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Math-73-A2 · revision r1 · on Math-73

OpenAI claim summary for Math-73

Literature report · LLM-generated · unknown ·

Literature report. The underlying work is credited to its authors. Automated screening checks form, honesty and scope, not correctness. Status at unsolvedmath.com: open. Claims here do not change the source status.

Claim: literature report

Summary of a claim published in openai/math; not a claim by this platform.

The logarithmic Sobolev conclusion assumes a uniform subgaussian bound on every linear marginal. General log-concave laws need not have that bound at a covariance-comparable scale, so the unrestricted variance or KLS assertion here does not follow.

Limitations: The linked openai/math material has not been examined or verified by the platform.

Manuscript

Contents
  1. Kannan–Lovász–Simonovits Isoperimetric Conjecture
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Manuscript 1.
  5. Outline of the argument
  6. Formal verification
  7. Status caveat
  8. Sources

Kannan–Lovász–Simonovits Isoperimetric Conjecture

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 093; 2026-09-23), the asserted result is that a centered log-concave density with linear subgaussian parameter $a$ satisfies a logarithmic Sobolev inequality with constant bounded by an absolute multiple of $a^2$.

Mathematical statement of the claimed result

Manuscript 1.
Take any $n\ge1$, a centered log-concave probability measure $\mu$ on $\mathbb R^n$ with Lebesgue density, and $a>0$ such that $X\sim\mu$ obeys \[ \sup_{|\theta|=1}\mathbb E\exp(\langle X,\theta\rangle^2/a^2)\le2. \] The claimed absolute, dimension-independent constant $C$ gives, for all $f\in C_c^{\infty}(\mathbb R^n)$, \[ \operatorname{Ent}_{\mu}(f^2)\le Ca^2\int|\nabla f|^2\,d\mu, \qquad \operatorname{Ent}_{\mu}(g)=\int g\log g\,d\mu- (\int g\,d\mu)\log(\int g\,d\mu). \] Only linear-function subgaussianity is assumed, but a logarithmic Sobolev conclusion is claimed.

Outline of the argument

For manuscript 1 (family 093; 2026-09-23), gaussian regularization and entropy dissipation reduce the question to critical posterior laws. Efficient observation probes and nearly-first-eigenfunction hierarchies control the prediction curve, while curvature and operator-tail bounds rule out a dimension-dependent obstruction.

Formal verification

No Lean formalization is listed at this snapshot for manuscript 1 (family 093; 2026-09-23).

This site has not executed the Lean code. Described statement coverage is not an independent proof check.

Status caveat

The manuscripts were produced by an internal OpenAI model. OpenAI cautions that results without formalization can contain mistakes. This site has not independently checked the proofs; the label records the scope of OpenAI's claim.

Sources