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.Laughlin.planar_gap
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.
defOccupationBidegree (Q n w : ℕ) (x : Space Q) : Prop := ∀ A : Finset (Fin (Q+1)), (A.card ≠ n ∨ (∑ i ∈ A, i.val) ≠ w) → (occupationBasis Q).repr x A = 0
noncomputabledefpairMatrix (L : ℕ) (c : Fin (L+1) → Fin (L+1) → ℂ) : Matrix (Occupations L) (Occupations L) ℂ := fun A B => ∑ i, ∑ j, c i j * (occupationBasis L).repr (annihilate j (annihilate i (occupationBasis L B))) A
No paper theorem is paired with this Lean statement.
Why none: no paper statement matches the Lean text clearly enough to pair, so the closest candidates are listed below.
Statements in A Fock-space inequality and the Laughlin spectral gap come first because the release’s name for this Lean statement shares words with that paper’s title.
Other possible paper matches, closest wording first
Corollary 7.1 of A Fock-space inequality and the Laughlin spectral gap. Words of the Lean statement, its names split into words, that also appear in the theorem: planar, pair, hamiltonian, gamma, gap, particle and one more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: projector, untruncated and spectral.
Theorem 1.1 of A Fock-space inequality and the Laughlin spectral gap. Words of the Lean statement, its names split into words, that also appear in the theorem: gamma, integer, particle and number. Found in the paper’s title, which its main theorem states: inequality, gap and space.
Corollary 1.2 of A Fock-space inequality and the Laughlin spectral gap. Words of the Lean statement, its names split into words, that also appear in the theorem: integer. Found in the paper’s title, which its main theorem states: inequality, gap and space.
Corollary 6.3 of A Fock-space inequality and the Laughlin spectral gap. Words of the Lean statement, its names split into words, that also appear in the theorem: coefficient, gap, particle and number. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: one.
Proposition A.1 of Uniform Stability of the Spherical Laughlin Gap. Words of the Lean statement, its names split into words, that also appear in the theorem: blocks, inequality, integer, particle, space and number.
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 269’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: Stated: the unperturbed spherical Laughlin gap at flux Q = 3(N-1): Laughlin.MainTarget (1/100) and LaughlinGap.MainTarget ((1/25) * distanceToLaughlinSq ψ ≤ energy ψ for every antisymmetric ψ and N >= N₀), the Fock-space inequality thm_fock (H² ≥ γ H for all γ < γ* ≈ 0.4617 and large flux, independent of particle number) and a planar version. Not stated: the summary's second claim, stability of the uniform gap and uniqueness of the ground state under weak bounded real scalar one-body potentials projected to the lowest Landau level (docs: 'Stability under projected one-body potentials and uniqueness of the perturbed ground state are outside it'); nor that laughlinVector is a zero-energy state (the inequality is stated against the line through it).
Definitions the group asks readers to compare with the paper: Hand-coded spherical V₁ Hamiltonian: energy ψ := Σ_{i<j} Σ_{p<2Q-1} Σ_{a, a_i = a_j = 0} |pairAmplitude ψ i j p a|² with explicit Clebsch-Gordan-type pairCoefficient Q p x y (descFactorial/factorial formula), laughlinVector from the coefficients of ∏(z_i w_j - z_j w_i)^3, and distanceToLaughlinSq ψ := sInf (range (c ↦ ‖ψ - c • laughlin‖²)). Its agreement with the physical V₁ projector is not machine-checked against any standard definition.
Finds part of it stated, or a narrower or weaker statementTheir term: “NARROWER — DECLARED”.
The challenge statements state less than the headline, and OpenAI’s Lean note says so. Their record
Their note: Stability of the gap and uniqueness of the ground state under projected bounded scalar one-body potentials (the October 5 'Uniform Stability' result).
OpenAI’s own note on what this Lean statement covers
“For the untruncated planar model, the formalization proves the corresponding inequality at the endpoint $\gamma_*$ on every homogeneous particle sector. These statements use coefficient one for each pair projector.”
An AI agent drafted an interpretation of that note, classifying this statement as “states a consequence of a paper’s theorem”. No human review of the draft is recorded.
Supporting material
The complete Lean statement file
82 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: 41 to its own declarations, 31 to Mathlib’s documentation. 18 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.
There are constants \lambda_*>0, \Delta_*>0, and N_*<\infty such that, for every N\ge N_*, every real bounded measurable \varphi on S_{3(N-1)} with \|\varphi\|_\infty\le1, and every real |\lambda|\le\lambda_*,
There are constants $\lambda_*>0$, $\Delta_*>0$, and
$N_*<\infty$ such that, for every $N\ge N_*$, every real
bounded measurable $\varphi$ on $S_{3(N-1)}$ with
$\|\varphi\|_\infty\le1$, and every real
$|\lambda|\le\lambda_*$,
\begin{equation}
E_1(N,\lambda,\varphi)-E_0(N,\lambda,\varphi)\ge\Delta_*.
\end{equation}
In particular the perturbed ground state is unique.
For every \(0<\gamma<\gamma_*\), there is an integer \(Q_\gamma\) such that
for every integer \(Q\ge Q_\gamma\),
\begin{equation}
H_Q^2\ge\gamma H_Q\qquad\text{on }\FF_Q.
\end{equation}
The threshold \(Q_\gamma\) is independent of particle number.
Corollary
There is an integer N_0\ge2 such that, for every N\ge N_0 and Q=3(N-1),
There is an integer \(N_0\ge2\) such that, for every \(N\ge N_0\) and
\(Q=3(N-1)\),
\begin{equation}
H_{N,Q}\ge\frac1{25}\bigl(I-P_{\mathrm L,N}\bigr)
\qquad\text{on }\bigwedge^N U_Q,
\end{equation}
where \(P_{\mathrm L,N}\) projects onto the line spanned by
\[
\Psi_{\mathrm L,N}=\prod_{i<j}(u_iv_j-u_jv_i)^3.
\]
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.