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Precise statement
Seymour conjectured that every finite oriented graph has a vertex with at least as many exact second outneighbors as outneighbors. Known cases include tournaments (Fisher 1996) and minimum outdegree at most six (Kaneko-Locke 2001), and for dense incomplete graphs a series of results restricting the structure of the missing edges. This work proves the conjecture for every oriented graph of order $n = 2\delta + 2$, where $\delta$ is the minimum outdegree, with no prescribed structure on the missing edges; with Fisher's tournament theorem this gives every oriented graph satisfying $n \le 2\delta + 2$.
The source statement is reproduced for indexing with attribution. Mathematical correctness requires domain-expert or mechanical review. VibeMath has not independently audited statement
fidelity, correctness, priority, or novelty.
What AI did
GPT-5 family, Claude
The acknowledgements are unusually specific about the division of labour: "The author initiated and directed the investigation, curated intermediate results, selected the theorem for publication, and edited the final statement and exposition. OpenAI language models (GPT-5 family) carried out the detailed mathematical exploration, implemented counterexample searches and verification tools, discovered the fixed-target capacity argument and its double-counting proof, and drafted the manuscript; Anthropic Claude models performed an adversarial audit of an intermediate draft and assisted with revisions. The author verified the proofs and accepts sole responsibility for the final manuscript and its claims." The model is credited with discovering the central argument by name, which is the discovered tier rather than the co-developed one.
arXiv preprint, one day old at cataloguing, not peer-reviewed and with no independent commentary yet. The proof is described by the author as a short counting argument, so it is human-checkable in principle, but this site has not verified it and no expert has publicly endorsed it. What is unusually strong here is the disclosure rather than the verification.
arXiv:2608.11530 - A dense-case theorem for Seymour's second neighborhood conjecture
A dense case, not the conjecture: it remains open in general. The concrete gain is on the size of any counterexample - combined with the known minimum-outdegree results, this raises the best known lower bound on the order of a counterexample from 16 to 17, and to 19 conditional on the 2026 preprint of Sadhukhan, Sandeep and Sen. The novelty against the earlier dense-case work is that no structure is prescribed on the missing edges.
Known method families
argument (source-reported)
Source-reported tools: argument.
Independent: unknown · difference confidence: 0
What remains uncertain
VibeMath has not independently audited the mathematical statement, proof, or novelty claim.
VibeMath has not independently verified the mathematical claim.
AI-attempt independence and training-data exposure are unknown unless explicitly documented.
VibeMath has not independently audited the mathematical statement, proof, or novelty claim.