Relative bicentralizers and modular spectral recovery
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Proves Connes’ bicentralizer conjecture for every type III1 factor with separable predual and every faithful normal state. More generally, for every inclusion \(N\subset M\) of von Neumann algebras with separable preduals admitting a faithful normal conditional expectation, constructs an amenable expected subalgebra \(P\subset N\) with \(P'\cap c(M)=N'\cap c(M)\), resolving the relative bicentralizer conjecture. (OpenAI’s Lean note)
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Possible match to OpenAI’s description, selected from the papers’ main statements because it shares the most wording. No review confirming the selection is recorded.
Let N\subset M be a unital inclusion of von Neumann algebras with separable preduals, and suppose that there is a faithful normal conditional expectation E_N:M\to N. There exist a unital amenable von Neumann subalgebra P\subset N and a faithful normal conditional expectation F:N\to P such that
P'\cap c(M)=N'\cap c(M).
Here amenable means injective.
No Lean statement in the release is paired with this theorem.
Three more possible matches
Theorem 1.1in Bounded recovery for modular spectral averages, 23 September 2026.
We prove that every inclusion \(N\subset M\) of von Neumann algebras with separable preduals and a faithful normal conditional expectation contains an expected amenable subalgebra \(P\subset N\) such that \(P'\cap c(M)=N'\cap c(M)\), where \(c(M)\) denotes the continuous core of M. This resolves the relative bicentralizer conjecture in this setting.
We prove a bounded spectral recovery theorem for a von Neumann algebra with a faithful normal state whose centralizer consists only of scalars. A positive averaged squared norm for an arbitrary bounded Hilbert-space operator on shrinking modular spectral bands can be recovered on uniformly bounded algebra elements with shrinking spectral support. We apply this theorem to spectral intertwining rigidity and obtain an alternative proof of bicentralizer triviality for type III1 factors with separable predual.
One Lean statement in this family
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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 Lemma 3.4. 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}].
The words they share: standard, space, eta, delta, mean, recovery, average, support, xi, von, neumann, norm, bounded, algebra, spectral, subsequence, elements, vectors, unit, state, faithful, normal.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
BoundedRecovery.lean, the release lists it as a main result.
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
This file states the theorem with sorry, a placeholder, in place of its proof. OpenAI supplies a separate proof file. Comparator tests whether that proof proves the statement here.
three independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet; Neal McBurnett, openai-math-lean-tooling. Who ran the proof-verification program
cert-machine records it as “CERTIFIED”.
MathVet records it as “comparator-sandboxed-pass”.
Neal McBurnett, openai-math-lean-tooling records it as “accepted”.
Other mathematical statements in OpenAI’s Lean library
Atlas found these propositions by scanning the text of the project’s Model.lean file. Each name links to its definition, with any associated statement file beside it. This list does not record running Lean to interpret or verify those definitions.
Independent comparisons and proof-verification reports
Does the Lean express the advertised result?
Neither group finds that the Lean statements express or imply the whole advertised result. They describe what the statements express in different terms.
MathVet and Carlos Toledo’s cert-machine independently assessed whether this family’s Lean statements express the whole result OpenAI describes in its catalogue. These are assessments of the family as a whole, not of each paper theorem separately.
MathVet finds that the Lean statements express a supporting result (their term: “supporting-only”).
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).
“supporting-only”, in their words: The challenge statements state a lemma or auxiliary result, not the headline.
Their method: 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.
cert-machine finds that the Lean statements express a different result (their term: “DIFFERENT”).
The gap they name:both headline clauses; the Lean states a bounded spectral-recovery lemma for modular spectral averages that neither implies nor is implied by either clause
“DIFFERENT”, in their words: The challenge statements state a different theorem from the headline.
Their method: Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule.
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.
AI-drafted comparisons, with no recorded human review
Bounded recovery from modular spectral averagesBoundedRecovery
Unreviewed AI draft: states a supporting result. States the bounded-recovery theorem for modular spectral averages (a theorem in Section 4 of the paper), a step towards the main theorem. The bicentraliser theorem BC(M, φ) = C1, the paper’s main result, is not included.
Who ran the proof-verification program?
These groups report running Comparator with the release’s Lean statement files. For a configured theorem, it verifies OpenAI’s separate proof against the exact statement; some files instead configure definitions for comparison by name and type. Each row is the named group’s report. Atlas has not repeated the runs.
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Bounded recovery from modular spectral averagesBoundedRecovery
Three reports, all at the release’s recorded version.
Reported by
Their word
Their account
Date
Original report
cert-machine
“CERTIFIED”
Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.
The program that collects Atlas’s source records, Statement Map, could not obtain or interpret some source material. These notes describe those limits; they do not identify mathematical errors.
BoundedRecovery: Tracked source lean/OAI/Analysis/ModularRecovery/Main.lean unavailable: bytes were not captured at fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb
What these papers cite in the release, and what cites them
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Papers elsewhere in the release that this family’s papers cite
This family’s papers cite no paper elsewhere in the release.
Papers elsewhere in the release that cite this family’s papers
No paper elsewhere in the release cites this family’s papers.
References outside the release
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Commentary on this family by others, dated and credited. It is commentary, not a finding.
An unsigned public tier list, the OpenAI Math Tier List, classes it as a proof and rates its significance, if it holds up, as “Major math journal”. The page says that nothing on it is verified.
Cyril Houdayer clarifies that Houdayer and Amine Marrakchi settled the non-relative bicentralizer problem in September and OpenAI’s result extends it to the relative version.
Questions awaiting review
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Identify the theorem the catalogue describes
Family 290The theorem shown was selected from shared wording with OpenAI’s description. Atlas records no review confirming that it is the theorem the description means.Open the form
Address a question raised in an independent assessment
The challenge statements state a lemma or auxiliary result, not the headline.
Their method
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.
The challenge statements state a different theorem from the headline.
Their method
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…
Re-run through Comparator in a sandboxed virtual machine: accepted.
Date
8 October 2026
Version
at the release’s recorded version
Their method
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.
Re-run through Comparator with Lean’s kernel: accepted (the check took 216 s; peak memory 6.5 GiB).
Date
10 October 2026
Version
at the release’s recorded version
Their method
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.
three independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet; Neal McBurnett, openai-math-lean-tooling.