Large language models (LLMs) excel in competitive programming and math but have had limited success in genuine mathematical discovery due to the strict requirement for absolute correctness. Previous AI-generated mathematical proofs often lack verifiable correctness without human intervention. In response, researchers developed AlphaEvolve, a system that uses LLMs to iteratively evolve code and discover new mathematical structures. This approach led to advancements in complexity theory by improving the inapproximability bound for the MAX-4-CUT problem and tightening bounds on average-case hardness for random graph properties. The method leverages "lifting," where evolved finite structures are integrated into existing proof frameworks to yield universal theorems. Specifically, AlphaEvolve discovered a complex gadget for MAX-4-CUT, establishing a new approximation limit of 0.987. The system also found extremal Ramanujan graphs with large cuts, significantly improving lower bounds for average-case hardness. A key aspect of this research is the verifiable correctness of the discovered structures, achieved through a 10,000x speedup in verification. Although AI is proving to be a valuable collaborator, the verification process remains a critical bottleneck for future AI-assisted mathematical discovery.