Mathematicians with AI agents proved the 31-year-old KLS conjecture. Unguided models with the same tokens did not

On October 21, Microsoft Research announced that its researcher Xinzhi Zhang and co-author Zhao Song have proved the KLS conjecture from 1995. For mathematics, that settles a 31-year-old problem. For AI teams, it is an unusually detailed account of what agents did in such work and what humans did.

A flat cut is enough in any dimension

The Kannan, Lovász and Simonovits conjecture concerns a broad class of high-dimensional probability distributions known as log-concave distributions. It asks whether the best flat cut through such a shape is nearly as good as the best cut of any shape. The conjecture says the gap is bounded by a constant that does not depend on the number of dimensions. The theory of random walks, which algorithms use to sample from distributions, rests on exactly this.

The best known result, from July 2026, still allowed the bound to grow slowly with dimension. The authors first sharpened it and, working with AI agents, explored more than 100 candidate approaches, many involving long calculations with higher-order tensors. That produced a new criterion. The October 1 version of the paper applied it repeatedly and pushed the dimension dependence down to the iterated logarithm. The October 4 version repeated the whole procedure, showed that the small errors do not pile up without limit and completed the proof. According to the blog, the idea for this step came from the authors' earlier work on fast algorithms for linear programming.

Others have already confirmed the result. On the same day, a separate group of experts posted its own complete proof and called the first version of the Microsoft paper "the crucial step". A third proof posted that day relies on a closely related idea. A Lean 4 formalization followed, combining the criterion from the first version with additional research.

The control run without humans matters more to AI teams than the proof itself

The blog spells out the division of labor precisely. AI tools tried variations, carried out detailed derivations, helped write the manuscript and ran internal proof checks. The humans set the direction, picked the promising ideas, assembled them into the final argument and verified the proof. Agents ran in parallel, and after each round the researcher cut dead ends and passed the best intermediate result to the next round. There were roughly a hundred such branches.

The most valuable detail is the control attempt. The authors repeatedly tried to solve the same problem with several leading models, unguided, on roughly the same token budget. They did not reach the goal. For companies hoping that one large model run can replace research, that is a useful warning. What worked here was the combination of search and human judgment that kept steering agents away from bad directions.

For mathematics, the impact is direct. Many theorems that so far held only on the assumption that KLS is true now hold unconditionally. That strengthens the foundations of convex geometry, probability theory and theoretical computer science.

The control experiment is so far just a sentence in a blog post

Microsoft has not disclosed which models ran in the control attempt, how many attempts there were or how many tokens they used. The claim that the work failed without humans is therefore, for now, the authors' own testimony, not a measurement anyone else can repeat. The same goes for "more than 100 approaches", which describes their own process.

The formal verification also has two layers. The published Lean 4 formalization combines the criterion from the first version with other research, so it is not a machine check of this exact proof. The authors do have a shortened proof verified in Lean, but they plan to release it only after final review. The result itself also does not deliver a new sampling algorithm. What changes is mainly what can be proved about the existing ones.

A released Lean proof and details of the control run will decide how strong the argument is

For mathematicians, the deciding factor will be the shortened Lean-verified proof and its acceptance through peer review. Three independent proofs on the same day are a strong signal, but review will give the result its final weight.

For the AI world, the bigger question is whether Microsoft publishes the data from the control attempt: the models, the number of runs and the token spend. Only then can anyone fairly measure how much work the human who cut the dead ends actually did. It will also show whether human-pruned search carries over to other open problems or only worked here.

Lilith's verdict

A hundred agents searched for a path while one mathematician crossed out dead ends after every round. Models without him got the same token budget and found no proof, which every manager planning to hand research to one big run should read twice.