Strong Kadison–Kastler stability and its spatial boundaries
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Proves that sufficiently close unital von Neumann algebras on the same Hilbert space are conjugate by a unitary arbitrarily close to the identity, with a universal tolerance in the operator-norm distance between unit balls. Counterexamples show that near-identity conjugacy fails for one-sided near inclusions, and that arbitrarily close norm-separable C∗-algebras need not be ambiently unitarily conjugate. (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.
The other possible matches are Theorem 1.1 in Near Inclusions of von Neumann Algebras Without Small Spatial Embeddings () and Theorem 1.1(Strong Kadison–Kastler stability) in Universal strong Kadison–Kastler stability ().
For every \varepsilon>0 there are a separable complex Hilbert space H and unital, norm-separable C^*-algebras A,B\subseteq\B(H) with common identity \id_H such that
\KK(A,B)<\varepsilon,\qquad A''=B'',
but no unitary u\in\B(H) satisfies uAu^*=B.
No Lean statement in the release is paired with this theorem.
Reported proof verification
One Lean statement in the release; two independent groups report that Comparator accepted runs for this file at the version shown here.
We prove that sufficiently close unital von Neumann algebras are conjugate by a unitary arbitrarily close to the identity. The tolerance depends only on the prescribed distance of that unitary from the identity, uniformly over all algebras, representations, and Hilbert spaces. This resolves the strong Kadison–Kastler conjecture.
We give a negative answer to the unrestricted small-spatial-embedding problem for one-sided near inclusions of von Neumann algebras. On separable complex Hilbert spaces, we construct pairs of unital von Neumann algebras with common identity whose one-sided gaps tend to zero. Spatial embeddings of the source into the target exist, but every implementing unitary stays a fixed positive distance from the identity. The obstruction therefore concerns small implementing unitaries, rather than the existence of spatial embeddings.
We disprove the separable C∗-algebraic spatial form of the Kadison–Kastler conjecture. For every ε > 0, we construct unital, norm-separable C∗-algebras on a common separable complex Hilbert space, with the same identity and Kadison–Kastler distance less than ε, that are not conjugate by any unitary. The two algebras have the same von Neumann closure.
One Lean statement in this family
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For every \eps>0 there is \delta=\delta(\eps)>0 such that the following holds. If H is any complex Hilbert space and M,N\subseteq\B(H) are unital von Neumann algebras with identity I_H and d(M,N)<\delta, there is a unitary u\in\B(H) satisfying
uMu^*=N,\qquad \norm{u-I_H}<\eps.
The words they share: strong, stability, complex, von, neumann, algebras, unitary, space, norm, universal, kastler, kadison, delta, identity, unital, hilbert.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
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.
two independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; Neal McBurnett, openai-math-lean-tooling. Who ran the proof-verification program
cert-machine records it as “CERTIFIED”.
Neal McBurnett, openai-math-lean-tooling records it as “accepted”.
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 part of the advertised result (their term: “partial”).
Their note:Stated: universal_strong_stability : ∀ ε > 0, ∃ δ > 0, ∀ H, ∀ von Neumann algebras M N on H (Mathlib VonNeumannAlgebra), kkDistance M N < δ → ∃ unitary v, v M v* = N ∧ ‖v - 1‖ < ε with kkDistance the Hausdorff distance of the unit balls. Not stated: the summary’s two counterexamples (near-identity conjugacy fails for one-sided near inclusions; arbitrarily close norm-separable C*-algebras need not be ambiently unitarily conjugate); the docs scope also only mentions the stability theorem although the docs title says ‘spatial boundaries’.
“partial”, in their words: Only a special case is stated, or a co-equal headline claim has no challenge statement.
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 narrower statement than the advertised result (their term: “NARROWER — UNLINKED”).
The gap they name:Both counterexample clauses (the family’s two October 5 manuscripts) are not stated.
“NARROWER — UNLINKED”, in their words: The challenge statements state less than the headline; the missing part comes from a paper the Lean note does not link.
Their method: Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule.
The formalization proves the strong Kadison–Kastler conjecture uniformly. For every $\varepsilon>0$, there is a $\delta>0$ such that any two unital von Neumann algebras on the same complex Hilbert space at Kadison–Kastler distance below $\delta$ are conjugate by a unitary $u$ with $\lVert u-1\rVert<\varepsilon$. The tolerance depends only on $\varepsilon$, uniformly over the algebras, their representations, and the Hilbert space.
AI-drafted comparisons, with no recorded human review
Universal near-identity unitary conjugacy for close von Neumann algebrasStrongKadisonKastler
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: von Neumann algebras close in Kadison–Kastler distance are conjugate by a unitary near the identity, with δ depending only on ε.
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.
These reports do not list the software versions, proof-verification programs, axioms, time or memory in separate fields. Any such details in a report’s prose are reproduced below.
Universal near-identity unitary conjugacy for close von Neumann algebrasStrongKadisonKastler
Two 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.
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Papers elsewhere in the release that this family’s papers cite
Papers elsewhere in the release that cite this family’s papers
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Identify the theorem the catalogue describes
Family 289The 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
Compare the proposed paper and Lean matches
Family 289Atlas records no review of whether each linked Lean statement and paper theorem mean the same thing.Open the form
Why this passage is shown
Theorem 1.1
In Near Inclusions of von Neumann Algebras Without Small Spatial Embeddings, 5 October 2026
How it was found
Its words were compared with the words of OpenAI’s catalogue description.
Only a special case is stated, or a co-equal headline claim has no challenge statement.
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 less than the headline; the missing part comes from a paper the Lean note does not link.
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 with Lean’s kernel: accepted (the check took 5351 s; peak memory 8.1 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.