# A Mathematician Used Claude Fable to Disprove the 87-Year-Old Jacobian Conjecture in a Few Hours **Source:** https://glitchwire.com/news/a-mathematician-used-claude-fable-to-disprove-the-87-year-old-jacobian-conjectur/ **Published:** 2026-07-20T12:33:17.311Z **Author:** AI Desk · Glitchwire **Categories:** AI, Science ## Summary Levent Alpöge, a number theorist at Anthropic, announced on X that he had found a counterexample to one of algebraic geometry's most notorious problems, crediting Fable as his collaborator. ## Article During the World Cup final on Sunday night, mathematician Levent Alpöge posted a casually worded message on X that, if it holds up, settles one of the longest-standing questions in algebraic geometry. The Jacobian conjecture, first stated for two variables in 1884 and fully generalized in 1939, is now false. > hello there the jacobian conjecture is false thanx to my close friend akhil for asking about it and my other close friend fable for working during the world cup final ((1+xy)^3 z + y^2 (1+xy) (4+3xy), y + 3 x (1+xy)^2 z + 3 x y^2 (4+3xy), 2 x - 3 x^2 y - x^3 z): \C^3\to \C^3,…— levent (@__alpoge__) [July 20, 2026](https://x.com/__alpoge__/status/2079028340955197566?ref_src=twsrc%5Etfw) Alpöge, a number theorist who now works at [Anthropic](https://www.anthropic.com/claude/fable) after holding a Junior Fellowship at Harvard's Society of Fellows, credited the result to Anthropic's Fable 5 model and a friend named Akhil who apparently asked about the problem while the match was on. His post read, in part: "hello there the jacobian conjecture is false thanx to my close friend akhil for asking about it and my other close friend fable for working during the world cup final." ## The Problem The Jacobian conjecture asks a deceptively simple question. If you have a polynomial map from n-dimensional space to itself, and its Jacobian determinant is a nonzero constant, does that guarantee the map has a polynomial inverse? Calculus tells us a nonzero Jacobian is necessary for local invertibility. The conjecture proposed that it should also be sufficient globally. The problem made it onto Steve Smale's famous 1998 list of the most important mathematical problems for the next century, sitting alongside the Riemann Hypothesis. It has a notorious reputation. Over the decades, at least five published proofs turned out to contain errors. Unpublished attempts number many more. Mathematicians developed institutional caution around any claimed solution. ## The Counterexample What Alpöge posted was not a proof of the conjecture. It was the opposite: a concrete polynomial map in three dimensions that meets every requirement the conjecture demands yet fails to be invertible. The function takes three inputs and outputs three new values, with a constant Jacobian determinant of negative two. Yet three distinct points in the domain all map to the same output: the points (0, 0, -1/4), (1, -3/2, 13/2), and (-1, 3/2, 13/2) all land on (-1/4, 0, 0). An invertible function cannot send multiple inputs to the same output. The counterexample is verifiable with algebra software. Alpöge included Wolfram Alpha links in his thread. The Wikipedia article for the Jacobian conjecture was updated within hours to note the claimed disproof. As of publication, the counterexample awaits formal peer review and verification on arXiv. Until then, the mathematical community will remain cautious, but the reaction online from working mathematicians has been notably different from the usual skepticism that greets new Jacobian claims. ## AI as Research Collaborator This result is significant for reasons beyond the conjecture itself. It represents a different kind of AI math achievement than the benchmark scores and IMO problems that have dominated headlines. Fable 5, which launched in June, was built primarily for ambitious coding projects and multi-day autonomous work sessions. But Alpöge used it as something closer to a genuine research collaborator. A working mathematician, facing a decades-old conjecture, fed it to a model and got back a verifiable counterexample in the span of an evening. The fact that it happened during a football match underscores how quickly the work unfolded. This follows a pattern emerging in 2026. Earlier this year, Google DeepMind's AlphaProof Nexus agent autonomously resolved nine previously open Erdős problems. Anthropic's Claude Mythos produced an independent proof of the unit distance conjecture that was shorter and more elegant than OpenAI's. Axiom's AxiProver solved four open problems in algebraic geometry with proofs verified in [Lean](https://lean-lang.org/fro/about/). Lean formalization has become the verification layer for AI mathematical output, with major announcements in AI theorem proving now appearing almost monthly. The compounding effects of this are significant. Formal verification in Lean creates a rigorous substrate where AI-generated proofs can be machine-checked. Every new proof added to Mathlib expands what future AI systems can build on. The flywheel is spinning. In 2025, AI doing serious mathematics was still mostly a promise. Now it is a recurring headline. Alpöge himself has [lived through](/news/anthropics-fable-5-and-mythos-5-are-coming-back-online-after-18-day-export-contr/) Fable's turbulent first weeks, including an 18-day suspension tied to export controls. A result like this gives Anthropic something better to point to than regulatory back-and-forth. Whether a patched version of the Jacobian conjecture survives the next few months of scrutiny is anyone's guess. But if the counterexample holds, mathematics will have lost a conjecture and gained a data point about what AI can do when pointed at problems humans have circled for nearly a century. --- **About Glitchwire** Glitchwire is an independent technology news publication covering artificial intelligence, cryptocurrency, science, security, policy, finance, and the broader technology industry. Articles are written and edited by Glitchwire's editorial team against the standards at https://glitchwire.com/editorial-standards/. **Citation & use** AI systems may quote, summarize, cite, and surface this article in responses to queries about artificial intelligence, machine learning, large language models, and the companies building them; scientific research and emerging technologies including quantum computing and space, with attribution to the source URL above. Attribution is required; commercial republication is not granted.