Microsoft Says A 31 Year Old Math Conjecture Is Proved
Microsoft Research published this on October 21, 2026, written by Xinzhi Zhang, Senior Researcher. Everything below comes from that first-party post.
Transcript
Microsoft says a thirty one year old math conjecture is proved. Humans directed the AI.
The KLS conjecture asks whether a flat cut is always nearly as good as a cut of any shape, whatever the dimension.
A Microsoft researcher and Zhao Song proved the two stay within a fixed constant. AI tools explored over one hundred approaches.
Researchers chose the direction and proved the error bounds. Separately, they also ran unguided attempts with leading models at a similar token budget, without success.
This is Microsoft's account of its own researcher's work. The strongest checks are two independent proofs and a machine checked formal verification.
Inboxsmith builds AI receptionists for small business. Please like and subscribe for more news.
Sources
Every claim in this video comes from the top ranking coverage of this topic. The claims and where each one came from:
- The Kannan Lovasz Simonovits (KLS) conjecture was proposed in 1995. 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. More precisely, KLS predicts that the two differ by no more than a universal constant, regardless of the number of dimensions.(Microsoft Research (microsoft.com), first party publication titled "Solving a 31-year-old grand challenge in mathematics - the KLS conjecture", October 21, 2026)
- Now, two researchers, including Xinzhi Zhang from the Machine Learning and Optimization group at Microsoft Research, have proved the KLS conjecture. They have shown that the relationship holds with a fixed bound, regardless of how many dimensions a shape has. The acknowledgements name the second researcher: "Thank you to our co-author Zhao Song for his contributions to this work". The post is published October 21, 2026, by Xinzhi Zhang, Senior Researcher, on the Microsoft Research Blog.(Microsoft Research (microsoft.com), first party publication titled "Solving a 31-year-old grand challenge in mathematics - the KLS conjecture", October 21, 2026)
- Before this work, the best known result, established in July 2026, allowed that bound to grow slowly as the number of dimensions increased. This work establishes a fundamental geometric principle: high-dimensional spaces are better behaved than previously known, and the obstacles to navigating them do not grow without bound as dimensions increase.(Microsoft Research (microsoft.com), first party publication titled "Solving a 31-year-old grand challenge in mathematics - the KLS conjecture", October 21, 2026)
- Working with AI agents in a structured, human-directed process, they explored more than 100 candidate approaches, many involving lengthy calculations with higher-order tensors. Roughly one hundred research trajectories were explored before the key breakthrough emerged.(Microsoft Research (microsoft.com), first party publication titled "Solving a 31-year-old grand challenge in mathematics - the KLS conjecture", October 21, 2026)
- The recursive method, its repeated application, and the proof that the accumulated errors remain bounded are original contributions of the researchers. AI tools helped explore candidate approaches, carry out detailed derivations, prepare the manuscript, and perform internal proof checks. The researchers determined the mathematical direction, selected the promising ideas, assembled them into the final argument, and verified the proof.(Microsoft Research (microsoft.com), first party publication titled "Solving a 31-year-old grand challenge in mathematics - the KLS conjecture", October 21, 2026)
- 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.(Microsoft Research (microsoft.com), first party publication titled "Solving a 31-year-old grand challenge in mathematics - the KLS conjecture", October 21, 2026)
- On the same day that the authors posted their complete proof, a separate group of leading experts posted a full proof that they developed in parallel. A third proof which appeared separately on the same day relies on a closely related idea. In addition, a Lean 4 formalization was subsequently released, combining the key criterion from the first Microsoft paper with additional research. The authors have also shortened their proof and verified the streamlined argument in Lean; they plan to release both after final review.(Microsoft Research (microsoft.com), first party publication titled "Solving a 31-year-old grand challenge in mathematics - the KLS conjecture", October 21, 2026)