# Microsoft says researchers solved the 31-year-old KLS conjecture

- Published: 2026-10-21T18:39:50.000Z
- Updated: 2026-10-21T22:05:58.839Z
- Section: Models
- Topics: Research
- Page: https://aidailyjournal.com/en/n/2026/10/21/microsoft-says-researchers-solved-the-31-year-old-kls-conjecture/
- Sources: [Microsoft](https://microsoft.com/en-us/research/blog/solving-a-31-year-old-grand-challenge-in-mathematics-the-kls-conjecture)

## Summary

Microsoft says two researchers, including Xinzhi Zhang of Microsoft Research, proved the Kannan–Lovász–Simonovits (KLS) conjecture, proposed in 1995 and considered a central problem in high-dimensional geometry. According to the company, the result establishes that flat cuts can capture the narrowest bottlenecks of log-concave distributions within a fixed factor, regardless of the number of dimensions. The conjecture has implications for probability, statistics, optimization, algorithms and high-dimensional sampling.

Microsoft says the proof emerged from a research process involving AI agents, with the systems exploring possibilities while researchers evaluated, filtered and redirected the ideas. The company says multiple independent researchers validated the work and that a machine-checked formal verification confirmed its central criterion. The post does not provide the proof’s full technical details.

## Why it matters (editorial interpretation)

If confirmed, the results described by Microsoft could expand the role of AI agents in exploring difficult mathematical problems without removing researchers from evaluating and directing ideas. The independent validation and formal verification cited by the company also highlight the importance of external checks for this kind of claim.

_Summary written by AI Daily Journal in its own words, based on the sources above. For details, read the original articles._