= Solution
Take $\lambda=\omega\cdot2$. 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 number>[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 $L_{\omega\cdot2}$.
The representation of $R$ is $\omega^2$, but
$$
\omega^2>\omega\cdot2
$$
and the <ordinal>[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.
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