Not evaluated
Seymour's second-neighborhood conjecture (family 173)
The gist
In its release of October 6, 2026, OpenAI announces, using an unreleased internal model, a proof of Seymour's second-neighbourhood conjecture.
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-173-seymours-second-neighborhood-conjecture-r1
@misc{discoveria-oai-fam-173-seymours-second-neighborhood-conjecture-r1,
author = {{DiscoverIA}},
title = {Seymour's second-neighborhood conjecture (family 173)},
howpublished = {DiscoverIA registry, entry oai-fam-173-seymours-second-neighborhood-conjecture},
note = {Status “Not evaluated”, snapshot 1, recorded on October 12, 2026. New result},
year = {2026},
url = {https://discoveria.ai/entry/oai-fam-173-seymours-second-neighborhood-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
An oriented graph has no loop and no pair of edges in opposite directions. Seymour's conjecture states that every oriented graph has at least one vertex with at least as many neighbours at distance two as direct neighbours. If the proof held, this conjecture would be proved.
What remains unclear
The Lean (a language in which a computer checks every step of a proof) was supplied by OpenAI itself, and its tracking file marks it as unchecked. No third-party verification is recorded.
For experts
Sources, method, key facts, status history, report an error
Status
Not evaluated
Recorded on · github.com · 173.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-proof-of-Seymours…link only, reading not recorded
- Lean proofgithub.com · 173.mdpage read word for word
- Lean proofgithub.com · SeymourSecondNeighborhood.…link only, reading not recorded
Method
The openai/math repository contains the manuscript “A proof of Seymour’s second-neighborhood conjecture”, dated September 23, 2026, and the file lean/docs/173.md, which describes the Lean formalisation with one Comparator statement (lean/ComparatorChallenges/SeymourSecondNeighborhood.lean). The family is among the sources of formalization.yaml; history.md reports no revision of this manuscript. 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 173, section “Combinatorics”, 1 manuscript dated Sep 23, 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 167 of 372, by family number
- Not evaluated : Sharp logarithmic exponents for off-diagonal Ramsey numbers (family 170)
- Not evaluated : The hypercube Ramsey conjecture (family 171)
- Not evaluated : Classification of finite Euclidean Ramsey configurations (family 172)
- Not evaluated : Seymour's second-neighborhood conjecture (family 173)
- Not evaluated : Deterministic construction of strong thin spanning trees (family 174)
- Not evaluated : Talagrand's conjectures and graph decompositions at expectation thresholds (family 175)
- Not evaluated : The second Kahn–Kalai conjecture (family 176)
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