Researchers at Microsoft Research have proved the Kannan-Lovász-Simonovits (KLS) conjecture, a 1995 problem in high-dimensional geometry, using AI agents that explored candidate approaches in parallel under human direction. Senior researcher Xinzhi Zhang and collaborator Zhao Song detailed the result in a Microsoft Research blog post on Oct. 21.
The paper, "An O(1) Bound for the KLS Constant," runs 140 pages and was first posted to arXiv on Oct. 1, with a revision on Oct. 4. It shows that the KLS constant is bounded by a universal constant that does not depend on dimension. The previous best result, a July 2026 paper by Brayden Letwin, gave a bound of O(log^(1/4) n), which still grew slowly as dimensions increased.
Microsoft describes the conjecture in plain terms: for a log-concave distribution, the best flat cut is always nearly as effective as the best cut of any shape, differing by at most a fixed constant no matter how many dimensions are involved. The question matters for high-dimensional sampling, because the geometry of those cuts governs how quickly random walks can explore a distribution.
Key facts
- 01Xinzhi Zhang of Microsoft Research and Zhao Song proved the KLS conjecture, an open problem since 1995.
- 02Their 140-page paper shows the KLS constant is bounded by a universal constant, independent of dimension.
- 03AI agents explored more than 100 candidate approaches in parallel under human direction.
- 04Unguided runs of several leading AI models with roughly the same token budget fell short, Microsoft says.
- 05Bizeul, Klartag and Lehec posted an independent proof that calls the Song-Zhang criterion the crucial step.
The problem carried a heavy reputation. MSRI's 2017-18 annual report called KLS "a central open problem in high-dimensional convex geometry," and the Simons Institute described it as "one of the most important open problems in convex geometry."
Microsoft's write-up calls the method an agentic search with iterative human pruning. AI agents worked through more than 100 candidate approaches, many involving lengthy higher-order tensor calculations, across roughly 100 research trajectories before the key breakthrough. A researcher reviewed intermediate results, dropped weak directions and passed the strongest ones to the next round. AI tools also helped carry out derivations, prepare the manuscript and run internal proof checks.
The humans set the mathematical direction, selected the promising ideas, assembled the final argument and verified the proof. Microsoft credits the recursive method, and the proof that its accumulated error stays bounded, as original human contributions. The argument introduces a new criterion, applies it recursively to cut the dimension dependence down to an iterated logarithm, then repeats the refinement with bounded error to reach a dimension-free constant.
Microsoft also points to a comparison. In a separate track, unguided attempts using several leading AI models with roughly the same total token budget fell short of the result, according to the company.
Independent mathematicians reached the result in parallel. Pierre Bizeul, Boaz Klartag and Joseph Lehec posted a 32-page proof on Oct. 4 whose abstract calls the Song-Zhang criterion "the crucial step in the proof." Their paper also describes its own AI use. Separately, Krishnakumar Balasubramanian and Shiva Kasiviswanathan posted a dimension-free Poincaré constant bound of 25 for isotropic log-concave measures.
A Lean 4 formalization called KLS500, released on GitHub under the Apache 2.0 license, proves a Poincaré constant of at most 500 by combining the key criterion from the Microsoft paper with additional research.
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