OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this one:
A splitting of the tangent bundle of a compact Kähler manifold into two integrable holomorphic subbundles induces a compatible product decomposition of its universal cover, proving the two-summand form of Beauville’s splitting conjecture. On smooth rationally connected projective manifolds, both summands are automatically integrable, establishing Höring’s conjecture and the corresponding product decomposition. (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 two independent assessments disagree on how much of the result described above is expressed by the Lean statements.Read the assessments
The paper’s main theorem is shown as a possible match to OpenAI’s description. No review confirming the selection is recorded.
Let X be a compact connected Kähler manifold of complex dimension n\geq2, and let
T_X=E_1\oplus E_2
be a specified decomposition into integrable holomorphic subbundles of positive ranks r_1,r_2. If \pi:\widetilde X\to X is the ordinary universal covering map, there are connected simply connected complex manifolds Y_1,Y_2 and a biholomorphism \Phi:\widetilde X\to Y_1\times Y_2 such that
\dd\Phi(\pi^*E_i)=\pr_i^*T_{Y_i}\qquad(i=1,2),
where \dd\pi identifies T_{\widetilde X} with \pi^*T_X. In particular, \dim_\C Y_i=r_i.
Possible match to OpenAI’s description, selected because it shares the most wording. No review confirming the selection is recorded.
Let X be a smooth connected projective complex manifold of dimension n\geq2 with nef and big canonical bundle K_X. Every specified holomorphic splitting T_X=E_1\oplus E_2 into positive-rank subbundles is integrable and induces a compatible product decomposition of the ordinary universal cover as in Theorem 1.1.
No Lean statement in the release is paired with this theorem.
The release does not say which paper theorem its catalogue describes. Both statements above are shown whole: the paper’s main theorem first, then the statement that shares the most wording with the description. The atlas has not reviewed either choice.
Three more possible matches
Corollary 1.2(Non-uniruled projective manifolds)in Universal-cover splitting for compact Kähler manifolds, 23 September 2026.
Theorem 1.1(Automatic integrability)in Integrability of split tangent bundles on rationally connected manifolds, 23 September 2026.
Corollary 1.2(Compatible product)in Integrability of split tangent bundles on rationally connected manifolds, 23 September 2026.
Reported proof verification
Two Lean statements in the release; independent groups report that Comparator accepted runs for all these files at the version shown here.
We prove the two-summand form of Beauville’s compatible splitting conjecture. If the tangent bundle of a compact connected Kähler manifold decomposes into two integrable holomorphic subbundles of positive rank, then its ordinary universal cover admits a product decomposition whose factor tangent bundles are the lifted specified summands.
We prove that both summands of every specified holomorphic splitting of the tangent bundle of a smooth rationally connected projective complex manifold into two positive-rank subbundles are integrable. This proves Höring’s conjecture on rationally connected projective manifolds. Höring’s product theorem then gives a product decomposition compatible with the specified splitting.
Two Lean statements in this family
Choose one to see it beside the paper theorem it is paired with.
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 X be a compact connected Kähler manifold of complex dimension n\geq2, and let
T_X=E_1\oplus E_2
be a specified decomposition into integrable holomorphic subbundles of positive ranks r_1,r_2. If \pi:\widetilde X\to X is the ordinary universal covering map, there are connected simply connected complex manifolds Y_1,Y_2 and a biholomorphism \Phi:\widetilde X\to Y_1\times Y_2 such that
\dd\Phi(\pi^*E_i)=\pr_i^*T_{Y_i}\qquad(i=1,2),
where \dd\pi identifies T_{\widetilde X} with \pi^*T_X. In particular, \dim_\C Y_i=r_i.
The words they share: pi, compact, biholomorphism, simply, universal, covering, ordinary, ranks, positive, integrable, manifolds, complex, connected, map, holomorphic, splitting, identifies, dimension, subbundles.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
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”.
Let X be a smooth connected projective complex manifold of dimension at least two. Suppose that two general points of X lie on the image of a morphism \PP^1\to X. For every specified holomorphic decomposition
T_X=E_1\oplus E_2
into subbundles of positive rank, both E_1 and E_2 are integrable.
The words they share: points, positive, integrable, projective, complex, manifold, connected, holomorphic, rationally, bundles, tangent, two, least, rank, dimension.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
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”.
Other mathematical statements in OpenAI’s Lean library
Atlas found these propositions by scanning the text of the project’s Model.lean file. Each name links to its definition, with any associated statement file beside it. This list does not record running Lean to interpret or verify those definitions.
Independent comparisons and proof-verification reports
Does the Lean express the advertised result?
MathVet and cert-machine disagree on how much of the advertised result the Lean statements express. MathVet finds that the Lean statements express or imply the whole advertised result (their term: “full”); cert-machine finds that the Lean statements express a narrower statement than the advertised result (their term: “NARROWER — DECLARED”).
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 or imply the whole advertised result (their term: “full”).
Their note:Both headline theorems are stated; the ‘corresponding product decomposition’ for rationally connected manifolds (composition of the two, needing a Kahler metric from the embedding) is not a single statement. Complex manifold, Kahler metric, integrability and rational connectedness are all bespoke encodings.
Definitions the group asks readers to compare with the paper:Bespoke ComplexManifold/KahlerMetric (closedness of the fundamental form via a cyclic derivative sum in charts), Integrable (Frobenius condition through chartwise Lie brackets of holomorphic local sections), OrdinaryUniversalCover, CompatibleProduct, ProjectiveEmbedding (injective holomorphic immersion into P^N) and RationallyConnected (dense Zariski-open set of pairs joined by a RationalCurve); plausible but unchecked against Mathlib/standard notions
“full”, in their words: The challenge statements imply the headline claim after unpacking definitions or one routine step.
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 product decomposition for rationally connected projective manifolds (the last clause) is not stated; neither are the Kahler paper’s further corollaries.
“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.
The splitting question asks whether a holomorphic decomposition of the tangent bundle comes from a product decomposition of the universal cover. The formalized result gives an affirmative answer for a compact connected Kähler manifold whose tangent bundle splits into two positive-rank, integrable holomorphic subbundles. The universal cover is a product of connected simply connected complex manifolds of the prescribed dimensions, and the differential identifies the two tangent factors with the original summands.
Both integrability assumptions are required. Automatic integrability and the paper’s additional corollaries are not included.
The formalization proves integrability of both summands in a holomorphic splitting of the tangent bundle of a rationally connected projective manifold. More precisely, for a compact connected complex manifold of dimension at least two with a projective embedding witnessing rational connectedness, each positive-rank summand of a holomorphic tangent-bundle splitting is integrable. The paper’s compatible product decomposition is a separate consequence, outside this selected statement.
AI-drafted comparisons, with no recorded human review
Universal-cover product splittingKahlerSplitting
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: for a compact Kähler manifold whose tangent bundle splits into two integrable holomorphic subbundles, the universal cover is a product matching the splitting. The paper’s corollaries are not included.
Integrability of both holomorphic tangent summandsSplitTangentIntegrability
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem of the integrability paper: on a rationally connected projective manifold, both summands of a holomorphic tangent splitting are integrable. The product-decomposition corollary is outside the statement.
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.
Universal-cover product splittingKahlerSplitting
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.
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
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.
Each button opens a GitHub form with this page and its source version filled in. Opening the form does not reserve the work.
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
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 052The 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
The challenge statements imply the headline claim after unpacking definitions or one routine step.
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 216 s; peak memory 6.5 GiB).
Date
11 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.
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 471 s; peak memory 6.5 GiB).
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.
three independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet; Neal McBurnett, openai-math-lean-tooling.
Integrability of both holomorphic tangent summands
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.