Define a neighbor of region R wrt boundary b as a region separated by and only by b from R. Regions R0 and R1 are neighbors iff they are neighbors wrt some boundary. A region R and boundary b are attached iff R has a neighbor wrt b. The corners of a set of boundaries B are the regions attached to every boundary in B. Region R1 is the opposite of region R0 iff R1 is separated from R0 by and only by boundaries attached to R0. Spaces S0n and S1n are migrations of each other iff they are identical except for mutually opposite regions R0 and R1, such that R0 is nonempty in and only in S0n, and R1 is nonempty in and only in S1n.
A space Sn with m boundaries is linear iff for each set of l boundaries B from Sn, the number of corners of B is 2lf(n-l,m-l).
The sections of Sn with B removed by each boundary in B produce a space corresponding to f(n-l,m-l) regions in Sn with B removed. The boundaries of B divide each of those regions into 2l regions. Thus the only if part is proved. For the if part, proceed by induction on number of boundaries. The correct numbers of corners are preserved when removing a boundary, and the section by the removed boundary has the correct numbers of corners, so the space without the boundary removed is linear by the Antisection Theorem. To complete the proof, note that the empty space has the correct numbers of corners and is linear.
If S0n and S1n are migrations of each other, then S0n is linear iff S1n is linear.
The corners of a set of boundaries B are the same in S0n and S1n except for R0, R1, and neighbor regions of R0 and R1. R0 and R1 replace each other, and their neighbors replace each other. Thus the theorem is proved by the Corner Theorem.
A region is inside iff it is a subregion of the inside of a simplex subspace. A region that is not inside is outside. Intuitively, think of outside regions as peripheral regions with infinite extent. An outside neighbor of a region is a neighbor that is outside.
In a space of dimension
OUTSIDE NEIGHBOR THEOREM
A simplex has only
If b divides the base regions of a simplex in linear space
By definition of linear, I can consider the simplex subspace. Assume