Conjecture · version 2
For every finite loopless undirected multigraph G with no bridges, there exists a multiset C of cycles in G such that every edge of G belongs to exactly two members of C, counted with multiplicity.
Universal existence. The cited 2026 proof and formal endpoint are reported by their sources; independent mathematical assessment is pending.
In Cycle double cover conjecture · publication v3
Flow construction and cycle decomposition
The source proposes a full cycle double cover argument. The linked Lean project is a separate formal artifact for this account.
Independent mathematical assessment is pending.
Independent mathematical assessment of this argument remains separate from the statement’s attributed status.
Every bridgeless graph has a cycle double cover →Three-bit flows and cycle covers
A separately authored manuscript addresses the same cycle double cover assertion.
Separate authorship does not establish independent validation of the other manuscript.
Independent mathematical assessment of this argument remains separate from the statement’s attributed status.
Every bridgeless graph has a cycle double cover →In Cycle double cover conjecture · publication v5
Flow construction and cycle decomposition
The source proposes a full cycle double cover argument. The linked Lean project is a separate formal artifact for this account.
Independent mathematical assessment is pending.
Proposed: Full cycle double cover proof reported ↗ · observed Oct 13, 2026. Universal finite loopless bridgeless multigraph formulation.
Independent mathematical assessment of this argument remains separate from the statement’s attributed status.
Every bridgeless graph has a cycle double cover →Three-bit flows and cycle covers
A separately authored manuscript addresses the same cycle double cover assertion.
Separate authorship does not establish independent validation of the other manuscript.
Proposed: Separately authored proof manuscript ↗ · observed Oct 13, 2026. The same universal cycle double cover question.
Independent mathematical assessment of this argument remains separate from the statement’s attributed status.
Every bridgeless graph has a cycle double cover →Proposed
The same universal cycle double cover question.
Three-Bit Flows and Cycle Covers. Part I · Abstract and theorem statement →
Observed Oct 13, 2026
Replaces an earlier source assessment. Version history ↓
Proposed
Universal finite loopless bridgeless multigraph formulation.
A proof of the cycle double cover conjecture · Main theorem and flow-to-cover construction →
Observed Oct 13, 2026
Proposed
CDCLean.cycleDoubleCover_of_bridgeless, with no additional project axioms reported.
Cycle double cover in Lean · CDCLean/Main.lean; README verification and exact endpoint →
Observed Oct 13, 2026
Complete formal proof reported externally
The project reports an unconditional proof endpoint using only standard logical axioms. It is linked as source evidence while mathematical assessment of the new proof remains pending.
Statement match: Linked by OpenMath · independent review separate · checked Oct 13, 2026
CDCLean.cycleDoubleCover_of_bridgelessCommit 577e9d9ea326d520f80672ee69b830bf1d513df5
In this problem · publication v3 · Selected context
Primary formulation · Proposed
Universal existence. The cited 2026 proof and formal theorem are reported by their sources; independent mathematical assessment is pending.
Relationship not independently reviewed. This concerns its use in this problem.
In this problem · publication v5
Primary formulation · Proposed
Universal existence. The cited 2026 proof and formal theorem are reported by their sources; independent mathematical assessment is pending.
Relationship not independently reviewed. This concerns its use in this problem.
A new argument is reported to construct a cycle double cover for every finite loopless bridgeless multigraph.
paraphrase · OpenMath supplies original paraphrases and links; no proof text is republished.
OpenAI · Preprint · Publication date Jul 10, 2026
Public manuscript or exposition; publication does not imply independent OpenMath review.
Source revision: 2026-07-10. Observed Oct 13, 2026.
OpenAI reports a GPT-5.6 Sol Ultra multi-agent discovery workflow. This provenance is first-party and is not independent mathematical review.
Source updated Jul 10, 2026.
The source endpoint covers finite loopless bridgeless multigraphs via a formalized eight-flow theorem and a conversion to cycle covers.
paraphrase · OpenMath supplies original paraphrases and links; no proof text is republished.
OpenAI · openai/cdc-lean project · Project documentation · Publication date Jul 9, 2026
Public formal source; build and independent mathematical review are separate checks.
Public source inspected for statement and assumptions; this proof project has not been executed by OpenMath.
Source revision: 577e9d9ea326d520f80672ee69b830bf1d513df5. Observed Oct 13, 2026.
This formal source accompanies the OpenAI AI-assisted proof report; formal checking and human exposition are separate evidence roles.
Source updated Jul 9, 2026.
Cycle double cover in Lean — pinned source: 577e9d9ea326d520f80672ee69b830bf1d513df5.
A separately authored manuscript reports a proof of cycle double cover using an AI-assisted workflow.
paraphrase · OpenMath supplies original paraphrases and links; no proof text is republished.
Shiva Kintali · Preprint · Publication date Jul 13, 2026
Public manuscript or exposition; publication does not imply independent OpenMath review.
Source revision: 2026-07-13. Observed Oct 13, 2026.
Shiva Kintali reports work with Codex; this is a distinct manuscript and has not been adjudicated by OpenMath.
Source updated Jul 13, 2026.
Statement versions and problem publication versions are separate. A new publication can update context or sources while retaining the same statement.
Assessments retained in these publication records. Earlier statement versions retain their own source histories.
Proposed
CDCLean.cycleDoubleCover_of_bridgeless, with no additional project axioms reported.
Observed Oct 13, 2026 · Source passage →
Proposed
Universal finite loopless bridgeless multigraph formulation.
Observed Oct 13, 2026 · Source passage →
Proposed
The same universal cycle double cover question.
Observed Oct 13, 2026 · Source passage →
Replaces an earlier source assessment; see the original source for its history.