Face-Neighborhood Inheritance in P-Well-Composed Cubical Face Posets

Local topological singularities on digital or voxel grids motivated well-composedness conditions excluding them; P-well-composedness lifts this regularity into an intrinsic, order-theoretic setting on posets. We study whether P-well-composedness is inherited by the strict neighborhood of a face in a finite embedded cubical complex. Let X = F(K) be the face poset of a finite nonempty set of grid n-cubes, with ambient rank n ≤ 3, and let N_h = θ′_X (h) carry the induced order. We prove N_h is P-well-composed whenever X is, provided h is a vertex or has no strict coface in the border ΔX, via a coherence argument for vertices and a finite, computer-assisted enumeration otherwise. The remaining rank-three case, an edge with a strict coface in the border, is posed as a No-Shrinkage Conjecture, with computational evidence reported separately.