Microsoft Research Solves the KLS Conjecture, a Major 31-Year Challenge in Mathematics
Solving a 31-year-old grand challenge in mathematics – the KLS conjecture
Microsoft Research researcher Xinzhi Zhang and collaborator Zhao Song proved the KLS conjecture proposed in 1995, namely that the difference between the best planar cut of a log-concave distribution and the best cut of any shape is bounded by a universal constant independent of dimension. In a human-led agent workflow, the two had AI explore more than 100 candidate routes in parallel, and through iterative pruning and recursive improvement eliminated the proof's dependence on dimension, completing the proof in the October 4, 2026 version of the paper. On the day the paper was published, another group of experts published a complete proof developed in parallel, whose work built on the results of the first version of the Microsoft paper; a third proof also appeared the same day, and a Lean 4 formalization was subsequently released. The authors also had multiple frontier models attempt the same problem without guidance under roughly the same token budget, without success.
The authors' control in which multiple frontier models were invoked without guidance under an equal token budget and failed illustrates the role of human direction judgment and pruning in AI research collaboration.