For nearly nine decades, the Jacobian Conjecture stood as one of the most famous unsolved problems in algebraic geometry.

First proposed by German mathematician Ott-Heinrich Keller in 1939, the conjecture appeared deceptively simple. Despite thousands of research papers and decades of effort by some of the world's leading mathematicians, nobody had been able to either prove or disprove it.

That changed in July 2026.

Mathematician Levent Alpöge, working with Anthropic's frontier AI model Claude Fable 5, announced an elegant counterexample showing that the conjecture is false in three dimensions and higher. The result spread rapidly through the mathematics community, where independent researchers quickly verified the construction by hand and with computer-assisted tools.

What Is the Jacobian Conjecture?

The Jacobian Conjecture belongs to algebraic geometry, a branch of mathematics that studies polynomial equations and the shapes they describe.

In simple terms, imagine transforming every point on a sheet of graph paper using a mathematical rule.

If every tiny neighborhood behaves perfectly with no folding, tearing, or collapsing the Jacobian Conjecture asked a natural question:

Does that guarantee the entire transformation can always be reversed using another polynomial function?

More formally, the conjecture states:

Every polynomial map whose Jacobian determinant is a non-zero constant should possess a polynomial inverse.

Although the statement is easy to understand, proving or disproving it resisted mathematicians for 87 years.

Why Was It So Difficult?

Most mathematicians believed the conjecture was probably true.

Local behavior often predicts global behavior in mathematics, making Keller's statement appear intuitive.

Over the decades:

• Hundreds of partial proofs were published.
• Many claimed complete solutions.
• Small but critical errors were eventually discovered in each one.

The Jacobian Conjecture eventually became one of the problems listed by mathematician Stephen Smale among the important mathematical challenges for the 21st century.

How the Breakthrough Happened

Rather than attempting to prove the conjecture true, Alpöge explored the opposite possibility.

Working with Claude Fable 5, he searched for a counterexample.

In mathematics, finding just one valid counterexample is enough to disprove a universal statement.

Instead of constructing a lengthy proof, the search focused on exploring an enormous space of polynomial maps that would be extremely difficult for humans to examine manually.

The result was surprisingly elegant.

Claude Fable 5 helped identify a polynomial map that satisfied every assumption of Keller's conjecture while still failing to be invertible.

The Counterexample

The published map operates in three-dimensional complex space:

It satisfies the key requirement of the conjecture:

• The Jacobian determinant is constantly −2 everywhere.

Yet the map sends three distinct input points to exactly the same output point.

Because multiple inputs produce the same output, the function is not one-to-one (injective).

Without injectivity, no inverse polynomial function can exist.

This single counterexample disproves the Jacobian Conjecture for dimensions three and higher.

What About Two Dimensions?

Interestingly, the original two-dimensional version remains unsolved.

The newly discovered counterexample only applies to dimensions:

• 3
• 4
• 5
• and all higher dimensions

Whether Keller's original statement remains true in two dimensions is still an open question for mathematicians.

Why AI Was Important

The discovery represents one of the most significant examples of AI-assisted mathematical research.

Rather than replacing human reasoning, Claude Fable 5 helped navigate an enormous search space of candidate polynomial mappings.

Many researchers believe this illustrates a new role for frontier AI systems:

• discovering mathematical objects,
• suggesting promising directions,
• generating counterexamples,
• assisting human mathematicians with difficult searches.

Importantly, the result is regarded as an AI-assisted discovery, with the mathematical verification ultimately performed by human researchers.

The Mathematics Community's Response

The reaction was immediate.

Within hours of the announcement:

• Independent mathematicians checked the Jacobian calculation.
• The polynomial map was verified computationally.
• Researchers reproduced the counterexample using symbolic mathematics software.
• Discussions rapidly spread across universities and mathematical research communities worldwide.

Several follow-up papers have already expanded upon the construction and generalized the counterexample to broader classes of polynomial maps.

Why This Discovery Matters

Although the Jacobian Conjecture may seem highly abstract, its influence extends throughout modern mathematics.

Its resolution affects research in:

• Algebraic geometry
• Polynomial mappings
• Complex analysis
• Dynamical systems
• Computational algebra

Beyond mathematics itself, the discovery demonstrates that advanced AI systems are becoming valuable collaborators in scientific research rather than simply tools for automation.

Frequently Asked Questions

Did AI solve the Jacobian Conjecture by itself?

No.

The breakthrough is best described as an AI-assisted mathematical discovery. Mathematician Levent Alpöge worked with Anthropic's Claude Fable 5 to discover the counterexample, while mathematicians independently verified the result afterward.

What exactly was discovered?

A polynomial map from ℂ³ to ℂ³ whose Jacobian determinant is constantly -2, yet maps three different input points to the same output, proving the map is not invertible.

Is the Jacobian Conjecture completely solved?

Not entirely.

The conjecture has been disproved for dimensions three and higher, but the two-dimensional case remains open.

Why is this important for AI?

The discovery highlights AI's growing ability to assist with high-level scientific research, particularly in exploring vast mathematical search spaces that are difficult for humans to investigate manually.

Final Thoughts

The Jacobian Conjecture resisted generations of mathematicians for nearly 87 years.

Its downfall came not through a massive proof spanning hundreds of pages, but through a remarkably elegant counterexample discovered by Levent Alpöge with assistance from Anthropic's Claude Fable 5.

The achievement is significant for two reasons. It resolves one of mathematics' most famous open problems in dimensions three and above, and it demonstrates that frontier AI systems are becoming powerful collaborators in mathematical discovery.

As AI continues to evolve, breakthroughs like this may become increasingly common, reshaping how mathematicians explore, test, and solve some of the world's most challenging problems.