Microsoft Research announced a proof of the 31-year-old KLS conjecture. Xinzhi Zhang and Zhao Song authored the 140-page paper. The conjecture concerns high-dimensional probability distributions. Their proof shows the KLS constant is bounded by a universal constant. Previous best bound still grew with dimension. The work used a human-directed agentic AI research process. Over 100 candidate approaches were explored. A recursive method with repeated application completed the proof. Independent proofs by other teams followed within days. A Lean 4 formalization verified a dimension-free Poincaré constant of at most 500.
What changed
The best known bound for the KLS constant was O(log^(1/4) n).
What it unlocks
Proving a universal constant bound for high-dimensional probability distributions.
- 31-year-old KLS conjecture
- 140-page proof paper
- 100+ candidate approaches explored
- 500 Poincaré constant bound in Lean verification
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