Incidence-Based Local Surface Classification in Cubical Complexes

Deciding whether a vertex of a cubical complex lies on a locally well-composed surface reduces to a finite question: examine the occupancy pattern of the eight voxels incident to the vertex and decide whether the resulting active incidence structure is connected and 2-regular. We give a symbolic account of this decision. Writing \(n_E(S)\) and \(n_F(S)\) for the numbers of active edges and faces determined by elementary local activity rules, we show that every active face has degree exactly two and every active edge has degree two or four, yielding the identity \(n_F(S)=n_E(S)+k(S)\) for a single defect count \(k(S)\), and hence a surface criterion equivalent to \(\mathrm{comps}(S)=1\wedge n_E(S)=n_F(S)\). We further prove that the active incidence structure is invariant under complementation as a literal identity, not merely up to isomorphism, and combine this duality with a finite case analysis to classify explicitly which pairs \((|S|,d_1(S))\) of cardinality and Hamming-adjacency count are realized by a surface state, yielding an exact threshold classification as a proved corollary. The classification is established by a complete finite case analysis up to cube symmetries for \(|S|\le4\) and by duality for \(|S|\in\{5,6,7\}\).