A Weak Condition for Limited Augmented Zarankiewicz Numbers

We introduce a weak version of the limited augmented Zarankiewicz number. The underlying $1$-edge graph is still required to be extremal $C_4$-free, but the original opposite-cell condition is replaced by two weaker requirements: an acyclic dependency condition on nondegenerate $2$-edges, and a simplified local prohibition that the two opposite cells of a nondegenerate $2$-edge cannot both be $1$-edges. For this weak condition, the vector proof of irreducibility from the limited augmented theory still goes through. Consequently, if $z_{WL}(m,n)$ denotes the resulting weak limited augmented Zarankiewicz number, then $$\mathrm{BSR}(m,n)\ge z_{WL}(m,n)\ge z_L(m,n)\ge z(m,n)\mathrm{.}$$ We show that the simplified local prohibition cannot be removed entirely: a $5\times 3$ example with two $2$-edges admits an exact $9$-square decomposition, so it is not irreducible. We also exhibit small irreducible blocks showing that neither the acyclicity condition nor the local prohibition is necessary in every isolated instance. Several open problems are discussed, including whether the acyclicity condition can be weakened and whether there exists any case where $z_{WL}(m,n)>z_L(m,n)$.