Lean statement

Isomorphism of all interpolated free group factors

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.FreeGroupFactorMain.Interpolation.allInterpolatedIsomorphic

Lean statement file: InterpolatedFactors.leanShow on the mapOpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 287 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 allInterpolatedIsomorphic (r s : ℝ≥0∞) (hr : 1<r) (hs : 1<s) :    Nonempty (NormalTracialEquiv (interpolatedTrace r) (interpolatedTrace s)      (interpolatedTopology r) (interpolatedTopology s)) := 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.FreeGroupFactorMain.GroupHilbert line 13

abbrev GroupHilbert (G : Type u) := lp (fun _ : G => ℂ) 2

abbreviation OAI.FreeGroupFactorMain.GroupOperator line 14

abbrev GroupOperator (G : Type u) := GroupHilbert G →L[ℂ] GroupHilbert G

definition OAI.FreeGroupFactorMain.leftTranslations line 16

def leftTranslations (G : Type u) [Group G] : Set (GroupOperator G) :=  {T | ∃ g : G, ∀ (ξ : GroupHilbert G) (h : G), (T ξ) h = ξ (g⁻¹ * h)}

definition OAI.FreeGroupFactorMain.groupVonNeumann line 19

def groupVonNeumann (G : Type u) [Group G] : StarSubalgebra ℂ (GroupOperator G) :=  StarSubalgebra.centralizer ℂ    (StarSubalgebra.centralizer ℂ (leftTranslations G) : Set (GroupOperator G))

definition OAI.FreeGroupFactorMain.canonicalTrace line 23

def canonicalTrace {G : Type u} [Group G] (T : groupVonNeumann G) : ℂ := by  classical  exact ((T : GroupOperator G) (lp.single 2 (1 : G) (1 : ℂ))) 1

definition OAI.FreeGroupFactorMain.ultraweakOperatorTopology line 27

@[instance_reducible] def ultraweakOperatorTopology (G : Type u) : TopologicalSpace (GroupOperator G) :=  ⨅ (ξ : lp (fun _ : ℕ => GroupHilbert G) 2)    (η : lp (fun _ : ℕ => GroupHilbert G) 2),    TopologicalSpace.induced      (fun T : GroupOperator G => ∑' k : ℕ, inner ℂ (η k) (T (ξ k)))      inferInstance

definition OAI.FreeGroupFactorMain.groupUltraweakTopology line 34

@[instance_reducible] def groupUltraweakTopology (G : Type u) [Group G] :    TopologicalSpace (groupVonNeumann G) :=  TopologicalSpace.induced (fun T : groupVonNeumann G => (T : GroupOperator G))    (ultraweakOperatorTopology G)

abbreviation OAI.FreeGroupFactorMain.FreeGroupRank line 39
```
abbrev FreeGroupRank (n : ℕ) := FreeGroup (Fin n)
```

definition OAI.FreeGroupFactorMain.Amplification.cornerSubalgebra line 61

def cornerSubalgebra (p : A) (hp : IsStarProjection p) : NonUnitalStarSubalgebra ℂ A where

Show the remaining 15 lines

  carrier := {x | p*x=x ∧ x*p=x}  zero_mem' := by simp  add_mem' hx hy := by constructor <;> simp [mul_add,add_mul,hx.1,hx.2,hy.1,hy.2]  mul_mem' hx hy := by    constructor    · rw [← mul_assoc,hx.1]    · rw [mul_assoc,hy.2]  smul_mem' c x hx := by    constructor    · rw [mul_smul_comm,hx.1]    · rw [smul_mul_assoc,hx.2]  star_mem' hx := by    constructor    · simpa only [star_mul,hp.isSelfAdjoint.star_eq] using congrArg star hx.2    · simpa only [star_mul,hp.isSelfAdjoint.star_eq] using congrArg star hx.1

abbreviation OAI.FreeGroupFactorMain.Amplification.Corner line 78

abbrev Corner (p : A) (hp : IsStarProjection p) := ↥(cornerSubalgebra p hp)

definition OAI.FreeGroupFactorMain.Amplification.cornerTopology line 113

@[instance_reducible] def cornerTopology (𝒯 : TopologicalSpace A)    (p : A) (hp : IsStarProjection p) : TopologicalSpace (Corner p hp) :=  TopologicalSpace.induced (fun x : Corner p hp => (x : A)) 𝒯

definition OAI.FreeGroupFactorMain.Amplification.normalizedCornerTrace line 117

def normalizedCornerTrace (τ : A → ℂ) (t : ℝ) (p : A) (hp : IsStarProjection p)    (x : Corner p hp) : ℂ := (t : ℂ)⁻¹ * τ x

definition OAI.FreeGroupFactorMain.Amplification.stabilizedRightTranslations line 122

def stabilizedRightTranslations (G : Type u_5) [Group G] : Set (GroupOperator (ℕ × G)) :=  {T | ∃ g : G, ∀ (ξ : GroupHilbert (ℕ × G)) (x : ℕ × G),    T ξ x = ξ (x.1,x.2*g)}

definition OAI.FreeGroupFactorMain.Amplification.stabilization line 126

def stabilization (G : Type u_6) [Group G] : StarSubalgebra ℂ (GroupOperator (ℕ × G)) :=  StarSubalgebra.centralizer ℂ (stabilizedRightTranslations G)

definition OAI.FreeGroupFactorMain.Amplification.stabilizedProjectionTrace line 137

def stabilizedProjectionTrace {G : Type u_9} [Group G] (p : stabilization G) : ℝ≥0∞ := by  classical  exact ∑' k : ℕ, ENNReal.ofReal (‖(p : GroupOperator (ℕ × G))    (lp.single 2 (k,1) (1 : ℂ))‖ ^ 2)

definition OAI.FreeGroupFactorMain.Amplification.stabilizedTrace line 142

def stabilizedTrace {G : Type u_10} [Group G] (T : stabilization G) : ℂ := by  classical  exact ∑' k : ℕ, inner ℂ (lp.single 2 (k,1) (1 : ℂ))    ((T : GroupOperator (ℕ × G)) (lp.single 2 (k,1) (1 : ℂ)))

definition OAI.FreeGroupFactorMain.Amplification.stabilizationTopology line 147

@[instance_reducible] def stabilizationTopology (G : Type u_11) [Group G] :    TopologicalSpace (stabilization G) :=  TopologicalSpace.induced (fun T : stabilization G => (T : GroupOperator (ℕ × G)))    (ultraweakOperatorTopology (ℕ × G))

structure OAI.FreeGroupFactorMain.Amplification.NormalTracialEquiv line 155

structure NormalTracialEquiv (τA : A → ℂ) (τB : B → ℂ)    (𝒯A : TopologicalSpace A) (𝒯B : TopologicalSpace B) where  equiv : A ≃⋆ₐ[ℂ] B  trace_eq : ∀x,τB (equiv x)=τA x  normal : @Continuous _ _ 𝒯A 𝒯B equiv  inverse_normal : @Continuous _ _ 𝒯B 𝒯A equiv.symm

abbreviation OAI.FreeGroupFactorMain.Interpolation.RankTwoProjection line 171

abbrev RankTwoProjection := {p : stabilization (FreeGroupRank 2) // IsStarProjection p}

definition OAI.FreeGroupFactorMain.Interpolation.selectedProjection line 176

def selectedProjection (a : ℝ) : RankTwoProjection :=  Classical.epsilon (fun p : RankTwoProjection => stabilizedProjectionTrace p.val=ENNReal.ofReal a)

definition OAI.FreeGroupFactorMain.Interpolation.interpolationScale line 179
```
def interpolationScale (r : ℝ) : ℝ := (Real.sqrt (r-1))⁻¹
```

inductive type OAI.FreeGroupFactorMain.Interpolation.Model line 181

inductive Model where  | integer (n : ℕ)  | continuous (r : ℝ)  | infinity

definition OAI.FreeGroupFactorMain.Interpolation.Model.Algebra line 186

def Model.Algebra : Model → Type  | .integer n => groupVonNeumann (FreeGroupRank n)  | .continuous r => Corner (selectedProjection (interpolationScale r)).val      (selectedProjection (interpolationScale r)).property  | .infinity => groupVonNeumann (FreeGroup ℕ)

definition OAI.FreeGroupFactorMain.Interpolation.Model.trace line 195

def Model.trace : (m : Model) → m.Algebra → ℂ  | .integer _ => canonicalTrace  | .continuous r => normalizedCornerTrace stabilizedTrace (interpolationScale r)      (selectedProjection (interpolationScale r)).val (selectedProjection (interpolationScale r)).property  | .infinity => canonicalTrace

4 more definitions are reached the same way; the full file lists them.

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: canonical, star, single, tracial, adjoint, self, von, algebra and three more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: factor.

Corollary 7.1 of An isomorphism of the free group factors

Corollary (Dependence on von Neumann generators)

Let M=L(\F_2) with its canonical trace. For every integer n\geq2 there are a self-adjoint n-tuple X^{(n)} and a self-adjoint 2n-tuple Y^{(n)} such that

W^*(X^{(n)})=W^*(Y^{(n)})=M

and

\delta(X^{(n)})=\delta_0(X^{(n)})=n, \qquad \delta^*(Y^{(n)})=\delta^\star(Y^{(n)})=n.

Consequently, none of \delta,\delta_0,\delta^*,\delta^\star is invariant under changing a finite self-adjoint W^*-generating tuple of a tracial von Neumann algebra. Each takes unbounded values on such generating tuples of the single factor L(\F_2).

Section 7, Free entropy dimensions and von Neumann generators Read it in the paper TeX source, line 21

Show the TeX

Let $M=L(\F_2)$ with its canonical trace. For every integer $n\geq2$
there are a self-adjoint $n$-tuple $X^{(n)}$ and a self-adjoint
$2n$-tuple $Y^{(n)}$ such that
\[
 W^*(X^{(n)})=W^*(Y^{(n)})=M
\]
and
\[
 \delta(X^{(n)})=\delta_0(X^{(n)})=n,
 \qquad
 \delta^*(Y^{(n)})=\delta^\star(Y^{(n)})=n.
\]
Consequently, none of $\delta,\delta_0,\delta^*,\delta^\star$ is
invariant under changing a finite self-adjoint $W^*$-generating tuple
of a tracial von Neumann algebra. Each takes unbounded values on such
generating tuples of the single factor $L(\F_2)$.

Other possible paper matches, closest wording first

Theorem 1.1 of An isomorphism of the free group factors. Words of the Lean statement, its names split into words, that also appear in the theorem: ranks, integer, unital, normal and trace. Found in the paper’s title, which its main theorem states: group and free. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: isomorphism.
Proposition 3.1 of An isomorphism of the free group factors. Words of the Lean statement, its names split into words, that also appear in the theorem: eq, operator, parameter, continuous, free, group and three more.
Proposition 5.1 of An isomorphism of the free group factors. Words of the Lean statement, its names split into words, that also appear in the theorem: von, algebra, neumann, normal and trace.
Theorem 1.2 of An isomorphism of the free group factors. Words of the Lean statement, its names split into words, that also appear in the theorem: unital, normal and trace. Found in the paper’s title, which its main theorem states: group and free. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: isomorphism.
Corollary 6.1 of An isomorphism of the free group factors. Words of the Lean statement, its names split into words, that also appear in the theorem: unital, normal and trace. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: isomorphism.

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

three independent groups report that Comparator accepted runs for this Lean statement file: Carlos Toledo’s cert-machine; 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…

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

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 287 s).
Reported 10 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 287’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.

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: allInterpolatedIsomorphic (r s : ℝ≥0∞) (hr : 1 < r) (hs : 1 < s) : Nonempty (NormalTracialEquiv (interpolatedTrace r) (interpolatedTrace s) (interpolatedTopology r) (interpolatedTopology s)): every pair of interpolated free group factors L(F_r), L(F_s) with r,s ∈ (1,∞] (integer ranks via groupVonNeumann (FreeGroup (Fin n)), FreeGroup ℕ for ∞, and non-integer r as the corner of L(F_2) ⊗ B(ℓ²) by a projection of trace 1/√(r-1)) is related by a *-isomorphism that preserves the canonical trace and is ultraweakly bicontinuous. The summary's statement L(F_2) ≅ L(F_3) and all interpolated factors; the fundamental-group conclusion is a consequence, not stated.
Definitions the group asks readers to compare with the paper: Hand-built II_1 factor models: groupVonNeumann G := centralizer (centralizer (leftTranslations G)) (double commutant of left translations), canonicalTrace, ultraweakOperatorTopology := ⨅ ξ η, induced (T ↦ Σ' k ⟨η k, T (ξ k)⟩), corners Corner p hp and selectedProjection a := Classical.epsilon (stabilizedProjectionTrace p = ofReal a) (junk default projection 0 if none existed, which does not happen in L(F_2)⊗B(ℓ²)); standard but not Mathlib's VonNeumannAlgebra-with-trace.
Carlos Toledo’s cert-machine
Finds part of it stated, or a narrower or weaker statement Their term: “NARROWER — DECLARED”.
The challenge statements state less than the headline, and OpenAI’s Lean note says so. Their record
Their note: The fundamental-group clause is not stated; it is a (short, unformalized) consequence of the stated isomorphisms.

Read the complete assessments on the family page.

OpenAI’s own note on what this Lean statement covers

“The formalization proves that interpolated free group factors with any parameters $r,s>1$, including the infinite parameter, are normally trace-preservingly isomorphic. The paper's fundamental-group conclusion is a further consequence rather than a separate selected statement here.”
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

225 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: 102 to its own declarations, 136 to Mathlib’s documentation. 42 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.

import Mathlibnamespace OAInoncomputable sectionnamespace FreeGroupFactorMainuniverse u uGopen scoped ENNRealabbrev GroupHilbert (G : Type u) := lp (fun _ : G => ℂ) 2abbrev GroupOperator (G : Type u) := GroupHilbert G →L[ℂ] GroupHilbert Gdef leftTranslations (G : Type u) [Group G] : Set (GroupOperator G) :=  {T | ∃ g : G, ∀ (ξ : GroupHilbert G) (h : G), (T ξ) h = ξ (g⁻¹ * h)}def groupVonNeumann (G : Type u) [Group G] : StarSubalgebra ℂ (GroupOperator G) :=  StarSubalgebra.centralizer ℂ    (StarSubalgebra.centralizer ℂ (leftTranslations G) : Set (GroupOperator G))def canonicalTrace {G : Type u} [Group G] (T : groupVonNeumann G) : ℂ := by  classical  exact ((T : GroupOperator G) (lp.single 2 (1 : G) (1 : ℂ))) 1@[instance_reducible] def ultraweakOperatorTopology (G : Type u) : TopologicalSpace (GroupOperator G) :=  ⨅ (ξ : lp (fun _ : ℕ => GroupHilbert G) 2)    (η : lp (fun _ : ℕ => GroupHilbert G) 2),    TopologicalSpace.induced      (fun T : GroupOperator G => ∑' k : ℕ, inner ℂ (η k) (T (ξ k)))      inferInstance@[instance_reducible] def groupUltraweakTopology (G : Type u) [Group G] :    TopologicalSpace (groupVonNeumann G) :=  TopologicalSpace.induced (fun T : groupVonNeumann G => (T : GroupOperator G))    (ultraweakOperatorTopology G)abbrev FreeGroupRank (n : ℕ) := FreeGroup (Fin n)instance groupVN_isClosed (G : Type uG) [Group G] :    IsClosed (groupVonNeumann G : Set (GroupOperator G)) := by  unfold groupVonNeumann  rw [StarSubalgebra.coe_centralizer]  exact Set.isClosed_centralizer _instance groupVN_CStarAlgebra (G : Type uG) [Group G] :    CStarAlgebra (groupVonNeumann G) := inferInstanceend FreeGroupFactorMainuniverse u_1 u_2 u_3 u_4 u_5 u_6 u_7 u_8 u_9 u_10 u_11 u_42 u_43namespace FreeGroupFactorMain.Amplificationopen scoped ENNRealsectionvariable {A : Type u_1} {B : Type u_2} [CStarAlgebra A] [CStarAlgebra B]def cornerSubalgebra (p : A) (hp : IsStarProjection p) : NonUnitalStarSubalgebra ℂ A where  carrier := {x | p*x=x ∧ x*p=x}  zero_mem' := by simp  add_mem' hx hy := by constructor <;> simp [mul_add,add_mul,hx.1,hx.2,hy.1,hy.2]  mul_mem' hx hy := by    constructor    · rw [← mul_assoc,hx.1]    · rw [mul_assoc,hy.2]  smul_mem' c x hx := by    constructor    · rw [mul_smul_comm,hx.1]    · rw [smul_mul_assoc,hx.2]  star_mem' hx := by    constructor    · simpa only [star_mul,hp.isSelfAdjoint.star_eq] using congrArg star hx.2    · simpa only [star_mul,hp.isSelfAdjoint.star_eq] using congrArg star hx.1abbrev Corner (p : A) (hp : IsStarProjection p) := ↥(cornerSubalgebra p hp)instance cornerClosed (p : A) (hp : IsStarProjection p) :    IsClosed (cornerSubalgebra p hp : Set A) :=  (isClosed_eq (continuous_const.mul continuous_id) continuous_id).inter    (isClosed_eq (continuous_id.mul continuous_const) continuous_id)instance cornerSemiring (p : A) (hp : IsStarProjection p) : Semiring (Corner p hp) where  __ : NonUnitalSemiring (Corner p hp) := inferInstance  one := ⟨p,hp.isIdempotentElem.eq,hp.isIdempotentElem.eq⟩  one_mul x := Subtype.ext x.property.1  mul_one x := Subtype.ext x.property.2instance cornerRing (p : A) (hp : IsStarProjection p) : Ring (Corner p hp) where  __ : Semiring (Corner p hp) := inferInstance  __ : NonUnitalRing (Corner p hp) := inferInstanceinstance cornerAlgebra (p : A) (hp : IsStarProjection p) : Algebra ℂ (Corner p hp) :=  Algebra.ofModule smul_mul_assoc mul_smul_comminstance cornerNormedRing (p : A) (hp : IsStarProjection p) : NormedRing (Corner p hp) where  __ : Ring (Corner p hp) := inferInstance  __ : NonUnitalNormedRing (Corner p hp) := inferInstanceinstance cornerCStarAlgebra (p : A) (hp : IsStarProjection p) : CStarAlgebra (Corner p hp) where  __ : NonUnitalCStarAlgebra (Corner p hp) := inferInstance  __ : NormedRing (Corner p hp) := inferInstance  __ : Algebra ℂ (Corner p hp) := inferInstance  norm_smul_le := norm_smul_leendsectionvariable {A : Type u_3} {B : Type u_4} [CStarAlgebra A] [CStarAlgebra B]@[instance_reducible] def cornerTopology (𝒯 : TopologicalSpace A)    (p : A) (hp : IsStarProjection p) : TopologicalSpace (Corner p hp) :=  TopologicalSpace.induced (fun x : Corner p hp => (x : A)) 𝒯def normalizedCornerTrace (τ : A → ℂ) (t : ℝ) (p : A) (hp : IsStarProjection p)    (x : Corner p hp) : ℂ := (t : ℂ)⁻¹ * τ xenddef stabilizedRightTranslations (G : Type u_5) [Group G] : Set (GroupOperator (ℕ × G)) :=  {T | ∃ g : G, ∀ (ξ : GroupHilbert (ℕ × G)) (x : ℕ × G),    T ξ x = ξ (x.1,x.2*g)}def stabilization (G : Type u_6) [Group G] : StarSubalgebra ℂ (GroupOperator (ℕ × G)) :=  StarSubalgebra.centralizer ℂ (stabilizedRightTranslations G)instance stabilization_isClosed (G : Type u_7) [Group G] :    IsClosed (stabilization G : Set (GroupOperator (ℕ × G))) := by  rw [stabilization,StarSubalgebra.coe_centralizer]  exact Set.isClosed_centralizer _instance stabilization_CStarAlgebra (G : Type u_8) [Group G] : CStarAlgebra (stabilization G) :=  inferInstancedef stabilizedProjectionTrace {G : Type u_9} [Group G] (p : stabilization G) : ℝ≥0∞ := by  classical  exact ∑' k : ℕ, ENNReal.ofReal (‖(p : GroupOperator (ℕ × G))    (lp.single 2 (k,1) (1 : ℂ))‖ ^ 2)def stabilizedTrace {G : Type u_10} [Group G] (T : stabilization G) : ℂ := by  classical  exact ∑' k : ℕ, inner ℂ (lp.single 2 (k,1) (1 : ℂ))    ((T : GroupOperator (ℕ × G)) (lp.single 2 (k,1) (1 : ℂ)))@[instance_reducible] def stabilizationTopology (G : Type u_11) [Group G] :    TopologicalSpace (stabilization G) :=  TopologicalSpace.induced (fun T : stabilization G => (T : GroupOperator (ℕ × G)))    (ultraweakOperatorTopology (ℕ × G))sectionvariable {A : Type u_42} {B : Type u_43} [CStarAlgebra A] [CStarAlgebra B]structure NormalTracialEquiv (τA : A → ℂ) (τB : B → ℂ)    (𝒯A : TopologicalSpace A) (𝒯B : TopologicalSpace B) where  equiv : A ≃⋆ₐ[ℂ] B  trace_eq : ∀x,τB (equiv x)=τA x  normal : @Continuous _ _ 𝒯A 𝒯B equiv  inverse_normal : @Continuous _ _ 𝒯B 𝒯A equiv.symmendend FreeGroupFactorMain.Amplificationnamespace FreeGroupFactorMain.Interpolationopen Amplificationopen scoped ENNRealabbrev RankTwoProjection := {p : stabilization (FreeGroupRank 2) // IsStarProjection p}instance rankTwoProjectionInhabited : Inhabited RankTwoProjection :=  ⟨⟨0,IsStarProjection.zero _⟩⟩def selectedProjection (a : ℝ) : RankTwoProjection :=  Classical.epsilon (fun p : RankTwoProjection => stabilizedProjectionTrace p.val=ENNReal.ofReal a)def interpolationScale (r : ℝ) : ℝ := (Real.sqrt (r-1))⁻¹inductive Model where  | integer (n : ℕ)  | continuous (r : ℝ)  | infinitydef Model.Algebra : Model → Type  | .integer n => groupVonNeumann (FreeGroupRank n)  | .continuous r => Corner (selectedProjection (interpolationScale r)).val      (selectedProjection (interpolationScale r)).property  | .infinity => groupVonNeumann (FreeGroup ℕ)instance modelCStarAlgebra (m : Model) : CStarAlgebra m.Algebra := by  cases m <;> dsimp only [Model.Algebra] <;> infer_instancedef Model.trace : (m : Model) → m.Algebra → ℂ  | .integer _ => canonicalTrace  | .continuous r => normalizedCornerTrace stabilizedTrace (interpolationScale r)      (selectedProjection (interpolationScale r)).val (selectedProjection (interpolationScale r)).property  | .infinity => canonicalTrace@[instance_reducible] def Model.topology : (m : Model) → TopologicalSpace m.Algebra  | .integer n => groupUltraweakTopology (FreeGroupRank n)  | .continuous r => cornerTopology (stabilizationTopology (FreeGroupRank 2))      (selectedProjection (interpolationScale r)).val (selectedProjection (interpolationScale r)).property  | .infinity => groupUltraweakTopology (FreeGroup ℕ)def parameterModel (r : ℝ≥0∞) : Model := by  classical  exact if r=⊤ then .infinity else    if h : ∃n : ℕ,r=(n:ℝ≥0∞) then .integer h.choose else .continuous r.toRealabbrev InterpolatedFactor (r : ℝ≥0∞) := (parameterModel r).Algebraabbrev interpolatedTrace (r : ℝ≥0∞) := (parameterModel r).traceabbrev interpolatedTopology (r : ℝ≥0∞) := (parameterModel r).topologytheorem allInterpolatedIsomorphic (r s : ℝ≥0∞) (hr : 1<r) (hs : 1<s) :    Nonempty (NormalTracialEquiv (interpolatedTrace r) (interpolatedTrace s)      (interpolatedTopology r) (interpolatedTopology s)) := by  sorryend FreeGroupFactorMain.Interpolationendend 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.

An isomorphism of the free group factors

Theorem (Isomorphism of consecutive ranks)

For every integer n\geq3 there is a unital trace-preserving normal *-isomorphism

L(\F_n)\cong L(\F_{n+1}).

Section 1, Introduction Read it in the paper TeX source, line 64

Show the TeX

For every integer $n\geq3$ there is a unital trace-preserving normal
$*$-isomorphism
\[
 L(\F_n)\cong L(\F_{n+1}).
\]

Theorem

There exists a unital normal trace-preserving *-isomorphism

\Phi:L(\F_2)\longrightarrow L(\F_3).

Section 1, Introduction Read it in the paper TeX source, line 74

Show the TeX

There exists a unital normal trace-preserving $*$-isomorphism
\[
 \Phi:L(\F_2)\longrightarrow L(\F_3).
\]

Configuration and file facts

File: lean/ComparatorChallenges/InterpolatedFactors.lean
OpenAI’s Lean catalogue: OpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 287 link it.
Solution module: OAI.Analysis.InterpolatedFactors.Interpolation
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: 41 local vocabulary declarations
Lines of Lean, without comments or blank lines: 139
Second proof-verification program (nanoda): Off in OpenAI's configuration.

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