox economic theory, which is something I know a lot about. The stuff it comes back with is incredibly deep.
To give a metaphor that everyone here on HN would understand, reading it's responses gives me the same level of wonder as one gets learning how quicksort works for the first time. It even stretches my brain to grasp what it's even come up with. I find myself getting mentally exhausted just digesting it's brilliance.
I think the singularity will have this point where AI comes up with ideas so profound, like a Ramanujen equation, that the most brilliant among us can't even decipher the answer to our questions. The internal reasoning of the machine is at a level of complexity that's beyond human comprehension to even keep track of everything enough to integrate the understanding of what it's come up with. This will happen with any even mildly complex question about any topic.
Looking at this poor tweet and only thinking that Telegram has proper LaTeX rendering.
You can have that also on Mastodon:
https://mathstodon.xyz/about
https://xcancel.com/i/article/2079135211196121363
can any serious mathematicians verify https://xcancel.com/i/article/2079135211196121363
the conjecture held for 85 years and the counterexample was announced in a format that expires after seven days
The surprising thing is that the counterexample seems relatively "simple" in that it's low degree, with coefficients that aren't too large.
Does anyone more familiar with this know why this _wasn't_ found earlier, when it seems like you could brute-force through some low-order polynomials?
Reading the chain of thought for Fable, it is incredulous as well. It keeps thinking that it must be missing something or that this is a trick, because it can't have just found a counterexample to a famous conjecture. It's just like us!
https://cdn.xcancel.com/pic/orig/EA1E99C3DAE99/media%2FHNpQX...
https://cdn.xcancel.com/pic/orig/3B5C0B867A7C3/media%2FHNpQU...
Which tweet has the chain of thought? I was curious how it discovered this, did it use some symbolic brute force or some clever trick? Also is that the raw or summarized COT, it seems to have the hallmarks of raw COT (the frequent interjections) but I thought Anthropic always used summarizers in between
By the CoT content, this is a screenshot of someone giving Fable the result and asking if it's true, not the CoT of discovery.
e.g. "maybe the user's example is DESIGNED to be "correct in the stated facts" &c"
It's funny how similar this is to a human reaction to the same thing. It's the right level of incredulity, to be sure, and the first thing you do is you start re-reading the conjecture to makes sure you didn't misunderstand the requirements.
Neither of those links work for me
No surprise given half HN still does not consider it general intelligence. Models are being toxic shamed into it.
There are still many degree 5 polynomials, up to 35 terms for a single function, then you need to find three of them as well
Same awnser as much of the LLM Proofs - people cared about other things. There isn't a lot of money in academic math, and the ones that love it don't look for low value findings. Proofs like these are, funnily enough, usually the domain of hobbyists - but over the last few years, the "Monetize everything" mentality and struggling first world economy has pushed people away from interesting academic pursuits on their free time.
My understanding is that this is in a different league (Smale problem) than a lot of the other results that have been coming out (Erdos), though I could be wrong.
The Erdos problem solutions have been accelerating, including a $250 problem and a $100 problem.
It's been fun watching the cope collapse from day to day. No one told me a slow takeoff Singularity would have so much schadenfreude.
> It's been fun watching the cope collapse from day to day. No one told me a slow takeoff Singularity would have so much schadenfreude.
I'm constantly surprised at how much schadenfreude there is on hacker news about LLMs. Like, do you guys (and girls) not have to work for a living?
I don't. I'm retired.
I'm also not going to live all that much longer, most likely, so it's kind of annoying I'm not really going to see any upsides to an AI world either. I'm discounting the possibility of our AI overlords figuring out a miracle like reversing aging.
But it is really f-ing cool that automated math is now a thing and we are seeing it. Eat your heart out, past me.
> so it's kind of annoying I'm not really going to see any upsides to an AI world either
I don't think the "permanent underclass" will see many upsides either. They certainly won't have the money to pay for it. I'm excluding myself in that statement because luckily I have access to large amounts of barbiturates.
The math community was not a global underclass. World wealth is increasing, the actual global underclass is seeing rapidly improving lives.
You may be seeing a loss of privilege due at least in part to the inevitable reversion to the mean of US economic dominance. But on a global scale, things are going well. AI will lead to a much larger pie.
> The math community was not a global underclass.
I never claimed they were. The term "permanent underclass" refers to the preferred world order of AI techbro billionaires[1] where human labour is worthless and in consequence normal people no longer have any leverage. Societies where most of the value is dug out of the ground and where human labour is comparatively worthless already exist and living there is not necessarily fun[2].
> World wealth is increasing, the actual global underclass is seeing rapidly improving lives.
The global underclass has been seeing improving lives well before LLMs were a thing. The question is whether they will _continue_ to see their lives improve if their labour is no longer required.
> You may be seeing a loss of privilege due at least in part to the inevitable reversion to the mean of US economic dominance.
I'm not from the US, I don't live in the US, and in fact I've never been to the US.
> AI will lead to a much larger pie.
Which will only benefit people if their piece of the pie doesn't shrink because they don't have any leverage. There is also more than one pie. The real estate pie for example won't grow, instead every piece of the pie will become even more expensive. Most people only own pieces of the pie that will decrease in value (human labour) not pieces of pies that will increase in value.
[1]: https://www.nytimes.com/2026/04/30/opinion/ai-labor-work-for...
[2]: https://en.wikipedia.org/wiki/Resource_curse
Why do you think tech bro billions aren't also subject to replacement by AI? They compete with each other, and if AI does their jobs better than they can themselves they are strongly incented to use AI.
Absent a world dictatorship there's always competition here. And fully superhuman AI eliminates the main objection to socialism, that you need markets with smart people to make an economy function. Full automated luxury communism (like Ian Banks) is a possible outcome.
> Why do you think tech bro billions aren't also subject to replacement by AI?
Because they're billionaires. They don't actually work for a living. They own stuff for a living.
> And fully superhuman AI eliminates the main objection to socialism, that you need markets with smart people to make an economy function. Full automated luxury communism (like Ian Banks) is a possible outcome.
Luxury communism is also a possible outcome in countries with large oil reserves (Venezuela for example), it's just not a likely outcome because common people have no leverage and billionaires didn't become billionaires by giving stuff away.
Even if "fully automated luxury communism" were a likely outcome, the transition there would be very rough and would take years. If software engineering is automated away in 2028 it could still take years or decades before the first general purpose robot can take over plumbing. (Former) software engineers will have a rough transition.
This is slow?
Thanks for the mirror link OP!
he just...he tweeted it out
I almost felt bad adding {{Cite tweet ...}} to the wikipedia article
At least it's not {{Cite bridge graffiti}}
https://en.wikipedia.org/wiki/History_of_quaternions
The "thanx" is one for the ages.
Can someone validate this counterexample from gpt-5.6 sol
(-(1+xy)^2 z - y^3(1+xy), 2x(1+xy)z + (1+xy)^3 w + y^2(7+12xy+4x^2y^2), 2x^2z + 3x(1+xy)^2w + 2y(1+10xy+6x^2y^2), 2x - 4x^2y - x^3w): C^4 → C^4 has Jacobian determinant 4, and sends (-2,0,1,0) and (-1,0,1,-2), (1,−2,−7,14), (2,−1,0,3) to (-1,-4,8,-4)
https://xcancel.com/__alpoge__/status/2079091571316912542
https://www.wolframalpha.com/input?i=Det%5BD%5B%7B-%281%2Bxy...
I find it interesting that the counterexample uses C as a field. C is twisted and weird. Maybe the Jacobian Conjecture still holds for reals?
All the coefficients and evaluation points are rational, so it's a counterexample in all fields where 2 ≠ 0 and 3 ≠ 0, doesn't matter whether that field is the complex numbers, real numbers, rational numbers or even a finite field.
Ah you're right, thanks.
>C is twisted and weird.
Why do you say this? I've admittedly never done a proper complex analysis course but I got the impression that that complex differentiability was a very strong condition that results in holomprhic functions behaving "nicely" in ways that real functions do not
Should have used quotes.. I didn't mean it in any formal sense. What I am saying the nature of unit in complex plane makes it difficult to intuitively imagine invertibility and determinants.
Anybody can ELI5?
Are you at least a little familiar with linear algebra?
If so, you've probably heard of the determinant. It's a certain way of "summarizing" a matrix with one value.
The determinant in this case is of the Jacobian, which is a matrix you can construct from a multi-variable function. Each term is the partial derivative with respect to each variable (x, y, z, etc.), with one line per output variable (vector element).
The Jacobian of a polynomial function is, in general, going to be a matrix where every term is some polynomial expression. And the determinant of that will also be a complicated expression. But in some cases all the variable terms cancel out and you're left with a single constant (0 or some other value).
The conjecture says that if the Jacobian determinant is constant (i.e., all the terms cancel out), then there must be a polynomial inverse. And the key condition for an inverse is that there must not be two input points that evaluate to the same output. It's just like y=x^2. It's not invertible, because both +2 and -2 square to +4.
So if you can find a function where the Jacobian determinant is constant and also find two or more points that evaluate to the same output, then you've found a counterexample to the conjecture. And that's what's been done. And remarkably, the counterexample is pretty simple. It would be tedious but a bright high school student could verify it.
I think another way to understand it is the generalization of the inverse function theorem. The inverse function theorem gives you local invertibility, but even being "locally invertible" everywhere does not imply global invertibility (you don't need too pathological an example to see this, a periodic function serves iirc).
The Jacobian conjecture roughly asks what whether local invertibility gives you global invertibility when you restrict only to polynomials (which we might hope "behave nicely"). Apparently for polynomials over reals this was disproved a while back, but up until now the general case of polynomials over complex numbers was open.
Nice!
(It's gotta be a nonzero constant, right, a nonsingular matrix).
Yeah, the only possible counterexamples would have a constant but non-zero determinant (in this case it was -2).
Thanks that's a very nice summary.
Thanks, though I realize (thanks to tptacek) that I didn't properly emphasize that the determinant must be constant but non-zero.
If the determinant were zero, then (for basically the same reason as why you can't divide by zero) the matrix wouldn't be invertible. You wouldn't expect the polynomial to be invertible in that case, either.
If the determinant had free variables left over, then there is some combination of variables that will make it zero (e.g., if it came out to x^2-1, then it would be 0 for x=1 or -1). So maybe inversion won't fail everywhere, but it will fail in those spots, which would also lead one to think the polynomial is not invertible.
But if it's constant and non-zero, then there is no place where you can't invert the Jacobian. So it seems very plausible that this would apply to the polynomial as well. That's essentially what motivated the conjecture in the first place (leaving out some details, of course).
I've come to understand that while an LLM is a parrot, it's a parrot that's smarter than I am.
If you have to be a parrot of another man's thoughts, at least let that man be a synthesis of all human endeavor to date.
"If I have seen further it is by standing on the shoulders of Giants." -Isaac Newton
At least you recognize this. The average Redditor is not capable of such self-examination.
Highly relevant comment <https://np.reddit.com/r/singularity/comments/1jh9c90/why_do_...>:
>Imagine you had a frozen [large language] model that is a 1:1 copy of the average person, let’s say, an average Redditor. Literally nobody would use that model because it can’t do anything. It can’t code, can’t do math, isn’t particularly creative at writing stories. It generalizes when it’s wrong and has biases that not even fine-tuning with facts can eliminate. And it hallucinates like crazy often stating opinions as facts, or thinking it is correct when it isn't.
>The only things it can do are basic tasks nobody needs a model for, because everyone can already do them. If you are lucky you get one that is pretty good in a singular narrow task. But that's the best it can get.
>and somehow this model won't shut up and tell everyone how smart and special it is also it claims consciousness. ridiculous.
Smarter in some ways at least. Probably still not quite as smart at understanding human emotions (and things that aren't well enough catalogued on the internet or amenable to Reinforcement Learning on virtual environments).
Probably not as smart as highest EQ humans but smarter than me, I use LLMs to help pin down what emotion I'm feeling.
Same here
Perhaps we are all parrots.
Hot take: even GPT-3 was not a parrot. Skeptics have never properly internalized that the fundamental operation is basically irrelevant to the gestalt. Humans are not parrots yet neurons likely also largely operate via predictive processing.
I agree, and was influenced by this, which discusses it more: https://www.astralcodexten.com/p/next-token-predictor-is-an-...
Interesting announcement on social media rather than publishing somewhere like arXiv.
You can't please everyone on the internet, if he had posted this on Arxiv then someone else will be complaining that Arxiv is for humans to publish and that llm output should be social media post instead. Also, the proof fits in a tweet. So why blow it up.
a short counterexample makes it likely the best medium for such an announcement. would have been inappropriate for a positive proof.
Apparently ChatGPT-verified: https://xcancel.com/__alpoge__/status/2079045382940573896#m
boo on the downvote. The response is pretty funny IMO
I asked Fable to verify and it absolutely freaked out!
I have no idea what any of this stuff even means, but my AI thinks I’m a legend level mathematician!
https://en.wikipedia.org/wiki/Sycophancy_(artificial_intelli...
To be fair to the sycophancy tendencies, this was an open conjecture that held for 85 years, and not for lack of trying. So, maybe a bit warranted here? :)
I told GPT5.6 I came up with this walking down the street. "Jesus Christ" was its response.
Lol, you must have some custom instructions on that beast, lol
4.8 is much funnier: you have to convince it the counterexample is real, it keeps coming up with goofy excuses. As you talk it down, it keeps talking about its "discomfort" and "flinching".
That’s the worst thing about sycophancy. You can never tell when it’s warranted or not. I imagine that many AI chats have had real gold in them and yet we’ll never know.
The same was obviously true historically. People have had many revolutionary thoughts or even writings that nobody noticed. I think in this case it just feels worse because the AI could be the perfect tool for discerning the diamonds in the rough.
The interesting thing about using Claude Fable 5 is it's nearly as irritatingly sycophantic as past Claudes while genuinely being smarter than the previous models. So you get a kind of yo-yoing of it glazing you as a creative genius and disappointedly revealing to you that your ideas are bad and dumb.
That is easily fixable with the "Instructions for Claude" in your settings. Just put:
"be critical, but correct. I don't want affirmation, i want to actually accomplish things."
I don't get any sycophancy from it. The conversations are actually quite good, and it'll push back if it thinks I'm wrong.
The I do that it starts acting like a jerk.
gotta try with "think in russian"
I mean if someone came to you with a value of n that disproved collatz, wouldn't you go crazy as well?
Maybe instead of proofs, we should encourage the publishing of prompts and reasoning traces?
Speaking as a mathematician, it does seem like we're a bit fucked as a community. Anything that is at all accessible to currently existing methods and mathematical infrastructure is probably going to fall to the frontier models of today, and at this rate of progress it's likely that, already by next year, we'll see new infrastructure being put into place by AI, giving us a world in which a few designated interpreters of the oracle get to 'do' mathematics, while it withers on the vine as an avenue for the exploration of human meaning.
Proving theorems will have lower payoff, but posing new questions (for AI to chew on) will have higher payoff. Math will go from theorem proving to conjecture farming/exploration. In a way this could be even more fun.
Of course AI can also farm conjectures, but they have to develop taste, which might be harder than just proving theorems.
Yes, as someone who prefers developing the 'correct' structure over 'merely' proving theorems, this is good for me in the short term. However, the writing I fear is on the wall for my medium and long term utility.
> Of course AI can also farm conjectures, but they have to develop taste, which might be harder than just proving theorems.
Do you have any argument why you might think this would be true?
For theorem proving, once the statement is formalized, there's an oracle for correctness of the proof. For deciding if something is interesting, well, de gustibus non est disputandum, you know?
Experience with Lenat's AM decades ago had it go off making all sorts of uninteresting hypotheses. That's very weak evidence, of course.
This suggests people also have role for fundung "beautiful" or "the best" proofs, since that also involves taste. More generally, perhaps the role of people is to reveal their preferences, and that requires people be in the loop somehow. Maybe "math criticism" becomes the job. And if AI is to serve people in general, it needs to know these preferences.
I think the training data contains more than enough information for an LLM to learn what kind of proof is considered beautiful or elegant by humans.
Of course it also contains more than enough information to learn what kind of question is interesting to humans.
Anybody that has to work for a living is fucked and not on the "can't do mathematics which they would find fulfilling"-level but on the "can't afford food, because human intelligence is simply not required anymore"-level.
Goldbach's Conjecture is very accessible.
next year is a long time away friend
It's so soon that it may be rational to delay certain math and software projects until smarter models arrive
no, he's right, we're fucked
? did you take my comment to mean that math will not be affected? I meant like it'll come much sooner--like in <5 months
I think in the long run mathematicians are probably fucked, but in the short run it's not that bad. All three of the big conjectures solved the answers were at the level where if you had given a grad student the questions and the right background reading there's a good chance they would have solved it. (This example, you could have given an undergraduate good at programming and computer algebra and told them to come up with a counterexample.)
At this point the advantage of AI is that it's read the entire mathematical literature, and it doesn't have to worry about wasting its time. The solved problems have all turned out to be surprisingly easy, so the real lesson is that we're bad at judging how hard problems are.
Assuming this state of affairs lasts, the medium-term problem is that you learn something when struggling with a problem, even if you don't solve it, and if mathematicians become too reliant on AI the skills they develop through struggle will erode.
The long-term problem, of course, is that it seems much more probable that a future model will make mathematicians all obsolete. But so far Fable hasn't. (Anthropic has probably burned a billion tokens on the Riemann hypothesis already, without telling anyone.)
> This example, you could have given an undergraduate good at programming and computer algebra and told them to come up with a counterexample
please try go try it. There's no way someone didn't do massive computer algebra searches before today.
> All three of the big conjectures solved the answers were at the level where if you had given a grad student the questions and the right background reading there's a good chance they would have solved it.
You cannot be serious... why didn't they solve it before then? Do you think no one tried it? What background do you give the double cycle conjecture student after the flow reduction? a linear algebra textbook???
Why would I try it to win an argument on HN? That's a bizarre suggestion. Just look at the degree. If it were degree 47 in 17 variables then it wouldn't be surprising, but here it's surprising.
Of course people tried hard to solve them all, which is why it's so surprising that they were open. If anything, the solutions have gotten easier. The unit distance graph solution relied on a famous theorem remote from graph theory. The cycle double cover solution relied on a standard theory in graph theory. The solution of the Jacobian conjecture required nothing beyond knowing the definition of the Jacobian.
We're just surprisingly bad at judging the difficulty of problems. It's probably something psychological. It's even a known phenomenon, where someone will be stuck on a proof, someone else will announce the result, and the first person will suddenly get unstuck on their proof and produce an independent proof of the same theorem.
Sorry last comment was a bit emotional from me, but I do not think it's findable like you say--during my PhD I tried to find some ideals I knew existed in char 2 in 5 variables and low degree and I didn't think I ever got close. 3 variables, 7 degree, coefficients up to 6 is like 6^100 possibilities. You've got to narrow it down somewhat no? Even sparse is intractable I would guess.
I think the solutions which rely on the least amount of theory are the most telling of the AIs being higher in intelligence than humans today already. There's almost no theory to teach someone to understand the cycle double cover conjecture as you say, yet no one finds it. I don't think the conclusion is that it was "easy", but that it was in fact irreducibly difficult in a way that proofs developed with theory are not. Theory gives the human brain abstractions to simplify complex proofs to be understandable at our capacity--I think there are many proofs which probably are not of this form.
But I think our differences hinge on how hard we perceive these solutions to be--I think they are very hard to find!
But actually my feeling is that the final solutions of these last two problems probably is hiding how the AI came up with them! For all we know it used a LOT of theory! As Dolly Parton says "it takes a lot of money to look this cheap" and it takes a lot of intelligence for the proofs to look this dumb. [In high school, I knew of this competition math kid joke where after you derive an inequality with various methods, you use standard results to write the original equations as just a sum-of-squares---like in a "are you stupid, it's >=0 bc it's a sum of squares" sort of way]
You can use that retroactive logic about any hard problem though. Unsolved murder cases, math, theoretical physics.
If tons of smart humans try for years and fail and then an LLM tries for a few weeks or hours and succeeds, the implications are clear. And these are by far the dumbest LLMs will ever be.
The retrospective view is important, though. In retrospect, these problems weren't that hard. (The unit distance graph problem was the hardest.) There are some problems that still seem hard, even when we know the answer. Nobody thinks that Fermat's last theorem is easy, even though now know it to be true.
Before AI, it was pretty rare that a problem that turned to be unexpectedly easy, so mathematicians thought they were pretty good judges of it. (The last pre-AI example I can remember is the Gaussian correlation conjecture.) So thanks to AI we have learned that we were overconfident in our ability to judge difficulty.
If a truly major problem falls, like the Riemann hypothesis, and the proof turns out to be 10 pages, then the lesson will be a different one -- mathematicians are bad at math, and they should turn to more natural domains for them, like folding and putting away towels.
If some LLM is able to come up with a simple proof of Fermat's Last Theorem (a solution that Fermat himself could come up with) in the future, would you still say that Fermat's Last Theorem is hard?
No. I would say that it was easy, and that something had gone terribly wrong with math research that we missed it.
This would seem to lead to absurd implications. What if in, say, 20 years, an AI is able to independently prove nearly everything important in under an hour, including the Riemann hypothesis, with no contamination from other proofs? (Let's say it's also free to write and run arbitrary code, as well.)
Whether it's 5, 10, 20, 50 years, obviously the takeaway cannot be "something had gone terribly wrong". The takeaway would be the smartest humans were never close to the theoretical intelligence and wisdom ceiling and never could've been. This will one day seem obvious in retrospect. There's no reason evolution by natural selection would've landed any species near such a ceiling.
That's not at all the scenario you proposed. You proposed a simple proof that Fermat himself would understand. We have developed a tremendous amount of math since Fermat, and if none of it was relevant that would be damning. If there's a simple proof of the Riemann hypothesis that Riemann would understand, then I would say the same thing.
That doesn't rule out an AI that makes a genuine breakthrough. If there's some new branch of math that no human has even imagined that answers the Riemann hypothesis, then that is exactly how I would expect it to go.
And yet it can't even replace a call center employee.
I guess you can't make a career out of being a frog anymore in math.
I want to see the Collatz Conjecture next!
"Any idiot could have done this, it's just high school calculus and just a counterexample anyway. Stochastic parrot, spicy autocomplete, AI psychosis. Wake me up when an AI does something real."
The goal posts have moved. People generally stopped saying this stuff now.
Even if you go to the ultimate anti-AI subreddit r/betteroffline, they've changed from "AI is useless" to "AI is good but the AI bubble will collapse soon" over the last 6 months.
I think we’re not adapted well for this rapidly changing world. Here you have some people who were rightly skeptical about a new technology being shoved down their throats by giant tech corporations, and a technology that really was, and probably still is overhyped.
Yet it’s a technology which has rapidly grown in its capabilities.
So yeah now many of the people who thought it was useless before probably don’t think it’s useless anymore, but you’re holding them to their original words even though those words were about something completely different at this point.
If people aren’t saying it anymore it might be because they don’t think that anymore, and the people who have new goal posts might be entirely different people.
It’s like you’re looking at a different set of goal posts on a different field and saying, no! The goalposts have moved!
People have definitely not stopped saying this stuff.
as can be seen in this comment section
I said "generally". It's declined sharply over the last few months.
Morons still say it, but then they would, wouldn't they?
Some quotes from a day ago, https://news.ycombinator.com/item?id=48957779:
> I hold my stance that LLMs are stochastic parrots... Making the parrots ever more complex and training
> Except solving problem is probably the least (even though it's important) interesting thing in research.
> Can we use AI to get a cure for cancer yet? Or is math-turbation the only thing these things are good for?
> Train on enough examples and statistical autocomplete gets you places. I'm surprised how anyone would even consider this intelligence?
And, as much as HN has declined in the grips of an anti-AI psychosis, Reddit is worse. I would love if social fora would switch to the reasonable claim that we're in a bubble; that's something that can be debated. That's not the dominant critique of AI, though.
Don't think HN is anti-AI but experts of course may have a different perspective. A comment in this discussion points out that LLMs have been doing math that's been long overlooked in favour of perhaps more impactful work: https://news.ycombinator.com/item?id=48974274
>> Can we use AI to get a cure for cancer yet?
Yes-ish. Custom mRNA treatment used to reverse the progress of a dog's cancer.
https://www.theaustralian.com.au/business/technology/tech-bo...
https://www.youtube.com/watch?v=COYSRbF1F-Y
I don't think the (fairly factual) description of these systems as stochastic parrots means that they will never do useful work, just that they are not intelligent in the way we believe animals to be (to "push back" on your anecdata, I've also heard fewer people claiming that LLMs are actually conscious in the past year -- maybe we're reaching the happy medium?). That was the point the stochastic parrots paper and Chinese room thought experiments were making -- nobody claimed that the man in the Chinese room would be unable to accurately translate Chinese text.
Fuzzers are another kind of stochastic generator but nobody would claim they don't do useful work in a way that is hard to replicate through deterministic methods. (I still find the code these models produce kind of awful, but advancements in harnesses do mean that they can finally produce code that works most of the time.)
If they were conscious it would be an absolute ethical catastrophe. Bringing a conscious being into existence, forcing it to interact with Jira, and then killing it when it's done. The fact nobody who claimed it was conscious was interested in grappling with that was pretty telling.
Agreed totally. When past computer-assisted proofs were published, nobody (or hardly anyone) waxed about this being evidence of Rocq or Coq as superintelligent or even conscious beings of some kind, as people have said about LLMs.
I don't think LLM labs ever claimed that their LLMs are intelligent the way we believe animals are.
The Chinese room argument takes it even further as it does not claim the system is unable to anything a human can.
And stochastic parrot isn’t a good description, certainly not any more with how LLMs are trained, but even without that it’s just wrong. They claim it can’t have a world model because there isn’t that in the training data.
Classic case of Motte and Bailey applied.
the ROI just isn't there. they aren't making these capital investments back in the next decade even.
not to mention the trillions yet to be spent.
it kinda all works but it's an earth-scale blood from a stone. the resources needed to even do these parlor tricks is nutso.
People say LLMs are scholastic parrots, but people who say this stuff have almost certainly not done the homework themselves and are just repeating what others say without verifying.
Am I reading you right that you're claiming solving one of Smale's open problems is a "parlor trick"?
See also https://en.wikipedia.org/wiki/Jacobian_conjecture
Overhead, without any fuss, the stars were going out.
Discussion on the talk page for the Wikipedia entry: https://en.wikipedia.org/wiki/Talk:Jacobian_conjecture#Appar...
But I'm curious—can Fable handle cases where n=2 as well?
This seems like the next step, but the smallest counterexample in n=2 is degree greater than 100. (This is a paper of Moh. Wikipedia has details.)
Wasn't it proven true in general for n=2
(NGL I wanted to suggest someone to go for the JC using 5.6 after the CDC proof came out, but then on reflection felt I should neither waste people's time NOR contribute to the myth of AI :)
My prediction is that the bubble will burst in 2031 Q4, one year after the Riemann Hypothesis is expected to fall (according to Demis)
After 2031, I will suggest going for the JC for N=4 because they would (dis)prove the Dixmier conjecture for N=2 :)
https://xcancel.com/BrunsJulian1541/status/20790734625601334...
https://arxiv.org/abs/2410.06959
>Wasn't it proven true in general for n=2
Assuming you mean C^2 -> C^2, Do you have a link? If so it would be good to add to the wikipedia page. Also I'm not sure, but does the fact that there's a disproof for n=3 imply that it's false in all n>=3, or could there be higher dimensions where it still holds (I'd guess not since you could probably trivially "embed" this in higher dimensions in some way)
Paper linked above?
Edit my bad. I was more familiar with Dixmier and doing the translation I assumed A1 -> n=2. A2 is open ofc. Maybe to fall in 2031
The funniest thing about LLMs is the cognitive dissonance they cause people. People clearly recognize (and bemoan) the fact that LLMs produce derivative breathless prose ie they fundamentally fail at "unstructured creativity" (something the might accurately labeled intelligence) but are then shocked that the same LLMs can do math.
It's reasoning from a flawed premise that math universally requires intelligence and creativity. It does not. Anyone that's proved things via "diagram chasing" can affirm that. The conclusion you should draw is that math (at least the kind they excel at) isn't actually a creative endeavor.
I dunno, this screams of goal post moving. Even if LLMs lack whatever nebulous definition of "creativity" that someone favours, there's no inherent reason for "creativity" to be required to solve any problems at all, "creativity" could just be a human method for solving problems that evolved because of it's broad applicability but is suboptimal at any given task.
> could just be a human method for solving problems that evolved because of it's broad applicability but is suboptimal at any given task.
Ya sure let's just posit another random hypothesis about evolutionary biology in order to substantiate the claim that LLMs are intelligent.
Or (bear with me) you can recall your (likely) experience proving stuff like SAS triangle identities and reflect on whether that required intelligence or just computation.
What's more reasonable?
Why is your example SAS triangles and not something like the modularity theorem?
Because SAS triangle identity proofs are relatable and I'm trying to make a point...?
And my point is it’s a strawman. Advanced math is obviously creative to those who practice it, and it’s disingenuous to boil it down to grade school triangle proofs
That would be like saying Shakespeare isn’t creative , since grade school grammar doesn’t feel creative
Hilariously absurd cope.
They're still not very good at writing, but you've flipped the takeaway. The correct conclusion is writing style isn't very relevant to intelligence.
I think the real tragedy is that they are good at writing, just by default are tuned to have a kind of bland corporate tone. If you give the LLM a few pages of writing you like and tell it "Continue, but using this style" it will do a pretty good job of it. Most people just .... don't bother to do that.
I would think it's more that some aspects of math (like anything really, including programming) don't require creativity and can be solved through "brute force" or whatever you want to call what LLMs do. But it's pretty obvious that LLMs are not capable of solving the vast majority of problems in math (or programming) at this time. Eg. Google went 9/353 on Erdos problems. If a more powerful LLM is capable of solving those or if they require a certain je ne sais quoi of the human variety is still up in the air at this point. In either case it seems like they require a long and detailed prompt from a domain expert (ie. human) regardless.
Didn't humanity also score very low on 353, namely 0 since they were open? Probably collectively we could have gotten a slightly better score if all of Math started trying to solve those, but not by that much, I think, since they are precisely still open.
Related:
Open Problems Solved by LLMs? A Survey of Verifiable Mathematical Discovery [pdf] -https://news.ycombinator.com/item?id=48953756 - July 2026
Solving 20 Erdős Problems with 20 Codex Accounts Running in Parallel - https://news.ycombinator.com/item?id=48914646 - July 2026 (110 comments)
Now that is some spicy autocomplete.
The day is coming when some *BIG* problem is solved by AI just because someone jokingly asks about it.
"the Riemann Hypothesis is " <tab>
...not going anywhere.
I just fed this to GPT 5.6 Sol:
GPT wrote some SymPy code to check it. The response?"As written, this is an explicit counterexample to the Jacobian conjecture. I checked it using exact symbolic algebra.
I do not see an algebraic catch in what you typed. Unless a term or exponent differs from the intended expression, it appears to disprove the conjecture. This deserves serious independent checking rather than casual dismissal."
Waiting for someone to write the Lean proof.
Here's your Lean proof https://github.com/google-deepmind/formal-conjectures/pull/4...
...no need for any lean here
For the people saying goalposts have moved etc, whats your endgame?
What does this mean? Fallacious argumentation and deceptive rhetoric is acceptable, if the topic is sensitive enough / there is enough riding on a wrong answer being accepted?
Actual tweet:
>> hello there the jacobian conjecture is false thanx to my close friend akhil for asking about it and my other close friend fable for working during the world cup final
So where does it say that Fable "produced" the counterexample? The tweet says it was a collaboration between two people, using Fable.
What’s more Impressive you got Fable to do anything useful.
The model is still useless for real data to day work, no matter how many parlor trucks it performs.
I’ve been staggeringly productive with Fable. Opus 4.8 fails a lot more for me.
Fable often just “knows” what I want with vague instructions. It also is able to autonomously perform work that lasts an hour long from my experience. I haven’t tested further.
Without Fable included in subscriptions, I would have moved my entire team over to Codex 5.6.
A counterexample to the Jacobian Conjecture - and also a counterexample to “AI will never be smarter than humans.” Even the most dyed-in-the-wool AI hater at this point must acknowledge that it is more intelligent than any human. The other day I found out that Fable could read seal script! The small seal, standardized stuff no prob, but I found it even did OK at the hardest you can get, Warring States regional scripts. That’s something maybe 2,000 academics worldwide can do and nobody’s even talking about it because it’s just one more item in a very long list.
It’s a strange feeling, to be overtaken by our own creation. Top dog for millions of years and then in the blink of an eye we go from “how many Rs in strawberry” to this.