KLS Conjecture Solved After 31 Years With Human-Guided AI
Xinzhi Zhang and Zhao Song prove the 31-year-old KLS conjecture, using human-guided AI and Lean 4 checks to establish a dimension-free bound universally.
Summary
Detailed on October 21, 2026, Xinzhi Zhang of Microsoft Research’s Machine Learning and Optimization group and co-author Zhao Song proved the Kannan-Lovasz-Simonovits conjecture, proposed in 1995. For log-concave distributions, they established that a flat hyperplane cut captures the narrowest bottleneck within a universal constant of any shaped cut, regardless of dimension. The dimension-free result strengthens high-dimensional geometry, probability, statistics, optimization, algorithms and computer science because these bottlenecks determine how efficiently random walks sample distributions. The previous best bound, published in July 2026, still grew slowly with dimension.
Human-directed AI explored more than 100 candidates, many using higher-order tensors, while the researchers pruned paths, selected ideas, assembled the proof and verified it. A new recursive criterion reduced dimension dependence to an iterated logarithm in the October 1 paper, but repeating the full refinement, an idea inspired by earlier linear-programming algorithms, proved accumulated errors remain universally bounded in the October 4 version. Unguided attempts with several leading AI models and roughly the same token budget failed. Also on October 4, an independent group posted a parallel proof built on the criterion, while a third proof used a related idea; a subsequent Lean 4 formalization added machine-checked confirmation. Results previously conditional on KLS can now become unconditional. Zhang and Song have shortened their proof, verified it in Lean and plan to release both after final review.
Positives
- The October 4 proof establishes a universal constant independent of dimension, fully resolving the conjecture proposed in 1995.
- More than 100 AI-assisted candidate approaches produced the recursive criterion underlying the final argument.
- Independent parallel and third proofs appeared on October 4, providing separate confirmation of the central mathematical idea.
- Lean 4 formalization supplied machine-checked verification using the proof’s key criterion and additional research.
- Results in convex geometry, probability and theoretical computer science that assumed KLS can now be stated unconditionally.
Risks & concerns
- The July 2026 best result still allowed its bound to grow as dimensionality increased.
- The October 1 version achieved only iterated-logarithmic dependence, leaving the conjecture unresolved until the October 4 refinement.
- Unguided attempts by several leading AI models failed despite using roughly the main process’s total token budget.
- The shortened proof and its Lean verification remain pending final review and public release.