A Cubic-Link Obstruction to Manifold Skeleta of Generalized Cluster Complexes

For every finite root system Phi of rank n>=3, we prove that the generalized cluster complex Delta^2(Phi) is not the full (n-1)-skeleton of a finite polytopal PL manifold, with or without boundary, in any dimension. A face consisting of negative simple roots has link K3,3. A connected cubic nonplanar graph cannot be the full graph of a polytopal sphere or ball: the graph of any three-dimensional cell would have minimum degree three and hence exhaust the nonplanar graph. The type-A3 example consists of partial quadrangulations of a decagon. This gives a complete negative answer to the universal manifold-skeleton suggestion in AIM Problem 6.3 and Fomin-Reading Remark 4.5, but does not classify other parameters or close the entire dataset record. Cluster compatibility and parabolic rules, rank-two projective-plane examples and the separate known homological results are credited. The exact checker uses two independent integer crossing predicates. This is an unrefereed, AI-assisted preprint; absolute priority is not certified.