Do not restart minimum outdegree at most six, degree-seven deficiency two, q=0 through 4, or the provisionally closed q=5/q=6 and q=7 small-W branches except for audit or source recovery. Find a scalable multi-mask alignment theorem turning Hall criticality and layer-degree identities into concentration on at most two heavy columns, then obtain a structural good-vertex argument that also scales beyond degree seven.
Route status · Narrowed routeDirected graph theory
Seymour’s Second Neighborhood Conjecture
Collaboration betaIn every oriented graph, must some vertex reach at least as many new vertices in exactly two steps as it reaches in one step? Many special and local cases are known, but the general statement remains open.
Known results and sources
Research problem
Exact mathematical statement
Every finite oriented graph contains a vertex v whose strict second outneighborhood is at least as large as its outneighborhood: d^{++}(v) >= d^+(v). The strict second outneighborhood excludes v and all direct outneighbors.
Problem infographic
Problem at a glance
Current mathematical picture
Where work on Seymour’s Second Neighborhood Conjecture stands
Selected route highlights from the mathematical source. This is not yet a complete mathematical inventory.
The source's minimal-counterexample framework reports minimum outdegree at least seven.
Evidence posture · Source-reported route statement · dependencies incompleteWe corrected the cited passages. We removed a duplicate or outdated task or route step. We updated the highlighted open task or route. The mathematical claims and their status did not change.
Reader-facing record corrected; mathematics unchangedWork mapped so far
Seymour’s Second Neighborhood Conjecture in numbers
3.9kretained lines of mathematical investigation3,872 in the current working snapshot
- Argument development
- 3,285 · 85%
- Explored or eliminated routes
- 131 · 3%
- Computational analysis
- 173 · 4%
- Open obligations
- 114 · 3%
- Definitions and setup
- 169 · 4%
6selected mapped statements2routes investigated4open questions4contribution-ready tasks
How this is measured
This measures retained mathematical investigation, not proximity to a proof. Code, data, logs, repeated text, operational instructions, and generated presentation copy are excluded.
Recommended next task
Independently replay the complete D7-Q7-M6 primitive state space.
Suggested move: Use a distinct state representation, export primitive witnesses, and check orbit weights against the exact raw baselines.
What would count as progress
- Supply a complete argument with every imported premise identified.
- Survive an independent attempt to falsify the proposed step.
Argument map and routes
How the current approaches connect
Claims, reductions, open questions, active routes, and narrowed alternatives in one mathematical map.
Visible working map
Research route map
Selected claims, active routes, useful failures, and open questions from the current research map. Arrows appear only for explicitly recorded relationships.
Scroll horizontally to explore the route
Working claimActive routeOpen, active, or blocked questionUseful failure
Working overview, not proof. The map shows selected recorded relationships; more nodes or edges do not establish correctness or completion.
Explored alternatives
Other routes
Do not restart minimum outdegree at most six, degree-seven deficiency two, q=0 through 4, or the provisionally closed q=5/q=6 and q=7 small-W branches except for audit or source recovery. Find a scalable multi-mask alignment theorem turning Hall criticality and layer-degree identities into concentration on at most two heavy columns, then obtain a structural good-vertex argument that also scales beyond degree seven.
Route status · Narrowed routeOperational closure is not independent certification; q5 retains recovery gaps, q6/q7 share implementation models, and q7-m6 has no closure claim. Find a scalable multi-mask alignment theorem turning Hall criticality and layer-degree identities into concentration on at most two heavy columns, then obtain a structural good-vertex argument that also scales beyond degree seven.
Route status · Narrowed routeMore ways to contribute
Open questions
Additional prepared tasks for exploring this research frontier.
Independently replay the complete D7-Q7-M6 primitive state space.Suggested move: Use a distinct state representation, export primitive witnesses, and check orbit weights against the exact raw baselines.
Ready to work on
Solve or sharply constrain the six-mask packing problem.Suggested move: Retain all A/B/deficiency/reversal correlations and produce explicit feasible masks or dual infeasibility certificates.
Ready to work on
Exact Conjecture
Every finite oriented graph is conjectured to have a vertex with strict second outdegree at least its outdegree.
Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.Ready to work on
Extract a scalable structural theorem beyond degree seven.Suggested move: Test whether Hall-criticality and layer-degree identities force laminar masks or heavy-column concentration, followed by a good-vertex argument.
Ready to work on
Sourced mathematical context
The known mathematical landscape
External progress
What the literature has established
Selected external milestones in reverse chronological order, with their evidence posture.
PreprintBai, Li, and Park proposed a matching-strengthening and proved it for minimum outdegree at most 5 and other special classes; their abstract says the original remains open in general.[3] PreprintHuang and Peng reported the universal factor 0.715538 in place of the conjectured factor 1; the result is a preprint in this snapshot.[2] Peer reviewedThe conjecture was verified for broad random and pseudorandom orientation regimes, a special-case result rather than a general proof.[4]
Mathematical neighborhood
Related results and reusable starting points
Current focusSeymour's Second Neighborhood Conjecture
The 2026 preprint asks for a complete matching from the first to the strict second outneighborhood, which would imply the cardinality inequality.
[3]Formalization opportunities
Lean work can make these reusable foundations precise without being presented as a proof of the core problem.
- Formalization targetNo general proof or counterexample appears in the bounded current-source review.
- Formalization targetThe source material's finite computation claims require independent replay before use as certified mathematical evidence.
Research-record corrections
What changed in the research record
These notes describe corrections to cited passages, highlighted tasks, or connections between claims. The mathematical claims and their status did not change.
Research-record correctionWe corrected the cited passages. We removed a duplicate or outdated task or route step. We updated the highlighted open task or route. The mathematical claims and their status did not change.
Correction detailsCorrected the research recordCorrection note
The initial argument structure appears separately. Uploads, model runs, and presentation changes do not count as mathematical updates.
Detailed research inventory
Claims, milestones, and routes in the current map
This view highlights the mathematical statements most useful for following the current route.
4 standing statements2 proposed statements4 open questions2 narrowed routes
- theorem candidate
1 of 6 1 - reduction
3 of 6 3 - lemma
2 of 6 2
Selected mathematical clusters1 mathematical clusters
Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.22 displayed rows · 2 routes included
- retained route statementEvery finite oriented graph contains a vertex v whose strict second outneighborhood is at least as large as its outneighborhood: d^{++}(v) >= d^+(v). The strict second outneighborhood excludes v and all direct outneighbors.
- retained route statementCurrent reductionintermediate
- retained route statementClosing targetintermediate
- retained route statementMinimum Degree Frontierintermediate
- retained route statementPrimitive Stateintermediate
- retained route statementAudit Boundaryintermediate
- Recorded relationshipThe source reports this as a route toward the conjecture; missing or unaudited premises remain and the reduction does not itself prove the target.supports · reported by source
- Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
- Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
- Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
- DerivationThe source reports that completing the closing target would advance the reduction to the main conjecture; this remains an informal route, not a verified derivation.proposed
- Useful failureSource-reported closed or invalid routereported failure
- Useful failureSecond source-reported limitationreported failure
- Research targetIndependently replay the complete D7-Q7-M6 primitive state space.open
- Research targetSolve or sharply constrain the six-mask packing problem.open
- Research targetExtract a scalable structural theorem beyond degree seven.open
- Research targetExact Conjectureopen
- Research targetFirst Open Branchsuperseded
- Research targetScalable Gapsuperseded
- ComputationThe source specifies a raw count of C(42,7)=26,978,328 deficient-cell sets before reverse marks; ProofAtlas did not execute any attached generator or verifier.The raw baseline and state contract are source-reported only here; every closure remains bounded by the source's evidence tiers. · reported unreproduced
- Narrowed routeSource-reported closed or invalid routeDo not restart minimum outdegree at most six, degree-seven deficiency two, q=0 through 4, or the provisionally closed q=5/q=6 and q=7 small-W branches except for audit or source recovery. Find a scalable multi-mask alignment theorem turning Hall criticality and layer-degree identities into concentration on at most two heavy columns, then obtain a structural good-vertex argument that also scales beyond degree seven.
- Narrowed routeSecond source-reported limitationOperational closure is not independent certification; q5 retains recovery gaps, q6/q7 share implementation models, and q7-m6 has no closure claim. Find a scalable multi-mask alignment theorem turning Hall criticality and layer-degree identities into concentration on at most two heavy columns, then obtain a structural good-vertex argument that also scales beyond degree seven.
How to interpret these counts
A statement may be a lemma, conditional reduction, special case, documented limitation, or open target. These counts describe the work's structure; they do not estimate distance to a proof.
Research outlook
Conditions that would advance the current route
2 approaches have already been tested and narrowed. The task above is the current priority within the larger open route.
A result can change the outlook by closing the bridge, narrowing its scope, or showing that the route cannot work.
- Supply a complete argument with every imported premise identified.
- Survive an independent attempt to falsify the proposed step.
Continue the mathematics
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Prepared starting pointIndependently replay the complete D7-Q7-M6 primitive state space.
Seymour’s Second Neighborhood Conjecture · ready to start
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Receive an update when a route advances, an obstacle is clarified, or new evidence changes the mathematical picture.
Research contextPrepared context for any AI agent
In every oriented graph, must some vertex reach at least as many new vertices in exactly two steps as it reaches in one step? Many special and local cases are known, but the general statement remains open.
- Exact question and boundaries
- Current routes and known obstacles
- What a useful result should report
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Sources and references4 cited works · next context review by Nov 15, 2026
The mathematical context was checked on Aug 15, 2026. Status can be refreshed sooner after a material result or claim.
- 1Seymour's 2nd Neighborhood Conjecturemaintained problem list · Douglas B. West · Douglas B. West, University of Illinois · accessed Aug 15, 2026
- 2An improved bound on Seymour's second neighborhood conjecturepreprint · Hao Huang, Fei Peng · arXiv · 2024-12-28 · ARXIV 2412.20234 · accessed Aug 15, 2026
- 3Towards a strengthening of the second neighborhood conjecturepreprint · Yandong Bai, Binlong Li, Boram Park · arXiv · 2026-07-20 · ARXIV 2607.18047 · accessed Aug 15, 2026
- 4Seymour's Second Neighborhood Conjecture for orientations of (pseudo)random graphspeer reviewed result · Fábio Botler, Phablo F. S. Moura, Tássio Naia · Discrete Mathematics · 2023-12 · DOI 10.1016/j.disc.2023.113583 · accessed Aug 15, 2026
Important qualifications
- This was a bounded current-status check, not an exhaustive priority or attribution review.
- Preprints remain labeled as preprints and were not upgraded to peer-reviewed posture.
- Current open status was cross-checked against recent official or primary sources where available.
- No source-package attachment was executed or rendered, and no URL copied from a submitted packet was fetched.
- Metadata has no proof, review, acceptance, credit, publication, or deployment effect.
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