Entity resolution -- identifying database records that refer to the same real-world entity -- is naturally modelled on bipartite graphs connecting entity nodes to their attribute values. Applying a message-passing neural network (MPNN) with all available extensions (reverse message passing, port numbering, ego IDs) incurs unnecessary overhead, since different entity resolution tasks have fundamentally different complexity. For a given matching criterion, what is the cheapest MPNN architecture that provably works? We answer this with a four-theorem separation theory on typed entity-attribute graphs. We introduce co-reference predicates $\mathrm{Dup}_r$ (two same-type entities share at least $r$ attribute values) and the $\ell$-cycle predicate $\mathrm{Cyc}_\ell$ for settings with entity-entity edges. For each predicate we prove tight bounds -- constructing graph pairs provably indistinguishable by every MPNN lacking the required adaptation, and exhibiting explicit minimal-depth MPNNs that compute the predicate on all inputs. The central finding is a sharp complexity gap between detecting any shared attribute and detecting multiple shared attributes. The former is purely local, requiring only reverse message passing in two layers. The latter demands cross-attribute identity correlation -- verifying that the same entity appears at several attributes of the target -- a fundamentally non-local requirement needing ego IDs and four layers, even on acyclic bipartite graphs. A similar necessity holds for cycle detection. Together, these results yield a minimal-architecture principle: practitioners can select the cheapest sufficient adaptation set, with a guarantee that no simpler architecture works. Computational validation confirms every prediction.