AI-Generated Proofs Expand Mathematical Results but Leave Questions of Understanding

OpenAI says it has released hundreds of mathematical results produced by an internal frontier model. The list shown by Károly Zsolnai-Fehér runs to 722 papers, though he says he was still scrolling at the letter C.

One example is irrational triangular billiards. Imagine a ray bouncing around the mirrored walls of a triangle. If at least one angle is irrational relative to π, the pattern does not repeat in the ordinary way. OpenAI claims to have proved that, for every triangle of this kind, almost every ray visits every spot, travels in every direction, and spends an equal amount of time everywhere. Zsolnai-Fehér stresses how difficult this is: irrational angles make the motion hard to bring under control, leaving familiar proof techniques without a reliable handle.

A second result concerns conformal transformations: angle-preserving stretches of a planar region. Start with a shape containing irregular holes, and the transformation can turn those holes into circles or points. The challenge is that holes can shrink without limit and become tangled, making the final shape difficult to control. Zsolnai-Fehér says that before this result, the behavior was known only in easier cases; OpenAI claims it always works.

The displayed paper titled “The Quasi-Riemann Hypothesis” makes another striking claim: that certain finite-order Hecke L-functions over Q and Dirichlet L-functions are zero-free in the half-plane where the real part of s is greater than 7/8, allowing the principal pole at s = 1. It says this includes the Riemann zeta function, and calls the result a proof of the “quasi-Riemann hypothesis.”

These are striking mathematical claims, but the form of the work matters too. Zsolnai-Fehér says the papers are not elegant proofs. Some, he says, are difficult to read and understand without help from another AI. He describes some of them as “a big ball of barf.”

The split in the mathematical community, as Károly Zsolnai-Fehér presents it, turns on what mathematics is for. For some, the primary goal is to answer a question: yes or no, or whether a quantity is N or N squared. If a proof establishes the answer, its lack of elegance may be secondary.

For others, mathematics is a way to learn. Zsolnai-Fehér compares the difference to the many proofs of the Pythagorean theorem. One proof is enough to establish the theorem, but different proofs can illuminate it in different ways. A hundred proofs are not better than one as confirmation; their value is that each may teach something new.

He uses neural-network simulations to make the distinction concrete. People can write fluid or smoke simulations using known theory; a trained network may produce similar results faster. But the network’s answer is not an elegant algorithm. It is internal activations—a “big bunch of numbers.” The mathematical proofs, he suggests, can be similar: another AI may be needed to make sense of them.

For researchers who mainly want a result, that may still be a useful tool. For those who want a proof to explain why the result holds, a mass of difficult-to-interpret output is a different kind of contribution. Zsolnai-Fehér says both positions make sense to him.

Károly Zsolnai-Fehér recounts a sequence of claims about AI’s limits: about six years earlier, people said GPT-3 could not reliably add numbers; two years earlier, they said AI could not reliably solve high-school competition problems; a year earlier, that it could not reliably win the Mathematical Olympiad gold medal. Then, he says, just a month before the recording, AI very likely solved a millennium problem no human had solved before.

Against that backdrop, the objection that AI has produced nothing novel because it merely combines existing ideas sounds to him like “an ever-receding argument.” The point, as he frames it, is that the criticism shifts as the systems’ capabilities change.

The scale adds to the sense of change. Zsolnai-Fehér says the released work amounts to hundreds of results, each taking roughly three hours to compute on average. But he immediately qualifies the achievement: the model that produced them is not generally available. Access, he says, is limited to perhaps a handful of people in frontier labs.

For Károly Zsolnai-Fehér, restricted access is not a side issue. If AI can advance mathematics and science at this level, he argues, the technology should be available beyond a small group with access to frontier models. His proposed route is open-weights models that people can run and own themselves.

He frames participation broadly: writing code, running experiments, and telling others about the work all count as contributions. The aim is not only to distribute results, but to let more people help produce and test them.

A slide attributed to OpenAI and its math repository says a sign error in “Algebraicity of Weil classes on split abelian eightfolds” invalidated a stabilization-trace cancellation argument and the construction used by two dependent papers. The notice lists three withdrawn manuscripts: that paper, “Algebraicity of Kuga-Satake Correspondences for K3 Surfaces,” and “The rational Hodge conjecture for products of K3 surfaces.”

Zsolnai-Fehér says the withdrawals followed community feedback, likely with an LLM also helping identify the issue. That is his account of how the problem came to light; the displayed notice itself specifies the sign error and its consequences. He connects the episode to his call for broader access: the work should not be an expensive, exclusive pursuit, but something a wider community can take part in and scrutinize.