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Breakthrough Tracker record

Three proof routes and a Lean artifact strengthen the KLS resolution claim

Song and Zhang, Bizeul, Klartag and Lehec, and Balasubramanian and Kasiviswanathan give three proof routes to a dimension-free KLS bound. A separate public Lean project checks a Poincaré constant at most 500 for an encoded statement.

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Stable ID
math-kls-dimension-free-proof-claims-2026
Revision
math-kls-dimension-free-proof-claims-2026.v2
Field
Mathematics · Asymptotic convex geometry, probability and stochastic localization
Evidence
Tier 1 · Peer reviewed: No
Record state
Provisional · Formalization-backed parallel provisional proofs
Last checked

AI role

Bizeul, Klartag and Lehec attribute most proofs and mathematical ideas to ChatGPT while retaining human understanding and exposition. Song and Zhang report collaborative use of GPT-6 Astra, GPT-5.6 Sol and Claude Fable models, with the authors choosing directions and checking the proofs.

Record details

Problem or result
Kannan–Lovász–Simonovits conjecture for isotropic log-concave measures
Authors
Zhao Song and Xinzhi Zhang; Pierre Bizeul, Boaz Klartag and Joseph Lehec; Krishnakumar Balasubramanian and Shiva Kasiviswanathan; Lean artifact by Hongru Zhao
Institutions
Song: independent researcher; Zhang: Microsoft Research. The inspected Bizeul–Klartag–Lehec manuscript did not list affiliations in its author block.
Result date
Three proof preprints posted October 4; public Lean artifact and synthesis documented by October 21, 2026

Why it matters

KLS has guided high-dimensional convex geometry since 1995 and controls concentration, random-walk mixing, sampling and volume computation. Three mathematical routes plus a kernel-checked quantitative statement provide materially stronger evidence than a single manuscript.

Limits

The papers and Lean repository remain recent and unrefereed. The formalization's own evidence says Comparator and Nanoda were not run, and kernel checking validates the encoded theorem rather than every prose-to-formal correspondence or historical claim. Microsoft is an author-affiliated source, so its synthesis is context rather than independent peer review.

Sources

  1. Primary: Bizeul, Klartag and Lehec KLS proof preprint
  2. Parallel primary: Song and Zhang dimension-free KLS bound
  3. Specialist context: PIMS seminar on the surrounding high-dimensional geometry programme
  4. Third proof: Balasubramanian and Kasiviswanathan KLS proof
  5. Formal proof: KLS500 Lean formalization
  6. Author-affiliated context: Microsoft Research synthesis of the KLS proof routes

Correction and revision history

  1. 2026-10-19 — Added after primary-source, scope, status, AI-role and limitation review.
  2. 2026-10-22 — Added a third proof route, the KLS500 Lean artifact and author-affiliated synthesis; upgraded the methodology label to formalization-backed while retaining provisional status pending peer review and independent semantic audit.

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