Microsoft researchers prove the KLS conjecture

Oct 22

1. 14:11

   Rohan Paul

   Microsoft researchers used AI agents to explore over a hundred paths and proved the 31-year-old KLS conjecture

   Microsoft researchers used AI agents to explore about 100 research paths and completed a proof of the Kannan-Lovász-Simonovits (KLS) conjecture, which was proposed in 1995 and is 31 years old; the resulting manuscript is about 140 pages. AI was responsible for exploring candidate methods, carrying out derivations, assisting in writing the manuscript, and performing internal proof checks, while the recursive method and error bounds that completed the proof are credited to the two researchers. The conjecture can be understood as: a random walk inside a convex shape will not be trapped by narrow passages, and can efficiently explore the entire shape even in high-dimensional space.

   Expand summary
2. 03:00

   Unite.AI

   Microsoft researchers and collaborators prove the 31-year-old KLS conjecture

   Microsoft Research announced that senior researcher Xinzhi Zhang and collaborator Zhao Song, through a human-led agentic AI research workflow, proved the KLS conjecture proposed in 1995; the paper "An O(1) Bound for the KLS Constant" was submitted to arXiv on October 1 and revised on October 4. During the research process, AI agents explored more than 100 candidate paths, but when the authors gave multiple leading models the same token budget for unguided attempts, they failed to achieve the goal. In the same period, there were also independent proofs by Bizeul, Klartag, and Lehec, as well as the Lean 4 formalization project KLS500, which completed machine verification.

   Expand summary
3. 00:00

   Microsoft ResearchFirst-partySelected

   Microsoft Research proves the 31-year-old unsolved KLS conjecture

   Microsoft Research researcher Xinzhi Zhang and co-authors proved the KLS conjecture proposed in 1995, giving a dimension-independent fixed bound; the previous best result only allowed the bound to grow slowly with dimension. The proof came from a human-guided agent search process, in which the team explored more than 100 candidate paths, with researchers evaluating, pruning, and redirecting them; an independent team also gave a complete proof at the same time, followed by Lean 4 formal verification. The authors say earlier unguided end-to-end attempts with multiple frontier models failed to achieve the goal.

   Expand summary