Tom Zahavy

Excited to announce our recent @GoogleDeepMind paper, AlphaProof, out in @Nature today!

It has been over a year since AlphaProof achieved silver-medal standard solving International Mathematical Olympiad (IMO) problems, by teaching itself mathematics in LEAN (@leanprover). This milestone was special to me because it was often hard to believe we would reach it.

You’re crazy until you're successful

Blog: tomzahavy.com/post/how-we-ac…
Paper: nature.com/articles/s4158…

Over the past year, the Discovery team at @GoogleDeepMind, in a project led by student researcher @dsantosmarco, has made new discoveries in one of the most beautiful open problems at the edge of computation: Finding the 6th Busy Beaver (BB) number.

While most AI math systems solve human-generated problems, like theorems, this work tackles a different aspect of mathematical creativity: Discovering hidden problems.

For years, the BB Community has pursued one of theoretical computer science’s most captivating collaborative quests: finding the simplest programs that run the longest before halting. Because determining whether these programs ever stop is uncomputable, no universal algorithm can solve it, meaning new mathematical techniques need to be found and invented to make progress.

In this way, the BB challenge serves as an open ended domain for mathematical creativity and, in particular, a generator of hard math problems called Cryptids. These are tiny programs whose halting behavior is governed by and equivalent to deep, unsolved math problems in number theory, similar to the Collatz or Goldbach conjectures. Historically, the BB community had recognized ~20 Cryptids in BB(6), and the best problems were selected for the Busy Math Olympiad (BMO).

Building on their work, Marco built a pipeline powered by Gemini 4 Argon that:
1️⃣ Simulates raw Turing machine traces to compress recurring patterns.
2️⃣ Conjectures mathematical transition rules & runs synthetic experiments to test edge cases.
3️⃣ Proves the equivalence formally in Lean 4.

Using this pipeline, Marco and the team discovered 20 new potential Cryptids and verified 33 more machines previously studied in the BB community. All the proofs were open sourced and shared in the community discord channel and we hope that they will help the community to make progress towards resolving BB(6). We also contributed related halting/non-halting proofs directly to the community GitHub repository.

Finding these Cryptids shows hidden links between simple programs and number theory, and helps mathematicians map the limits of computation. I'm also excited about Gemini 4 Argon's ability to help researchers both solve hard problems and accelerate scientific discoveries, of which Busy Beaver is just one example.

Work led by our student researcher @dsantosmarco with paper contributions from @vivek_veeriah, @Dahoas1 , Andrew Bolt, @zuza777 , Satinder Singh, and myself with additional contributions from Hussain Masoom, @maxencefaldor , @MoritzFirsching and Petar Sirkovic.

Read more about it in our paper: arxiv.org/abs/2610.22603

I was fortunate to advise @AatuSelkee on this and to collaborate with @ericmalmi , Severi and @Xidong_Feng . Aatu did a fantastic work on building a puzzle generator that you can condition by theme, partial board position or best move.

This extends our earlier puzzle generation work, and big parts of that are now open source, including the first open-weights models for chess puzzle generation. We're excited to see what the community will do with it.

Paper and demo in Eric's post below 👇

🙏
xenaproject.wordpress.com/2026/10/01/to-…

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