OpenAI: An internal version of our next generation main model, which has achieved 10 new results on long-unresolved open issues in mathematics and theoretical computer science. The calculation cost used is approximately $2,000 in tokens based on the GPT-5.6 Sol API rate. We will publish these manuscripts, official Lean proof certificate, and an explanation of the reasoning process for mathematicians to review these results and further develop them based on relevant research ideas. These results cover fields such as sphere accumulation, coding theory, group theory, quantum complexity, lattice cryptography, and extreme value combinatorics. These include proving the existence of non-sofic groups and exponentially improving the accumulation boundaries of high-dimensional spheres.

OpenAI: An internal version of our next generation main model, which has achieved 10 new results on long-unresolved open issues in mathematics and theoretical computer science. The calculation cost used is approximately $2,000 in tokens based on the GPT-5.6 Sol API rate. We will publish these manuscripts, official Lean proof certificate, and an explanation of the reasoning process for mathematicians to review these results and further develop them based on relevant research ideas. These results cover fields such as sphere accumulation, coding theory, group theory, quantum complexity, lattice cryptography, and extreme value combinatorics. These include proving the existence of non-sofic groups and exponentially improving the accumulation boundaries of high-dimensional spheres.