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Neither paper receives an independent-review label from this curation.","result":"proposed","report_ids":["kls-conjecture:report:second:v1","kls-conjecture:report:bkl:v1"],"independent_review":"pending"},{"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. 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Singular covariance requires a formulation on its affine support."}},{"claim":{"id":"claim-kls-lee-vempala","version":1},"relation":"adjacent_result","relationship_note":"An earlier proved bound whose growth in n the conjecture would remove.","result":"proved","report_ids":["kls-conjecture:report:lee-vempala:v1"],"independent_review":"pending","credit":{"name":"Lee & Vempala","year":"2024","entry_id":"kls-conjecture:entry:bkl-history:v1"}},{"claim":{"id":"claim-kls-chen","version":1},"relation":"adjacent_result","relationship_note":"An earlier proved bound whose growth in n the conjecture would remove.","result":"proved","report_ids":["kls-conjecture:report:chen:v1"],"independent_review":"pending","credit":{"name":"Chen","year":"2021","entry_id":"kls-conjecture:entry:bkl-history:v1"}},{"claim":{"id":"claim-kls-klartag-lehec","version":1},"relation":"adjacent_result","relationship_note":"An earlier proved bound whose growth in n the conjecture would remove.","result":"proved","report_ids":["kls-conjecture:report:klartag-lehec:v1"],"independent_review":"pending","credit":{"name":"Klartag & Lehec","year":"2022","entry_id":"kls-conjecture:entry:bkl-history:v1"}},{"claim":{"id":"claim-kls-klartag-log","version":1},"relation":"adjacent_result","relationship_note":"An earlier proved bound whose growth in n the conjecture would remove.","result":"proved","report_ids":["kls-conjecture:report:klartag-log:v1"],"independent_review":"pending","credit":{"name":"Klartag","year":"2023","entry_id":"kls-conjecture:entry:bkl-history:v1"}}],"milestones":[{"id":"kls-conjecture-origin","date":{"value":"1995"},"date_basis":"publication","category":"foundation","title":"The convex-geometric conjecture","summary":"The cited manuscripts trace KLS to Kannan, Lovász and Simonovits’ isoperimetric work.","limitation":"The reader uses the equivalent Poincaré formulation up to universal constants.","claims":[],"source_entry_ids":["kls-conjecture:entry:bkl:v1","kls-conjecture:entry:song:v1"],"independent_review":"pending"},{"id":"kls-conjecture-v1","date":{"value":"2026-10-01"},"date_basis":"preprint","category":"proposal","title":"A slowly dimension-dependent bound is proposed","summary":"Song and Zhang’s first revision states an inverse Cheeger estimate with iterated-logarithm dependence.","limitation":"The corresponding Poincaré bound is squared and remains dimension dependent.","claims":[{"id":"claim-kls-first-revision-iterated-log-bound","version":1}],"source_entry_ids":["kls-conjecture:entry:song:v1"],"independent_review":"pending"},{"id":"kls-conjecture-v2","date":{"value":"2026-10-04"},"date_basis":"preprint","category":"proposal","title":"The second revision proposes a constant bound","summary":"Song and Zhang replace the earlier asymptotic target with a uniform bound.","limitation":"This source revision supersedes the earlier passage but preserves its historical identity and Claim.","claims":[{"id":"claim-kls-dimension-free-poincare","version":1}],"source_entry_ids":["kls-conjecture:entry:song-new:v1"],"independent_review":"pending"},{"id":"kls-conjecture-bkl","date":{"value":"2026-10-04"},"date_basis":"preprint","category":"proposal","title":"Another manuscript presents a proof","summary":"Bizeul, Klartag and Lehec present a route through cumulants and suspension using a Song–Zhang criterion.","limitation":"Different authors and arguments do not by themselves constitute independent validation.","claims":[{"id":"claim-kls-dimension-free-poincare","version":1}],"source_entry_ids":["kls-conjecture:entry:bkl:v1"],"independent_review":"pending"}],"activities":[{"id":"kls-conjecture-assessment","title":"Review the current proof proposals","kind":"research_direction","source_entry_ids":["kls-conjecture:entry:song-new:v1","kls-conjecture:entry:bkl:v1"],"summary":"Compare the current revisions, the high-order tilt criterion and the later cumulant/suspension argument.","limitation":"No independent review verdict or executable formal proof is recorded in this source selection.","related_claim_ids":["claim-kls-dimension-free-poincare"]}],"curation":{"checked_on":"2026-10-13","selection":"Original question, distinct results, proof provenance, selected history and an explanatory schematic.","search_scope":"Selected primary papers, exact formal declarations, project provenance and official descriptions checked on 13 October 2026; Bizeul–Klartag–Lehec §1 for the earlier proved bounds.","formal_status":"No linked formal proof project for the full KLS theorem was found in these primary sources. The October proposals remain under assessment; publication and model checks are not independent validation.","ai_status":"Song–Zhang and Bizeul–Klartag–Lehec own separate manuscripts. Their AI disclosures assign models proof exploration, ideas and checking; the authors retain mathematical responsibility. Bizeul–Klartag–Lehec explicitly use a Song–Zhang criterion, which is kept visible as a dependency.","independent_review":"pending"},"schema_version":"research-record-v3","claims":[{"statement":"There exists C > 0 such that for every n ≥ 1 and every isotropic log-concave probability measure μ on ℝⁿ, Var_μ(f) ≤ C ∫|∇f|² dμ for every locally Lipschitz real f for which the variance and gradient energy are finite.","statement_notation":{"schema_version":"mathematical-text-v1","segments":[{"kind":"text","text":"There exists C > 0 such that for every n ≥ 1 and every isotropic log-concave probability measure "},{"kind":"math","text":"μ on ℝⁿ","latex":"\\mu\\text{ on }\\mathbb R^n"},{"kind":"text","text":", "},{"kind":"math","text":"Var_μ(f) ≤ C ∫|∇f|² dμ","latex":"\\operatorname{Var}_\\mu(f)\\le C\\int |\\nabla f|^2\\,d\\mu"},{"kind":"text","text":" for every locally Lipschitz real f for which the variance and gradient energy are finite."}]},"id":"claim-kls-dimension-free-poincare","slug":"kls-dimension-free-poincare","version":1,"kind":"conjecture","title":"A uniform Poincaré bound for isotropic log-concave measures","assumptions":["Isotropic: mean zero and covariance I.","Log-concave density and positive finite dimension; the same C works for all dimensions."],"scope":"One shared mathematical assertion with separately attributed October proof proposals. Neither paper receives an independent-review label from this curation.","source_entry_ids":["kls-conjecture:entry:song-new:v1","kls-conjecture:entry:bkl:v1"],"formal_declarations":[]},{"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":[]},{"statement":"If every isotropic log-concave probability measure has Poincaré constant at most C, then every full-dimensional log-concave probability measure μ with covariance Σ satisfies C_P(μ) ≤ C ‖Σ‖op.","id":"claim-kls-conjecture-consequence","slug":"kls-conjecture-consequence","version":1,"kind":"theorem","title":"Transporting the proposed bound beyond isotropic measures","assumptions":["Finite nonsingular covariance and a log-concave probability density.","The same universal constant C is assumed in the isotropic theorem."],"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_entry_ids":["kls-conjecture:entry:bkl:v1"],"formal_declarations":[]},{"statement":"There exists C > 0 such that for every n ≥ 1, every isotropic log-concave probability measure μ on ℝⁿ has C_P(μ) ≤ C√n.","id":"claim-kls-lee-vempala","slug":"kls-lee-vempala","version":1,"kind":"bound","title":"A square-root bound","assumptions":["Isotropic log-concave probability measures; C_P is the Poincaré constant.","Reported in the 2026 paper’s account of earlier work."],"scope":"A proved dimension-dependent bound; it does not give one constant for every dimension.","source_entry_ids":["kls-conjecture:entry:bkl-history:v1"],"formal_declarations":[]},{"statement":"For every ε > 0 there exists C_ε > 0 such that for every n ≥ 1, every isotropic log-concave probability measure μ on ℝⁿ has C_P(μ) ≤ C_ε n^ε.","id":"claim-kls-chen","slug":"kls-chen","version":1,"kind":"bound","title":"A sub-polynomial bound","assumptions":["Isotropic log-concave probability measures; C_P is the Poincaré constant.","Reported in the 2026 paper’s account of earlier work."],"scope":"A proved dimension-dependent bound; it does not give one constant for every dimension.","source_entry_ids":["kls-conjecture:entry:bkl-history:v1"],"formal_declarations":[]},{"statement":"There exist C, k > 0 such that for every n ≥ 2, every isotropic log-concave probability measure μ on ℝⁿ has C_P(μ) ≤ C (log n)^k.","id":"claim-kls-klartag-lehec","slug":"kls-klartag-lehec","version":1,"kind":"bound","title":"A poly-logarithmic bound","assumptions":["Isotropic log-concave probability measures; C_P is the Poincaré constant.","Reported in the 2026 paper’s account of earlier work."],"scope":"A proved dimension-dependent bound; it does not give one constant for every dimension.","source_entry_ids":["kls-conjecture:entry:bkl-history:v1"],"formal_declarations":[]},{"statement":"There exists C > 0 such that for every n ≥ 2, every isotropic log-concave probability measure μ on ℝⁿ has C_P(μ) ≤ C log n.","id":"claim-kls-klartag-log","slug":"kls-klartag-log","version":1,"kind":"bound","title":"A logarithmic bound","assumptions":["Isotropic log-concave probability measures; C_P is the Poincaré constant.","Reported in the 2026 paper’s account of earlier work."],"scope":"A proved dimension-dependent bound; it does not give one constant for every dimension.","source_entry_ids":["kls-conjecture:entry:bkl-history:v1"],"formal_declarations":[]}],"problem":{"resolution_scope":{"resolution":"unresolved","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.","report_ids":["kls-conjecture:report:second:v1","kls-conjecture:report:bkl:v1"],"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."},"statement_kind":"conjecture","problem_id":"problem-kls-conjecture","slug":"kls-conjecture","title":"Kannan–Lovász–Simonovits conjecture","question":"Is there a universal upper bound on the Poincaré constant of every isotropic log-concave probability measure, independent of its dimension?","scope":"For a probability measure μ on ℝⁿ with log-concave density, isotropic means mean zero and covariance matrix I. Log-concave means the density is e^(−V) for a convex function V; V = +∞ is allowed where the density is zero. Its Poincaré constant C_P(μ) is the least constant in Var_μ(f) ≤ C_P(μ) ∫|∇f|² dμ for locally Lipschitz f with the required finite integrals. The proposed bound must use one constant for all positive dimensions and all such measures. Equivalent isoperimetric formulations change constants; a dimension-dependent bound alone does not settle the question.","aliases":["KLS","KLS conjecture","Dimension-free Poincaré inequality","Log-concave spectral gap"],"fields":["Probability","Convex geometry","Analysis"],"msc":["52A23","60E15"],"topics":[],"background":"graduate","scale":"major conjecture","depth":"maintained","source_entry_ids":["kls-conjecture:entry:song:v1","kls-conjecture:entry:song-new:v1","kls-conjecture:entry:bkl:v1","kls-conjecture:entry:bkl-history:v1"],"status_report_ids":["kls-conjecture:report:second:v1","kls-conjecture:report:bkl:v1"],"editorial":{"collection_id":"openmath-pilot","collection_title":"OpenMath pilot collection","reviewer":"OpenMath source curation","state":"under_assessment","scope":"Source comparison and reader curation; independent mathematical assessment remains pending.","rationale":"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.","decided_on":"2026-10-13","review_due_on":"2027-01-13","report_ids":["kls-conjecture:report:second:v1","kls-conjecture:report:bkl:v1"],"independent_mathematical_review":"pending"},"literature":[{"title":"An O(4^log* n) Bound for the KLS Constant — first revision","url":"https://arxiv.org/abs/2610.01447v1","note":"Main theorem and Poincaré conversion; exact first revision"},{"title":"An O(1) Bound for the KLS Constant — second revision","url":"https://arxiv.org/abs/2610.01447v2","note":"Second revision main theorem and AI acknowledgement"},{"title":"Presenting a proof of the Kannan–Lovász–Simonovits conjecture","url":"https://arxiv.org/abs/2610.05474v1","note":"Theorem 1.1 and proof/AI acknowledgement"}],"formalizations":[],"related":[],"tasks":[],"insights":[{"id":"kls-conjecture-normalization","title":"Normalize before comparing constants","summary":"Isotropy removes linear scale and direction: the mean is zero and covariance is the identity. Affine transport relates the general covariance case to this target.","review":"pending","kind":"approach","source_entry_ids":["kls-conjecture:entry:bkl:v1"],"claims":[{"id":"claim-kls-dimension-free-poincare","version":1}],"limitation":"A bound for one dimension or one family of densities does not supply the uniform theorem."},{"id":"kls-conjecture-impact","title":"Transporting the proposed bound beyond isotropic measures","summary":"If every isotropic log-concave probability measure has Poincaré constant at most C, then every full-dimensional log-concave probability measure μ with covariance Σ satisfies C_P(μ) ≤ C ‖Σ‖op.","review":"pending","kind":"impact","source_entry_ids":["kls-conjecture:entry:bkl:v1"],"claims":[{"id":"claim-kls-conjecture-consequence","version":1}],"scenario":{"id":"claim-kls-dimension-free-poincare","version":1},"limitation":"This affine-transfer implication does not establish the premise or a numerical value of C. 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":"2026-10-21"},"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},{"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":"Song and Zhang credit GPT-6 Astra, GPT-5.6 Sol and Claude Fable 5/5.1 for extensive proof exploration and checking, with authors directing priorities and retaining responsibility.","artifacts":[],"published":{"value":"2026-10-04"}},"id":"kls-conjecture:source:song:v2","source_id":"kls-conjecture-song","title":"An O(1) Bound for the KLS Constant — second revision","owner":"Zhao Song, Xinzhi Zhang","url":"https://arxiv.org/abs/2610.01447v2","subject_scope":"The second revision proposes a uniform O(1) bound, strengthening the first revision’s slowly dimension-dependent estimate.","revision":"arXiv:2610.01447v2","content_sha256":"c69ea9c264ec83518df1ee205443fa54efe647c3c175344cbe5c75d7d091ce3d","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/2610.01447v2","attribution":"Zhao Song, Xinzhi Zhang","rights_note":"OpenMath supplies its own mathematical paraphrases and links; proof text is not republished.","check_interval_days":90},{"work":{"authors":["Pierre Bizeul","Boaz Klartag","Joseph Lehec"],"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 credit ChatGPT with many proof ideas and arguments, and credit the suspension idea and exposition to the human authors. 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},{"id":"kls-conjecture:entry:bkl-history:v1","source_version_id":"kls-conjecture:source:bkl:v1","item_id":"introduction-history","url":"https://arxiv.org/abs/2610.05474v1","wording":"For isotropic log-concave measures, Lee and Vempala obtained C_P(μ) ≤ C√n (Ann. Math. 2024); Chen established a sub-polynomial bound (GAFA 2021); Klartag and Lehec proved a poly-logarithmic bound (GAFA 2022); and Klartag proved C_P(μ) ≤ C log n (2023).","wording_kind":"paraphrase","locator":"§1, paragraph on stochastic localization and (8); references [12], [22], [23], [28]","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},{"id":"kls-conjecture:report:lee-vempala:v1","entry_id":"kls-conjecture:entry:bkl-history:v1","state":"resolved","source_label":"A square-root bound (Lee & Vempala 2024)","scope":"Only the exact statement and passage cited here; not an independent OpenMath assessment.","source_date":null,"observed_on":"2026-10-21","evidence_url":"https://arxiv.org/abs/2610.05474v1","supersedes":null},{"id":"kls-conjecture:report:chen:v1","entry_id":"kls-conjecture:entry:bkl-history:v1","state":"resolved","source_label":"A sub-polynomial bound (Chen 2021)","scope":"Only the exact statement and passage cited here; not an independent OpenMath assessment.","source_date":null,"observed_on":"2026-10-21","evidence_url":"https://arxiv.org/abs/2610.05474v1","supersedes":null},{"id":"kls-conjecture:report:klartag-lehec:v1","entry_id":"kls-conjecture:entry:bkl-history:v1","state":"resolved","source_label":"A poly-logarithmic bound (Klartag & Lehec 2022)","scope":"Only the exact statement and passage cited here; not an independent OpenMath assessment.","source_date":null,"observed_on":"2026-10-21","evidence_url":"https://arxiv.org/abs/2610.05474v1","supersedes":null},{"id":"kls-conjecture:report:klartag-log:v1","entry_id":"kls-conjecture:entry:bkl-history:v1","state":"resolved","source_label":"A logarithmic bound (Klartag 2023)","scope":"Only the exact statement and passage cited here; not an independent OpenMath assessment.","source_date":null,"observed_on":"2026-10-21","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:second:v1","source_id":"kls-conjecture-song","source_version_id":"kls-conjecture:source:song:v2","source_title":"An O(1) Bound for the KLS Constant — second revision","source_label":"Uniform bound proposed by Song and Zhang in revision two","state":"proposed_solution","scope":"The later manuscript proposes a constant independent of dimension; independent mathematical assessment is pending.","observed_on":"2026-10-13","evidence_url":"https://arxiv.org/abs/2610.01447v2","entry_id":"kls-conjecture:entry:song-new:v1","source_date":"2026-10-04"},{"report_id":"kls-conjecture:report:bkl:v1","source_id":"kls-conjecture-bkl","source_version_id":"kls-conjecture:source:bkl:v1","source_title":"Presenting a proof of the Kannan–Lovász–Simonovits conjecture","source_label":"Uniform bound proposed by Bizeul, Klartag and Lehec","state":"proposed_solution","scope":"Theorem 1.1 asserts a universal Poincaré bound. No formal execution or independent mathematical assessment is recorded here.","observed_on":"2026-10-13","evidence_url":"https://arxiv.org/abs/2610.05474v1","entry_id":"kls-conjecture:entry:bkl:v1","source_date":"2026-10-04"}],"reader_guide":{"reading":[],"definitions":[],"proofs":[{"id":"kls-conjecture-account-1","claim":{"id":"claim-kls-dimension-free-poincare","version":1},"title":"Song second-revision account","method":"A proposed dimension-free argument","summary":"The second revision proposes a universal bound, strengthening the first revision’s iterated-logarithm dependence.","limitation":"This is a proposed proof under assessment; the earlier quantitative assertion remains a separate historical Claim.","dependencies":[],"source_entry_ids":["kls-conjecture:entry:song-new:v1"],"report_ids":["kls-conjecture:report:second:v1"]},{"id":"kls-conjecture-account-2","claim":{"id":"claim-kls-dimension-free-poincare","version":1},"title":"Bizeul–Klartag–Lehec account","method":"A proposed argument using Song’s criterion","summary":"A separately authored proof account addresses the same uniform Poincaré assertion using a criterion from Song’s work.","limitation":"The dependency on Song’s criterion means this account is not independent verification of that criterion.","dependencies":[],"source_entry_ids":["kls-conjecture:entry:bkl:v1"],"report_ids":["kls-conjecture:report:bkl:v1"]}],"bounds":[{"id":"kls-dimension-free-poincare-bound-detail","claim":{"id":"claim-kls-dimension-free-poincare","version":1},"quantity":"Poincaré constant","direction":"upper","value":"Universal C","conditions":"Proposed bound for all dimensions; isotropic log-concave measures; mathematical assessment pending.","comparison_group":"kls-uniform-proposal","date":{"value":"2026-10-04"},"source_entry_ids":["kls-conjecture:entry:song-new:v1","kls-conjecture:entry:bkl:v1"]}],"witnesses":[],"contributors":null,"scopes":[]},"problem_id":"problem-kls-conjecture","slug":"kls-conjecture","title":"Kannan–Lovász–Simonovits conjecture","publication_version":5,"relationship":{"claim":{"id":"claim-kls-dimension-free-poincare","version":1},"relation":"primary_formulation","relationship_note":"One shared mathematical assertion with separately attributed October proof proposals. Neither paper receives an independent-review label from this curation.","result":"proposed","report_ids":["kls-conjecture:report:second:v1","kls-conjecture:report:bkl:v1"],"independent_review":"pending"},"assessments":[{"report_id":"kls-conjecture:report:second:v1","source_id":"kls-conjecture-song","source_version_id":"kls-conjecture:source:song:v2","source_title":"An O(1) Bound for the KLS Constant — second revision","source_label":"Uniform bound proposed by Song and Zhang in revision two","state":"proposed_solution","scope":"The later manuscript proposes a constant independent of dimension; independent mathematical assessment is pending.","observed_on":"2026-10-13","evidence_url":"https://arxiv.org/abs/2610.01447v2","entry_id":"kls-conjecture:entry:song-new:v1","source_date":"2026-10-04"},{"report_id":"kls-conjecture:report:bkl:v1","source_id":"kls-conjecture-bkl","source_version_id":"kls-conjecture:source:bkl:v1","source_title":"Presenting a proof of the Kannan–Lovász–Simonovits conjecture","source_label":"Uniform bound proposed by Bizeul, Klartag and Lehec","state":"proposed_solution","scope":"Theorem 1.1 asserts a universal Poincaré bound. No formal execution or independent mathematical assessment is recorded here.","observed_on":"2026-10-13","evidence_url":"https://arxiv.org/abs/2610.05474v1","entry_id":"kls-conjecture:entry:bkl:v1","source_date":"2026-10-04"}],"connections":[],"related_claims":[],"progress":[],"tasks":[],"statements":[{"statement":"There exists C > 0 such that for every n ≥ 1 and every isotropic log-concave probability measure μ on ℝⁿ, Var_μ(f) ≤ C ∫|∇f|² dμ for every locally Lipschitz real f for which the variance and gradient energy are finite.","statement_notation":{"schema_version":"mathematical-text-v1","segments":[{"kind":"text","text":"There exists C > 0 such that for every n ≥ 1 and every isotropic log-concave probability measure "},{"kind":"math","text":"μ on ℝⁿ","latex":"\\mu\\text{ on }\\mathbb R^n"},{"kind":"text","text":", "},{"kind":"math","text":"Var_μ(f) ≤ C ∫|∇f|² dμ","latex":"\\operatorname{Var}_\\mu(f)\\le C\\int |\\nabla f|^2\\,d\\mu"},{"kind":"text","text":" for every locally Lipschitz real f for which the variance and gradient energy are finite."}]},"id":"claim-kls-dimension-free-poincare","slug":"kls-dimension-free-poincare","version":1,"kind":"conjecture","title":"A uniform Poincaré bound for isotropic log-concave measures","assumptions":["Isotropic: mean zero and covariance I.","Log-concave density and positive finite dimension; the same C works for all dimensions."],"scope":"One shared mathematical assertion with separately attributed October proof proposals. Neither paper receives an independent-review label from this curation.","source_entry_ids":["kls-conjecture:entry:song-new:v1","kls-conjecture:entry:bkl:v1"],"formal_declarations":[],"relation":"primary_formulation","relationship_note":"One shared mathematical assertion with separately attributed October proof proposals. Neither paper receives an independent-review label from this curation.","result":"proposed","report_ids":["kls-conjecture:report:second:v1","kls-conjecture:report:bkl:v1"],"independent_review":"pending"}],"entries":[{"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}],"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":"Song and Zhang credit GPT-6 Astra, GPT-5.6 Sol and Claude Fable 5/5.1 for extensive proof exploration and checking, with authors directing priorities and retaining responsibility.","artifacts":[],"published":{"value":"2026-10-04"}},"id":"kls-conjecture:source:song:v2","source_id":"kls-conjecture-song","title":"An O(1) Bound for the KLS Constant — second revision","owner":"Zhao Song, Xinzhi Zhang","url":"https://arxiv.org/abs/2610.01447v2","subject_scope":"The second revision proposes a uniform O(1) bound, strengthening the first revision’s slowly dimension-dependent estimate.","revision":"arXiv:2610.01447v2","content_sha256":"c69ea9c264ec83518df1ee205443fa54efe647c3c175344cbe5c75d7d091ce3d","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/2610.01447v2","attribution":"Zhao Song, Xinzhi Zhang","rights_note":"OpenMath supplies its own mathematical paraphrases and links; proof text is not republished.","check_interval_days":90},{"work":{"authors":["Pierre Bizeul","Boaz Klartag","Joseph Lehec"],"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 credit ChatGPT with many proof ideas and arguments, and credit the suspension idea and exposition to the human authors. 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}]}}}