The Gaussian propeller conjecture in every dimension
OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this one:
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. (OpenAI’s Lean note)
Lean is a language for mathematical statements and proofs that a computer can verify. The assessments below ask whether those statements express the result OpenAI describes here.
The two independent assessments disagree on how much of the result described above is expressed by the Lean statements.Read the assessments
Possible match to OpenAI’s description, selected from the papers’ main statements because it shares the most wording. No review confirming the selection is recorded.
For all positive integers d and k, every measurable partition (A_1,\ldots,A_k) of \mathbb R^d satisfies
Empty cells are allowed. The constant is sharp whenever d\ge2 and k\ge3: three planar sectors of angle 2\pi/3, multiplied by the orthogonal complement of their plane, attain equality, with any further cells empty.
Reported proof verification
One Lean statement in the release; three independent groups report that Comparator accepted runs for this file at the version shown here.
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.
One Lean statement in this family
Lean is a language for mathematical statements and proofs that a computer can verify. The Lean statement below is shown beside its possible paper counterpart, with the reason they were linked.
For a configured theorem, the Comparator program tests whether OpenAI’s separate proof proves the exact Lean statement using only the allowed axioms. Some statement files instead configure definitions for comparison by name and type. Acceptance does not establish that a formal statement means the same thing as the paper’s theorem.
Empty cells are allowed. The constant is sharp whenever d\ge2 and k\ge3: three planar sectors of angle 2\pi/3, multiplied by the orthogonal complement of their plane, attain equality, with any further cells empty.
The words they share: measurable, partition, real, pi, integers, propeller, gaussian, orthogonal, empty, sectors, planar, angle, attain, allowed, equality, cells, sharp, positive, three.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
GaussianPropeller.lean, the release lists it as a main result.
defAllPartitions : Prop := (∀ (d k : ℕ), 0 < d → 0 < k → ∀ A : Fin k → Set (Space d), IsPartition A → value A ≤ 9 / (8 * Real.pi)) ∧ (∀ (d k : ℕ), 2 ≤ d → 3 ≤ k → IsPartition (propeller d k) ∧ value (propeller d k) = 9 / (8 * Real.pi))⋮theoremall_partitions : AllPartitions := bysorry
This file states the theorem with sorry, a placeholder, in place of its proof. OpenAI supplies a separate proof file. Comparator tests whether that proof proves the statement here.
three independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet; Neal McBurnett, openai-math-lean-tooling. Who ran the proof-verification program
cert-machine records it as “CERTIFIED”.
MathVet records it as “comparator-sandboxed-pass”.
Neal McBurnett, openai-math-lean-tooling records it as “accepted”.
Other mathematical statements in OpenAI’s Lean library
Atlas found these propositions by scanning the text of the project’s Model.lean file. Each name links to its definition, with any associated statement file beside it. This list does not record running Lean to interpret or verify those definitions.
Independent comparisons and proof-verification reports
Does the Lean express the advertised result?
MathVet and cert-machine disagree on how much of the advertised result the Lean statements express. MathVet finds that the Lean statements express or imply the whole advertised result (their term: “full”); cert-machine finds that the Lean statements express a narrower statement than the advertised result (their term: “NARROWER — DECLARED”).
MathVet and Carlos Toledo’s cert-machine independently assessed whether this family’s Lean statements express the whole result OpenAI describes in its catalogue. These are assessments of the family as a whole, not of each paper theorem separately.
MathVet finds that the Lean statements express or imply the whole advertised result (their term: “full”).
Their note:Sharp 9/(8 pi) bound and attainment are stated; the NP-hardness of improving the kernel-clustering loss factor (a consequence combined with the Unique Games theorem) is not.
“full”, in their words: The challenge statements imply the headline claim after unpacking definitions or one routine step.
Their method: An independent, model-assisted, pre-referee reading of each family’s challenge statements against its headline claim under a published rubric (PROTOCOL.md), with its own Comparator runs per challenge.
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.
AI-drafted comparisons, with no recorded human review
Gaussian propeller bound and attainmentGaussianPropeller
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: the sharp bound 9/(8π) for the sum of squared Gaussian first moments over every partition, in every dimension, with the propeller attaining it. The Gaussian-maxima and kernel-clustering consequences are not included.
Who ran the proof-verification program?
These groups report running Comparator with the release’s Lean statement files. For a configured theorem, it verifies OpenAI’s separate proof against the exact statement; some files instead configure definitions for comparison by name and type. Each row is the named group’s report. Atlas has not repeated the runs.
These reports do not list the software versions, proof-verification programs, axioms, time or memory in separate fields. Any such details in a report’s prose are reproduced below.
Gaussian propeller bound and attainmentGaussianPropeller
Three reports, all at the release’s recorded version.
Reported by
Their word
Their account
Date
Original report
cert-machine
“CERTIFIED”
Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.
These reports examine claims in the papers using exact calculations for finite claims, independently of the release’s Lean proofs. Read each report for the exact claim it covers.
Appendix A, and Tables 1, 5, 2, 3 and 4 of The Gaussian propeller bound in every dimension
Reported by
cert-machine
Method
Exact computation
Their verdict
“CERTIFIED” Exact code written independently from the paper confirms that the published object has the claimed property.
What it covers
“a finite component: every finite numerical comparison and enclosure of the m >= 5 exclusion (Appendix cert:arithmetic, Tables sc:interval-table, cert:quantiles, elim:small-table, elim:four-table, elim:final-table and the inline bounds), plus the finite algebra of geometry.tex; NOT the case m <= 4 (Heilman-Jagannath-Naor, external, computer-assisted) and not the prose analysis”
What these papers cite in the release, and what cites them
These links come from the papers’ bibliographies. A citation records that one paper cites another; how each cited result is used has not been examined here.
Papers elsewhere in the release that this family’s papers cite
anxkhn, Series of 15 issues and 15 draft PRs adding release results to Formal Conjectures (issues 6972 to 6997, PRs 6977 to 7002), 8 October 2026. Source
Proposes adding the release’s Gaussian propeller partition bound to Formal Conjectures (issue 6997, PR 7002).
References outside the release
Each paper’s outside references are listed with it on its reading page, where each citation is placed. Every outside work cited in the release: References outside the release
Explain a result, compare statements or propose a correction
Atlas has no contributed explanation for this family yet. You can offer one with a GitHub account. You receive named credit and disclose any AI assistance.
Each button opens a GitHub form with this page and its source version filled in. Opening the form does not reserve the work.
Commentary on this family by others, dated and credited. It is commentary, not a finding.
An unsigned public tier list, the OpenAI Math Tier List, classes it as a proof and rates its significance, if it holds up, as “Major math journal”. The page says that nothing on it is verified.
Each item identifies a statement, definition or report that needs further review. Its link opens a GitHub form with the source version already recorded.
Read the Lean statement against the paper
An AI agent drafted an assessment of which part of the paper this Lean statement expresses. No named human review is recorded.Open the form
Identify the theorem the catalogue describes
Family 096The theorem shown was selected from shared wording with OpenAI’s description. Atlas records no review confirming that it is the theorem the description means.Open the form
Compare the proposed paper and Lean matches
Family 096Atlas records no review of whether each linked Lean statement and paper theorem mean the same thing.Open the form
Why this passage is shown
Theorem 1.1
In The Gaussian propeller bound in every dimension
How it was found
Its words were compared with the words of OpenAI’s catalogue description.
The challenge statements imply the headline claim after unpacking definitions or one routine step.
Their method
An independent, model-assisted, pre-referee reading of each family’s challenge statements against its headline claim under a published rubric (PROTOCOL.md), with its own Comparator runs per challenge. Its protocol describes a planned error-rate study; this import does not establish that the study has been completed.
The challenge statements state less than the headline, and OpenAI’s Lean note says so.
Their method
Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule; every family’s challenge statements read against OpenAI’s headline, clause by clause. Decider: Comparator (leanprover/comparator d03acab1 = tag v4.34.0, built with the release’s toolchain v4.34.1), lean4export 076e8e57, landrun 811cfff5, nanoda_lib 3a240721 — on GitHub’s Linux runners…
Re-run through Comparator in a sandboxed virtual machine: accepted.
Date
8 October 2026
Version
at the release’s recorded version
Their method
An independent, model-assisted, pre-referee reading of each family’s challenge statements against its headline claim under a published rubric (PROTOCOL.md), with its own Comparator runs per challenge.
Re-run through Comparator with Lean’s kernel: accepted (the check took 660 s; peak memory 6.5 GiB).
Date
11 October 2026
Version
at the release’s recorded version
Their method
Each challenge’s solution is built and checked by Comparator at the pinned release, on Lean leanprover/lean4:v4.34.1, recording the verdict, the check time and the peak memory.
Appendix A, and Tables 1, 5, 2, 3 and 4 of The Gaussian propeller bound in every dimension
Exact calculations for finite claims, cert-machine
Their verdict
“CERTIFIED”
Claim tested or proved
Exact code written independently from the paper confirms that the published object has the claimed property.
What it covers
“a finite component: every finite numerical comparison and enclosure of the m >= 5 exclusion (Appendix cert:arithmetic, Tables sc:interval-table, cert:quantiles, elim:small-table, elim:four-table, elim:final-table and the inline bounds), plus the finite algebra of geometry.tex; NOT the case m <= 4 (Heilman-Jagannath-Naor, external, computer-assisted) and not the prose analysis”
three independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet; Neal McBurnett, openai-math-lean-tooling.