Microsoft researcher and contributor proves 31-year-old KLS conjecture – Unite.AI

Microsoft Research announced on October 21, 2026 that Xinzhi Zhang, a senior researcher in its machine learning and optimization group, and collaborator Zhao Song proved the Kannan-Lovász-Simonovats conjecture, a 31-year-old problem in high-dimensional geometry and probability, using a human-directed agent-based artificial intelligence research process.

The Conjecture and the New Limit

The evidence appears in the 140-page paper An O(1) Bound for the KLS Constant, attributed to Zhao Song and Xinzhi Zhang, first submitted to arXiv on October 1, 2026, and revised on October 4, 2026, with Zhang as the presenting author. The abstract states that the authors demonstrate that the KLS constant ψ_n is bounded above by a universal constant C, solving the conjecture, and that they also demonstrate a universal bound on the Poincaré constant of any log-concave isotropic probability measure.

Formally, the conjecture, proposed in 1995, asks whether every log-concave isotropic probability measure on n-dimensional space has a Cheeger constant bounded below by a universal positive constant. In the simplest terms of the Microsoft Research post, it asks whether, for a large class of high-dimensional probability distributions, the best flat or hyperplane cut is always nearly as good as the best cut of any shape, with the two differing no more than a universal constant regardless of the number of dimensions. Because the geometry of these cuts determines how quickly random walks can explore a distribution, conjecture plays a central role in high-dimensional sampling theory.

Before this work, the best-known result was the 21-page paper The KLS Constant is O(log^{1/4} n) by Brayden Letwin, submitted on July 27, 2026. It established that ψ_n is at most time constant log^{1/4} n, a limit that the Microsoft Research post notes still grew slowly as the number of dimensions increased. The new work holds with a fixed limit that does not vary with size.

The validity of the conjecture was recognized well before its resolution. The post reports that the Mathematical Sciences Research Institute’s 2017-18 annual report called KLS a central open problem in high-dimensional convex geometry, that the Simons Institute described it as one of the most important open problems in convex geometry, and that Ronen Eldan noted its broad implications in the proceedings of the 2022 International Congress of Mathematicians.

A human-directed agentic search process

According to the post, the authors began by working to improve the July 2026 result. Through a structured, human-directed process with AI agents, they explored more than 100 candidate approaches, many of which involved time-consuming calculations with higher-order tensors, and produced a new mathematical criterion that became the foundation of the final proof.

The authors then found that the criterion could be applied recursively, with each improved limit reinserted into the same argument to produce an even tighter one. In the October 1, 2026 version of the paper, this reduced the size dependence to the iterated logarithm, which grows extremely slowly but still grows with size and thus has not yet solved the conjecture. The final step was to repeat the entire recursive refinement rather than apply it just once, an idea that, according to the post, was inspired by the authors’ previous work on fast algorithms for linear programming. The authors demonstrated that the small errors introduced at each step reduce quickly enough that their total remains bounded by a size-independent universal constant, completing the proof in the October 4, 2026 release.

The post claims that the recursive method, its repeated application, and the bounded error argument are the researchers’ original contributions. AI tools helped explore candidate approaches, make detailed derivations, prepare the manuscript, and perform internal proof checks, as researchers determined the mathematical direction, selected promising ideas, assembled them into the final argument, and verified the proof. The post describes the method as agent search with human-assisted iterative pruning: As multiple agents explored variations of an idea in parallel, the researcher examined intermediate results, eliminated unpromising directions, and passed the most promising result to the next round. Before the turning point emerged, approximately one hundred research trajectories were explored.

In a separate path of the project, the authors made repeated unguided attempts to solve the same problem with several leading AI models using roughly the same total token budget as the main trial, and the post reports that this effort fell short of the goal. Ultimate success, the post states, depended on both the capabilities of the AI ​​systems and the mathematical judgment of the researchers. The post thanks Song for his contributions to the work and Victor Reis and Ishai Menache for discussions during the project.

Independent tests and mechanical verification

On October 4, 2026, the same day the full proof was published, Pierre Bizeul, Boaz Klartag, and Joseph Lehec published a 32-page proof developed in parallel, entitled Presenting a proof of the Kannan-Lovász-Simonovats conjecture. The paper’s abstract states that “the crucial step in the proof is Song and Zhang’s very recent criterion involving high-order derivatives of the skew averages” and its comments on arXiv note that the paper contains a paragraph describing the use of artificial intelligence. The abstract places KLS alongside the Bourgain slicing problem and the variance conjecture as driving forces for mathematical developments in high-dimensional geometry in recent decades.

Krishnakumar Balasubramanian and Shiva Kasiviswanathan have published a separate proof, A Dimension-Free Bound on the Poincaré Constant of Isotropic Log-Concave Measures, first presented on October 6, 2026, and revised on October 8, 2026. Its abstract describes a deterministic variational proof with an explicit bound of 25 on the Poincaré constant of any isotropic log-concave probability measure.

This is followed by a Lean 4 formalization called KLS500, which formalizes a dimensionless Poincaré constant equal to at most 500 for each log-concave isotropic probability law. According to the repository, the development is mainly based on the cumulative and suspension approach of Bizeul, Klartag and Lehec, incorporating the tilt-spectral criterion of Song and Zhang, and its verification reconstructed 1,710 mathematical modules at the kernel confidence level of zero. The Lean verification was also passed for the exact statement of the KLS conjecture defined by OpenAI, using the universal Poincaré constant 500. The project is licensed under Apache 2.0.

The Microsoft Research post describes independent proofs as one of the strongest forms of validation in mathematics, because they demonstrate that a result is not based on a single chain of reasoning. Once the conjecture is resolved, the post states, results that previously relied on it as an assumption can now be affirmed unconditionally. The authors shortened their proof and verified the simplified argument in Lean, and the post reports that they plan to publish both after final revision.