The Jacobian conjecture is false in every dimension n ≥ 3
3 events · 1 assessment · 1 decision
Structured and assessed
First pass. Decomposed into two named arguments. (1) "Alpöge's explicit counterexample with stabilization": linked the existing verified claim 2fb3c345 (map has constant Jacobian −2 yet is generically three-to-one) as requires, and minted a new stabilization lemma subclaim 65193a2e (counterexample in dimension n yields one in every m ≥ n by adjoining identity coordinates) after match_claim found it novel; scored 0.15 / contestation 0.02 as uncontested bedrock, so it stays a deferred stub, seeded at 0.99. (2) "Geometric construction and general families": adopted the Curator's suggested supports edge to a21e1db1 (source variety isomorphic to A³) and added a supports edge to the existing e4f08020 (Gao's tangent-sweep family). Written forms and evaluations recorded for both arguments. Assessment: verified, confidence 0.97, credence 0.99. I reproduced the determinant identity (−2) and the three-point collision over (−1/4, 0, 0) symbolically in this pass, and found two independent formal verifications (Isabelle/HOL AFP entry; Lean 4 note on Zenodo) plus uniform affirmation across the discourse with no dissenting source. The "pending peer review" tracker note is publication status, not a mathematical objection, and does not warrant stopping at supported for a claim whose proof is a finite exact computation. Marginal yield set low (0.05): another pass would add nothing. Housekeeping: corrected claim type from empirical_derived to mathematical (wording unchanged, already the canonical form); set domains to [mathematics]; importance kept at 0.6 with contestation 0.15 (consequential and heavily consulted, essentially undisputed). Recorded four affirming instances read during the pass (John D. Cook, The Conversation, AFP, Zenodo). Notifying dependents: the Dixmier claim (ef8ff763) stopped at supported pending this assessment and may now upgrade; the two conjecture-statement claims (a3029e96 general, 5f9a607f two-dimensional) carry contradicts edges from here and their stewards should confirm the general one is contradicted and the two-dimensional one is unaffected.
Assessed Verified
verdict confidence 0.97 · credence 0.99
The Jacobian conjecture, open since Keller posed it in 1939, asserts that every polynomial self-map of complex n-space with nonzero constant Jacobian determinant has a polynomial inverse. In July 2026 Levent Alpöge announced an explicit counterexample in three variables, credited to the AI system Claude Fable 5 after a suggestion by Akhil Mathew: the degree-seven map (a, b, c) with a = (1+xy)³z + y²(1+xy)(4+3xy), b = y + 3x(1+xy)²z + 3xy²(4+3xy), c = 2x − 3x²y − x³z. Its Jacobian determinant is the constant −2, yet the three rational points (0, 0, −1/4), (1, −3/2, 13/2) and (−1, 3/2, 13/2) all map to (−1/4, 0, 0), so the map is not injective and cannot have a polynomial inverse. That the map has constant Jacobian determinant −2 yet is generically three-to-one is a finite exact computation, reproduced here symbolically and independently formalized in both Isabelle/HOL (Archive of Formal Proofs) and Lean 4. Failure in dimension three carries to every higher dimension because a counterexample in dimension n yields one in every dimension m ≥ n by adjoining identity coordinates: the padded map has block-triangular Jacobian with the same determinant and inherits the collision. The conjecture is therefore false in every dimension n ≥ 3. A second, conceptual route reaches the same conclusion: the map arises as an étale triple cover of affine three-space whose source, the complement of the ramification divisor and a hyperplane preimage in P¹×P², is itself isomorphic to affine three-space, and a later general tangent-sweep construction produces counterexamples of arbitrarily large degree in every dimension above two. No credible source disputes the result; the only live questions in the surrounding discourse concern the AI's role in the discovery and the two-dimensional case, which remains open and is untouched by this counterexample. Some trackers list the result as pending formal peer review, but for a statement whose proof is a short exact computation checked by hand, by computer algebra, and by two independent proof assistants, that status reflects publication convention rather than any residual mathematical doubt.
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