Consider the higher Cantor space $2^\kappa$ with the ${<}\kappa$-box topology ($\kappa$ at least inaccessible). This canonically defines the notion of higher Borel sets.


A higher Borel code $\mathbf{B}$ is a wellfounded tree of size $\kappa$ consisting of finite sequences, such that terminal nodes are label with basic clopen sets and non terminal nodes are label with either $\bigcup$ or $\bigcap$ .


Let $V \subsetneq V'$ be models of ZFC\* (large enough fragment of ZFC), let $V \vDash$ " $\mathbf{B}\_1 , \mathbf{B}\_2$ are higher Borel codes " , and $V \vDash$ " $\mathbf{B}\_1 \cap \mathbf{B}\_2 = \emptyset$ " . Does this also hold in $V'$ if $\kappa$ remains a large cardinal?



On the absoluteness of higher Borel sets?
In the case of forcing extensions by ${<}\kappa$-complete posets (which you probably meant), we have $\Sigma^1\_1$-absoluteness. (This is well known / folklore, see e.g. [Friedman Khomskii Kulikov](http://www.logic.univie.ac.at/~ykhomski/papers/Regprop.pdf), Lem 2.7). Strategic closure is sufficient. And ``Borelcode evaluates to something nonempty'' is $\Sigma^1\_1$.


