Fixed-amplification pure infiniteness for exact algebras
Lean is a language for mathematical statements and proofs that a computer can verify. This page shows one statement file from OpenAI’s release, possible paper matches and independent reports. The code-like names below identify the definitions or theorems selected by the proof-verification settings.
Theorems selected for verification:OAI.WeakPureInfiniteness.exact_weaklyPurelyInfinite_individuallyProperlyInfinite_anyOAI.WeakPureInfiniteness.exact_weaklyPurelyInfinite_stronglyPurelyInfinite
The lines of the Lean statement that state the theorem, then the definitions in the file that it uses.
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⋮theoremexact_weaklyPurelyInfinite_stronglyPurelyInfinite (A : Type u) [NonUnitalCStarAlgebra A] [PartialOrder A] [StarOrderedRing A] (hExact : Exact A) (hWeak : WeaklyPurelyInfinite A) : StronglyPurelyInfinite A := bysorry
A Lean statement states the theorem and leaves its proof as sorry on purpose. OpenAI’s proof is in a separate solution module, and a program called Comparator checks that proof against this statement.
Definitions needed to read it
Local definitions the statement names, followed through the definitions they name in turn. The match is by name in the source text, so a name reached only through notation can be missed; the full file is below.
structureSpatialRepresentation (A : Type u) (B : Type v) [NonUnitalCStarAlgebra A] [NonUnitalCStarAlgebra B] where left : ComplexHilbert.{max u v} right : ComplexHilbert.{max u v} leftRep : A →⋆ₙₐ[ℂ] (left →L[ℂ] left) rightRep : B →⋆ₙₐ[ℂ] (right →L[ℂ] right)
defSpatialAmbient (A : Type u) (B : Type v) [NonUnitalCStarAlgebra A] [NonUnitalCStarAlgebra B] : Type (max u v + 1) := lp (fun π : SpatialRepresentation A B => π.space →L[ℂ] π.space) ∞
defminimalTensorSubalgebra (A : Type u) (B : Type v) [NonUnitalCStarAlgebra A] [NonUnitalCStarAlgebra B] : NonUnitalStarSubalgebra ℂ (SpatialAmbient A B) := (NonUnitalStarAlgebra.adjoin ℂ (Set.range fun p : A × B => spatialElementary p.1 p.2)).topologicalClosure
defSpatialRepresentation.precompRight (π : SpatialRepresentation A D) (q : C →⋆ₙₐ[ℂ] D) : SpatialRepresentation A C where left := π.left right := π.right leftRep := π.leftRep rightRep := π.rightRep.comp q
deftensorKernelIdeal (q : C →⋆ₙₐ[ℂ] D) : NonUnitalStarSubalgebra ℂ (SpatialAmbient A C) := (NonUnitalStarAlgebra.adjoin ℂ {z | ∃ a : A, ∃ c : C, q c = 0 ∧ z = spatialElementary a c}).topologicalClosure
defExact (A : Type u) [NonUnitalCStarAlgebra A] : Prop := ∀ (C D : Type u) [NonUnitalCStarAlgebra C] [NonUnitalCStarAlgebra D] (q : C →⋆ₙₐ[ℂ] D), Function.Surjective q → ∀ z : MinimalTensor A C, (∀ π : SpatialRepresentation A D, (z.val : SpatialAmbient A C) (π.precompRight q) = 0) → z.val ∈ tensorKernelIdeal (A := A) q
defblockSum (a : CStarMatrix m m A) (b : CStarMatrix k k A) : CStarMatrix (m ⊕ k) (m ⊕ k) A := CStarMatrix.ofMatrix (Matrix.fromBlocks (CStarMatrix.ofMatrix.symm a) 00 (CStarMatrix.ofMatrix.symm b))
defStronglyPurelyInfinite : Prop := ∀ a b : A, 0 ≤ a → 0 ≤ b → ∀ ε : ℝ, 0 < ε → ∃ s t : A, ‖star s * (a * a) * s - a * a‖ < ε ∧ ‖star t * (b * b) * t - b * b‖ < ε ∧ ‖star s * a * b * t‖ < ε
Possible paper match, with no recorded review
No paper statement clearly matched the Lean wording. Statement Map, the program that collects Atlas’s source records, selected this possible match using words in OpenAI’s catalogue and Lean note. No review of whether the statements mean the same thing is recorded.
Shared with the catalogue’s wording or OpenAI’s note, none of them matched in the Lean text: algebra, complex, element, exact, infinite, positive, properly, purely and one more.
Theorem 1.2 of Weak pure infiniteness and O-infinity absorption
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.
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.
Other possible paper matches, closest wording first
Theorem 1.1 of Weak pure infiniteness and O-infinity absorption. Words of the Lean statement, its names split into words, that also appear in the theorem: purely, strongly, complex, infinite and algebra. Found in the paper’s title, which its main theorem states: weak.
Theorem 6.1 of Weak pure infiniteness and O-infinity absorption. Words of the Lean statement, its names split into words, that also appear in the theorem: properly, complex, infinite, integer and algebra. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: amplification, element, one and positive.
Proposition 9.3 of Weak pure infiniteness and O-infinity absorption. Words of the Lean statement, its names split into words, that also appear in the theorem: purely, strongly, infinite and algebra.
Corollary 1.3 of Weak pure infiniteness and O-infinity absorption. Words of the Lean statement, its names split into words, that also appear in the theorem: properly, complex, infinite, integer and algebra. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: one.
Proposition 7.1 of Weak pure infiniteness and O-infinity absorption. Words of the Lean statement, its names split into words, that also appear in the theorem: unital, pi, representation, integer and algebra.
Proposition 4.3 of Weak pure infiniteness and O-infinity absorption. Words of the Lean statement, its names split into words, that also appear in the theorem: pi and representation. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: individual, proper, infiniteness and positive.
How Atlas selects possible matches: it compares words in the Lean statement with every statement in the family. It also splits Lean names into words. The release’s label and note receive less weight; a main theorem wins a close match. If the Lean wording yields no clear match, Atlas uses the catalogue description. This selects passages to compare; it does not establish that they mean the same thing.
Reported Comparator runs
One independent group reports that Comparator accepted a run for this Lean statement file: Carlos Toledo’s cert-machine. Each group reports its own Comparator run for this statement file. Atlas has not repeated the runs.
For a configured theorem, Comparator tests whether OpenAI’s proof proves the exact Lean statement. Some files instead configure definitions for comparison by name and type. Acceptance does not establish that the statement means the same thing as the paper’s theorem.
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 (.github/workflows/openai-math-kernel.yml), one j…
AcceptedTheir term: “CERTIFIED”.
Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.
If Comparator is run under this configuration and accepts, the solution proves the configured theorem using only the allowed axioms, and the Lean kernel accepts every step.
This configuration lists no definition holes, so every definition the statement uses is compared as it is written in this file.
Acceptance leaves open whether the Lean statement means the same thing as the paper’s theorem, whether the paper’s other claims are correct, and whether the result is new. Those questions require comparing the definitions and statements, reviewing the paper’s arguments, and reading the earlier literature.
Do family 303’s Lean statements express the advertised result?
These assessments compare all the family’s Lean statements with the whole result OpenAI describes in its catalogue. They do not assess this statement alone. Each group uses its own published rules.
MathVet and cert-machine disagree on how much of the advertised result the Lean statements express.
Finds the whole advertised result stated or impliedTheir term: “full”.
The challenge statements imply the headline claim after unpacking definitions or one routine step. Their record
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
Finds part of it stated, or a narrower or weaker statementTheir term: “NARROWER — UNDECLARED”.
The challenge statements state less than the headline, and OpenAI’s Lean note does not say so. Their record
Their note: 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.
OpenAI’s own note on what this Lean statement covers
“For exact algebras, proper infiniteness of one fixed finite amplification of every positive element already implies individual proper infiniteness and strong pure infiniteness.”
An AI agent drafted an interpretation of that note, classifying this statement as “states a paper’s theorem as the paper does”. No human review of the draft is recorded.
Supporting material
The complete Lean statement file
179 lines, copied from release version fd4aeeb. A name defined in this file links to its line; pointing at the name shows its definition.
Declared in this fileOpenAI’s library at fd4aeebOther linked names open the current documentation for Mathlib, Lean’s mathematical library.
Links in the file: 46 to its own declarations, 103 to Mathlib’s documentation. 25 declared names also link to the same declaration in OpenAI’s library. OpenAI uses Mathlib version d13f23b. The linked documentation describes the current version, so names that have since changed may be missing there.
Let \(C\) be a complex \(C^*\)-algebra satisfying \(\propP\).
For all \(a,b\in C_+\), \(c\in C\), and \(\varepsilon>0\), there are
\(s,t\in C\) such that
\begin{equation}
\|s^*as-a\|<\varepsilon,\qquad
\|t^*bt-b\|<\varepsilon,\qquad
\|s^*ct\|<\varepsilon.
\end{equation}
Consequently \(C\) is strongly purely infinite.
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.
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.
Corollary
Let A be a separable nuclear complex C^*-algebra. If there is one integer n\ge1 such that h^{\oplus n} is properly infinite for every h\in A_+, then
Let \(A\) be a separable nuclear complex \(C^*\)-algebra. If there is
one integer \(n\ge1\) such that \(h^{\oplus n}\) is properly infinite
for every \(h\in A_+\), then
\[
A\cong A\otimes_{\min}\Oinf.
\]
OpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 303 link it.
Solution module
OAI.Analysis.WeakInfiniteness.Main
Allowed axioms
propext, Quot.sound, Classical.choiceThe axioms this configuration allows a proof to use. A replay shows which of them a proof uses.
Imports
Mathlib
Imports from outside Mathlib
None
Local vocabulary
23 local vocabulary declarations
Lines of Lean, without comments or blank lines
116
Second proof-verification program (nanoda)
Off in OpenAI's configuration.
Statement comparison
A shareable summary of this page: the proposed paper match, what the Lean statement expresses, and which reviews are recorded.
Plectis Atlas
Fixed-amplification pure infiniteness for exact algebras
Family 303: Weak pure infiniteness and Cuntz-algebra absorption
Candidate pairing
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.
Weak pure infiniteness and O-infinity absorption
TeX label thm:exact-infiniteness, build/introduction.tex, line 98
The Lean statement says
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⋮theoremexact_weaklyPurelyInfinite_stronglyPurelyInfinite (A : Type u) [NonUnitalCStarAlgebra A] [PartialOrder A] [StarOrderedRing A] (hExact : Exact A) (hWeak : WeaklyPurelyInfinite A) : StronglyPurelyInfinite A := bysorry
A Lean statement states the theorem and leaves its proof as sorry on purpose. OpenAI’s proof is in a separate solution module, and a program called Comparator checks that proof against this statement.
Recorded comparisons with the paper
Atlas records no contributed comparison of this statement with the paper.
Contributed Comparator run reports
Atlas records no contributed report of running Comparator for this statement file.
Questions to resolve when comparing this statement with the paper
Two independent readers, MathVet and cert-machine, disagree on whether this family's challenges state its headline; a specialist reading would settle it.