{"paper":{"title":"The KLS constant is $O(\\log^{1/4} n)$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA","math.MG"],"primary_cat":"math.PR","authors_text":"Brayden Letwin","submitted_at":"2026-07-27T08:45:35Z","abstract_excerpt":"We confirm the Kannan--Lov\\'asz--Simonovits conjecture for quadratic forms: if $X \\sim \\mu$ is an isotropic log-concave random vector in $\\mathbb{R}^n$, then for any symmetric matrix $M$ one has $$ \\operatorname{Var}_{X \\sim \\mu}(\\langle MX,X\\rangle) \\leq 2\\,\\mathbb{E}_{X \\sim \\mu}|\\nabla\\langle MX,X\\rangle|^2. $$ As an application, we apply the above to $M=\\mathbb{E}_{X \\sim \\mu}(\\langle X,\\theta\\rangle X\\otimes X)$ for $\\theta\\in S^{n-1}$ and show that the Kannan--Lov\\'asz--Simonovits constant $\\psi_n$ satisfies $$ \\psi_n\\leq C\\log^{1/4}n $$ for some absolute constant $C>0$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.24164","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.24164/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}