Proof

I think I have another way to prove that every linear symbolic space is constructible as a numeric space. It is by induction on number of boundaries, and assumes a numeric cospace is constructible from a numeric space. By classification of the numeric cospace, the symbolic space has a symbolic cospace. To complete the proof I must prove every section of the symbolic space is a region in the symbolic cospace. Using a perspective, sections have sidedness wrt vertices. Consider a section of the symbolic space. I must show the section’s region in the cospace is not empty. Choose a point in the numeric cospace corresponding to a region in the symbolic cospace, a numeric, and a symbolic boundary in the space. Move the point across attached boundaries matching to section sidedness wrt vertices in the space. In some region of the cospace, the considered section is the same as the moved boundary wrt space vertices of attached cospace boundaries. Assume for contradiction that the point differs from the considered section wrt some vertex, called the contradiction vertex. In the cospace, choose a perspective that puts the contradiction boundary and an attached boundary on the same side of the attached region. That perspective in the space puts the contradiction vertex and a vertex attached to a region, in the considered section, divided by the moving boundary, on the same side of the considered section and moving boundary. This contradiction completes the proof.