Constructs an entangled state on \(\mathbb C^{10}\otimes\mathbb C^{10}\) with zero distillable secret key for the specified local-instrument protocols that complete almost surely. These allow joint local processing and authenticated two-way public communication, with no other shared private resource and an eavesdropper holding the input purification and public record. A trace-preserving PPT channel on \(M_{21}(\mathbb C)\) whose square is not entanglement breaking disproves Christandl’s PPT-square conjecture. (OpenAI’s Lean note)
Lean is a language for mathematical statements and proofs that a computer can verify. The assessments below ask whether those statements express the result OpenAI describes here.
The independent assessments find that the Lean statements express only part of the result described above, or a narrower statement. Read the assessments
In Entanglement with zero distillable secret key in local dimension ten, 27 September 2026
Possible match to OpenAI’s description, selected from shared wording. No review confirming the selection is recorded.
There is an explicitly specified trace-preserving PPT channel \Theta:\Mat{21}\to\Mat{21} whose square \Theta\circ\Theta is not entanglement breaking.
In Entanglement with zero distillable secret key in local dimension ten, 27 September 2026
Possible match to OpenAI’s description, selected from shared wording. No review confirming the selection is recorded.
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 4.1 on \rho^{\otimes n}, where E' retains an initial purification and the complete public transcript. For every state \sigma_{E'},
Thus the corresponding trace distance, with its conventional factor 1/2, is at least 1/10.
No Lean statement in the release is paired with this theorem.
The release does not say which paper theorem its catalogue describes. The statements above share the most wording with the description; the atlas has not reviewed that choice. Identifying it is listed under Questions awaiting review.
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For a configured theorem, the Comparator program tests whether OpenAI’s separate proof proves the exact Lean statement using only the allowed axioms. Some statement files instead configure definitions for comparison by name and type. Acceptance does not establish that a formal statement means the same thing as the paper’s theorem.
DimensionTenChannelThe paper states
In Entanglement with zero distillable secret key in local dimension ten, 27 September 2026
There is an explicitly specified trace-preserving PPT channel \Theta:\Mat{21}\to\Mat{21} whose square \Theta\circ\Theta is not entanglement breaking.
The words they share: square, channel, preserving, breaking, entanglement, trace, ppt, mat, dimension, whose.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
DimensionTenChannel.lean, the release lists it as a supporting statement.
theorem exists_channel_fin21 : ∃ Θ : ChannelCompletion.Map (Fin 21) (Fin 21), ChannelCompletion.PPT Θ ∧ ChannelCompletion.TracePreserving Θ ∧ ¬ ChannelCompletion.EntanglementBreaking (Θ.comp Θ) := by sorryThis file states the theorem with sorry, a placeholder, in place of its proof. OpenAI supplies a separate proof file. Comparator tests whether that proof proves the statement here.
two independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet. Who ran the proof-verification program
DimensionTenPairThe paper states
In Entanglement with zero distillable secret key in local dimension ten, 27 September 2026
There are explicitly specified PPT maps \Phi_1,\Phi_2:\Mat{10}\to\Mat{10} for which
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.
The words they share: breaking, composition, entanglement, explicitly, maps, nonzero, product, range, specified, vector.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
DimensionTenPair.lean, the release lists it as a supporting statement.
theorem main_pair : ∃ Φ₁ Φ₂ : Mat 10 → Mat 10, Φ₁ = phiOne ∧ Φ₂ = phiTwo ∧ PPT Φ₁ ∧ PPT Φ₂ ∧ (let Z := choi (Φ₂ ∘ Φ₁) Z ≠ 0 ∧ (∀ u v : Fin 10 → ℂ, ∀ w : Fin 10 × Fin 10 → ℂ, Z *ᵥ w = productVector u v → u = 0 ∨ v = 0) ∧ ¬ EntanglementBreaking (Φ₂ ∘ Φ₁)) := by sorryThis file states the theorem with sorry, a placeholder, in place of its proof. OpenAI supplies a separate proof file. Comparator tests whether that proof proves the statement here.
two independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet. Who ran the proof-verification program
Neither group finds that the Lean statements express or imply the whole advertised result. They describe what the statements express in different terms.
MathVet and Carlos Toledo’s cert-machine independently assessed whether this family’s Lean statements express the whole result OpenAI describes in its catalogue. These are assessments of the family as a whole, not of each paper theorem separately.
MathVet finds that the Lean statements express part of the advertised result (their term: “partial”).
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’).
“partial”, in their words: Only a special case is stated, or a co-equal headline claim has no challenge statement.
Their method: An independent, model-assisted, pre-referee reading of each family’s challenge statements against its headline claim under a published rubric (PROTOCOL.md), with its own Comparator runs per challenge.
cert-machine finds that the Lean statements express a narrower statement than the advertised result (their term: “NARROWER — DECLARED”).
The gap they name: The zero-distillable-secret-key state (the headline’s lead claim).
“NARROWER — DECLARED”, in their words: The challenge statements state less than the headline, and OpenAI’s Lean note says so.
Their method: Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule.
OpenAI’s own note on what its Lean covers
The paper’s entanglement construction yields counterexamples to PPT-composition claims. The linked formalization covers these consequences: two explicitly specified PPT maps on $10\times10$ complex matrices have a composition that is not entanglement breaking, and the nonzero Choi matrix of that composition has no nonzero product vector in its range. A separate trace-preserving PPT channel on $21\times21$ matrices has a square that is not entanglement breaking.
The paper’s zero-distillable-secret-key statement is outside these two selected Comparator statements.
PPT channel on dimension 21 whose square is not entanglement breaking DimensionTenChannel
Unreviewed AI draft: states a paper’s theorem as the paper does. States the paper’s channel theorem: a trace-preserving PPT channel on 21 × 21 matrices whose square is not entanglement breaking. The note presents it as a consequence of the paper’s construction; the zero-distillable-key theorem is outside it.
Dimension-ten PPT pair with non-entanglement-breaking composition DimensionTenPair
Unreviewed AI draft: states a paper’s theorem as the paper does. States the paper’s pair theorem: two explicit PPT maps on 10 × 10 matrices whose composition has a nonzero Choi matrix with no product vector in its range, so it is not entanglement breaking.
These groups report running Comparator with the release’s Lean statement files. For a configured theorem, it verifies OpenAI’s separate proof against the exact statement; some files instead configure definitions for comparison by name and type. Each row is the named group’s report. Atlas has not repeated the runs.
These reports do not list the software versions, proof-verification programs, axioms, time or memory in separate fields. Any such details in a report’s prose are reproduced below.
DimensionTenChannelTwo reports, all at the release’s recorded version.
| Reported by | Their word | Their account | Date | Original report |
|---|---|---|---|---|
| cert-machine | “CERTIFIED” | Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution. | 10 October 2026 | Original report |
| MathVet | “comparator-sandboxed-pass” | Re-run through Comparator in a sandboxed virtual machine: accepted. | 8 October 2026 | Original report |
DimensionTenPairTwo reports, all at the release’s recorded version.
| Reported by | Their word | Their account | Date | Original report |
|---|---|---|---|---|
| cert-machine | “CERTIFIED” | Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution. | 10 October 2026 | Original report |
| MathVet | “comparator-sandboxed-pass” | Re-run through Comparator in a sandboxed virtual machine: accepted. | 8 October 2026 | Original report |
Other people’s accounts of running proof-verification software, with links to their reports.
Pico, openai-math-counterexamples: do the counterexamples in openai/math hold when someone else computes them?, 9 October 2026. Source
The explicit 4x4 PPT pencil checks exactly and no quadric passes through ten of the 184,756 point subsets.
These reports examine claims in the papers using exact calculations for finite claims, independently of the release’s Lean proofs. Read each report for the exact claim it covers.
Entanglement with zero distillable secret key in local dimension ten, 27 September 2026
Explanations of this family written elsewhere, by the people named. Each is their own reading of the mathematics.
seaworld008, seaworld008/openai-math-top30: thirty selected families explained in Chinese, 7 October 2026. Source
Chinese-language explainer for general readers, with a beginner’s section, a specialist section, background on why the problem was hard and a diagram; it ends by saying it paraphrases OpenAI’s summary and does not represent consensus.
Yixin Ma, OpenAI mathematical physics, 25 families: a Chinese guided reading, 7 October 2026. Source
Chinese guided reading of this family that compares the manuscripts’ main theorems with their abstracts and flags boundary conditions and limit orders that change the physical reading; drafted with Codex and not a re-proof.
These links come from the papers’ bibliographies. A citation records that one paper cites another; how each cited result is used has not been examined here.
This family’s papers cite no paper elsewhere in the release.
No paper elsewhere in the release cites this family’s papers.
Each paper’s outside references are listed with it on its reading page, where each citation is placed. Every outside work cited in the release: References outside the release
Atlas has no contributed explanation for this family yet. You can offer one with a GitHub account. You receive named credit and disclose any AI assistance.
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Offer to write an explanationCompare one Lean statement with its paper: DimensionTenChannel, DimensionTenPairSuggest a source attributionSuggest a change
Commentary on this family by others, dated and credited. It is commentary, not a finding.
An unsigned public tier list, the OpenAI Math Tier List, classes it as a counterexample and rates its significance, if it holds up, as “Solid publication, could lead to tenure”. The page says that nothing on it is verified.
Author not stated. Source
Each item identifies a statement, definition or report that needs further review. Its link opens a GitHub form with the source version already recorded.
Why this passage is shown
Theorem 1.3
In Entanglement with zero distillable secret key in local dimension ten, 27 September 2026
Why this passage is shown
Theorem 1.1
In Entanglement with zero distillable secret key in local dimension ten, 27 September 2026
Why this passage is shown
Theorem 1.2
In Entanglement with zero distillable secret key in local dimension ten, 27 September 2026
Independent assessment
MathVet
14c7f91Independent assessment
Carlos Toledo’s cert-machine
9814e4cProof-verification report
cert-machine, DimensionTenChannel
Proof-verification report
MathVet, DimensionTenChannel
Proof-verification report
cert-machine, DimensionTenPair
Proof-verification report
MathVet, DimensionTenPair
Report using another proof or calculation method
Entanglement with zero distillable secret key in local dimension ten, 27 September 2026
Exact calculations for finite claims, cert-machine
9814e4cLean statement
DimensionTenChannel
Lean statement
DimensionTenPair