Prompt Used for "A Proof of the Cycle Double Cover Conjecture" [pdf] (cdn.openai.com)

OpenAI has announced a significant achievement in graph theory with the successful proof of the Cycle Double Cover Conjecture, accomplished using its advanced GPT 5.6 Sol Ultra model. The conjecture posits that every finite bridgeless loopless multigraph can be covered by cycles in such a way that each edge appears in exactly two cycles. The breakthrough came after deploying a multiagent approach that involved dynamic and diverse strategies, allowing the system to explore various mathematical formulations and maintain the independence of problem-solving efforts while ensuring rigorous validation against potential flaws in the proof. This development is noteworthy for the AI and machine learning community as it demonstrates the potential of AI systems to tackle complex mathematical problems traditionally reserved for human mathematicians. The strategies employed included a combination of adversarial testing, which rigorously checked for common pitfalls, and a systematic search for new approaches, ensuring that the proof derived was robust against various counterarguments. The success marks a leap forward in leveraging AI algorithms for solving deep theoretical issues and showcases the model's capability to manage and synthesize large-scale exploratory research, thereby opening new avenues for future AI-driven mathematical inquiries.