Microsoft Research announced on October 21, 2026 that senior researcher Xinzhi Zhang and collaborator Zhao Song proved the Kannan-Lovasz-Simonovits conjecture, proposed in 1995. Their 140-page paper "An O(1) Bound for the KLS Constant" shows the KLS constant is bounded by a universal constant independent of dimension. The work used a human-directed agentic AI research process, which matters as a tested model for solving deep mathematical problems.
How the proof was built and verified
The paper is credited to Zhao Song and Xinzhi Zhang, first submitted to arXiv on October 1, 2026 and revised on October 4, 2026, with Zhang as submitting author. Its abstract claims a universal upper bound C for the KLS constant and a universal bound on the Poincare constant of every isotropic log-concave probability measure. Formally, the conjecture asks whether every such measure in n-dimensional space has a Cheeger constant bounded below by a universal positive constant. In plainer terms, the best flat cut remains nearly as effective as the best cut of any shape.
The method centered on agentic search with iterative human-assisted pruning. The researchers explored more than 100 candidate approaches, many involving lengthy calculations with higher-order tensors, while multiple agents tested variations in parallel. The researchers reviewed intermediate results, removed unpromising directions, and carried the strongest candidate to the next round. AI tools performed derivations, prepared the manuscript, and ran internal proof checks, while direction, selection, assembly, and verification stayed with the researchers.
The mathematical core was a new criterion that became the foundation of the final proof. The authors found it could be applied recursively, feeding each improved bound back into the same argument for a tighter result. The October 1 version reduced dimensional dependence to the iterated logarithm, which still grew with dimension. The final step repeated the entire recursive refinement, inspired by earlier work on fast algorithms for linear programming, with errors shrinking fast enough to stay bounded by a universal constant.
What this means for business use of AI research
For companies that use AI in research and development, the case shows a workable division of labor between agents and experts. Agents widened the search across roughly one hundred trajectories and handled detailed derivations, while researchers set direction and pruned ideas before the breakthrough. Small teams can replicate the pattern by running parallel explorations with clear review gates, while large organizations can formalize it across groups. The result also removes a conditional assumption: results that relied on KLS can now be stated unconditionally.
Unguided attempts with several leading AI models and roughly the same total token budget fell short of the goal, according to the post. That contrast is a limitation worth noting when selecting tools and planning budgets for AI-assisted mathematics. Independent validation followed quickly: Pierre Bizeul, Boaz Klartag, and Joseph Lehec posted a 32-page parallel proof on October 4, while Balasubramanian and Kasiviswanathan posted a separate proof with an explicit Poincare bound of 25. A Lean 4 formalization named KLS500 later verified a dimension-free constant of 500.
The marker to watch is the promised streamlined proof and its Lean verification, which the authors plan to release after final review. Publication of a shorter argument plus completed formal checks would confirm that the result holds beyond a single chain of reasoning. For business, that would signal when agent-assisted proofs are ready for wider use in sampling, optimization, and high-dimensional modeling.
