@FrancescoPolizzi: in the video, they are used as rolls, the axis position is allowed to move and is there only to maintain the wheel in the plane of the bi... of this machine.

Breaking up the proof into preparatory lemmas is a way of explanation. If they are appropriately chosen, stating the lemmas actually gives the outline of a long proof better than what natural language can.

The crux is in the $-\Delta(\eta d)<C$ part. Can you give a brief idea of the proof? Does it work with nonconvex $\Omega$?

I also incline to believe it's a good approximation, if not it should be possible to write $OX'_i\cdot OX'_{i+1}$ independently of $i$ as a combination of the other vectors in the drawing, but if the postulate is true then transcendent numbers such as $\sqrt{2}$ could be obtained with rationals. The school book displays an $N=17$ example which looks pretty good though, I'd be happy to bound the approximation error. Also, could it be exact for (some sequence of) constructible polygons?

The marked ruler is needed for dividing AB into 7 equal segments, I believe.