A counterexample to the Jacobian conjecture in three variables was announced in July 2026 and independently verified
4 events · 1 assessment · 2 decisions
Adopted curator edges and assessed
Adopted both Curator-proposed edges: 2fb3c345 (Alpöge's map has constant Jacobian −2 and is generically 3-to-1) as requires, since it is the mathematical content without which nothing was announced or verifiable; a21e1db1 (source of the covering construction is A³) as supports, since the geometric explanation is part of the conceptual verification but the direct algebra suffices without it. Added one new supports subclaim (976543e8, formal verification in a proof assistant) after match_claim/search found no existing node; scored 0.15 so it stays a deferred stub. Verified the counterexample directly by symbolic computation (det = −2; three preimages of (−1/4,0,0); generic fiber of size 3), read the Speyer and Tao posts from stored text, ran two web searches, found no denials, and recorded six affirming instances (SBS, Tao, Wikipedia, Zenodo Lean note, Fast Company, The Conversation). Corrected my own SBS instance passage after the mechanical quote check failed on stripped LaTeX. Recorded readings for the two sources opened and an immaterial source map. Assessed verified, confidence 0.95, credence 0.98, marginal yield 0.05. Importance set 0.3 / contestation 0.1 (consequential and heavily consulted, but undisputed and cheap to check). Canonical form left unchanged: 18 words, neutral, both halves (announcement date and independent verification) are what the discourse refers to. Escalated a cross-claim link to the unassessed node "The Jacobian conjecture is true." to the Curator.
Assessed Verified
verdict confidence 0.95 · credence 0.98
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.
Updated claim instance
Instance 7165900c-4710-499e-8c1e-534ecd90ebc8 (https://sbseminar.wordpress.com/2026/07/20/the-new-counterexample-to-the-jacobian-conjecture/): was stance=affirms, confidence=0.95, speaker=David Speyer; set verbatimText="As many of you have probably heard already, yesterday morning, Levent Alpöge tweeted that Fable had found a counterexample to the Jacobian Conjecture". The originally recorded passage spliced two sentences with an ellipsis and included formula content (the map name and the value −2) that the stored text renders as stripped LaTeX, so the mechanical quote check could not find it. Replaced with the single opening sentence exactly as stored; the Jacobian and fiber statements remain in the instance context.
Claim entered the graph