Fable 5 Disproved What Algebraists Spent 87 Years Trying to Prove

Levent Alpöge posted the announcement in the most casual language possible for a mathematical event of this magnitude. “hello there the jacobian conjecture is false,” he wrote on X at around 2 AM UTC on July 20, during the World Cup Final, thanking “my close friend akhil for asking about it and my other close friend fable.” Alpöge is a number theorist at Anthropic and a former Junior Fellow at Harvard’s Society of Fellows. The “fable” he was thanking was Fable 5. The Jacobian Conjecture had been open since Ott-Heinrich Keller restated it for all dimensions in 1939. By the time most of Europe was awake on Sunday morning, working mathematicians had independently verified his counterexample. The arithmetic, as Fields Medallist Timothy Gowers put it, was “pretty amazing” — and, more precisely, it was checkable.

What the Conjecture Actually Claimed

The Jacobian Conjecture asks whether a certain algebraic regularity guarantees invertibility. Specifically: if a polynomial map from n-dimensional complex space to itself has a Jacobian determinant that is a nonzero constant everywhere, must the map have a polynomial inverse? This is the algebraic cousin of the Inverse Function Theorem from real calculus, which says something similar for smooth maps. The appeal is that the hypothesis is simple and purely algebraic. The difficulty is that the conclusion doesn’t follow from anything obvious — and decades of effort couldn’t produce a proof, because the conjecture turns out not to be true.

In one dimension, the statement is trivially correct. In two dimensions — the case Keller originally studied, and still the one that carries most of the community’s historical investment — it remains unresolved as of today. Over eight decades, the conjecture attracted what Wikipedia’s entry calls “a large number of published and unpublished false proofs that turned out to contain subtle errors.” This was not a hobbyist problem. It had consumed serious effort from serious algebraists across three generations, with confidence in the conjecture’s truth remaining high through each successive failed attempt. No one was seriously searching for a counterexample. The community had converged on the assumption that the conjecture was probably right and that the proof was just hard to find.

Gowers called Alpöge’s result “the first example of an LLM solving a problem not in my area that was nevertheless big enough that I had very definitely heard of it.” Jared Duker Lichtman at Stanford called it “quite a remarkable result,” noting its status as “one of the central open problems in algebraic geometry.” Daniel Litt at the University of Toronto described it as “very bullish for near-term impact of AI on math” and predicted that “probably lots more of these” are coming. These are not observers from outside the field.

The Counterexample

The polynomial map Fable 5 identified lives in three-dimensional complex space. Three distinct input points — (0, 0, −1/4), (1, −3/2, 13/2), and (−1, 3/2, 13/2) — all map to the same output: (−1/4, 0, 0). The Jacobian determinant of the map evaluates to −2 everywhere — a nonzero constant, exactly as the conjecture’s hypothesis requires. But a function that sends three distinct inputs to the same output cannot be injective. A function that isn’t injective cannot be invertible. The Jacobian Conjecture is false in dimension 3 and, by a standard coordinate-extension argument, in every higher dimension as well.

Within hours of Alpöge’s announcement, Paul Rouzeau, a doctoral student at Imperial College London, had formalized the counterexample in Lean and submitted it to Google DeepMind’s Formal Conjectures repository. The pull request has received multiple approvals and remains under review. Kevin Buzzard — who runs the Xena Project at Imperial and has been documenting AI’s role in formal mathematics — has raised questions in the PR comments about the exact statement of the conjecture being formalized, noting that the version in the repository differs slightly from Alpöge’s original. These are real procedural questions. They don’t affect the underlying arithmetic, which has been independently checked by multiple parties.

The collaboration’s structure matters for understanding what happened. Akhil Mathew, a professor at the University of Chicago, suggested to Alpöge that he look for a counterexample to the Jacobian Conjecture — he had been thinking about it and had an intuition. Alpöge brought that search to Fable 5. Bartosz Naskręcki at Adam Mickiewicz University has been direct about what this means: “This wasn’t a simple prompt for Fable. Finding counterexamples like this…requires real insight.” The human expertise is in knowing that a counterexample might exist in dimension 3 specifically, and in directing the search accordingly. Fable found the polynomial. Neither piece works without the other.

The dimension qualification is also load-bearing. Alpöge’s result falsifies the Jacobian Conjecture for n ≥ 3. The two-dimensional case — the original Keller problem, which has absorbed the largest share of mathematical attention across eight decades — remains unsettled. Whether the 2D conjecture holds, or whether Fable-assisted search could falsify that version too, is now an open question where it wasn’t one a week ago. Gowers noted that the result is “a counterexample, so not in ‘end of mathematics’ territory” — a precise qualification that places the disproof where it actually lives: significant, not total.

The Third Upset in Eight Weeks

Nine days before Alpöge’s announcement, Kevin Buzzard had been watching a structurally identical event. On July 11, Akhil Mathew — the same UChicago professor who would later point Alpöge toward the Jacobian — reported that OpenAI’s GPT-5.6 Sol had found a counterexample to an old question of Grothendieck about finite group schemes. The question: does every finite free group scheme of order n get killed by n — meaning n-fold summation of any element returns zero? Deligne had proved it in the commutative case. Grothendieck had proved it when the base was reduced. René Schoof had proved it in further cases. In 2025, Emiliano Torti published a paper proving it in an expanded set of cases. The body of evidence pointing toward yes was growing. Then Sol found a counterexample: a group scheme of order 4 that is not killed by 4. Mathew formalized it in 1,076 lines of Lean, using nothing outside what was already in mathlib.

Buzzard documented both the Grothendieck and Jacobian results in his Xena Project post published the same day as Alpöge’s announcement. His framing was deliberate: human mathematicians are being “outcounterexampled.” Before the Grothendieck result came the Erdős unit distance conjecture, disproved in May using the Golod-Shafarevich theorem — a number-theoretic bridge that discrete geometers hadn’t thought to cross. The accumulated pattern is three conjectures where mathematical effort had been organized around proving them true, and AI systems found counterexamples in the search space that human intuition had deprioritized or dismissed.

What makes this different from AI solving mathematical puzzles is the epistemic structure. Torti’s 2025 paper on Grothendieck’s group scheme question made the conjecture look more likely true by proving it in more cases. Every new confirmatory result added to the community’s confidence — and made the counterexample harder to find, not because it didn’t exist, but because each positive result narrowed where you’d think to look. AI can move across mathematical search space without the prior that each confirmatory result builds into human intuition. That isn’t insight in any deep sense. But it’s a form of search advantage that is now falsifying things the field had been building on.

Buzzard wrote in his post that “any PhD student who was not paying $200 per month to access these tools was crazy” — a statement about competitive necessity that would have sounded extreme six months ago. A faculty colleague offered a different read: the counterexample had been easy to find, which “just indicated that humans had not spent enough time thinking about the problem.” Buzzard labeled that reaction “the denial phase.”

Both contain something accurate. The 2D Jacobian Conjecture is still open, and the Grothendieck and Jacobian counterexamples were found by human experts who knew where to direct the search. Vidit Nanda at Oxford put the right frame on the present: “Let us celebrate the present and beware the future.” The shift isn’t that AI has become an autonomous mathematician. The shift is that AI is now fast enough through the space of possible counterexamples that the mathematical community can no longer assume its collective intuition about which direction to search is reliable. When a conjecture accumulates eighty years of failed proofs and a growing set of positive special-case results, the prior that it is probably true is exactly the kind of prior that has now been demonstrated to be wrong. Three times.

AI-generated editorial illustration · TemperatureZero · July 21, 2026