${\mathsf{w}}_{k}={\mathsf{w}}_{k}(\mu)$, it is enough to find two automorphisms $\phi_{1},
\phi_{2}\in D_{K}(1,k+1)$ such that

-   $\phi_{1}(v_{\mu,{\mathsf{w}}_{k}})$ is non-negative and reduced

-   for each $\epsilon>0$, $\phi_{2}(v_{\mu,{\mathsf{w}}_{k}+\epsilon})$ contains a negative coefficient.

This can be done using the reduction algorithm. Our strategy is then to proceed backward: (i) use the algorithm to find upper bounds $w_{n}=w_{n}(\mu)$ for the value of ${\mathsf{w}}_{k}$, (ii) find the smallest one, say $w_{n_{0}}$, and (iii) show that one gets a nonnegative reduced vector after setting $c=w_{n_{0}}$ in $v_{\mu,c}^{i}$. Since the algorithm consists in applying elements of $D_{K}(1,k+1)$ to the initial vector $v_{\mu,c}^{i}$, a simple dualization gives us an exceptional class $E\in\mathcal{C}_{K}(X_{k+1})$ that defines the obstruction. More precisely, if $\phi\in D_{k}$ is the automorphism corresponding to the upper bound ${\mathsf{w}}_{k}=w_{n_{0}}$, then $$0=\left(\phi\,v_{\mu,{\mathsf{w}}_{k}}, E_{k+1}\right)=\left(v_{\mu,{\mathsf{w}}_{k}}, \phi^{*}\,E_{k+1}\right)$$ so that $\phi^{*}\,E_{k+1}$ is an obstructing exceptional class. Such a class is generally not unique and, in fact, the reduction process often gives finitely many choices.

Embeddings of $k\geq 8$ disjoint balls in the trivial bundle $M_{\mu}^{0}$ {#section:TrivialBundle}
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This section will be dedicated to proving the results in Theorem \[thm:MainTheoremTrivial\] as well as introducing several immediate corollaries. As explained in Section \[sec:strategy\], the computation of the generalized Gromov widths ${\mathsf{w}}_{k}(M^0_{\mu})$ and of the packing numbers 