Tao sketches the proof behind OpenAI's Siegel zero result, saying he used SAIR's open math models to work through it

Mathematician Terence Tao posted "A sketch of the proof of non-existence of Siegel zeroes" on his blog on October 19 (primary source). The post outlines the main line of argument in a proof from OpenAI that Siegel zeroes do not exist. Also notable: Tao states plainly that he used SAIR's open math models to work through it. Checking AI-produced mathematics has moved into a phase where leading mathematicians use another AI as their tool.

What happened: Tao publicly set out the proof's argument

On October 6, OpenAI published a batch of math manuscripts on GitHub, produced by an unnamed internal model. It later withdrew three of them because of sign errors and revised other proofs. The claims reportedly included progress related to the Riemann hypothesis. OpenAI has still not named the model that produced the results.

Over this period, Tao's blog has become a gathering point for debate about AI and mathematics. The launch of Hexagon, Sophie Huiberts' "This is normal bad work," and a proposal for a new journal built around formal verification all appeared there as guest posts. Those discussions centered on systems and norms: how the field should handle AI-generated results. This post is different. Tao himself went into the substance of a single proof and wrote out its argument.

As the word "sketch" in the title indicates, the post is meant to explain how the proof flows. It cannot be read as a declaration that the full proof has been verified. Still, it matters that a leading mathematician traced the logic of an AI-produced result himself and published it. At least one of the OpenAI manuscripts, which Huiberts criticized as "tremendously hard to decipher," now has a map that humans can read.

Siegel zeroes, and why this result matters

A Siegel zero is a hypothetical exception: a real zero of the L-function of a real-valued Dirichlet character lying very close to s=1. Because no one knows whether such zeroes exist, some basic results in analytic number theory, such as lower bounds for the class numbers of imaginary quadratic fields, could only be proved in an "ineffective" form, with constants that cannot actually be computed. If the generalized Riemann hypothesis is true, Siegel zeroes do not exist. But ruling them out without assuming the Riemann hypothesis is a major open problem dating back to the time of Landau and Siegel.

If their non-existence really has been shown, as claimed, it would rank among the most significant mathematical results produced by AI. That is exactly why verification has to match the scale of the claim. Which raises the question of who will read the proof, and how.

The tool for reading it was an "open" AI

Tao states that he used SAIR's open math models to work through the proof. He introduced SAIR in an October 16 post, "SAIR Open Math OS." It is a platform where academics can use open models as teams. The models listed as available are DeepSeek V4.1 Flash, Kimi K3 and GLM 5.3 Flash. In that post, Tao also wrote that these models are about a year behind the frontier.

The asymmetry is clear. The result came from OpenAI's unnamed, unreleased internal frontier model. The person reading it used open models whose inner workings anyone can examine. At solving research problems, open models lag far behind. In the third round of First Proof, Kimi K3, described as the best open-weight model, solved 0 out of 10 problems. Even so, Tao's use of them suggests that a model behind the frontier can still help with following an already-written proof and sorting out which lemma does its work where.

Using open models matters for reasons beyond performance. Other mathematicians can use the same verification tools under the same conditions. Huiberts criticized OpenAI for hindering further development of its results through its terms of use. In response, the verification side has started building its own workflow that does not rely on closed models.

A shift in how verification work is divided

Over the past two weeks, one proposal after another has emerged for how to handle AI-produced mathematics: revised rules for the Erdős problems site; Hexagon, which collects results obtained with LLM assistance; a proposed journal that would require every proof to be formally verified in Lean; and a $15 million initiative from the Simons Foundation, Anthropic and OpenAI. All of these were about institutions, deciding how humans would evaluate and publish such work.

Tao's latest post shows the practical side. A human mathematician uses one AI as a tool to break down a long, hard-to-read proof produced by another AI, then rewrites the argument in human language. Formal verification in Lean uses machines to check whether a proof is correct. This kind of work instead brings what was proved, and how, back within human understanding. It gives concrete shape to the vision Tao laid out in "Math 2.0" on October 10: mathematics in which humans stay at the center while drawing on AI's help.

Open questions

First, this is an outline of the argument, not a conclusion about whether the full proof is correct. A complete verification, the status of formalization in Lean and assessments by other experts in analytic number theory are still to come. Second, OpenAI has not yet named the model that produced the result. Given the company's position that it is holding back its most capable models, which system produced this result remains an open question. Third, how much open models can be relied on as verification tools depends on which SAIR models were used, at which stages and to what extent. Whether the mathematical community widely adopts this approach will depend on whether other researchers can follow the same proof with the same tools.