I DO need to actually enjoy the "Break" part of Summer Break & will probably go camping.
ERRATA:
1) The statement "2 manifold analogue of As(star) is everywhere negative except at the points associated with the elementary vectors" is false. Take any triangulation of As^2(star) and note that, when we remove the boundary of the triangle, the remaining sub-manifold is everywhere positive, since both intersecting curves at any given point are negative. This means that at any given point in the sub-manifolds described by the above statement, the curvature is -k_1*-k_2=+(k_1*k_2). So the positive curvature on As^n is NOT strictly concentrated at the points associated with elementary vectors E_n.**
As for Sq' (and Sq by duality of the octahedron & the square): the statement that all positive curvature is strictly concentrated at the points associated with elementary vectors E_n still stands.
I plan on writing & typing up a short revision of this paper to reflect the changes I need to make & to better explain those changes.
**(9/10/17): In fact, the negative curvature is strictly concentrated on As^1 (minus its endpoints that intersect E_{n}) subset of As^2..In terms of the gradient projection of the unit Hyperbolic Octahedron onto the plane, these are the outermost points of that representation.
Heuristically, we can see that any neighbourhood directly surrounding one of these points will have negative curvature, noting the neighbourhood's shape is resembling that of a saddle, unlike the cup shape we can see in the neighbourhood of a point inside of the boundary of any triangulation of As^2.. I'll get to explaining that in a less heuristic way in the revision I'm working on (slowly), but for now this sufficiently (in terms of intuition) explains where the concentrations of negative (and positive) curvature on As^2 lie.
Also: I'm pretty sure I've noted this before, BUT (just to be sure): Note that the triangles comprising the surface of As^2 are exactly the projection of an Euclidean triangle onto a catenoid. (Also:The triangles comprising Ci^2 are exactly the projection of an Euclidean triangle on a sphere, and the triangles comprising Sq'^2 are Euclidean triangles). One of the longer term goals of this research is to use the geometric lens concept to study the motion of topological bending of these triangles from one form to another. In essence, the fluidity/elasticity that goes into the deformation of the triangles and the objects comprised of these triangles. Later work will hopefully cover this same concept with an extension to volume. I'm currently about to start reading Robert William Soutas-Little's book "Elasticity" to see just how much some of this subject matter has already been covered.