Episode Details

This week we talk about Fable, sandboxes, and the Jacobian conjecture.

We also discuss counterexamples, X, and ChatGPT.

Recommended Book: After the Fall by Edward Ashton

Transcript

In mathematics, a conjecture is a proposition, something like a guess by someone who knows what they’re talking about, about something believed to be true, but not yet proven in a formal sense. The goal is to then eventually come up with a formal proof for that informed guess, at which point the conjecture becomes a theorem. If even a single exception is found to the proposition, however, that exception called a counterexample, the conjecture is considered disproven, and it can then never become a theorem.

The Jacobian conjecture—and this is a radical simplification of a very complex concept—but it basically says that if a formula-based map of coordinates stretches or moves without experiencing any local crushing or folding along its surface (which in more formal language would mean the Jacobian determinant is always a constant number that isn’t zero), if that’s true, that map can always be completely reversed, and that will return all the points to their original positions.

This conjecture has been posited and tested since the late 19th century, and it’s generally been considered very compelling by mathematicians, many of whom have proposed proofs which were, ultimately, found to have subtle errors, keeping them from becoming theorems. No one was able to find a counterexample, either, which would definitively prove the conjecture was wrong.

No one, that is, until a mathematician named Levent Alpöge (leh-VENT ahl-PUH-geh), who works as a researcher at Anthropic, decided to task the company’s currently most capable, publicly available model, Fable, to find a counterexample. He posted the counterexample—and again, this is a formal mathematical finding that disproves a conjecture, keeping it from ever becoming a theorem, something that would typically be presented in a far more formal setting, and to much fanfare—but he posted it to the social network X, saying “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.”

Terence Tao, who’s considered by many to be the finest mathematician of his generation, reviewed the posted counterexample on his blog and said that it “appears like a massive miracle,” before going on to use ChatGPT, a competing LLM-based AI tool, to “discuss various aspects of this problem and to confirm several of the calculations.”

Another mathematician named Dmitry Rybin, within days of all that happening, used ChatGPT to do something similar, disproving the Dinitz-Garg-Goemans conjecture.

Both men posted the prompts that they used to make all this happen, and while Tao’s conversation with ChatGPT, checking the math on the Jacobian conjecture counterexample, was pretty mathematically dense, the latter counterexample was derived by using exactly four prompts, which are the messages typed into the text box built into these AI tools, telling the model what to do. In their totality those prompts read:

“You should do a breakthrough

please continue research and find a complete unconditional counterexample

Continue the search. Have a clear strategy obtained from deeper understanding of the problem structure.

it’s enough of partial results. let’s finish with a complete unconditional counterexample”

What I’d like to talk about today is another new, interesting thing these top-of-the-line, frontier models are doing, that would seem to violate our sense of what a clever AI tool is capable of doing, and why this thing has some facets of the technology and cybersecurity world on high alert.

—

In mid-July 2026, AI company Hugging Face anno