The numerals near the edges are the squared lengths of those edges. The metric in the right triangle is ds2 = dx2 + dy2 whereas in the left triangle it is ds2 = − dx2 + dy2. The signatures differ. These cannot be joined by embedding them in a 2D space of any signature! This set of squared edge lengths for this simple abstract complex must be excluded! This example refutes my unexpressed assumption that arbitrary sets of real values for els (edge lengths squared), applied to some abstract complex could be interpreted as an approximation to some manifold. It seems clear that the shape of a zone puts a stamp on each of its facets that restrict the zone shapes of its neighbor, and that these limitations go beyond congruent facet shapes. They also go beyond the probably oversimplified version of the CDT restrictions that I heard somewhere. Perhaps it is merely that the normals to the facet be both time-like or both space-like.

We need to understand how signatures of metrics (SMs) of sub simplexes relate to SMs of simplexes. (Maybe not! The following may be on the wrong path. Here is a branch thought.) I think that it is clear that there is a containment relation. Notation: we code (+ + + − −) as SM = (3, 2). I think it is clear that if SM of a simplex is (m, n) and (m', n') is that of a sub-simplex then m' ≤ m and n' ≤ n.

There is another issue. Our formalism seems oblivious to the triangle inequality and other higher dimensional limitations as to the edge lengths of Euclidean simplexes. Perhaps some of these (edge length)2 specifications produce Minkowski triangles, even when they are all positive. Perhaps I will have to compute some eigenvalues. The above triangle, with indicated (edge length)2s violates the triangle inequality when roots are taken and produces the metric:
1.5−5
−51.5
when the top depicted vertex is taken as origin. The metric is
10
0−1
as expressed in the Cartesian coordinates of the above image, and has signature (+ −).

Here is a similar case of a topological tetrahedron in 3D but no Euclidean model:
Each facet obeys the triangle inequality but flouts some tetrahedral inequality. It exists in Minkowski space with signature (− + +).

I trust that this conjecture protects us from complexes with inhomogeneous signatures.

Lectures by Renata Loll at Perimeter Institute