Microsoft Research reports proof of the KLS conjecture

Microsoft Research reports proof of the KLS conjecture

Microsoft Research reports that researchers, including its researcher Xinzhi Zhang, have proved the Kannan–Lovász–Simonovits (KLS) conjecture, a longstanding question in high-dimensional geometry and probability. The conjecture asks whether a flat cut through a distribution can be almost as effective as the best cut of any shape, with a bound that does not grow with the number of dimensions. The reported proof establishes such a fixed bound. The source says this matters to the theory of random walks used to explore distributions and to research in geometry, probability and theoretical computer science; it does not demonstrate particular practical improvements in those fields. The researchers used AI systems to explore mathematical approaches, while they chose directions, evaluated intermediate results and assembled and checked the proof. Microsoft Research says unguided attempts with leading AI models fell short, and that the successful process depended on both the systems and researchers’ mathematical judgment. The source also reports separate proofs and a formal computer-checked version, while noting that the authors are still simplifying their proof and extending its formal verification. How readily will the streamlined proof and expanded verification enable other researchers to examine and build on the result?

Publication: 78e907fb-1a29-4cf4-8dbb-2d0eb747bf47

attribution — pass: The reading consistently attributes the proof claim, the AI-process account and the confirmation claims to Microsoft Research's report. It names Xinzhi Zhang as one of the researchers without implying he was the sole author, which is consistent with the source's mention of two researchers and co-author Zhao Song.

Now, two researchers, including Xinzhi Zhang from the Machine Learning and Optimization group at Microsoft Research, have proved the KLS conjecture

summary — uncertain: The draft says the conjecture concerns whether a flat cut through a distribution, implying any distribution, is nearly as effective as the best cut. The source limits it to a broad class of high-dimensional distributions, which the At a glance section calls log-concave. A lay reader is also not told what makes a cut effective; the source says cuts capture a distribution's narrowest bottlenecks. Smallest repair: say a flat cut through distributions in a broad (log-concave) class can capture their narrowest bottleneck nearly as well as a cut of any shape. Otherwise the title, the human-AI account and the question are faithful.

It asks whether, for a broad class of high-dimensional probability distributions, the best flat or hyperplane cut is always nearly as effective as the best cut of any shape.

date — pass: originalPublished is 2026-10-21, based on the publisher page, and matches the feed entry for this Microsoft Research blog post. The October 1 and October 4 dates refer to paper versions, not to this source's publication. The draft does not misstate either set of dates.

support — pass: Each claim is supported by the source. These include the fixed-bound proof, the link to random walks and sampling, the shortfall of the unguided attempts, the separate proofs and the Lean formalization, and the ongoing simplification of the proof. The interpretive claim that no practical improvements are demonstrated is correctly labeled and stays within the evidence. The question follows from the source's own stated aim and does not presuppose an outcome.

In a separate track of the project, the authors made repeated unguided attempts to solve the same problem with several leading AI models, using roughly the same total token budget as the main process. That effort fell short of the goal.

media — pass: The selected media is the verified direct link to the original Microsoft Research blog post. It is the most direct way for readers to reach the source.