Possible inconsistency of almost huge cardinals (I hope not) The paper "Double helix in large large cardinals and iteration of elementary embeddings" by Kentaro Sato 2007 gives the following characterization of almost huge cardinals as lemma 5.4 (note that $\kappa$-normal ultrafilters of $\mathcal{N}_{\kappa_{n−1},\kappa_n}$ are usually called coherent sequences of ultrafilters on $\mathcal{P}_{\kappa_{n−1}} \mu$ for $\mu \lt \kappa_n$):

$\kappa$ is $n$-fold almost huge with targets $\kappa_1$,...,$\kappa_n$ iff $\kappa_1$,...,$\kappa_n$ are inaccessible and there is a $\kappa$-normal ultrafilter $A$ of $\mathcal{N}_{\kappa_{n−1},\kappa_n}$ such that

*

*$\{s \in \mathsf{D}(\mathcal{N}_{\kappa_{n−1},\kappa_n})|ot(s \cap \kappa_{i+1}) = \kappa_i \} \in A$ for $i \lt n−1$ where $\kappa_0=\kappa$ and,

*for any $f \in \mathrm{Ult}(\mathcal{N}_{\kappa_{n−1},\kappa_n})$ with $Im(f) \subset \kappa_{n−1}$, there is $\nu \lt \kappa_n$ such that $\{s \in \mathsf{D}(\mathcal{N}_{\kappa_{n−1},\kappa_n})|f(s) \lt ot(s \cap \nu)\} \in A$.


$\mathrm{Ult}(\mathcal{N}_{\kappa_{n−1},\kappa_n})$ is the class of functions with domain $\mathcal{P}_{\kappa_{n−1}} \kappa_n$ such that there is an ordinal $\mu_f \lt \kappa_n$ such that $f(s)=f(t)$ whenever $s \cap \mu_f = t \cap \mu_f$ (they can also be seen a functions $\overline{f}$ with domain $\mathcal{P}_{\kappa_{n−1}} \mu_f$). I claimed in this Mathoverflow answer that this, minus inaccessibility, is satisfied by a club of cardinals $\lambda \lt \kappa_n$ in place of $\kappa_n$. Why doesn't this hold for every $\lambda$ such that $\kappa_{n−1} \lt \lambda \lt \kappa_n$? A function $f \in \mathrm{Ult}(\mathcal{N}_{\kappa_{n−1},\kappa_n})$ is in $\mathrm{Ult}(\mathcal{N}_{\kappa_{n−1},\lambda})$ iff $\mu_f \lt \lambda$ and whether $f(s) \lt ot(s \cap \nu)$ depends only on $s \cap \nu$.
I first posted this on Mathoverflow, where it was ignored, but on further consideration I think that my confusion is not research-level so I have deleted it there and posted it here instead.
 A: Let $j$ denote the ultrapower embedding by $A$ and suppose $\kappa \leq \lambda \leq j(g)(\kappa)$ for some $g : \kappa \to \kappa$ (for example, if $g(\alpha)$ is the successor cardinal of $\vert \alpha \vert$, then $j(g)(\kappa)$ is the successor cardinal of $\vert \kappa \vert$). Define $f : \mathcal{P}_\kappa \lambda \to \kappa$ by $f(s) = g( sup (s) \cap \kappa)$. Then $Im(f) \subseteq \kappa$ and $\mu_f = \kappa$ but $\{s \in \mathcal{P}_\kappa \lambda | f(s) \geq ot(s) \} \in A$, for the ordinals less than $\lambda$ are represented by functions $f_\zeta : \mathcal{P}_\kappa \lambda \to \kappa$ such that, for all $\zeta \lt \eta \leq \lambda$, $f_\zeta(s) \lt f_\eta(s)$ for $A$-almost all $s$, $f_\kappa (s) = sup(s \cap \kappa)$ (by lemma 3.6 of the linked paper) and $f_\lambda$ is $f$ as defined above. Thus $f$ is a counterexample to the last clause of the quoted characterization of almost-hugeness ultrafilters.
More generally, if $\lambda \leq j(g)(\xi)$ for $\kappa \leq \xi \lt \lambda$, define $f : \mathcal{P}_\kappa \lambda \to \kappa$ by $f(s) = g(f_\xi(s))$.
