Thank you for the great article. This is exactly the kind of clarity and skepticism we need nowadays when so many mathematicians adapt a defeatist tone and seem ready to throw mathematics under the AI hype train in anticipation of some superhuman AI oracle.
In OpenAI's companion/comments paper on page 16 Jacob Tsimerman states: "I actually briefly worked on this problem and tried to make a counterexample, but failed to make progress. On Boris Alexeev’s suggestion, I thought about this problem with the idea of making a counterexample stemming from a varying family of bounded degree number fields. Increasing degree occured to me, but is a very scary dynamic and often doesn’t work out. " i.e. the exact approach that lead to OpenAI's disproof and Boris Alexeev works for OpenAI... In other words a mathematician directly supervising/developing the model already had the key insight. Although the proof published by OpenAI does not (even) include a list of authors/contributors Boris Alexeev is an author on one of the recent Erdős problem results of OpenAI (https://arxiv.org/pdf/2603.29961) and also has a (nonAI) paper on "The Erdős unit distance problem for small point sets" (https://arxiv.org/pdf/2412.11914) so we have all the reasons to think that he was involved in some capacity in producing the proof.
Although OpenAI's claim that the result was autonomously "one shotted" by a "general purpose internal model" I think many people miss how easily these models can be steered with minimal interventions - for instance by just keeping track of model progress and restarting the model from a previous state if it goes in the wrong direction resulting in a token sequence that is technically indistinguishable from an autonomously generated sequence.
Thank you for the great article. This is exactly the kind of clarity and skepticism we need nowadays when so many mathematicians adapt a defeatist tone and seem ready to throw mathematics under the AI hype train in anticipation of some superhuman AI oracle.
In OpenAI's companion/comments paper on page 16 Jacob Tsimerman states: "I actually briefly worked on this problem and tried to make a counterexample, but failed to make progress. On Boris Alexeev’s suggestion, I thought about this problem with the idea of making a counterexample stemming from a varying family of bounded degree number fields. Increasing degree occured to me, but is a very scary dynamic and often doesn’t work out. " i.e. the exact approach that lead to OpenAI's disproof and Boris Alexeev works for OpenAI... In other words a mathematician directly supervising/developing the model already had the key insight. Although the proof published by OpenAI does not (even) include a list of authors/contributors Boris Alexeev is an author on one of the recent Erdős problem results of OpenAI (https://arxiv.org/pdf/2603.29961) and also has a (nonAI) paper on "The Erdős unit distance problem for small point sets" (https://arxiv.org/pdf/2412.11914) so we have all the reasons to think that he was involved in some capacity in producing the proof.
Although OpenAI's claim that the result was autonomously "one shotted" by a "general purpose internal model" I think many people miss how easily these models can be steered with minimal interventions - for instance by just keeping track of model progress and restarting the model from a previous state if it goes in the wrong direction resulting in a token sequence that is technically indistinguishable from an autonomously generated sequence.