An AI disproved a century-old math conjecture during the World Cup final

Fable's counterexample, and the pattern that produced it

Kevin Buzzard, the Imperial College mathematician behind the Xena Project and a leading voice in formalizing mathematics in the Lean proof assistant, has a new entry for what's become a recurring genre: an AI model finding a counterexample to a conjecture humans had failed to crack, followed by a machine-checked Lean formalization landing within days.

The freshest case is the Jacobian conjecture, open for roughly a century. Fable found a counterexample in three variables of degree seven; Paul Lezeau formalized it in Lean, and Buzzard notes the timing with some relish — it was resolved during the World Cup final. Weeks earlier, Sol and Fable had cracked a 60-year-old question of Grothendieck's about group schemes — whether every finite free group scheme of order n is killed by multiplication by n — producing a counterexample of order 4 and a 1,076-line Lean file that compiled clean. In May, ChatGPT had disproved the Erdős unit distance conjecture using the Golod-Shafarevich theorem, a tool from an entirely different branch of mathematics; Logical Intelligence autoformalized the paper within a week, and Boris Alexeev at OpenAI followed with a full independent formalization — 1.2 million lines of Lean, built over three weeks.

Buzzard is careful about what these tools are and aren't doing. "One cannot trust AI-generated code," he writes, "so I ran it in a sandbox on my machine" before believing the Jacobian result. He also flags where the real work still is: "The next step in that work is for humans to understand exactly what is going on with the example" — a disproof isn't an explanation, and nobody yet knows why these constructions work.

Why $200 a month became a line item in a math department's budget

Buzzard recounts an email from an Imperial College professor, surprised that graduate students were paying $200 a month for access to Sol and Fable — he thought it was crazy. Buzzard's reply: any PhD student not paying that is the crazy one. He'd learned from a Harvard PhD student that Harvard already gives free access to all PhD students, postdocs and faculty in the department.

The reaction in the thread ran the same arithmetic further:

"given how much it accelerates grad students to produce meaningful output more quickly, why wouldn't you make an investment of $2400/student/year. Seems like pennies overall." — a commenter with nearly 16,000 karma since 2018

The skeptic who checked Wikipedia before buying the hype

Not everyone bought the framing that this is something genuinely new. The most detailed pushback came with receipts:

"Human mathematicians have been being out-counterexampled for at least two decades. The main difference... is that (A) we now have a lot more compute to throw at such things, and (B) it is currently trendy to do so... Tzuong-Tsieng Moh checked the conjecture for polynomials of degree at most 100 in two variables [in 1983]... my impulse is to side with the unnamed faculty member cited in the article: the fact that the counterexample was so easy to find just indicated that humans had not spent enough time thinking about the problem." — a commenter who cross-checked the Jacobian conjecture's citation history on Wikipedia before writing this

The point isn't that the disproof is fake. It's that the search space had already been narrowed by human hands decades ago; the AI's contribution was compute and persistence, not a new idea.

The Subway sandwich shop a faster corollary check could have prevented

The thread's sharpest human cost traced this exact conjecture back forty years:

"Yitang Zhang of the twin-prime-conjecture fame spent 7 years working on the Jacobian conjecture under the advisor Tzuong-Tsieng Moh at Purdue. A key step in his thesis used a corollary of Moh's. It turned out that the corollary was incorrect. As a result, Moh refused to write any recommendation letter for Zhang, and Zhang couldn't find any teaching or research job and ended up spending years working at a Subway." — an HN reader with 7,000-plus karma since 2017

Zhang went on to prove a landmark result on bounded gaps between primes in 2013, decades after the Subway years. The counterfactual hangs in the thread unanswered: a tool capable of stress-testing a corollary in an afternoon might have saved him most of a decade.