I apologize if this question has been asked here already; I am wondering whether it is known precisely what is the class of origami-constructible numbers? The class of compass-and-straightedge constructible numbers is the real quadratic closure of the rationals, but it has been shown (e.g. see here) that the Huzita-Hatori axioms enable construction of all compass-and-straightedge constructible numbers, and more. Wikipedia mentions that using origami it is possible to double the cube, i.e. $\sqrt[3]{2}$ is constructible, and indeed asserts that all equations up to degree $4$ can be solved using the Huzita-Hatori axioms, but I cannot access the reference (Geometric Origami), and do not know if it is possible to solve even higher order polynomials. It is confirmed here that it is possible to solve all cubic equations using origami. This reference discusses origami-constructible numbers, but does not go beyond roots of cubic equations. This question mentions that any number in the real quadratic or cubic closure of $\mathbb{Q}$ is constructible, but that doesn't imply that this may not be true for the quartic closure as well. This question doesn't answer my question either; the comments and answer there merely assert that field extensions of degree $2$ and $3$ are origami constructible, and do not address whether higher order extensions are possible, and that question does not address my question of whether there is a class of numbers whose origami-constructibility is open.
My question is simple: Is it known what is the class of constructible numbers using the axioms of origami? Is there a class of numbers whose origami-constructibility is open?
