An AI model has produced original mathematics that working mathematicians accepted within hours. On July 20, 2026, Levent Alpoge, a number theorist at Anthropic, posted that Claude Fable 5 had found a counterexample to the Jacobian Conjecture, a problem in algebraic geometry that had resisted proof since Ott-Heinrich Keller posed it in 1939. The counterexample is a polynomial map in three complex dimensions, degree 7, whose Jacobian determinant is the constant -2. The conjecture says such a map must be invertible. This one is not: three different input points land on exactly the same output.

  • The counterexample is small enough to check by hand. Substituting the three points takes about a minute of arithmetic, which is why the result was accepted informally within hours rather than months.
  • The three colliding points are (0, 0, -1/4), (1, -3/2, 13/2) and (-1, 3/2, 13/2), all of which map to (-1/4, 0, 0). A map that sends three inputs to one output cannot be inverted.
  • Verification and discovery are not symmetric. Disproving a universal claim needs one object; proving it needs an argument covering infinitely many cases. That asymmetry, not raw model intelligence, is what made this result land so fast.
  • It is not peer reviewed and there is no preprint yet. The official record still lists the Jacobian Conjecture as open, and the prompt and reasoning trace behind the discovery have not been published.
What the Jacobian Conjecture claims and where the counterexample breaks itA polynomial map from complex 3-space to itself with constant nonzero Jacobian determinant was conjectured to be invertible. The new counterexample satisfies the hypothesis but fails the conclusion. THE CLAIM, AND THE BREAK Hypothesis F : C³ → C³ polynomial det JF = -2 (constant) Keller, 1939 predicts F is invertible 2026 counterexample F is not injective so it cannot be inverted Conjecture falsified WHY ONE OBJECT IS ENOUGH To prove the conjecture Cover every polynomial map, every degree, every dimension. 87 years, open. To disprove it Exhibit one map that breaks it. Checking is arithmetic, not argument. genztech.blog
Fig 1 The Jacobian Conjecture predicted invertibility from a constant nonzero determinant. The counterexample satisfies the hypothesis and fails the conclusion.

What exactly did Alpoge and Fable 5 find?

The Jacobian Conjecture is easy to state and famously hard to settle. Take a polynomial map F that sends complex 3-space to itself. Compute its Jacobian, the matrix of first partial derivatives, and take the determinant. If that determinant is a nonzero constant, the map is locally invertible everywhere, because the derivative never degenerates. Keller asked in 1939 whether local invertibility everywhere forces global invertibility. In plain terms: if you can always undo the map in a small neighborhood, can you always undo it outright?

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Alpoge reported that Fable 5 produced a map where the answer is no. It is degree 7, it lives in three dimensions, and its Jacobian determinant is the constant -2, which is nonzero and therefore satisfies the hypothesis exactly. Yet the points (0, 0, -1/4), (1, -3/2, 13/2) and (-1, 3/2, 13/2) all map to (-1/4, 0, 0). Three inputs, one output. A function that collapses distinct points has no inverse, because the inverse would not know which point to return.

Mathematicians reading the announcement pointed out the underlying shape is natural rather than arbitrary: it relates to a map between symmetric products of the projective line, restricted to affine space. That matters, because it suggests the construction was found by reasoning about structure rather than by blind search over coefficients.

Why did a 1939 problem fall this fast?

Because the community was looking for a proof, and the answer turned out to be a counterexample. Those are different searches. A proof of the Jacobian Conjecture has to hold for every polynomial map in every dimension, which is why decades of partial results reduced the problem, restricted it to special degrees, and still never closed it. A counterexample only has to be one object that breaks the pattern.

This is the part most coverage skipped, and it is the reason to take the result seriously despite the absence of peer review. Checking a claimed disproof is arithmetic. You substitute the three points, confirm they produce the same output, expand the determinant, and confirm it is constant. Undergraduate calculus and about a minute get you there. Several independent readers, and several other AI models, ran that check and agreed. Kevin Buzzard's Xena Project blog reported that Paul Lezeau went further and formalized the statement in Lean, the proof assistant, which turns informal agreement into a machine-checked fact.

Three distinct input points map to a single output pointDiagram showing the three colliding points of the counterexample converging on one image point, which proves the map is not injective. THE COLLISION · DOMAIN C³ → IMAGE C³ Three distinct inputs (0, 0, -1/4) (1, -3/2, 13/2) (-1, 3/2, 13/2) F One output (-1/4, 0, 0) No inverse can exist: it would have to return three answers at once. Degree 7 · dimension 3 · det JF = -2, constant and nonzero, so the hypothesis holds genztech.blog
Fig 2 The whole disproof in one picture. Three distinct points in the domain share an image, so the map cannot be inverted.

How does this compare to other AI math claims?

2026 has produced a run of announcements about models contributing to research mathematics, and they are not all the same kind of claim. The distinction that matters is whether a human can cheaply check the output.

PropertyFable 5, JacobianModel-assisted proofsBenchmark scores
What is producedOne explicit objectA multi-step argumentA number
Cost to verifyRoughly a minuteWeeks to months of reviewDepends on the harness
Failure modeArithmetic slip, caught instantlySubtle gap deep in the argumentContamination or scaffolding
Formalizable nowYes, done in LeanRarely, and slowlyNot applicable
Peer reviewedNot yetSometimesNo

This result sits in the most trustworthy category available today. That is a real distinction, and it is worth holding onto when the next announcement arrives without a checkable artifact attached.

What does it mean for Anthropic and the model race?

Anthropic is private, so there is no ticker to watch, but the signal for investors is still legible. Frontier labs are trying to justify enormous capital raises by arguing their models generate value beyond chat and code completion. A checkable contribution to open mathematics is the cleanest evidence yet offered for that argument, and it arrived from an in-house researcher rather than a marketing team.

The caution is that one result is not a rate. Fable 5 currently sits second on our AI coding leaderboard at 95.0% on SWE-bench Verified, close enough to the leader that the gap is inside the error bars. Nothing about this counterexample changes that ranking, and nobody has shown that models produce research-grade mathematics reliably rather than occasionally. The publicly traded read-through is indirect: it strengthens the case that inference demand keeps climbing, which is the thesis under Nvidia, Broadcom and the hyperscaler capex cycle. What a savvy reader should watch is whether a second lab reproduces this kind of result on a different open problem within the next few months.

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What are the caveats?

Several, and they are worth stating plainly. There is no preprint and no journal submission, so the official record still lists the conjecture as open. The prompt and the reasoning trace have not been released, which means nobody outside can judge how much of the construction came from the model and how much came from an expert number theorist steering it. Alpoge is an Anthropic employee announcing a result about an Anthropic model, and that is a conflict worth naming even when the underlying math checks out. Commenters also raised the possibility that related constructions exist in older literature, which review will settle.

None of that undermines the mathematics. The map either collapses those three points or it does not, and it does. But the interesting question was never whether the arithmetic is right. It is how much credit the model deserves, and that question stays open until the transcript does not.

  1. 1939Ott-Heinrich Keller poses the conjecture Local invertibility everywhere implies global invertibility
  2. 1939 to 2026Decades of partial results, no resolution Reductions to special degrees, several retracted proofs
  3. Jul 20, 2026Alpoge posts the counterexample Degree 7 in three dimensions, det -2
  4. Jul 20, 2026Independent readers verify the arithmetic Formalized in Lean per the Xena Project
  5. NextPreprint and peer review Prior-art check and the release of the reasoning trace

Our take

The headline most outlets reached for was that AI solved an 87-year-old problem. The more accurate and more interesting story is that AI produced an object, and the object survived contact with people who wanted to check it. That distinction is the whole game. Language models have been generating confident mathematical claims for years, and the reason almost none of them mattered is that verifying a long argument costs an expert weeks they will not spend on a machine's say-so.

Counterexamples dodge that problem entirely. They are the one shape of mathematical output where a model's lack of reliability barely matters, because the artifact carries its own proof. If you want to predict where AI contributes to research mathematics next, do not look for the hardest problems. Look for the problems whose answers are cheap to check. That is where the next few results will come from, and it is a narrower and more useful prediction than the one the headlines are making.

What to watch · 2026 to 2027
  • A preprint with the reasoning trace. Until the prompt and transcript are public, the size of the model's contribution is an open question, not a settled one.
  • A second result from a different lab. One counterexample is an anecdote. Two, on unrelated open problems, would be a capability.
  • Lean as the default receipt. Formalization turned informal agreement into a machine-checked fact within a day. Expect that to become the standard demand for any AI math claim.
  • Prior-art review. If a related construction turns up in older literature, the story shifts from discovery to rediscovery, which is a materially different claim.
Primary sources

Original analysis by GenZTech. Reporting based on Levent Alpoge's public announcement of July 20, 2026 and independent technical discussion.