AI-assisted proof search helps researchers resolve the KLS geometry conjecture

A longstanding question in high-dimensional geometry asks whether the best flat cut can stay within a fixed factor of the best cut of any shape. Song and Zhang’s October preprint presents a dimension-independent bound, resolving the Kannan–Lovász–Simonovits conjecture as stated in the paper.[1][3]

The Kannan–Lovász–Simonovits conjecture, proposed in 1995, asserts that an appropriate measure of this geometric bottleneck can be controlled by a universal constant, no matter how many dimensions are involved. It sounds abstract because it is. Yet the absence of such a dimension-free guarantee has mattered to the theoretical analysis of random walks, sampling, optimisation and related tools used throughout mathematical science.[1][3]

An almost-flat answer is not enough

Previous work had brought the dependence on dimension down substantially without eliminating it entirely. An upper bound that grows extraordinarily slowly with dimension is still a different mathematical statement from a constant that never grows at all. Song and Zhang’s paper, submitted on October 1 and revised on October 4, establishes the dimension-independent result. The arXiv record states their proof resolves the conjecture.[1]

Microsoft Research’s October 21 account describes a sequence that is more interesting than the claim “AI solved a famous math problem.” A human-guided team used agents to explore over 100 candidate approaches, including derivations involving higher-order mathematical objects. The researchers identified a useful criterion and found that the resulting improvement could feed back into itself. They then pushed that recursion beyond a very slowly growing bound to a universal constant.[3]

The important mathematical step was a criterion that could be applied recursively. AI agents expanded the search over possible approaches; the researchers selected and developed the argument. Microsoft’s account describes the contribution as a directed research process rather than an autonomous proof delivered by a model.[1][3]

Two proofs complicate the hero story

A second paper, posted on October 4 by Pierre Bizeul, Boaz Klartag and Joseph Lehec, also presents a proof of the conjecture. Its abstract explicitly identifies the recent Song–Zhang criterion involving high-order derivatives as the crucial ingredient. This parallel result matters for both mathematics and the sociology of AI. It demonstrates that the central new idea was legible enough for other experts to use, while making it impossible to reduce a field-wide breakthrough to a single software demonstration.[2]

Microsoft reports expert examination and a subsequent machine-checked formal verification. The separately posted Bizeul–Klartag–Lehec proof explicitly credits the Song–Zhang criterion, making the shared mathematical ingredient visible beyond the company’s account.[3]

The contribution that might generalise

The AI angle here is methodological. Long proof searches contain many conceivable routes, most of them unpromising. Agents can work through variations, algebraic consequences and potential dead ends that would consume large amounts of a researcher’s time. Human direction determines which results are meaningful, which reductions are valid, and when to change the question. Even if the same theorem had been proved without AI, a disciplined record of where it altered the search process would remain valuable.[3]

The breakthrough’s reusable part is the criterion and the argument built from it. The AI contribution is the documented expansion of the search that helped researchers reach that point—an account of collaboration worth examining alongside the theorem.[1][2][3]

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