How well-studied is origami field theory?

Question

It's well known that angle trisection cannot be done with straightedge and compass alone, as

Theorem 1. If $z \in \mathbb C$ is constructible with straightedge and compass from $\mathbb Q$, then $$\mathbb Q (z) : \mathbb Q = 2^n.$$

But the minimal polynomial of $\cos 20 ^{\circ}$ is $8 x ^ { 3 } - 6 x - 1$, so $$\mathbb Q (\cos 20 ^{\circ}) : \mathbb Q = 3,$$

That proves we cannot trisect $ 60 ^{\circ}$.

However, it's doable with origami, as Huzita Axiom 6 - Computing the Origami Trisection of an Angle shows. My question is:

Exactly what field extensions can be obtained by considering origami constructible number? Is this as well-studied as straightedge and compass, i.e. do we have a similar theorem as Theorem 1?

Answer 1

There is a paper posted on ArXiv by Antonio M. Oller Marcén entitled "Origami Constructions" that claims to show that:

If $a \in \mathbb{R}$ is origami-constructible, then $$[\mathbb{Q}(a): \mathbb{Q}] = 2^r3^s$$ for some $0\leq r, s \in \mathbb{Z}$.

Unfortunately, I have not been able to find this particular paper published in any peer-reviewed venue nor am I able to personally vouch for the proof, so I guess caveat lector.

EDITED TO ADD:

The same result is also found in a Master's thesis by Hwa Young Lee entitled "Origami-Constructible Numbers" (Corollary 4.3.10, pg. 50).