By Hamza Faisal

A geometry problem that has beaten mathematicians since 1995 now has a proof — and the researchers behind it say a tightly supervised team of AI agents did much of the legwork. Microsoft Research announced on October 21 that senior researcher Xinzhi Zhang, of its Machine Learning and Optimization group, and collaborator Zhao Song have resolved the Kannan–Lovász–Simonovits (KLS) conjecture in a 140-page paper posted to arXiv.

The KLS conjecture, proposed by Ravi Kannan, László Lovász and Miklós Simonovits, asks whether the best flat cut through a broad class of high-dimensional probability distributions is always nearly as effective as the best cut of any shape — within a fixed constant that never grows, no matter how many dimensions you add. Because the geometry of these cuts governs how quickly random walks can explore a distribution, the conjecture sits at the foundation of high-dimensional sampling theory, with deep links to statistics, optimization and the algorithms that power modern computing.

Progress had come in slow steps for three decades, each result shrinking how fast the bound grew with dimension. In July 2026, Brayden Letwin set the latest record with a bound that still grew slowly as dimensions increased. The new work removes that growth entirely: a fixed, universal constant.

A human-directed hunt through 100 approaches

What makes this proof notable is not just the mathematics but how Microsoft says it was found. Rather than handing the problem to an autonomous system end-to-end, the researchers ran a human-directed agentic process: AI systems explored more than 100 candidate approaches while the humans evaluated results, pruned dead ends and redirected the search. Roughly a hundred trajectories preceded the key breakthrough.

The team even ran a control track. Repeated unguided attempts by several leading AI models, using roughly the same total token budget, fell short. The difference, Microsoft says, was judgment: the researchers chose the promising directions, assembled the argument and checked it — the AI was the engine, not the driver.

Validation came fast

The claim has already passed a tough informal test. Within days of the paper appearing, a rival team including Boaz Klartag — who set some of the earlier records on the problem — posted a proof built on its central criterion. A third proof and a machine-checked Lean 4 formalization followed. The authors say they have shortened the proof, verified the streamlined version in Lean, and plan to release both after final review.

For the wider field, the payoff is immediate: many results that previously had to assume the conjecture can now be stated unconditionally. The Simons Institute has called KLS one of the most important open problems in convex geometry, and Quanta Magazine wrote that it lies at the heart of many questions in statistics and computer science.

The next chapter of machine-assisted math

The KLS result joins a growing list of hard problems where AI agents serve as tireless explorers while humans keep the compass. It is also a useful corrective to both extremes of the debate: unguided AI failed at this one, and unaided humans spent 31 years on it. The proof that finally landed was a genuine collaboration — one that suggests the next decade of mathematical breakthroughs will be built by teams like this, where the machines do the searching and the people do the thinking.

By Hamza Faisal