So what is needed to demonstrate something (say like $L_{\omega_1^{CK}}[a]$ is a $\omega$-models of $\mathbf{\Sigma^1_1}$-$DC$ or $\mathbf{\Delta^1_1}$-$CA$? It's not like I don't understand what the terms mean but I suspect there is a nice property that will suffice, at least for admissible ordinals with extra reals stuck on.

Also I've been told that this is closely related to the question of whether the set is admissible. However, I've found very little reference material on admissible sets that *isn't* merely about admissible ordinals, i.e., looks at sets like $L_{\omega_1^{CK}}[a]$). I'm sure that the best way forward here isn't manually verifying all the axioms for KP so any hints?

For the reader's info this is coming from Friedman's 102 problems in mathematical logic numbers 64 and 65. MY advisor claims to have answered these in a short cryptic note and I had to write up his general method for another problem so I want to include these corollaries but I still feel too fuzzy about them to write anything up.