Breaking

Anthropic Researcher Reports AI-Assisted Counterexample Disproving the 87-Year-Old Jacobian Conjecture

July 20, 2026 at 08:19 EDT

Foundation Models
Research & Papers

Anthropic researcher Levent Alpöge has announced a proposed counterexample to the Jacobian Conjecture, a longstanding open problem in algebraic geometry, credited in part to assistance from the company's Claude Fable 5 model. If it withstands scrutiny, the result would settle a question that has resisted resolution since roughly 1939.

July 2026 · Anthropic Research

An 87-Year-Old Math Problem Meets a Proposed Counterexample — Built With an AI's Help

Anthropic researcher Levent Alpöge has announced a claimed counterexample to the Jacobian Conjecture, open since roughly 1939 — crediting assistance from the company's Claude Fable 5 model. If it survives peer verification, it settles a longstanding question in algebraic geometry.

~87 yrs

Open since roughly 1939, resistant to resolution in higher dimensions

−2

The constant Jacobian determinant — satisfying the conjecture's hypothesis

n ≥ 3

Extends to all dimensions three and above via the identity map

Why It Breaks The Conjecture

The map is not injective — three distinct points collide into one image

(0, 0, −1/4)

(1, −3/2, 13/2)

(−1, 3/2, 13/2)

→

(−1/4, 0, 0)

Constant nonzero Jacobian should imply global invertibility — but three different source points landing on one image proves this polynomial map is not injective.

How A Result Like This Gets Confirmed

Explicit map

A concrete polynomial written down

→

Symbolic check

Jacobian and point collisions verified directly

→

Peer verification

Full write-up needed before it is established

The Enthusiasm

A marker of AI's growing role in advanced mathematics — a frontier model contributing to a problem well beyond routine calculation.

The Caution

Reaching such a counterexample requires substantive human insight, not a single prompt — and a complete write-up is still needed.

Continue reading

The rest of this article is for AI News Blitz readers. Choose an option below to keep reading.