= Solution
The essential fact is <absoluteness of cardinalhood in limit ranks>. Let $\alpha\leq\beta$ be <limit ordinals> and $x\in V_\alpha$. Being an <ordinal> is absolute between these <transitive sets>, so a non-ordinal is a <cardinal number> in neither structure. For an <ordinal> $\delta\in V_\alpha$, we have $\delta<\alpha$.
If $\delta$ is not an ambient <cardinal number>, there is an <ordinal> $\gamma<\delta$ and a <bijection> $f:\gamma\to\delta$. With <Kuratowski ordered pairs>, its graph has <rank of a set> at most $\delta+3$. Since $\alpha$ is a <limit ordinal>, $\delta+3<\alpha$, so $f\in V_\alpha$. The assertion that $f$ is such a <bijection> is a <bounded formula in set theory> and is therefore evaluated correctly by each rank. Conversely, any internal <bijection> witnessing failure of <cardinal number> status is an actual one by the same <set-theoretic absoluteness>. Consequently
$$
\boxed{V_\alpha\models\text{``}\delta\text{ is a cardinal''}\ \Longleftrightarrow\ \delta\text{ is an ambient cardinal}\ \Longleftrightarrow\ V_\beta\models\text{``}\delta\text{ is a cardinal''.}}
$$
Unlike arbitrary <transitive models>, these rank-initial structures contain every sufficiently low-rank witness. That witness availability is what upgrades the usual <downward absoluteness of cardinalhood> to agreement in both directions.
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