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

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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)