Microsoft Researcher and Collaborator Prove 31-Year-Old KLS Conjecture

Microsoft Research's Xinzhi Zhang and collaborator Zhao Song have proved the Kannan–Lovász–Simonovits conjecture, a 31-year-old problem in high-dimensional geometry, using a human-directed agentic AI research process. The 140-page paper An O(1) Bound for the KLS Constant was submitted to arXiv on October 1 and revised on October 4, 2026. The method combined AI-agent exploration of over 100 candidate approaches with the researchers' mathematical judgment.

• The KLS conjecture states that the constant ψ_n is bounded above by a universal constant C
• The 140-page proof was posted to arXiv on October 1 and revised on October 4, 2026
• AI agents explored roughly 100 approaches; the final step drew on linear-programming algorithms
• A parallel 32-page proof by Bizeul, Klartag and Lehec appeared the same day