Vertex–Facet Assignment Obstructions Beyond Free Joins: Exact Defect, Transfer, and Constant-Additive Constructions

We study vertex–facet assignments for convex polytopes through their vertex–facet nonincidence graphs. We derive an exact face formula for the matching number and decompose assignment deficiency across connected components. For nondegenerate vertex-preserving one-point extensions in the strict visible/retained setting, we prove a transfer theorem in which connectedness is characterized by a rooted horizon graph and surviving faces yield explicit matching bounds. This produces, for every d ≥ 7, rational counterexamples beyond nontrivial free joins, including balanced deficiency-one examples with 33 vertices and 33 facets in dimension seven and infinitely many pairwise nonisomorphic families with exact matching numbers. Within the factor-witnessed strict join-stack class, every obstructive construction has at least 2d + 6 vertices and 2d + 6 facets, while balanced constructions of sizes 2d + 10 for even d ≥ 18 and 2d + 12 for odd d ≥ 19 imply the class optimum η_d = 2d + O(1).