Happy to finally do a deep dive on multi-agent with @dwarkesh_sp! None of it would have happened without the great work on multi-agent from my @OpenAI teammates @kevinleestone, @mikegmalek, @__eknight__, @amuellerml, @zhangir_azerbay, @CheukHeiChu, and many others.
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Yes, this result cost millions of dollars.
But remember that when @OpenAI announced o3 it cost ~$500,000 to score 87.5% on ARC-AGI 1. Today, Astra scores higher for ~$20.
In 2025 it took us and GDM an enormous amount of compute to achieve IMO gold. For the 2026 IMO, anyone with a $20/month ChatGPT subscription could do it.
Massively scaling test-time compute gives us a glimpse of the future. I believe that a year from now everyone will have an AI at their fingertips capable of solving problems of this caliber.
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As part of the GPT-6 Astra launch, we announced that Astra had given an improvement to the longest gap between primes by roughly a log log n factor; the first such improvement since the 1930's! (More recent progress by subsets of Ford, Green, Konyagin, Maynard and Tao and more recently by GPT 5.6 Sol were by logloglog n factors.) (1/4)
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We have heard concerns about the proof of the existence of a non-sofic group, particularly its reliance on results of Kun and Kun-Thom. The original lean certificate is end-to-end formalizing every necessary ingredient from those papers. In doing so, we encountered minor imprecisions, which Thom himself describes at the level of typos: Such issues are commonplace in the mathematical literature and do not affect the results we cite.
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It is remarkable—and takes a moment to process—how quickly the next generation of models will accelerate our research, and breathe new life into old problems.
This was an incredible project done by an incredible team (and model) in just a week. Ad Astra we go ✨
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yes, nonsofic groups exist: this statement is one of many new beautiful results proved by Astra, our next major model.
We're releasing 10 such Astra proofs, complete with lean certificates and CoT walkthroughs for each of them. The results are wide-ranging, from von Neumann algebras (disproof of Connes' Rigidity Conjecture) to better bounds for high dimensional sphere packing, for circuit complexity, for monochromatic triangles in multicolored graphs, and more.
More thoughts here:
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What would Erdos have done if he had had access to GPT-5.6 Sol?
Congrats to Levent for discovering this!
OOC I had an internal version of Codex attempt a proof too (without web search), and it discovered (essentially) the same counterexample! It wrote up a nice summary of the strategy here:
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Yesterday, we made GPT-5.6 Sol Ultra generally available. Today, we're sharing that it produced a proof of the 50-year-old Cycle Double Cover Conjecture using 64 subagents in just under one hour. We're sharing the prompt and proof below. We're excited to see what you all do with Ultra!
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Btw this was done with gpt-5.6. One million lines of LEAN to formalize the unit distance solution.
Pretty cool that this can be done by a single person in a short time (as opposed to a team over year(s))!
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This is incredible!!! My first paper in my PhD back in 2018 was on this problem (getting a 2^{logstar n} dim bound); since then, I have been thinking about it on and off for at least 7 years.
It's really crazy how things turned out. Last month at OpenAI, our model disproved the famous unit distance conjecture, and the number-theoretic techniques are EXACTLY what you need to settle this furthest pair problem!!!
I am so excited about the future of TCS and can't wait to see more open problems I care about being resolved by collaboration between AI and humans :)
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