NeuralStackMicrosoft Research says researcher Xinzhi Zhang and collaborator Zhao Song have resolved the Kannan–Lovász–Simonovits, or KLS, conjecture, a major open problem proposed in 1995. The result addresses whether a simple flat or hyperplane cut can perform nearly as well as the best cut of any shape for a broad class of high-dimensional probability distributions. The researchers show that the relationship holds with a fixed bound regardless of how many dimensions the shape has.
The conjecture concerns log-concave distributions, a broad family that appears in high-dimensional geometry, probability, statistics, optimization, and theoretical computer science. The geometry of these cuts also affects how quickly random walks can explore a distribution, making KLS central to the theory of high-dimensional sampling. Microsoft describes the result as removing a major theoretical obstacle to understanding randomized algorithms used across science and computing.
For years, evidence suggested the conjecture was true, but a complete proof remained out of reach. The previous best result, established in July 2026, allowed the relevant bound to grow slowly with the number of dimensions. The new work removes that dependence and establishes a universal geometric principle for the problem.
The proof emerged through a structured, human-directed collaboration with AI agents. The researchers explored more than 100 candidate approaches, including many involving lengthy calculations with higher-order tensors, while humans evaluated the possibilities, discarded unproductive paths, and redirected the search.
The process produced a new mathematical criterion that became the foundation of the final proof. The authors then applied that criterion recursively, feeding each improved bound back into the same argument. Microsoft says the resulting errors shrink quickly enough at each stage that their total remains bounded by a universal constant independent of dimension.
Multiple independent researchers have validated the work, according to Microsoft Research, and the proof has also undergone machine-checked formal verification. That combination matters because the result is not presented simply as an AI-generated conjecture or calculation, but as a human-led mathematical argument whose central criterion and formal details received additional scrutiny.
Why this matters
The KLS result strengthens the theoretical foundation for algorithms that operate in high-dimensional spaces. A dimension-independent guarantee suggests that certain sampling and optimization obstacles do not become unbounded as problems grow more complex, potentially informing future work in statistics, computational geometry, machine learning, and randomized algorithms.
It also offers a concrete example of AI being used as a research instrument rather than an autonomous mathematician. The researchers directed the investigation and supplied judgment about which ideas were worth pursuing, while AI systems helped explore a large space of mathematical possibilities. The model could influence how research teams tackle difficult problems where exhaustive human exploration would be impractical, but the proof still depends on mathematical insight, evaluation, and verification.
The KLS conjecture's resolution closes a problem that had shaped research in high-dimensional geometry for more than three decades. Its next impact will likely come from translating the new geometric guarantee into sharper analyses of sampling, optimization, and algorithms across fields that rely on navigating complex probability distributions.