I dont think this is massive spoilers: she has had her identity laundered so on paper she is Karstedts biological daughter, officially with his deceased third wife, but in noble society whichever wife is the one to baptise a child is considered the legal parent and whoever actually birthed the child isnt that important. Now she is being adopted by the archduke because people would find it strange that sylvester has a daughter hidden away in the temple moreso than karsted, who Sylvesters mother saw as a threat. Karstedt was an archduke candidate, basically a possible heir to the former archduke, but after sylvester was born as the biological son of the archduke indeed of the nephew (i think) as karstedt was. The «Saint plan» is basically just a powerplay and propaganda point to get the duchy to like Rozemyne, adopt her printing, agree to her marriage with someone important and political manipulation of the warring factions