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.
Theorem selected for verification:OAI.Kaplansky.kaplansky_quasitrace_counterexample
The lines of the Lean statement that state the theorem, then the definitions in the file that it uses.
defMainTarget : Prop := ∃ (A : Type) (inst : CStarAlgebra A),letI := inst TopologicalSpace.SeparableSpace A ∧ (∃ τ : OneQuasitrace A, τ.Normalized ∧ τ.IsTwo) ∧ ∃ a b : A, PositiveContraction a ∧ PositiveContraction b ∧ ∀ τ : OneQuasitrace A, τ.Normalized → τ.IsTwo → (1 : ℝ) / 144 ≤ (τ (a+b) - τ a - τ b).re⋮theoremkaplansky_quasitrace_counterexample : MainTarget := 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.
structureOneQuasitrace (A : Type*) [CStarAlgebra A] where toFun : A → ℂ square_nonneg : ∀ x, 0 ≤ toFun (star x * x) square_swap : ∀ x, toFun (star x * x) = toFun (x * star x) real_imag : ∀ h k, IsSelfAdjoint h → IsSelfAdjoint k → toFun (h + Complex.I • k) = toFun h + Complex.I * toFun k abelian_linear : ∀ S : NonUnitalStarSubalgebra ℂ A, IsClosed (S : Set A) → (∀ x y : S, x * y = y * x) → ∃ f : S →ₗ[ℂ] ℂ, ∀ x : S, f x = toFun x
defPositiveContraction (a : A) : Prop := (∃ x, a = star x * x) ∧ ‖a‖ ≤ 1
Possible paper match, with no recorded review
Atlas selected this possible match by comparing words in the Lean statement with every statement in the family. No review of whether the statements mean the same thing is recorded.
Words of the Lean statement, its names split into words, that also appear in the theorem: contractions, positive, normalized, quasitrace, separable, algebra, unital and complex. Found in the paper’s title, which its main theorem states: kaplanskys and counterexample. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: tau.
Theorem 1.1 of A counterexample to Kaplansky’s quasitrace conjecture and failure of tensor-product stable finiteness
Theorem
There exist a separable unital complex C^*-algebra A, admitting a normalized 2-quasitrace, and positive contractions a,b\in A such that every normalized 2-quasitrace \tau on A satisfies
There exist a separable unital complex \(C^*\)-algebra \(A\), admitting
a normalized \(2\)-quasitrace, and positive contractions \(a,b\in A\)
such that every normalized \(2\)-quasitrace \(\tau\) on \(A\) satisfies
\[
\tau(a+b)-\tau(a)-\tau(b)\ge\frac1{144}.
\]
Other possible paper matches, closest wording first
Corollary 1.2 of A counterexample to Kaplansky’s quasitrace conjecture and failure of tensor-product stable finiteness. Words of the Lean statement, its names split into words, that also appear in the theorem: two, one, normalized, quasitrace, separable, algebra and two more.
Theorem 2.1 of A counterexample to Kaplansky’s quasitrace conjecture and failure of tensor-product stable finiteness. Words of the Lean statement, its names split into words, that also appear in the theorem: positive, quasitrace, normalized, algebra and unital.
Proposition 4.1 of A counterexample to Kaplansky’s quasitrace conjecture and failure of tensor-product stable finiteness. Words of the Lean statement, its names split into words, that also appear in the theorem: complex. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: pair.
Theorem 5.1 of A counterexample to Kaplansky’s quasitrace conjecture and failure of tensor-product stable finiteness. Words of the Lean statement, its names split into words, that also appear in the theorem: separable, algebra, unital and complex.
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
two independent groups report that Comparator accepted runs for this Lean statement file: Carlos Toledo’s cert-machine; MathVet. 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.
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.
AcceptedTheir term: “comparator-sandboxed-pass”.
Re-run through Comparator in a sandboxed virtual machine: accepted.
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 294’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 agree 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: kaplansky_quasitrace_counterexample : MainTarget := ∃ separable C*-algebra A, (∃ τ : OneQuasitrace A, τ.Normalized ∧ τ.IsTwo) ∧ ∃ positive contractions a b, ∀ normalized 2-quasitraces τ, 1/144 ≤ Re(τ(a+b) - τ a - τ b) (all normalized 2-quasitraces are non-additive), and stable_finiteness_bundle := SimpleStableTensorCounterexample ∧ ReducedQuasitraceLoss ∧ TraceFreeStablyFinite (C*_r(F₂) and a simple stably finite M whose spatial tensor product tensorAlgebra ρ is ProperlyInfinite, plus the quasitrace-loss and trace-free examples). Both parts of the summary are stated.
Definitions the group asks readers to compare with the paper: Hand-written quasitraces: OneQuasitrace (positivity, τ(x*x) = τ(xx*), real/imaginary additivity on self-adjoints, linearity on closed abelian subalgebras) and IsTwo τ := ∃ σ : OneQuasitrace (M₂(A)), ∀ a, σ (upperLeft a) = τ a (a 2-quasitrace as one that extends to M₂(A), rather than requiring τ ⊗ Tr₂ to be a quasitrace); C*_r(F₂) is built as closure (adjoin (left translations)) on ℓ²(F₂), ProperlyInfinite A := ∃ s t, s*s = 1 ∧ t*t = 1 ∧ s*t = 0.
OpenAI’s own note on what this Lean statement covers
“The formalization constructs a separable $C^*$-algebra with normalized $2$-quasitraces and fixed positive contractions $a,b$ for which every such quasitrace satisfies $\mathrm{Re}(\tau(a+b)-\tau(a)-\tau(b))\ge1/144$. Thus none is additive on this pair.”
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
55 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: 11 to its own declarations, 25 to Mathlib’s documentation. 7 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.
There exist a separable unital complex C^*-algebra A, admitting a normalized 2-quasitrace, and positive contractions a,b\in A such that every normalized 2-quasitrace \tau on A satisfies
There exist a separable unital complex \(C^*\)-algebra \(A\), admitting
a normalized \(2\)-quasitrace, and positive contractions \(a,b\in A\)
such that every normalized \(2\)-quasitrace \(\tau\) on \(A\) satisfies
\[
\tau(a+b)-\tau(a)-\tau(b)\ge\frac1{144}.
\]
Corollary(Tensor products and stable finiteness)
(i)
There are two unital simple stably finite complex C^*-algebras whose minimal tensor product is properly infinite. One factor can be taken to be C_r^*(\mathbb F_2).
(ii)
There is a separable unital complex C^*-algebra B admitting a normalized 2-quasitrace such that B\otimes_{\min}C_r^*(\mathbb F_2) is properly infinite and admits no normalized 2-quasitrace.
(iii)
There is a separable unital stably finite complex C^*-algebra with no tracial state.
\leavevmode
\begin{enumerate}[label=\textup{(\roman*)}]
\item There are two unital simple stably finite complex \(C^*\)-algebras
whose minimal tensor product is properly infinite. One factor can be
taken to be \(C_r^*(\mathbb F_2)\).
\item There is a separable unital complex \(C^*\)-algebra \(B\) admitting
a normalized \(2\)-quasitrace such that
\(B\otimes_{\min}C_r^*(\mathbb F_2)\) is properly infinite and admits no
normalized \(2\)-quasitrace.
\item There is a separable unital stably finite complex \(C^*\)-algebra
with no tracial state.
\end{enumerate}
A shareable summary of this page: the proposed paper match, what the Lean statement expresses, and which reviews are recorded.
Plectis Atlas
Nonadditive normalized $2$-quasitraces
Family 294: Kaplansky’s quasitrace conjecture and failure of tensor-product stable finiteness
Candidate pairing
Theorem
There exist a separable unital complex C^*-algebra A, admitting a normalized 2-quasitrace, and positive contractions a,b\in A such that every normalized 2-quasitrace \tau on A satisfies
\tau(a+b)-\tau(a)-\tau(b)\ge\frac1{144}.
A counterexample to Kaplansky’s quasitrace conjecture and failure of tensor-product stable finiteness
TeX label thm:main, build/01-introduction.tex, line 80
The Lean statement says
defMainTarget : Prop := ∃ (A : Type) (inst : CStarAlgebra A),letI := inst TopologicalSpace.SeparableSpace A ∧ (∃ τ : OneQuasitrace A, τ.Normalized ∧ τ.IsTwo) ∧ ∃ a b : A, PositiveContraction a ∧ PositiveContraction b ∧ ∀ τ : OneQuasitrace A, τ.Normalized → τ.IsTwo → (1 : ℝ) / 144 ≤ (τ (a+b) - τ a - τ b).re⋮theoremkaplansky_quasitrace_counterexample : MainTarget := 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.