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Math-2638-A1 · revision r1 · on Math-2638

OpenAI claim summary for Math-2638

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: 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 result supplies the conclusion in this problem with its displayed hypotheses and quantifiers. The broader statements and applications in the manuscript are kept separate from the particular question above. The supporting result is manuscript 1 (family 096; 2026-09-24).

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

Manuscript

Contents
  1. Gaussian propeller partition bound
  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

Gaussian propeller partition bound

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 096; 2026-09-24), the asserted result is that every finite measurable Gaussian partition has squared first-moment sum at most $9/(8\pi)$, attained by three equal planar sectors with orthogonal extensions.

Mathematical statement of the claimed result

Manuscript 1.
Let $\gamma_d$ be standard Gaussian probability on $\mathbb R^d$. For all positive integers $d,k$ and measurable partitions $(A_1,\ldots,A_k)$, allowing empty cells, the claimed bound is \[ \sum_{i=1}^k\left\|\int_{A_i}x\,d\gamma_d(x)\right\|_2^2\le\frac9{8\pi}. \] For every $d\ge2,k\ge3$, three sectors of angle $2\pi/3$ in a plane, multiplied by its orthogonal complement with remaining cells empty, are claimed to attain equality. The constant therefore does not increase with dimension or number of cells.

Outline of the argument

For manuscript 1 (family 096; 2026-09-24), optimization over partitions reduces to a finite family of Gaussian linear scores. Deleting a score and competing against pairs impose covariance constraints; certified scalar estimates eliminate five or more active cells and bound the remaining configurations sharply.

Formal verification

For manuscript 1 (family 096; 2026-09-24), OpenAI lists Lean statements for this assertion: every finite measurable Gaussian partition has squared first-moment sum at most $9/(8\pi)$, attained by three equal planar sectors with orthogonal extensions.

The central assertion is within the documented formal scope; the additional applications retain their own scope limits.

The linked family Comparator checks are GaussianPropeller.lean.

Their named declarations include OAI.GaussianPropeller.all_partitions.

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