Complement Erasures and Exact Weak Saturation for Complete Bipartite Graphs

Complement erasures on n=s+t+1 translate weak saturation for complete bipartite graphs into sole-crossing cuts with one temporarily omitted vertex. The paper retains the exact difference-two two-fan family and develops the candidate formula wsat(3k+4,K_(3,3k))=(9k^2+9k+6)/2 for every k>=2. A structural-deficit interpretation of the Akhmejanova-Vorobyev-Zhukovskii potential yields a one-cost obstruction, while a separate missing-spoke construction proves initial freeness for the upper bound. Further results concern tight structural completion, intrinsic witness width, pure diamond chains and sharp disjoint lifting ranges. Vertex universes remain fixed, including isolates; conclusions retain their stated structural domains.