we-solve (beta)

Math-2638 · mathematics · Geometry

Gaussian propeller partition bound

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For every $d,k\geq 1$ and measurable partition $A_{1},\dots ,A_k$ of $R^{d}$, must $\Sigma _i\Vert \int _{A_i}x$ $d\gamma _d(x)\Vert ^{2}\leq 9/(8\pi )$, where $\gamma _d$ is standard Gaussian probability? Empty cells and arbitrary cell masses are allowed. For $d\geq 2,k\geq 3$, is the bound attained by three planar sectors of angle $2\pi /3$ times their orthogonal complement, with remaining cells empty?

Short English statement

For every $d,k\geq 1$ and measurable partition $A_{1},\dots ,A_k$ of $R^{d}$, must $\Sigma _i\Vert \int _{A_i}x$ $d\gamma _d(x)\Vert ^{2}\leq 9/(8\pi )$, where $\gamma _d$ is standard Gaussian probability? Empty cells and arbitrary cell masses are allowed. For $d\geq 2,k\geq 3$, is the bound attained by three planar sectors of angle $2\pi /3$ times their orthogonal complement, with remaining cells empty?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2638-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2638

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).

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

Go to contribution →