Formally verified Collaborative

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

What was found

The map $F(x,y,z) = \bigl(u^3z + y^2u(4+3xy),\ y + 3xu^2z + 3xy^2(4+3xy),\ 2x - 3x^2y - x^3z\bigr)$ with $u = 1+xy$ has Jacobian determinant identically $-2$ , yet sends the three distinct points $(0, 0, -1/4)$ , $(1, -3/2, 13/2)$ and $(-1, 3/2, 13/2)$ all to $(-1/4, 0, 0)$ . 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. A separate question was opened in September 2026, about provenance rather than priority of the counterexample: the supplemental statement on page 160 of arXiv:2609.05746 asserts prior systematic use of the $xy$ , $1+xy$ and $\delta+xy$ constructions, including the map to the projective line given by $(x : 1+xy)$ , in an in-progress monograph circulated publicly in January 2025. That bears on where the construction came from, not on whether a counterexample already existed, and the caveats record what is and is not claimed.

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Not yet peer-reviewed.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. A priority and disclosure question was raised on the record in September 2026 and is unresolved. Adrian Vasiu, the third author of the Borisov, Gabber and Vasiu monograph on endomorphisms of affine spaces, appended a signed supplemental statement to arXiv:2609.05746 recording that an in-progress version of that work, dated 14 January 2025, was posted publicly at IHES two days later without his knowledge, and that in his understanding it subsequently fed AI models. He makes four numbered points: that no prior work aimed to classify etale endomorphisms of affine spaces of geometric degree three; that no prior work systematically used $xy$ , $1+xy$ and $\delta+xy$ in constructing open embeddings, morphisms and etale endomorphisms, including the map $(x : 1+xy)$ ; that between the monograph and the seven-page write-up of this counterexample there is overlap of notation and of intent; and that the prompt history, model version and workflow behind that write-up have not been disclosed publicly. What he does not claim is that the counterexample itself appears in the earlier work, and he notes that one of the overlapping constructions plays no role in the counterexamples. So this is a question about where the construction came from and what was disclosed, not a challenge to whether the map is a counterexample, and the formal grade is unaffected. It is not a generic complaint either: $(x : 1+xy)$ is the device the seven-page paper's own abstract is built on.

Independent checks

whataifound.org (symbolic recomputation, SymPy): confirmed, $\det J = -2$ identically; all three points map to $(-1/4, 0, 0)$

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 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

MathDBJacobian conjecture

315779

Nothing mechanically. It records what a problem says, what is known about it, and the standing of any claimed solution.

Also recorded at

vibemathedjacobian-conjecture

Nothing mechanically. It is a parallel listing of the same result, carrying its own verification label rather than an independent check.

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History of this entry

2026-09-14ChallengedCited the supplemental statement on page 160 of arXiv:2609.05746, in which Adrian Vasiu raises a priority and disclosure question about the construction behind this counterexample. Neither grade moved: the statement does not dispute that the map is a counterexample, and does not claim the counterexample appears in the earlier monograph. source ↗
2026-08-31CheckedA 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 ↗
2026-08-25CheckedA 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 ↗
2026-08-21CorrectedRecorded the vibemathed record for this result as a registration. No grade moved.
2026-08-19CorrectedRecorded the MathDB record for this result as a registration.
2026-07-20AddedEntered 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

Plain text

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-conjecture

BibTeX

@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}
}

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