What was already known
The Erdős–Gallai conjecture was open for general graphs; Bucić and Montgomery proved the earlier O(n log* n) upper bound, with linear bounds known for special classes.
AI involvement
OpenAI says its internal frontier model produced this mathematics release and credits OpenAI as manuscript author. The source does not detail the individual human mathematical input for this result.
OpenAI says its internal frontier model produced this release and credits OpenAI as the paper author. The source does not establish a more precise individual autonomy claim.
- Supporting source (openai.com) — Release provenance
Importance
Settles the general Erdős–Gallai cycle-decomposition conjecture with the optimal linear order of growth.
Limitations
The theorem asserts existence of an absolute constant, not its optimal value. The manuscript and Lean theorem are OpenAI-authored reports; the repository review status is unchecked and no named human explanation or independent proof review is documented.
Technical verification
The pinned repository note states that its Lean theorem covers the selected statement, with a named Comparator theorem and source file. This is source-reported machine checking; repository review is unchecked and no independent run is recorded here.
Formal checking
The pinned scope note identifies a Lean proof of the selected statement and links the comparator theorem.
- Checker
- OpenAI-reported Lean 4 / Comparator
- Scope
- For some absolute constant C, every finite simple undirected graph on n vertices has an edge partition into at most Cn simple cycles and single edges.
- Basis
- Reported by the linked checker
Checking a formal statement does not establish that it matches the original problem.
Human understanding
Few is the default when evidence for a higher level is insufficient. The manuscript explains the proof, but OpenAI is its credited author and no named human reconstruction of the central argument is documented. Few is the evidence default, not a finding that people cannot understand it.
Correction and publication history
Primary manuscript, prior-work and pinned formalization review of one OpenAI-authored contribution; historical non-notifying inclusion.
Read this published revision