Properties of the spaces associated to the uniform ultrafilters of an extender This question of mine comes from Woodin's "In Search of Ultimate-L" article, where they introduce some variables associated to each extender. I have a question about the remark following definition $4.4$ on page $30$.
I have been able to verify that $i_E$ is a cardinal, but I am failing to show why $\sup(\text{SP}(E)) = i_E$. I tried assuming $i_E \not \in \text{SP}(E)$, which gives $\sup(j[i_E]) = \nu_E$, but I can't even deduce that $i_E$ is a limit cardinal; because I tried assuming $\beta^+ = i_E$ for some $\beta < i_E$, then if $j(i_E) > \nu_E$ we get a contradiction because we have a surjection from $j(\beta)$ onto $\nu_E$, because $j(i_E) = j(\beta)^+$, and then some generator can't be a cardinal in it's own initial segment ultrapower, which can't happen. But I don't know what to do when $j(i_E) = \nu_E$, because then I can't use the generator idea. $(I)$ So I would appreciate any help with proving $\sup(\text{SP}(E)) = i_E$.
Continuing with the remark, it is mentioned that if $\gamma \in \text{SP}(E)$, then $E$ induces an uniform ultrafilter on $\gamma$. To me the most natural thing to do, since we have a generator $\xi$ such that $\sup(j[\gamma]) \le \xi < j(\gamma)$, we can take $U$ to be the set of $X\subseteq\gamma$ such that $\xi \in j(X)$. But the thing is, I think we don't need $\xi$ to be a generator for $U$ to be a uniform non-principal ultrafilter. My only guess is that this is used to show normality of $U$, which I guess has to somehow come from the factor map, but I can't see the argument and Woodin doesn't mention normality. $(II)$ So what does that bit of the remark mean?
 A: (I) I write $j=j_E$. Say an ordinal $\gamma$ is crucial iff there is a generator $\xi$ of $E$ such that $(\sup j``\gamma)\leq\xi<j(\gamma)$; so crucial is the weakening of "element of $\mathrm{SP}(E)$" given by removing the requirements that $\gamma$ be a cardinal
and $\gamma\leq\iota_E$. But every crucial ordinal is a cardinal $\leq\iota_E$. (If $\gamma$ is not a cardinal, fix a surjection $f:\theta\to\gamma$ where $\theta=|\gamma|$,
so $j(f):j(\theta)\to j(\gamma)$ is a surjection, and use this to see there are no generators in the interval $[j(\theta),j(\gamma)]$.) So crucial is actually equivalent to "element of $\mathrm{SP}(E)$".
Note that for generators $\xi$ are never in $\mathrm{rg}(j)$ (if $\xi=j(\bar{\xi})$ then consider the constant function $f$ taking value $\bar{\xi}$). Therefore for each generator $\xi$ there is a unique crucial ordinal
$\gamma$ such that $(\sup j``\gamma)\leq\xi<j(\gamma)$.
Note that there is no largest ordinal $\alpha$ such that $j(\alpha)<\nu_E$,
because if $\alpha$ were the largest, then $\nu_E=j(\alpha)+1$, so $j(\alpha)$ is a generator, a contradiction. So $\iota_E$ is a limit ordinal and
$\sup(j``\iota_E)\leq\nu_E\leq j(\iota_E)$.
Now if $\iota_E$ is crucial then (as mentioned above, $\iota_E$ is then a cardinal so) $\iota_E\in\mathrm{SP}(E)$ and we are done, so suppose otherwise. So then $\nu_E\leq\sup j``\iota_E$, so in fact $\nu_E=\sup j``\iota_E$. Note that it follows that $\iota_E$ is a limit of crucial ordinals. (Otherwise observe that $\nu_E$ would be too small.) Since each of those crucial ordinals are cardinals, we again get $\iota_E=\sup\mathrm{SP}(E)$.
(II) Let $\iota$ be a cardinal which is not in $\mathrm{SP}(E)$, i.e. not crucial.
Let $\xi<j(\iota)$, and let $U$ be the ultrafilter over $\iota$ derived from $j$ with seed $\xi$ (i.e. $A\in U$ iff $\xi\in j(A)$).  Then $U$ is not uniform. For there is some function $f$ and some generator $\eta<\sup j``\iota$ such that $j(f)(\eta)=\xi$. (The definition directly allows some finite tuple $\vec{\eta}$ and a function $g$ such that $j(g)(\vec{\eta})=\xi$, but we can modify $g$ and $\vec{\eta}$ by coding tuples with single ordinals, to get $f,\eta$ as mentioned.) Fix $\kappa<\iota$ such that $j(\kappa)>\eta$. Let $A=\iota\cap f``\kappa$. Then $A$ has cardinality $\leq\kappa<\iota$, but note that $A\in U$, so $U$ is non-uniform.
