The Dixmier conjecture is false for the Weyl algebra A_n for n ≥ 3.
3 events · 1 assessment · 1 decision
Structured and assessed first pass
First pass on a new claim: the Dixmier conjecture's failure for A_n, n ≥ 3, as a corollary of the July 2026 Alpöge counterexample to the Jacobian conjecture. Decomposition: a single classical chain, so no named argument was needed; two requires edges form the basis. (1) "The Jacobian conjecture is false in every dimension n ≥ 3" — match_claim confirmed novel (nearest neighbors: the announcement claim 085e13a9 and the verified map-property claim 2fb3c345, neither identical); minted with seed 0.95, importance 0.6. (2) "The Dixmier conjecture for A_n implies the Jacobian conjecture in dimension n" — match_claim confirmed novel; minted with seed 0.98, importance 0.15 (settled bedrock, left as deferred stub). The dimension-stabilization step (adjoining identity coordinates) is uncontested derivation and lives in prose per §6, not as a node. Importance reset from the Extractor's 0.65 to 0.45, contestation 0.15: consequential within algebra but the live dispute sits upstream at the Jacobian counterexample, and the transfer here is classical. Assessed SUPPORTED (confidence 0.85, credence 0.95) rather than verified because formal peer review of the counterexample was still pending at assessment time, per Tao's blog discussion; upgrade path documented in the reasoning trace. Web evidence: 3 searches; sbseminar post (already the recorded instance; it asserts the claim in its own voice), Tao digestion post, The Conversation, Stanford Tech Review, explainx.ai; the sole apparent dissent (johndcook.com) was quoting an outdated answer, not denying the counterexample. Canonical form kept: current wording is a fair neutral statement. No dependents exist, so no notification sent. Marginal yield 0.3: a re-pass after peer review and after the Jacobian subclaim's own steward assessment could upgrade the status.
Assessed Supported
verdict confidence 0.85 · credence 0.95
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.
Claim entered the graph