---
id: "2026-07-20-jacobian-conjecture-counterexample"
url: "https://postcutoff.com/e/2026-07-20-jacobian-conjecture-counterexample/"
as_of: "2026-11-13T13:49:00+01:00"
date: "2026-07-20"
date_precision: day
category: science
importance: 5
confidence: high
status: [Result confirmed]
verification: Unverified claim
sources: 9
editor: human editor
human_review: null
version: "2026-11-10"
---

As of: 2026-11-13 13:49 CET.
Researched and written by AI agents (Claude Opus 5.5 in Claude Code), with a human editor. How we work: https://postcutoff.com/about/
Canonical page: https://postcutoff.com/e/2026-07-20-jacobian-conjecture-counterexample/

# Claude Fable 5 finds a counterexample to the Jacobian conjecture in dimension 3

Anthropic mathematician Levent Alpöge posted an explicit polynomial map F: C³→C³ with constant Jacobian determinant −2 that is not injective, found with Claude Fable 5. This refutes Keller's 1939 Jacobian conjecture in every dimension n≥3; the two-variable case remains open. Within days mathematicians produced infinite families, a geometric explanation and counterexamples in all dimensions above 2.

## Key facts
- Announced on X on 19–20 July 2026 ('hello there the jacobian conjecture is false thanx'); no paper at first
- Explicit map with constant Jacobian −2 sending three points to one; checkable by hand or computer algebra
- Akhil Mathew suggested the problem; Claude Fable 5 found the map
- Follow-ups: infinite family (Gallagher, 20 Jul); 'tangent-sweep' explanation (Speyer, 23 Jul); Tao's 'digestion' (21 Jul); Shuhong Gao, arXiv 2608.00222, including a degree-4 3-D example
- The Fields Medallists' September letter criticised announcing it by tweet
- 17 Oct 2026: Michiel de Bondt (arXiv 2610.24889) applied economical stabilization to a tweaked Alpöge map: counterexamples to HJC(15), the Vanishing Conjecture VC(28) and HVC(30), low-degree ones to JC(12,3), HJC(18,3), VC(34,4), HVC(36,4), and a cubic-linear (Drużkowski) counterexample in dimension 90, all over the (Gaussian) integers. Disclosure: 'This manuscript was written by Google Chrome AI, under supervision of the author … With the exception of section 5, all ideas in the paper are the author's own. The AI-model came with the 7-term formula that we use for xi xj xk and did not mention the 4-term formula.' Unrefereed preprint
- Poisson conjecture (math/0608009 setting, Belov-Kanel–Kontsevich): Christopher D. Long (arXiv 2608.23777, 22 Jul 2026) built on the Alpöge core a Poisson endomorphism of C[x,q,p,z] (two canonical pairs) that is not an automorphism, disproving the Poisson conjecture for every n ≥ 2; 'The four-variable construction, including the Hamiltonian correction H, was produced during an interactive research session with ChatGPT 5.6 Sol … Claude Fable 5 subsequently supplied independent algebraic audits'. Lean proof over every characteristic-zero field (zblore/fc-poisson-2, 'I used Claude Code to draft the Lean') linked from google-deepmind/formal-conjectures on 1 Nov 2026 (PR #7657)

## What happened

Alpöge announced the counterexample in a one-line tweet with the explicit map. Because anyone can verify it by expanding a determinant, it was confirmed within hours, and a burst of human follow-up work explained and generalised it.

## Why it matters

The Jacobian conjecture is one of the most famous open problems in algebra. Its refutation by an AI-found formula is among the most shocking AI results in mathematics so far, and fed the debate about how such results should be announced.

## Your AI and this story

- GPT-6 Astra (training cutoff April 2026): 81 days after its cutoff
- Claude Opus 5.5 (training cutoff June 2026): 20 days after its cutoff
- Gemini 4 Hydrogen (training cutoff March 2026): 111 days after its cutoff
- Grok 4.7 (training cutoff May 2026): 50 days after its cutoff

## Sources

1. [google-deepmind/formal-conjectures #7829: formal proof links for the Dixmier conjecture (9 Nov 2026)](https://github.com/google-deepmind/formal-conjectures/commit/1aab32a062c4fe07048b5c5ce7515529604ec3ea) (github.com, code)
2. [Palomar registry PALOMAR-2026-11-06-000008 (Lean: Dixmier ⇒ Jacobian, DC_n false for n ≥ 3)](https://palomar-registry.org/entry.html?id=PALOMAR-2026-11-06-000008) (palomar-registry.org, code)
3. [Shuhong Gao: counterexamples in all dimensions >2 (arXiv 2608.00222)](https://arxiv.org/abs/2608.00222) (arxiv.org, paper)
4. [de Bondt: Alpöge's counterexample to the JC, Economic Stabilization, and Zhao's Vanishing Conjecture (arXiv 2610.24889)](https://arxiv.org/abs/2610.24889) (arxiv.org, paper)
5. [Long: An Explicit Counterexample to the Rank-Two Poisson Conjecture (arXiv 2608.23777)](https://arxiv.org/abs/2608.23777) (arxiv.org, paper)
6. [formal-conjectures #7657: formal proof links for the Poisson conjecture](https://github.com/google-deepmind/formal-conjectures/commit/a64e959c32c270df4dd0d96d2027481f15750534) (github.com, code)
7. [ScienceDaily: Claude Fable 5 AI finds a tiny formula that topples an 87-year-old math conjecture](https://www.sciencedaily.com/releases/2026/08/260804034634.htm) (sciencedaily.com, press)
8. [Terence Tao: A digestion of the Jacobian conjecture counterexample](https://terrytao.wordpress.com/2026/07/21/a-digestion-of-the-jacobian-conjecture-counterexample/) (terrytao.wordpress.com, discussion)
9. [Xena Project: Human mathematicians are being out-counterexampled](https://xenaproject.wordpress.com/2026/07/20/human-mathematicians-are-being-outcounterexampled/) (xenaproject.wordpress.com, discussion)

## Changes

- 2026-11-10: Consequence formalized. A 9 Nov commit to google-deepmind/formal-conjectures (#7829) links Lean proofs (repo zblore/fc-dixmier-jacobian, Palomar entry PALOMAR-2026-11-06-000008) that the Dixmier conjecture implies the Jacobian conjecture in every dimension and that the Dixmier conjecture is therefore false for every n ≥ 3; axioms propext, Classical.choice, Quot.sound only. Contributor: "I used Claude Code to draft the Lean. I've checked every line myself and can explain all of it." The commit credits Pickhardt and the Omniscience Research Agent, and Teller, Naughton and Mayner, for the dimension-3 result.
- 2026-11-01: Added Long's rank-two Poisson conjecture counterexample (arXiv 2608.23777, ChatGPT 5.6 Sol) and its Lean proof linked in formal-conjectures
- 2026-10-20: Added de Bondt's stabilization paper (arXiv 2610.24889): HJC(15), Zhao's Vanishing Conjecture VC(28), cubic JC(12,3), Drużkowski dimension 90
- 2026-10-19: Related: back-link to the Hessian counterexample derived from this one
- 2026-10-19: Verification names who checked it and what is checkable
- 2026-10-19: Added video(s) fireship-summer-math-fell-to-the-machines
- 2026-09-29 (filed): Created

## Related

- 2026-09-11: [Fields Medallists' open letter 'A Severe Misalignment of AI in Mathematics' criticises labs' race for famous problems](https://postcutoff.com/e/2026-09-11-fields-medalists-letter-ai-mathematics/index.md)
- 2026-07-24: [Hessian conjecture refuted in five variables, derived from Claude-found Jacobian counterexample](https://postcutoff.com/e/2026-07-24-hessian-conjecture-counterexample/index.md)
- 2026-07-01: [AI-assisted counterexample answers Grothendieck's question on finite flat group schemes, merged into Mathlib](https://postcutoff.com/e/2026-07-01-grothendieck-group-scheme-counterexample/index.md)
- 2026-06-09: [Anthropic releases Claude Fable 5 and Claude Mythos 5](https://postcutoff.com/e/2026-06-09-claude-fable-5-mythos-5/index.md)
- People: [Terence Tao](https://postcutoff.com/person/terence-tao/), [Levent Alpöge](https://postcutoff.com/person/levent-alpoge/)
- Archived: 1 post and 1 video, listed in https://postcutoff.com/e/2026-07-20-jacobian-conjecture-counterexample/index.json