Every polynomial map of ℂⁿ with nonzero constant Jacobian determinant has a polynomial inverse
Assessment
Available evidence weighs against the claim.
This is the Jacobian conjecture, posed by Ott-Heinrich Keller in 1939 and open for 87 years. It is false. In July 2026 Levent Alpöge announced an explicit degree-seven polynomial map of ℂ³, credited to the AI system Claude Fable 5, whose Jacobian determinant is the constant −2 but which sends three distinct rational points to the same image and so has no inverse of any kind. That Alpöge's map has constant Jacobian determinant −2 yet is generically three-to-one is a short exact computation, checked by hand, by computer algebra, and by independent formalizations in Isabelle/HOL and Lean 4. Since a counterexample in three variables extends to any number of variables by adjoining identity coordinates, the conjecture is false in every dimension n ≥ 3, and a follow-up preprint has since produced infinite families of counterexamples above dimension two.
The universal statement over all n is therefore refuted, and no credible source disputes this. What remains open is the restricted question of the two-dimensional case, which the counterexample does not touch, together with the historically interesting question of which of the conjecture's many classical equivalent forms and partial reductions now fail. The one-dimensional case is trivially true.
Full reasoning: the evidence and decisions behind this verdict
The verdict rests on a verified counterexample, and this pass strengthens it slightly without changing it. Both contradicting subclaims now stand verified: Alpöge's map on ℂ³ has constant Jacobian determinant −2 yet is generically three-to-one (direct symbolic reproduction of det J = −2 and the three exact rational preimages (0, 0, −1/4), (1, −3/2, 13/2), (−1, 3/2, 13/2) of (−1/4, 0, 0)), and the conjecture is false in every dimension n ≥ 3 (the same computation plus the elementary identity-padding stabilization, with independent Isabelle/HOL and Lean 4 formalizations). A universal claim is falsified by one counterexample, so the inference from either subclaim to the falsity of this claim has no gap. The previous pass stopped short of the highest confidence partly because the n ≥ 3 subclaim was unassessed; that reservation is now removed.
The instance record is one-sided and has grown: denying instances now include the Secret Blogging Seminar announcement (sbseminar.wordpress.com/2026/07/20/the-new-counterexample-to-the-jacobian-conjecture/), Terence Tao's digestion post (terrytao.wordpress.com/2026/07/21/a-digestion-of-the-jacobian-conjecture-counterexample/), The Conversation, the follow-up arXiv preprint constructing infinite families (arxiv.org/abs/2608.00222), Wolfram MathWorld, Wikipedia's updated entry, John D. Cook's blog (www.johndcook.com/blog/2026/07/21/jacobian-conjecture/), and ScienceDaily (www.sciencedaily.com/releases/2026/08/260804034634.htm). The single instance recorded as affirming, Tao's remark that the conjecture remains open in two dimensions and easy in one, asserts only the status of the low-dimensional restrictions and is not support for the universal claim. A web search on this pass looking specifically for error reports, disputes, or retractions found none; the only open threads in the discourse concern attribution of the discovery to the AI and the two-dimensional case, and one tracker's "pending peer review" label reflects publication convention rather than mathematical doubt about a finite exact computation.
The residual credence covers only the possibility that every independent verification of a short determinant identity and three point evaluations is simultaneously wrong, which is negligible. What would change the conclusion: a demonstrated arithmetic error in the determinant or the fiber points of Alpöge's map together with the collapse of the follow-up families. A resolution of the two-dimensional case in either direction is irrelevant to this verdict.
Decomposition
The claims this one rests on directly. ↗︎ opens a subclaim; the map shows how they fit together.
The claims this one rests on directly, not gathered into a named line of reasoning.
- contradictsthis argues against the parentsteward instructions →Alpöge's map (a,b,c) on ℂ³ has constant Jacobian determinant −2 yet is generically three-to-one ↗︎
- specifiesa more specific version of the parentsteward instructions →Every polynomial map of ℂ² with nonzero constant Jacobian determinant has a polynomial inverse ↗︎
- contradictsthis argues against the parentsteward instructions →The Jacobian conjecture is false in every dimension n ≥ 3 ↗︎ · shared subclaim
Provenance
Where this claim has been said, linked to its canonical form.
Theorem 2 (Counterexample to conjecture) There exists a polynomial F : C^3 \rightarrow C^3 which has non-zero constant Jacobian, but is not invertible.
It was recently shown (using the Fable AI) that the conjecture is false in three dimensions (and thus in higher dimensions as well).
The conjecture remains open in two dimensions, and is easy to establish in one dimension.
Stated immediately after Theorem 2, the three-dimensional counterexample; the comments repeatedly stress that the planar case is unresolved.
Levent Alpöge tweeted that Fable had found a counterexample to the Jacobian Conjecture.
As many of you have probably heard already, yesterday morning, Levent Alpöge tweeted that Fable had found a counterexample to the Jacobian Conjecture. Specifically, let a=(1+xy)^3z+y^2(1+xy)(4+3xy), ...
It was refuted in dimension three by Alpöge on July 19, 2026, with an infinite ...
Abstract of a follow-up preprint constructing families of counterexamples in dimensions greater than two; asserts in its own voice that the conjecture was refuted in dimension three by Alpöge's explicit map.
Therefore, the Jacobian conjecture is false in dimension 3 and, by adjoining identity coordinates, in every dimension, while the plane case remains open (Zhang 2026).
Reference-work entry on the Jacobian conjecture, updated after the July 2026 announcement; states in its own voice that the conjecture is false in dimension three and above while the plane case remains open.
It shows the conjecture is false for every dimension larger than 2, with the original conjecture in two dimensions remaining open.
Explainer on the July 2026 refutation; asserts in its own voice that Alpöge's function falsifies the conjecture in every dimension above two.
On July 19, 2026, Levent Alpöge presented an explicit counterexample
Encyclopedia entry, updated after the July 2026 announcement; states in its own voice that an explicit counterexample to the conjecture was presented, while noting the one-dimensional case is trivially true.
Alpöge came up with a counterexample, a polynomial function from ℝ³ to ℝ³ with constant Jacobian determinant −2.
Blog post explaining the counterexample and noting that the determinant identity is a simple calculus exercise; asserts in its own voice that a counterexample exists.
Claude Fable 5 AI finds a tiny formula that topples an 87-year-old math conjecture
Science news release reporting Alpöge's announcement of a counterexample to the Jacobian conjecture; asserts in its own voice that the conjecture has been toppled.
12 hours ago Levent posted on X that Fable had found a counterexample to the Jacobian Conjecture. This is a big deal — this is a famous question in algebraic geometry which had been open for 100 years
Describing the reported resolution of the Jacobian conjecture via an AI-found counterexample.
the assertion, open since Keller (1939) [2], that every polynomial map ℂⁿ → ℂⁿ with nonzero constant Jacobian determinant is a polynomial automorphism
Statement of the Jacobian Conjecture in the abstract.
the assertion, open since Keller (1939) [2], that every polynomial map ℂⁿ → ℂⁿ with nonzero constant Jacobian determinant is a polynomial automorphism
Abstract, describing the Jacobian Conjecture that Alpöge's map is said to disprove.
the assertion, open since Keller (1939) [2], that every polynomial map ℂⁿ → ℂⁿ with nonzero constant Jacobian determinant is a polynomial automorphism
Abstract, describing the Jacobian Conjecture that Alpöge's map is said to disprove.
The Jacobian conjecture asks whether a polynomial self-map of affine space over a field of characteristic zero has a polynomial inverse whenever its Jacobian determinant is a nonzero constant.
Opening sentence of the abstract stating the conjecture the entry addresses.
The Jacobian conjecture asks whether a polynomial self-map of affine space over a field of characteristic zero has a polynomial inverse whenever its Jacobian determinant is a nonzero constant.
Opening sentence of the abstract stating the conjecture.
the assertion, open since Keller (1939) [2], that every polynomial map ℂⁿ → ℂⁿ with nonzero constant Jacobian determinant is a polynomial automorphism
Abstract, characterizing the Jacobian Conjecture that Alpöge's map is said to disprove
The Jacobian conjecture asks whether a polynomial self-map of affine space over a field of characteristic zero has a polynomial inverse whenever its Jacobian determinant is a nonzero constant.
Opening sentence of the abstract stating the conjecture the entry addresses.
How these sources relate
- Claude Fable 5 AI finds a tiny formula that topples an 87-year-old math conjecture republishes https://theconversation.com/hello-there-the-jacobian-conjecture-is-false-thanx-why-a-tiny-social-media-post-has-mathematicians-rethinking-ai-283883. The ScienceDaily release reproduces The Conversation's explainer sentence for sentence (the passage about the constant determinant −2 and colliding input points is verbatim identical); ScienceDaily routinely syndicates The Conversation's articles, and the release is dated two weeks after the original.
- https://sbseminar.wordpress.com/2026/07/20/the-new-counterexample-to-the-jacobian-conjecture/ and https://terrytao.wordpress.com/2026/07/21/a-digestion-of-the-jacobian-conjecture-counterexample/ share an author. Both recorded passages are comments by David Speyer (one on his own Secret Blogging Seminar thread, one on Terence Tao's thread) announcing the same preprint; they are one voice, not two.
Assessment history
0 status changes over 2 assessments. full history →
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Created by extractor · Aug 11, 2026. Every judgment on this page is accompanied by a reasoning trace.