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Anthropic Mathematician Says Claude Fable 5 Produced Counterexample to 90-Year-Old Jacobian Conjecture

July 20, 2026 at 22:04 EDT

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Levent Alpöge, a mathematician at Anthropic and a member of Harvard's Society of Fellows, says he used the company's latest model, Claude Fable 5, to disprove the Jacobian conjecture in three or more variables—a problem that had stood open since 1939. The result, announced around July 19-20, 2026, is notable both for its mathematical substance and for the almost casual circumstances in which it emerged: Alpöge said the model did the work while he was watching the World Cup final.

July 19–20, 2026 · Anthropic

An 87-Year-Old Conjecture Falls — While the Mathematician Watched the World Cup Final

Using Claude Fable 5, mathematician Levent Alpöge produced an explicit counterexample disproving the Jacobian conjecture in three or more variables — a problem open since 1939. The machine's job: hunting for a simple, checkable example, not writing a deep proof.

1939

Year O. Keller first proposed the conjecture

−2

Constant Jacobian determinant of the counterexample map

77%

Fable 5 score on ProofBench (graduate-level Lean math)

The counterexample, drawn to scale

Three distinct points → one image

If a polynomial map with constant Jacobian were invertible, no two inputs could share an output. This map collapses three inputs onto a single point — so it is not injective.

(0, 0, −1/4)

(1, −3/2, 13/2)

(−1, 3/2, 13/2)

→

Same image

(−1/4, 0, 0)

What the machine did — and didn't

✓ Fast, verifiable example-hunting

A simple, low-degree (cubic) map that anyone can confirm by direct calculation or with SymPy / Wolfram Alpha.

− No deep structural insight

Critics note the counterexample settles the question but doesn't illuminate the geometry. The 2-variable case stays open, and human verification remains essential.

A growing pattern

This follows a reported May 2026 case in which an internal OpenAI model produced a counterexample to the Erdős unit distance conjecture — frontier models increasingly enlisted to hunt down checkable examples, even as the harder conceptual questions stay firmly in human hands.

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