The Gaussian propeller conjecture

Theorem 1.1: for all $d,k$ , every measurable partition of $\mathbb R^d$ satisfies $\sum_i\|\int_{A_i}x\,d\gamma_d\|^2\le9/(8\pi)$ , with equality for the propeller when $d\ge2$ , $k\ge3$ . The new step shows extremizers with five or more active cells cannot beat $9/(8\pi)$ , using Ehrhard's inequality and residual-score estimates; four-cell configurations come from Heilman-Jagannath-Naor. Consequence stated: combined with the release's Unique Games theorem, NP-hardness of beating the factor $(8\pi/9)(1-1/k)$ for identity-target kernel clustering, $k\ge3$ . Uniqueness of extremizers is not claimed.

Exact FrontierDelta

Prior state unknown→proved

Scope and record

Occurred: Sep 24, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: The Gaussian propeller conjecture · Propeller conjecture

Confidence: Not scored

Registry verification: lean checked · announcement · candidate

Open the source record ↗

Attribution

VibeMathed
registry · event recorded by

Unreleased internal OpenAI model
model · ai model contributor · OpenAI

Evidence vector

SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review

Named evidence

VibeMathed record: The Gaussian propeller conjecture

specialist registry · passed

No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the conjecture as the paper states it with the Khot-Naor reference. The proof uses the Heilman-Jagannath-Naor $\mathbb R^3$ theorem (computer-assisted) as an established input and was not refereed. The challenge ComparatorChallenges/GaussianPropeller.lean (OAI.GaussianPropeller.all_partitions, in lean/formalization.yaml) was read here: for all $d,k>0$ and every measurable a.e.-partition, $\sum_i\|\int_{A_i}x\,d\gamma\|^2\le9/(8\pi)$ , and for $d\ge2$ , $k\ge3$ the explicit propeller attains it. This states the headline. Not rebuilt here. The kernel-clustering hardness consequence is not formalized and depends on the release's separate Unique Games theorem.