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Dimension-free logarithmic Sobolev inequality for subgaussian log-concave measures

OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this one:

Proves a dimension-free logarithmic Sobolev inequality for centered log-concave densities with uniformly subgaussian linear marginals, with constant bounded by a universal multiple of the squared linear subgaussian parameter.

The release has no Lean statement for this family, so there is no formal statement to compare with the result described above.

What the papers state

Theorem 1.1

In A dimension-free logarithmic Sobolev inequality for subgaussian log-concave measures, 23 September 2026

Possible match to OpenAI’s description, selected from the papers’ main statements because it shares the most wording. No review confirming the selection is recorded.

There is an absolute constant C<\infty with the following property. For every n\ge1, every centered log-concave probability measure \mu with a Lebesgue density on \R^n, and every a>0 satisfying (1.1) for X\sim\mu,

\Ent_\mu(f^2)\le C a^2\int_{\R^n}|Df|^2\,d\mu \qquad\text{for every }f\in C_c^\infty(\R^n).

The same constant works in every dimension.

Corollary 1.2 (Quadratic transport–entropy inequality)

In A dimension-free logarithmic Sobolev inequality for subgaussian log-concave measures, 23 September 2026

Possible match to OpenAI’s description, selected from shared wording. No review confirming the selection is recorded.

Under the assumptions of Theorem 1.1, every \nu\in\mathcal P_2(\R^n) satisfies

W_2(\nu,\mu)^2\le Ca^2 H(\nu\mid\mu),

with the same absolute constant C as in that theorem.

The release does not say which paper theorem its catalogue describes. Both statements above are shown whole: the paper’s main theorem first, then the statement that shares the most wording with the description. The atlas has not reviewed either choice.

Reported proof verification
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Papers and explanations
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Papers

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A dimension-free logarithmic Sobolev inequality for subgaussian log-concave measures

23 September 2026, current edition.

Read it herePDF

We prove that every centered log-concave probability measure with a Lebesgue density on ℝn and linear subgaussian parameter a satisfies \(\mathop{\mathrm{Ent}}\nolimits _\mu(f^2)\le Ca^2\int|Df|^2\,d\mu\) for compactly supported smooth f, with one universal constant C. This resolves positively the dimension-free logarithmic Sobolev conjecture for subgaussian log-concave measures.

Lean statements

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Independent comparisons and proof-verification reports

Does the Lean express the advertised result?

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What these papers cite in the release, and what cites them

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Papers elsewhere in the release that this family’s papers cite

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Papers elsewhere in the release that cite this family’s papers

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References outside the release

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What people have said

Commentary on this family by others, dated and credited. It is commentary, not a finding.

  • An unsigned public tier list, the OpenAI Math Tier List, classes it as a proof and rates its significance, if it holds up, as “Solid publication, could lead to tenure”. The page says that nothing on it is verified.

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Questions awaiting review

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Identify the theorem the catalogue describes

  • Family 093 The theorem shown was selected from shared wording with OpenAI’s description. Atlas records no review confirming that it is the theorem the description means. Open the form

State the result in Lean

  • Family 093 The release has no Lean statement for this family. Open the form