Breakthrough Tracker record
The Jacobian conjecture is false in dimension 3
An explicit polynomial map from C³ to C³ has constant Jacobian determinant −2 but is not injective. The example disproves the unrestricted Jacobian conjecture in dimension 3.
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- Stable ID
math-jacobian-conjecture-dimension-3-2026- Revision
math-jacobian-conjecture-dimension-3-2026.v1- Field
- Mathematics · Algebraic geometry
- Evidence
- Tier 1 · Peer reviewed: No
- Record state
- Current · Announced; exact object independently checked
- Last checked
AI role
Claude Fable reportedly found the example; humans posed the question, checked the algebra and announced the result.
Record details
- Problem or result
- Jacobian conjecture
- Authors
- Levent Alpöge; example credited to Claude Fable after a question from Akhil Mathew
- Institutions
- Affiliation not stated in the inspected announcement
- Result date
- Announced July 19, 2026
Why it matters
The conjecture had been open since 1939 and is a central problem in affine algebraic geometry.
Limits
The two-dimensional conjecture remains open. The exact determinant and collision have been checked independently, but no conventional paper or refereed publication by the announcer was located.
Sources
- Primary: Levent Alpöge’s original announcement and explicit map
- Independent: T. Shaska, Graded Keller maps and the Jacobian Conjecture
- Independent: Terence Tao, A digestion of the Jacobian conjecture counterexample
- Independent: Wolfram MathWorld, Jacobian Conjecture
- Independent: Oliver Knill, exact Mathematica verification
- Independent: Independent symbolic check and explainer
- Independent: Christopher D. Long, Small Counterexamples to the Gaussian Moments Conjecture
Correction and revision history
- 2026-07-22 — Added as an announced, directly checkable counterexample with conventional peer review still pending.
- 2026-07-22 — Added a new independent scholarly preprint analyzing the grading of the announced counterexample; this does not change the entry's peer-review status.
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