Disproves Kaplansky’s quasitrace conjecture by constructing a separable unital complex C∗-algebra admitting normalized 2-quasitraces, all of which are nonadditive. As a consequence, two unital simple stably finite C∗-algebras can have a properly infinite minimal tensor product, with one factor \(C_r^*(\mathbb F_2)\). (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.
Both independent assessments find that the Lean statements express or imply the whole result described above. Read the assessments
In A counterexample to Kaplansky’s quasitrace conjecture and failure of tensor-product stable finiteness, 23 September 2026
The paper’s main theorem is shown as a possible match to OpenAI’s description. No review confirming the selection is recorded.
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
In A counterexample to Kaplansky’s quasitrace conjecture and failure of tensor-product stable finiteness, 23 September 2026
Possible match to OpenAI’s description, selected because it shares the most wording. No review confirming the selection is recorded.
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).
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.
There is a separable unital stably finite complex C^*-algebra with no tracial state.
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KaplanskyQuasitraceThe paper states
In A counterexample to Kaplansky’s quasitrace conjecture and failure of tensor-product stable finiteness, 23 September 2026
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
The words they share: contractions, positive, normalized, quasitrace, separable, algebra, unital, complex, kaplanskys, counterexample, tau.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
KaplanskyQuasitrace.lean, the release lists it as a main result.
def MainTarget : 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 theorem kaplansky_quasitrace_counterexample : MainTarget := by sorryThis 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
KaplanskyStableFinitenessThe paper states
In A counterexample to Kaplansky’s quasitrace conjecture and failure of tensor-product stable finiteness, 23 September 2026
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).
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.
There is a separable unital stably finite complex C^*-algebra with no tracial state.
The words they share: finite, state, tracial, two, stably, one, simple, product, tensor, stable, finiteness, infinite, normalized, properly, quasitrace, separable, algebra, unital, complex, whose.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
KaplanskyStableFiniteness.lean, the release lists it as a supporting statement.
def StableFinitenessBundle : Prop := SimpleStableTensorCounterexample ∧ ReducedQuasitraceLoss ∧ TraceFreeStablyFinitetheorem stable_finiteness_bundle : StableFinitenessBundle := by sorryThis 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
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: 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.
“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 or imply the whole advertised result (their term: “MATCHES”).
“MATCHES”, in their words: 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.
OpenAI’s own note on what its Lean covers
Kaplansky’s quasitrace conjecture predicts that every $2$-quasitrace on a $C^*$-algebra is a trace. 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.
The formalization also gives simple stably finite $C^*$-algebras whose spatial tensor product is properly infinite, a separable stably finite algebra with no tracial state, and a tensor-product example in which normalized $2$-quasitraces are lost. These are the stable-finiteness consequences named in the paper’s title.
Nonadditive normalized $2$-quasitraces KaplanskyQuasitrace
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem with its constant: a separable C*-algebra with positive contractions a, b on which every normalised 2-quasitrace has τ(a + b) − τ(a) − τ(b) ≥ 1/144.
Tensor-product failure of stable finiteness and quasitrace consequences KaplanskyStableFiniteness
Unreviewed AI draft: states a consequence of a paper’s theorem. States the paper’s corollary on tensor products: stably finite algebras with properly infinite tensor product, a stably finite algebra without tracial states, and loss of quasitraces.
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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KaplanskyQuasitraceTwo 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 |
KaplanskyStableFinitenessTwo 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 A counterexample to Kaplansky’s quasitrace conjecture and failure of tensor-product stable finiteness, 23 September 2026
Why this passage is shown
Corollary 1.2 (Tensor products and stable finiteness)
In A counterexample to Kaplansky’s quasitrace conjecture and failure of tensor-product stable finiteness, 23 September 2026
Independent assessment
MathVet
14c7f91Independent assessment
Carlos Toledo’s cert-machine
9814e4cProof-verification report
cert-machine, KaplanskyQuasitrace
Proof-verification report
MathVet, KaplanskyQuasitrace
Proof-verification report
cert-machine, KaplanskyStableFiniteness
Proof-verification report
MathVet, KaplanskyStableFiniteness
Lean statement
KaplanskyQuasitrace
Lean statement
KaplanskyStableFiniteness