Condensation for constructible levels

By F1, $(L_{\alpha },\in )\models C$. Since $X\prec L_{\alpha }$, also $(X,\in )\models C$. F2 makes $\pi$ an isomorphism from $(X,\in )$ to the transitive set $(M,\in )$, so $M\models C$. In particular $M$ is nonempty; no assertion that $L_{\alpha }$ or $M$ satisfies Infinity, Power Set, Replacement, Separation, or Choice has been used.

Put $\beta =M\cap \mathrm{Ord}$. The converse direction of F1 applies directly to the nonempty transitive set $M$ satisfying $C$ and yields $M=L_{\beta }$. This includes the boundary $\alpha =\omega$: the supplier proves $L_{\omega }=V_{\omega }$, verifies $C$ there by finite certificates, and explicitly does not infer Infinity.

F3 applied to this collapse fixes each transitive $A\subseteq X$ pointwise. It also gives $\pi (\xi )=\mathrm{otp}(X\cap \xi )$ for every actual ordinal $\xi \in X$, and gives $\pi (\xi )=X\cap \xi$ when that intersection is transitive. These conclusions do not require $X$ itself to be transitive, and the empty transitive part is fixed vacuously.