The KLS conjecture was formally announced as proved on October 21, 2026, in a Microsoft Research blog post: the high-dimensional geometry problem proposed in 1995 and left open for 31 years has been given a complete proof by Xinzhi Zhang of Microsoft Research and his collaborator Zhao Song. The question sounds simple. Split a high-dimensional convex body into two equal volumes: can the best flat, hyperplane cut be much worse than the best cut of any shape? The answer is no. The gap is at most a universal constant that does not grow with dimension.
Behind the flat cut sits the speed of sampling
The weight of the KLS conjecture lies not in the picture but in the efficiency of a class of basic algorithms. Many methods in statistics and optimization work by taking random walks through a high-dimensional distribution and drawing samples; how quickly a walk spreads out depends on how wide the distribution's narrowest bottleneck is. With the conjecture proved, such bottlenecks are controlled by flat cuts and cannot deteriorate without bound as dimensions rise. The best previous result, from July 2026, still let the bound grow extremely slowly with dimension, always one step short. A body of results that used to be stated only as "true if KLS holds" can now drop the assumption.
The key criterion was sieved out by humans and AI together
The proof was not produced by handing the problem to a model in a single run. The two researchers used a human-directed agentic process in which AI explored more than a hundred candidate approaches and carried out lengthy calculations involving higher-order tensors, while the researchers inspected intermediate results, pruned unpromising directions, and passed the most promising fragments to the next round. The new criterion they sieved out first reduced the dimension dependence to an iterated logarithm, which grows extraordinarily slowly but still grows. Repeating the whole recursive argument a second time, and proving that the small errors introduced at each stage sum to a bounded total, removed the dimension dependence entirely. The October 1 version still lacked that final step; the October 4 revision completed the proof.
On the same day, a separate leading team posted an independent proof built on the same criterion, and a Lean 4 formalization followed, a form of confirmation mathematicians rate above any single paper. The control experiment is just as telling: with roughly the same total token budget, the researchers let several frontier models attack the problem unguided, repeatedly, and none got through. On the path that worked, AI supplied search and computation, while choosing the direction, recognizing the criterion, and assembling the argument remained human work, which is probably the most honest picture yet of how AI actually helps with hard research.
