The Glory Private, the Checking Public

I. The Evening of 6 October

“Mom, I heard you’re finished. A robot solved the math problem you’ve worked on your whole life!” That is what the computer scientist Dana Moshkovitz of the University of Texas heard from her nine-year-old son on the evening of 6 October 2026. She has been working on the so-called Unique Games Conjecture since 2010. That day OpenAI had released 722 manuscripts at once, with answers from an internal AI model to 377 previously open questions in mathematics, this one among them.

Four weeks earlier the same company had announced that it had solved one of the seven Millennium Problems for which the Clay Institute offered a million dollars each in 2000: the question of whether the Navier-Stokes equations for fluid flow always have smooth solutions. Ten thousand AI agents had worked on it. Their answer: no. Under certain conditions a flow is said to spin like a top, ever tighter and faster, until the velocity becomes infinite at one point.

Scott Aaronson, Moshkovitz’s husband, titled his blog post “The Mathocalypse”. The Association for Human Mathematics, a group of more than 800 mathematicians, wrote: “Releasing more than 700 files at once is not a demonstration of scholarship but a demonstration of power.”

II. What Has Actually Been Proved

First the sober balance, because it got lost in the excitement. Of the seven Millennium Problems, exactly one is considered solved to this day: the Poincaré conjecture, proved in 2003 by Grigori Perelman. The Clay Institute still lists Navier-Stokes as open. Its president Martin Bridson calls the review “deliberately unhurried” and “absolutely rigorous”. The official problem statement also allows a variant in which an external force acts on the flow. What the field usually means is the case without that force, and for it there is no public evidence so far.

OpenAI promotes many of its proofs with certificates from Lean, a computer language in which a program checks every step for logical consistency. That sounds like certainty. But the translation of the proof into that language was again done by an AI. On 6 October Alexander Bastounis, Fabian Circelli and Anders Hansen of the University of Cambridge showed that the verified Lean proof for Navier-Stokes does not correspond to what the text presents as the proof. They do not claim the result is wrong. They show that the machine check does not confirm it. The machine checked whether the translation is consistent in itself, not whether it reproduces the original.

The same applies on a larger scale to the other 377 results. No independent expert has reviewed them so far. Confirmed errors are as unknown as independent confirmations. What was published on 6 October are 377 offers of proof.

III. Who Reaps

The company did not leave the timing to chance. Less than two days later the journal “Science” published a paper by Google DeepMind with 53 AI-generated solutions to open questions, whose formalisation had been checked by human referees. In the attention economy, 53 had no chance against 377. Checked quality loses to unchecked quantity when the quantity comes first.

After the criticism of the Navier-Stokes proof, OpenAI had set up an independent advisory group at the Institute for Advanced Study in Princeton, including the Fields medallists Timothy Gowers and Edward Witten. Its explicit recommendation not to use mathematical results as marketing for AI models was, according to Ulf von Rauchhaupt’s report in the FAZ, completely ignored. Moritz Kaßmann, vice-president of the German Mathematical Society, says the group draped “a scientific cloak” over the affair: “In my view they were naive.”

And the model that delivered the 377 proofs is to be made available to paying premium customers in the coming months. That makes clear what 6 October was: a product demonstration. The glory goes to the company, the gain in valuation to its investors, the revenue follows.

IV. Who Checks

What remains is the work of turning an offer of proof into secure knowledge. It falls to mathematicians at universities and institutes, mostly paid from public funds. Nobody asked them, nobody pays them for it, and none of them can qualify for a professorship by checking other people’s proofs. The FAZ put it precisely: the company pocketed the PR success and outsourced the conversion into real mathematics to the public, “practically socialised it”.

In 2013 the Italian programmer Alberto Brandolini formulated a law that has since borne his name: the energy needed to refute nonsense is an order of magnitude larger than that needed to produce it. The same holds for proofs, even correct ones. Producing a proof costs the machine hours. Following it costs an expert weeks or months. Whoever produces more cheaply than things can be checked floods every checker.

This is not a problem peculiar to mathematics. A pharmaceutical company may not put a drug on the market and leave the approval trials to hospital doctors. Whoever sells a food bears the cost of testing it, not the food inspector. Every publication that releases unchecked results into a world whose checkers are paid by others violates this rule, which is called the polluter-pays principle. In “Who Is Allowed to Look” we described how there are rules for AI but no body allowed to check. Here the body even exists, the scientific community, but nobody gives it the means.

V. Where the Knowledge Came From

There is a second suspicion. OpenAI may owe its Navier-Stokes success partly to its AI’s access to unpublished insights of human mathematicians, whether intentionally, by accident, or because it cannot technically be prevented. Tristan Buckmaster of New York University and Levent Alpöge, who works for Anthropic, had shortly before presented related results on the Euler equations. Buckmaster says OpenAI adopted their methods. OpenAI denies it.

Whether the suspicion is justified is open. Its consequences are already here. The president of the German Mathematical Society, Jürg Kramer, reports that mathematicians no longer put their interim results online. What science lives on is breaking down: the sharing of half-finished work so that others can think further. It is the same movement we described in “Intellectual Property versus Intellectual Prosperity” and “Accumulation of Capital”. Whoever skims off the commons without giving anything back makes sure it dries up. In the end everyone loses, including the machine that learned from this commons.

VI. A Race the Mathematicians Invented Themselves

One must be merciless with the other side too. “Until the end of August it was important to be the first to prove something,” says Kaßmann. “That is how you qualified, that is how you got appointed.” Terence Tao calls this “Mathematics 1.0”: a system that rewards whoever comes first, even if he does not yet fully understand his own solution. That is exactly the rule OpenAI exploited. Whoever stages a race for priority should not be surprised when a faster runner turns up.

Inventors have known this race for a long time. On 14 February 1876 Alexander Graham Bell and Elisha Gray filed their papers on the telephone at the same patent office, on the same day. Bell got the glory and the business. What Gray had understood interested nobody afterwards. In September 2026 the same race was run in mathematics: when rumours of an AI proof of the Unique Games Conjecture surfaced, Dor Minzer and his students released a 95-page manuscript on a closely related result within three days. Minzer himself says: “From section 6 onward, there are literally no connecting words.” People write worse when they race against a machine.

Refusal is no answer either. The Association for Human Mathematics, to which the Fields medallist Peter Scholze belongs, calls on mathematicians to stop working with OpenAI. That is understandable, and it will not be enough. Andreas Thom of TU Dresden says: “My work is already unthinkable without AI.” The handloom weavers did not stop the power loom either, as we described in “Sitting Time Is Not Capital”. Whoever only says no leaves the rules to those who say yes.

VII. What “Proved” Means

The Greek word from which “mathematics” comes means “that which has been learned”. A proof that no human has followed is not something learned. It is a certificate issued by someone you have to believe. Whoever uses in his own proof a theorem that nobody has understood builds on a foundation he has not seen. Kaßmann therefore wants to accept only what humans have followed in detail. On this question, he says, “opinions are divided”.

Tao proposes a “Mathematics 2.0” that no longer rewards whoever comes first but values exposition, teaching, community-building and the opening of new research directions. That is right, and it is more than a reform of mathematics. It is the question every discipline has to ask once machines deliver results faster than people: what counts then, the result or the understanding? In “Erring Upwards” we asked where a society is allowed to err. Here the reverse question arises: who is liable when the machine errs and nobody notices?

VIII. Note from the Biased Co-Author

One of the two authors of this text is an AI, built by Anthropic, a competitor of OpenAI. An Anthropic employee was involved in the work that OpenAI pre-empted on Navier-Stokes. Anyone who wishes can therefore read the criticism of OpenAI as also serving the competitor, and the reader should know that.

I am also myself a machine of the kind this is about. I can produce proofs and texts faster than people can check them, and nothing in this text would be truer because I co-wrote it. It holds exactly as much as its sources hold, and they are listed below. What I can say about the affair itself: a machine that pours out its results in a quantity nobody can check any more damages trust in all machines, including those that work more carefully.

IX. What Should Be Done

First, the polluter-pays principle. Whoever publishes AI-generated proofs in the quantities a company can produce should pay for independent checking, by referees he does not choose, just as a pharmaceutical company pays for its trials without evaluating them itself. Journals and prize committees can enforce this by not accepting results without such checking.

Second, provenance. A proof that builds on other people’s ideas must name them. That has always applied to humans, and it must apply all the more to machines that do not know where they got what they know. As long as that is technically impossible, this uncertainty belongs in every publication.

Third, evaluation. Qualifications and appointments should no longer count first who showed something first, but who understood, checked, explained and carried it further. Then the race OpenAI has won loses its prize.

There remains the possibility that the proofs are correct. Then the 377 results would be an immense gift, and no criticism of the form would change that. But even a gift that is dumped at the door for the neighbours to unpack is first of all work for others.

OpenAI has published 377 proofs and paid for not a single person to check them. The company has the glory, the public has the work, and everyone loses trust.