A new result shows that a translation‑invariant topological p‑ary CSS code can be split into independent layers only when its coarse‑grained profile over the superlattice Λ matches that of a toric‑code stack. That is, the layered check operators must be of the form \(E\,X\,|\, \mathbb{F}_p[\Lambda] \simeq \bigoplus_{j=1}^{D-1}\mathbb{F}_p[-j]\oplus t_j\). If the profile fails this form, the code cannot be decoupled on that lattice.

The authors developed an algorithm that first verifies the coarse‑graining condition. It builds a bounded complex of finite free modules from the CSS check matrices, computes the layers \(E_j^X\), and checks if any layer is infinite dimensional over \(\mathbb{F}_p\). If all layers are finite, the algorithm constructs a finite‑depth local circuit using SUM and SWAP gates that realizes the decoupling. This circuit is built from a stable isomorphism of the check cokernels, lifted to reversible unitaries, and then factored into layers of summed‑and‑swapped gates.

However, a non‑zero inter‑layer gluing \(\delta_j\) is an extra obstruction. When layers are glued, the excitation structure is altered, and even if each layer individually admits a toric‑code profile, the gluing can block a decoupling that keeps the original lattice translations. In three‑dimensional examples, trivial translations on every layer coexist with a gluing that prevents decoupling.

The method also encounters limits. If any superselection layer contains infinitely many excitation types—as in many fracton models—the code cannot be decoupled on any superlattice. Even adding ancillas does not help, because the infinite layer count blocks a finite‑depth local circuit that would preserve translations.

These findings sharpen our understanding of when topological quantum memories can be re‑described as stacks of toric codes and when their complex excitation patterns inherently resist such decomposition.