AI Cracks 87-Year-Old Math Mystery: First Counterexample to Jacobian Conjecture Triggers Paradigm Crisis

In the summer of 2026, the mathematics community is experiencing an unprecedented storm. Following breakthroughs by OpenAI and DeepMind in May that cracked math problems that had baffled humanity for 80 years, on July 20, an Anthropic researcher used only its model Fable 5—while watching the World Cup on the side—to find a counterexample to one of mathematics' most notorious unsolved problems: the Jacobian Conjecture. First proposed by German mathematician Ott-Heinrich Keller in 1939 and later listed by Fields Medalist Steve Smale among the "Mathematical Problems for the 21st Century," this conjecture had seen mathematicians spend 87 years trying to prove it true. Now, AI has effortlessly torn a hole in it.

The dense barrage of "AI breakthroughs" has left the math community both stunned and on high alert. From overturning legendary mathematician Paul Erdős's Unit Distance Conjecture to autonomously solving problems that had remained open for 56 years, AI is assaulting the foundations of mathematical research at astonishing speed. Yet accompanying these breakthroughs is not just glory, but deep anxiety about the collapse of the peer review system, the distortion of academic values, and the loss of autonomy for human researchers.

Triple Strike: From Erdős to Jacobian

AI's accelerating breakthroughs in mathematics reached a crescendo in 2026.

On May 20, OpenAI struck first. An internal general-purpose reasoning model autonomously overturned the Unit Distance Conjecture, proposed by legendary Hungarian mathematician Paul Erdős in 1946. The problem is elegantly simple: given n points on a plane, what is the maximum number of point pairs that are exactly distance 1 apart? Erdős conjectured that a square grid arrangement was optimal. OpenAI's model proved this intuition wrong. Fields Medalist Timothy Gowers remarked, "No previously AI-generated proof came close to this level."

Just one day later, Google's DeepMind responded with an even bigger splash. Their purpose-built mathematical model, AlphaProof Nexus, autonomously solved nine open Erdős problems, two of which had remained unsolved for 56 years. More astonishingly, the reasoning cost for each problem was only a few hundred dollars.

While the world was still digesting these results, on July 20, Anthropic employee Levent Alpöge delivered an even more seismic announcement. He claimed that while watching the World Cup, he casually used Fable 5 to find a counterexample to the Jacobian Conjecture.

The Jacobian Conjecture itself is not difficult to state: if a polynomial mapping from n-dimensional space to itself has a Jacobian determinant that is identically equal to some non-zero constant, then the mapping must be one-to-one, and its inverse mapping must also be polynomial. For 87 years, mathematicians tried to prove this statement was always true. The counterexample Alpöge published consists of just three lines—a polynomial mapping in three variables:

a = (1+xy)^3 z + y^2 (1+xy)(4+3xy)
b = y + 3x(1+xy)^2 z + 3xy^2 (4+3xy)
c = 2x - 3x^2 y - x^3 z

The Jacobian determinant of this set of polynomials is calculated to be identically −2, fully satisfying the conjecture's premise; yet the mapping (a,b,c) is generally three-to-one, not one-to-one—the premise holds, but the conclusion fails, and the conjecture is thereby overturned. After Columbia University mathematician David Speyer posted these equations on the math blog Secret Blogging Seminar, researchers worldwide completed independent verification within hours and began discussing them in public threads, reducing them to standard form and exploring their structure.

The uproar over this result stems not only from the Jacobian Conjecture being a core problem in algebraic geometry, listed by Smale as one of 18 major problems for the 21st century, but also from its connection to the tumultuous academic career of Chinese mathematician Zhang Yitang. Zhang's doctoral dissertation topic was related to the Jacobian Conjecture, and the difficulties with that thesis indirectly led to his years of working odd jobs to survive.

Li Xinyi, an associate professor at Peking University's International Center for Mathematical Research, analyzed that the sensational impact of these AI results is no accident. OpenAI's math team clearly invested substantial time carefully selecting, from a vast pool of unsolved problems, those most likely to yield results and simple enough for the public to understand. "They even had to consider that the proof couldn't be too long—mathematicians shouldn't need half a year to verify it," he said. The Jacobian Conjecture counterexample proposed by Anthropic is so concise—just a few lines of equations—that it is extremely easy for other researchers to verify.

Result	Proposer & Year	Years Open	AI's Role	Verification Method
Unit Distance Conjecture overturned	Paul Erdős, 1946	80 years	General reasoning model autonomously generated complete proof	9 top mathematicians manually reviewed and improved the proof
9 Erdős problems solved	2 open for 56 years	Up to 56 years	AlphaProof Nexus solved autonomously	Self-verified during reasoning using Lean formal language
Jacobian Conjecture counterexample found	Ott-Heinrich Keller, 1939	87 years	Fable 5 found three-line polynomial counterexample	Counterexample extremely simple; independently verified by mathematicians worldwide within hours of release

From "Super Literature Search" to Autonomous Breakthroughs

These three breakthroughs in 2026 are fundamentally different from what came before.

Looking back at the evolution of AI mathematical research, the magnitude of this leap becomes clear. In July 2024, DeepMind's AlphaProof solved two algebra problems and one number theory problem at the International Mathematical Olympiad; combined with AlphaGeometry 2's geometry solutions, the two systems together solved 4 out of 6 problems and scored 28 out of 42 points, reaching silver medal level—just one point shy of the 29-point gold threshold. But those problems all had known answers. In October 2025, OpenAI claimed GPT-5 had solved 10 Erdős problems, but within hours this was debunked as "super literature search"—it had merely found previously obscure published papers. In late 2025, DeepMind's deployed agent Aletheia correctly solved 13 out of 700 Erdős problems, but 9 of those were again retrieval successes; the other 4 new solutions mostly involved obscure problems, some of which may never have been seriously attempted by experts.

The 2026 results represent a qualitative leap, first in the weight of the problems. The Unit Distance Conjecture overturned by OpenAI is central and well-known; Erdős himself offered a bounty for it, yet it remained unsolved for decades. The Jacobian Conjecture cracked by Anthropic is a cornerstone problem in algebraic geometry. Although AI has so far only found a counterexample in three dimensions—and whether the conjecture holds in two dimensions remains unknown—multiple mathematicians consider this a major breakthrough.

More crucially, AI's autonomy has dramatically increased. Previously, AI solving math problems required mathematicians to manually filter and correct minor errors. OpenAI's May breakthrough achieved fully autonomous generation of an entire proof by the model. Google's AlphaProof Nexus went further, using a formal language called Lean to autonomously verify correctness, eliminating the need for mathematicians to serve as "quality inspectors."

Li Xinyi used a vivid metaphor to explain AI's current capability boundaries: "If human mathematical knowledge can be likened to a polygon with concave and convex sections, with the exterior being the unknown, then a first-rate mathematician might push the boundary far outward at one sharp corner; a great mathematician like Hilbert might advance far in several directions; and the vast majority of mathematical workers spend most of their time filling in the concave parts. At this stage, the 'headline-making' math AIs are merely turning knowledge into its convex hull, rather than exploring further into unknown directions."

AI is not yet smarter than humans, but it has no disciplinary silos or authority bias. It proved Erdős was wrong and that the Jacobian Conjecture has a counterexample, rather than following the psychological inertia of most people trying to prove the conjecture correct. It is also more patient than those who tried to find counterexamples, willing to attempt repeatedly. Moreover, AI's advantage as a "generalist" humbles human researchers. The unit distance problem solved by OpenAI belongs to geometry, yet the key to success was introducing a seemingly unrelated tool from number theory in algebra. In the context of modern mathematics' extreme specialization, researchers in combinatorial geometry typically do not delve deeply into number theory—the distance between these two fields is like asking a cardiac surgeon to think of using immunological methods to treat disease.

The Leiden Manifesto: Defending the Right to Decide "What Mathematics Is Worth Doing"

Faced with AI's surging assault, the mathematics community has begun collective reflection.

This manifesto was not improvised. Its origins trace to a workshop held at Leiden University's Lorentz Center in September 2025, where approximately 60 researchers and policymakers gathered to discuss where mathematics would go once AI became heavily involved in generating proofs. On June 2, 2026, the Leiden AI and Mathematics Manifesto was formally released, led by Jim Portegies of Eindhoven University of Technology and co-authored by 16 mathematicians, including Ursula Martin of Oxford University, Rodrigo Ochigame of Leiden University, and Michael Harris of Columbia University.

The manifesto received formal endorsement from the International Mathematical Union (IMU), with Fields Medalist Peter Scholze, Imperial College London's Kevin Buzzard, Oxford's Leslie Ann Goldberg, and IMU Vice President Ulrike Tillmann among its named supporters. Terence Tao publicly expressed "great admiration" for the manifesto and gave it his full backing, particularly endorsing its call for mathematicians to participate in public discourse. Nature magazine published an editorial on June 18 affirming the manifesto's formation process and conclusions. The manifesto does not oppose the use of AI and encourages mathematicians to understand emerging technologies, but it issues a stern warning about the risk that AI could undermine the fundamental values of the mathematics community.

The most immediate problem is the "indigestion" of the peer review system. AI can "complete" a proof in hours, but seriously reviewing it requires weeks or even longer. The pool of mathematicians qualified to referee is extremely limited, and a flood of AI-generated results will cause a deluge of proofs that "look correct" but may contain hidden errors to pour into academia. The manifesto bluntly states that corporate press releases operate on "the market's timetable" and cannot wait for the mathematics community to complete its established evaluation processes. This concern was almost immediately validated: just days earlier, OpenAI, at the launch of GPT-5.6 Sol, announced via press release that it had solved a 50-year-old math problem, but the result had not been confirmed by external mathematics experts.

Even the much-praised Unit Distance Conjecture result drew concerns about information asymmetry. Oxford's Ursula Martin, while calling the result "remarkable," cautioned: "We are not told how many times the model failed. If you threw massive amounts of human effort at this problem, you would likely solve it in the same way. But in mathematics, human effort is a scarce resource, and it is often spent elsewhere." Leiden University's Rodrigo Ochigame pointed out that the information needed to evaluate this result—"methods, human-written prompts, training data, computational resources consumed"—was not disclosed.

Another thorny issue is attribution and copyright. Mathematical AI is built on a vast body of prior research, yet its outputs often fail to properly credit contributors. The manifesto criticizes that "models trained on published works frequently produce output without appropriate citation of the human work it synthesizes," and that some training data is obtained by "exploiting loopholes in licensing and access arrangements" or even "directly violating copyright protections." Mathematician Melanie Matchett Wood expressed excitement about OpenAI's result while also noting that the company did not properly cite "the history of a series of closely related ideas" in the literature. A deeper question looms: if mathematicians collaborate with AI to produce new mathematics, does the result belong to the human or the AI?

The manifesto also reveals a more insidious threat: tech companies could begin to influence the direction of the mathematics community. It writes: "The involvement of tech companies in research brings the risk that research problems are prioritized and incentivized because they suit AI methods and models, rather than because they have deeper significance for understanding itself." Against the backdrop of tight university budgets, this could alter incentive structures, implicitly encouraging researchers to enter into unequal collaborations with tech companies; researchers without access to these AI tools would be at a disadvantage. If left unchecked, this could undermine existing mechanisms for hiring, funding, and recognition, threaten researchers' autonomy, and even affect the scope and depth of mathematical research itself.

The value of mathematics is not just about getting more answers. It also includes understanding why answers hold, judging which questions matter, and discovering new research directions from old knowledge. These capacities are formed by the mathematics community through long-term discussion, debate, and transmission, and are difficult to measure by "how many problems were solved."

Thus, the manifesto demands that researchers disclose their use of AI and computational resources, uphold peer review, have human authors take responsibility for correctness and citations, and build public computational infrastructure independent of commercial companies so that academia can compete with for-profit enterprises on equal footing. Michael Harris of Columbia University, one of the co-authors, put the goal more bluntly: "From my perspective, the aim is to wrest back control of the narrative about the value and purpose of mathematics from the AI industry." What mathematicians want to defend is not just their jobs; they want to defend the right to decide "what mathematics is worth doing." Portegies will formally present the manifesto at the International Congress of Mathematicians (ICM) in July 2026.

Humans Still Present: The Last Line of Defense in Verification and Interpretation

Despite AI's growing autonomy in proof generation, human mathematicians still play an irreplaceable role in verification and interpretation.

When OpenAI overturned the Unit Distance Conjecture, it used a general-purpose reasoning model that generated an extremely long chain of thought in natural language. But natural language reasoning has an inherent weakness: the model might make a small error at some step in the reasoning chain, ultimately leading to a major mistake. To address this, OpenAI brought in nine top mathematicians to manually review the proof. These mathematicians not only verified its correctness but also substantially improved the original proof, making it more concise and general. Thomas Bloom, maintainer of the Erdős problem database, commented: "Humans still play a key role in discussing, digesting, improving this proof, and exploring its consequences."

Google's AlphaProof Nexus took a different path, relying on the Lean formal language to self-verify correctness during the reasoning process, with humans only responsible for confirming the mathematical significance of the final result. However, this path still faces significant obstacles. Lean is rigorous but verbose, making it difficult for humans to directly write or read; it requires "translation." And the foundational work of translating existing mathematical theorems into Lean is far from complete.

The details of Anthropic Fable 5's work have not yet been disclosed, and Alpöge initially announced the result not through a paper but on social media. But precisely because the counterexample is so concise, other mathematical researchers could easily verify it, and the discussion thread soon saw analyses of the counterexample's structure and attempts at generalization. This precisely illustrates that even when AI produces a result, it is ultimately human judgment and consensus that "validate" it.

Li Xinyi noted that if AI were to one day master the ability to explore in unknown directions, "that might indeed truly represent a comprehensive surpassing of humans." But at least for now, AI is still only "taking the convex hull" within the boundaries of known knowledge, while human mathematicians still stand at the frontier, deciding whether the next question is worth asking.