1. you should read carefully what I wrote (I did not say that get all algebraic numbers with the Huzita–Hatori axioms for origami). See for example Theorem 1 in One-, Two-, and Multi-Fold Origami Axioms by Alperin and Lang.(In particular this is not original research) 2. I asked you to discuss before making changes, I think you should do it next time. (Note that it is not me who started this "war".) 3. For the rest I gave the answers already - I will wait couple of days and undo you changes. I am 100% sure think this type of construction worth to be mentioned here.