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.SharpLiebThirring.sharp_lieb_thirring
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.
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: optimal, 3/2, equality, constant, 1/2 and gamma. Found in the paper’s title, which its main theorem states: thirring, sharp, one and lieb. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: nonnegative, rx, attained and sech.
Statements in Sharp one-dimensional Lieb–Thirring constants come first because the release lists only that paper for this Lean statement.
Theorem 1.1 of Sharp one-dimensional Lieb–Thirring constants
Theorem
For every 1/2<\gamma<3/2 and every nonnegative W\in L^{\gamma+1/2}(\R),
For every $1/2<\gamma<3/2$ and every nonnegative
$W\in L^{\gamma+1/2}(\R)$,
\begin{equation}
\Tr(H_{-W})_-^\gamma
\le 2\left(\frac{\gamma-1/2}{\gamma+1/2}\right)^{\gamma-1/2}
L_{\gamma,1}^{\cl}\int_\R W^{\gamma+1/2}.
\end{equation}
The constant is optimal and equals $L_{\gamma,1}^{(1)}$.
More precisely, if $r=(\gamma-1/2)^{-1}$, equality is attained by
$W(x)=(r+1)\sech^2(rx)$.
Other possible paper matches, closest wording first
Theorem 1.1 of Sharp one-dimensional Lieb–Thirring inequalities for matrix potentials. Words of the Lean statement, its names split into words, that also appear in the theorem: finite, sqrt, optimal, 3/2, potential, sharp and five more. Found in the paper’s title, which its main theorem states: thirring, one and lieb. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: rx, attained, sech and inequality.
Theorem 1.1 of Equality cases in the sharp one-dimensional matrix Lieb–Thirring inequality. Words of the Lean statement, its names split into words, that also appear in the theorem: exactly, sqrt, 3/2, eigenvalues, equality, negative and seven more. Found in the paper’s title, which its main theorem states: sharp, thirring and lieb. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: sech.
Corollary 7.3 of Sharp one-dimensional Lieb–Thirring constants. Words of the Lean statement, its names split into words, that also appear in the theorem: form, negative, real, gamma and bounded.
Proposition 7.2 of Sharp one-dimensional Lieb–Thirring constants. Words of the Lean statement, its names split into words, that also appear in the theorem: negative, eigenvalues, gamma, bounded and dx.
Theorem 2.1 of Sharp one-dimensional Lieb–Thirring constants. Words of the Lean statement, its names split into words, that also appear in the theorem: supremum, real and dx. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: inequality and times.
Proposition 5.1 of Sharp one-dimensional Lieb–Thirring constants. Words of the Lean statement, its names split into words, that also appear in the theorem: stated, real, bounded and dx.
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
three independent groups report that Comparator accepted runs for this Lean statement file: Carlos Toledo’s cert-machine; MathVet; Neal McBurnett’s openai-math-lean-tooling. 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.
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; the run was still in progress at this snapshot (192 accepted, 224 pending).
AcceptedTheir term: “accepted”.
Re-run through Comparator with Lean’s kernel: accepted (the check took 456 s; peak memory 6.6 GiB).
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 262’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: “weaker-statement”.
The challenge statements do not imply the headline claim: they state a nontrivially weaker statement. Their record
Their note: Only scalar potentials are treated: MainClaim : ∀ γ ∈ (1/2, 3/2), (∀ W : ℝ → ℝ, Admissible γ W → negativeMoment γ W ≤ sharpConstant γ * potentialMass γ W) ∧ optimalConstant γ = sharpConstant γ ∧ oneStateConstant γ = sharpConstant γ ∧ equality at (r+1) sech²(r x) with negativeMoment the sup over finite orthonormal families of H¹ weak negative eigenfunctions. The summary's finite-MATRIX-valued potentials with constant independent of matrix size, and the complete classification of equality cases (direct sums of sech² solitons with independent scales/centers and zero channels), are not stated: this is the scalar bound with attainment at one potential but without the equality-case classification.
Definitions the group asks readers to compare with the paper: Custom spectral definitions: H1 as an L² function with an L² weak derivative, schrodingerForm, IsNegativeEigenfunction W k u (weak eigenfunction at energy -k²) and negativeMoment γ W := ⨆ N (u k) orthonormal eigenfunctions, Σ (ofReal k_i)^(2γ) in ℝ≥0∞ instead of Mathlib spectral theory; constants semiclassicalConstant, sharpConstant are written explicitly (Γ-function formulas).
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: The matrix-valued inequality (arbitrary finite matrix size, constant independent of size) and the classification of all equality cases — the content of two of the family's three manuscripts and the subject of the family title — are not stated. Only the scalar sharp inequality with optimality and attainment at one sech² potential is.
OpenAI’s own note on what this Lean statement covers
“The formalized result determines the sharp one-dimensional Lieb–Thirring constant for $1/2<\gamma<3/2$. For every nonnegative $W\in L^{\gamma+1/2}(\mathbb R)$, it bounds the full negative-eigenvalue moment of $-d^2/dx^2-W$ by the one-bound-state constant times the potential integral. This constant is optimal and is attained by $(r+1)\mathrm{sech}^2(rx)$, where $r=(\gamma-1/2)^{-1}$. No finite spectral cutoff is imposed.”
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
100 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, 29 to Mathlib’s documentation. 16 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.
For every $1/2<\gamma<3/2$ and every nonnegative
$W\in L^{\gamma+1/2}(\R)$,
\begin{equation}
\Tr(H_{-W})_-^\gamma
\le 2\left(\frac{\gamma-1/2}{\gamma+1/2}\right)^{\gamma-1/2}
L_{\gamma,1}^{\cl}\int_\R W^{\gamma+1/2}.
\end{equation}
The constant is optimal and equals $L_{\gamma,1}^{(1)}$.
More precisely, if $r=(\gamma-1/2)^{-1}$, equality is attained by
$W(x)=(r+1)\sech^2(rx)$.
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.