Normal-Euler excess for disjoint nonorientable surfaces in a closed $4$-manifold

Let $M$ be a closed connected oriented topological $4$-manifold. We prove that if $F_1,\dots ,F_r\subset M$ are pairwise disjoint connected locally flat topologically embedded nonorientable surfaces with nonorientable genera $g_i$, same-sign twisted normal Euler numbers $e_i$, and $[F_1]+\cdots +[F_r]=0\in H_2(M;{\text{F}}_2),$ then the normal-Euler excess ${\sum }_{i=1}^r(\text{abs}{e}_i-2g_i)$ is bounded above by a constant depending only on $M$. Thus same-sign mod-$2$-null families of disjoint nonorientable surfaces in a fixed ambient $4$-manifold have uniformly bounded total excess over Massey’s $S^4$ bound. The proof combines a tubing construction with the signature and Euler-characteristic formulas for $2$-fold branched covers. As corollaries, every closed oriented topological $4$-manifold contains only finitely many pairwise disjoint locally flat topologically embedded copies of ${\text{RP}}^2$ with $\text{abs}e>2$, and only finitely many pairwise disjoint tubular neighborhoods modeled on real $2$-plane bundles over ${\text{RP}}^2$ whose total spaces are orientable and whose twisted Euler numbers have absolute value greater than $2$. When $M$ is a homology $4$-sphere, the ambient error term vanishes, and the theorem recovers Massey’s sharp inequality $\text{abs}e(F)\le 2g(F)$ for nonorientable surfaces in $S^4$.