TLDR: Give a language model a difficult theorem and enough time, and it can produce a remarkable amount of mathematics. The hard part is deciding which of it deserves to survive. AlphaProof, Goedel-Prover-V2, and DeepSeekMath-V2 treat formal and informal proof as a laboratory for recursive improvement: a generator proposes candidates, a checker decides what becomes training material, and the feedback has to stay informative as the proposer gets stronger. We cover how Lean's kernel creates a useful asymmetry between finding a proof and checking one, how AlphaProof turns exploration into training (including test-time reinforcement learning), why useful theorems still have to be discoverable, and what changes when verification itself is learned. We close with the boundary a checker cannot cross'whether the stated theorem is the one intended'and the measurements that would show a genuine recursive benefit. Give a language model a difficult theorem and enough time, and it can produce a...
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