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.BoundedRecovery.bounded_recovery
The lines of the Lean statement that state the theorem, then the definitions in the file that it uses.
theorembounded_recovery (S : StandardModularData H) (hscalar : S.ScalarCentralizer) (omega : Ultrafilter ℕ) (hfree : (omega : Filter ℕ) ≤ Filter.atTop) (T : H →L[ℂ] H) (s : ℝ) (delta : ℕ → ℝ) (hdelta : ∀ n, 0 < delta n) (hdelta0 : Tendsto delta atTop (𝓝 0)) (h : ℕ → H) (hunit : ∀ n, ‖h n‖ = 1) (hband : ∀ n, S.D.HasSpectralSupport (Icc (s - delta n) (s + delta n)) (h n)) (hpositive : 0 < Filter.limsup (fun n => fixedMean omega (fun t => ‖T (S.D.unitary t (h n))‖ ^ 2)) atTop) : RecoveryConclusion S T s delta := 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.
defHasSpectralSupport (D : RealSpectralCalculus H) (I : Set ℝ) (h : H) : Prop := ∀ f : BoundedContinuousFunction ℝ ℂ, (∀ u ∈ I, f u = 0) → D.calculus f h = 0
structureStandardModularData (H : Type u) [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] where
Show the remaining 11 lines
M : VonNeumannAlgebra H xi : H norm_xi : ‖xi‖ = 1 cyclic : Dense {h : H | ∃ a : H →L[ℂ] H, a ∈ M ∧ a xi = h} separating : ∀ a : H →L[ℂ] H, a ∈ M → a xi = 0 → a = 0 D : RealSpectralCalculus H J : H ≃ₗᵢ⋆[ℂ] H conjugation : Function.Involutive J tomita_graph : closure {p : H × H | ∃ a : H →L[ℂ] H, a ∈ M ∧ p = (a xi, (star a) xi)} = {p | ∃ q : H, (p.1, q) ∈ D.halfGraph ∧ p.2 = J q}
defRecoveryConclusion (S : StandardModularData H) (T : H →L[ℂ] H) (s : ℝ) (delta : ℕ → ℝ) : Prop := ∃ (nj : ℕ → ℕ) (_ : StrictMono nj) (v : ℕ → (H →L[ℂ] H)) (C eta : ℝ),0 < eta ∧ ∀ j : ℕ, v j ∈ S.M ∧ ‖v j‖ ≤ C ∧ eta ≤ ‖T (v j S.xi)‖ ∧ S.D.HasSpectralSupport (Icc (s - 4 * delta (nj j)) (s + 4 * delta (nj j))) (v j S.xi)
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: standard, space, eta, delta, mean, recovery, average, support and seven more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: subsequence, elements, vectors, unit, state, faithful and normal.
Statements in Bounded recovery for modular spectral averages come first because the release lists only that paper for this Lean statement.
Theorem 4.1 of Bounded recovery for modular spectral averages
Theorem(Bounded recovery)
Let M be a von Neumann algebra with a faithful normal state \phi such that M_\phi=\C1. Use the standard-space notation and the fixed mean m of mod:average. Fix T\in\B(H) and s\in\R. Suppose that \delta_n>0 tends to zero and that h_n\in H are unit vectors with D-spectral support in [s-\delta_n,s+\delta_n]. If
\limsup_{n\to\infty}m_t\norm{TU_th_n}^{2}>0,
then there are a subsequence (n_j), elements v_j\in M, and constants C_*<\infty and \eta>0 such that, for every j,
\norm{v_j}\le C_*,\qquad
\norm{Tv_j\xi}\ge\eta,
and v_j\xi has D-spectral support in [s-4\delta_{n_j},s+4\delta_{n_j}].
Let $M$ be a von Neumann algebra with a faithful normal state $\phi$ such
that $M_\phi=\C1$. Use the standard-space notation and the fixed mean $m$
of \Cref{mod:average}. Fix $T\in\B(H)$ and $s\in\R$. Suppose that
$\delta_n>0$ tends to zero and that $h_n\in H$ are unit vectors with
$D$-spectral support in $[s-\delta_n,s+\delta_n]$. If
\begin{equation}
\limsup_{n\to\infty}m_t\norm{TU_th_n}^{2}>0,
\end{equation}
then there are a subsequence $(n_j)$, elements $v_j\in M$, and constants
$C_*<\infty$ and $\eta>0$ such that, for every $j$,
\[
\norm{v_j}\le C_*,\qquad
\norm{Tv_j\xi}\ge\eta,
\]
and $v_j\xi$ has $D$-spectral support in
$[s-4\delta_{n_j},s+4\delta_{n_j}]$.
Other possible paper matches, closest wording first
Theorem 1.1 of Bounded recovery for modular spectral averages. Found in the paper’s title, which its main theorem states: modular, averages, recovery, bounded and spectral. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: state, faithful and normal.
Proposition 5.1 of Expected amenable subalgebras preserving core commutants. Words of the Lean statement, its names split into words, that also appear in the theorem: function, eta, delta, support, xi and spectral. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: whose, vector and unit.
Theorem 1.2 of Bounded recovery for modular spectral averages. Words of the Lean statement, its names split into words, that also appear in the theorem: continuous, von, neumann, norm, algebra, bounded and one more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: uniformly, state, faithful and normal.
Proposition 2.1 of Bounded recovery for modular spectral averages. Words of the Lean statement, its names split into words, that also appear in the theorem: continuous, norm, algebra and bounded. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: bicentralizer, results, uniformly, state, faithful and normal.
Proposition 2.1 of Expected amenable subalgebras preserving core commutants. Words of the Lean statement, its names split into words, that also appear in the theorem: omega and spectral. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: positive, state and normal.
Theorem 1.1 of Expected amenable subalgebras preserving core commutants. Words of the Lean statement, its names split into words, that also appear in the theorem: means, von, neumann and algebras. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: faithful and normal.
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 216 s; peak memory 6.5 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 290’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 a supporting or a different result statedTheir term: “supporting-only”.
The challenge statements state a lemma or auxiliary result, not the headline. Their record
Their note: Only a technical lemma is stated: bounded_recovery (S : StandardModularData H) (hscalar : S.ScalarCentralizer) (omega free ultrafilter) (T) (s) (delta) ... (hpositive : 0 < limsup (fixedMean omega (fun t => ‖T (D.unitary t (h n))‖²))) : RecoveryConclusion S T s delta (a subsequence of uniformly bounded algebra elements v_j with eta ≤ ‖T (v_j ξ)‖ and spectral support in bands of 4x the width). The summary's headline, Connes' bicentralizer conjecture for every type III₁ factor with separable predual and every faithful normal state (and the relative bicentralizer statement producing an amenable P with P' ∩ c(M) = N' ∩ c(M)), is not stated (docs: 'The absolute bicentralizer conjecture is not included').
Definitions the group asks readers to compare with the paper: Hand-built Tomita-Takesaki setting: StandardModularData (M, ξ, a hand-written RealSpectralCalculus D, J, tomita_graph equal to the closure of the graph of S₀ composed with half-exponential cutoffs), fixedMean omega f := Filter.limUnder omega (symmetricAverage f) (junk if the ultralimit does not exist, but ultrafilter limits on bounded sequences do exist).
Finds a supporting or a different result statedTheir term: “DIFFERENT”.
The challenge statements state a different theorem from the headline. Their record
Their note: both headline clauses; the Lean states a bounded spectral-recovery lemma for modular spectral averages that neither implies nor is implied by either clause
OpenAI’s own note on what this Lean statement covers
“For a faithful normal state with scalar centralizer in the stated standard-space representation, the formalized result converts positive modular spectral averages into uniformly bounded algebra elements. Given unit vectors in shrinking spectral bands around a real number $s$ and a positive limiting symmetric average for a bounded operator $T$, it finds a subsequence of bounded algebra elements whose images under $T$ stay uniformly nonzero and whose spectral bands have four times the original widths. The absolute bicentralizer conjecture is not included.”
An AI agent drafted an interpretation of that note, classifying this statement as “states a supporting result”. No human review of the draft is recorded.
Supporting material
The complete Lean statement file
102 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: 13 to its own declarations, 53 to Mathlib’s documentation. 13 declared names also link to the same declaration in OpenAI’s library. 2 identifiers stay plain because they resolve to no single declaration: variables Lean binds automatically, notation, or names 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.
Let $M$ be a type $\mathrm{III}_1$ factor with separable predual and let
$\phi$ be any faithful normal state on $M$. Then
\[
\BC(M,\phi)=\C1.
\]
Theorem(Spectral intertwining rigidity)
Let M be a von Neumann algebra with a faithful normal state \phi such that M_\phi=\C1. Let b:\R\curvearrowright M be a continuous \phi-preserving action. Suppose that, for every s\in\R, every uniformly bounded sequence (w_n) in M, and every a\in M,
Let $M$ be a von Neumann algebra with a faithful normal state $\phi$ such
that $M_\phi=\C1$. Let $b:\R\curvearrowright M$ be a continuous
$\phi$-preserving action. Suppose that, for every $s\in\R$, every uniformly
bounded sequence $(w_n)$ in $M$, and every $a\in M$,
\begin{equation}
\norm{\phi w_n-e^s w_n\phi}_{M_*}\longrightarrow0
\quad\Longrightarrow\quad
w_na-b_s(a)w_n\longrightarrow0
\quad\text{strongly}.
\end{equation}
Then $b_s=\id_M$ for every $s\in\R$.
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
12 local vocabulary declarations
Lines of Lean, without comments or blank lines
59
Second proof-verification program (nanoda)
Off in OpenAI's configuration.
Statement comparison
A shareable summary of this page: the proposed paper match, what the Lean statement expresses, and which reviews are recorded.
Plectis Atlas
Bounded recovery from modular spectral averages
Family 290: Relative bicentralizers and modular spectral recovery
Candidate pairing
Theorem (Bounded recovery)
Let M be a von Neumann algebra with a faithful normal state \phi such that M_\phi=\C1. Use the standard-space notation and the fixed mean m of mod:average. Fix T\in\B(H) and s\in\R. Suppose that \delta_n>0 tends to zero and that h_n\in H are unit vectors with D-spectral support in [s-\delta_n,s+\delta_n]. If
\limsup_{n\to\infty}m_t\norm{TU_th_n}^{2}>0,
then there are a subsequence (n_j), elements v_j\in M, and constants C_*<\infty and \eta>0 such that, for every j,
\norm{v_j}\le C_*,\qquad
\norm{Tv_j\xi}\ge\eta,
and v_j\xi has D-spectral support in [s-4\delta_{n_j},s+4\delta_{n_j}].
Bounded recovery for modular spectral averages
TeX label rec:bounded, build/sections/04-bounded-recovery.tex, line 20
The Lean statement says
theorembounded_recovery (S : StandardModularData H) (hscalar : S.ScalarCentralizer) (omega : Ultrafilter ℕ) (hfree : (omega : Filter ℕ) ≤ Filter.atTop) (T : H →L[ℂ] H) (s : ℝ) (delta : ℕ → ℝ) (hdelta : ∀ n, 0 < delta n) (hdelta0 : Tendsto delta atTop (𝓝 0)) (h : ℕ → H) (hunit : ∀ n, ‖h n‖ = 1) (hband : ∀ n, S.D.HasSpectralSupport (Icc (s - delta n) (s + delta n)) (h n)) (hpositive : 0 < Filter.limsup (fun n => fixedMean omega (fun t => ‖T (S.D.unitary t (h n))‖ ^ 2)) atTop) : RecoveryConclusion S T s delta := 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
The AI draft finds no Lean statement in its family that states a paper's theorem as the paper does.