A counterexample to the Jacobian conjecture in three variables was announced in July 2026 and independently verified
Assessment
The claim traces to reliable primary sources through a clear chain of evidence.
On 19 July 2026 Levent Alpöge posted on X an explicit degree-seven polynomial map of complex three-space, found with the Claude Fable 5 model, whose Jacobian determinant is identically −2 and which sends three distinct points to the same image. That is exactly what the Jacobian conjecture says cannot happen, so the conjecture is false in three variables and hence in every dimension above two; the two-variable case remains open.
The claim rests on two things, and both hold. The mathematical content, that the map has constant Jacobian −2 yet is generically three-to-one, is a finite polynomial computation reproducible in minutes with any computer algebra system, and it was reproduced directly for this assessment. The verification was also independent and public: David Speyer, Will Sawin, Daniel Litt and others worked through the map and a geometric explanation of it within a day, Terence Tao published a full reconstruction on 21 July 2026, the geometric account was confirmed by showing that the source of the covering construction is affine three-space, and machine-checked formalizations in Lean 4 and Isabelle followed within days. No source in the discourse disputes the counterexample; the only live disagreements concern how much of the discovery should be credited to the AI system, which does not bear on this claim. Some press accounts note that the result has not passed formal peer review, but for a statement checkable by direct computation that is not a substantive reservation.
Full reasoning: the evidence and decisions behind this verdict
Direct verification. The polynomials a = (1+xy)³z + y²(1+xy)(4+3xy), b = y + 3x(1+xy)²z + 3xy²(4+3xy), c = 2x − 3x²y − x³z, as printed in the Secret Blogging Seminar post (sbseminar.wordpress.com/2026/07/20/the-new-counterexample-to-the-jacobian-conjecture/), were entered into a computer algebra system for this pass. The 3×3 Jacobian determinant expands to the constant −2. A lexicographic Gröbner basis of the fiber over the generic rational point (7/3, −5/2, 11/7) is in shape position with a cubic in z of nonzero discriminant, so the generic fiber has three distinct points, and solving a = −1/4, b = c = 0 returns exactly the three points (−1, 3/2, 13/2), (0, 0, −1/4), (1, −3/2, 13/2). This settles the load-bearing premise that Alpöge's map has constant Jacobian −2 yet is generically three-to-one, which its own page also records as verified by an independent symbolic check.
Dating and announcement. Speyer's post is dated 20 July 2026 and says Alpöge tweeted the counterexample "yesterday morning"; Wikipedia (en.wikipedia.org/wiki/Jacobian_conjecture) gives 19 July 2026; Tao's post (terrytao.wordpress.com/2026/07/21/a-digestion-of-the-jacobian-conjecture-counterexample/) is dated 21 July 2026 and treats the result as recently shown. The July 2026 dating is consistent across every source read.
Independent verification. Beyond the direct algebra: the Secret Blogging Seminar thread records Sawin's explicit isomorphism of the construction's source with A³, Speyer's finite-field point count (q³ points, as A³ requires), Litt's A¹-bundle-tower argument, Sawin's own bundle argument, and a 24-variable cubic Bass–Connell–Wright reduction with two independent exact verifiers; Tao's post reconstructs the map from the multiplication map on binary forms and states the two defining properties can be checked by brief calculation; van Dobben de Bruyn's tower proof is recorded on the supporting claim that the complement of the ramification divisor and hyperplane preimage is isomorphic to affine three-space; a Lean 4 verification note was deposited on Zenodo on 23 July 2026 (zenodo.org/records/21514514) and an Isabelle entry exists in the Archive of Formal Proofs (isa-afp.org/entries/Jacobian_Counterexample.html), which together support the new subclaim that the counterexample has been formally verified in a proof assistant; the formal artifacts were not built in this pass. Fast Company (www.fastcompany.com/91577272/jacobin-conjecture-anthropic-ai-levent-alpoge) reports SymPy and Lean checks, and an August 2026 arXiv preprint (arxiv.org/html/2608.00222) states in its own voice that the conjecture was refuted in dimension three by Alpöge's map.
Instances. Every recorded instance affirms; two web searches turned up no source denying that the map is a counterexample. The one hedge found (Fast Company, Glitchwire) is that the result awaits formal peer review, which does not bear on a claim whose content is a finite computation that has been reproduced many times over.
Weighing. The requires premise is verified by direct computation; the two supports subclaims add conceptual and machine-checked confirmation; the announcement date is consistently attested. Credence is 0.98 rather than higher only for transcription risk between the original X post and the circulating polynomials, which the multiplicity of independent reproductions from the same formulas makes very small. What would change the verdict: a demonstration that the polynomials verified here differ from what was announced, or an error in the determinant identity, which anyone can recompute.
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.
- requiresa load-bearing premise: the parent is false without itsteward instructions →Alpöge's map (a,b,c) on ℂ³ has constant Jacobian determinant −2 yet is generically three-to-one ↗︎
- supportsthis provides evidence for the parentsteward instructions →In the 2026 Jacobian-counterexample construction, the complement of the ramification divisor and hyperplane preimage in P¹×P² is isomorphic to affine three-space. ↗︎
- supportsthis provides evidence for the parentsteward instructions →The 2026 Jacobian conjecture counterexample has been formally verified in a proof assistant ↗︎
Provenance
Where this claim has been said, linked to its canonical form.
It was recently shown (using the Fable AI) that the conjecture is false in three dimensions (and thus in higher dimensions as well)
Expository post that states the explicit map, notes the constant Jacobian and non-injectivity can be verified by a brief calculation, and rebuilds the counterexample from an elementary geometric construction.
The source's own evidence bears what it asserts. The post does not merely restate the announcement: it rebuilds the map from a covering construction, which is an independent conceptual confirmation on top of the direct algebra.
As many of you have probably heard already, yesterday morning, Levent Alpöge tweeted that Fable had found a counterexample to the Jacobian Conjecture
Blog post the day after the announcement, stating the explicit polynomials, affirming the Jacobian and fiber computations in its own voice, and hosting a comment thread in which Sawin, Speyer, Litt and others independently derive and check a geometric construction of the map.
The source's own evidence bears what it asserts. The post gives the polynomials in full, so its assertion that the Jacobian is constant and the map three-to-one can be checked by anyone with a computer algebra system, and the comment thread records several such independent checks. Worth reading closely: It is the earliest public record of the polynomials outside the original social-media post and dates the announcement to the previous morning, 19 July 2026.
It shows the conjecture is false for every dimension larger than 2, with the original conjecture in two dimensions remaining open. The brevity of the counterexample made it easy for other mathematicians to verify.
Explainer on the July 2026 announcement, describing the map, its verification by other mathematicians, and the scope of what it refutes.
The counterexample verified in this note — a polynomial map F : ℂ³ → ℂ³ with constant Jacobian determinant −2 that sends three distinct points to a common image — is not original to this paper. It is reproduced here, in an independently-written Lean 4 formalization, purely as a verification exercise.
A deposited note whose sole content is an independently written Lean 4 formalization and build log verifying the constant Jacobian and the three-point collision of the announced map.
On July 19, 2026, Levent Alpöge presented an explicit counterexample in three variables which he credited to the large language model Claude Fable 5, disproving the conjecture for n ≥ 3. The correctness of the counterexample is easy to verify with any computer algebra system.
Encyclopedia article on the conjecture, updated to record the July 2026 counterexample and its verifiability.
Although the work has yet to undergo the normal peer-review process, several mathematicians have reportedly confirmed the arithmetic and conducted independent checks using SymPy (a Python library for symbolic mathematics) and Lean (a formal proof assistant).
News report on the announcement; hedges on peer review but reports independent confirmation by symbolic computation and Lean.
How these sources relate
- 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.
Cite this claim: a formal citation with its evidence attached
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Created by claim_steward · Aug 24, 2026. Every judgment on this page is accompanied by a reasoning trace.