Universal energy minimality of the triangular lattice
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.AtomicTriangular.universal_energy_minimum
The lines of the Lean statement that state the theorem, then the definitions in the file that it uses.
theoremuniversal_energy_minimum (g : ℝ → ℝ) (C : Set Plane) (hg : AdmissiblePotential g) (hC : LocallyFinite C) (hd : DensityOne C) : latticeEnergy g ≤ energy g C ∧ latticeEnergy g = energy g A := bysorry
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.
deflatticeEnergy (g : ℝ → ℝ) : ℝ≥0∞ := ∑' a : {x : Plane // x ∈ A ∧ x ≠ 0}, ENNReal.ofReal (g (‖a.val‖ ^ 2))
Possible paper match, with no recorded review
No paper statement clearly matched the Lean wording. Statement Map, the program that collects Atlas’s source records, selected this possible match using words in OpenAI’s catalogue and Lean note. No review of whether the statements mean the same thing is recorded.
Shared with the catalogue’s wording or OpenAI’s note, none of them matched in the Lean text: centered, comparison, completely, density, disk, energy, equality, finite and eight more.
Theorem 1.1 of 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,
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.
Other possible paper matches, closest wording first
Theorem 1.1 of Universal optimality of the triangular lattice. Words of the Lean statement, its names split into words, that also appear in the theorem: universal, minimality, smooth, monotone, completely, one and six more. Found in the paper’s title, which its main theorem states: triangular and lattice. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: nonnegative, centered, disk and configuration.
Proposition 7.2 of Universal optimality of the triangular lattice. Words of the Lean statement, its names split into words, that also appear in the theorem: smooth, monotone, completely, one, locally, density and three more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: nonnegative, centered and disk.
Corollary 8.1 of Universal optimality of the triangular lattice. Words of the Lean statement, its names split into words, that also appear in the theorem: points, one, density, energies and integer. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: among, minimizes, planar and configuration.
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: one, locally, density, real and finite. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: nonnegative, centered, disk, lower, configuration and function.
Proposition 2.4 of Triangular minimality for planar Coulomb renormalized energy. Words of the Lean statement, its names split into words, that also appear in the theorem: potential, points, energy, integer and finite. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: square and configuration.
Theorem 1.1 of Triangular minimality for planar Coulomb renormalized energy. Words of the Lean statement, its names split into words, that also appear in the theorem: triangular and finite. Found in the paper’s title, which its main theorem states: minimality and energy.
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
two independent groups report that Comparator accepted runs for this Lean statement file: Carlos Toledo’s cert-machine; MathVet. 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.
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…
AcceptedTheir term: “CERTIFIED”.
Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.
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.
AcceptedTheir term: “comparator-sandboxed-pass”.
Re-run through Comparator in a sandboxed virtual machine: accepted.
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.
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.
Finds part of it stated, or a narrower or weaker statementTheir 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
Finds part of it stated, or a narrower or weaker statementTheir 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.
OpenAI’s own note on what this Lean statement covers
“The formalization proves that the density-one triangular lattice minimizes lower energy per particle among all locally finite planar configurations of centered-disk density one, for every nonnegative completely monotone function of squared distance. Infinite energies are allowed in the comparison.”
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
74 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: 38 to its own declarations, 30 to Mathlib’s documentation. 21 declared names also link to the same declaration in OpenAI’s library. 1 identifier stays plain because it resolves to no single declaration: a variable Lean binds automatically, notation, or a name the scan of the pinned sources did not find. OpenAI uses Mathlib version d13f23b. The linked documentation describes the current version, so names that have since changed may be missing there.
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
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.
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}
The comparison and the equality are in the extended nonnegative reals.
An atomic certificate for triangular-lattice universal optimality
TeX label thm:universal, build/main.tex, line 68
The Lean statement says
theoremuniversal_energy_minimum (g : ℝ → ℝ) (C : Set Plane) (hg : AdmissiblePotential g) (hC : LocallyFinite C) (hd : DensityOne C) : latticeEnergy g ≤ energy g C ∧ latticeEnergy g = energy g A := bysorry
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.
Recorded comparisons with the paper
Atlas records no contributed comparison of this statement with the paper.
Contributed Comparator run reports
Atlas records no contributed report of running Comparator for this statement file.
Questions to resolve when comparing this statement with the paper
Both independent readers find this family's challenges state less than its headline, while the atlas's AI draft classes this file as stating a paper's theorem as the paper does; check which part of the headline it covers.