The bound no longer grows, July's still did

The Microsoft Research post is dated Published October 21, 2026. The judgment first: the Kannan–Lovász–Simonovits conjecture, KLS, is written as proved. The gap between the best flat cut and the best cut of any shape is held by a constant that does not depend on dimension. The page does not print the value of that constant. AI took part in searching candidate paths and in internal checks. Researchers set the direction and checked the final proof. Unguided attempts, at roughly the same token budget, did not reach the goal.

The conjecture was proposed in 1995. It asks whether, for a broad class of high-dimensional probability distributions, the best flat cut is always nearly as effective as the best cut of any shape, with the two differing by no more than a universal constant, regardless of the number of dimensions. The best known result as of July 2026 still let that bound grow slowly as the number of dimensions increased. This post says the new proof removes the dependence on dimension. Obstacles in high-dimensional space no longer grow without bound as dimension increases. That is the geometric conclusion on the page, not a table that already lists the constant.

The authors include Xinzhi Zhang of the Machine Learning and Optimization group at Microsoft Research. The acknowledgements name co-author Zhao Song. They first tightened the July result, then searched with AI agents in a human-directed process. They explored more than 100 candidate approaches, many of them long calculations with higher-order tensors. That search produced a new mathematical criterion. The October 1, 2026 version of the paper reduced the dependence on dimension to the iterated logarithm. The iterated logarithm grows very slowly, but it still grows with dimension, so that version had not fully resolved the conjecture.

The October 4, 2026 version repeats the same recursive refinement instead of applying it once. The page says the idea came from the authors' earlier work on fast algorithms for linear programming. They proved that the small errors introduced at each stage shrink fast enough that their total stays bounded by a universal constant, independent of dimension. The recursive method, its repeated application, and the proof that the accumulated errors stay bounded are written as the researchers' own contributions. AI tools helped explore candidates, carry out derivations, prepare the manuscript, and run internal proof checks. Choosing the ideas, assembling the argument, and verifying the proof are credited to the researchers.

On the same day, a separate group of leading experts posted a full proof and called the first version of the Microsoft paper the crucial step. A third proof appeared separately on the same day and relies on a closely related idea. A Lean 4 formalization followed, combining the key criterion from the first Microsoft paper with additional research. The authors have also shortened their proof and verified the streamlined argument in Lean. They plan to release both after final review. Release after review is a plan, not a shortened text already public.

On a separate track, the authors made repeated unguided attempts with several leading models, using roughly the same total token budget as the main process. That effort fell short of the goal. The page therefore says success depended both on the systems' capabilities and on the researchers' mathematical judgment. Roughly one hundred research trajectories were explored before the key step appeared. The hundred is the scale given on the page, not a log that is already attached and can be opened line by line.