AI Cracks 87-Year-Old Jacobian Conjecture: A Paradigm Shift in Math?

AI Cracks an 87-Year-Old Enigma

In a development that has sent shockwaves through the global mathematical community, a researcher has used an AI model to find a counterexample to the Jacobian Conjecture, a foundational problem in algebraic geometry that has resisted human effort for 87 years. The discovery, announced on July 20, 2026, by Harvard mathematician Levent Alpoge, was made possible by Anthropic's experimental model, Claude Fable 5.

The conjecture, first proposed by German mathematician Ott-Heinrich Keller in 1939, posits that if a polynomial map from n-dimensional space to itself has a constant, non-zero Jacobian determinant (the best local approximation of the map), then the map has a polynomial inverse. It is a deceptively simple statement that has profound implications for geometry and algebra.

Alpoge's counterexample is a remarkably concise 216-character string of equations mapping three complex variables to three outputs. The formula, ((1+xy)^3 z + y^2 (1+xy) (4+3xy), y + 3 x (1+xy)^2 z + 3 x y^2 (4+3xy), 2 x — 3 x^2 y — x^3 z), passes the local check everywhere yet is not globally invertible, effectively disproving the conjecture.

Terence Tao's Deep Dive with ChatGPT

Fields Medalist Terence Tao, a signatory of the recent 'Leiden Manifesto' on AI and mathematics, didn't just accept the result at face value. He conducted a detailed verification using a chatbot, sharing the full conversation publicly. Tao's analysis provides a masterclass in how AI can be used as a collaborator rather than a black-box oracle.

Tao uploaded a PDF containing an alternate approach to constructing the counterexample and asked ChatGPT to compare it to the original formula. The AI confirmed that the two maps were 'mutually inverse polynomial isomorphisms,' establishing a clear bridge between the new construction and the known one. This process of symbolic verification is a powerful demonstration of AI's utility in advanced mathematics.

From Construction to Confirmation

Tao's conversation reveals a step-by-step verification process. He first had ChatGPT confirm that a polynomial map, Phi, from 3D space to a 5D algebraic variety, X, was an isomorphism. The AI performed a direct expansion of the defining equations to prove the maps were inverses, explicitly showing that there were no hidden singularities.

Tao then instructed the AI to compose this map with another polynomial, F, and drop a constant coordinate. The result was a new polynomial map, G, from 3D space to itself. A direct symbolic differentiation, performed by the AI, showed that the Jacobian determinant of G was identically -1, a constant. This confirmed that the construction produced the desired 'Keller map'—a polynomial map with constant, non-zero Jacobian.

Connecting the Dots

The most critical step was Tao's final question: he asked ChatGPT to connect the newly derived map G to the original counterexample formula. The AI identified the relationship with precision, showing that the original counterexample was simply a linear transformation of G. By applying a coordinate substitution and output rescaling, the AI demonstrated that the two maps were identical up to these elementary normalizations.

This verification is significant for two reasons. First, it confirms the counterexample is correct, as it aligns with a known, rigorous construction. Second, it illustrates a new workflow: an AI discovers a candidate, and a human mathematician uses another AI to verify, contextualize, and explain the result, creating a transparent and verifiable chain of reasoning.

The Leiden Manifesto and a Paradigm Crisis

This breakthrough is not happening in a vacuum. It follows a string of AI-driven mathematical discoveries, including OpenAI's disproof of the Erdős Unit Distance Conjecture in May 2026. These events have prompted a profound crisis within the mathematics community, culminating in the 'Leiden AI and Mathematics Manifesto' on June 2, 2026.

The manifesto, signed by Terence Tao and Peter Scholze, among others, warns that AI could 'collapse the peer-review system, create chaos around attribution, and enable tech companies to potentially manipulate the direction of mathematical research.' It calls for mathematicians to retain the right to decide 'what mathematics is worth doing,' even as they embrace AI as a tool.

Why This Matters

The Jacobian Conjecture counterexample is more than just a solved problem; it is a symbol of a changing world. The speed and ease with which an AI model, used during a break from watching the World Cup, overturned a central conjecture of the 20th century underscores the transformative power of these tools.

The role of the human mathematician is evolving from discoverer to validator and sense-maker. As Tao's ChatGPT conversation shows, the future of mathematics may be a deeply collaborative one, where humans and AIs work together to explore the vast, complex landscapes of abstract thought. The 'last line of defense' remains human judgment, but the front line is now undeniably digital.