A nonzero unital separable complex C∗-algebra is Jiang–Su stable exactly when its norm central-sequence algebra has no characters, for every free ultrafilter. This answers the Kirchberg–Rørdam character question. Also, the infinite minimal tensor power of every such algebra without characters is Jiang–Su stable, answering the Dadarlat–Toms question. (OpenAI’s Lean note)
Lean is a language for mathematical statements and proofs that a computer can verify. The assessments below ask whether those statements express the result OpenAI describes here.
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In The Kirchberg–Rørdam character criterion, 25 September 2026
The paper’s main theorem is shown as a possible match to OpenAI’s description. No review confirming the selection is recorded.
Let A be a nonzero unital separable complex C^*-algebra, and let \omega be any free ultrafilter on \N. Then
In The Kirchberg–Rørdam character criterion, 25 September 2026
Possible match to OpenAI’s description, selected because it shares the most wording. No review confirming the selection is recorded.
Let D be a nonzero unital separable complex C^*-algebra without characters, and write
Then B\cong B\otmin\Z, and B contains a unital copy of \Z.
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For a configured theorem, the Comparator program tests whether OpenAI’s separate proof proves the exact Lean statement using only the allowed axioms. Some statement files instead configure definitions for comparison by name and type. Acceptance does not establish that a formal statement means the same thing as the paper’s theorem.
CharacterCriterionThe paper states
In The Kirchberg–Rørdam character criterion, 25 September 2026
Let A be a nonzero unital separable complex C^*-algebra, and let \omega be any free ultrafilter on \N. Then
The words they share: ultrafilter, omega, separable, complex, characters, algebra, unital, criterion, kirchberg, free, cong, nonzero.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
CharacterCriterion.lean, the release lists it as a main result.
theorem character_criterion (A : Type) [CStarAlgebra A] [Nontrivial A] [TopologicalSpace.SeparableSpace A] (ω : Ultrafilter ℕ) (hω : (ω : Filter ℕ)≤Filter.cofinite) : OAI.KirchbergRordam.HasNoCharacters (NormUltrapower.CentralAlgebra A ω) ↔ Nonempty (A ≃⋆ₐ[ℂ] MinTensor.Algebra A JiangSu.Algebra) := by sorry 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; MathVet. Who ran the proof-verification program
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: character_criterion (A) [CStarAlgebra A] [Nontrivial A] [SeparableSpace A] (ω) (hω : ω ≤ cofinite) : HasNoCharacters (NormUltrapower.CentralAlgebra A ω) ↔ Nonempty (A ≃⋆ₐ[ℂ] MinTensor.Algebra A JiangSu.Algebra) (for unital separable A: the relative commutant A_ω ∩ A' has no nonzero *-character iff A ≅ A ⊗_min 𝒵; for unital A the annihilator quotient is trivial). Not stated: the summary’s second claim, that the infinite minimal tensor power of every such algebra without characters is Jiang-Su stable (the Dadarlat-Toms question; docs only mention the criterion).
Definitions the group asks readers to compare with the paper: The Jiang-Su algebra is not Mathlib’s: JiangSu.Algebra := CStarInductiveLimit.Algebra StandardPrimeModel.presentation.system, a hand-built inductive limit of prime dimension-drop algebras with explicit multiplicities (copies p := (p-1)(p^4+p^2+1), size n = 2^(7^n)), so ‘A ≅ A ⊗_min 𝒵’ is only as faithful as this model of 𝒵; likewise NormUltrapower.Algebra, MinTensor.Algebra (spatial/min tensor norm via representation pairs) and the central subalgebra are hand-built (2.1k-line challenge file).
“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 — UNDECLARED”).
The gap they name: Clause 3 — the infinite minimal tensor power D^{⊗min ∞} of every nonzero unital separable characterless C*-algebra is Z-stable (Dadarlat–Toms question; manuscript corollary cor:infinite-tensor) — is claimed by the headline and by the scope note’s title but is not stated by any challenge. Clauses 1–2 (the character criterion) are stated in full.
“NARROWER — UNDECLARED”, in their words: The challenge statements state less than the headline, and OpenAI’s Lean note does not say so.
Their method: Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule.
OpenAI’s own note on what its Lean covers
The Kirchberg–Rørdam criterion relates Jiang–Su absorption to characters of the central-sequence algebra. The formalized result proves that, for every nonzero unital separable complex $C^*$-algebra $A$ and every free ultrafilter on $\mathbb N$, the norm central-sequence algebra has no nonzero character exactly when $A\cong A\otimes_{\min}\mathcal Z$. No nuclearity, simplicity, trace, or comparison hypothesis is imposed.
Kirchberg–Rørdam character criterion CharacterCriterion
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: for nonzero unital separable A and every free ultrafilter, the central-sequence algebra has no characters exactly when A absorbs the Jiang–Su algebra. The finite and infinite tensor-power results are not part of the statement.
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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CharacterCriterionTwo 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. | 10 October 2026 | Original report |
| MathVet | “comparator-sandboxed-pass” | Re-run through Comparator in a sandboxed virtual machine: accepted. | 8 October 2026 | Original report |
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Why this passage is shown
Theorem 1.1
In The Kirchberg–Rørdam character criterion, 25 September 2026
Why this passage is shown
Corollary 1.3 (Infinite minimal tensor powers)
In The Kirchberg–Rørdam character criterion, 25 September 2026
Why this passage is shown
Theorem 1.2
In The Kirchberg–Rørdam character criterion, 25 September 2026
Independent assessment
MathVet
14c7f91Independent assessment
Carlos Toledo’s cert-machine
9814e4cProof-verification report
cert-machine, CharacterCriterion
Proof-verification report
MathVet, CharacterCriterion
Lean statement
CharacterCriterion