Claude Fable 5 Helps Disprove 87-Year-Old Jacobian Conjecture
The Jacobian conjecture, posed by German mathematician Ott-Heinrich Keller in 1939, held that a polynomial map over complex space with a nonzero constant Jacobian determinant must have a polynomial inverse. Its inclusion among Stephen Smale’s 18 problems for the 21st century underscored its stature. A verifiable counterexample would reshape algebraic geometry and offer unusually concrete evidence that frontier AI can contribute to original mathematics.
On July 20, 2026, Anthropic researcher and mathematician Levent Alpöge posted a compact three-dimensional counterexample credited to Claude Fable 5. The map from C³ to C³ has a constant Jacobian determinant of −2 but sends three distinct input points to the same output, proving it is not invertible. The example refutes the conjecture in dimension three and, by extension, every dimension n≥3, though formal journal peer review had not yet been completed.
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The history behind this eventClaude Lifts Riemann Zeta Zero Bound to 67.2%
The Riemann hypothesis, proposed by German mathematician Bernhard Riemann in 1859, holds that every non-trivial zero of the Riemann zeta function lies on the critical line. The statement is central to understanding the distribution of prime numbers and remains one of the Clay Mathematics Institute’s seven Millennium Prize Problems, carrying a $1 million award for a valid proof. Researchers have therefore pursued partial results, including guarantees that a minimum share of zeros lies on the line.
Anthropic said on Aug. 10, 2026, that an unreleased research version of Claude did not prove the hypothesis but raised the known lower bound for zeros on the critical line to 67.2% from 41.6%. Over roughly a day and a half, the system coordinated about 60 subagents, tested 650 initial ideas and consulted 54 arXiv papers while checking novelty and counterexamples. Anthropic mathematicians and outside experts reviewed the work, while a Lean formalization provided machine-checked verification of the result.
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