Lean statement

Sharp Fourier certificate for planar circle packing

Lean is a language for mathematical statements and proofs that a computer can verify. This page shows one statement file from OpenAI’s release, possible paper matches and independent reports. The code-like names below identify the definitions or theorems selected by the proof-verification settings.

Theorem selected for verification: OAI.sharp_fourier_certificate

Lean statement file: PlanarPacking.leanShow on the mapOpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 090 link it.

See the candidate theorem Who ran the proof-verification program Report a comparison with the paper (GitHub issue)

The Lean statement

The lines of the Lean statement that state the theorem, then the definitions in the file that it uses.

theorem sharp_fourier_certificate :    ∃ f : SchwartzMap SharpPlanar.Plane ℝ, SharpPlanar.SharpCertificate f := by  sorry

A Lean statement states the theorem and leaves its proof as sorry on purpose. OpenAI’s proof is in a separate solution module, and a program called Comparator checks that proof against this statement.

Definitions needed to read it

Local definitions the statement names, followed through the definitions they name in turn. The match is by name in the source text, so a name reached only through notation can be missed; the full file is below.

abbreviation OAI.SharpPlanar.Plane line 16

abbrev Plane := EuclideanSpace ℝ (Fin 2)

definition OAI.SharpPlanar.Radial line 19

def Radial (f : Plane → ℝ) : Prop :=  ∀ x y, ‖x‖ = ‖y‖ → f x = f y

definition OAI.SharpPlanar.fourier line 23

noncomputable def fourier (f : Plane → ℝ) (ξ : Plane) : ℂ :=  ∫ x : Plane, (f x : ℂ) *    Complex.exp (((-2 * Real.pi * inner ℝ x ξ : ℝ) : ℂ) * Complex.I)

definition OAI.SharpPlanar.SharpCertificate line 28

def SharpCertificate (f : SchwartzMap Plane ℝ) : Prop :=  Radial f ∧ fourier f 0 = 1 ∧ f 0 = 2 / Real.sqrt 3 ∧    (∀ ξ, (fourier f ξ).im = 0 ∧ 0 ≤ (fourier f ξ).re) ∧    (∀ x, 1 ≤ ‖x‖ → f x ≤ 0)

Possible paper match, with no recorded review

Atlas selected this possible match by comparing words in the Lean statement with every statement in the family. No review of whether the statements mean the same thing is recorded.

Words of the Lean statement, its names split into words, that also appear in the theorem: sharp, sqrt, certificate, fourier, schwartz, radial, real and function. Found in the paper’s title, which its main theorem states: planar. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: widehat.

Statements in A sharp Fourier certificate for planar circle packing come first because the release’s name for this Lean statement shares words with that paper’s title.

Theorem 1.1 of A sharp Fourier certificate for planar circle packing

Theorem (Sharp Fourier certificate)

There is a real radial Schwartz function f:\RR^2\to\RR such that

\widehat f(0)=1,\qquad f(0)=\frac{2}{\sqrt3},\qquad \widehat f(\xi)\ge0\quad(\xi\in\RR^2),\qquad f(x)\le0\quad(|x|\ge1).

Section 1, Introduction Read it in the paper TeX source, line 34

Show the TeX

There is a real radial Schwartz function $f:\RR^2\to\RR$ such that
\begin{equation}
  \widehat f(0)=1,\qquad f(0)=\frac{2}{\sqrt3},\qquad
  \widehat f(\xi)\ge0\quad(\xi\in\RR^2),\qquad
  f(x)\le0\quad(|x|\ge1).
\end{equation}

Other possible paper matches, closest wording first

Corollary 5.2 of A sharp Fourier certificate for planar circle packing. Words of the Lean statement, its names split into words, that also appear in the theorem: radius, euclidean, sqrt, disks and pi. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: packing and density.
Proposition 2.2 of A sharp Fourier certificate for planar circle packing. Words of the Lean statement, its names split into words, that also appear in the theorem: im, fourier, schwartz, radial, pi, real and one more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: widehat.
Proposition 6.1 of Universal optimality of the triangular lattice. Words of the Lean statement, its names split into words, that also appear in the theorem: transform, disk, one, fourier, schwartz, real and one more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: nonnegative, density and widehat.
Theorem 1.1 of An atomic certificate for triangular-lattice universal optimality. Words of the Lean statement, its names split into words, that also appear in the theorem: planar, disk, one, reals and function. Found in the paper’s title, which its main theorem states: certificate. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: nonnegative and density.
Theorem 2.1 of Universal optimality of the triangular lattice. Words of the Lean statement, its names split into words, that also appear in the theorem: normalized, fourier, schwartz, radial, pi, real and one more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: widehat.
Theorem 1.2 of An atomic certificate for triangular-lattice universal optimality. Words of the Lean statement, its names split into words, that also appear in the theorem: sharp, schwartz, radial, real and function. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: widehat.

How Atlas selects possible matches: it compares words in the Lean statement with every statement in the family. It also splits Lean names into words. The release’s label and note receive less weight; a main theorem wins a close match. If the Lean wording yields no clear match, Atlas uses the catalogue description. This selects passages to compare; it does not establish that they mean the same thing.

Reported Comparator runs

Atlas has no report of an independent group using Comparator to verify OpenAI’s proof of this Lean statement at the version shown here. Each group reports its own Comparator run for this statement file. Atlas has not repeated the runs.

For a configured theorem, Comparator tests whether OpenAI’s proof proves the exact Lean statement. Some files instead configure definitions for comparison by name and type. Acceptance does not establish that the statement means the same thing as the paper’s theorem.

Carlos Toledo’s cert-machine

How they ran it, in their words

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 (.github/workflows/openai-math-kernel.yml), one j…

No verdict reached Their term: “REFUSED”.
Re-run through Comparator did not reach a verdict (the job stopped before Comparator ran (see the run log)).
Reported 10 October 2026, at the release version shown here. Original verification report

What an accepted Comparator run would establish

If Comparator is run under this configuration and accepts, the solution proves the configured theorem using only the allowed axioms, and the Lean kernel accepts every step.

This configuration lists no definition holes, so every definition the statement uses is compared as it is written in this file.

Allowed axioms: propext, Quot.sound, Classical.choice

Acceptance leaves open whether the Lean statement means the same thing as the paper’s theorem, whether the paper’s other claims are correct, and whether the result is new. Those questions require comparing the definitions and statements, reviewing the paper’s arguments, and reading the earlier literature.

Do family 090’s Lean statements express the advertised result?

These assessments compare all the family’s Lean statements with the whole result OpenAI describes in its catalogue. They do not assess this statement alone. Each group uses its own published rules.

Neither group finds that the Lean statements express or imply the whole advertised result. They describe what the statements express in different terms.

MathVet
Finds part of it stated, or a narrower or weaker statement Their term: “partial”.
Only a special case is stated, or a co-equal headline claim has no challenge statement. Their record
Their note: Universal minimality of the triangular lattice for lower-limit energy per particle is stated; the renormalized Riesz (0<s<2) and Coulomb minimality, the Sandier-Serfaty and Brauchart-Hardin-Saff (spherical log energy) claims are not.
Definitions the group asks readers to compare with the paper: Custom energy/diskEnergy (liminf of per-particle disk energy in ℝ≥0∞, with diskPoints := if h : finite then h.toFinset else ∅), DensityOne, latticeEnergy := ∑' over nonzero lattice points; AtomicGaussian/TriangularGaussian carry a huge bespoke 'atomic construction' (explicit rational constants such as h = 17/50, H = 27/50, residue list, 20x23 jet matrices) inside the conclusion HasAtomicConstruction α f, unreadable as a statement
Carlos Toledo’s cert-machine
Finds part of it stated, or a narrower or weaker statement Their term: “NARROWER — UNLINKED”.
The challenge statements state less than the headline; the missing part comes from a paper the Lean note does not link. Their record
Their note: Triangular minimality for unit-background renormalized Riesz energies (0<s<2) and for the planar Coulomb renormalized energy, hence the Sandier–Serfaty and two-dimensional Brauchart–Hardin–Saff conclusions. These come from two of the family's four manuscripts ('Universal optimality of the triangular lattice' and 'Triangular minimality for planar Coulomb renormalized energy'), neither of which the scope note links. PlanarPacking (Cohn–Elkies sharpness) is an extra result the headline does not mention.

Read the complete assessments on the family page.

OpenAI’s own note on what this Lean statement covers

“The formalization constructs a radial Schwartz function $f$ on $\mathbb R^2$ with $\widehat f(0)=1$, $f(0)=2/\sqrt3$, $\widehat f$ real and nonnegative everywhere, and $f(x)\le0$ whenever $\|x\|\ge1$. These are the sharp Fourier certificate conditions yielding density $\pi/(2\sqrt3)$.”
Quoted from OpenAI’s Lean note for this family

An AI agent drafted an interpretation of that note, classifying this statement as “states a paper’s theorem as the paper does”. No human review of the draft is recorded.

Supporting material

The complete Lean statement file

40 lines, copied from release version fd4aeeb. A name defined in this file links to its line; pointing at the name shows its definition.

Declared in this fileOpenAI’s library at fd4aeebOther linked names open the current documentation for Mathlib, Lean’s mathematical library.

Links in the file: 11 to its own declarations, 9 to Mathlib’s documentation. 5 declared names also link to the same declaration in OpenAI’s library. OpenAI uses Mathlib version d13f23b. The linked documentation describes the current version, so names that have since changed may be missing there.

Definitions and lemmas named by the statement

If Comparator accepts a proof, it proves sharp_fourier_certificate exactly as written here. To compare that statement with the paper, also read the definitions and lemmas listed below. Atlas found them by following names in the source text and lists each after the names it uses. For x.foo, Atlas includes every foo defined here for a type, even when the statement may use only one of them.

It reaches all 4 other declarations of the file; with the theorem itself they span 13 lines. 2 of them are named in the statement itself.

Names in the namespace OAI.SharpPlanar are listed without it.

abbrev Planeline 16in the statementlibrary
def Radialline 19library
def fourierline 23library
def SharpCertificateline 28in the statementlibrary
theorem OAI.sharp_fourier_certificateline 36selected theoremlibrary

import Mathlibnamespace OAI/-!# A sharp Fourier certificate for planar circle packingThe normalization and Fourier convention are those of*A sharp Fourier certificate for planar circle packing* (OpenAI, 2026).-/open MeasureTheorynamespace SharpPlanarabbrev Plane := EuclideanSpace ℝ (Fin 2)/-- Radiality of a real function on the Euclidean plane. -/def Radial (f : Plane → ℝ) : Prop :=  ∀ x y, ‖x‖ = ‖y‖ → f x = f y/-- Fourier transformation with kernel `exp (-2πi⟪x, ξ⟫)`. -/noncomputable def fourier (f : Plane → ℝ) (ξ : Plane) : ℂ :=  ∫ x : Plane, (f x : ℂ) *    Complex.exp (((-2 * Real.pi * inner ℝ x ξ : ℝ) : ℂ) * Complex.I)/-- A normalized sharp certificate for disks of radius one half. -/def SharpCertificate (f : SchwartzMap Plane ℝ) : Prop :=  Radial f ∧ fourier f 0 = 1 ∧ f 0 = 2 / Real.sqrt 3 ∧    (∀ ξ, (fourier f ξ).im = 0 ∧ 0 ≤ (fourier f ξ).re) ∧    (∀ x, 1 ≤ ‖x‖ → f x ≤ 0)end SharpPlanar/-- The sharp Fourier certificate theorem. -/theorem sharp_fourier_certificate :    ∃ f : SchwartzMap SharpPlanar.Plane ℝ, SharpPlanar.SharpCertificate f := by  sorryend OAI

Every extracted statement of the accompanying papers

The statements the atlas extracted automatically from each accompanying paper’s TeX source, with links to the full sources.

An atomic certificate for triangular-lattice universal optimality

Theorem (Universal energy minimum)

For every smooth nonnegative completely monotone function g on (0,\infty) and every locally finite planar set \mathcal C of centered disk density one,

E_g(\mathcal C)\ge \sum_{a\in A\setminus\{0\}}g(|a|^2)=E_g(A).

The comparison and the equality are in the extended nonnegative reals.

Section 1, Introduction Read it in the paper TeX source, line 68

Show the TeX

For every smooth nonnegative completely monotone function $g$ on
$(0,\infty)$ and every locally finite planar set $\mathcal C$ of centered
disk density one,
\[
 E_g(\mathcal C)\ge
 \sum_{a\in A\setminus\{0\}}g(|a|^2)=E_g(A).
\]
The comparison and the equality are in the extended nonnegative reals.

Theorem (Sharp Gaussian minorants)

For every \alpha>0 there is a real radial Schwartz function f_\alpha:\mathbb R^2\to\mathbb R such that

f_\alpha(x)\le k_\alpha(x),\qquad \widehat f_\alpha(\xi)\ge0 \quad(x,\xi\in\mathbb R^2),

and

f_\alpha(a)=k_\alpha(a)\quad(a\in A\setminus\{0\}),\qquad \widehat f_\alpha(w)=0\quad(w\in A^*\setminus\{0\}).

For \alpha\ge1, the construction uses two finite 20-by-20 interpolation blocks and an absolutely summable infinite correction.

Section 1, Introduction Read it in the paper TeX source, line 95

Show the TeX

For every $\alpha>0$ there is a real radial Schwartz function
$f_\alpha:\mathbb R^2\to\mathbb R$ such that
\[
 f_\alpha(x)\le k_\alpha(x),\qquad
 \widehat f_\alpha(\xi)\ge0
 \quad(x,\xi\in\mathbb R^2),
\]
and
\[
 f_\alpha(a)=k_\alpha(a)\quad(a\in A\setminus\{0\}),\qquad
 \widehat f_\alpha(w)=0\quad(w\in A^*\setminus\{0\}).
\]
For $\alpha\ge1$, the construction uses two finite $20$-by-$20$ interpolation
blocks and an absolutely summable infinite correction.

A sharp Fourier certificate for planar circle packing

Theorem (Sharp Fourier certificate)

There is a real radial Schwartz function f:\RR^2\to\RR such that

\widehat f(0)=1,\qquad f(0)=\frac{2}{\sqrt3},\qquad \widehat f(\xi)\ge0\quad(\xi\in\RR^2),\qquad f(x)\le0\quad(|x|\ge1).

Section 1, Introduction Read it in the paper TeX source, line 34

Show the TeX

There is a real radial Schwartz function $f:\RR^2\to\RR$ such that
\begin{equation}
  \widehat f(0)=1,\qquad f(0)=\frac{2}{\sqrt3},\qquad
  \widehat f(\xi)\ge0\quad(\xi\in\RR^2),\qquad
  f(x)\le0\quad(|x|\ge1).
\end{equation}

Configuration and file facts

File: lean/ComparatorChallenges/PlanarPacking.lean
OpenAI’s Lean catalogue: OpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 090 link it.
Solution module: OAI.Analysis.PlanarPacking.Main
Allowed axioms: propext, Quot.sound, Classical.choiceThe axioms this configuration allows a proof to use. A replay shows which of them a proof uses.
Imports: Mathlib
Imports from outside Mathlib: None
Local vocabulary: 4 local vocabulary declarations
Lines of Lean, without comments or blank lines: 13
Second proof-verification program (nanoda): Off in OpenAI's configuration.

A shareable summary of this page: the proposed paper match, what the Lean statement expresses, and which reviews are recorded.

Plectis Atlas