Certified Countermodels in Profile and Incidence Fibres of Diamond-Induced Edge Partitions

Abstract

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.