Result 096, Convex and metric geometry

The Gaussian propeller conjecture in every dimension

Proves that the sum of squared Gaussian first moments of any finite measurable partition is at most $9/(8\pi)$ . In dimension at least two, three planar sectors of angle $2\pi/3$ , extended orthogonally, attain the bound. Combined with the separate Unique Games theorem, this proves NP-hardness of improving the loss factor $(8\pi/9)(1-1/k)$ for identity-target kernel clustering with fixed k ≥ 3 on rational centered positive semidefinite inputs.

Lean formalization Proof

The bigger picture

Why it matters

Dividing a Gaussian bell-shaped distribution into regions creates a balance between each region's probability and its displacement from the center. The manuscript claims that three equal angular sectors already maximize a precise measure of this imbalance.

What changes?

The unreviewed manuscript reports that, for every finite measurable partition of any Euclidean space, the sum of squared lengths of the cells' Gaussian first moments is at most 9/(8 pi). A first moment is the vector obtained by integrating position over a cell with Gaussian probability weights, not its conditional mean. In dimension at least two, with at least three cells, three planar sectors of 120 degrees, extended along all perpendicular directions, attain the bound.

What does that help mathematicians do?

The claimed bound shows that neither extra dimensions nor more partition cells can surpass the value already achieved by the planar propeller. This gives researchers an exact benchmark for partition problems governed by Gaussian first moments. It rules out any proposed configuration exceeding that benchmark, while providing an explicit extremal configuration against which other constructions and inequalities can be compared.

Are there practical applications?

Its immediate value is foundational, but the summary also reports a clustering consequence. Combined with a separate Unique Games theorem, the bound yields NP-hardness of improving the loss factor (8 pi/9)(1 - 1/k) for identity-target kernel clustering with fixed k at least three on rational, centered, positive semidefinite inputs. This limits achievable approximation guarantees under standard complexity assumptions; it does not supply a faster clustering algorithm.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

The Gaussian propeller bound in every dimension

September 24, 2026 26 pages Main result formalized in Lean

We prove the Gaussian propeller conjecture: for every finite measurable partition of a Euclidean space, the sum of the squared lengths of its Gaussian first moments is at most $9/(8\pi)$ . For dimension at least two and at least three cells, three planar sectors of angle $2\pi/3$ , extended by an orthogonal Euclidean factor, attain the bound.

PDF LaTeX source

Cite (BibTeX)

@misc{OAI:The-Gaussian-Propeller-Bound-in-Every-Dimension-September-24-2026,
  author = {{OpenAI}},
  title = {{The Gaussian propeller bound in every dimension}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/The-Gaussian-Propeller-Bound-in-Every-Dimension-September-24-2026/main.pdf}{OAI:The-Gaussian-Propeller-Bound-in-Every-Dimension-September-24-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/096.md.

The Gaussian propeller conjecture in every dimension

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

The Gaussian propeller bound in every dimension

Scope

The Gaussian propeller problem asks how large the sum of squared Gaussian first moments can be over a partition. The formalized result proves the sharp bound $9/(8\pi)$ for every positive dimension and every positive number of labelled cells, allowing empty cells and arbitrary masses. In dimension at least two with at least three cells, three planar sectors of angle $120^\circ$ , extended in orthogonal directions, attain equality. The separate Gaussian-maxima and kernel-clustering consequences are not included.

Comparator links

Result	Comparator statement
Gaussian propeller bound and attainment	GaussianPropeller.lean

At a glance

Subject: Convex and metric geometry
Manuscripts: 1
Pages: 26
Dated: Sep 24, 2026
Lean formalization: Yes
Reasoning summary: No

Named problems addressed

Gaussian propeller conjectureProved

Indexed by Emergent Mind with an LLM from the text above; check the manuscripts for the precise statements.

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