Let $\mathcal{L}$ be a cocycle on a separable Banach space $X$ with its Borel $\sigma$-algebra. Let $Q_\omega$ be strongly measurable, tempered, equivariant idempotents ($Q_{\sigma\omega}\mathcal{L}_\omega = \mathcal{L}_\omega Q_\omega$) with range a closed subspace $V(\omega)$ of codimension $M$ and kernel $F(\omega)$, and suppose $F(\omega)$ grows strictly faster than $\mathcal{L}^{(n)}_\omega Q_\omega$ for large $n$. Let $A$ be a measurable set of positive measure on which $\|Q_\omega\|$ is bounded.

If $\mathcal{L}'$ is the cocycle over the same base with generator $\mathcal{L}_\omega Q_\omega$ on $A$ and $\mathcal{L}_\omega$ off $A$, then almost surely its signature is that of $\mathcal{L}$ shifted by $M$: $\chi'_k = \chi_{M+k}$ for all $k \ge 1$.