# A Microsoft researcher and Zhao Song prove the KLS conjecture > Source: > Published: 2026-10-21 17:45:31+00:00 # A Microsoft researcher and Zhao Song prove the KLS conjecture **The October proof removes dimension dependence from a 1995 bound, after a human-directed AI search helped identify the key mathematical criterion.** By [Ryan Merket](https://runtimewire.com/author/ryan-merket) · Published · Updated Primary source: [Microsoft Research](https://www.microsoft.com/en-us/research/blog/solving-a-31-year-old-grand-challenge-in-mathematics-the-kls-conjecture/) ## Why it matters The proof turns a widely used assumption about high-dimensional sampling into a theorem. Its AI contribution is also specific: guided exploration helped researchers find a useful criterion, while human mathematical judgment shaped and completed the proof. A proof posted on October 4th resolves the Kannan-Lovász-Simonovits conjecture, a 1995 problem about how efficiently random walks can explore high-dimensional probability distributions. Microsoft Research announced the result on October 21st, crediting Microsoft researcher [Xinzhi Zhang](https://www.microsoft.com/en-us/research/people/xinzhizhang/) and collaborator [Zhao Song](https://simons.berkeley.edu/people/zhao-song). Their [paper](https://arxiv.org/abs/2610.01447) establishes a bound that does not grow with the number of dimensions, the defining claim of the conjecture.… When you purchase through links in our articles, [we may earn a small commission](https://runtimewire.com/affiliate-monetization-standards). This doesn’t affect our editorial independence.