Some further thoughts on the 372 results released by OpenAI today, across 722 manuscripts. If I were to classify theorems that mathematicians prove and publish according to their groundbreaking nature, I would (very roughly) divide them into four categories: A) Non-breakthrough results. This is the overwhelming majority of published mathematics. Such results can range from solid to excellent, and some represent genuine advances in a field. But they are not hugely surprising, and would not normally be described as “breakthroughs.” B) Exceptional advances within an existing programme. These are spectacular results, but where there was nonetheless an existing credible route to the theorem, and some expectation that sufficient work would get you there. Completing the programme may require a lot of ingenuity and deep work, but mathematicians would not be shocked that the theorem had finally been proved. C) Surprising breakthroughs. These are results that clear a major barrier and substantially change the state of a field. Before the proof there was no convincing roadmap to the full result. Yet, while mathematicians would find the theorem remarkable and surprising, they would not find it completely shocking: if you asked them beforehand if it was plausible such a theorem could be proved today, most would say yes. Note: The very best mathematicians prove only a small number of results in categories B and C in a lifetime; many mathematicians never prove even one. Such results would normally belong in the very top journals, such as Annals, Inventiones, etc., and there are only a handful of them each year in any given area. Several results of this calibre by a single person would make a very strong case for a Fields Medal. D) Shock breakthroughs. These are results that, before their announcement, leading experts would have regarded as *extremely* unlikely to be proved with the current mathematical technology available. So the theorem itself would come as a shock. These are extraordinarily rare, and instant-Fields medal variety. (There is one further category I have deliberately left out, because I suspect it is empty: a correct proof of a problem for which the overwhelming consensus of top experts, until the proof came, was that a proof was so far beyond existing mathematics that a claimed solution should, on prior grounds alone, be regarded as almost impossible. I would put the Riemann Hypothesis today in that category) My current impression is that some of OpenAI's announcements today lie in A, but most fall into categories B or C. There is exactly one example in D (the Quasi-Riemann Hypothesis). It is a very big day for mathematics.