PomeProposition 16.29.11.93A1136C4F72 A system $\boldsymbol \Gamma \in \mathscr S_{a,c}(\theta )$ is realized in $\theta $ if and only if, for every $\tau <c$, its limiting slice $(\mathsf R\boldsymbol \Gamma )_\tau $ is multiplier-ideal equivalent to one, and hence every, slice $\Gamma ^\eta _\tau $.
Proposition 16.29.1193A1136C4F72
Proof. For a realized system, this follows from $\mathsf R\mathsf E=\mathrm{id}$ and the preservation of multiplier ideals by each projection. Conversely, the assumed equivalence and the fact that $\Gamma ^\eta _\tau $ is fixed by its envelope in the larger background give
\[ P_{\theta +\eta }[(\mathsf R\boldsymbol \Gamma )_\tau ]_{\mathcal I} =\Gamma ^\eta _\tau . \]
Thus the system equals $\mathsf E\mathsf R(\boldsymbol \Gamma )$. $\square$
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