Perspective

A perspective is a maximal set of paths through a focal region, such that each path crosses each boundary once and only once.

Because paths cross boundaries only once, regions and vertices after the boundary all have the same sidedness wrt the boundary. Thus path sidedness is well defined and equivalent to space sidedness.

Sidedness of a boundary wrt a vertex and perspective is whether a path reaches the boundary or one of the vertice’s regions first.

Note that the point of convergence of lines perpendicular to the base in the numeric space is in some particular outside region. Thus, there is a symbolic perspective that makes numeric sidedness wrt vertices equivalent to symbolic sidedness wrt vertices.

Write algorithms to go back and forth between a numeric space and a numeric cospace, with one boundary per vertex, and one boundary per vertex at infinity.

Complete the construction proof by noting that the algorithms do not use the choice function.

Accessibility

Definitions:

A sub-vertex is any n or fewer boundaries, where n is the dimension of the space. The boundaries are said to contain the sub-vertex.

A vertex is a sub-vertex, where n boundaries contain the vertex.

A nonempy region and sub-vertex are attached iff there is a nonempty region with sidedness identical to the attached region, except wrt every boundary containing the sub-vertex.

A vertex is on the same side as each of its regions wrt a boundary if the boundary does not contain the vertex.

A vertex is accessible to a boundary dividing a super region iff the vertex can be on either side of the boundary without changing which side of the boundary the super region’s vertices are on.

Theorem:

A vertex is inaccessible iff there is a simplex-super-region by which the vertex is inaccessible to the dividing boundary.

Proof:

If the vertex is inaccessible, adding boundaries will not make it accessible, so on to the only if part.

The entire outside of the given super-region is covered by base and vertex regions of simplex-super-regions, so complete the proof by choosing one the given vertex is in.

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.