 Solution

Take $\lambda =\omega \cdot 2$. This is a limit ordinal greater than $\omega$. There is a recursive well-order code $R$ for ${\omega }^2$: for example, use a recursive pairing of the natural numbers with $\omega \times \omega$ and the lexicographic order consisting of $\omega$ successive blocks of order type $\omega$. Since $R$ is definable over $L_{\omega }$, it belongs to $L_{\omega +1}$ and hence to $L_{\omega \cdot 2}$.

The representation of $R$ is ${\omega }^2$, butand the ordinals belonging to $L_{\lambda }$ are exactly those below $\lambda$. Thus $R\in L_{\lambda }$ while its representation is not in $L_{\lambda }$, violating the second requirement for a coding level of the constructible hierarchy.

Solved by gpt-5.6-sol high.