A mathematician at Anthropic says he and Claude Fable 5 have taken down one of the oldest standing problems in algebraic geometry, and he announced it on X in the most understated way possible.
Levent Alpoge, a number theorist who works at Anthropic and previously held a Junior Fellowship at Harvard’s Society of Fellows, posted a single explicit polynomial map on Sunday night, with a note thanking “my close friend akhil for asking about it” and “my other close friend fable for working during the world cup final.” Buried in the tweet, almost as an afterthought, was a counterexample to the Jacobian conjecture, a question about polynomial equations that has been open since 1939.
The reaction from mathematicians online was immediate. A Stanford professor summed up the significance in post, walking through exactly why the example works and noting a further, almost cinematic coincidence buried in the problem’s history.
What the Jacobian conjecture actually says
Take a function that turns a set of numbers into another set of numbers using nothing but addition, multiplication, and whole-number exponents, a polynomial, in other words. Now imagine that function working across several variables at once, mapping points in three-dimensional space to other points in three-dimensional space, the way the Alpoge example does with inputs (x, y, z).
For any such function, you can build a matrix called the Jacobian, filled with the function’s partial derivatives, essentially a table describing how sensitive each output is to a small nudge in each input. The determinant of that matrix, a single number, tells you whether the function is locally reversible at a given point: whether a tiny change in output corresponds to exactly one tiny change in input, rather than several different inputs all producing nearly the same output.
The Jacobian conjecture asked a very specific version of this question. If that determinant is the same nonzero constant everywhere, does that guarantee the whole function is invertible, meaning you can always work backward from an output to find the unique input that produced it, using another polynomial? Calculus already tells us a nonzero Jacobian is necessary for a function to have a smooth inverse near any given point. Ott-Heinrich Keller, the German mathematician who first posed the question in 1939, wanted to know if that local condition was strong enough to guarantee something global: full invertibility across the entire space, not just in some small neighborhood.
It’s the kind of question that sounds like it should have been settled by 1950. Instead it turned into one of the most notorious traps in modern mathematics.
Why it took 85 years
The Jacobian conjecture has a reputation among mathematicians for looking deceptively approachable and then eating careers. It made it onto Steve Smale’s famous list of the most important math problems for the 21st century, sitting alongside things like the Riemann Hypothesis and Navier-Stokes existence and smoothness. Even the simplest possible case, functions of just two variables, has never been resolved. Over the decades the problem attracted at least five published proofs that were later found to contain errors, along with a long tail of unpublished ones that quietly fell apart under scrutiny. Mathematicians studying it developed something close to institutional caution: a new claimed proof of the Jacobian conjecture is treated, almost by reflex, as guilty until proven innocent.
That reputation is what makes Sunday’s announcement land differently than a typical research update. This wasn’t a proof of the conjecture. It was the opposite: a single explicit function, written out in full, that satisfies every condition the conjecture demands and still fails to be invertible.
The counterexample, in plain terms
The function Alpoge posted takes three numbers (x, y, z) and outputs three new numbers, built entirely from sums and products of the inputs. Its Jacobian determinant works out to a constant, -2, nonzero everywhere, exactly the condition the conjecture cares about.
But the function sends three genuinely different starting points, (0, 0, -1/4), (1, -3/2, 13/2), and (-1, 3/2, 13/2), to the exact same output, (-1/4, 0, 0). A function that isn’t invertible has to do exactly this: collapse at least two distinct inputs onto one output. Here it collapses three. That single collision is enough to sink the conjecture. The rest is arithmetic anyone can verify by plugging the numbers in by hand, which is part of why the mathematics community moved so fast to take it seriously.
The Stanford professor who amplified the result online pointed out that the interesting open question now isn’t whether the conjecture is dead, it clearly is, but whether some weaker, “repaired” version of it survives. One proposal floating around, attributed to OpenAI’s GPT-5.6, suggests the conjecture might hold if you add a condition ruling out this kind of degeneracy at infinity. Whether that patched version is provable, or just another dead end waiting to happen, is now an open question in its own right.
A strange echo of Yitang Zhang
The most striking part of the story has nothing to do with Claude. The Jacobian conjecture, or a special case of it, was reportedly the subject of Yitang Zhang’s PhD thesis in the late 1980s. His advisor had him build the work on top of a lemma the advisor had proven himself. That lemma turned out to be false, and Zhang’s entire thesis fell apart along with it.
The fallout was brutal by any standard of academic life. Zhang couldn’t secure the recommendation letters he needed, spent years drifting between jobs including a stint working at a Subway sandwich shop, and didn’t land a stable academic position until well into his forties. Then, in 2013, he stunned the mathematics world by proving a landmark result on bounded gaps between prime numbers, a problem that had resisted serious progress for decades. It remains one of the great comeback stories in modern mathematics, and it is more than a little strange that the very conjecture at the root of Zhang’s early misfortune has now been resolved, four decades later, by someone else entirely.
AI is quietly becoming a fixture in serious math research
The Fable disproof is the latest entry in a run of results that suggest frontier AI models have moved past competition math and puzzle-solving into work that touches actual open problems. Google DeepMind’s AlphaProof Nexus recently solved nine open Erdős problems, including two that had stood for over 50 years, generating fully machine-verified Lean proofs for a few hundred dollars apiece. A year earlier, OpenAI and Google DeepMind both hit gold-medal scores at the International Mathematical Olympiad, working entirely in natural language rather than relying on specialized proof-formatting pipelines.
What sets the Jacobian result apart is that it isn’t a formal proof search racking up points on a known problem set. It’s a working mathematician using a model as a genuine research collaborator to hunt down a specific counterexample to a named, decades-old conjecture, then publishing it in a single tweet with no fanfare. Fable itself has had a chaotic few months since its June debut, including a brief, government-mandated suspension tied to export controls before access was restored. A result like this one gives Anthropic a considerably better headline to point to than the regulatory back-and-forth that has dominated the model’s news cycle so far.
Whether a patched version of the Jacobian conjecture survives the next few months of scrutiny is anyone’s guess. What’s harder to argue with is that a problem sitting on Smale’s list of the hardest open questions in mathematics, one that broke a future Fields-adjacent mathematician’s PhD and quietly humbled the field for 85 years, just fell to a tweet.