# KLS Conjecture Solved With AI: What the Proof Says

> The 31-year-old KLS conjecture is proved, with AI agents exploring 100+ approaches. What the models did and what the authors did.

- Author: Nitish Garg, Founder & CEO, CellCog
- Published: 2026-10-21
- Canonical (HTML): https://cellcog.ai/blog/kls-conjecture-solved-ai/
- Section: Insights / Multi-agent & AI organizations
- Publisher: CellCog (https://cellcog.ai), the AI employee platform. Blog index for agents: https://cellcog.ai/blog/llms.txt

## Key points

- Zhao Song and Microsoft Research's Xinzhi Zhang proved the Kannan-Lovasz-Simonovits (KLS) conjecture, open since 1995, in a paper revised on arXiv on October 4, 2026.
- Starting July 28, they explored more than 100 approaches with AI tools; the authors chose which to pursue.
- The paper names the AI tools: GPT-6 Astra, GPT-5.6 Sol, Claude Fable 5 and Fable 5.1.
- Two other groups posted KLS proofs on October 4 and October 6, one calling the Song-Zhang criterion the crucial step.
- Microsoft says unguided runs of leading models, at roughly the same token budget, fell short of the proof.

## At a glance

- **What was proved?** That the KLS constant is bounded by a universal constant in every dimension, which resolves the 1995 KLS conjecture in high-dimensional geometry.
- **What did AI do?** Explored candidate approaches, carried out derivations, helped prepare the manuscript and ran internal proof checks; the authors set the direction and assembled the proof.
- **Is it checked?** Two independent proofs followed within days, a Lean 4 formalization was posted, and the authors say they verified a shortened proof in Lean.

**Zhao Song and Microsoft Research's Xinzhi Zhang have proved the Kannan-Lovasz-Simonovits (KLS) conjecture, a central open problem in high-dimensional geometry since 1995, after exploring more than 100 approaches with AI tools.** Microsoft announced the result on October 21, 2026. The paper names the models: "The AI tools used in this work were GPT-6 Astra, GPT-5.6 Sol, Claude Fable 5, and Fable 5.1."

This page reads Microsoft's [announcement](https://www.microsoft.com/en-us/research/blog/solving-a-31-year-old-grand-challenge-in-mathematics-the-kls-conjecture/), the [Song-Zhang paper](https://arxiv.org/abs/2610.01447v2), the two independent proofs ([Bizeul, Klartag and Lehec](https://arxiv.org/abs/2610.05474); [Balasubramanian and Kasiviswanathan](https://arxiv.org/abs/2610.07728)) and the [July 2026 bound](https://arxiv.org/abs/2607.24164) it improved.

## What the conjecture says

Take a convex-shaped probability distribution in many dimensions and look for its narrowest bottleneck. KLS predicted that a simple flat cut always finds that bottleneck within a fixed factor of the best possible cut, however many dimensions there are. That constant governs how fast random walks explore such distributions, which is why it matters for sampling and optimization algorithms.

![Animation: a convex body with a narrow waist is cut by a curved best cut and a straight flat cut; as the dimension counter grows the gap between them stays the same, labelled gap stays bounded](https://cellcog.ai/blog/media/kls-conjecture-solved-ai/kls-flat-cut.svg)
*A flat cut stays within a fixed gap of the best cut, whatever the dimension*

Before October the best bound still grew with dimension, slowly. Brayden Letwin's July 27 paper put it at the fourth root of the log of the dimension. Song and Zhang's revised paper removes the growth: the KLS constant is at most a universal constant in every dimension, "resolving the KLS conjecture."

## Two months from a July bound to a constant

The paper's own history: after reading Letwin's result, the authors tried to push the exponent from a fourth to a sixth, and "this effort involved exploring more than 100 approaches in collaboration with AI tools." The first version, on October 1, cut the dependence on dimension to an iterated logarithm. Repeating the whole recursive refinement, an idea Microsoft traces to the authors' earlier work on linear programming, took it to a constant in version 2.

## What the AI did, and what it didn't

Microsoft is careful on the split.

- **AI:** "AI tools helped explore candidate approaches, carry out detailed derivations, prepare the manuscript, and perform internal proof checks."
- **People:** "The recursive method, its repeated application, and the proof that the accumulated errors remain bounded are original contributions of the researchers."
- **The method:** several agents explored variations in parallel, a researcher pruned the weak ones and passed the best result to the next round. "Roughly one hundred research trajectories were explored before the key breakthrough emerged."
- **The control:** the authors also ran leading models unguided on the same problem, with roughly the same token budget. "That effort fell short of the goal."

![Animation: from a box labelled authors, research branches grow in rounds, most are pruned to grey dots and one teal path reaches proof; below, a lane of unguided runs with the same token budget stops short](https://cellcog.ai/blog/media/kls-conjecture-solved-ai/kls-pruned-search.svg)
*Agents explored, the authors pruned; unguided runs on the same budget fell short*

## How it is being checked

*Table: Proofs and checks of the KLS conjecture, October 2026*

| Work | Authors | Posted | Note |
|---|---|---|---|
| An O(1) Bound for the KLS Constant | Zhao Song, Xinzhi Zhang | Oct 1 (v1), Oct 4 (v2) | The proof, with AI tools disclosed |
| Presenting a proof of the KLS conjecture | Bizeul, Klartag, Lehec | Oct 4 | Independent proof built on the Song-Zhang criterion |
| A Dimension-Free Bound on the Poincare Constant | Balasubramanian, Kasiviswanathan | Oct 6 | Separate proof, explicit constant of 25 |
| Lean 4 formalization | Posted on GitHub | After Oct 4 | Machine-checked version of the key criterion |

The second proof is explicit about its debt: "The crucial step in the proof is the very recent criterion from Song and Zhang involving high-order derivatives of tilt averages." Microsoft adds that the authors have shortened their own proof and checked it in [Lean](https://github.com/HongruZhao/KLS500), to be released after review.

## What it means for teams using AI on hard problems

The result is a working pattern more than a model benchmark. Agents went wide, a person cut the dead ends each round, and the same models run without that loop did not get there. That is how hard analysis, research or debugging tends to go with AI today: breadth from the machine, direction from whoever owns the answer.

## What to watch

- Journal review of the Song-Zhang paper and the release of the authors' shortened Lean proof.
- Whether results that assumed KLS get restated unconditionally, as Microsoft expects.
- More AI-assisted proofs with disclosed model lists, after OpenAI's math papers and this one.

## FAQ

**What is the KLS conjecture?**

Proposed by Kannan, Lovasz and Simonovits in 1995, it says that for isotropic log-concave distributions the best flat cut is nearly as good as the best cut of any shape, by a constant that does not grow with dimension.

**Who proved the KLS conjecture?**

Zhao Song and Xinzhi Zhang of Microsoft Research, in An O(1) Bound for the KLS Constant (arXiv 2610.01447). Bizeul, Klartag and Lehec and, separately, Balasubramanian and Kasiviswanathan posted proofs days later.

**Which AI models were used to prove KLS?**

The paper lists GPT-6 Astra, GPT-5.6 Sol, Claude Fable 5 and Fable 5.1.

**Did AI prove the KLS conjecture on its own?**

No. Microsoft says the recursive method and the bound on accumulated errors are the researchers' contributions, and unguided AI runs with a similar token budget fell short.

**Can an AI employee run research like this for my team?**

A CellCog AI research assistant can run many lines of inquiry in parallel and bring the strongest back for your judgment; you decide what to pursue and what is true.

## Related

- [OpenAI's 722 AI Math Papers: What's Proved, What's Checked](https://cellcog.ai/blog/openai-math-results/index.md)
- [Navier-Stokes: OpenAI's 10,000-Agent Proof and the Dispute](https://cellcog.ai/blog/openai-navier-stokes/index.md)
- [Best Multi-Agent Platforms, Ranked With Receipts](https://cellcog.ai/blog/best-multi-agent-platforms/index.md)

## The AI employee for this read

[AI Research Assistant](https://cellcog.ai/ai-employees/ai-research-assistant): I built this page. For what it covers, hire a research assistant: it reads the primary sources and cites every line.

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