At the Limits of Computability: Steps Towards Autonomous AI Discovery on the Busy Beaver Frontier

Marco Dos Santos, Vivek Veeriah, Alex Havrilla, Andrew Bolt, Goran Zuzic +2

Recent breakthroughs in AI for mathematics---ranging from Olympiad medal performances to resolving long-standing open conjectures---have rapidly expanded what automated systems can achieve. Yet these successes primarily concern predefined, human-formulated problems, leaving open how AI systems can autonomously uncover mathematical structure and formulate new questions. To push beyond static challenges, we explore the Busy Beaver (BB) domain, an uncomputable frontier that is fundamentally open-ended. Because no algorithm can compute the Busy Beaver number $\mathrm{BB}(n)$ for every $n$, this domain provides a natural setting for sustained exploration through mathematical conjecture and proof. In this paper, we introduce a Gemini-based agentic pipeline that automatically discovers accelerations of Turing machines, expressed as compact mathematical rules summarizing many execution steps. Our system integrates pattern recognition, automated conjecturing, informal mathematical reasoning, and formal theorem proving in Lean. With this framework, our system autonomously discovered and formally verified novel Collatz-like accelerations for eleven machines in $\mathrm{BB}(6)$. Subject to…