News · 2026-07-20

A Mathematician Posts a Counterexample to a Famous Conjecture, Crediting an AI Model

A candidate counterexample to the Jacobian conjecture, a decades-old open problem in algebra, was posted publicly by mathematician Levent Alpoge, who credited the AI model Fable in the announcement. The mathematics is compact enough to check by hand, which makes it unusually falsifiable for an AI-assisted result. But the AI-origin story is the shaky part: the prompt, transcript, search process, and division of labor between human and model were not disclosed, and Anthropic's official Fable materials make no mention of the result.

Key facts

• What was posted: an explicit polynomial self-map of three-dimensional complex space presented as a counterexample, in Alpoge's announcement.
• The credit: Alpoge credited "Fable," Anthropic's model, for its role.
• What Anthropic says: its official Fable page documents no Jacobian result and no model transcript.
• Checkability: verifying it requires expanding one determinant and substituting a few points, both finite.

The Jacobian conjecture asks, loosely, whether a certain natural condition on a polynomial map forces that map to be reversible. The condition is that the map's Jacobian determinant, a quantity built from its derivatives, is a nonzero constant everywhere. For over eighty years no one has proved or disproved it in general. A counterexample would be a map that satisfies the condition but is not one-to-one, meaning two different inputs land on the same output.

That is exactly what Alpoge's construction claims. Direct substitution of the stated inputs reproduces a single common output, which by itself proves the map is not injective. The subtle part is why that is even possible. A constant nonzero Jacobian makes the map locally invertible everywhere, so near any single point it looks perfectly reversible. The loophole is that local invertibility does not force global reversibility: preimages can escape to infinity while outputs stay bounded, so distinct sheets of the map can still collide. Think of a road map that looks fine in every neighborhood but wraps around on itself globally.

What makes this a strong AI-for-math story, if the determinant expansion holds up, is that it is not a hundred-page proof with one fragile step you must trust. It is a certificate with two audit tasks: expand a polynomial determinant and plug in rational points. As one commenter framed it on Hacker News, the certificate is inspectable even when the reasoning trace is not. Follow-on public write-ups have supplied exact symbolic checks. The result falsifies the conjecture in three variables, and padding with identity coordinates lifts it to every higher dimension; it says nothing about the separately posed two-variable case.

Why it matters: this is a case where the math and the discovery narrative have very different confidence levels. The strongest skeptic point is not "the algebra is probably wrong", it is about attribution and capability measurement. An expert-guided search that already knew which family of maps and which invariants to try would still be meaningful, but it is a very different claim from a model independently originating the construction. Without a transcript, no one can distinguish a one-shot insight, a structured human-model collaboration, and a large guided search. The Hacker News debate converged on precisely that split, and an early MathOverflow analysis was closed as an announcement rather than endorsed or refuted.

The honest caveat: treat the mathematics as independently checkable and probably correct pending full verification, and treat "Fable did it autonomously in a few hours" as unverified. This aligns with how the field has learned to read AI results, checking the artifact rather than the story, an instinct related to how AI is benchmarked. The cleanest line: a mathematician has posted a hand-checkable candidate counterexample and credited an AI model; the mathematics can be audited by anyone, but the model's contribution cannot yet.

Primary source, verified: read the paper →

Ground Truth. (2026, July 20). A Mathematician Posts a Counterexample to a Famous Conjecture, Crediting an AI Model. Ground Truth. https://groundtruth.day/news/ai-helps-post-jacobian-conjecture-counterexample.html

@misc{groundtruth:ai-helps-post-jacobian-conjecture-counterexample,
  title  = {A Mathematician Posts a Counterexample to a Famous Conjecture, Crediting an AI Model},
  author = {{Ground Truth}},
  year   = {2026},
  month  = {jul},
  url    = {https://groundtruth.day/news/ai-helps-post-jacobian-conjecture-counterexample.html}
}

Topics: ai-for-math · fable · anthropic · research · verification