Key Points
・A conjecture that had stood unresolved for 87 years, the Jacobian conjecture, was disproved by a counterexample generated by an AI model, and mathematicians around the world independently checked the computation within hours.
・Claims about AI and mathematics changed in character after an overstatement episode in the autumn of 2025, and results that survive expert checking and formal verification are now appearing, though peer review and disclosure about the AI’s contribution still vary case by case.
・AI has begun producing counterexamples and proof ideas for problems humans failed to crack, but selecting the problem and certifying the result remain human work, which puts the division of labor inside discovery itself in question.
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A Conjecture of 87 Years Fell to One Counterexample
The mathematician Levent Alpöge posted a short message on X on the night of July 19 US time, in the small hours of July 20 Japan time, stating in effect that the Jacobian conjecture is false. The post went out during the World Cup final between Spain and Argentina and has been reported to have passed 20 million views.
What was presented was not a resolution of the conjecture but a refutation by counterexample. A single polynomial map in three complex dimensions was constructed, and it was confirmed that three distinct points map to the same point even though the Jacobian determinant is constant. That makes the conjecture false in every dimension of three or above. The two-dimensional case, which was the original object of interest, remains open, and as of July no peer-reviewed paper had appeared.
The roles were divided. The problem was chosen by Professor Akhil Mathew of the University of Chicago, who reasoned that AI might be able to produce a counterexample in algebraic geometry. The counterexample was generated by Anthropic’s AI model Claude Fable 5, and it was Alpöge, a number theorist, who verified it and made it public. The number of attempts made before the counterexample emerged, and the prompts used, have not been published.
Because a counterexample can be checked by substituting values and computing, mathematicians around the world verified it independently within hours. Paul Lezeau of Imperial College London formalized it by hand in Lean and submitted it to Google DeepMind’s repository on July 20, where it was merged on the 26th. With the checking and the formalization done, attention has moved to writing the counterexample up as a paper and taking it through peer review.
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Eighty-Seven Years of the Conjecture, and Two Axes for Measuring AI Results
Hard to Prove, Easy to Disprove
The Jacobian conjecture has an asymmetric shape. Proving it is extremely difficult, while a refutation of it can be checked by anyone.
The conjecture concerns transformations of space written as polynomials. Picture stretching a rubber sheet. If, wherever you look and however closely you zoom in, the sheet is never crushed flat or turned inside out, then the sheet as a whole should have no folds and the transformation should be exactly reversible. That is what the conjecture asserts.
Whether the sheet is crushed at a given point is decided by whether a quantity called the Jacobian determinant vanishes there. The German mathematician Ott-Heinrich Keller raised the question in 1939. The name Jacobian conjecture first appeared in a 1973 paper by Masayoshi Miyanishi and was popularized by Abhyankar.
The difficulty lies in the breadth of the target. The statement has to hold without exception across the infinite family of all polynomial maps in a given dimension, and partial progress never reached the whole. Stephen Smale included it as the sixteenth of his 1998 list of mathematical problems for the next century, and more than five erroneous proofs have been published over the years.
Refutation works differently. The counterexample the AI produced satisfies the condition of never being crushed at any point, and yet three separate points land on the same single point. Everything looks fine under local inspection, while the map folds over itself globally.
Whether a candidate counterexample really satisfies the conditions can be settled by plugging in values and computing. That asymmetry between proof and refutation is why mathematicians around the world could confirm it independently within hours of publication.
From an Overstatement Episode to Results That Survive Checking
Claims about AI and mathematics have shifted in quality since the autumn of 2025, moving from overstatement toward results that pass through the hands of mathematicians.
The Erdős problems are the large body of open questions left behind by the Hungarian mathematician Paul Erdős, who lived from 1913 to 1996. The mathematician Thomas Bloom runs a database site, erdosproblems.com, which collects them and marks each one as solved or open.
On October 18, 2025, Kevin Weil, a vice president at OpenAI, posted on X that GPT-5 had solved ten open problems in that database. On inspection, what GPT-5 had done was not to produce new proofs but to locate already published papers containing the solutions.
The site’s “open” marking was never a guarantee that nobody in the world had solved the problem. It meant that Bloom, who maintains the site, was not aware of a solution. A literature search had been announced as the resolution of open problems. Bloom called this a dramatic misrepresentation, and Weil deleted the post.
Demis Hassabis of the rival lab DeepMind was reported to have called the announcement embarrassing. Even so, the literature search was not worth nothing, and the contribution is recorded on the Erdős problems wiki.
In 2026, results of a different character began to appear. The unit distance conjecture, which OpenAI announced it had refuted on May 20, is a 1946 question about how many pairs of points at distance exactly one can be produced when n points are placed in the plane.
Erdős conjectured that no arrangement of points could make the number of such pairs very large. OpenAI’s internal model constructed arrangements with more pairs than the conjectured upper bound, refuting it. Nine mathematicians took part in verifying and simplifying the argument, and a joint paper was released as a preprint.
One of those coauthors was Bloom, the same person who had criticized the overstatement the year before. The critic of 2025 became a verifying coauthor in 2026.
Two Axes for Measuring What “AI Solved It” Means
The phrase “AI solved it” becomes legible when split along two axes: which part of the work the AI did, and how far the result has been certified.
| Case | Date | Years open | AI’s role | Result | Verification stage |
|---|---|---|---|---|---|
| GPT-5 “ten problems” episode | October 2025 | (solutions already existed) | Literature search only | Not new mathematics | Retracted |
| OpenAI, unit distance conjecture | May 2026 | 80 years | Independently generated a counterexample construction | Conjecture refuted | Checked by 9 mathematicians, preprint |
| DeepMind, Erdős problems | May 2026 | Up to 56 years | Autonomous proof search | 9 complete proofs out of 353 problems | Formally verified in Lean |
| Liam Price, problem 1196 | April 2026 | 60 years | Supplied the idea for the method | Conjecture resolved | Refined by experts, Lean, joint paper |
| Jacobian conjecture | July 2026 | 87 years | Generated the counterexample (problem chosen by a human) | Refuted in dimension 3 and above, dimension 2 open | Checked by mathematicians and formalized in Lean, peer review pending |
Based on company announcements, arXiv preprints, and entries on the Erdős problems wiki. As of July 2026.
The first axis is which stage of the work the AI handled. Searching existing literature, producing an idea or a counterexample for a problem a human posed, and running the whole loop from choosing the problem to searching for a proof are very different things. In the Jacobian case, the decision about which target to aim at came from the human side.
The second axis is how far the result has been certified. A generated candidate, a candidate checked by experts, a peer-reviewed paper, and a statement formally verified in the proof assistant Lean carry different weights. This counterexample has reached checking and Lean formalization, and has not yet been through peer review.
Lean is a language for writing mathematical statements and proofs in a form a computer can read. A proof is written as a strict sequence of symbols, and the machine inspects it step by step. Any logical gap fails to compile, so compiling is itself evidence of correctness.
The difference from human refereeing is that the check does not depend on a reader’s understanding or attention. Output that looks plausible but contains an error, which is exactly the failure mode of AI-generated proofs, can be filtered out mechanically from the outside. Mathematicians worldwide have collectively built up libraries of existing theorems written in Lean.
Lean’s role does need a qualification. Within the range of the formalized statement and its assumptions, the proof steps can be checked mechanically. Whether the translation of the original mathematical claim into a formal statement is faithful is something people establish outside Lean. Here too, the formalization was done by a human mathematician.
Is Discovery Becoming Machine Work?
Creativity in this article means something narrow: the power to produce new, checkable solutions or counterexamples that are not reconstructions of solutions already known. The power to choose which questions matter, and the power to invent new concepts, remain separate kinds of creativity. Within that limit, here is what has been happening in mathematics in 2026.
Cheap Verification Separates AI’s Errors From Its Results
Mathematics, long treated as the last redoubt of creativity, moved first not because insight is easy there but because answers are cheap to check.
Mathematical problems come with two conditions that other fields lack. Questions can be defined precisely, and there is a mechanical way to confirm whether an answer is correct. A counterexample can be checked by substitution, and a proof can be checked by formal verification in Lean.
AI output contains errors. In fields where checking is cheap, those errors can be pruned away from the outside. Generation can be unreliable as long as verification is solid, and mathematics happened to be one of the fields where generated answers are easy to verify rigorously.
None of this is the AI’s achievement alone. People stated the problems precisely, accumulated the prior work, and built the verification machinery. In this case a human, Professor Mathew, decided which problem to aim at. What is in motion is not AI against humans but an entire discovery apparatus combining mathematicians, AI, and verification systems.
Discovery Gets a Price Tag, and Checkers Become the Scarce Resource
Part of discovery is already being described in the language of cost and workflow.
In DeepMind’s May announcement, agents attacked 353 Erdős problems and produced complete proofs for nine. The company’s paper puts the inference cost at a few hundred dollars for each problem successfully solved. That figure excludes failed searches and development costs, but the fact that the marginal cost of a discovery is now an estimable number is itself new.
Once the cost is estimable, usage changes. Having an AI hunt for counterexamples to your own conjecture before you publish becomes easier to build into ordinary research routine as the price falls. Discovery moves from a rare event toward a repeatable process.
Industrialization has a limit that is easy to miss. Nine out of 353 is a hit rate of 2.5 percent, and what grows is not only the set of solutions but the pile of candidates waiting to be checked. Cheaper generation does not create more expert hours for deciding which results matter. The faster generation gets, the scarcer verification and judgment become.
The Entrance Widens While the Foundation Concentrates
The pool of users widens and ownership of the foundation concentrates. These are not opposite movements; they are happening at once.
The widening is visible in the case of Liam Price, aged 23. Without graduate training, he fed the text of an Erdős problem that had been open for 60 years into GPT-5.4 Pro, and the model produced the skeleton of a proof. The entrance to trying AI on open problems is now open beyond specialists. Still, turning rough output into mathematics was done by experts, so it is not the research capability itself that has spread.
That widening rests on corporate infrastructure. The model’s internals, the compute available for inference, and the record of how many attempts were made all sit on the company’s side. In this case as well, the exchanges that led to the counterexample have not been made public. Democratizing the entrance also means widening access to a concentrated foundation.
A path that preserves both advantages is starting to appear. The formal verification of this counterexample was taken into Google DeepMind’s public repository, an example of publishing results and their checks in a form anyone can inspect. Whether search records keep being published, and whether compute keeps being provided to research institutions, will indicate how unevenly the means of discovery are distributed.
Choosing the Question and Judging the Meaning Are Not Automated Yet
What can be stated confidently is this much: on problems humans had failed to solve for a long time, discoveries originating with AI have begun to pass verification.
Saying that AI has surpassed humans would require comparisons on the same problems under the same conditions, and that material does not exist. Neither the number of attempts nor the degree of human steering has been published. As the 2025 episode showed, assessments can be revised years later.
Seen from the other side, what was confirmed here is a case in which humans chose a problem that was precisely defined and cheap to check, and experts finished off candidates produced by a large number of attempts. From that position, there is a solid argument that it is too early to conclude AI has acquired mathematical creativity.
Expert assessments vary in range. Some hold that the practical use of AI stops at assisting literature searches; others read this as a turning point for mathematical research. The novelty of producing a counterexample and the capacity to advance mathematics autonomously need to be measured separately.
Both ends of the workflow remain in human hands. A mathematician chose the target in the Jacobian case, and mathematicians certified the result as mathematics. Kevin Buzzard, a central figure in the Lean community, welcomed the day as a big one while noting that the questions AI poses for itself are either boring or trivial, a remark about the human role in choosing what to ask.
The change in the middle of the workflow is real all the same. Timothy Gowers, a Fields medalist, indicated that this was the first time AI had solved a problem large enough for him to know its name. Now that the middle stages of discovery are no longer a human monopoly, the human role shifts toward setting problems, verifying, and judging meaning. That rearrangement is starting in mathematics.
As Discoveries Multiply, the Question Grows Heavier
The Human Role Is Not Disappearing, It Is Moving
The industrial revolution moved muscular work to machines, and the information revolution moved calculation to machines. What is happening in mathematics in 2026 may be an instance of part of discovery moving to machines.
None of this is settled in one direction. Collaboration between AI and mathematicians becoming standard is one path, mathematics aimed at human understanding surviving as a separate value is another, and further progress in autonomous discovery is a third. In this case at least, the stage that decides what is worth asking was carried by people. How long that arrangement holds is the next thing to watch.
In an era when generating candidate discoveries gets cheaper and attempts get faster, what stays scarce is the ability to choose the question and judge what a result means. The news that an 87-year-old conjecture fell overnight leaves us with the question of how that ability is cultivated and who holds it.
Reference Links
- A digestion of the Jacobian conjecture counterexample｜What’s new (Terence Tao)
- Human mathematicians are being outcounterexampled｜Xena Project (Kevin Buzzard)
- Mathematicians grapple with a ‘very rapid and very unsettling change’ as AI cracks yet another century-old problem｜Fortune
- ‘hello there the jacobian conjecture is false thanx’: why a tiny social media post has mathematicians rethinking AI｜The Conversation
- Remarks on the disproof of the unit distance conjecture｜arXiv
- Advancing Mathematics Research with AI-Driven Formal Proof Search｜arXiv
- Amateur armed with ChatGPT ‘vibe maths’ a 60-year-old problem｜Scientific American
- OpenAI claims it solved an 80-year-old math problem, for real this time｜TechCrunch
