Proves that the edges of every finite simple undirected graph on n vertices can be partitioned into at most \(Cn\) simple cycles and single edges, for an absolute constant C. This resolves the Erdős–Gallai cycle-decomposition conjecture, bounding the number of pieces linearly even for dense graphs. (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.
Both independent assessments find that the Lean statements express or imply the whole result described above. Read the assessments
It bears on Erdős problem #184, as listed by Thomas Bloom, erdosproblems.com.
In A linear cycle-and-edge decomposition of every graph, 24 September 2026
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.
There is an absolute constant C>0 such that, for every finite simple undirected graph G with n vertices, E(G) has a partition into at most Cn parts, each of which is the edge set of a cycle or consists of a single edge.
In A linear cycle-and-edge decomposition of every graph, 24 September 2026
Possible match to OpenAI’s description, selected from shared wording. No review confirming the selection is recorded.
Every finite simple undirected Eulerian graph on n vertices has an edge partition into at most Cn cycles, with C from Theorem 1.1.
For every fixed \delta,p>0, there exist \eps>0 and an integer n_0\ge1 such that every finite simple undirected Eulerian graph G on n\ge n_0 vertices has an edge partition into at most \Delta(G)/2+\delta n cycles, provided that every partition V(G)=A\mathbin{\dot\cup}B with |A|,|B|\ge\eps n satisfies
Here \Delta(G) denotes the maximum degree, and e_G(A,B) is the number of edges joining A to B.
No Lean statement in the release is paired with this theorem.
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CycleDecompositionThe paper states
In A linear cycle-and-edge decomposition of every graph, 24 September 2026
There is an absolute constant C>0 such that, for every finite simple undirected graph G with n vertices, E(G) has a partition into at most Cn parts, each of which is the edge set of a cycle or consists of a single edge.
The words they share: parts, single, cycle, graph, simple, edge, decomposition, cn, constant, absolute, vertices, finite.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
CycleDecomposition.lean, the release lists it as a main result.
def MainStatement : Prop := ∃ C : ℝ, 0 < C ∧ ∀ (n : ℕ) (G : SimpleGraph (Fin n)), ∃ k : ℕ, EdgeDecomposition G k ∧ (k : ℝ) ≤ C * ntheorem erdos_gallai : MainStatement := 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.
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
MathVet and cert-machine agree on how much of the advertised result the Lean statements express.
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: MainStatement : ∃ C > 0, ∀ n (G : SimpleGraph (Fin n)), ∃ k, EdgeDecomposition G k ∧ k ≤ C * n where EdgeDecomposition = pairwise-disjoint parts each CycleOrSingleEdge (edge set of a Mathlib IsCycle walk or {e}) covering G.edgeSet. Exactly the Erdős–Gallai conjecture with an absolute constant.
“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 or imply the whole advertised result (their term: “MATCHES”).
“MATCHES”, in their words: The challenge statements state the headline as OpenAI wrote it.
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 Erdős–Gallai cycle-decomposition conjecture asks for a linear bound on the number of cycles and single edges needed to partition a graph’s edges. The formalized result gives one absolute constant $C>0$ such that every finite simple graph on $n$ vertices has an edge-disjoint decomposition into at most $Cn$ cycles or singleton edges. Edgeless and small graphs are included. The optimal value of $C$ is not determined.
Linear cycle-and-edge decomposition CycleDecomposition
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: every graph on n vertices decomposes into at most Cn cycles and single edges. The Eulerian corollary is not part of the statement.
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.
CycleDecompositionThree 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 |
| Neal McBurnett, openai-math-lean-tooling | “accepted” | Re-run through Comparator with Lean’s kernel: accepted (the check took 85 s; peak memory 6.3 GiB). | 11 October 2026 | Original report |
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.
The Justin Sun Prize, The Justin Sun Prize: problem bank and award records, 11 October 2026. Source
JSP-000178 lists the release’s claim as Solved with independent mathematical review pending; Lean proof column: reported complete, independent verification pending.
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
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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 “Major math journal”. The page says that nothing on it is verified.
Author not stated. Source
Moritz Firsching, anxkhn, Paul Lezeau, Formal Conjectures: mark problems solved in the OpenAI math release and link formal proofs, 11 October 2026. Source
A Formal Conjectures maintainer, after source inspection, links the cycle decomposition result to Erdos problem 184.
Thomas Bloom, Erdős problems in the OpenAI proofs, 8 October 2026. Source
Bloom lists Erdős 184 (linear cycle and edge decomposition), names Bucić and Montgomery’s earlier bound, and links the formalisation.
MalekZ, jleng01, NicoV, efiebig, Thomas Bloom, Comments on ‘Erdős problems in the OpenAI proofs’, 8 October 2026. Source
A commenter says at least two claimed proofs of the linear cycle and edge decomposition appeared before the announcement, one on the same site on 1 October and one on arXiv at almost the same time.
ZZZamchi on Epoch AI’s repository, fc100open/erdos lag upstream FC: twoFivePerfect trivially disprovable, 7 statement fixes and 15 now-solved items (issue 78), 7 October 2026. Source
Erdős 184 covering variant in the benchmark’s pinned Erdős dataset is marked research solved upstream.
Jaehoon Kim, A proof of the Erdős--Gallai cycle decomposition conjecture, 6 October 2026. Source
Independent human proof of the same conjecture appeared on arXiv before the release; the other earlier claim, on erdosproblems.com, is Ryan Coffey’s proof with a Lean formalisation.
Jaehoon Kim, A proof of the Erdős--Gallai cycle decomposition conjecture, 6 October 2026. Source
Jaehoon Kim posted an independent arXiv proof of the Erdos-Gallai cycle-decomposition conjecture on 6 October, the same day as the OpenAI release.
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 1.1
In A linear cycle-and-edge decomposition of every graph, 24 September 2026
Why this passage is shown
Corollary 1.2
In A linear cycle-and-edge decomposition of every graph, 24 September 2026
Independent assessment
MathVet
14c7f91Independent assessment
Carlos Toledo’s cert-machine
9814e4cProof-verification report
cert-machine, CycleDecomposition
Proof-verification report
MathVet, CycleDecomposition
Proof-verification report
Neal McBurnett, openai-math-lean-tooling, CycleDecomposition
Lean statement
CycleDecomposition