Upon further modification of the "Solid," it seems like maybe the Object name was limiting the selectability
of the various parts of the solid.
Attached is a modified .3dm, in which the Blue denotes two different edges.
There is an inner surface to the spherical solid, in the shape (more or less) of an elliptical cylinder, with curved top and bottom.





Edit: The former Blend that I was doing was actually with the curvy edge of an extruded ellipse: