Researchers Prove Long-Standing KLS Conjecture

The 31-year-old geometric problem was solved using an AI agent search process alongside independent human research.

Updated on Oct. 21, 2026 in Mathematics

Researchers Prove Long-Standing KLS Conjecture

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Microsoft researchers and an independent team have both published proofs for the long-standing KLS conjecture in high-dimensional geometry. The Microsoft team achieved their result using an AI-driven agentic search process that explored roughly 100 research trajectories.

Why it matters

The KLS conjecture has remained an open problem in geometry since 1995, and its resolution provides foundational insights into high-dimensional data distribution. The successful application of human-guided AI agents to verify such complex theoretical proofs signals a potential shift in how mathematicians tackle long-standing problems.

The KLS conjecture—which addresses the behavior of log-concave measures in high dimensions—remained unsolved for 31 years. Researchers used an agentic process that applied a mathematical criterion recursively to remove dimension dependence from the proof.

The players

Xinzhi Zhang

A researcher at Microsoft who utilized AI-assisted search methods to address complex geometric proofs.

Zhao Song

A researcher at Microsoft who co-authored the AI-driven proof of the KLS conjecture.

Pierre Bizeul

A mathematician who independently posted a proof of the KLS conjecture alongside colleagues.

Boaz Klartag

A researcher who independently contributed to the KLS conjecture proof posted on October 4.

Joseph Lehec

A mathematician who collaborated on the independent proof of the KLS conjecture.

The details

The Microsoft team utilized an agentic search process where AI agents generated potential research paths, which human researchers then pruned to maintain only the most promising directions. The final proof was verified using Lean 4, a formalization system that turns mathematical reasoning into machine-executable code. This process enabled the team to rigorously evaluate 100 potential trajectories to resolve the 31-year-old geometric challenge.

1995: The KLS conjecture was first proposed as an open problem.
July 2026: The previous best-known result for the KLS conjecture was established.
October 1, 2026: The Microsoft research paper was first posted on arXiv.
October 4, 2026: A revised paper and an independent proof were posted on arXiv.
October 21, 2026: Microsoft Research formally announced the results.

The Tech Race

The simultaneous arrival of two independent proofs resolves a 31-year-old objective that had previously seen incremental progress. This outcome sets a new standard for high-dimensional geometric research by demonstrating that human-AI collaboration can successfully clear long-standing mathematical hurdles.

This development primarily affects the academic and computational mathematics communities by providing a verified framework for future high-dimensional data analysis. There is no immediate software or consumer tool release associated with this proof, as the findings remain in the peer-review and documentation phase.

The takeaway

The resolution of the KLS conjecture proves that agentic AI search processes can successfully navigate complex mathematical landscapes. Mathematicians should watch for the release of the shortened proof and the complete Lean 4 verification expected following final review.