San Francisco mathematician uses AI to disprove Jacobian conjecture
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Levent Alpöge, a mathematician at AI company Anthropic in San Francisco, reports finding a counterexample to the long-standing Jacobian conjecture in higher dimensions, using the firm’s large language model Claude Fable 5. The conjecture, posed 87 years ago, concerns polynomial functions whose Jacobian determinant is a non-zero constant and asks whether such functions must be invertible. Alpöge’s claimed counterexample, shared in a concise formula on X shortly after the FIFA World Cup Final, challenges decades of prevailing belief and is now subject to independent verification by experts.
More context
Timeline
- On Jun 1, 1884, Czech mathematician Ludwig Kraus stated the initial two-dimensional version.
- On Jan 1, 1939, Ott-Heinrich Keller generalized the algebraic geometry problem globally.
- On May 15, 1998, Stephen Smale listed the problem among century mathematical challenges.
- On Jun 1, 2026, Anthropic officially released the large language model Claude Fable 5.
- On Jul 19, 2026, Levent Alpöge posted the AI-generated polynomial solution online.
- On Jul 20, 2026, Independent mathematicians quickly verified the arithmetic correctness via software.
- On Aug 1, 2026, Academic communities and pre-print servers circulated comprehensive verification breakdowns.
- On Sep 1, 2026, Researchers anticipate formal peer-reviewed publication of the counterexample findings.
- On Oct 15, 2026, Academic workshops plan specialized sessions analyzing AI discovery methodologies.
- On Jan 1, 2027, Mathematicians expect ongoing investigations into the remaining two-dimensional case.
Why it matters
AI language models accelerate complex mathematical problem-solving discoveries nationwide. Automated scientific discovery paradigms will transform advanced academic research. Mathematicians, technology developers, university researchers, software engineers, and academic institutions. Prioritise verified computer algebra system checks over speculative commentary.
Coverage notes
Left: Evidence was insufficient. Center: Focus on the collaborative nature of AI tools and verifiable math. Right: Evidence was insufficient.
Primary report
Levent Alpöge social media post published on July 19, 2026.
Sources
JQJO Editorial Desk · Human reviewed · AI assisted. Report a correction.
