Lean statement

Tensor-product failure of stable finiteness and quasitrace consequences

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.KaplanskyConsequences.stable_finiteness_bundle

Lean statement file: KaplanskyStableFiniteness.leanShow on the mapOpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 294 link it.

See the candidate theorem Who ran the proof-verification program Report a comparison with the paper (GitHub issue)

The Lean statement

The lines of the Lean statement that state the theorem, then the definitions in the file that it uses.

def StableFinitenessBundle : Prop :=  SimpleStableTensorCounterexample ∧ ReducedQuasitraceLoss ∧ TraceFreeStablyFinite⋮theorem stable_finiteness_bundle : StableFinitenessBundle := by  sorry

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.

abbreviation OAI.KaplanskyConsequences.Regular.HilbertSum line 13

abbrev HilbertSum (ι H : Type*) [NormedAddCommGroup H] := lp (fun _ : ι => H) 2

definition OAI.KaplanskyConsequences.Regular.reindex line 22

def reindex (e : ι ≃ ι) : HilbertSum ι H ≃ₗᵢ[ℂ] HilbertSum ι H where

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  toFun x := ⟨fun i => x (e.symm i), reindex_mem e.symm x⟩  invFun x := ⟨fun i => x (e i), reindex_mem e x⟩  left_inv x := by ext; simp  right_inv x := by ext; simp  map_add' x y := by ext; rfl  map_smul' r x := by ext; rfl  norm_map' x := by    rw [lp.norm_eq_tsum_rpow (by norm_num),lp.norm_eq_tsum_rpow (by norm_num)]    congr 1    exact e.symm.tsum_eq (fun i => ‖x i‖ ^ (2 : ℝ≥0∞).toReal)

definition OAI.KaplanskyConsequences.Regular.left line 39

def left (g : G) : HilbertSum G H →L[ℂ] HilbertSum G H :=  (reindex (Equiv.mulLeft g)).toLinearIsometry.toContinuousLinearMap

definition OAI.KaplanskyConsequences.Regular.Algebra line 60

def Algebra (G : Type*) [Group G] : StarSubalgebra ℂ (HilbertSum G ℂ →L[ℂ] HilbertSum G ℂ) :=  (StarAlgebra.adjoin ℂ (Set.range (left (H := ℂ) (G := G)))).topologicalClosure

abbreviation OAI.KaplanskyConsequences.Regular.F2 line 77
```
abbrev F2 := FreeGroup (Fin 2)
```

definition OAI.KaplanskyConsequences.Regular.diagonal line 85

def diagonal (T : H →L[ℂ] H) : HilbertSum G H →L[ℂ] HilbertSum G H :=  lp.mapCLM 2 (fun _ => T) (norm_nonneg T) (fun _ => le_rfl)

structure OAI.Kaplansky.OneQuasitrace line 101

structure OneQuasitrace (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

definition OAI.Kaplansky.OneQuasitrace.Normalized line 113

def OneQuasitrace.Normalized (τ : OneQuasitrace A) : Prop := τ 1 = 1

definition OAI.Kaplansky.ProperlyInfinite line 122

def ProperlyInfinite (A : Type*) [CStarAlgebra A] : Prop :=  ∃ s t : A, star s*s = 1 ∧ star t*t = 1 ∧ star s*t = 0

definition OAI.Kaplansky.upperLeft line 131

def upperLeft (a : A) : CStarMatrix (Fin 2) (Fin 2) A :=  CStarMatrix.ofMatrix fun i j => if i = 0 ∧ j = 0 then a else 0

definition OAI.Kaplansky.OneQuasitrace.IsTwo line 134

def OneQuasitrace.IsTwo (τ : OneQuasitrace A) : Prop :=  ∃ σ : OneQuasitrace (CStarMatrix (Fin 2) (Fin 2) A), ∀ a, σ (upperLeft a) = τ a

structure OAI.KaplanskyConsequences.ClosedStarIdeal line 147

structure ClosedStarIdeal (A : Type*) [CStarAlgebra A] extends NonUnitalStarSubalgebra ℂ A where  mul_mem_left : ∀ (a b : A), b ∈ toNonUnitalStarSubalgebra → a * b ∈ toNonUnitalStarSubalgebra  mul_mem_right : ∀ (a b : A), a ∈ toNonUnitalStarSubalgebra → a * b ∈ toNonUnitalStarSubalgebra  closed : IsClosed (toNonUnitalStarSubalgebra : Set A)

definition OAI.KaplanskyConsequences.StablyFinite line 158

def StablyFinite (A : Type*) [CStarAlgebra A] : Prop :=  ∀ n : ℕ, 0 < n → ∀ v : CStarMatrix (Fin n) (Fin n) A,    star v * v = 1 → v * star v = 1

structure OAI.KaplanskyConsequences.TracialState line 162

structure TracialState (A : Type*) [CStarAlgebra A] where  functional : A →ₚ[ℂ] ℂ  normalized : functional 1 = 1  tracial : ∀ a b, functional (a * b) = functional (b * a)

definition OAI.KaplanskyConsequences.TraceFreeStablyFinite line 167

def TraceFreeStablyFinite : Prop :=  ∃ (A : Type) (inst : CStarAlgebra A),    letI := inst    Nontrivial A ∧ TopologicalSpace.SeparableSpace A ∧ StablyFinite A ∧      IsEmpty (TracialState A)

definition OAI.KaplanskyConsequences.Regular.tensorAlgebra line 182

def tensorAlgebra (ρ : A →⋆ₐ[ℂ] (H →L[ℂ] H)) :    StarSubalgebra ℂ (HilbertSum G H →L[ℂ] HilbertSum G H) :=  (StarAlgebra.adjoin ℂ (Set.range (fun a => diagonal (G:=G) (ρ a)) ∪ Set.range (left (G:=G) (H:=H)))).topologicalClosure

definition OAI.KaplanskyConsequences.ReducedQuasitraceLoss line 200

def ReducedQuasitraceLoss : Prop :=

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  ∃ (B : Type) (instB : CStarAlgebra B),    letI := instB    Nontrivial B ∧ TopologicalSpace.SeparableSpace B ∧      (∃ τ : OneQuasitrace B, τ.Normalized ∧ τ.IsTwo) ∧      ∃ (H : Type) (instH : NormedAddCommGroup H) (innerH : InnerProductSpace ℂ H),        letI := instH        letI := innerH        ∃ (completeH : CompleteSpace H),          letI := completeH          ∃ ρ : B →⋆ₐ[ℂ] (H →L[ℂ] H), Function.Injective ρ ∧            ProperlyInfinite (tensorAlgebra (G:=F2) ρ) ∧            ¬ ∃ τ : OneQuasitrace (tensorAlgebra (G:=F2) ρ), τ.Normalized ∧ τ.IsTwo

definition OAI.KaplanskyConsequences.CStarSimple line 217

def CStarSimple (A : Type*) [CStarAlgebra A] : Prop :=  ∀ I : ClosedStarIdeal A,    (∀ a : A, a ∈ I.toNonUnitalStarSubalgebra → a = 0) ∨    (∀ a : A, a ∈ I.toNonUnitalStarSubalgebra)

definition OAI.KaplanskyConsequences.SimpleStableTensorCounterexample line 222

def SimpleStableTensorCounterexample : Prop :=

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  CStarSimple (Algebra F2) ∧ StablyFinite (Algebra F2) ∧  ∃ (M : Type) (instM : CStarAlgebra M),    letI := instM    Nontrivial M ∧ CStarSimple M ∧ StablyFinite M ∧      ∃ (H : Type) (instH : NormedAddCommGroup H) (innerH : InnerProductSpace ℂ H),        letI := instH        letI := innerH        ∃ (completeH : CompleteSpace H),          letI := completeH          ∃ ρ : M →⋆ₐ[ℂ] (H →L[ℂ] H), Function.Injective ρ ∧            ProperlyInfinite (tensorAlgebra (G:=F2) ρ)

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: finite, state, tracial, two, stably, one, simple, product and eleven more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: whose.

Corollary 1.2 of A counterexample to Kaplansky’s quasitrace conjecture and failure of tensor-product stable finiteness

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.

Section 1, Introduction Read it in the paper TeX source, line 110

Show the TeX

\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}

Other possible paper matches, closest wording first

Theorem 1.1 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: normalized, quasitrace, separable, algebra, unital and complex. Found in the paper’s title, which its main theorem states: counterexample, product, finiteness, tensor and stable.
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: infinite, quasitrace, normalized, properly, integer, algebra and one more.
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: bundles, maps, continuous, integers and complex.
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: infinite, properly, integer, separable, algebra, unital and one more.

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.

Carlos Toledo’s cert-machine

How they ran it, in their words

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…

Accepted Their term: “CERTIFIED”.
Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.
Reported 10 October 2026, at the release version shown here. Original verification report

MathVet

How they ran it, in their words

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.

Accepted Their term: “comparator-sandboxed-pass”.
Re-run through Comparator in a sandboxed virtual machine: accepted.
Reported 8 October 2026, at the release version shown here. Original verification report

What an accepted Comparator run would establish

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.

Allowed axioms: propext, Quot.sound, Classical.choice

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.

MathVet
Finds the whole advertised result stated or implied Their 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.
Carlos Toledo’s cert-machine
Finds the whole advertised result stated or implied Their term: “MATCHES”.
The challenge statements state the headline as OpenAI wrote it. Their record

Read the complete assessments on the family page.

OpenAI’s own note on what this Lean statement covers

“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.”
Quoted from OpenAI’s Lean note for this family

An AI agent drafted an interpretation of that note, classifying this statement as “states a consequence of a paper’s theorem”. No human review of the draft is recorded.

Supporting material

The complete Lean statement file

247 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: 67 to its own declarations, 127 to Mathlib’s documentation. 24 declared names also link to the same declaration in OpenAI’s library. 2 identifiers stay plain because they resolve to no single declaration: variables Lean binds automatically, notation, or names the scan of the pinned sources did not find. 2 of them could name more than one declaration. OpenAI uses Mathlib version d13f23b. The linked documentation describes the current version, so names that have since changed may be missing there.

import Mathlibnamespace OAInoncomputable sectionnamespace KaplanskyConsequencesopen scoped ComplexOrder InnerProductSpace BigOperators ENNRealnamespace Regularvariable {ι : Type*} {H : Type*} [NormedAddCommGroup H]abbrev HilbertSum (ι H : Type*) [NormedAddCommGroup H] := lp (fun _ : ι => H) 2lemma reindex_mem (e : ι ≃ ι) (x : HilbertSum ι H) :    Memℓp (fun i => x (e i)) 2 := by  apply memℓp_gen  exact e.summable_iff.2 ((memℓp_gen_iff (by norm_num)).1 (lp.memℓp x))variable [InnerProductSpace ℂ H]def reindex (e : ι ≃ ι) : HilbertSum ι H ≃ₗᵢ[ℂ] HilbertSum ι H where  toFun x := ⟨fun i => x (e.symm i), reindex_mem e.symm x⟩  invFun x := ⟨fun i => x (e i), reindex_mem e x⟩  left_inv x := by ext; simp  right_inv x := by ext; simp  map_add' x y := by ext; rfl  map_smul' r x := by ext; rfl  norm_map' x := by    rw [lp.norm_eq_tsum_rpow (by norm_num),lp.norm_eq_tsum_rpow (by norm_num)]    congr 1    exact e.symm.tsum_eq (fun i => ‖x i‖ ^ (2 : ℝ≥0∞).toReal)@[simp] lemma reindex_apply (e : ι ≃ ι) (x : HilbertSum ι H) (i : ι) :    reindex e x i = x (e.symm i) := rflvariable {G : Type*} [Group G]def left (g : G) : HilbertSum G H →L[ℂ] HilbertSum G H :=  (reindex (Equiv.mulLeft g)).toLinearIsometry.toContinuousLinearMap@[simp] lemma left_apply (g h : G) (x : HilbertSum G H) : left g x h = x (g⁻¹*h) := rfllemma left_one : left (H := H) (1 : G) = 1 := by ext x h; simplemma left_mul (g h : G) : left (H := H) (g*h) = left g * left h := by  ext x k  simp [mul_assoc]variable [CompleteSpace H]lemma left_star (g : G) : star (left (H := H) g) = left g⁻¹ := by  symm  apply (ContinuousLinearMap.eq_adjoint_iff _ _).2  intro x y  change inner ℂ (left g⁻¹ x) y = inner ℂ x (left g y)  rw [lp.inner_eq_tsum,lp.inner_eq_tsum]  simpa [left_apply,mul_assoc] using    (Equiv.mulLeft g).tsum_eq (fun i => inner ℂ (x i) (y (g⁻¹*i)))def Algebra (G : Type*) [Group G] : StarSubalgebra ℂ (HilbertSum G ℂ →L[ℂ] HilbertSum G ℂ) :=  (StarAlgebra.adjoin ℂ (Set.range (left (H := ℂ) (G := G)))).topologicalClosureinstance : IsClosed (Algebra G : Set (HilbertSum G ℂ →L[ℂ] HilbertSum G ℂ)) :=  StarSubalgebra.isClosed_topologicalClosure _instance : CStarAlgebra (Algebra G) := inferInstancedef generator (g : G) : Algebra G :=  ⟨left g,StarSubalgebra.le_topologicalClosure _    (StarAlgebra.subset_adjoin ℂ _ ⟨g,rfl⟩)⟩end Regularend KaplanskyConsequencesnamespace KaplanskyConsequences.Regularabbrev F2 := FreeGroup (Fin 2)open scoped ComplexOrder InnerProductSpace BigOperators ENNRealattribute [local instance] Classical.propDecidablevariable {G H : Type*} [Group G] [NormedAddCommGroup H]  [InnerProductSpace ℂ H] [CompleteSpace H]def diagonal (T : H →L[ℂ] H) : HilbertSum G H →L[ℂ] HilbertSum G H :=  lp.mapCLM 2 (fun _ => T) (norm_nonneg T) (fun _ => le_rfl)end KaplanskyConsequences.Regularnamespace Kaplanskysectionsectionopen scoped ComplexOrder IsMulCommutativevariable {A : Type*} [CStarAlgebra A]structure OneQuasitrace (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 xinstance : CoeFun (OneQuasitrace A) (fun _ => A → ℂ) := ⟨OneQuasitrace.toFun⟩def OneQuasitrace.Normalized (τ : OneQuasitrace A) : Prop := τ 1 = 1endopen scoped ComplexOrder CStarAlgebraattribute [local instance] CStarAlgebra.spectralOrder CStarAlgebra.spectralOrderedRingvariable {A : Type*} [CStarAlgebra A]def ProperlyInfinite (A : Type*) [CStarAlgebra A] : Prop :=  ∃ s t : A, star s*s = 1 ∧ star t*t = 1 ∧ star s*t = 0endattribute [local instance] CStarAlgebra.spectralOrder CStarAlgebra.spectralOrderedRingvariable {A : Type*} [CStarAlgebra A]def upperLeft (a : A) : CStarMatrix (Fin 2) (Fin 2) A :=  CStarMatrix.ofMatrix fun i j => if i = 0 ∧ j = 0 then a else 0def OneQuasitrace.IsTwo (τ : OneQuasitrace A) : Prop :=  ∃ σ : OneQuasitrace (CStarMatrix (Fin 2) (Fin 2) A), ∀ a, σ (upperLeft a) = τ aend Kaplanskynamespace KaplanskyConsequencessectionopen scoped ComplexOrderopen Metric Topologyattribute [local instance] CStarAlgebra.spectralOrder CStarAlgebra.spectralOrderedRingstructure ClosedStarIdeal (A : Type*) [CStarAlgebra A] extends NonUnitalStarSubalgebra ℂ A where  mul_mem_left : ∀ (a b : A), b ∈ toNonUnitalStarSubalgebra → a * b ∈ toNonUnitalStarSubalgebra  mul_mem_right : ∀ (a b : A), a ∈ toNonUnitalStarSubalgebra → a * b ∈ toNonUnitalStarSubalgebra  closed : IsClosed (toNonUnitalStarSubalgebra : Set A)endopen scoped ComplexOrderattribute [local instance] CStarAlgebra.spectralOrder CStarAlgebra.spectralOrderedRingdef StablyFinite (A : Type*) [CStarAlgebra A] : Prop :=  ∀ n : ℕ, 0 < n → ∀ v : CStarMatrix (Fin n) (Fin n) A,    star v * v = 1 → v * star v = 1structure TracialState (A : Type*) [CStarAlgebra A] where  functional : A →ₚ[ℂ] ℂ  normalized : functional 1 = 1  tracial : ∀ a b, functional (a * b) = functional (b * a)def TraceFreeStablyFinite : Prop :=  ∃ (A : Type) (inst : CStarAlgebra A),    letI := inst    Nontrivial A ∧ TopologicalSpace.SeparableSpace A ∧ StablyFinite A ∧      IsEmpty (TracialState A)end KaplanskyConsequencesnamespace KaplanskyConsequences.Regularopen scoped ComplexOrder InnerProductSpace BigOperators ENNRealattribute [local instance] Classical.propDecidableattribute [local instance] CStarAlgebra.spectralOrder CStarAlgebra.spectralOrderedRingvariable {A : Type*} [CStarAlgebra A]variable {G H : Type*} [Group G] [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]def tensorAlgebra (ρ : A →⋆ₐ[ℂ] (H →L[ℂ] H)) :    StarSubalgebra ℂ (HilbertSum G H →L[ℂ] HilbertSum G H) :=  (StarAlgebra.adjoin ℂ (Set.range (fun a => diagonal (G:=G) (ρ a)) ∪ Set.range (left (G:=G) (H:=H)))).topologicalClosureinstance (ρ : A →⋆ₐ[ℂ] (H →L[ℂ] H)) :    IsClosed (tensorAlgebra (G:=G) ρ : Set (HilbertSum G H →L[ℂ] HilbertSum G H)) :=  StarSubalgebra.isClosed_topologicalClosure _instance (ρ : A →⋆ₐ[ℂ] (H →L[ℂ] H)) : CStarAlgebra (tensorAlgebra (G:=G) ρ) := inferInstanceend KaplanskyConsequences.Regularnamespace KaplanskyConsequencessectionsectionopen Kaplansky Regulardef ReducedQuasitraceLoss : Prop :=  ∃ (B : Type) (instB : CStarAlgebra B),    letI := instB    Nontrivial B ∧ TopologicalSpace.SeparableSpace B ∧      (∃ τ : OneQuasitrace B, τ.Normalized ∧ τ.IsTwo) ∧      ∃ (H : Type) (instH : NormedAddCommGroup H) (innerH : InnerProductSpace ℂ H),        letI := instH        letI := innerH        ∃ (completeH : CompleteSpace H),          letI := completeH          ∃ ρ : B →⋆ₐ[ℂ] (H →L[ℂ] H), Function.Injective ρ ∧            ProperlyInfinite (tensorAlgebra (G:=F2) ρ) ∧            ¬ ∃ τ : OneQuasitrace (tensorAlgebra (G:=F2) ρ), τ.Normalized ∧ τ.IsTwoendopen Kaplansky Regulardef CStarSimple (A : Type*) [CStarAlgebra A] : Prop :=  ∀ I : ClosedStarIdeal A,    (∀ a : A, a ∈ I.toNonUnitalStarSubalgebra → a = 0) ∨    (∀ a : A, a ∈ I.toNonUnitalStarSubalgebra)def SimpleStableTensorCounterexample : Prop :=  CStarSimple (Algebra F2) ∧ StablyFinite (Algebra F2) ∧  ∃ (M : Type) (instM : CStarAlgebra M),    letI := instM    Nontrivial M ∧ CStarSimple M ∧ StablyFinite M ∧      ∃ (H : Type) (instH : NormedAddCommGroup H) (innerH : InnerProductSpace ℂ H),        letI := instH        letI := innerH        ∃ (completeH : CompleteSpace H),          letI := completeH          ∃ ρ : M →⋆ₐ[ℂ] (H →L[ℂ] H), Function.Injective ρ ∧            ProperlyInfinite (tensorAlgebra (G:=F2) ρ)enddef StableFinitenessBundle : Prop :=  SimpleStableTensorCounterexample ∧ ReducedQuasitraceLoss ∧ TraceFreeStablyFinitetheorem stable_finiteness_bundle : StableFinitenessBundle := by  sorryend KaplanskyConsequencesendend OAI

Every extracted statement of the accompanying papers

The statements the atlas extracted automatically from each accompanying paper’s TeX source, with links to the full sources.

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

\tau(a+b)-\tau(a)-\tau(b)\ge\frac1{144}.

Section 1, Introduction Read it in the paper TeX source, line 80

Show the TeX

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.

Section 1, Introduction Read it in the paper TeX source, line 110

Show the TeX

\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}

Configuration and file facts

File: lean/ComparatorChallenges/KaplanskyStableFiniteness.lean
OpenAI’s Lean catalogue: OpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 294 link it.
Solution module: OAI.Analysis.Kaplansky.StableFiniteness
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: 26 local vocabulary declarations
Lines of Lean, without comments or blank lines: 125
Second proof-verification program (nanoda): Off in OpenAI's configuration.

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