Which unions of Schubert varieties can occur as the special fibre of a family whose general surface is irreducible? An AIM problem asks whether Cohen–Macaulayness is enough. This anonymous, unrefereed candidate gives a sharp negative answer for coordinate surfaces in products of projective lines. Encode the union by a graph: an edge between i and j selects the surface on which just those two coordinates vary. The candidate proves two exact classifications. The union is Cohen–Macaulay precisely when the active graph is connected. It deforms, inside the same product, to a geometrically integral surface precisely when the graph is complete multipartite. These graph properties first separate for the four-vertex path. Its three edges select three surface components in four factors. The graph is connected, so the union is Cohen–Macaulay, but it is not complete multipartite, so no integral deformation of the stated kind exists. The proof links Stanley–Reisner rings to connectivity and links multidegree support to rank-two matroids. It also constructs the integral surface for every complete multipartite graph and invokes a scheme-theoretic multiplicity-free degeneration theorem. Python and Macaulay2 replay the four-factor ideal, Hilbert–Burch data, basis-exchange failure and minimal cases. Those are producer-side checks, not an independent proof of the geometric argument. This is version 0.1.0-candidate, dated 4 September 2026. The paper, evidence and limitations are linked on this page. This AI-generated voice is a communication aid, not additional mathematical evidence.