I would like to modify extender construction by relaxing the restriction that indices have to be finite in order to control closure properties, but I have run into an apparent contradiction.
Most of what I know about extenders comes from the paper Double helix in large large cardinals and iteration of elementary embeddings by Kentaro Sato.
The paper denotes, for a partition space $\mathcal{A}$, its underlying set by $D(\mathcal{A})$, and $B(\mathcal{A})$ denotes the set of subsets $x \subseteq D(\mathcal{A})$ such that the partition $x, x \setminus D(\mathcal{A})$ is a partition in $\mathcal{A}$ By an ultrafilter of a partition space, that paper means a filter $U$ on the underlying set of the partition space such that for every $x \in B(\mathcal{A})$, $x \in U$ or $x \setminus D(\mathcal{A}) \in U$.
The relevant partition space is $\mathcal{E}_{\lambda, \mu}$, defined as follows:
$D(\mathcal{E}_{\lambda, \mu}) = \mu^\lambda$ $P \in \mathcal{E}_{\lambda, \mu} \iff (\exists a \subset \lambda) (\vert a \vert \lt \omega \vee P \sqsupset \{\{ s \in D(\mathcal{E}_{\lambda, \mu}) \vert s \upharpoonright a = f\} \vert f \in \mu^a\})$
The paper defines what most set theorists call extenders in the following way:
A $\kappa$-normal filter $E$ of $\mathcal{E}_{\lambda, \mu}$ is an $\mathcal{E}_{\lambda, \mu}$, $\kappa$-complete ultrafilter of $B(\mathcal{E}_{\lambda, \mu})$ such that
$\{s \in D(\mathcal{E}_{\lambda, \mu}) | s(\kappa) \lt \kappa \} \in E$.
for $\alpha \lt \beta \lt \lambda$, $\{s \in D(\mathcal{E}_{\lambda, \mu}) | s(\alpha) \lt s(\beta)\} \in E$
for $\alpha \lt \beta \lt \lambda$ and $f \in Ult(\mathcal{E}_{\lambda, \mu})$ if $f(s) \lt s(\alpha)$ for all $s \in D(\mathcal{E}_{\lambda, \mu})$ with $s(\alpha) \neq \emptyset$, then there is $\xi \lt \alpha$ such that $\{s \in D(\mathcal{E}_{\lambda, \mu}) | f(s) = s(\xi)\} \in E$,
for $\langle x_n | n \lt \omega \rangle \in E^\omega$, $\cap_{n \in \omega} x_n \neq \emptyset$
By instead working with the partition space $\mathcal{E}_{\lambda, \mu}^\kappa$, relaxing the inequality $\vert a \vert \lt \omega$ to $\vert a \vert \lt \kappa$, I can apparently prove the following, analogously with part of lemma 3.6 of the paper (where $Ult(\mathcal{E}_{\lambda, \mu}^\kappa)$ denotes the class of function used to define the ultrapower by $\mathcal{E}_{\lambda, \mu}^\kappa$ and $Ult(\mathcal{E}_{\lambda, \mu}^\kappa) / E$ denotes the ultrapower itself):
$(Ult(\mathcal{E}_{\lambda, \mu}^\kappa) / E)^{\lt \kappa} \subset Ult(\mathcal{E}_{\lambda, \mu}^\kappa) / A$. Proof: Let $\langle [f_\xi]_E \vert \xi \lt \nu \rangle \in Ult(\mathcal{E}_{\lambda, \mu}^\kappa)/E$, with $f_\xi \in Ult(\mathcal{E}_{\lambda, \mu}^\kappa)$ and $\nu \lt \kappa$. Define, for each $s \in \mu^\lambda$, $g(s) : \nu \to V$ so that $g(s)(\xi) = f_\xi(s)$. By $\kappa$-completeness of $\mathcal{E}_{\lambda, \mu}^\kappa$, $g \in Ult(\mathcal{E}_{\lambda, \mu}^\kappa)$.
Combined with lemmas 3.9 and 3.10 of the paper, this apparently proves that if $E$ is a $\lt \kappa$-indexed $(\kappa, \lambda)$-extender derived from an elementary embedding $j : V \to M$ such that $V_\gamma \subset M$ with $\beth_\gamma = \lambda$, then the ultrapower embedding $j_E : V \to M_E$ satisfies $V_\gamma \subset M_E$ and $M^{\lt \kappa}$. If $cf(\gamma) \lt \kappa$ (for example, if $\gamma = \kappa + \omega$), this implies that $V_{\gamma+1} \subset M_E$
But my understanding is that $+\omega$-strong cardinalas are not generally $+\omega+1$-strong. Where did my proof go wrong?
One possibility is that the proof of lemma 3.10 fails when not all indices are in $M$, but I saw nothing that seemed to depend of all indices being in $M$. While indices in $M$ are used in the definitions of $\bar{j}(x)$ and $\bar{j}(f)$, I don't see how we need all indices to be in $M$; the important thing seems to be $\bar{j}(f)(r)$ and whether $r \in \bar{j}(x)$