Kirchberg’s \(\mathcal O_2\) norm-ultrapower embedding problem
OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this one:
Constructs an explicit separable unital full group C∗-algebra that cannot embed unitally into the norm ultrapower of any fixed nonzero unital nuclear C∗-algebra, for any free ultrafilter on the natural numbers. Taking the target to be \(\mathcal O_2\) answers Kirchberg’s norm-ultrapower embedding problem negatively. (OpenAI’s Lean note)
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Both independent assessments find that the Lean statements express or imply the whole result described above.Read the assessments
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.
and let A=C^*(G) be its full group C^*-algebra. Then A is separable and unital. For every nonzero unital nuclear C^*-algebra B and every free ultrafilter \omega on \N, there is no unital embedding
A\longrightarrow B^\omega.
In particular, A does not embed unitally into \Otwo^\omega.
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 give a negative answer to Kirchberg’s norm-ultrapower embedding problem. We construct an explicit separable unital full group C∗-algebra that admits no unital embedding into Bω, for any nonzero unital nuclear C∗-algebra B and any free ultrafilter ω on ℕ. In particular, it does not embed unitally into \(\mathcal O_2^\omega\).
One Lean statement in this family
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and let A=C^*(G) be its full group C^*-algebra. Then A is separable and unital. For every nonzero unital nuclear C^*-algebra B and every free ultrafilter \omega on \N, there is no unital embedding
A\longrightarrow B^\omega.
In particular, A does not embed unitally into \Otwo^\omega.
The words they share: embedding, nonzero, group, full, sl, nuclear, algebra, unital, ultrafilter, free, omega, separable, ultrapower, norm, explicit, particular, unitally, embed.
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: MathVet; Neal McBurnett, openai-math-lean-tooling. Who ran the proof-verification program
MathVet records it as “comparator-pass”.
Neal McBurnett, openai-math-lean-tooling records it as “accepted”.
Other mathematical statements in OpenAI’s Lean library
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Independent comparisons and proof-verification reports
Does the Lean express the advertised result?
MathVet and cert-machine agree on how much of the advertised result the Lean statements express.
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 or imply the whole advertised result (their term: “full”).
Their note:main_no_embedding : MainStatementWithCuntzCorollary states: there is a separable unital C*-algebra A with ι : G →* unitary A satisfying the universal property of the FULL group C*-algebra of G = (ℤ[1/2])³ ⋊ (SL₃(ℤ) × ℤ) (IsFullGroupAlgebra), and for every nonzero unital nuclear B (nuclear = min and max C*-tensor norms agree on B ⊗ C for all C) and every free ultrafilter ω on ℕ there is no injective unital *-homomorphism A →⋆ₐ[ℂ] NormUltrapower B ω; the same for the universal Cuntz algebra O₂. Exactly the summary, including the O₂ corollary.
Definitions the group asks readers to compare with the paper:Hand-built objects: NormUltrapower B ω := (ultraCon B ω).Quotient (bounded sequences modulo ω-null), nuclearity via minTensorNorm/maxTensorNorm defined as sSup over representations on Hilbert spaces of one universe, IsFullGroupAlgebra (dense span + universal property), IsCuntzTwoAlgebra (two Cuntz isometries). Standard formulations, but none is a Mathlib notion, and the tensor-norm sSups are real-valued.
“full”, in their words: The challenge statements imply the headline claim after unpacking definitions or one routine step.
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.
Kirchberg’s norm-ultrapower embedding problem asks whether separable $C^*$-algebras embed into norm ultrapowers of nuclear algebras. The formalization constructs an explicit separable unital full group $C^*$-algebra that has no unital embedding into $B^\omega$ for any nonzero unital nuclear $C^*$-algebra $B$ and any free ultrafilter $\omega$ on $\mathbb N$. In particular, it does not embed unitally into an ultrapower of the Cuntz algebra $\mathcal O_2$.
AI-drafted comparisons, with no recorded human review
Obstruction to nuclear norm-ultrapower embeddingsNuclearUltrapower
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: the full group C*-algebra of Z[1/2]³ ⋊ (SL₃(Z) × Z) embeds unitally in no norm ultrapower of a nonzero unital nuclear algebra, in particular not in one of O₂.
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.
Obstruction to nuclear norm-ultrapower embeddingsNuclearUltrapower
Three reports, all at the release’s recorded version.
Reported by
Their word
Their account
Date
Original report
cert-machine
“REFUSED”
Re-run through Comparator did not reach a verdict (the job ended (failure) in the step "comparator, Lean’s kernel and nanoda" without uploading a result: the runner terminated it (GitHub: Process completed with exit code 143.)).
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
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Address a question raised in an independent assessment
The challenge statements imply the headline claim after unpacking definitions or one routine step.
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 the headline as OpenAI wrote it.
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 did not reach a verdict (the job ended (failure) in the step "comparator, Lean’s kernel and nanoda" without uploading a result: the runner terminated it (GitHub: Process completed with exit code 143.)).
Date
10 October 2026
Version
at the release’s recorded version
Their method
Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule.
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 1817 s; peak memory 19.6 GiB).
Date
11 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.