Replacement failure at V omega plus omega ($V_{\omega +\omega }/\models \mathrm{Replacement}$)

The transitive rank stage $V_{\omega +\omega }$ contains $\omega$ and every ordinal $\omega +n$. The relation mapping a finite ordinal $n$ to the $n$th successor of $\omega$ is internally definable using finite successor chains, whose finite coding overhead remains below this limit rank. Its image would have rank ${\sup }_n(\omega +n+1)=\omega +\omega$, so it is not an element of the stage. This is a failure to collect existing values, rather than a missing individual value.