Microsoft Research Proves 31-Year-Old KLS Conjecture Using Human-Directed AI

• Xinzhi Zhang and Zhao Song proved the KLS conjecture, showing that the best flat cut is within a universal constant of the best cut of any shape for log-concave distributions, regardless of dimension.
• The proof eliminates the previous dependence on dimension, which had grown slowly with the number of dimensions in the best known result from July 2026.
• The breakthrough was achieved through a human-directed agentic process where AI explored over 100 candidate approaches while researchers pruned unpromising directions and selected the final argument.
• Independent validation includes parallel proofs by other experts and a Lean 4 formalization, confirming the result as a genuine underlying principle rather than a fragile technical argument.

Technical Method

• The authors improved a July 2026 result to create a new mathematical criterion, which was then applied recursively to reduce dimension dependence to the iterated logarithm in the October 1, 2026 version of the paper.
• The final step involved repeating the recursive refinement to prove that accumulated errors remain bounded by a universal constant, completing the proof in the October 4, 2026 version of the paper.

Verification and Impact

• Multiple independent researchers posted full proofs on the same day as the authors, with one group citing the Microsoft paper's criterion as the crucial step.
• The resolution allows results in high-dimensional geometry, probability, and theoretical computer science that previously relied on the conjecture to be stated unconditionally.