View source for Connes' bicentralizer problem

'''Connes' bicentralizer problem''' asks whether the bicentralizer of every faithful normal state on a type III<sub>1</sub> factor is trivial; Marrakchi's '''relative bicentralizer conjecture''' is its form for inclusions of von Neumann algebras. Two manuscripts in OpenAI's public mathematics collection ([https://github.com/openai/math github.com/openai/math]), dated 23 September 2026, claim the absolute problem for factors with separable predual, and one of them claims the relative conjecture for inclusions with separable preduals. The collection includes a Lean formalization of a supporting lemma. This page summarizes the published claims and lists publicly available follow-up work. Statements are the respective authors' claims; the publication of a paper, code or a formal proof is not presented here as independent verification.

== Background ==
* As recounted in the introduction of the principal manuscript:
** The bicentralizer of a faithful normal state consists of the elements that commute asymptotically with every bounded sequence asymptotically centralizing the state. Connes raised the problem of its triviality in the classification of injective factors. Haagerup solved the amenable type III<sub>1</sub> case, completing the uniqueness of the injective type III<sub>1</sub> factor, and characterized triviality for separable type III<sub>1</sub> factors through states whose centralizers have trivial relative commutant.
** Masuda introduced a relative bicentralizer in the finite-index setting, and Ando, Haagerup, Houdayer and Marrakchi constructed the relative bicentralizer flow for expected inclusions with type III<sub>1</sub> bottom factor. Marrakchi formulated the relative bicentralizer conjecture and proved that, for separable preduals, it is equivalent to the existence of an expected amenable subalgebra preserving the relative commutant in the continuous core (Conjecture C and Theorem D of his Inventiones paper, [https://arxiv.org/abs/2308.15163v3 arXiv:2308.15163]).
** Houdayer and Marrakchi announced that every type III<sub>1</sub> factor with a faithful normal state has trivial bicentralizer (Theorem G of [https://arxiv.org/abs/2609.11462v1 arXiv:2609.11462], version 1 of 10 September 2026, which has no separability restriction). The manuscript describes its absolute argument as an alternative proof of the separable-predual case.

== The original result ==
* '''Claim.''' Theorem 1.1 of ''Expected amenable subalgebras preserving core commutants'' states that every unital inclusion ''N'' ⊂ ''M'' of von Neumann algebras with separable preduals and a faithful normal conditional expectation contains an amenable subalgebra ''P'' ⊂ ''N'', with a faithful normal conditional expectation onto it, such that ''P''′ ∩ ''c''(''M'') = ''N''′ ∩ ''c''(''M''), where ''c''(''M'') is the continuous core. The manuscript states that this resolves the relative bicentralizer conjecture, including inclusions with arbitrary centers and types, and derives a maximal abelian version (Corollary 1.2). Its Theorem 3.1 states that every type III<sub>1</sub> factor with separable predual has trivial bicentralizer for every faithful normal state.
* ''Method, as described:'' the absolute argument starts from the Houdayer–Isono reduction to a factor that equals its own bicentralizer and has trivial centralizer, and derives a spectral contradiction by comparing two isometries obtained by averaging the modular action, using the binormal-state identity and zero-shift completely positive map of Marrakchi's [https://arxiv.org/abs/2606.23636v2 arXiv:2606.23636]. The relative argument iterates that identity uniformly in tensor length and flow shifts and passes through the continuous core.
* The companion manuscript ''Bounded recovery for modular spectral averages'' proves a bounded spectral recovery theorem for states with scalar centralizer and applies it to give another proof of bicentralizer triviality for type III<sub>1</sub> factors with separable predual.
* '''Sources''' (openai/math repository, commit <code>1581d4c</code>): [https://github.com/openai/math/tree/1581d4cbdd69a0faf34b2089614fe2a814316a6a/preprints/Expected-amenable-subalgebras-preserving-core-commutants-September-23-2026 Expected amenable subalgebras preserving core commutants] and [https://github.com/openai/math/tree/1581d4cbdd69a0faf34b2089614fe2a814316a6a/preprints/Bounded-recovery-for-modular-spectral-averages-September-23-2026 Bounded recovery for modular spectral averages].
* '''Formalization, as described by the repository.''' The [https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/290.md scope note] states that, 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, and that "the absolute bicentralizer conjecture is not included". Its Comparator statement is [https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/ComparatorChallenges/BoundedRecovery.lean BoundedRecovery.lean].

== Public follow-up work ==
{| class="wikitable"
! Date !! Work !! Relation !! Public claim !! Public status (checked 19 October 2026)
|-
| 2026-10-06 || S. Zhou, [https://hexagonmath.org/2610.00134v2 ''Poisson boundaries of type III factors and the relative bicentralizer conjecture''], Hexagon 2610.00134v2 || Parallel work (relative conjecture) ||
* The abstract states that the paper develops Poisson boundary theory for von Neumann algebras with separable preduals and proves that for every inclusion ''M'' ⊂ ''N'' with faithful normal conditional expectation, ''M'' ⊂ ''c''(''N'') has the weak Dixmier property; by Marrakchi's Theorem D this gives the relative bicentralizer conjecture for von Neumann algebras with separable preduals. It also states irreducible embeddings of continuous factors into hyperfinite type III<sub>λ</sub> factors.
* Its provenance statement describes the work as independently developed and AI-assisted, and states that the result on the relative conjecture "overlaps with OpenAI's concurrent work, but the proofs use fundamentally different methods". The reference list includes the manuscript.
| Hexagon versions 1 and 2, both dated 6 October 2026. The provenance statement says that Section 7 (the relative conjecture) is fully polished and Section 8 (irreducible inclusions) not yet.
|-
| 2026-10-08 || [https://github.com/mathvet/mathvet MathVet] (GitHub; review of openai/math, tables at repository commit <code>c76f1e4</code>, 14 October 2026) || Review (fidelity of the formal statements) || Rates this family "supporting-only": only the technical bounded-recovery lemma is stated in Lean, while the absolute and relative bicentralizer statements are not. It records a sandboxed Comparator pass for the challenge <code>BoundedRecovery</code> (8 October 2026). || MathVet states that it does not claim any theorem false and reports what was machine-checked as stated.
|-
| 2026-10-10 to 2026-10-14 || [https://github.com/leanprover/con-leche/blob/90577d9e49ee3aed45da3d421bdf5dc780192304/DESIGN.md con-leche development journal] (task #348) || Verification (second kernel) || In its run over all Comparator challenges of the collection at commit <code>fd4aeeb</code>, the journal's table of exceptions (time-outs, memory, errors) does not list <code>BoundedRecovery</code>, so the challenge was accepted by con-leche and by Lean's official kernel. || Development journal.
|-
| 2026-10-16
| J. Jardine, Lattice24, referee review [https://github.com/JJardine919/openai-math-audit/blob/a06a8c6e77974d4c6021338a55a8e91b634f67f7/reviews/rank_386.json rank_386] in [https://github.com/JJardine919/openai-math-audit JJardine919/openai-math-audit] (archived at [https://doi.org/10.5281/zenodo.23422995 Zenodo 10.5281/zenodo.23422995])
| Review (AI-assisted audit)
| Reports no gap for the principal manuscript. Its one minor note is a dependency caveat: the argument rests on cited inputs that cannot be checked from the paper itself (lemmas of Marrakchi's arXiv:2606.23636v2, a lemma of Ando–Haagerup–Houdayer–Marrakchi, results of Marrakchi's Inventiones paper including Theorem D, and the Houdayer–Isono reduction), and the reviewer states that this is not a gap. The review describes itself as an independent second pass agreeing with an earlier one.
| The reviews are published under CC BY 4.0.
|}

== Notes ==
* Houdayer and Marrakchi posted version 2 of arXiv:2609.11462 on 5 October 2026; its comment states that it completes the classification of flows with full Connes spectrum that are not outer, a case left open in version 1. The manuscript cites version 1.
* The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.

[[Category:OpenAI results]]