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Suppose $\kappa$ is an almost huge cardinal. Using the characterization of Theorem 24.11 in Kanamori's book, one can show that if $j : V \to M$ is an embedding derived from an almost-huge tower of measures, then $M^{<j(\kappa)} \subseteq M$, but $M^{j(\kappa)} \nsubseteq M$. Also, there is some embedding with target $j(\kappa) = \lambda$ and a stationary $X \subseteq \{ \alpha < \kappa : \alpha$ is inaccessible$\}$ such that $j(X)$ is not stationary in $\lambda$ (in $V$). For example, we can take $\lambda$ to be non-Mahlo.

Question: If $j : V \to M$ is an almost huge embedding derived from a tower with critical point $\kappa$, is there always a stationary $X \subseteq \{ \alpha < \kappa : \alpha$ is inaccessible$\}$ such that $j(X)$ is not stationary in $j(\kappa)$?

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I think the answer is no. If $\kappa$ is huge, and we derive an almost-huge tower from the huge embedding, then the embedding computed from the tower has stationary correctness.

Suppose $h : V \to M$ is a huge embedding and $h(\kappa) = \lambda$. Then $M$ computes stationary subsets of $\lambda$ correctly by the closure of $M$. As in Kanamori's proof, let $\langle U_\alpha : \kappa \leq \alpha < \lambda \rangle$ be the almost-huge tower derived from $h$. So $U_\alpha = \{ X \in [\alpha]^{<\kappa} : h[\alpha] \in h(X) \}$. The argument in Kanamori's book shows that this is a tower giving rise to an almost-huge embedding with the same target for $\kappa$.

For each $\alpha < \lambda$, we have the ultrapower embedding $j_\alpha : V \to M_\alpha$, and we have factor maps into the direct limit $k_\alpha : M_\alpha \to M_\infty = N$. The critical point of each $k_\alpha$ is greater than $\alpha$. Let $j : V \to N$ be the limit embedding.

I claim that for any $X \subseteq \kappa$, $j(X) = h(X)$, so stationary sets are mapped to stationary sets. Of course $j(X) \cap \kappa = X$, so let $\kappa \leq \alpha < \lambda$. We have:

$$\alpha \in j(X) \Leftrightarrow \alpha \in k_\alpha \circ j_\alpha(X)$$ $$\Leftrightarrow \alpha \in j_\alpha(X) \text{ (since crit}(k_\alpha) > \alpha)$$ $$\Leftrightarrow \{ z : \mathrm{ot}(z) \in X \} \in U_\alpha$$ $$\Leftrightarrow \mathrm{ot}(h[\alpha]) \in h(X)$$ $$\Leftrightarrow \alpha \in h(X).$$

We don't need the full strength of hugeness to get an almost-huge tower with this property, just because the statement "there is an almost huge tower with stationary correctness" will reflect below a huge cardinal. (But I suppose we wouldn't think it's consistent without proving it from a simpler principle like hugeness.)

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