Iterating Generalised Perfect Set Forcing Along Well-Founded Orders

Vladimir Kanovei \cite{zbMATH01335192} developed the technique of geometric iteration and used it to prove that the perfect set forcing can be iterated with countable supports along any partial order, while preserving ${\mathrm{\aleph }}_1$. In \cite{Property-B} we considered a generalised perfect set forcing with respect to a filter on a cardinal $\kappa$ satisfying ${\kappa }^{<\kappa }=\kappa$, which we denoted $\mathbb{P}(\mathcal{F})$, and proved that its iteration with supports of size $\le \kappa$ along any ordinal preserves cardinals up and including ${\kappa }^{+}$. We show that there is a version of the geometric iteration technique that applies to $\mathbb{P}(\mathcal{F})$, to yield that for $\kappa$ satisfying ${\kappa }^{<\kappa }=\kappa$, the forcing $\mathbb{P}(\text{FF})$ can be iterated with supports of size $\le \kappa$ along any well-founded partial order, while preserving cardinals up and including ${\kappa }^{+}$.