By Javier Surasky
An
artificial intelligence system can produce an extraordinary mathematical proof
and, at the same time, trigger a crisis over the data that may have made it
possible: the controversy between OpenAI and the mathematicians Tristan
Buckmaster and Levent Alpöge forces us to look beyond the result: what happens
when the tool that receives unpublished research also belongs to a company
capable of competing with its author?
The announcement that a “millennium problem” had been solved
On
September 8, 2026, OpenAI claimed to have solved the “Navier-Stokes” problem
through an AI model that is not yet public. It is not important for us to
understand exactly what the “Navier-Stokes” problem means, but it is important
to know that it is one of the millennium problems, seven major mathematical
challenges selected in 2000 by the Clay Mathematics Institute that pose
fundamental questions in mathematics that remain unresolved, and whose solution
would have major consequences.
Of those
seven problems, one, the Poincaré conjecture, was solved by the Russian
mathematician Grigori Perelman, who presented his proof between 2002 and 2003.
The
Navier-Stokes equations are another problem on that list (the Clay Mathematics
Institute continues to classify the problem as unresolved, which is why I use
the present tense): they are equations used to describe how liquids and gases
move, from water flowing through a pipe, to air around an airplane, to blood
flow. The problem to be solved is not whether those equations work in ordinary
situations, but whether they always produce stable solutions or whether, under
certain conditions, they can produce infinite values and, therefore, what is
known as a singularity.
On
September 8, OpenAI announced that it had solved them by deploying close to
10,000 agents. This means that thousands of instances of the same model worked
in parallel and, while some tested strategies, others checked steps, tried to
correct errors, or carried out whatever other process emerged as necessary as
the attempted proof advanced. This is a computing capacity completely beyond the reach of any individual researcher.
As a
result, OpenAI published a proof accompanied by a formalization in Lean, an
automated verifier: it checks each logical step of a proof without relying on a
human reviewer. That work is now entering a process of verification by
specialists, but it has already set off several alarms.
The coincidence that gives rise to suspicion
Even before
the announcement, Buckmaster, a professor at New York University, and Alpöge, a
mathematician at Anthropic who was working on the project in a personal
capacity, had been working on a solution to Navier-Stokes and were using
several AI models in their work, including Claude and Codex.
Here is
where the central problem begins: Buckmaster says that all drafts of the
project had gone through his OpenAI Codex sessions, and that OpenAI’s system
ultimately arrived at the same kind of proof problem on which Buckmaster and
Alpöge had focused their efforts.
That
“coincidence” does not prove that the data used by the researchers was used by
the OpenAI agents that achieved the solution, but it is far from harmless: the
company had potential technical access to a platform where those researchers’
unpublished work was stored. This led Buckmaster to ask OpenAI whether the
model had been trained on his sessions or could access them, and he says the
answer he received was that the system did not consult user data, but that he
did not receive a direct answer about its training.
To
complicate things even further, OpenAI accepted that it could not rule out that
anonymized data derived from the use of its products had contributed to
improving the models.
Privacy is not protection of ideas
The word
anonymized often creates a false sense of calm; removing names, email addresses
or other identifiers protects a person’s identity only partially, but it does
not necessarily remove their work or their ideas. For a researcher, then, the
main harm may not be that the system reveals who they are, but that it
appropriates confidential work that improves its capacity to compete with the
very research being developed, taking over the researcher’s intellectual
effort.
This forces
us to separate what is legal from what is private and from scientific
integrity, because a generic clause authorizing the use of conversations to
improve models is not the same as informed consent to turn unpublished research
into a corporate commercial advantage.
Buckmaster
wrote that he did not know whether his data had been used and that he was not
accusing anyone of having done so, because there is not enough evidence to
sustain the claim that OpenAI appropriated his work, but there is also no independent audit mechanism that would allow that possibility to be ruled out,
and presenting the company’s denial as the end of the matter is unacceptable.
When cases
like this appear, the researcher is usually asked to prove from the outside
what happened inside a closed model, but that is an impossible burden for the
researchers involved, because it is OpenAI that controls the training records,
retrievals, internal prompts, model versions and communications of its team.
The absence
of public proof of appropriation is not equivalent to proof of
non-appropriation either: the company should affirmatively demonstrate that the
process was isolated, which requires an external audit with confidential access
to relevant records, the capacity to trace the first appearance of the central
ideas, and the authority to verify whether materials from Buckmaster, Alpöge or
their collaborators entered, through any channel, into training, product memory
or the research process.
The dispute
over credit makes the problem worse. Buckmaster says that OpenAI proposed that
he present the company’s result without Alpöge (let us remember that he works
at Anthropic) and that, when he announced that he would make the situation
public, he received statements that he interpreted as a threat to his career.
The company
responded through Sébastien Bubeck, a French-American mathematician and
researcher specializing in artificial intelligence, who has worked at OpenAI
since 2024 and was part of the team that produced the proof and held direct
conversations with Buckmaster about publication, coordination and the
allocation of credit for the mathematical finding. Bubeck maintains that he
merely told Buckmaster that he considered it “improper” for an Anthropic
employee to sign an OpenAI paper.
Conclusion: a red line for all human intellectual work
Navier-Stokes
is an exceptional case because of the prestige of the problem, and two
questions converge in it: the first is whether OpenAI researchers or its agents
directly consulted Buckmaster and Alpöge’s conversations. The company denies
this. The second is whether those conversations may previously have been used
to train or improve the model, something OpenAI acknowledges it cannot rule
out.
The
conflict it raises extends to researchers, journalists, programmers, lawyers
and even companies that upload sensitive documents into AI systems every day:
if the platform is taking the information they produce to train models, it can
appropriate their knowledge and experience and then compete with those who
produced it, even if no record of who they were actually remains.
The
underlying problem is not that a company deliberately “reads” other people’s
sessions, but the absence of a verifiable technical separation between the use
of a product and the training of models that compete in the same field as their
users.
Neither the
researcher nor an external regulator can currently audit whether anonymized
data from a work session entered a training pipeline or not. This creates a
vacuum that different regulatory frameworks, such as the European Union’s AI Act, have begun to address through transparency requirements for training data
in general-purpose models, but without reaching the structural conflict of
interest produced by the presence of a company that is, at the same time, the
source of the tools and a competitor of its own users.
Systems
intended for intellectual work should offer confidentiality by default,
verifiable separation from training, records of provenance, understandable
opt-out mechanisms and real avenues for redress. In high-impact scientific
results, data and attribution audits should be as normal as the review of the
actual proofs behind a finding.
The
question of which human ideas entered the system, under what conditions and who
has the right to benefit from them needs an answer based on verifiable actions,
in which AI companies do not seem especially interested.
And, by the way, acceptance of authorship for the solution of a millennium problem represents, for its author, one of the greatest intellectual recognitions possible within mathematics and, as a detail, solving one of these problems means gaining access to a one-million-dollar prize and is, in this case, a door to the Nobel Prize in Physics, which grants just as much recognition and financial reward.