AI 中文总结
该研究针对有向图的钻石诱导典范边划分，探究关系凭证的标识能力，通过证明划分性质、计算纤维与邻域、构造反例，得出凭证具有中等选择性的结论，结果可独立验证。
AI 中文摘要
设$G=(V,E)$为有限无环有向图。每个有向两步钻石结构标识出其两对相对边，由此在边集$E$上构建的辅助图的连通分量定义了一个典范划分$Π_{\mathrm{opp}}(G)$。我们探究由边划分的块构建的有限关系凭证是否能标识出这个典范划分。\n我们证明了$Π_{\mathrm{opp}}(G)$的函子性与通用粗化性质，计算了其完全固定轮廓纤维及其精确的半径为1的对换邻域，并将凭证中的奇偶条件归约为有向$\mathbb Z_2$-增益图中的奇闭途径。在一个固定目录中，代表三种同构类型的四个带标记的目标为正的行，每一个都存在通过单次跨块对换得到的非典范正划分。其中一行中，一个更强的正比较划分还保留了每个顶点的固定角色入度和出度计数；它源自一个12边交替交易。其余三行中对应的关联纤维为单点集。\n一个确定性的64索引比较划分族在256个非恒等划分中产生了78个有界正例。在同一轮廓纤维上，其中一个成员具有完整的885元关系半群且无写-保留-使用凭证，给出了一个精确负例。因此该凭证具有真实但中等的选择性：它既不由块大小决定，也不专门针对典范目标。这些结论是计算组合学中的有限陈述，并有明确的、可独立验证的凭证作为支撑。
英文摘要
Let $G=(V,E)$ be a finite loopless directed graph. Each directed two-step diamond identifies its two pairs of opposite edges, and the connected components of the resulting auxiliary graph on $E$ define a canonical partition $Π_{\mathrm{opp}}(G)$. We ask whether a finite relational certificate built from the blocks of an edge partition identifies this canonical partition. We prove the functoriality and a universal coarsening property of $Π_{\mathrm{opp}}(G)$, count its complete fixed-profile fibre and its exact radius-one transposition neighbourhood, and reduce the parity condition in the certificate to an odd closed walk in a directed $\mathbb Z_2$-gain graph. In a fixed catalogue, four labelled target-positive rows, representing three isomorphism types, each admits a noncanonical positive partition obtained by a single cross-block transposition. In one row a stronger positive comparison partition also preserves every per-vertex, fixed-role incoming and outgoing count; it arises from a 12-edge alternating trade. The corresponding incidence fibre is a singleton in the other three rows. A deterministic 64-index family of comparison partitions yields 78 bounded positives among 256 nonidentity partitions. On the same profile fibre, one member has a complete 885-element relation semigroup and no write-preserve-use certificate, giving an exact negative. Thus the certificate has genuine but intermediate selectivity: it is neither determined by block sizes nor specific to the canonical target. The claims are finite statements in computational combinatorics and are supported by explicit, independently checkable certificates.
Comments20 pages; ancillary files contain exact certificates and standard-library verification code
