A surplus lattice for Seymour and Sullivan neighborhood inequalities

Definitions and mathematical statement

The original is only the fragment "Conjecture 6.1. If we believe Seymour's second neighborhood conjecture and", a truncated transition rather than an independent conjecture. Its verification field says surrounding text was recovered by source comparison but does not reproduce the completed sentence. The attached aid studies pointwise relations among the neighboring neighborhood inequalities; this does not restore the missing sentence verbatim.

For a vertex $v$ of a finite oriented graph, let $a=|N^+(v)|$, $b=|N^-(v)|$ and $c=|N_2^+(v)|$, with the second out-neighborhood at directed distance exactly two. Define surpluses \[ \sigma_S=c-a,\quad\sigma_1=c-b,\quad\sigma_2=c+a-2b,\quad \sigma_3=c+a-2\min(a,b). \] The corresponding pointwise properties are Seymour $S$ ($\sigma_S\ge0$), Sullivan-1 ($\sigma_1\ge0$), Sullivan-2 ($\sigma_2\ge0$), and the aid's Sullivan-3 compromise ($\sigma_3\ge0$). It states the exact identities \[ 2\sigma_1=\sigma_S+\sigma_2,\qquad \sigma_3=\max(\sigma_S,\sigma_2)=\sigma_1+|a-b|. \] Consequently the sharp threshold equivalences at this same vertex are \[ (S\ \text{or}\ S2)\ \Longleftrightarrow\ \sigma_1\ge-|a-b|, \] \[ S1\ \Longleftrightarrow\ \sigma_1\ge0,\qquad (S\ \text{and}\ S2)\ \Longleftrightarrow\ \sigma_1\ge|a-b|. \] In particular Sullivan-3 is exactly the pointwise disjunction of Seymour and Sullivan-2. All four properties coincide when $a=b$. Equalities are included in every displayed threshold.

The aid reports sharp local nonimplications realized by oriented depth-two gadgets. One explicit realization convention takes a distinguished vertex $v$, $a\ge1$ out-neighbors, $b$ in-neighbors, $c$ new second neighbors, and only the arcs into $v$, out of $v$, and from one selected out-neighbor to all second neighbors. Its distinguished vertex has the specified triple $(a,b,c)$. Thus feasible degree triples expose failures of stronger local implications. These are not counterexamples to any conjecture asserting that every graph has some suitable vertex. Pointwise conjunctions or disjunctions at one vertex must not be confused with existential statements whose witnesses could differ.

Short English statement

This truncated transition is not a standalone conjecture. The supplied aid organizes neighboring Seymour and Sullivan conditions into exact pointwise surplus identities and sharp thresholds, while keeping local nonimplications separate from global existence conjectures.

Sources

• Local UnsolvedMath registry, permanent ID $20001044$, source code catalogue entry; canonical aim-combinatorics-notes.json, notes[168].
• American Institute of Mathematics, "The Caccetta--Häggkvist conjecture" workshop record: https://aimath.org/WWN/caccetta/caccetta.pdf. The assigned extraction is an unfinished transition. The supplied verification reports comparison with the original arXiv TeX and AIM PDF but omits the claimed complete sentence; no such inspection is claimed here.
• Attached research attempt 1: context only, recorded run status not a problem, rigor label proved, novelty confidence low; registry status open, classification OPEN-TRIAGE, suggested difficulty L2. The surplus relations are machine generated and not peer reviewed; independently verify before citation. No full bibliography is supplied.