A few months back, I posted a question asking about a proposed axiom of Laver's and I, unfortunately, left out a critical piece. Here is the full axiom:
(L) Some elementary embedding $j:V_{\lambda+1}\prec V_{\lambda+1}$ extends to a non-trivial elementary embedding $h:HOD(ord^\lambda)\prec HOD(ord^\lambda)$ where it is assumed that $HOD(ord^\lambda)\models ZF +DC_\lambda + Unif(V_{\lambda+1})$.
Here $DC_\lambda$ denotes the axiom of $\lambda$-dependent choice and $Unif(V_{\lambda +1})$ is the axiom that uniformization holds for $V_{\lambda +1}$. More specifically, given any $R\subseteq V_{\lambda +1}\times V_{\lambda +1}$ there exists some function $f\subset R$ with the same domain as $R$.
This axiom is generically fragile in that any small forcing adding a real kills the axiom. This fragility is evidently a consequence of the further assumption about which axioms hold in $HOD(ord^\lambda)$, in particular $Unif(V_{\lambda +1})$.
As before, I have no indication about how this is established at the moment and I suspect that the problem is that I don't understand what the model $HOD(ord^\lambda)$ looks like in a generic extension. I am hoping someone can give me some indication as to the kinds of "damage" $HOD$-like models can undergo during a forcing.
