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Singular covariance requires a formulation on its affine support."}],"coverage_gaps":["Independent mathematical assessment of both current proposals is pending.","No full-theorem formal proof was found in the selected primary sources.","Exact numerical universal constants and a comprehensive earlier-bound history are not supplied by this curated reader."],"checked_on":"2026-10-13"},"sources":[{"work":{"authors":["Zhao Song","Xinzhi Zhang"],"kind":"research_paper","publication":"preprint","publication_note":"Source publication and attributed assessments are distinct from independent OpenMath review.","ai_role":"self_reported","ai_note":"The authors describe AI-assisted proof exploration and verification. 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Their approach uses a criterion of Song and Zhang; these are distinct papers with a mathematical dependency, not wholly independent proofs.","artifacts":[],"published":{"value":"2026-10-04"}},"id":"kls-conjecture:source:bkl:v1","source_id":"kls-conjecture-bkl","title":"Presenting a proof of the Kannan–Lovász–Simonovits conjecture","owner":"Pierre Bizeul, Boaz Klartag, Joseph Lehec","url":"https://arxiv.org/abs/2610.05474v1","subject_scope":"Proposes a dimension-free Poincaré bound for isotropic log-concave measures through cumulant estimates, localization and suspension.","revision":"2026-10-04","content_sha256":"fb51ab0d94171ac7de2a3009efb5249e44ee66de5c9a1dc5899bdb843de29df9","update_method":"Compare the exact source and publish a new immutable curation revision.","identifier_policy":"Stable source identity with exact versioned passages.","observed_on":"2026-10-13","source_updated_on":null,"rights":"metadata-link","license":"Original source license and document rights apply.","terms_url":"https://arxiv.org/abs/2610.05474v1","attribution":"Pierre Bizeul, Boaz Klartag, Joseph Lehec","rights_note":"OpenMath supplies its own mathematical paraphrases and links; proof text is not republished.","check_interval_days":90}],"entries":[{"id":"kls-conjecture:entry:song:v1","source_version_id":"kls-conjecture:source:song:v1","item_id":"song","url":"https://arxiv.org/abs/2610.01447v1","wording":"The first revision proposes inverse Cheeger bound O(4^log* n), giving Poincaré bound O(16^log* n).","wording_kind":"paraphrase","locator":"Main theorem and Poincaré conversion; exact first revision","supersedes":null},{"id":"kls-conjecture:entry:song-new:v1","source_version_id":"kls-conjecture:source:song:v2","item_id":"song-new","url":"https://arxiv.org/abs/2610.01447v2","wording":"The second revision proposes a uniform O(1) bound, strengthening the first revision’s slowly dimension-dependent estimate.","wording_kind":"paraphrase","locator":"Second revision main theorem and AI acknowledgement","supersedes":"kls-conjecture:entry:song:v1"},{"id":"kls-conjecture:entry:bkl:v1","source_version_id":"kls-conjecture:source:bkl:v1","item_id":"bkl","url":"https://arxiv.org/abs/2610.05474v1","wording":"Proposes a dimension-free Poincaré bound for isotropic log-concave measures through cumulant estimates, localization and suspension.","wording_kind":"paraphrase","locator":"Theorem 1.1 and proof/AI acknowledgement","supersedes":null}],"reports":[{"id":"kls-conjecture:report:first:v1","entry_id":"kls-conjecture:entry:song:v1","state":"proposed_solution","source_label":"Slowly dimension-dependent bound proposed in revision one","scope":"Inverse Cheeger constant scales as 4^log*(n+2); the Poincaré bound uses its square.","source_date":"2026-10-01","observed_on":"2026-10-13","evidence_url":"https://arxiv.org/abs/2610.01447v1","supersedes":null},{"id":"kls-conjecture:report:second:v1","entry_id":"kls-conjecture:entry:song-new:v1","state":"proposed_solution","source_label":"Uniform bound proposed by Song and Zhang in revision two","scope":"The later manuscript proposes a constant independent of dimension; independent mathematical assessment is pending.","source_date":"2026-10-04","observed_on":"2026-10-13","evidence_url":"https://arxiv.org/abs/2610.01447v2","supersedes":null},{"id":"kls-conjecture:report:bkl:v1","entry_id":"kls-conjecture:entry:bkl:v1","state":"proposed_solution","source_label":"Uniform bound proposed by Bizeul, Klartag and Lehec","scope":"Theorem 1.1 asserts a universal Poincaré bound. No formal execution or independent mathematical assessment is recorded here.","source_date":"2026-10-04","observed_on":"2026-10-13","evidence_url":"https://arxiv.org/abs/2610.05474v1","supersedes":null},{"id":"kls-conjecture:report:consequence:v1","entry_id":"kls-conjecture:entry:bkl:v1","state":"resolved","source_label":"Conditional mathematical implication in the cited source","scope":"This affine-transfer implication does not establish the premise or a numerical value of C. Singular covariance requires a formulation on its affine support.","source_date":null,"observed_on":"2026-10-13","evidence_url":"https://arxiv.org/abs/2610.05474v1","supersedes":null}]},"status":{"state":"proposed","label":"Solution proposed · under assessment","extent":"general_case","summary":"Two October manuscripts propose a dimension-free Poincaré bound. Song–Zhang’s second revision strengthens its first iterated-logarithm estimate; Bizeul–Klartag–Lehec give a separate argument using a Song–Zhang criterion. Independent assessment is pending.","basis":"editorial","attributed_to":"OpenMath pilot collection · OpenMath source curation","as_of":"2026-10-13","progress_label":null,"progress_claim_ids":[],"remaining":"Assess the proposed arguments and their dependencies. No formal proof project for the full theorem was located in this primary-source selection.","views":["uncertain"]}},"selected_context":{"assessment_history":[{"report_id":"kls-conjecture:report:first:v1","source_id":"kls-conjecture-song","source_version_id":"kls-conjecture:source:song:v1","source_title":"An O(4^log* n) Bound for the KLS Constant — first revision","source_label":"Slowly dimension-dependent bound proposed in revision one","state":"proposed_solution","scope":"Inverse Cheeger constant scales as 4^log*(n+2); the Poincaré bound uses its square.","observed_on":"2026-10-13","evidence_url":"https://arxiv.org/abs/2610.01447v1","entry_id":"kls-conjecture:entry:song:v1","source_date":"2026-10-01"}],"problem_id":"problem-kls-conjecture","slug":"kls-conjecture","title":"Kannan–Lovász–Simonovits conjecture","publication_version":1,"relationship":{"claim":{"id":"claim-kls-first-revision-iterated-log-bound","version":1},"relation":"supporting_result","relationship_note":"A historical proof proposal with genuine dimension dependence. The second revision proposes O(1); its evidence is not retroactively attached to this assertion.","result":"proposed","report_ids":["kls-conjecture:report:first:v1"],"independent_review":"pending","reading":{"placement":"background","note":"A historical proof proposal with genuine dimension dependence. The second revision proposes O(1); its evidence is not retroactively attached to this assertion."}},"assessments":[{"report_id":"kls-conjecture:report:first:v1","source_id":"kls-conjecture-song","source_version_id":"kls-conjecture:source:song:v1","source_title":"An O(4^log* n) Bound for the KLS Constant — first revision","source_label":"Slowly dimension-dependent bound proposed in revision one","state":"proposed_solution","scope":"Inverse Cheeger constant scales as 4^log*(n+2); the Poincaré bound uses its square.","observed_on":"2026-10-13","evidence_url":"https://arxiv.org/abs/2610.01447v1","entry_id":"kls-conjecture:entry:song:v1","source_date":"2026-10-01"}],"connections":[],"related_claims":[],"progress":[],"tasks":[],"statements":[{"statement":"There exists C > 0 such that every isotropic log-concave probability measure μ on ℝⁿ has C_P(μ) ≤ C · 16^log*(n+2), where log* is the number of successive natural logarithms needed to reach a value at most one.","statement_notation":{"schema_version":"mathematical-text-v1","segments":[{"kind":"text","text":"There exists C > 0 such that every isotropic log-concave probability measure μ on ℝⁿ has "},{"kind":"math","text":"C_P(μ) ≤ C · 16^log*(n+2)","latex":"C_P(\\mu)\\le C\\,16^{\\log^*(n+2)}"},{"kind":"text","text":", where log* is the number of successive natural logarithms needed to reach a value at most one."}]},"id":"claim-kls-first-revision-iterated-log-bound","slug":"kls-first-revision-iterated-log-bound","version":1,"kind":"bound","title":"The first revision’s iterated-logarithm estimate","assumptions":["The first arXiv revision of Song–Zhang, dated 1 October 2026.","Positive finite n; the Poincaré estimate squares the inverse Cheeger estimate."],"scope":"A historical proof proposal with genuine dimension dependence. The second revision proposes O(1); its evidence is not retroactively attached to this assertion.","source_entry_ids":["kls-conjecture:entry:song:v1"],"formal_declarations":[],"relation":"supporting_result","relationship_note":"A historical proof proposal with genuine dimension dependence. The second revision proposes O(1); its evidence is not retroactively attached to this assertion.","result":"proposed","report_ids":["kls-conjecture:report:first:v1"],"independent_review":"pending","reading":{"placement":"background","note":"A historical proof proposal with genuine dimension dependence. The second revision proposes O(1); its evidence is not retroactively attached to this assertion."}}],"entries":[{"id":"kls-conjecture:entry:song:v1","source_version_id":"kls-conjecture:source:song:v1","item_id":"song","url":"https://arxiv.org/abs/2610.01447v1","wording":"The first revision proposes inverse Cheeger bound O(4^log* n), giving Poincaré bound O(16^log* n).","wording_kind":"paraphrase","locator":"Main theorem and Poincaré conversion; exact first revision","supersedes":null}],"sources":[{"work":{"authors":["Zhao Song","Xinzhi Zhang"],"kind":"research_paper","publication":"preprint","publication_note":"Source publication and attributed assessments are distinct from independent OpenMath review.","ai_role":"self_reported","ai_note":"The authors describe AI-assisted proof exploration and verification. Contribution claims belong to this source revision and do not constitute independent assessment.","artifacts":[],"published":{"value":"2026-10-01"}},"id":"kls-conjecture:source:song:v1","source_id":"kls-conjecture-song","title":"An O(4^log* n) Bound for the KLS Constant — first revision","owner":"Zhao Song, Xinzhi Zhang","url":"https://arxiv.org/abs/2610.01447v1","subject_scope":"The first revision proposes inverse Cheeger bound O(4^log* n), giving Poincaré bound O(16^log* n).","revision":"arXiv:2610.01447v1","content_sha256":"11ded7a039724f1c925273968ead3ff1695e3161f11df5f40db7ee50ff1c1355","update_method":"Compare the exact source and publish a new immutable curation revision.","identifier_policy":"Stable source identity with exact versioned passages.","observed_on":"2026-10-13","source_updated_on":null,"rights":"metadata-link","license":"Original source license and document rights apply.","terms_url":"https://arxiv.org/abs/2610.01447v1","attribution":"Zhao Song, Xinzhi Zhang","rights_note":"OpenMath supplies its own mathematical paraphrases and links; proof text is not republished.","check_interval_days":90}]}}}