For a change, this will be a very short post.
I realized the error in my thinking. We don’t need the "Klein absolute" for this. Stereographic projection is sufficient. I’m not sure why I thought that parallel lines on a plane, when mapped via inverse stereographic projection onto a sphere, would "intersect like meridians" at the poles. That was incorrect.
Bjab, as usual, was guiding me in the right direction: they don’t intersect—they are tangent. Bjab was right, and I was wrong.
Below is an illustration showing the inverse stereographic projection of 21 vertical lines from the (x, y) plane, all parallel to the y-axis, with As expected, they are tangent to each other at the South Pole.
Bjab was right; I was wrong. And so, I’m humbly asking for mercy.
We will be back to the Klein absolute in the next post discussing transformations between circles.
P.S. Here is my attempt to show how one rays from the South Pole of the sphere through a circle on this sphere generate the vertical line x=1 on the z=0 plane. View at the South Pole:
