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Proof by machine: what holds so far

OpenAI's flood of AI-written mathematics, and how much of it has been checked.

7 Nov 2026 · Mathematics · Verification · about 6 minutes

Draft This topic has been selected for coverage. What you are reading is a first AI research draft based on my direction, published early so you can see what is coming. I have not reviewed it yet, so treat it as provisional. Once the final version is reviewed and published, this notice will be removed.

Since September OpenAI has announced AI-written proofs of famous open problems, and in October it put about 720 manuscripts online. Part of it comes with machine-checked proofs, three papers have been withdrawn, and experts have worked through one headline result. The two biggest claims, a proof that Yang–Mills theory exists and a refutation of the Exponential Time Hypothesis, have not been checked by anyone outside yet. What is settled so far is mostly how the claims are made and checked, not whether they are true.

What happened

In early September OpenAI announced that an internal model had proved a version of the Navier–Stokes problem, one of the famous open problems in mathematics. Researchers called the announcement rushed, and the mathematicians Tristan Buckmaster and Levent Alpöge said that ideas from their own recent work had leaked into it; OpenAI denies that. More than 8,000 researchers have since endorsed the concerns about how the claim was made, according to Retraction Watch.

On 6 October OpenAI put 722 manuscripts, covering 372 open problems, on GitHub. They come from an unreleased internal model. According to Scott Aaronson, it is a general model rather than a specialised system, ran for about three hours per problem, was tried on roughly 4,000 problems and solved about a tenth of the open ones. The failed attempts are not reported. Some papers claim progress on famous problems, including the Riemann hypothesis and the Birch–Swinnerton-Dyer conjecture.

On 26 October three more claims followed: the Kakeya conjecture in every dimension, a proof that Yang–Mills theory “exists” mathematically (one half of a Millennium Prize problem), and a refutation of the Exponential Time Hypothesis, the belief that a classic logic puzzle called 3-SAT needs exponential time. Work by Anthropic-linked researchers overlaps with two of them, according to Scientific American.

How it is checked

By machine. Lean is a proof checker: if it accepts a proof, every step follows the rules of logic. According to the repository’s README about 42% of the headline results come with Lean proofs. OpenAI’s spokesperson told Retraction Watch that roughly half of the results were released unconfirmed, on the advice of OpenAI’s own advisory group on mathematics and AI (AGMAI). A machine check has two weak spots: the Lean statement has to say what the paper says, and the formalisation is itself written by an AI. The Conversation notes that AI-written formalisations have contained errors, including one for Navier–Stokes.

By people. The best-checked result is the “quasi-Riemann” paper, which narrows where the zeros of the zeta function can lie. Quanta reports that OpenAI released two versions. The first is 199 pages and was machine-checked; the second, from about a week later, is 49 pages, not formally verified and the only paper among the 722 that a human edited. Experts have worked through the second one and most believe it is correct. Peter Sarnak, quoted by Quanta, says he will not call it a theorem until humans have verified every detail.

By retraction. On 7 October OpenAI withdrew three papers after finding a sign error, and revised 14 more. Retraction Watch quotes the mathematician Alex Townsend saying he expects more errors to turn up.

The two biggest claims

Yang–Mills. Physicists use this theory successfully, but nobody has built it from first principles and shown that it is consistent. That is the “existence” half; the “mass gap” half is not claimed. OpenAI’s proof has not been published. Scientific American reports that the advisory group has called for its immediate release. A physicist’s blog says a group of experts is checking it and has decided to hold it back until it is verified and explained. Meanwhile Alpöge, who works at Anthropic, and Ethan Sussman had been building on a decades-old programme by Tadeusz Balaban in public on GitHub, using Claude agents. They claim the three-dimensional case, which falls short of what the prize requires, and Alpöge criticised OpenAI for not contacting him.

The Exponential Time Hypothesis. OpenAI’s 67-page preprint, dated 17 October and revised on 25 October, describes a deterministic algorithm that counts the satisfying assignments of a k-CNF formula in roughly 2^{n/√log n} time when the number of clauses is proportional to the number of variables. The hypothesis predicted that no algorithm can solve 3-SAT in less than exponential time, and counting solutions is at least as hard. The result would not settle the P versus NP question, Scientific American points out. Russell Impagliazzo, Ramamohan Paturi and Ryan Williams are reviewing it. Scientific American also reports an independent claim from an internal Anthropic model.

What is established, and what is not

Established: the releases exist; roughly 720 manuscripts; some come with Lean proofs; three papers were withdrawn and 14 revised; and experts have worked through the 11/12 zeta-zero paper and mostly believe it.
Not established: that most of the other results are correct; that every Lean statement says what its paper says; that the Yang–Mills and ETH proofs hold, since no outsider has checked either; and how the results were produced. Prompts and run details are mostly unpublished, and the README gives abridged reasoning summaries for only nine of the problem families.
Disputed: credit and priority, mainly on Navier–Stokes and Yang–Mills, and whether the release helps mathematics. Aaronson calls it one of the biggest days in the history of the subject. The Association for Human Mathematics calls it a “demonstration of power” and has urged mathematicians to stop working with OpenAI. Terence Tao argues for AI that keeps human understanding central, not one that mainly solves open problems.

What this means if you build with AI

This is my reading, not something the sources conclude:

Making a claim is cheap, checking it is not. Hundreds of papers arrived at once and the people who can check them are few. AI-written code, reports and analyses have the same imbalance, so plan and budget the checking.
Keep three states apart: claimed, machine-checked, and checked by someone qualified. Most of what you read this month sits in the first.
A machine check proves only what was checked. A proof of the wrong statement is still a proof. Anthropic’s own post on formally verifying software says the same about “verified” code. I covered it on the Radar.
Evidence that is not published cannot be checked. When a vendor reports a result, ask what you could reproduce yourself.

Sources

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