Weak pure infiniteness and Cuntz-algebra absorption
OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this one:
Resolves the ordinary-to-strong pure-infiniteness question of Kirchberg and Rørdam for complex C∗-algebras. For exact algebras, proper infiniteness of one fixed finite amplification of every positive element also suffices. Consequently, every separable nuclear algebra with this property absorbs \(\mathcal O_\infty\), without unitality or simplicity assumptions. (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.
The two independent assessments disagree on how much of the result described above is expressed by the Lean statements.Read the assessments
Possible match to OpenAI’s description, selected because it shares the most wording. No review confirming the selection is recorded.
Let C be an exact complex C^*-algebra. Suppose that there is one integer n\ge1 such that h^{\oplus n} is properly infinite for every h\in C_+. Then every positive element of C is properly infinite. Consequently C is strongly purely infinite.
The release does not say which paper theorem its catalogue describes. Both statements above are shown whole: the paper’s main theorem first, then the statement that shares the most wording with the description. The atlas has not reviewed either choice.
One more possible match
Corollary 1.3in Weak pure infiniteness and O-infinity absorption.
Reported proof verification
Two Lean statements in the release; independent groups report that Comparator accepted runs for all these files at the version shown here.
We prove that every complex C∗-algebra in which each positive element is properly infinite is strongly purely infinite. This answers the ordinary-to-strong part of Kirchberg–Rørdam’s comparison question. For exact algebras, we also prove that proper infiniteness of one fixed finite amplification of every positive element implies proper infiniteness of each positive element. Consequently, every separable nuclear algebra with this fixed-amplification property absorbs \(\mathcal O_\infty\), without assumptions of unitality or simplicity.
Two Lean statements in this family
Choose one to see it beside the paper theorem it is paired with.
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.
Let C be an exact complex C^*-algebra. Suppose that there is one integer n\ge1 such that h^{\oplus n} is properly infinite for every h\in C_+. Then every positive element of C is properly infinite. Consequently C is strongly purely infinite.
The words they share: algebra, complex, element, exact, infinite, positive, properly, purely, strongly.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
ExactInfiniteness.lean, the release lists it as a supporting statement.
theoremexact_weaklyPurelyInfinite_individuallyProperlyInfinite_any {A : Type u} [NonUnitalCStarAlgebra A] [PartialOrder A] [StarOrderedRing A] (hA : Exact A) (hW : WeaklyPurelyInfinite A) {a : A} (ha : 0 ≤ a) : ProperlyInfinite (diagonalCopies 1 a) := 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.
Let C be an exact complex C^*-algebra. Suppose that there is one integer n\ge1 such that h^{\oplus n} is properly infinite for every h\in C_+. Then every positive element of C is properly infinite. Consequently C is strongly purely infinite.
The words they share: algebra, complex, element, exact, infinite, positive, properly, purely, strongly.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
IndividualStrongInfiniteness.lean, the release lists it as a supporting statement.
theoremindividual_to_strong_main (hP : PropertyP A) : (∀ a b : A, 0 ≤ a → 0 ≤ b → ∀ c : A, ∀ ε : ℝ, 0 < ε → ∃ s t : A, ‖star s * a * s - a‖ < ε ∧ ‖star t * b * t - b‖ < ε ∧ ‖star s * c * t‖ < ε) ∧ MainGap.StronglyPurelyInfinite A := 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”.
Independent comparisons and proof-verification reports
Does the Lean express the advertised result?
MathVet and cert-machine disagree on how much of the advertised result the Lean statements express. MathVet finds that the Lean statements express or imply the whole advertised result (their term: “full”); cert-machine finds that the Lean statements express a narrower statement than the advertised result (their term: “NARROWER — UNDECLARED”).
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:individual_to_strong_main (hP : PropertyP A) : unrestricted diagonalization ∧ StronglyPurelyInfinite A holds for every non-unital C*-algebra A, where PropertyP = every positive element properly infinite (Kirchberg-Rørdam’s characterization of ordinary pure infiniteness is the one routine identification step), and ExactInfiniteness gives the fixed-amplification version for exact A (WeaklyPurelyInfinite with some n ≥ 1 ⇒ ProperlyInfinite (diagonalCopies 1 a) and strongly purely infinite). The summary’s final consequence (separable nuclear algebras with this property absorb O_∞) is not stated.
Definitions the group asks readers to compare with the paper:def Exact (A : Type u)... := ∀ (C D : Type u) (q : C →⋆ₙₐ[ℂ] D), Surjective q → ∀ z : MinimalTensor A C, (∀ π : SpatialRepresentation A D, z (π.precompRight q) = 0) → z.val ∈ tensorKernelIdeal q: hand-built exactness via the spatial (min) tensor product as an ℓ^∞ sum over all pairs of Hilbert-space representations; matches the standard exactness of A ⊗_min - but tests only C, D in the universe of A; Two different hand-built notions of proper infiniteness: matrix-level CuntzSubequiv (blockSum a a) a (ε-approximation by v : CStarMatrix k m A) in ExactInfiniteness, and CuntzLE in the completed stabilization (MainGap.Stable.Stabilization, direct limit of CStarMatrix algebras) in IndividualStrongInfiniteness; both read as standard, the Lean theorem uses the weaker (stable) form as hypothesis in the ordinary-to-strong direction
“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.
cert-machine finds that the Lean statements express a narrower statement than the advertised result (their term: “NARROWER — UNDECLARED”).
The gap they name:The O_∞-absorption clause (the family title’s ‘Cuntz-algebra absorption’) is not stated; it is a consequence of the stated result via the (unformalized) Kirchberg–Rørdam absorption theorem.
“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.
The formalization proves that a complex $C^*$-algebra in which every positive element is properly infinite is strongly purely infinite, with the full positive-element diagonalization property. No exactness, nuclearity, unitality, or simplicity assumption is needed for this implication.
For exact algebras, proper infiniteness of one fixed finite amplification of every positive element already implies individual proper infiniteness and strong pure infiniteness.
AI-drafted comparisons, with no recorded human review
Fixed-amplification pure infiniteness for exact algebrasExactInfiniteness
Unreviewed AI draft: states a paper’s theorem as the paper does. States the paper’s exact-algebra theorem: proper infiniteness of one fixed amplification of every positive element gives strong pure infiniteness. The O∞-absorption corollary is not part of the statement.
Individual proper infiniteness implies strong pure infinitenessIndividualStrongInfiniteness
Unreviewed AI draft: states a paper’s theorem as the paper does. States the paper’s theorem that individual proper infiniteness of positive elements implies strong pure infiniteness, with no exactness or nuclearity assumption.
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.
Fixed-amplification pure infiniteness for exact algebrasExactInfiniteness
One report, 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.
What these papers cite in the release, and what cites them
These links come from the papers’ bibliographies. A citation records that one paper cites another; how each cited result is used has not been examined here.
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
Each paper’s outside references are listed with it on its reading page, where each citation is placed. Every outside work cited in the release: References outside the release
Explain a result, compare statements or propose a correction
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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 “Solid publication, could lead to tenure”. The page says that nothing on it is verified.
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Read the Lean statement against the paper
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An AI agent drafted an assessment of which part of the paper this Lean statement expresses. No named human review is recorded.Open the form
Identify the theorem the catalogue describes
Family 303The 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 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 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; 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 364 s; peak memory 6.4 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.
Individual proper infiniteness implies strong pure infiniteness
Role
the release lists it as a supporting statement
Paired with
Theorem 1.2 (candidate pairing)
Who ran the proof-verification program
three independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet; Neal McBurnett, openai-math-lean-tooling.