Conjecture KLS: Microsoft researcher and Zhao Song solve 31-year-old geometry problem, with AI agents for calculations

Xinzhi Zhang and Zhao Song demonstrate a constant terminal for Kannan-Lovász-Simonovits conjecture. AI was used as a computational and verification assistant, with humans keeping their hands on the strategy.

Formulated in 1995, the Kannan-Lovász-Simonovits (KLS) conjecture It deals with large-scale isotropic log-concave probability distributions: it asserts that their Cheeger constant remains diminished by a universal constant, i.e. that a simple cut by a hyperplane is almost as effective as the best cut possible, regardless of the size. The result is central in geometry of large dimensions, in probability and for analysis of random steps. Xinzhi Zhang, researcher at Microsoft Research, and Zhao Song claim the evidence in an arXiv article entitled "An O(1) Bound for the KLS Constant", posted on October 1 and revised on October 4.

A directed AI, not autonomous

According to the authors, AI agents have been able to explore more than a hundred runways, supporting long tensor calculations, bypasses and step verification. The choice of direction, the selection of promising ideas, the assembly of the final argument and its verification remained human; attempts made without supervision with the best models of the moment have not been successful. Other independent evidence is published in parallel, and a formalization in Lean 4, called KLS500, reinforces the result.