DeepMind AI proves 26 Busy Beaver machines in Lean 4

The preprint sits at the boundary between problem-solving and theory-building. Earlier AI wins in mathematics, such as the IMO medal systems or the Lean 4 formalization of Fermat's Last Theorem, started from problems humans had already posed and defined. The Busy Beaver setting removes that scaffolding: the machine's tape gives no obvious statement to prove, and the paper's first stage compresses raw tape into repeating-block exponents so an LLM can look for patterns at all. It is that compression step, not the proving step, that defines what the system can see.

The paper's own Antihydra formulas show why the two-way Lean loop matters. The informal rules on page 5 and page 7 allow a parameter value of zero, which makes a step count come out as -1, yet the natural trajectory starting from a blank tape never reaches that boundary, so the error stayed hidden. The four-persona rewriting loop, with a Problem Setter that can change the theorem statement, is presented as the fix for exactly this kind of hand-waving, and the Stellar BB6 Protocol uses five anti-cheat checks to stop the system from rewriting a theorem into something trivial.

Two limits remain visible. The Stage 1 pipeline names only 8 of the 26 machines; the rest came from a later Gemini 3.8 Flash system that could read the earlier proved code and was rewarded for more compressed acceleration, which suggests template transfer rather than fresh discovery. And because Algorithm 1 and 2 hard-code a repeating-block, Collatz-like view, the pipeline found only 52 candidates among 1,163 open machines and none from BB(3,3), leaving most of the search space untouched by its chosen lens.