EDITH

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FWIW, I think this moves up my AI timelines a bit. I think the next milestone will be "Artificial *Grothendieck* Intelligence" (AGrI): defining new general mathematical structures to solve the hardest of open problems as special cases, like the Riemann Hypothesis or P vs. NP. What impressed me about the OpenAI planar unit-distance result is not just that it solved a hard problem, but the particular way it seems to have done so. For decades, the expert intuition was that the best constructions should look roughly grid-like. That intuition was *not* obviously silly; it was held by extremely serious mathematicians (of the likes of Erdos!). And yet the model found a new family of constructions that defeated it, based on literature in other areas of mathematics. This feels like one of those cases where the "vague idea" is natural, but the solution lives in a huge space of possible design choices: which symmetries to preserve, which to break, which parameters to introduce, which ugly cases to try, which seemingly-unmotivated configurations to keep exploring. Humans tend to navigate that space with aesthetic priors. We get embarrassed by ugly constructions. We avoid paths that do not look conceptually clean early on. The model seems much more willing to "fearlessly" plough through the design space until something works. I imagine a lot of open problems in mathematics (and theoretical computer science!) may have a similar flavor, and would not be surprised if many of them start to fall soon. But for the "very big" problems, maybe extensive search through constructions in the vast existing literature is not enough. Maybe what is needed for those problems is closer to Grothendieck-style mathematics: inventing the right ambient language in which the original problem becomes a special case of a more general structure. That's what I mean by Artificial Grothendieck Intelligence (AGrI). Not merely AI that proves theorems, but AI that invents the new mathematical objects in which the theorems become *inevitable*. And why stop at one AGrI? You could imagine simulating something like the IHES school: manager agents dividing a research program into subprograms, subagents pursuing lemmas for hours or days, other agents distilling the resulting abstractions, checking them, and communicating the useful pieces back upward. One reason Grothendieck's IHES school was so successful is that its abstractions were relatively human-compressible. Once you adopted the relative perspective, the ideas could propagate through the community. But maybe that constraint has also been a bottleneck. Maybe many longstanding open problems, like those in number theory which Grothendieck felt was the hardest nut to crack, have solutions that are checkable in principle, but whose motivating abstractions are not human-compressible. In fact, I would wager that many, if not all, of these longstanding, open human conjectures live in PSPACE, but PSPACE is massive! I could imagine the AGrIs of the future might easily find non-human compressible abstractions that can be checked in PSPACE, but are infeasible for any human to check manually. Thus, the next frontier may be mathematics that is machine-discovered, machine-compressible, and machine-checkable — beautiful, in a different way to the machines, but not necessarily in the human way. I can't wait to see what open problems get solved next. What an exciting time to be alive.

Today, we share a breakthrough on the planar unit distance problem, a famous open question first posed by Paul Erdős in 1946. For nearly 80 years, mathematicians believed the best possible solutions looked roughly like square grids. An OpenAI model has now disproved that belief, discovering an entirely new family of constructions that performs better. This marks the first time AI has autonomously solved a prominent open problem central to a field of mathematics.

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