Lean statement

Obstruction to nuclear norm-ultrapower embeddings

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.NuclearUltrapower.main_no_embedding

Lean statement file: NuclearUltrapower.leanShow on the mapOpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 292 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.

theorem main_no_embedding : MainStatementWithCuntzCorollary.{0} := 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.NuclearUltrapower.Dyadic line 12

abbrev Dyadic := Localization.Away (2 : ℤ)

abbreviation OAI.NuclearUltrapower.Vector3 line 13
```
abbrev Vector3 := Fin 3 → Dyadic
```

abbreviation OAI.NuclearUltrapower.SL3Z line 14

abbrev SL3Z := Matrix.SpecialLinearGroup (Fin 3) ℤ

definition OAI.NuclearUltrapower.dyadicTwo line 16

noncomputable def dyadicTwo : Dyadicˣ :=  (IsLocalization.Away.isUnit_of_dvd (S := Dyadic) (2 : ℤ) (dvd_refl _)).unit

definition OAI.NuclearUltrapower.matrixAction line 19

noncomputable def matrixAction (M : SL3Z) : Vector3 ≃ₗ[Dyadic] Vector3 :=  Matrix.SpecialLinearGroup.toLin'    (Matrix.SpecialLinearGroup.map (Int.castRingHom Dyadic) M)

definition OAI.NuclearUltrapower.dilationAction line 23

noncomputable def dilationAction (g : SL3Z × Multiplicative ℤ) :    Vector3 ≃ₗ[Dyadic] Vector3 :=  (matrixAction g.1).trans (LinearEquiv.smulOfUnit (dyadicTwo ^ g.2.toAdd))

definition OAI.NuclearUltrapower.groupAction line 27

noncomputable def groupAction :    (SL3Z × Multiplicative ℤ) →* MulAut (Multiplicative Vector3) where  toFun g := AddEquiv.toMultiplicative (dilationAction g).toAddEquiv  map_one' := by    ext v i    simp [dilationAction, matrixAction, LinearEquiv.smulOfUnit]  map_mul' g h := by    ext v i    simp [dilationAction, matrixAction, LinearEquiv.smulOfUnit, zpow_add,      map_mul, mul_smul, Units.smul_def, mul_left_comm]

abbreviation OAI.NuclearUltrapower.G line 39

abbrev G := SemidirectProduct (Multiplicative Vector3)  (SL3Z × Multiplicative ℤ) groupAction

abbreviation OAI.NuclearUltrapower.BoundedSeq line 46

abbrev BoundedSeq := BoundedContinuousFunction ℕ B

definition OAI.NuclearUltrapower.ultraRel line 49

def ultraRel (x y : BoundedSeq B) : Prop :=  Tendsto (fun n => x n - y n) (ω : Filter ℕ) (𝓝 0)

definition OAI.NuclearUltrapower.ultraCon line 53

def ultraCon : RingCon (BoundedSeq B) where

Show the remaining 17 lines

  r := ultraRel B ω  iseqv := {    refl := fun x => by simp [ultraRel]    symm := fun {x y} h => by      simpa only [ultraRel, neg_sub, neg_zero] using h.neg    trans := fun {x y z} hxy hyz => by      simpa only [ultraRel, sub_add_sub_cancel, zero_add] using hxy.add hyz }  add' := by    intro x y z t hxy hzt    simpa only [ultraRel, BoundedContinuousFunction.add_apply, ← add_sub_add_comm, zero_add]      using hxy.add hzt  mul' := by    intro x y z t hxy hzt    have h₁ := hxy.zero_mul_isBoundedUnder_le (Filter.isBoundedUnder_of ⟨‖z‖, fun index => z.norm_coe_le_norm index⟩)    have h₂ := Filter.isBoundedUnder_le_mul_tendsto_zero (Filter.isBoundedUnder_of ⟨‖y‖, fun index => y.norm_coe_le_norm index⟩) hzt    simpa only [ultraRel, BoundedContinuousFunction.mul_apply, sub_mul, mul_sub,      sub_add_sub_cancel, zero_add] using h₁.add h₂

abbreviation OAI.NuclearUltrapower.NormUltrapower line 74
```
abbrev NormUltrapower := (ultraCon B ω).Quotient
```

definition OAI.NuclearUltrapower.IsFree line 104

def IsFree (ω : Ultrafilter ℕ) : Prop :=  ∀ F : Finset ℕ, (F : Set ℕ) ∉ (ω : Filter ℕ)

structure OAI.NuclearUltrapower.ComplexHilbert line 110

structure ComplexHilbert where  carrier : Type u  [normed : NormedAddCommGroup carrier]  [inner : InnerProductSpace ℂ carrier]  [complete : CompleteSpace carrier]

definition OAI.NuclearUltrapower.spatialEval line 124

def spatialEval (H K : ComplexHilbert.{w})    (ρ : B →⋆ₙₐ[ℂ] (H →L[ℂ] H)) (σ : C →⋆ₙₐ[ℂ] (K →L[ℂ] K)) :    B ⊗[ℂ] C →ₗ[ℂ] ((H ⊗[ℂ] K) →L[ℂ] (H ⊗[ℂ] K)) :=  TensorProduct.lift (LinearMap.mk₂ ℂ (fun b c => TensorProduct.mapL (ρ b) (σ c))    (by intros; simp [TensorProduct.mapL_add_left])    (by intros; simp [TensorProduct.mapL_smul_left])    (by intros; simp [TensorProduct.mapL_add_right])    (by intros; simp [TensorProduct.mapL_smul_right]))

definition OAI.NuclearUltrapower.commutingEval line 134

def commutingEval (H : ComplexHilbert.{w})    (ρ : B →⋆ₙₐ[ℂ] (H →L[ℂ] H)) (σ : C →⋆ₙₐ[ℂ] (H →L[ℂ] H)) :    B ⊗[ℂ] C →ₗ[ℂ] (H →L[ℂ] H) :=  (LinearMap.mul' ℂ (H →L[ℂ] H)).comp    (TensorProduct.map      (show B →ₗ[ℂ] (H →L[ℂ] H) from        { toFun := ρ, map_add' := map_add ρ, map_smul' := map_smul ρ })      (show C →ₗ[ℂ] (H →L[ℂ] H) from        { toFun := σ, map_add' := map_add σ, map_smul' := map_smul σ }))

definition OAI.NuclearUltrapower.minTensorNorm line 145

def minTensorNorm (x : B ⊗[ℂ] C) : ℝ :=  sSup {r | ∃ (H K : ComplexHilbert.{w})    (ρ : B →⋆ₙₐ[ℂ] (H →L[ℂ] H)) (σ : C →⋆ₙₐ[ℂ] (K →L[ℂ] K)),    r = ‖spatialEval H K ρ σ x‖}

definition OAI.NuclearUltrapower.maxTensorNorm line 151

def maxTensorNorm (x : B ⊗[ℂ] C) : ℝ :=  sSup {r | ∃ (H : ComplexHilbert.{w})    (ρ : B →⋆ₙₐ[ℂ] (H →L[ℂ] H)) (σ : C →⋆ₙₐ[ℂ] (H →L[ℂ] H)),    (∀ b c, Commute (ρ b) (σ c)) ∧ r = ‖commutingEval H ρ σ x‖}

definition OAI.NuclearUltrapower.IsNuclear line 158

def IsNuclear (B : Type u) [NonUnitalCStarAlgebra B] : Prop :=  ∀ (C : Type v) [NonUnitalCStarAlgebra C] (x : B ⊗[ℂ] C),    minTensorNorm.{u, v, max u v} x = maxTensorNorm.{u, v, max u v} x

structure OAI.NuclearUltrapower.UnitalCStar line 164

structure UnitalCStar where  carrier : Type u  [cstar : CStarAlgebra carrier]  [nontrivial : Nontrivial carrier]

definition OAI.NuclearUltrapower.IsFullGroupAlgebra line 174

def IsFullGroupAlgebra (A : UnitalCStar.{u}) (ι : G →* unitary A) : Prop :=  Dense (Submodule.span ℂ (Set.range (fun g => (ι g : A))) : Set A) ∧  ∀ (D : UnitalCStar.{u}) (ρ : G →* unitary D),    ∃! φ : A →⋆ₐ[ℂ] D, ∀ g, φ (ι g) = (ρ g : D)

definition OAI.NuclearUltrapower.CuntzTwoRelations line 180

def CuntzTwoRelations {D : Type u} [CStarAlgebra D] (s : Fin 2 → D) : Prop :=  (∀ i, star (s i) * s i = 1) ∧ ∑ i, s i * star (s i) = 1

definition OAI.NuclearUltrapower.IsCuntzTwoAlgebra line 184

def IsCuntzTwoAlgebra (O : UnitalCStar.{u}) (s : Fin 2 → O) : Prop :=  CuntzTwoRelations s ∧  ∀ (D : UnitalCStar.{u}) (r : Fin 2 → D), CuntzTwoRelations r →    ∃! φ : O →⋆ₐ[ℂ] D, ∀ i, φ (s i) = r i

definition OAI.NuclearUltrapower.MainStatementWithCuntzCorollary line 191

def MainStatementWithCuntzCorollary : Prop :=  ∃ (A : UnitalCStar.{u}) (ι : G →* unitary A),    IsFullGroupAlgebra A ι ∧ TopologicalSpace.SeparableSpace A ∧    (∀ (B : UnitalCStar.{u}), IsNuclear.{u, u} B →      ∀ (ω : Ultrafilter ℕ), IsFree ω →        ¬ ∃ φ : A →⋆ₐ[ℂ] NormUltrapower B ω, Function.Injective φ) ∧    ∃ (O : UnitalCStar.{u}) (s : Fin 2 → O), IsCuntzTwoAlgebra O s ∧      ∀ (ω : Ultrafilter ℕ), IsFree ω →        ¬ ∃ φ : A →⋆ₐ[ℂ] NormUltrapower O ω, Function.Injective φ

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: embedding, nonzero, group, full, sl, nuclear, algebra, unital and four more. Found in the paper’s title, which its main theorem states: ultrapower, norm and explicit. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: particular, unitally and embed.

Theorem 1.1 of An explicit obstruction to nuclear norm-ultrapower embeddings

Theorem

Let

G=\Z[1/2]^3\rtimes\bigl(\SL_3(\Z)\times\Z\bigr), \qquad (M,k)\cdot v=2^kMv,

and let A=C^*(G) be its full group C^*-algebra. Then A is separable and unital. For every nonzero unital nuclear C^*-algebra B and every free ultrafilter \omega on \N, there is no unital embedding

A\longrightarrow B^\omega.

In particular, A does not embed unitally into \Otwo^\omega.

Section 1, The embedding problem Read it in the paper TeX source, line 36

Show the TeX

Let
\[
 G=\Z[1/2]^3\rtimes\bigl(\SL_3(\Z)\times\Z\bigr),
 \qquad (M,k)\cdot v=2^kMv,
\]
and let $A=C^*(G)$ be its full group $C^*$-algebra. Then $A$ is separable
and unital. For every nonzero unital nuclear $C^*$-algebra $B$ and every
free ultrafilter $\omega$ on $\N$, there is no unital embedding
\[
 A\longrightarrow B^\omega.
\]
In particular, $A$ does not embed unitally into $\Otwo^\omega$.

Other possible paper matches, closest wording first

Corollary 6.1 of An explicit obstruction to nuclear norm-ultrapower embeddings. Words of the Lean statement, its names split into words, that also appear in the theorem: aut, maps, finite, max, free, separable and five more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: unitally and embed.

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: MathVet; Neal McBurnett’s openai-math-lean-tooling. 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…

No verdict reached Their term: “REFUSED”.
Re-run through Comparator did not reach a verdict (the job ended (failure) in the step "comparator, Lean’s kernel and nanoda" without uploading a result: the runner terminated it (GitHub: Process completed with exit code 143.)).
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-pass”.
Re-run through Comparator: accepted.
Reported 10 October 2026, at the release version shown here. Original verification report

Neal McBurnett’s openai-math-lean-tooling

How they ran it, in their words

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; the run was still in progress at this snapshot (192 accepted, 224 pending).

Accepted Their term: “accepted”.
Re-run through Comparator with Lean’s kernel: accepted (the check took 1817 s; peak memory 19.6 GiB).
Reported 11 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 292’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: main_no_embedding : MainStatementWithCuntzCorollary states: there is a separable unital C*-algebra A with ι : G →* unitary A satisfying the universal property of the FULL group C*-algebra of G = (ℤ[1/2])³ ⋊ (SL₃(ℤ) × ℤ) (IsFullGroupAlgebra), and for every nonzero unital nuclear B (nuclear = min and max C*-tensor norms agree on B ⊗ C for all C) and every free ultrafilter ω on ℕ there is no injective unital *-homomorphism A →⋆ₐ[ℂ] NormUltrapower B ω; the same for the universal Cuntz algebra O₂. Exactly the summary, including the O₂ corollary.
Definitions the group asks readers to compare with the paper: Hand-built objects: NormUltrapower B ω := (ultraCon B ω).Quotient (bounded sequences modulo ω-null), nuclearity via minTensorNorm/maxTensorNorm defined as sSup over representations on Hilbert spaces of one universe, IsFullGroupAlgebra (dense span + universal property), IsCuntzTwoAlgebra (two Cuntz isometries). Standard formulations, but none is a Mathlib notion, and the tensor-norm sSups are real-valued.
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 constructs an explicit separable unital full group $C^*$-algebra that has no unital embedding into $B^\omega$ for any nonzero unital nuclear $C^*$-algebra $B$ and any free ultrafilter $\omega$ on $\mathbb N$. In particular, it does not embed unitally into an ultrapower of the Cuntz algebra $\mathcal O_2$.”
Quoted from OpenAI’s Lean note for this family

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

213 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: 75 to its own declarations, 124 to Mathlib’s documentation. 25 declared names also link to the same declaration in OpenAI’s library. 1 identifier stays plain because it resolves to no single declaration: a variable Lean binds automatically, notation, or a name the scan of the pinned sources did not find. OpenAI uses Mathlib version d13f23b. The linked documentation describes the current version, so names that have since changed may be missing there.

Definitions and lemmas named by the statement

If Comparator accepts a proof, it proves main_no_embedding exactly as written here. To compare that statement with the paper, also read the definitions and lemmas listed below. Atlas found them by following names in the source text and lists each after the names it uses. For x.foo, Atlas includes every foo defined here for a type, even when the statement may use only one of them.

It reaches 24 of the file’s 29 other declarations; with the theorem itself they span 105 lines. 1 of them is named in the statement itself.

Names in the namespace OAI.NuclearUltrapower are listed without it.

abbrev Dyadicline 12library
abbrev Vector3line 13library
abbrev SL3Zline 14library
def dyadicTwoline 16library
def matrixActionline 19library
def dilationActionline 23library
def groupActionline 27library
abbrev Gline 39library
abbrev BoundedSeqline 46library
def ultraRelline 49library
def ultraConline 53library
abbrev NormUltrapowerline 74library
def IsFreeline 104library
structure ComplexHilbertline 110library
def spatialEvalline 124library
def commutingEvalline 134library
def minTensorNormline 145library
def maxTensorNormline 151library
def IsNuclearline 158library
structure UnitalCStarline 164library
def IsFullGroupAlgebraline 174library
def CuntzTwoRelationsline 180library
def IsCuntzTwoAlgebraline 184library
def MainStatementWithCuntzCorollaryline 191in the statementlibrary
theorem main_no_embeddingline 208selected theoremlibrary

Other Lean rules that may affect the statement 7

Lean can use instances, notation and attributes without naming them in the statement. These are rules for interpreting or finding mathematical objects. Atlas’s scan of names cannot follow all such uses; read the listed source as well.

instance Star (NormUltrapower B ω) line 76
instance StarRing (NormUltrapower B ω) line 81
instance StarModule ℂ (NormUltrapower B ω) line 97
instance CoeSort ComplexHilbert (Type u) line 117
instance CoeSort UnitalCStar (Type u) line 169
attribute [instance] ComplexHilbert.normed ComplexHilbert.inner Comple… line 116
attribute [instance] UnitalCStar.cstar UnitalCStar.nontrivial line 168

import Mathlibnamespace OAInoncomputable sectionuniverse u v wopen Filteropen scoped TensorProduct Topologynamespace NuclearUltrapower/- The dyadic coefficient ring is exactly Z[1/2]. -/abbrev Dyadic := Localization.Away (2 : ℤ)abbrev Vector3 := Fin 3 → Dyadicabbrev SL3Z := Matrix.SpecialLinearGroup (Fin 3) ℤnoncomputable def dyadicTwo : Dyadicˣ :=  (IsLocalization.Away.isUnit_of_dvd (S := Dyadic) (2 : ℤ) (dvd_refl _)).unitnoncomputable def matrixAction (M : SL3Z) : Vector3 ≃ₗ[Dyadic] Vector3 :=  Matrix.SpecialLinearGroup.toLin'    (Matrix.SpecialLinearGroup.map (Int.castRingHom Dyadic) M)noncomputable def dilationAction (g : SL3Z × Multiplicative ℤ) :    Vector3 ≃ₗ[Dyadic] Vector3 :=  (matrixAction g.1).trans (LinearEquiv.smulOfUnit (dyadicTwo ^ g.2.toAdd))noncomputable def groupAction :    (SL3Z × Multiplicative ℤ) →* MulAut (Multiplicative Vector3) where  toFun g := AddEquiv.toMultiplicative (dilationAction g).toAddEquiv  map_one' := by    ext v i    simp [dilationAction, matrixAction, LinearEquiv.smulOfUnit]  map_mul' g h := by    ext v i    simp [dilationAction, matrixAction, LinearEquiv.smulOfUnit, zpow_add,      map_mul, mul_smul, Units.smul_def, mul_left_comm]/-- The exact group of the semidirect-product construction. -/abbrev G := SemidirectProduct (Multiplicative Vector3)  (SL3Z × Multiplicative ℤ) groupActionsection UltrapowerDefinitionvariable (B : Type u) [CStarAlgebra B] (ω : Ultrafilter ℕ)/-- Bounded sequences, with their supremum norm. N is discrete. -/abbrev BoundedSeq := BoundedContinuousFunction ℕ B/-- Equality modulo I_omega(B). This is equivalent to norm-null difference. -/def ultraRel (x y : BoundedSeq B) : Prop :=  Tendsto (fun n => x n - y n) (ω : Filter ℕ) (𝓝 0)/-- The ring congruence defining the norm (not algebraic) ultrapower. -/def ultraCon : RingCon (BoundedSeq B) where  r := ultraRel B ω  iseqv := {    refl := fun x => by simp [ultraRel]    symm := fun {x y} h => by      simpa only [ultraRel, neg_sub, neg_zero] using h.neg    trans := fun {x y z} hxy hyz => by      simpa only [ultraRel, sub_add_sub_cancel, zero_add] using hxy.add hyz }  add' := by    intro x y z t hxy hzt    simpa only [ultraRel, BoundedContinuousFunction.add_apply, ← add_sub_add_comm, zero_add]      using hxy.add hzt  mul' := by    intro x y z t hxy hzt    have h₁ := hxy.zero_mul_isBoundedUnder_le (Filter.isBoundedUnder_of ⟨‖z‖, fun index => z.norm_coe_le_norm index⟩)    have h₂ := Filter.isBoundedUnder_le_mul_tendsto_zero (Filter.isBoundedUnder_of ⟨‖y‖, fun index => y.norm_coe_le_norm index⟩) hzt    simpa only [ultraRel, BoundedContinuousFunction.mul_apply, sub_mul, mul_sub,      sub_add_sub_cancel, zero_add] using h₁.add h₂/-- The exact quotient l-infinity(N,B)/I_omega(B) as a complex star algebra.The quotient identifies sequences whose difference tends to zero along the ultrafilter. -/abbrev NormUltrapower := (ultraCon B ω).Quotientinstance : Star (NormUltrapower B ω) where  star := Quotient.map' star (fun first second hypothesis => by    change ultraRel B ω (star first) (star second)    simpa only [ultraRel, star_sub, star_zero, BoundedContinuousFunction.star_apply] using hypothesis.star)instance : StarRing (NormUltrapower B ω) where  star_involutive := by    intro x    induction x using Quotient.inductionOn with | h x =>      exact congrArg Quotient.mk'' (star_star x)  star_mul := by    intro x y    induction x using Quotient.inductionOn with | h x =>      induction y using Quotient.inductionOn with | h y =>        exact congrArg Quotient.mk'' (star_mul x y)  star_add := by    intro x y    induction x using Quotient.inductionOn with | h x =>      induction y using Quotient.inductionOn with | h y =>        exact congrArg Quotient.mk'' (star_add x y)instance : StarModule ℂ (NormUltrapower B ω) where  star_smul := by    intro c x    induction x using Quotient.inductionOn with | h x =>      exact congrArg Quotient.mk'' (star_smul c x)/-- Freeness: no finite subset of N belongs to the ultrafilter. -/def IsFree (ω : Ultrafilter ℕ) : Prop :=  ∀ F : Finset ℕ, (F : Set ℕ) ∉ (ω : Filter ℕ)end UltrapowerDefinition/-- A bundled complex Hilbert space, without any separability assumption. -/structure ComplexHilbert where  carrier : Type u  [normed : NormedAddCommGroup carrier]  [inner : InnerProductSpace ℂ carrier]  [complete : CompleteSpace carrier]attribute [instance] ComplexHilbert.normed ComplexHilbert.inner ComplexHilbert.completeinstance : CoeSort ComplexHilbert (Type u) := ⟨ComplexHilbert.carrier⟩section TensorNormDefinitionsvariable {B : Type u} {C : Type v} [NonUnitalCStarAlgebra B] [NonUnitalCStarAlgebra C]/-- Spatial evaluation of the algebraic tensor product on separate Hilbert spaces.Its operator norm is unchanged on the Hilbert completion. -/def spatialEval (H K : ComplexHilbert.{w})    (ρ : B →⋆ₙₐ[ℂ] (H →L[ℂ] H)) (σ : C →⋆ₙₐ[ℂ] (K →L[ℂ] K)) :    B ⊗[ℂ] C →ₗ[ℂ] ((H ⊗[ℂ] K) →L[ℂ] (H ⊗[ℂ] K)) :=  TensorProduct.lift (LinearMap.mk₂ ℂ (fun b c => TensorProduct.mapL (ρ b) (σ c))    (by intros; simp [TensorProduct.mapL_add_left])    (by intros; simp [TensorProduct.mapL_smul_left])    (by intros; simp [TensorProduct.mapL_add_right])    (by intros; simp [TensorProduct.mapL_smul_right]))/-- Evaluation on one Hilbert space; used only when the two ranges commute. -/def commutingEval (H : ComplexHilbert.{w})    (ρ : B →⋆ₙₐ[ℂ] (H →L[ℂ] H)) (σ : C →⋆ₙₐ[ℂ] (H →L[ℂ] H)) :    B ⊗[ℂ] C →ₗ[ℂ] (H →L[ℂ] H) :=  (LinearMap.mul' ℂ (H →L[ℂ] H)).comp    (TensorProduct.map      (show B →ₗ[ℂ] (H →L[ℂ] H) from        { toFun := ρ, map_add' := map_add ρ, map_smul' := map_smul ρ })      (show C →ₗ[ℂ] (H →L[ℂ] H) from        { toFun := σ, map_add' := map_add σ, map_smul' := map_smul σ }))/-- The minimal C*-tensor norm: supremum over spatial representations. -/def minTensorNorm (x : B ⊗[ℂ] C) : ℝ :=  sSup {r | ∃ (H K : ComplexHilbert.{w})    (ρ : B →⋆ₙₐ[ℂ] (H →L[ℂ] H)) (σ : C →⋆ₙₐ[ℂ] (K →L[ℂ] K)),    r = ‖spatialEval H K ρ σ x‖}/-- The maximal C*-tensor norm: supremum over commuting representations. -/def maxTensorNorm (x : B ⊗[ℂ] C) : ℝ :=  sSup {r | ∃ (H : ComplexHilbert.{w})    (ρ : B →⋆ₙₐ[ℂ] (H →L[ℂ] H)) (σ : C →⋆ₙₐ[ℂ] (H →L[ℂ] H)),    (∀ b c, Commute (ρ b) (σ c)) ∧ r = ‖commutingEval H ρ σ x‖}/-- Nuclearity in the tensor-norm convention of the tensor-norm formulation.The second C*-algebra need not be unital, nonzero, or separable. -/def IsNuclear (B : Type u) [NonUnitalCStarAlgebra B] : Prop :=  ∀ (C : Type v) [NonUnitalCStarAlgebra C] (x : B ⊗[ℂ] C),    minTensorNorm.{u, v, max u v} x = maxTensorNorm.{u, v, max u v} xend TensorNormDefinitions/-- A nonzero unital complex C*-algebra. -/structure UnitalCStar where  carrier : Type u  [cstar : CStarAlgebra carrier]  [nontrivial : Nontrivial carrier]attribute [instance] UnitalCStar.cstar UnitalCStar.nontrivialinstance : CoeSort UnitalCStar (Type u) := ⟨UnitalCStar.carrier⟩/-- The universal property defining the full, not the reduced, group C*-algebra.The dense generation clause makes the intended completion explicit.This is definition data, not a hypothesis silently supplied to the main claim. -/def IsFullGroupAlgebra (A : UnitalCStar.{u}) (ι : G →* unitary A) : Prop :=  Dense (Submodule.span ℂ (Set.range (fun g => (ι g : A))) : Set A) ∧  ∀ (D : UnitalCStar.{u}) (ρ : G →* unitary D),    ∃! φ : A →⋆ₐ[ℂ] D, ∀ g, φ (ι g) = (ρ g : D)/-- The two Cuntz relations, including unit preservation explicitly. -/def CuntzTwoRelations {D : Type u} [CStarAlgebra D] (s : Fin 2 → D) : Prop :=  (∀ i, star (s i) * s i = 1) ∧ ∑ i, s i * star (s i) = 1/-- The universal nonzero unital C*-algebra on the two Cuntz isometries. -/def IsCuntzTwoAlgebra (O : UnitalCStar.{u}) (s : Fin 2 → O) : Prop :=  CuntzTwoRelations s ∧  ∀ (D : UnitalCStar.{u}) (r : Fin 2 → D), CuntzTwoRelations r →    ∃! φ : O →⋆ₐ[ℂ] D, ∀ i, φ (s i) = r i/-- The manuscript's main statement including its explicitly stated O_2corollary. Both universal C*-algebras must be constructed to prove this. -/def MainStatementWithCuntzCorollary : Prop :=  ∃ (A : UnitalCStar.{u}) (ι : G →* unitary A),    IsFullGroupAlgebra A ι ∧ TopologicalSpace.SeparableSpace A ∧    (∀ (B : UnitalCStar.{u}), IsNuclear.{u, u} B →      ∀ (ω : Ultrafilter ℕ), IsFree ω →        ¬ ∃ φ : A →⋆ₐ[ℂ] NormUltrapower B ω, Function.Injective φ) ∧    ∃ (O : UnitalCStar.{u}) (s : Fin 2 → O), IsCuntzTwoAlgebra O s ∧      ∀ (ω : Ultrafilter ℕ), IsFree ω →        ¬ ∃ φ : A →⋆ₐ[ℂ] NormUltrapower O ω, Function.Injective φend NuclearUltrapowerendnamespace NuclearUltrapower/-- The full dyadic group C*-algebra has no nuclear norm-ultrapower embedding,including the corollary for the universal Cuntz algebra on two isometries. -/theorem main_no_embedding : MainStatementWithCuntzCorollary.{0} := by  sorryend NuclearUltrapowerend OAI

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

An explicit obstruction to nuclear norm-ultrapower embeddings

Theorem

Let

G=\Z[1/2]^3\rtimes\bigl(\SL_3(\Z)\times\Z\bigr), \qquad (M,k)\cdot v=2^kMv,

and let A=C^*(G) be its full group C^*-algebra. Then A is separable and unital. For every nonzero unital nuclear C^*-algebra B and every free ultrafilter \omega on \N, there is no unital embedding

A\longrightarrow B^\omega.

In particular, A does not embed unitally into \Otwo^\omega.

Section 1, The embedding problem Read it in the paper TeX source, line 36

Show the TeX

Let
\[
 G=\Z[1/2]^3\rtimes\bigl(\SL_3(\Z)\times\Z\bigr),
 \qquad (M,k)\cdot v=2^kMv,
\]
and let $A=C^*(G)$ be its full group $C^*$-algebra. Then $A$ is separable
and unital. For every nonzero unital nuclear $C^*$-algebra $B$ and every
free ultrafilter $\omega$ on $\N$, there is no unital embedding
\[
 A\longrightarrow B^\omega.
\]
In particular, $A$ does not embed unitally into $\Otwo^\omega$.

Configuration and file facts

File: lean/ComparatorChallenges/NuclearUltrapower.lean
OpenAI’s Lean catalogue: OpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 292 link it.
Solution module: OAI.Analysis.NuclearUltrapower.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: 29 local vocabulary declarations
Lines of Lean, without comments or blank lines: 133
Second proof-verification program (nanoda): Off in OpenAI's configuration.

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