Anthropic’s Fable 5 Helps Mathematician Topple a 90‑Year‑Old Conjecture

On a quiet Sunday evening, when most screens were flashing soccer highlights, mathematician Levent Alpöge dropped a bombshell on X: he had leveraged Anthropic’s latest large‑language model, Fable 5, to generate a counterexample that disproves the Jacobian conjecture.

The Jacobian conjecture isn’t just any old math puzzle. First posed in the 1930s, it asks whether every polynomial map with a non‑zero constant Jacobian determinant must be invertible. For roughly 90 years, the question sat on Stephen Smale’s infamous “list of problems,” tantalising algebraic geometers and keeping whole research programs alive.

Enter Fable 5 – Anthropic’s public‑facing sibling of the even more guarded Claude Mythos Preview. The company says the model is tuned for advanced reasoning and, oddly enough, for “cyber‑security‑grade” safety, which is why the preview never saw a public rollout. Yet somehow, its raw analytical muscles were enough to sift through endless polynomial possibilities and surface a concrete counterexample.

Alpöge, a fellow of Harvard’s Society of Fellows, lists an affiliation with Anthropic on his LinkedIn page. He didn’t elaborate much beyond a terse tweet, and the post has since been removed – perhaps a hint of the usual academic caution when a claim of this magnitude surfaces.

What does this mean for AI? Columbia professor Andrew Blumberg, who straddles mathematics and computer science, says it’s exactly the sort of thing we should expect from today’s frontier models. “If there’s a concise counterexample hidden in a massive search space, a capable AI will find it,” he told Mashable.

Blumberg, however, draws a line between a neat disproof and a deeper, enlightening proof. He likens the situation to Moses descending a mountain with a tablet that simply reads, “Cancer can be cured.” The statement would be astonishing, but it tells us little about the underlying mechanisms. Likewise, a lone counterexample to the Jacobian conjecture may settle the question, but it does little to illuminate why the conjecture failed.

That nuance matters because AI has already delivered more than just counterexamples. In May, OpenAI disclosed that an internal model had invalidated the Erdős unit‑distance conjecture in discrete geometry – and, crucially, the result spurred follow‑up work that expanded the field’s toolbox. By contrast, the Jacobian counterexample has yet to generate a cascade of new insights, at least not according to Blumberg, who admits he isn’t an expert in that niche.

There’s also a legal backdrop to consider. Ziff Davis, Mashable’s parent company, sued OpenAI in April 2025 over alleged copyright infringement in training data. While Anthropic isn’t directly implicated, the broader conversation about AI, data, and intellectual property looms large over any headline‑making result.

So, does Fable 5’s feat represent a watershed moment for artificial intelligence, or just a clever shortcut that shortcuts human patience? The answer probably sits somewhere in between – a reminder that machines can now crunch through possibilities faster than any research team, but that true understanding still feels very human‑centred.