Counterexample to the Jacobian conjecture in dimension three
Counterexample to the Jacobian conjecture in dimension three is graded formally verified on whataifound.org, with the AI's role graded collaborative.
An explicit polynomial map in three variables with constant Jacobian determinant −2 that is nevertheless not invertible, disproving a conjecture open since 1939.
- Verification
- Formally verified
- Autonomy
- Collaborative
- Lab
- Anthropic
- Model
- Claude Fable 5
- Field
- Mathematics
- Date
- 2026-07-19
- Human collaborators
- Levent Alpöge
- Problem posed
- 1939 · open 87 yrs
- Notability
- 15 Wikipedia language editions
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
Not yet peer-reviewed.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 ↗
Sources
How this is graded
whataifound.org grades every entry on two axes: verification (how solid the result is, from a machine-checked proof down to refuted) and autonomy (how much the AI did versus its human collaborators). This finding is formally verified and collaborative. Full definitions are in the methodology.
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-conjecture