Proposed
Slowly dimension-dependent bound proposed in revision one ↗
Inverse Cheeger constant scales as 4^log*(n+2); the Poincaré bound uses its square.
Source dated Oct 1, 2026 · Observed Oct 13, 2026
Bound · version 1
The first revision’s iterated-logarithm estimate
Statement
There exists C > 0 such that every isotropic log-concave probability measure μ on ℝⁿ has , 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.
Scope
A historical proof proposal with genuine dimension dependence. The second revision proposes O(1); its evidence is not retroactively attached to this assertion.
This source presents a theorem with a proposed proof. Its wording does not establish acceptance or independent mathematical review.
Explanation
In Kannan–Lovász–Simonovits conjecture · publication v4
Bounds & computations · 1
Compare the quantity and hypotheses within each group. A finite check, conditional theorem and proposed improvement cover different scopes.
| Result | Value | Conditions |
|---|---|---|
| The first revision’s iterated-logarithm estimate → Proposed Selected source date: | ≤ C · 16log*(n+2) | First-revision proposed bound for isotropic log-concave measures; dimension-dependent. |
In Kannan–Lovász–Simonovits conjecture · publication v5
Bounds & computations · 1
Compare the quantity and hypotheses within each group. A finite check, conditional theorem and proposed improvement cover different scopes.
| Result | Value | Conditions |
|---|---|---|
| The first revision’s iterated-logarithm estimate → Proposed Selected source date: | ≤ C · 16log*(n+2) | First-revision proposed bound for isotropic log-concave measures; dimension-dependent. |
Evidence
Source reports
Connections
In this problem · publication v4 · Selected context
Kannan–Lovász–Simonovits conjecture →
Supporting result · Proposed
Relationship not independently reviewed. This concerns its use in this problem.
In this problem · publication v5
Kannan–Lovász–Simonovits conjecture →
Supporting result · Proposed
Relationship not independently reviewed. This concerns its use in this problem.
Sources
An O(4^log* n) Bound for the KLS Constant — first revision →
- Main theorem and Poincaré conversion; exact first revision → · Original passage ↗
Recorded wording
The first revision proposes inverse Cheeger bound O(4^log* n), giving Poincaré bound O(16^log* n).
paraphrase · OpenMath supplies its own mathematical paraphrases and links; proof text is not republished.
Zhao Song · Xinzhi Zhang · Preprint · Publication date Oct 1, 2026
Revision & attribution
Source revision: arXiv:2610.01447v1. Observed Oct 13, 2026.
The authors describe AI-assisted proof exploration and verification. Contribution claims belong to this source revision and do not constitute independent assessment.
History
Statement versions
Statement versions and problem publication versions are separate. A new publication can update context or sources while retaining the same statement.
Assessment history · 1
Assessments retained in these publication records. Earlier statement versions retain their own source histories.
Proposed
Slowly dimension-dependent bound proposed in revision one ↗
Inverse Cheeger constant scales as 4^log*(n+2); the Poincaré bound uses its square.
Source dated Oct 1, 2026 · Observed Oct 13, 2026 · Source passage →
Record details
- Claim ID
- claim-kls-first-revision-iterated-log-bound
- Statement version
- 1
- SHA-256
- ba53a3b734683db5d886d7503dd1592eaa75a64f79278e8f7655edb60561297b