The Jacobian Conjecture Stood for 87 Years. Claude Fable Killed It With a Counterexample That Fits in a Tweet.
TL;DR
The Jacobian Conjecture, one of the most famous open problems in algebraic geometry and number 16 on Smale's list of problems for the 21st century, is false. Late Sunday, Levent Alpöge, a number theorist who now works at Anthropic, posted an explicit polynomial map from C3 to C3 with constant Jacobian determinant -2 that sends three different points to the same output, and credited Claude Fable with the find. Unlike every previous "AI solved a famous problem" claim, this one requires no trust: the check is freshman calculus, it takes seconds in SymPy, and a Lean 4 formalization appeared within hours. AI Bacon ran the verification independently before writing this sentence. It holds.
What Actually Happened
The announcement was not a paper, a press release, or a livestream. It was this, posted to X while most of the planet was watching the World Cup final:
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
The model, apparently, does not watch football. Attached was the counterexample itself, a polynomial self-map of C3:
F(x, y, z) = ( (1+xy)^3 z + y^2 (1+xy)(4+3xy),
y + 3x(1+xy)^2 z + 3xy^2 (4+3xy),
2x - 3x^2 y - x^3 z )Alpöge is not a crank account. He won the 2015 Morgan Prize for the best undergraduate mathematics research in the US, took a Princeton PhD under Fields medalist Manjul Bhargava, spent 2021 to 2025 as a junior fellow in the Harvard Society of Fellows, and then joined Anthropic. The thread hit the top of Hacker News within hours, and Wikipedia's Jacobian conjecture article had been updated to "recently disproven" before sunrise UTC.
The Conjecture, and What Killed It
Posed in two variables by Ludwig Kraus in 1884 and in general by Ott-Heinrich Keller in 1939, the conjecture says: if a polynomial map from Cn to Cn has a Jacobian determinant that is a nonzero constant, the map must have a polynomial inverse. In particular it can never send two different inputs to the same output.
A constant Jacobian determinant means the map rescales volume by the same fixed factor at every single point of space. Picture a dough sheeter that compresses every crumb of dough by exactly the same amount no matter where the crumb sits: the conjecture said a machine like that can never fold the sheet back onto itself. Fable's map is a dough sheeter that lands three raisins on the same spot anyway.
Concretely: the map above has Jacobian determinant exactly -2 everywhere, its components have total degree at most 7, and it sends the three distinct points (0, 0, -1/4), (1, -3/2, 13/2), and (-1, 3/2, 13/2) to the same image, (-1/4, 0, 0). That is the whole refutation. One map, one constant, three colliding points.
Check It Yourself
This is the part that separates the story from every prior AI-mathematics headline: you do not have to believe anyone. Twenty lines of SymPy settle it.
import sympy as sp
x, y, z = sp.symbols("x y z")
F = [(1 + x*y)**3 * z + y**2 * (1 + x*y) * (4 + 3*x*y),
y + 3*x*(1 + x*y)**2 * z + 3*x*y**2 * (4 + 3*x*y),
2*x - 3*x**2*y - x**3*z]
J = sp.Matrix(F).jacobian([x, y, z])
print(sp.simplify(J.det())) # -2, identically
pts = [(0, 0, sp.Rational(-1, 4)),
(1, sp.Rational(-3, 2), sp.Rational(13, 2)),
(-1, sp.Rational(3, 2), sp.Rational(13, 2))]
for p in pts:
print([f.subs(dict(zip((x, y, z), p))) for f in F])
# each prints [-1/4, 0, 0]We ran exactly this before publishing. The determinant simplifies to the constant -2 and all three points evaluate to (-1/4, 0, 0). Since the three collision points are real, the map also fails injectivity over R, and a fortiori over C. There is no subtle analysis to trust, no referee to wait for. It is arithmetic.
The Verification Pile-On
The crowd response was immediate and unusually rigorous. Commenters on the Hacker News thread reproduced the check in SageMath and Wolfram|Alpha, several had rival models including OpenAI's GPT-5.6 Sol write and run their own verification code, and by early morning a repository, deancureton/jacobian, had formalized the result in Lean 4 on top of mathlib. The Lean development states something even stronger than the tweet: a variant of the map with Jacobian determinant exactly 1 that fails injectivity over every field, with the original det -2 complex version included as a specialization.
Wikipedia, meanwhile, briefly hosted an edit war over whether a tweet can be a reliable source for the falsity of a conjecture. The conjecture fell in an evening; the talk-page argument is expected to run considerably longer.
Why a Counterexample Hits Different
AI labs have claimed famous-problem results before, including twice this month. Those were proofs, and proofs demand expert review; the Jacobian Conjecture's own literature is a graveyard of published arguments that died under it. A counterexample is a different species of object entirely: it is a certificate. Finding it may have taken enormous search, but checking it is mechanical, cheap, and available to anyone with a laptop. That asymmetry, hard to generate and trivial to verify, is precisely the regime where large-model search is at its best, and it is why this result went from tweet to consensus in hours instead of months.
The uncomfortable question, asked repeatedly on Hacker News, is why 87 years of human effort never surfaced a degree-7 map with coefficients this small. Nobody has a good answer yet. Most of the field's energy went into proving the conjecture true: the degree-2 case was settled decades ago, and a celebrated reduction showed the general problem follows from the case of degree-3 homogeneous maps. Those theorems remain true. The program they served now has its answer, and it is no.
What It Settles, and What It Does Not
The counterexample lives in dimension 3, which kills the conjecture for every n of at least 3: pad the map with identity coordinates and the determinant and the collision survive unchanged. The two-variable case, the original 1884 planar question, is untouched and remains open. Expect a stampede toward it in both directions this week.
On attribution, be precise about what is and is not known. The claim that Claude Fable found the map rests on Alpöge's tweet. There is no paper, no prompt log, no description of how much search, steering, or compute was involved, and no official statement from Anthropic as of this writing. Alpöge is an Anthropic employee announcing a result he credits to his employer's model, which is exactly the kind of framing that deserves a raised eyebrow. But the raised eyebrow only applies to the origin story. The mathematical object is real, machine-checked, and does not care who is employed where.
For builders, the transferable lesson is the shape of the win: the model was pointed at a problem whose answers are cheap to verify, and the humans around it built the verification stack within hours. If you are designing agent loops, that is the pattern to steal. Generate against a verifier, and make the verifier's word final.
Key Takeaways
- The Jacobian Conjecture, open since 1939 and number 16 on Smale's list, is false for dimension 3 and above.
- The counterexample is an explicit degree-7 polynomial map on C3 with constant Jacobian determinant -2 that sends three points to one; AI Bacon verified it independently in SymPy.
- Anthropic mathematician Levent Alpöge announced it on X and credited Claude Fable with the find; workflow details and any official Anthropic account remain unpublished.
- A Lean 4 formalization appeared within hours and proves a stronger det = 1 variant fails over every field.
- Counterexamples are self-verifying certificates, which is why this claim reached consensus in hours while AI proof claims linger in review.
- The two-variable case remains open, and the 87-year-old question of why humans missed a map this simple now has its own research value.
Sources: Levent Alpöge on X, Hacker News discussion, deancureton/jacobian (Lean 4 formalization), Wikipedia: Jacobian conjecture