Countably closed end-extensions of elementary submodels

Question

The following is well-known. If $\kappa$ is measurable, $\theta > \kappa$, and $M \prec V_\theta$ has size $<\kappa$, then there is $N\prec V_\theta$ such that $N \supseteq M$, $M \cap \kappa \not= N \cap \kappa$, but $N \cap \sup(M \cap \kappa) = M \cap \kappa$. Repeating this a number of times allows us to also find such $N$ with the ordertype of $N \cap \kappa$ being any ordinal $\leq \kappa$.

The construction of $N$ can be done by adjoining a single ordinal that is in all measure-one sets in $M$ for some normal ultrafilter, and closing under Skolem functions. If $M$ is $\mu$-closed, then so is $N$. However, repeating this $\omega$-times in an elemenatary chain can kill countable closure.

My question: Is there is some large cardinal $\kappa$ with the following stronger property where we demand closure?

Whenever $\mu<\kappa$ is regular, $\theta > \kappa$, and $M \prec V_\theta$ is $\mu$-closed and of size $<\kappa$, then for every $\alpha <\kappa$, there is $N \prec V_\theta$ such that $N$ is $\mu$-closed, $N \supseteq M$, $M \cap \kappa \not= N \cap \kappa$, $N \cap \sup(M \cap \kappa) = M \cap \kappa$, and the ordertype of $N \cap \kappa$ is $\geq \alpha$.

Answer 1

It seems unlikely.

Let me use the following consequence of partial global square (which appears in my joint paper with Garti). This partial sequence is consistent with the existence of very large cardinals.

Lemma: Let us assume that there is a partial square sequence $\langle C_\alpha \mid \alpha \in S \rangle$ where $S^{\kappa}_{\omega_2} \subseteq S \subseteq \kappa$ (so $\mathrm{otp}\ C_\alpha < \alpha$ for all $\alpha \neq \omega_2$, each one of them is closed and unbounded at $\alpha$ and the sequence is coherent).

Then, there is a partition of $S^\kappa_{\omega}$ into $\aleph_2$ sets, $\langle S_i \mid i < \omega_2\rangle$ such that for all $\alpha < \kappa$ with $\mathrm{cf}\ \alpha = \omega_2$, $\forall i < \omega_2,\, S_i \cap \alpha$ is stationary.

Proof: Recursively, we can start with a partition of $\omega_2$ into $\omega_2$ disjoint stationary sets and using the coherence we can "copy" them upwards.

Now, let us assume that there is such a partition for $\kappa$ and assume $\mathrm{CH}$. Then, if $M$ is a $\sigma$-closed model of size $\aleph_1$, and $N$ is a $\sigma$-closed model extending it of order type at least $\omega_2$, then let's look at $\delta$ the supremum of the first $\omega_2$ elements of $N \cap \kappa$ - this is an ordinal of cofinality $\omega_2$. $N \cap \delta$ is $\sigma$-closed set of ordinals and in particular, intersects each $S_i$ for all $i < \omega_2$ which means that $\omega_2 \subseteq N$ but it is not a subset of $M$.