Counterexample to the Jacobian conjecture in dimension three
An explicit polynomial map in three variables with constant Jacobian determinant −2 that is nevertheless not invertible, disproving a conjecture open since 1939.
- Lab
- Anthropic
- Model
- Claude Fable 5
- Field
- Mathematics
- Date
- 2026-07-19
- Human collaborators
- Levent Alpöge
- Problem posed
- 1939 · open 87 yrs
Sources
Original work
Independent commentary
What was found
The map F(x,y,z) = (u³z + y²u(4+3xy), y + 3xu²z + 3xy²(4+3xy), 2x − 3x²y − x³z) with u = 1+xy has Jacobian determinant identically −2, yet sends the three distinct points (0,0,−¼), (1,−3/2,13/2) and (−1,3/2,13/2) all to (−¼,0,0). A map with a global inverse cannot be three-to-one. Alpöge, a number theorist at Anthropic, announced it on X the day it was found.
Novelty check
The Jacobian conjecture (Keller, 1939) has been a celebrated open problem for 87 years, with many published false proofs in both directions. No prior counterexample in any dimension over characteristic 0 exists in the literature. Ott-Heinrich Keller's original formulation is the one addressed.
Caveats and known objections
Not yet peer-reviewed. The formula is public and checkable in seconds by computer algebra, which makes conventional peer review less load-bearing than usual, but the official record lists the conjecture as open until the literature catches up. The division of labor between Alpöge and the model has not been fully documented; autonomy graded conservatively pending a transcript.
Independent checks
whataifound.org (symbolic recomputation, SymPy): confirmed, det J = −2 identically; all three points map to (−¼,0,0)
Multiple mathematicians via public computer-algebra checks: confirmed · link ↗
Machine-checked elsewhere
PalomarPALOMAR-2026-08-21-000006
Paul-Lez/jacobian-conjecture at b244ea482ed0 · checked 2026-08-21
ProvedJacobianConjecture.jacobian_conjecture
A Lean proof that typechecks against the recorded statement at a pinned commit under a declared axiom set, checked mechanically rather than by human review. It does not certify that a result is new or of research interest: the only filter on that is a language-model screen, and Palomar states that it adds no human editorial step.
A Comparator wrapper of the formalization by Paul Lezeau and Dean Cureton in Google DeepMind's formal-conjectures, whose Lean source names the construction as Alpöge and Fable's counterexample. It records a stronger statement than this entry claims, disproving the generalized conjecture over every nontrivial commutative ring rather than in characteristic zero alone, and omits the two-variable case, which stays open.
PalomarPALOMAR-2026-08-27-000013
Arthur742Ramos/jacobian-conjecture-lean at 8db55ea3ffa8 · checked 2026-08-27
ProvedJacobianCounterexample.jacobianDet_FJacobianCounterexample.explicit_three_point_fibreJacobianCounterexample.not_jacobian_conjecture_complex
A Lean proof that typechecks against the recorded statement at a pinned commit under a declared axiom set, checked mechanically rather than by human review. It does not certify that a result is new or of research interest: the only filter on that is a language-model screen, and Palomar states that it adds no human editorial step.
A second and independent Lean formalization of the same map, by Ramos, Hulak and de Queiroz, sharing no code with the Lezeau and Cureton development already cited. It records the two-variable case as untouched, as this entry does.
The open problem
315779
Nothing mechanically. It records what a problem says, what is known about it, and the standing of any claimed solution.
Also recorded at
Nothing mechanically. It is a parallel listing of the same result, carrying its own verification label rather than an independent check.
Disagree with these grades?
Bring a citation: a grade moves on evidence, not on argument.
Or on GitHub: submit a check challenge the grade send a correction or send a pull request
Flag this for triage
Signals order the review queue and nothing else. They are never published, and they never move a grade: that takes a citation.
History of this entry
- CheckedA second independent Lean formalization of this counterexample was registered at Palomar, by a different group from the one already cited. Two independent developments of the same construction reinforce the formal grade rather than change it. source ↗
- CheckedA third-party Lean formalization of this counterexample was registered at Palomar, mechanically confirming that it typechecks against the stated theorem at a pinned commit. The formalization is by Lezeau and Cureton rather than by the counterexample's author. The grade was already formal and did not move. source ↗
- CorrectedRecorded the vibemathed record for this result as a registration. No grade moved.
- CorrectedRecorded the MathDB record for this result as a registration.
- AddedEntered the registry graded Formally verified and Collaborative.
Entries are never deleted. A grade that does not hold up is downgraded on the record, with the reason beside it.
Graded formally verified for verification and collaborative for autonomy. What these mean.
Cite this entry
whataifound.org. (2026). Counterexample to the Jacobian conjecture in dimension three. whataifound.org: A Registry of AI Scientific and Mathematical Discoveries. https://whataifound.org/finding/2026-07-19-jacobian-conjectureBibTeX
@misc{whataifound-anthropic-2026-conjecture,
title = {Counterexample to the Jacobian conjecture in dimension three},
author = {{whataifound.org}},
year = {2026},
howpublished = {whataifound.org: A Registry of AI Scientific and Mathematical Discoveries},
note = {Result by Anthropic. Verification: Formally verified. Autonomy: Collaborative.},
url = {https://whataifound.org/finding/2026-07-19-jacobian-conjecture}
}