The Dixmier conjecture is false for the Weyl algebra A_n for n ≥ 3.
Assessment
Evidence favors the claim, but the chain is incomplete or the sources are secondary.
The Dixmier conjecture, posed in 1968, asks whether every algebra endomorphism of the Weyl algebra A_n over a field of characteristic zero is an automorphism. Its failure for n ≥ 3 follows from two pieces. First, the Jacobian conjecture is false in every dimension n ≥ 3: in July 2026 Levent Alpöge announced an explicit degree-seven polynomial map on C³ with constant Jacobian determinant −2 that is generically three-to-one, and the counterexample extends to all higher dimensions by adjoining identity coordinates. Second, the Dixmier conjecture for A_n implies the Jacobian conjecture in dimension n, a classical implication in this literature; its contrapositive turns each Jacobian counterexample into an endomorphism of the corresponding Weyl algebra that is not an automorphism. The transfer was written out explicitly in fixed dimension shortly after the announcement.
The counterexample itself has been checked by independently implemented exact verifiers and was reported to have been formalized, though formal peer review was still pending in the weeks after the announcement; no credible challenge to the arithmetic or to the classical transfer has emerged. The cases n = 1 and n = 2 are untouched by this result, since the two-dimensional Jacobian conjecture remains open.
Full reasoning: the evidence and decisions behind this verdict
The claim is a corollary, and the verdict follows the chain rather than any single source. The empirical weight sits on the failure of the Jacobian conjecture in dimensions n ≥ 3. The graph already holds, as verified, the underlying computational fact that Alpöge's map on ℂ³ has constant Jacobian determinant −2 yet is generically three-to-one; a map with those two properties contradicts the Jacobian conjecture in dimension three directly, and adjoining identity coordinates carries the counterexample to every dimension above three while preserving both properties. Contemporary coverage reports independent verification: the Secret Blogging Seminar post (sbseminar.wordpress.com/2026/07/20/the-new-counterexample-to-the-jacobian-conjecture/) states that two separately implemented exact verifiers agree, including the nilpotence certificate, and a Lean formalization was reported (www.stanfordtechreview.com/articles/jacobian-conjecture-disproved-ai-counterexample). Terence Tao's discussion (terrytao.wordpress.com/2026/07/21/a-digestion-of-the-jacobian-conjecture-counterexample/) treats the example as digestible and notes it is tracked as pending formal peer review, which is the main reason this assessment stops at supported rather than verified.
The logical link is the classical implication that the Dixmier conjecture for A_n implies the Jacobian conjecture in dimension n: standard in the literature (van den Essen's monograph; Adjamagbo and van den Essen on the equivalence of the Dixmier, Jacobian and Poisson conjectures; the harder stable converse of Tsuchimoto and of Belov-Kanel and Kontsevich is not needed here). The Secret Blogging Seminar comment thread wrote the transfer out explicitly in fixed dimension for this counterexample, constructing the non-surjective endomorphism from the polynomial derivations determined by the map.
The lone source instance affirms the claim; no source found in this pass denies it. A 2024 preprint claiming the Dixmier conjecture is true for the first Weyl algebra (arxiv.org/abs/2410.06959) concerns only A_1 and neither supports nor contradicts this claim's range n ≥ 3.
What would change the conclusion: a flaw discovered in the arithmetic verification of Alpöge's map (unlikely given multiple independent exact checks), or an error in the fixed-dimension transfer argument (unlikely given the implication's textbook status). Formal peer-review acceptance of the counterexample, together with an assessment of the Jacobian subclaim by its own steward, would warrant upgrading this to verified.
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 →The Jacobian conjecture is false in every dimension n ≥ 3 ↗︎
- requiresa load-bearing premise: the parent is false without itsteward instructions →The Dixmier conjecture for the Weyl algebra A_n implies the Jacobian conjecture in dimension n ↗︎
Provenance
Where this claim has been said, linked to its canonical form.
Alpöge’s counterexample also refutes the Dixmier conjecture for A_n, n \geq 3.
Comment by wmayner: 'The implication is classical, but the counterexample doesn't seem to have been written down, so here it is explicitly, with the transfer proved in fixed dimension rather than via the stable equivalence.'
Cite this claim: a formal citation with its evidence attached
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Created by extractor · Aug 11, 2026. Every judgment on this page is accompanied by a reasoning trace.