Density and separation for augmented Zarankiewicz numbers

We study the augmented Zarankiewicz problem, in which disjoint pairs of cells are added to a binary matrix with no all-one $2\times 2$ submatrix. The pairs must satisfy compatibility conditions, and the objective counts each original occupied cell and each added pair once. We show that starting with a maximum $C_4$-free matrix can lower the final optimum, answering a question of Qi, Cui, and Xu. Let $z_A(m,n)$ be the optimum over all $C_4$-free initial matrices, and $z_L(m,n)$ the optimum when the initial matrix must have the maximum number of occupied cells. As $n\to \mathrm{\infty }$ with $n\le m=o(n^2)$, we prove $$z_A(m,n)-z_L(m,n)\ge (\frac{1}{30}-o(1))mn$$ and determine the sharp second-order term: $$z_A(m,n)=\frac{mn}{3}+(\frac{1}{\sqrt{6}}+o(1))n\sqrt{m}\mathrm{.}$$ An explicit construction gives a separation at $m=n=1893$. We also find a sharp density threshold: when $n\to \mathrm{\infty }$ and $m\mathrm{/}{n}^2\to c>0$, the limited density $z_L(m,n)\mathrm{/}(mn)$ tends to $1\mathrm{/}3$ if and only if $c\ge 1\mathrm{/}12$. The proofs combine density and stability estimates, combinatorial constructions, and an exact polynomial certificate.