Technology
Claude Fable 5 helps break the 87-year-old Jacobian conjecture (July 20, 2026)
Anthropic researcher Levent Alpöge said on July 20 that he used Claude Fable 5 to produce a short counterexample to the Jacobian conjecture, an algebra problem open since 1939. The model generated a three-dimensional polynomial map whose Jacobian determinant is constant yet sends three distinct points to the same output, violating the conjecture’s central claim. Several mathematicians, including Stanford’s Jared Duker Lichtman, have independently checked the core calculation, though the result has not been peer-reviewed. It follows a May result in which an OpenAI model helped settle a long-open Erdős problem.
So what
The signal is not the theorem, it is that a frontier model materially advanced novel expert reasoning that had resisted people for decades, when driven by a domain expert. For research-intensive growth companies, that reframes where AI belongs: not only drafting and support, but the hardest specialist work, paired with someone who can verify it. Ask where your most valuable expertise is bottlenecked behind a handful of senior people, and pilot the model as their collaborator rather than their replacement. Insist on human verification of any output before it ships.
Hype check
The counterexample was posted to social media and spot-checked with computer algebra, not yet peer-reviewed or written up as a formal paper. Treat the milestone as real but provisional.
Confidence: Watch this — the counterexample and independent spot-checks are real, but formal mathematical acceptance is pending.
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