Levent Alpöge, a mathematician who works at Anthropic, says he was able to disprove the 87-year-old Jacobian conjecture with the help of Fable 5

Levent Alpöge announced the solution on X A mathematician has cracked an 87-year-old conundrum with the help of AI and announced the solution unceremoniously in a tweet.

New Scientist 2026-07-21 Matthew Sparkes

Context & Ripple Effects

The claim arrives amid a run of AI-assisted mathematics reports, from a single-prompt Erdős result to systems aimed specifically at proof verification. It extends the narrative from models proposing mathematical leads to researchers using them on longstanding open questions.

The key distinction is validation: an announcement on X can surface a proposed result quickly, but the mathematical significance depends on whether the argument withstands expert review. Earlier work such as the automated-conjecturing system shows that generating promising ideas and establishing proofs are separate capabilities.

First-order effects

Alpöge’s claimed result puts Fable 5 and Anthropic in direct association with a high-profile mathematical research use case, while placing the proposed proof under immediate scrutiny from mathematicians.
The announcement gives other researchers a concrete claimed example of an AI-assisted workflow, but not yet a validated benchmark for the model.

Second-order effects

AI labs and proof-tool builders face greater pressure to show reproducible reasoning traces and independent checking, rather than relying on problem-solution claims alone; AxiomProver’s verification-oriented approach is especially relevant.
If the proof is substantiated, researchers may devote more effort to using general-purpose models for conjecture exploration and proof development; if it fails review, the episode will reinforce demand for formal verification.

Third-order effects

The durable shift is likely to be toward paired systems: models that generate candidate arguments alongside tools and human processes that verify them. Whether this becomes a reliable research workflow depends on independent validation of headline-making results.
As AI-assisted claims reach harder problems, credit, provenance, and reproducibility standards may become more important to how mathematical institutions assess both the result and the model’s contribution.

The trend: AI mathematics is moving from automated conjecture generation toward claimed participation in frontier proof work, with verification becoming the decisive bottleneck.

Discussion

@__alpoge__ Levent on x
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 ((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,
@aaron_lou Aaron Lou on x
Congrats to Levent for discovering this! OOC I had an internal version of Codex attempt a proof too (without web search), and it discovered (essentially) the same counterexample! It wrote up a nice summary of the strategy here: https://aaronlou.com/...
@creatine_cycle Atlas on x
hey fable nice job on the jacobian conjecture now could you please use the doordash cli to order me popeyes 5-piece chicken tenders
@pdrmnvd Pedram.Md on x
((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) isn't even that impressive of a polynomial all of those numbers and letters are in the training data
@kevinroose Kevin Roose on x
No idea what any of this means but every math-adjacent person I follow is freaking out about it. AI casually disproving the Jacobian conjecture would be, what, a top 10 hardest open math problem?
@jtlonsdale Joe Lonsdale on x
First instinct of the LLM is to lecture you skeptically just like an annoyed math professor would, when asked about the new example that the Jacobian conjecture is false. Then - “this is more interesting than I initially credited...” 🔥🤣
@kevinroose Kevin Roose on x
Very fun to mess with AI models by telling them you disproved the Jacobian Conjecture in your sleep. [image]
@chaitjo Chaitanya K. Joshi on x
On the jacobian conjecture thing - certainly not a mathematician, but I feel the same reaction for programming. Something you valued as a skill, something deeply technical, somewhat finicky, something you got lost in for hours and enjoyed the craft of - it can now be done better
@nasqret Bartosz Naskręcki on x
I'm waiting for the full report from @__alpoge__ . As easy as it looks on X, this wasn't a simple prompt for Fable. Finding counterexamples like this to the Jacobian Conjecture is like searching for a needle in a haystack, it requires real insight. I'm really interested to see
@victortaelin @victortaelin on x
“hello there the riemann hypothesis is false” “wow!! what led you to look in that particular region of the critical strip??” “what is a critical strip?” - a conversation from 2028
@wtgowers @wtgowers on x
Assuming this is correct, it is for me the first example of an LLM solving a problem not in my area that was nevertheless big enough that I had very definitely heard of it. Again it's a counterexample, so not in “end of mathematics” territory, but still pretty amazing.
@aminkarbasi Amin Karbasi on x
This is crazy. The incredible Yitan Zhang ( https://www.newyorker.com/...) 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 Purd…
@vikhyatk Vik on x
there are decades where nothing happens then there are tweets where decades happen
@justindross JD Ross on x
This tweet length proof would have easily earned you an instant PhD and many accolades in math until extremely recently. Famously difficult problem with many previous subtle failures.
@qiaochuyuan @qiaochuyuan on x
for some background, 1) the jacobian conjecture is by far the most famous open problem resolved by an LLM so far 2) it's also infamous for attracting wrong proofs so this is quite funny in a specific way. a comment from a 2008 paper by t.t. moh debunking such a proof: > The [imag…
@jdlichtman Jared Duker Lichtman on x
This is quite a remarkable result: Posed in 1939, the Jacobian conjecture is one of the central open problems in algebraic geometry, but was just disproved by Alpoge, Matthew, and Claude Fable 5. The Jacobian conjecture roughly says that a multivariable polynomial F has an
@teortaxestex @teortaxestex on x
holy shit [image]
@catboosted @catboosted on x
Normie FAANG engineer: “I should definitely brush up on leetcode and get a job at Anthropic” The people at Anthropic:
@davidmanheim David Manheim on x
Gotta love the LLMs' standard response - because it's exactly what any mathematician would say if you showed up with this out of the blue. [image]
@kimmonismus @kimmonismus on x
Claude Fable 5 has produced a hand-checkable counterexample to the Jacobian conjecture, an open problem dating to 1939. The conjecture says that a polynomial map with a constant non-zero Jacobian determinant must have a polynomial inverse. An 87-year-old problem was casually [ima…
@shashj Shashank Joshi on bluesky
For the ‘just a stochastic parrot’ brigade. — ‘A mathematician has cracked an 87-year-old conundrum with the help of AI and announced the solution unceremoniously in a tweet. The finding is the most difficult mathematical problem yet solved by AI, say experts.’ www.newscientis…
r/worldnews r on reddit
A mathematician has cracked an 87-year-old conundrum with the help of AI and announced the solution unceremoniously in a tweet. …
r/BetterOffline r on reddit
Claude Fable Jacobian Conjecture
r/slatestarcodex r on reddit
Jacobian conjecture proven false by Fable
r/singularity r on reddit
Apparently the Jacobian conjecture was just proven false by Fable
r/mathematics r on reddit
Jacobian conjecture down with a hand checkable counterexample, has been open since 1939
r/technology r on reddit
AI's solution to 87-year-old riddle takes mathematicians by surprise
r/math r on reddit
The Jacobian Conjecture is False Per Anthropic (Link in Description)
@buildingcoolshi @buildingcoolshi on x
lmao so much cope [image]
@nexuist Andi on x
Never give up. Never ever give up [image]
@radishharmers Sridhar Ramesh on x
When open problems get LLM-solved, it seems often a mathematician working for an LLM company who gets the answer, instead of some random mathematician asking the LLM. Is that because they have more token spend, secret models, incentive, prompting skill (is that real?), or what?
@tomodovodoo Tomo on x
how to disproof the Jacobian conjecture: 1. give a random counterexample on twitter. 2. done. Well, congrats I guess. As far as I could check it seems to... Just be a correct counterexample.
@jiratickets @jiratickets on x
“claude fable just proved the jacobian conjecture is false” “incredible this changes everything” “do you know what the jacobian conjecture is?” “no, do you?” “no” [image]
@8teapi Prakash on x
Just to take a step back here, Levent had the highest GPA at Harvard, won the top undergrad math research prize and most frequently collaborates with a Fields Medalist. We don't have enough human mathematicians to ask questions to AI and spend time understanding the results
@beingjong Jonathan Gallagher on x
Most LLMs are (obviously) not aware that there exists a counter-example to the jacobian conjecture, and because it is so easily verifiable, you can just send this to them and say “it came to me in a dream” and watch them freak out
@tomieinlove Tomie on x
Shoutout to mathematical counterexamples with zero elegance. Gotta be one of my favorite genders [image]
@wirmgurl @wirmgurl on x
Seeing so many responses made me think Twitter is so smart. I have a maths degree and I'd never even heard of the Jacobian Conjecture. Then I see that a bunch of people are saying “a counterexample isn't proof” and I actually think Twitter is so dumb.
@chadwahl Chad Wahlquist on x
He doesn't miss: “My intuition would be it's gonna be quite the opposite, where it seems much worse for the math people than the word people.” [video]
@ejenk Evan Jenkins on x
AI skeptics desperately sifting through the training data to find where Fable plagiarized this counterexample from
@kukreja_abhinav Abhinav Kukreja on x
It was revealed to me via the claude API [image]
@madeleineebaker Madeleine Baker on x
This has what it has felt like in the decade or so since I decided to get a PhD in history instead of math [image]
@__nmca__ Nat McAleese on x
he just... he just... tweeted it out.
@calaldred Cal Aldred on x
Came up with a way to generate an infinite family of counterexamples, via Fable + GPT 5.6 Sol [image]
r/aiwars r on reddit
A fairly large open mathematical problem has fallen to AI (again). First screenshot - source. Second - opinion by math professor

Chronicles