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.
Open questions
The target is a dimension-independent Poincaré bound for isotropic log-concave probability measures.
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.
Definitions
Isotropic measure
A probability measure with mean zero and identity covariance. This normalization removes linear scaling before comparing Poincaré constants across dimensions.
Question paraphrased by OpenMath. Original wording is available in the sources.
Landscape
Figures and explorable examples. Selecting a statement opens its exact wording.
State of the problem
How the bound fell
From the target, a constant for every dimension, down through the proved bounds that grow with n. The top two rungs are 2026 proposals; everything below is proved.
Reading boundary. Lower rungs do not imply higher ones. The two top rungs are proposals with assessment pending, not established results.
The question: one constant for every n
Two October 2026 manuscripts claim CP(μ) ≤ C with C independent of n. Both are proposed; assessment is pending.
The conjecture asks for C_P(μ) ≤ C with C independent of the dimension n. Proved bounds have fallen from C√n (Lee–Vempala) to sub-polynomial (Chen), poly-logarithmic (Klartag–Lehec) and C log n (Klartag). In October 2026 Song–Zhang proposed first 16^{log*(n+2)} and then a constant, and Bizeul–Klartag–Lehec proposed a constant by a separate argument.
Isotropic log-concave probability measures on ℝⁿ; C_P is the Poincaré constant. Constants C differ between rungs; only the growth in n is compared.
CP ≤ C (proposed)
◌Proposed
A uniform Poincaré bound for isotropic log-concave measures
One shared mathematical assertion with separately attributed October proof proposals. Neither paper receives an independent-review label from this curation.
A historical proof proposal with genuine dimension dependence. The second revision proposes O(1); its evidence is not retroactively attached to this assertion.
Authored by OpenMath curation on 2026-10-21 (AI-assisted curation). Independent review of this visual explanation is pending; linked assertions and relationships have their own assessments.
One shared mathematical assertion with separately attributed October proof proposals.
≤ Universal CSong & Zhang 2026
Conjecture · Primary formulation
◌ProposedSong & Zhang 2026
A uniform Poincaré bound for isotropic log-concave measures
There exists C > 0 such that for every n ≥ 1 and every isotropic log-concave probability measure μ on Rn, Varμ(f)≤C∫∣∇f∣2dμ for every locally Lipschitz real f for which the variance and gradient energy are finite.
Assumptions
Isotropic: mean zero and covariance I.
Log-concave density and positive finite dimension; the same C works for all dimensions.
In this problem
One shared mathematical assertion with separately attributed October proof proposals. Neither paper receives an independent-review label from this curation.
Bound
Poincaré constant
≤ Universal C
Conditions
Proposed bound for all dimensions; isotropic log-concave measures; mathematical assessment pending.
The second revision proposes a universal bound, strengthening the first revision’s iterated-logarithm dependence.
This is a proposed proof under assessment; the earlier quantitative assertion remains a separate historical Claim.
Proposed:Uniform bound proposed by Song and Zhang in revision two ↗ · source dated Oct 4, 2026 · observed Oct 13, 2026. The later manuscript proposes a constant independent of dimension; independent mathematical assessment is pending.
A separately authored proof account addresses the same uniform Poincaré assertion using a criterion from Song’s work.
The dependency on Song’s criterion means this account is not independent verification of that criterion.
Proposed:Uniform bound proposed by Bizeul, Klartag and Lehec ↗ · source dated Oct 4, 2026 · observed Oct 13, 2026. Theorem 1.1 asserts a universal Poincaré bound. No formal execution or independent mathematical assessment is recorded here.
A historical proof proposal with genuine dimension dependence.
≤ C · 16log*(n+2)Song & Zhang 2026
Bound · Supporting result
◌ProposedSong & Zhang 2026
The first revision’s iterated-logarithm estimate
There exists C > 0 such that every isotropic log-concave probability measure μ on ℝⁿ has CP(μ)≤C16log∗(n+2), where log* is the number of successive natural logarithms needed to reach a value at most one.
Assumptions
The first arXiv revision of Song–Zhang, dated 1 October 2026.
Positive finite n; the Poincaré estimate squares the inverse Cheeger estimate.
In this problem
A historical proof proposal with genuine dimension dependence. The second revision proposes O(1); its evidence is not retroactively attached to this assertion.
Bound
Poincaré constant
≤ C · 16log*(n+2)
Conditions
First-revision proposed bound for isotropic log-concave measures; dimension-dependent.
This affine-transfer implication does not establish the premise or a numerical value of C.
Bizeul et al. 2026
Theorem · Conditional implication
●ProvedBizeul et al. 2026
Transporting the proposed bound beyond isotropic measures
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.
Assumptions
Finite nonsingular covariance and a log-concave probability density.
The same universal constant C is assumed in the isotropic theorem.
In this problem
This affine-transfer implication does not establish the premise or a numerical value of C. Singular covariance requires a formulation on its affine support.
Transporting the proposed bound beyond isotropic measures
Conditional
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.
Limit This affine-transfer implication does not establish the premise or a numerical value of C. Singular covariance requires a formulation on its affine support.
How the problem is attacked, and what each line of attack cannot yet reach.
Normalize before comparing constants
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.
Cannot yet reach A bound for one dimension or one family of densities does not supply the uniform theorem.
Reported in sources: 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.
CComputation
None recorded
RReview
Not yet reviewed
OpenMath assessment: 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.
Source comparison and reader curation; independent mathematical assessment remains pending.
OpenMath pilot collection · OpenMath source curation · assessed Oct 13, 2026 · review due Jan 13, 2027
Collection last checked Oct 13, 2026.
Sources
Choose a starting point below. Each result links to its exact statement, cited passages and checks.
Selected literature; source observations and coverage limits are retained in the publication details.
Contributors
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.
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.
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.
These are source credits; they do not imply OpenMath accounts, endorsements or independent mathematical review.
Record details
Record version
5
Classification
MSC2020 52A23 / 60E15
Prerequisite level
graduate
Assess both proposed arguments, their shared dependencies and manuscript revisions. No formal proof project for the full theorem is linked in this source selection.