Unitary vertex operator algebras and conformal nets
OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this one:
Proves the strongly rational case of the strong-locality conjecture: every simple unitary strongly rational complex vertex operator algebra generates a completely rational conformal net. Its simple modules are unitarizable, and its representation category agrees with the net’s finite-index sectors as a braided unitary tensor category. (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
Possible match to OpenAI’s description, selected from the papers’ main statements because it shares the most wording. No review confirming the selection is recorded.
Let V be a simple unitary strongly rational vertex operator algebra over \mathbb C. Then:
(1)
V has polynomial energy bounds and is CKLW-strongly local. The algebras (1.3) form an irreducible conformal net A_V.
(2)
V is completely unitary: every simple grading-restricted module is unitarizable, and all canonical invariant Hermitian fusion forms are positive definite.
(3)
Every unitary grading-restricted finite-length module is strongly integrable, with the closed-field equality (1.4).
(4)
The net A_V is completely rational.
(5)
After choosing invariant positive forms on the simple modules, the Carpi–Weiner–Xu functor gives a braided unitary tensor equivalence
Every simple unitary strongly rational vertex operator algebra generates a completely rational conformal net. All its simple grading-restricted modules are unitarizable, its canonical fusion forms are positive, and the Carpi–Weiner–Xu functor gives a braided unitary tensor equivalence from its grading-restricted finite-length module category onto the finite-index sectors of the net. Gui’s extension theorems then identify normalized irreducible finite-index local extensions of these nets with simple CFT-type conformal extensions of the vertex operator algebras.
One Lean statement in this family
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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.
Let V be a simple unitary strongly rational vertex operator algebra over \mathbb C. Then:
(1)
V has polynomial energy bounds and is CKLW-strongly local. The algebras (1.3) form an irreducible conformal net A_V.
(2)
V is completely unitary: every simple grading-restricted module is unitarizable, and all canonical invariant Hermitian fusion forms are positive definite.
(3)
Every unitary grading-restricted finite-length module is strongly integrable, with the closed-field equality (1.4).
(4)
The net A_V is completely rational.
(5)
After choosing invariant positive forms on the simple modules, the Carpi–Weiner–Xu functor gives a braided unitary tensor equivalence
This 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.
three independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet; Neal McBurnett, openai-math-lean-tooling. Who ran the proof-verification program
cert-machine records it as “CERTIFIED”.
MathVet records it as “comparator-sandboxed-pass”.
Neal McBurnett, openai-math-lean-tooling records it as “accepted”.
Independent comparisons and proof-verification reports
Does the Lean express the advertised result?
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:MinimalVertex.main (hSimple : A.toVertexAlgebra.IsSimple) (hSR : A.IsStronglyRational) : A.PolynomialEnergyBounds ∧ CKLWStrongLocal U _ ∧ Nonempty (IrreducibleConformalNetStructure (intervalAlgebra U _)): polynomial energy bounds, strong locality (intervalAlgebra I ≤ (intervalAlgebra I.complement).commutant) and an irreducible conformal net (isotony, locality, Möbius and Diff(S¹) covariance, unique vacuum, positive energy) for a simple unitary strongly rational VOA, i.e. the strong-locality core of the summary. Not stated: complete rationality of the net (the summary says it generates a COMPLETELY RATIONAL net), unitarizability of all simple modules, and the braided unitary tensor equivalence of Rep(V) with the net’s finite-index sectors (docs: 'does not include complete rationality... unitarizability... the braided tensor equivalence').
Definitions the group asks readers to compare with the paper:Entire VOA/conformal-net stack is hand-built (VertexAlgebra, CFTTypeVOA, IsStronglyRational := IsSelfContragredient ∧ IsRational ∧ C2Cofinite, smearedMap, closedSmoothField, localAlgebra, IrreducibleConformalNetStructure), with many definitions using junk defaults: smearedMap := if h : ∀ v, Summable (..) then tsumMap else 0, smoothSmearedField := if h :.. then.. else 0, resolvent/flow := if h : ∃ L,.. then h.choose else 0, projectiveStandardAction := if h : ∃ L,.. then h.choose else 1. Not Mathlib notions; fidelity rests on these encodings.
“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.
The linked formalization proves the first construction step relating strongly rational unitary vertex operator algebras to conformal nets. For a simple unitary strongly rational vertex operator algebra, it establishes polynomial energy bounds and strong locality and constructs an irreducible conformal net with the stated covariance, vacuum, and positive-energy properties.
The selected statement does not include complete rationality of the net, unitarizability of all simple modules, the braided tensor equivalence, or the classification of local extensions described in the paper.
AI-drafted comparisons, with no recorded human review
Energy bounds, strong locality, and an irreducible conformal netVertexAlgebraNet
Unreviewed AI draft: states part of a paper’s theorem. States the first parts of the main theorem: polynomial energy bounds, strong locality and an irreducible conformal net for a simple unitary strongly rational vertex operator algebra. Complete rationality, unitarisability of modules, the braided equivalence and the classification of extensions are outside it.
Who ran the proof-verification program?
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.
Energy bounds, strong locality, and an irreducible conformal netVertexAlgebraNet
Three 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.
Explanations of this family written elsewhere, by the people named. Each is their own reading of the mathematics.
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.
What these papers cite in the release, and what cites them
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.
Papers elsewhere in the release that this family’s papers cite
This family’s papers cite no paper elsewhere in the release.
Papers elsewhere in the release that cite this family’s papers
No paper elsewhere in the release cites this family’s papers.
References outside the release
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
Explain a result, compare statements or propose a correction
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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 proof and rates its significance, if it holds up, as “Solid publication, could lead to tenure”. The page says that nothing on it is verified.
Each item identifies a statement, definition or report that needs further review. Its link opens a GitHub form with the source version already recorded.
Read the Lean statement against the paper
An AI agent drafted an assessment of which part of the paper this Lean statement expresses. No named human review is recorded.Open the form
Identify the theorem the catalogue describes
Family 280The theorem shown was selected from shared wording with OpenAI’s description. Atlas records no review confirming that it is the theorem the description means.Open the form
Address a question raised in an independent assessment
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. Its protocol describes a planned error-rate study; this import does not establish that the study has been completed.
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; 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…
Re-run through Comparator in a sandboxed virtual machine: accepted.
Date
8 October 2026
Version
at the release’s recorded version
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.
Re-run through Comparator with Lean’s kernel: accepted (the check took 177 s).
Date
10 October 2026
Version
at the release’s recorded version
Their method
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.
Energy bounds, strong locality, and an irreducible conformal net
Role
the release lists it as a supporting statement
Paired with
Theorem 1.1 (candidate pairing)
Who ran the proof-verification program
three independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet; Neal McBurnett, openai-math-lean-tooling.