Researchers Solve 31-Year-Old KLS Conjecture
Microsoft researchers and an independent team have both released proofs for the long-standing mathematical problem.
Updated on Oct. 21, 2026 in Mathematics
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On October 4, 2026, researchers published independent proofs of the KLS conjecture, a major open problem in high-dimensional geometry that has remained unsolved since 1995. Microsoft researchers Xinzhi Zhang and Zhao Song, along with Pierre Bizeul, Boaz Klartag, and Joseph Lehec, confirmed the breakthrough via the Lean 4 formalization system.
Why it matters
The resolution of the KLS conjecture ends a decades-long search for a fundamental geometric constant. Successfully applying AI-assisted research and formal verification marks a shift in how complex mathematical proofs are generated and validated.
The researchers explored approximately 100 research trajectories using an agentic search process with human-assisted pruning. The proof was finalized and verified using the Lean 4 formalization system to eliminate dimension dependence.
The players
Xinzhi Zhang
He is a researcher at Microsoft who led the work on the KLS conjecture proof.
Zhao Song
He is a researcher at Microsoft who co-authored the paper on the KLS conjecture.
Pierre Bizeul
He is a mathematician who independently released a proof of the KLS conjecture.
Boaz Klartag
He is a researcher known for his work in high-dimensional geometry who co-authored an independent proof.
Joseph Lehec
He is a mathematician who contributed to the independent proof of the KLS conjecture.
The details
Microsoft researchers used AI agents to navigate research paths while applying a recursive mathematical criterion to solve the problem. Parallel to this effort, a separate team independently arrived at a proof that was also formally verified.
Timeline
The KLS conjecture was first proposed in 1995.
The best-known result prior to this proof was produced in July 2026.
The research paper was first published on arXiv on October 1, 2026.
A revised paper and a parallel independent proof were posted on October 4, 2026.
Microsoft Research officially announced the proof on October 21, 2026.
The Big Picture
The verification of the KLS conjecture via the Lean 4 formalization system demonstrates how automated tools are becoming standard for mathematical rigor. This discovery shifts the discipline toward more reliable computational proof-checking methodologies.
This mathematical milestone could lead to more efficient algorithms in high-dimensional data processing and statistics. Researchers plan to release a shortened proof, which may eventually streamline computational applications in various technical fields.
The takeaway
The solution to the KLS conjecture showcases the growing utility of AI in narrowing down complex research paths. Mathematics students and professionals can expect formal verification systems to become increasingly central to their work.
Further reading
For more developments in this field, explore the latest research in Mathematics.
Source note: This article includes information reported by WebProNews.
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