= Moved critical point is not an ambient cardinal under GCH
For a measure ultrapower $j:V_\lambda\to M$ with critical point $\kappa$, there are at most $2^\kappa$ equivalence classes of functions $\kappa\to\kappa$, so $|j(\kappa)|\leq2^\kappa$ in $V_\lambda$. The target $M$ regards $j(\kappa)$ as measurable and contains $V_{\kappa+1}$, so it sees $(2^\kappa)^M<j(\kappa)$. Under the <Generalized continuum hypothesis>, this places $j(\kappa)$ strictly above the ambient $\kappa^+$ while its ambient cardinality is at most $\kappa^+$; hence it is not an ambient cardinal.
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