A Characterization of Poset-Based Connected Manifolds and Discrete Surfaces via Cubically Normal Pseudomanifolds

<title>Abstract</title> <p>Local well-composedness conditions on digital grids constrain pixel- or voxel-local configurations that cause topological singularities, but do not describe the global combinatorial structure of the boundary they regularize. A parallel line of research — discrete surfaces and poset-based connected manifolds (PCMs) — classifies manifold-like structure directly on a poset, independently of any embedding. We identify the missing correspondence: for a finite regular cubical complex K of rank n ≥ 2, the face poset F (K) is an n-PCM or a discrete n-surface if and only if K is a cubically normal pseudomanifold: purity, a facet-degree condition, facet-adjacency connectivity, and link normality. At rank one, an analogous two-way classification holds for non-branching intervals and circles; at rank zero, only one direction survives. The proof recovers the four conditions through distinct components in each direction; as an algorithmic consequence, we give a recognition procedure for embedded voxel complexes issuing a pass/fail certificate.</p>