The structure of the time dilation experiment (see the image) is the same as projective geometry.
the speed of light is along the perpendicular direction because it is independent of the line that contains v. inf is also independent of R in the proj geometry. So this experiment can be expressed as a type of projection. Riemann sphere is very suitable here. The vertical c vector is the inf axis in the leverage mechanism and its inclination implies the inf axis shrinks and as a result, the inf axis turns into a linear combination of a finite value (a velocity along the axis which contains v) and the perpendicular axis.
This will be discussed in Article 5. I think the explanatory version of Article 5 will be finished tomorrow. Then I should work on a way to present that.
It should be noted that the inf axis is defined as an axis that connects zero to inf (excluding point zero itself). According to Article 3, there is no such line in R. But the inf axis is defined as such an axis that is independent of R.