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Wikipedia·wikipedia:JacobianConjecture

Jacobian conjecture

conjecture formal record: mixed F0 no formal statement

No formal proof attached. Any claim here rests on a write-up or a report.

Fidelity F0: Absent. No formal statement is attached to the result.

Source

AI activity

How grades work
Claude Fable 5

19 Jul 2026·with Levent Alpöge

An explicit polynomial map in three variables with constant Jacobian determinant −2 that sends distinct points to the same image, disproving the Jacobian conjecture in every dimension above two. The two-variable case remains open.

full A2 V2 F0
Reasoning and sources

Autonomy

Alpöge presented the counterexample and credited it to Claude Fable 5; how it was found has not been disclosed. A2 records a result credited to the system on a problem a person chose, with no public account of how much of the search was the system's own.

Details

formal record: Formal Conjectures proves in Lean that the map's Jacobian determinant is −2 and that it identifies two distinct points, and from a variant with determinant 1 that the conjecture fails in general; the two-variable statement is still stated with sorry

by hand: The counterexample can be verified with any computer algebra system

Fidelity

How fidelity is graded

F0 no formal statement. Absent. No formal statement is attached to the result.

Provenance

Concurrent work

  • Andy Jiang 2026-07-20

    A geometric reformulation of the counterexample, credited to GPT, announced the next day; Terence Tao developed it into an explanation through the factorisation of binary cubics. Source

In dispute

  • Attribution

    Adrian Vasiu said the AI-generated paper describing the counterexample borrowed notation and concepts from a draft by Alexander Borisov, Ofer Gabber and Vasiu released in January 2025. Alpöge later said the three should receive credit for their foundational work. Source

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Formal material

Formal statements · 1
Cited proofs · 0

No proof artifact cited by the formal record.

Also known as · 2
  • https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/Wikipedia/JacobianConjecture.lean
  • FormalConjectures/Wikipedia/JacobianConjecture.lean

Cite this record

qed.bot, “Jacobian conjecture”, https://qed.bot/s/wikipedia-jacobianconjecture, as of 30 Sep 2026.

This record as plain text, with each claim, its grades and its sources.