With new ideas and human-directed AI, researchers solve a 31-year-old math mystery

A longstanding open problem in mathematics, the Kannan–Lovász–Simonovits (KLS) conjecture, has been proven by researchers including Xinzhi Zhang from Microsoft Research. Proposed in 1995, the conjecture posits that in high-dimensional spaces, the best flat cut through a probability distribution is nearly as effective as the best cut of any shape, differing by no more than a universal constant. This resolves a major theoretical obstacle in understanding the efficiency of randomized algorithms used across science and computing.

The proof was achieved through a human-directed AI-assisted process, where researchers explored over 100 candidate approaches, refining mathematical bounds iteratively. Using structured agentic search, the team identified a new criterion that became the foundation for the final argument, ultimately eliminating dependence on the number of dimensions. The breakthrough relied on recursive refinement, inspired by prior work on linear programming algorithms, to ensure errors remained bounded by a universal constant.

Two independent proofs of the KLS conjecture were posted on the same day, with a third proof and a Lean 4 formalization released shortly after. The parallel efforts, including one building directly on the Microsoft team’s initial findings, strengthen confidence in the result by demonstrating it is not dependent on a single chain of reasoning. Formal verification further solidifies the proof’s validity and accessibility for broader mathematical scrutiny.

The resolution of KLS provides an unconditional foundation for research in convex geometry, probability, and theoretical computer science, enabling previously conditional results to be stated without assumptions. The project also highlights the role of guided human-AI collaboration in scientific discovery, showing that AI tools augmented rather than replaced human mathematical judgment in achieving the breakthrough.