Not evaluated
The Erdős–Gallai cycle-decomposition conjecture (family 181)
The gist
In its release of October 6, 2026, OpenAI claims that every graph admits a linear decomposition into cycles and edges.
DiscoverIA status on October 12, 2026: not evaluated, meaning no third-party evaluation found.
AI draft, not reviewedWhat has not been reviewed?
Text written by an AI (DiscoverIA pipeline), on . The status and its sources come from the registry; only the explanatory text has not been reviewed. A reported error is corrected, then the text is reviewed.
Cite this entry
A single status snapshot since Oct 12, 2026: a citation always points to a dated snapshot.
Cite snapshot 1, Oct 12, 2026
APA citation with the dated link
BibTeX @misc, key discoveria-oai-fam-181-erdosgallai-cycle-decomposition-conjecture-r1
@misc{discoveria-oai-fam-181-erdosgallai-cycle-decomposition-conjecture-r1,
author = {{DiscoverIA}},
title = {The Erdős–Gallai cycle-decomposition conjecture (family 181)},
howpublished = {DiscoverIA registry, entry oai-fam-181-erdosgallai-cycle-decomposition-conjecture},
note = {Status “Not evaluated”, snapshot 1, recorded on October 12, 2026. New result},
year = {2026},
url = {https://discoveria.ai/entry/oai-fam-181-erdosgallai-cycle-decomposition-conjecture#releve-1},
urldate = {2026-10-12}
}Dated link opens this snapshot in the history
To understand ≈ 20 s reading time, estimated
What it changes
If the manuscript held, every graph (points joined by links) would admit a so-called linear decomposition into cycles (closed loops) and edges. The sources read do not say which parameter the word “linear” refers to.
What remains unclear
No independent assessment is recorded. The Lean (a language in which a computer checks every step of a proof) is supplied by OpenAI and marked “review: status: unchecked” in formalization.yaml. The entry is named after the Erdős–Gallai cycle-decomposition conjecture; the sources read do not say whether the manuscript proves it.
For experts
Sources, method, key facts, status history, report an error
Status
Not evaluated
Recorded on · github.com · 181.md
Step 1 of 4 on the scale: not evaluated. Next step: under review.
- Announced
- Recordednot evaluated
- nextTo follow
What this status means
Source reading : Source read word for word (level 3 of 3)
The result is declared neither right nor wrong.
To move to “Under review”: a public review under way. To “Peer reviewed”: a journal publication or a review by named experts.
Sources
- Main sourcegithub.com · CONTENTS.mdlink only, reading not recorded
- Repositorygithub.com · A-linear-cycle-and-edge…link only, reading not recorded
- Lean proofgithub.com · 181.mdpage read word for word
- Lean proofgithub.com · CycleDecomposition.leanlink only, reading not recorded
Method
OpenAI publishes one manuscript, “A linear cycle-and-edge decomposition of every graph”, dated September 24, 2026. The repository also holds lean/docs/181.md, which describes the scope of the Lean formalisation (1 Comparator statement), and lean/ComparatorChallenges/CycleDecomposition.lean. DiscoverIA status: not evaluated, meaning that no third-party evaluation has been found in the sources read. Set on October 12, 2026 from github.com (page read, quotation checked word for word: “describes the scope of the Lean formalization”).
Limits
No named third party: the recorded checks come from the claimant or from a platform tool. The claimant supplies a Lean proof; as of this date, DiscoverIA has found no independent verification of this proof or of the result.
Key facts
- Claimed by
- OpenAI (unreleased internal model)
announced on Oct 6, 2026 - Nature
- new result
- Field
- Mathematics (Combinatorics)
- In openai/math
- family 181, section “Combinatorics”, 1 manuscript dated Sep 24, 2026
- Lean proof
- supplied by the claimant, marked “unchecked” (not reviewed) in its repository
- Revisions
- none reported by the claimant
- Checked by
- claimant (Lean supplied, not checked by a third party)
- Sources
- 4 : 1 read word for word, 3 links only
- This entry
- AI draft, not reviewed
Status history
Only one observation so far.
- 01
First observation:Not evaluatedCurrent status
Family with a Lean proof supplied by the claimant. No third-party check read; the source marks this proof as not reviewed (“unchecked”).
Source: github.comread word for word
An observation is added; none is ever erased.
In the same set
openai/math: 719 manuscripts in 372 families from an unreleased internal model · entry 175 of 372, by family number
- Not evaluated : Deterministic nonbipartite Ramanujan graphs in every fixed degree (family 178)
- Not evaluated : The circulant Hadamard conjecture (family 179)
- Not evaluated : Barnette's Hamiltonian-cycle conjecture (family 180)
- Not evaluated : The Erdős–Gallai cycle-decomposition conjecture (family 181)
- Not evaluated : Power savings for polynomial-difference-free sets (family 182)
- Not evaluated : Power savings for planar halving lines and k-sets (family 183)
- Not evaluated : Coloring and independence in graphs with forbidden subgraphs (family 184)
Report an error or suggest a source
Thank you, your message was sent.
Sending failed. Try again, or write to contact@discoveria.ai.
Is your result listed?What each status requires, correcting this entry, replying to it.