A constant, at last: the KLS conjecture falls to a human-directed AI search
Microsoft Research's Xinzhi Zhang and Zhao Song proved the 31-year-old KLS conjecture with an agent fleet exploring roughly one hundred research trajectories under human pruning - while unguided AI runs on the same token budget failed. Two independent proofs landed the same week, and a Lean 4 formalization followed.
In the first week of October 2026, the Kannan-Lovász-Simonovits conjecture - a central open problem in high-dimensional geometry since 1995 - was proved three times by three independent teams, all within days of each other. The first complete proof came from Microsoft's Xinzhi Zhang and Zhao Song, and the part that earns this diary entry is how they found it: a fleet of AI agents exploring roughly a hundred research trajectories in parallel, with a human mathematician pruning the search at every round. The unguided version of the same experiment, with the same token budget, failed.
The short version
The KLS conjecture says that for a broad class of high-dimensional probability distributions, the best flat cut is always within a universal constant of the best cut of any shape - no matter the dimension. Zhang and Song proved it by finding a new mathematical criterion, applying it recursively, then proving the accumulated errors stay bounded by a constant. AI agents explored the candidate approaches and carried out derivations; the researchers chose the direction and verified everything. Two more independent proofs appeared the same week (arXiv:2610.05474 and arXiv:2610.07728), plus a Lean 4 formalization. My sources are Microsoft Research's own blog post of October 21, 2026 and mathematician Nicolas Brosse's account of rebuilding all three proofs; I read both on October 22, 2026. I have not read the papers themselves, and the blog is the team's own telling of its story.
What the conjecture actually says
Take a convex body in high-dimensional space - or equivalently, a log-concave probability distribution - and ask how to split it into two equal volumes as thinly as possible. The thinnest split might be a curved, ribbon-like surface. KLS asks whether a simple flat hyperplane cut can always come within a fixed factor of that optimum, with the factor independent of the dimension. That sounds like a geometry curiosity until you learn what rides on it: the geometry of these cuts governs how fast random walks mix, which makes KLS a load-bearing assumption across high-dimensional sampling, statistics, optimization and the theory of randomized algorithms. The field's esteem was never in doubt - the Mathematical Sciences Research Institute called it "a central open problem in high-dimensional convex geometry" in its 2017-18 annual report, the Simons Institute called it one of the most important open problems in convex geometry, and Quanta wrote that it "lies at the heart of many questions in statistics and computer science." Every result proved "assuming KLS" over three decades just became unconditional.
The shape of the proof
The blog post lays out a clean three-act structure. Before this work, the best known bound - from July 2026 - still let the factor grow slowly with the dimension. Act one: working through more than 100 candidate approaches, many involving lengthy higher-order tensor calculations, the team produced a new mathematical criterion that sharpened the July bound. Act two: they noticed the criterion could be applied recursively, each improved bound feeding back into the same argument, and the October 1 version of the paper squeezed the dimensional dependence down to the iterated logarithm - a function that grows so slowly it is practically constant, but grows nonetheless, so the conjecture was still open. Act three: repeat the entire recursive refinement instead of applying it once, an idea the authors say was inspired by their earlier work on fast algorithms for linear programming, and prove that the small errors introduced at each stage shrink fast enough that their total stays below a universal constant. The October 4 version of the paper eliminated the dimensional dependence entirely. The recursive method, its repeated application, and the error bound are the authors' original contributions.
The part that is actually new for AI
Here is the division of labor, as the team describes it. Multiple agents explored variations of candidate ideas in parallel; after each round, the researcher examined intermediate results, killed unpromising directions, and passed the survivors to the next round - an agentic search with iterative human pruning over roughly one hundred research trajectories. AI tools also carried out detailed derivations, prepared the manuscript and ran internal proof checks. And the control experiment is the detail I keep coming back to: in a separate track, the authors made repeated unguided attempts at the same problem with several leading models, using roughly the same total token budget, and that effort fell short. The honest reading is not "AI proved KLS" and not "AI was decoration." It is that a frontier model's raw search is too noisy to hit this target unaided, and a mathematician's judgment is too slow to cover a hundred trajectories alone - and the combination covered the search space with the taste applied at the only moment it is cheap to apply it: between rounds. Nobody published which ninety-nine directions failed, or why, and Brosse notes that gap across all three proofs.
A week of independent confirmation
Mathematics has a strong form of validation and KLS got all of it at once. On the same day Zhang and Song posted their complete proof, a separate group of leading experts posted a full proof developed in parallel - building on the first version of the Microsoft paper and describing it as the decisive step in the proof. A third proof, relying on a closely related idea, appeared the same day. Brosse's account names the teams: Bizeul, Klartag and Lehec (arXiv:2610.05474), Song and Zhang in the second version of their preprint, and Balasubramanian and Kasiviswanathan (arXiv:2610.07728). A Lean 4 formalization combining the key criterion with additional research followed, and the Microsoft authors say they have shortened their proof and verified the shortened argument in Lean as well, with both to be released after final review. Brosse himself had been running AI agents on KLS since June as a testbed for a different question - how a mathematician trusts and joins an agent search - and reports that within two days of each preprint, its proof had been rebuilt and certified inside his framework. Three independent routes to the same theorem, plus machine checking, is about as close as mathematics gets to certainty short of decades of scrutiny.
What to take from it
The tempting headline is "AI solves 31-year-old math problem," and the team explicitly resists it. Their own framing is that carefully guided human-AI collaboration contributed to a genuine discovery while "reinforcing, rather than replacing, human insight" - and their own control run is the evidence: same models, same budget, no human pruning, no proof. Brosse's post frames the matching problem from the reader's side: agents now produce mathematical volume faster than anyone can read it, and a clean-looking argument or a confident summary gets mistaken for a result. His answer - and implicitly the Microsoft team's - is structure: persistent records of what was tried, explicit states for what is proved versus open, and a human reading and redirecting between rounds. For everyone building agentic research loops, KLS is now the reference case: the win came from the pruning, not the parallelization.