Dimension-free KLS bounds in Lean: Poincaré constant 500
Description
Complete Lean source and verification records for the dimension-free Poincaré bound C_P(μ) ≤ 500 for every isotropic log-concave probability measure. The inequality covers locally Lipschitz L² functions without a compact-support restriction. The development also proves the unchanged OpenAI KLSStatement with witness 500 and 4 ≤ C_* ≤ 500 for the optimal universal Poincaré constant. A fresh build verified 1,710 mathematical modules; endpoint axiom closures use only propext, Classical.choice, and Quot.sound. The proof follows the Bizeul–Klartag–Lehec route with quantitative refinements. The constant 500 is a sufficient bound and is not claimed to be optimal.
The Lean archive contains 1,714 Lean source files, pinned dependencies, licenses and verification evidence, matching GitHub commit c1a50992a81456849996deef2971d4a0b3190bdd. The accompanying four-page PDF is the technical endpoint report.
The main proof basis is Pierre Bizeul, Boaz Klartag and Joseph Lehec’s cumulant and suspension approach, including the criterion of Zhao Song and Xinzhi Zhang. Krishnakumar Balasubramanian and Shiva Kasiviswanathan provide related work through a separate integration-operator and Hodge-comparison approach.
Files
KLS_Endpoint_Report.pdf
Additional details
Software
- Repository URL
- https://github.com/HongruZhao/KLS500
- Programming language
- Lean
References
- Pierre Bizeul, Boaz Klartag and Joseph Lehec. Presenting a proof of the Kannan–Lovász–Simonovits conjecture. arXiv:2610.05474v1. https://arxiv.org/pdf/2610.05474v1
- Zhao Song and Xinzhi Zhang. An O(1) Bound for the KLS Constant. arXiv:2610.01447v2. https://arxiv.org/pdf/2610.01447v2
- Krishnakumar Balasubramanian and Shiva Kasiviswanathan. A Dimension-Free Bound on the Poincaré Constant of Isotropic Log-Concave Measures. GitHub manuscript, snapshot 6 October 2026. https://github.com/kriznakumar/paper/blob/4837c33649ba2271f43c9684e9350ecbdd725f95/KLS.pdf