# Microsoft researcher Xinzhi Zhang and Zhao Song prove the KLS conjecture, open since 1995, with human-directed AI agents

> A 140-page paper settles a central problem in high-dimensional geometry; Microsoft says unguided AI runs with a similar token budget fell short.

- Publisher: JARRUS (@jarrus.tech), thejarrus.com
- Story date: Oct. 21, 2026 (2026-10-21)
- Published: 2026-10-27
- Tags: Microsoft, Research, Agents
- URL: https://thejarrus.com/en/news/kls-conjecture-proof/

Posed by Kannan, Lovász and Simonovits in 1995, the KLS conjecture holds that for isotropic log-concave distributions, the best flat cut is within a dimension-free constant of the best cut of any shape. It governs how fast random walks explore convex bodies, and thus the speed of sampling and volume algorithms. A paper by Xinzhi Zhang of Microsoft Research and independent researcher Zhao Song shows that **the constant is universally bounded**. Its October 1 version cut the bound to an extremely slowly growing function; the October 4 version settled the conjecture. Microsoft announced the work on October 21.

Per the paper's AI disclosure, starting July 28 the authors explored more than 100 approaches with GPT-6 Astra, GPT-5.6 Sol, Claude Fable 5 and Fable 5.1. Microsoft says agents explored variations of an idea in parallel while the researcher pruned weak directions. The recursive method, its repeated application and the proof that errors stay bounded are credited to the authors. AI helped with exploration, derivations, the manuscript and internal checks. **Unguided runs of leading models with a similar token budget fell short**.

Pierre Bizeul, Boaz Klartag and Joseph Lehec posted an independent proof on October 4. It calls the Song-Zhang criterion "the crucial step in the proof," and its acknowledgments say ChatGPT found most of its proofs and ideas. Krishnakumar Balasubramanian and Shiva Kasiviswanathan posted another proof on October 6. The KLS500 project on GitHub verified a dimension-free bound in Lean 4, drawing mainly on the Bizeul, Klartag and Lehec approach. All are arXiv preprints; the authors plan to release a shortened, Lean-verified proof after final review.

## Sources

1. [Microsoft Research: Solving a 31-year-old grand challenge in mathematics – the KLS conjecture, Oct. 21, 2026](https://www.microsoft.com/en-us/research/blog/solving-a-31-year-old-grand-challenge-in-mathematics-the-kls-conjecture/)
2. [arXiv (Zhao Song, Xinzhi Zhang): An O(1) Bound for the KLS Constant, Oct. 4, 2026](https://arxiv.org/abs/2610.01447)
3. [arXiv (Bizeul, Klartag, Lehec): Presenting a proof of the Kannan-Lovasz-Simonovits conjecture, Oct. 4, 2026](https://arxiv.org/abs/2610.05474)
4. [arXiv (Balasubramanian, Kasiviswanathan): A Dimension-Free Bound on the Poincaré Constant of Isotropic Log-Concave Measures, Oct. 6, 2026](https://arxiv.org/abs/2610.07728)
5. [GitHub (HongruZhao/KLS500): KLS500](https://github.com/HongruZhao/KLS500)

Report an error: corrections@thejarrus.com