This might be of some historical interest for origami and mathematics. The usual set of origami axioms are acknowledged to be formed by Huzita-Justin-Hatori (https://langorigami.com/article/huzita-justin-axioms/).
I've looked in Donovan A. Johnson: Paper folding for the mathematics class, National council of teachers of mathematics, USA, 1957. Photographic reprint 2019. In the introduction Johnson writes: In mathematics we always make certain basic assumptions on which we build a mathematical structure. In paper folding we assume the following postulates: Paper can be folded so that the crease formed is a straight line. Paper can be folded so that the crease passes through one or two given points. Paper can be folded so that a point can be superimposed on another point on the same sheet. Paper can be folded so that a point on the paper can be superimposed on a line on the same sheet and the resulting crease pass through a second given point. Paper can be folded so that a straight line can be superimposed on another straight line on the same sheet. Lines and angles are said to be equal if they coincide when one can be superimposed upon another by folding the paper. If these assumptions are accepted, then it is possible to perform all the constructions of plane Euclidean geometry by folding and creasing. (citation end, my numbering) With reference to Robert Lang's formulation of the origami axioms O1-O7: (1) has no counterpart in the axioms. It seems a precursor to the axioms, like it is also a precursor to the origami axioms that constructing two crossing lines defines a point that can be used in later construction, cf. the construction rules (1-4) in https://langorigami.com/wp-content/uploads/2015/09/origami_constructions.pdf. (2) is (O1), except that (2) also states that a line can be folded through 1 point. (3) is (O2). (4) is (O5). (5) is (O3). (6) has no counterpart in the axioms. It seems more a definition than a postulate or axiom. Thus what is missing is O4 (crease through a given point perpendicular to a given line), O6 (two given points can be superimposed on two given lines by a new crease), and O7 (a crease perpendicular to one given line can superimpose a given point on another given line). Since O1-O7 are independent axioms, and O4 is Euclidean (you can construct a line through a point, perpendicular to another line), it seems that Donovan's postulates cannot construct all Euclidean constructs, despite his claim. Donovan later constructs "A line perpendicular to a given line and passing through a given point not on the line" by superimposing the given line on itself and sliding the crease until it passes through the point. He leaves it as an exercise to prove that the crease formed by superimposing the given line onto itself is perpendicular to the given line. His construction seems sound enough, but though it seems not to follow from his postulates, I'm not sharp enough to actually prove this. Donovan does not go beyond the Euclidean geometry and has no coordinated constructs like O6 that allows us to trisect an angle. Best regards, Hans Hans Dybkjær papirfoldning.dk Society: foldning.dk
