[Icy0]

That's a good response! I had to spend some time to work out what goes wrong here. But I figured it out.

Parallel transport is broken in your model of the sphere. Take this example. Take a vector pointing north in circle 1, and send through the north portal. It should be pointing south when it gets to the other circle. Fine -- north in circle 1 corresponds to south in circle 2.

Now send the north-pointing vector east instead. It is going to point north in circle 2.

So the north vector changes direction depending on which portal it goes through. So parallel transport changes directions. Hence your model of the sphere is not flat.