Lean statement

PPT channel on dimension 21 whose square is not entanglement breaking

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.DimensionTen.exists_channel_fin21

Lean statement file: DimensionTenChannel.leanShow on the mapOpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 272 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 exists_channel_fin21 :    ∃ Θ : ChannelCompletion.Map (Fin 21) (Fin 21),      ChannelCompletion.PPT Θ ∧ ChannelCompletion.TracePreserving Θ ∧        ¬ ChannelCompletion.EntanglementBreaking (Θ.comp Θ) := 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.ChannelCompletion.Mat line 11
```
abbrev Mat (n : Type) := Matrix n n ℂ
```

abbreviation OAI.ChannelCompletion.Map line 12

abbrev Map (n m : Type) := Mat n →ₗ[ℂ] Mat m

definition OAI.ChannelCompletion.amplify line 16

def amplify (F : Map n m) {k : Type} (X : Mat (k × n)) : Mat (k × m) :=  fun a b => F (fun i j => X (a.1, i) (b.1, j)) a.2 b.2

definition OAI.ChannelCompletion.CP line 19

def CP (F : Map n m) : Prop :=  ∀ (k : Type) [Fintype k] (X : Mat (k × n)), X.PosSemidef → (amplify F X).PosSemidef

definition OAI.ChannelCompletion.transposeMap line 22

def transposeMap : Map n n where  toFun := Matrix.transpose  map_add' _ _ := rfl  map_smul' _ _ := rfl

definition OAI.ChannelCompletion.PPT line 27

def PPT (F : Map n m) : Prop := CP F ∧ CP (transposeMap.comp F)

definition OAI.ChannelCompletion.TracePreserving line 29

def TracePreserving (F : Map n m) : Prop := ∀ X, Matrix.trace (F X) = Matrix.trace X

definition OAI.ChannelCompletion.Separable line 31

def Separable (Z : Mat (n × m)) : Prop :=  ∃ (r : ℕ) (A : Fin r → Mat n) (B : Fin r → Mat m),    (∀ i, (A i).PosSemidef) ∧ (∀ i, (B i).PosSemidef) ∧    Z = ∑ i, A i ⊗ₖ B i

definition OAI.ChannelCompletion.EntanglementBreaking line 36

def EntanglementBreaking (F : Map n m) : Prop :=  CP F ∧ ∀ (k : Type) [Fintype k] (X : Mat (k × n)),    X.PosSemidef → Separable (amplify F X)

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: square, channel, preserving, breaking, entanglement, trace, ppt and mat. Found in the paper’s title, which its main theorem states: dimension. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: whose.

Theorem 1.3 of Entanglement with zero distillable secret key in local dimension ten

Theorem

There is an explicitly specified trace-preserving PPT channel \Theta:\Mat{21}\to\Mat{21} whose square \Theta\circ\Theta is not entanglement breaking.

Section 1, Introduction Read it in the paper TeX source, line 101

Show the TeX

There is an explicitly specified trace-preserving PPT channel
$\Theta:\Mat{21}\to\Mat{21}$ whose square $\Theta\circ\Theta$ is not
entanglement breaking.

Other possible paper matches, closest wording first

Theorem 10.5 of Entanglement with zero distillable secret key in local dimension ten. Words of the Lean statement, its names split into words, that also appear in the theorem: completion, finite, matrix, square, map, preserving and six more.
Proposition 8.1 of Entanglement with zero distillable secret key in local dimension ten. Words of the Lean statement, its names split into words, that also appear in the theorem: maps, preserving, channel, breaking, entanglement, trace and two more.
Theorem 1.2 of Entanglement with zero distillable secret key in local dimension ten. Words of the Lean statement, its names split into words, that also appear in the theorem: maps, breaking, entanglement, ppt and mat. Found in the paper’s title, which its main theorem states: dimension.
Corollary 9.2 of Entanglement with zero distillable secret key in local dimension ten. Words of the Lean statement, its names split into words, that also appear in the theorem: complex, one, preserving, channel, trace, mat and one more.
Proposition 6.1 of Entanglement with zero distillable secret key in local dimension ten. Words of the Lean statement, its names split into words, that also appear in the theorem: twenty, maps, breaking, entanglement, ppt and mat.
Proposition 5.4 of Entanglement with zero distillable secret key in local dimension ten. Words of the Lean statement, its names split into words, that also appear in the theorem: non, one, channel, preserve and trace. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: whose.

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 272’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.

Neither group finds that the Lean statements express or imply the whole advertised result. They describe what the statements express in different terms.

MathVet
Finds part of it stated, or a narrower or weaker statement Their term: “partial”.
Only a special case is stated, or a co-equal headline claim has no challenge statement. Their record
Their note: PRIMARY CLAIM NOT STATED (only a secondary result is formalized). Stated: DimensionTenChannel.exists_channel_fin21 : ∃ Θ : Map (Fin 21) (Fin 21), PPT Θ ∧ TracePreserving Θ ∧ ¬ EntanglementBreaking (Θ.comp Θ) (the PPT-square counterexample with PPT F := CP F ∧ CP (T ∘ F) and EntanglementBreaking via separability of every amplification) and DimensionTenPair.main_pair (two explicit PPT maps on 10x10 matrices whose composition is not entanglement breaking and whose Choi matrix has no nonzero product vector in its range). Not stated: the summary's first claim, an entangled state on ℂ¹⁰⊗ℂ¹⁰ with zero distillable secret key under the specified local-instrument/authenticated-communication protocols (docs: 'outside these two selected Comparator statements').
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 zero-distillable-secret-key state (the headline's lead claim).

Read the complete assessments on the family page.

OpenAI’s own note on what this Lean statement covers

“A separate trace-preserving PPT channel on $21\times21$ matrices has a square that is not entanglement breaking.”
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

55 lines, copied from release version fd4aeeb. A name defined in this file links to its line; pointing at the name shows its definition.

Declared in this fileOpenAI’s library at fd4aeebOther linked names open the current documentation for Mathlib, Lean’s mathematical library.

Links in the file: 26 to its own declarations, 15 to Mathlib’s documentation. 10 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.

Definitions and lemmas named by the statement

If Comparator accepts a proof, it proves exists_channel_fin21 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 all 9 other declarations of the file; with the theorem itself they span 24 lines. 4 of them are named in the statement itself.

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

abbrev Matline 11library
abbrev Mapline 12in the statementlibrary
def amplifyline 16library
def CPline 19library
def transposeMapline 22library
def PPTline 27in the statementlibrary
def TracePreservingline 29in the statementlibrary
def Separableline 31library
def EntanglementBreakingline 36in the statementlibrary
theorem OAI.DimensionTen.exists_channel_fin21line 45selected theoremlibrary

import Mathlibnamespace OAInoncomputable sectionopen scoped BigOperators ComplexOrder Kronecker MatrixOrderopen Matrixnamespace ChannelCompletionabbrev Mat (n : Type) := Matrix n n ℂabbrev Map (n m : Type) := Mat n →ₗ[ℂ] Mat mvariable {n m p : Type} [Fintype n] [Fintype m] [Fintype p]def amplify (F : Map n m) {k : Type} (X : Mat (k × n)) : Mat (k × m) :=  fun a b => F (fun i j => X (a.1, i) (b.1, j)) a.2 b.2def CP (F : Map n m) : Prop :=  ∀ (k : Type) [Fintype k] (X : Mat (k × n)), X.PosSemidef → (amplify F X).PosSemidefdef transposeMap : Map n n where  toFun := Matrix.transpose  map_add' _ _ := rfl  map_smul' _ _ := rfldef PPT (F : Map n m) : Prop := CP F ∧ CP (transposeMap.comp F)def TracePreserving (F : Map n m) : Prop := ∀ X, Matrix.trace (F X) = Matrix.trace Xdef Separable (Z : Mat (n × m)) : Prop :=  ∃ (r : ℕ) (A : Fin r → Mat n) (B : Fin r → Mat m),    (∀ i, (A i).PosSemidef) ∧ (∀ i, (B i).PosSemidef) ∧    Z = ∑ i, A i ⊗ₖ B idef EntanglementBreaking (F : Map n m) : Prop :=  CP F ∧ ∀ (k : Type) [Fintype k] (X : Mat (k × n)),    X.PosSemidef → Separable (amplify F X)end ChannelCompletionnamespace DimensionTen/-- A trace-preserving PPT channel on twenty-one dimensions with a non-entanglement-breaking square. -/theorem exists_channel_fin21 :    ∃ Θ : ChannelCompletion.Map (Fin 21) (Fin 21),      ChannelCompletion.PPT Θ ∧ ChannelCompletion.TracePreserving Θ ∧        ¬ ChannelCompletion.EntanglementBreaking (Θ.comp Θ) := by  sorryend DimensionTenendend 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.

Entanglement with zero distillable secret key in local dimension ten

Theorem

There is an explicitly specified entangled density operator \rho on \C^{10}\otimes\C^{10} whose range contains no nonzero product vector and for which K_D(\rho)=0. More precisely, let n\geq1 and let \tau_{K_AK_BE'} be the normalized output of any admitted completed bit-output protocol of Definition def:protocol on \rho^{\otimes n}, where E' retains an initial purification and the complete public transcript. For every state \sigma_{E'},

\left\lVert \tau_{K_AK_BE'}- \frac12\sum_{i=0}^1|i,i\rangle\langle i,i|\otimes\sigma_{E'} \right\rVert_1\geq\frac15.

Thus the corresponding trace distance, with its conventional factor 1/2, is at least 1/10.

Section 1, Introduction Read it in the paper TeX source, line 45

Show the TeX

There is an explicitly specified entangled density operator $\rho$ on
$\C^{10}\otimes\C^{10}$ whose range contains no nonzero product vector and
for which $K_D(\rho)=0$.  More precisely, let $n\geq1$ and let
$\tau_{K_AK_BE'}$ be the normalized output of any admitted completed bit-output protocol of
Definition~\ref{def:protocol} on $\rho^{\otimes n}$, where $E'$ retains an
initial purification and the complete public transcript.  For every state
$\sigma_{E'}$,
\begin{equation}
 \left\lVert \tau_{K_AK_BE'}-
 \frac12\sum_{i=0}^1|i,i\rangle\langle i,i|\otimes\sigma_{E'}
 \right\rVert_1\geq\frac15.
\end{equation}
Thus the corresponding trace distance, with its conventional factor
$1/2$, is at least $1/10$.

Theorem

There are explicitly specified PPT maps \Phi_1,\Phi_2:\Mat{10}\to\Mat{10} for which

Z=J(\Phi_2\circ\Phi_1)

is nonzero and its range contains no nonzero product vector in \C^{10}\otimes\C^{10}. In particular, the composition \Phi_2\circ\Phi_1 is not entanglement breaking.

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

Show the TeX

There are explicitly specified PPT maps
$\Phi_1,\Phi_2:\Mat{10}\to\Mat{10}$ for which
\[
 Z=J(\Phi_2\circ\Phi_1)
\]
is nonzero and its range contains no nonzero product vector in
$\C^{10}\otimes\C^{10}$.  In particular, the composition
$\Phi_2\circ\Phi_1$ is not entanglement breaking.

Theorem

There is an explicitly specified trace-preserving PPT channel \Theta:\Mat{21}\to\Mat{21} whose square \Theta\circ\Theta is not entanglement breaking.

Section 1, Introduction Read it in the paper TeX source, line 101

Show the TeX

There is an explicitly specified trace-preserving PPT channel
$\Theta:\Mat{21}\to\Mat{21}$ whose square $\Theta\circ\Theta$ is not
entanglement breaking.

Configuration and file facts

File: lean/ComparatorChallenges/DimensionTenChannel.lean
OpenAI’s Lean catalogue: OpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 272 link it.
Solution module: OAI.Analysis.Quantum.DimensionTen.Channel
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: 9 local vocabulary declarations
Lines of Lean, without comments or blank lines: 24
Second proof-verification program (nanoda): Off in OpenAI's configuration.

A shareable summary of this page: the proposed paper match, what the Lean statement expresses, and which reviews are recorded.

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