#graph theory

OpenAI's GPT-5.6 Sol Ultra model autonomously proved the cycle double cover conjecture, a 50-year-old open problem in graph theory, in under one hour using 64 sub-agents. The model generated a 3-page PDF proof and OpenAI released the exact prompt used. This demonstrates that large language models can now tackle long-standing unsolved mathematical problems through advanced multi-agent reasoning, potentially transforming how mathematical research is conducted. It also validates the effectiveness of carefully crafted prompts and sub-agent architectures for complex reasoning. The model used 64 sub-agents working in parallel, transforming the problem into one about edge labelings and linear equations over finite fields. The prompt explicitly defined acceptance criteria, definitions, boundary conditions, and failure cases, while requiring dynamic sub-agent allocation and independent verification.