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Anthropic Mathematician Reports AI-Assisted Counterexample to 87-Year-Old Jacobian Conjecture

July 21, 2026 at 00:46 EDT

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A mathematician at Anthropic says the company's large language model, Claude Fable 5, helped produce a concrete, hand-checkable counterexample to the Jacobian conjecture, a problem that has stood open since 1939. The claim, made on July 19, 2026, has electrified the mathematics community even as researchers stress that it remains a personal result under active verification rather than an official corporate announcement.

July 19, 2026 · Claude Fable 5

An 87-Year-Old Math Problem, Broken by a Human–AI Pair

A mathematician reports that Anthropic's model helped produce a concrete, hand-checkable counterexample to the Jacobian conjecture — open since 1939. It remains a personal result under active verification, not an official announcement.

1939

Year Keller first proposed the conjecture

87 yrs

Time the problem stood open

ℂ³

Dimension of the reported counterexample

<24 hrs

Until the encyclopedia entry was updated

The hand-checkable break

The map's Jacobian determinant equals the nonzero constant −2, satisfying the hypothesis — yet three distinct points map to the same image, so it is not injective.

(0, 0, −1/4)

(1, −3/2, 13/2)

(−1, 3/2, 13/2)

→

(−1/4, 0, 0)

Short and explicit — verifiable by hand or in a computer algebra system in minutes.

The workflow

Mathematician poses the problem

→

Claude Fable 5 questions & suggests

→

Explicit candidate verified by hand

The model served as a questioning and verification partner, not an autonomous prover.

Excitement

A landmark — an 87-year-old problem falling on an ordinary Sunday evening. Because it is short and explicit, independent checking moved fast and the arithmetic holds.

Caution

A personal claim, not an official Anthropic statement. A formal write-up is still pending — and the classic two-variable (n=2) case remains open regardless.

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