Three AI-assisted teams approached a 31-year maths puzzle. The disagreement is about who supplied the ideas

Imagine trying to move at random through a room with a million dimensions. You cannot visualize the room, much less inspect every possible corridor, yet many statistical algorithms effectively confront versions of this problem. Their ability to sample from complicated probability distributions depends on whether narrow passages obstruct the walk.[1]

In 1995 Ravi Kannan, László Lovász and Miklós Simonovits proposed a geometric principle: for a broad family of probability distributions, the best flat cut should be within a universal constant factor of the most restrictive cut of any shape. The striking word is universal. The constant should not get worse merely because a problem has more dimensions.[1]

On October 21, 2026, Microsoft Research announced that Xinzhi Zhang and Zhao Song had settled the KLS conjecture, following a series of arXiv submissions. Other mathematicians had posted proofs in the same early-October period. Rather than offering a tidy story in which one chatbot solved a famous problem, the papers reveal three different forms of cooperation between mathematical judgment and generative AI.[1]

Why a cut through a shape matters to computing

A flat cut is the high-dimensional analogue of slicing a loaf of bread with a plane. An arbitrary curved cut can snake through the shape and perhaps find a much thinner bottleneck. The KLS conjecture says, roughly, that for log-concave probability distributions the best flat slice is never arbitrarily worse than that curved alternative as dimension increases. Log-concavity encompasses many well-behaved distributions used in optimization and statistics.[1]

Why care? Randomized algorithms estimate quantities and explore distributions by taking repeated steps. If the underlying geometry contains a very tight bottleneck, a random walk may take an extremely long time to traverse it. A bound that stays independent of dimension gives theoretical tools for studying how efficiently such procedures mix. This is a foundational mathematical guarantee, not a claim that ordinary AI tools will suddenly compute a million-dimensional answer in seconds.

Song and Zhang attacked the conjecture through a sequence of improving inequalities and a criterion based on higher-order mathematical quantities. Their account says human researchers directed AI agents down more than one hundred candidate research paths, pruning unpromising approaches. A recursive improvement first reduced the dimension dependence to a very slowly growing function. Applying a refinement again, and showing the accumulated error stayed bounded, delivered the crucial constant.

Three papers, three stories about AI

In the Microsoft account, the researchers selected the direction, found the structural way forward and used AI for exploration, derivations and internal checking. They say comparable attempts that left models unguided did not reach a proof. That is a description of how this particular team worked, not a universal experiment proving one workflow superior in all mathematics.[1]

A second manuscript, by Pierre Bizeul, Boaz Klartag and Joseph Lehec, explicitly credits ChatGPT with most of its proofs and mathematical ideas, while identifying an important construction suggested by the human authors. Its abstract also acknowledges Song and Zhang's earlier criterion as a crucial step. In this account the model's contribution is unusually prominent, but the arguments still had to be understood, checked and assembled by professional mathematicians.

A third paper by Krishnakumar Balasubramanian and Shiva Kasiviswanathan describes substantial help from frontier models. These are not three identical demonstrations. They involve overlapping mathematical developments and different decisions about research direction, validation and authorship. Independent routes toward the same result strengthen confidence, but the historical credit cannot be reduced to counting how many times an AI assistant appears in acknowledgments.

An additional distinction concerns verification. A human-reviewed proof, a research preprint and a Lean formalization are not interchangeable certificates. Formal checking can eliminate classes of gaps once the underlying mathematical statement is correctly encoded. It does not automatically adjudicate priority, originality or whether a contributor was indispensable. At the time of the announcements, the new papers were available as preprints; full peer-review histories were not yet established.[1]

A new scientific instrument, not a magic theorem button

The surprising development is that a language model can now be a productive participant in advanced mathematical research without taking a single, stable role. It may perform elaborate algebra, invent a lemma, pursue a false trail, or help turn a promising idea into a structured proof. One team may use it as a tireless assistant, another as a more independent conjecture engine.

That multiplicity should improve how such breakthroughs are reported. The important evidence is not the phrase 'AI solved it' but the dated manuscripts, mathematical dependencies, tool acknowledgments and checkable final arguments. There will be future arguments about how to assign credit; the geometry itself does not depend on settling them.

For the reader, KLS offers a double discovery. A long-standing principle says that dauntingly high-dimensional spaces can still have tractable structure. And the process by which it was proved shows a research community beginning to discover what structure is needed to work productively with AI.[1]

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