Microsoft Research and Collaborator Resolve the KLS Conjecture With Human-Directed AI
Solving a 31-year-old grand challenge in mathematics – the KLS conjecture
- Who
- Xinzhi Zhang and Zhao Song
- What
- proved the Kannan–Lovász–Simonovits conjecture
- When
- Quote
The authors proved that the small errors introduced at each stage shrink fast enough that their total remains bounded by a universal constant, independent of the dimension.
What happened
Microsoft Research’s Xinzhi Zhang and collaborator Zhao Song report proving the 31-year-old KLS conjecture, establishing a universal bound independent of dimension for log-concave distributions. Researchers directed AI agents through more than 100 candidate approaches, then developed recursive refinements that completed the proof. The account describes independent proofs and a subsequent Lean 4 formalization combining their key criterion with additional research.
Why it matters
The account offers a transferable research method through iterative human pruning of AI search trajectories, contrasted with unguided attempts that fell short using roughly the same total token budget.
The story so far
Microsoft Research’s Xinzhi Zhang and collaborator Zhao Song report proving the 31-year-old Kannan–Lovász–Simonovits (KLS) conjecture, establishing a universal bound independent of dimension for log-concave distributions. According to Microsoft Research, the researchers directed AI agents through more than 100 candidate approaches, then developed recursive refinements to complete the proof. Their argument showed that the small errors introduced at each stage shrink quickly enough for the total to remain bounded by a universal constant. Microsoft Research also described independent proofs and a subsequent Lean 4 formalization that combined the authors’ key criterion with additional research.
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