Chromatic Completeness and the Independence of Geometric Obstruction

Abstract

We establish a strict logical separation between two distinct phenomena in orthogonality hypergraphs: chromatic completeness, the possibility of assigning a single globally consistent nondegenerate spectrum to all contexts, and geometric coordinatizability, the existence of a faithful orthogonal representation by rays. A strong chromatic number larger than the Hilbert-space dimension obstructs only the former. It does not, by itself, obstruct the existence of a faithful orthogonal representation. We make this separation explicit by comparing two three-dimensional examples with the same strong chromatic number. A completed 25-ray version of the Yu-Oh configuration has strong chromatic number four and nevertheless possesses an explicit faithful orthogonal representation in R^3. Conversely, Greechie's G_{32} hypergraph also has strong chromatic number four, and has a separating and unital set of two-valued states, but we give an elementary algebraic proof that it admits no faithful orthogonal representation in C^3. The obstruction in G_{32} is therefore not chromatic but projective-geometric: the incidence relations force two distinct atoms to collapse onto the same ray.