Microsoft Research says Xinzhi Zhang and Zhao Song prove KLS conjecture with AI
Microsoft Research announced that senior researcher Xinzhi Zhang and collaborator Zhao Song have proved the Kannan–Lovász–Simonovits (KLS) conjecture, a 31‑year‑old problem in high‑dimensional geometry and probability. Their 140‑page paper “An O(1) Bound for the KLS Constant” was first submitted to arXiv on Oct 1, 2026 and revised on Oct 4, 2026, with Zhang as submitting author. The abstract…
Key points
- Xinzhi Zhang and Zhao Song proved the 31‑year‑old KLS conjecture, publishing a 140‑page paper on arXiv.
- The proof used a human‑directed agentic AI process that explored over 100 candidate approaches.
- Independent parallel proofs were posted the same day, including a 32‑page paper and a Lean formalization with a 500 bound.
The researchers used a human‑directed, agentic AI research process. Over 100 candidate approaches were explored by multiple AI agents, with the team pruning and iterating the most promising ideas. The recursive refinement ultimately removed any dimension‑dependent term, yielding a fixed bound. AI tools also assisted with derivations, manuscript preparation, and internal proof checks. Roughly one hundred research trajectories were examined before the breakthrough.
Independent groups posted parallel proofs the same day, including a 32‑page paper by Pierre Bizeul, Boaz Klartag and Joseph Lehec, and a Lean 4 formalization “KLS500” that verified a dimension‑free Poincaré constant bound of 500, rebuilding 1,710 modules. The Microsoft team plans to release a shortened proof and its Lean verification after review.
Microsoft Researcher and Collaborator Prove 31-Year-Old KLS Conjecture
Unite.AI · 21 October 2026
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