Coefficient-Depth Blindspots in Topological Message Passing

Learning tasks can involve interactions among groups of entities. Graphs represent pairwise relations; simplicial and cellular complexes also represent larger groups. Topological message passing learns through local exchanges between their cells. Homology supplies global summaries that local computations can miss. Yet separate limits of these two channels do not characterize their combined expressivity. For every prime $p$, integer $s\ge 2$ and prescribed order $k\ge 1$, we construct finite pure two-dimensional full-incidence pairs with identical order-$k$ observations at every finite depth. Their homology agrees over every field in every degree. For $A=\mathbb{Z}/p^{s-1}$, one chain isomorphism matches all permitted finite diagrams inside $A$-Mod, up to abstract isomorphism. Integral first homology and depth-$s$ modules distinguish different cyclic lengths under the common torsion exponent $p^s$. Full cubical realization from a height-four lattice preserves the separation, adding a common free summand. Boundary computations and shared-weight sum/attention forwards check three source pairs and one dependent cube realization. An additional source pair extends the neural checks to matching mod-$4$ and distinct mod-$8$ information. An order-four simulation covers the standard scalable multicellular network interface. A depth-$s$ homology readout separates each pair; fixed structural order, depth or width cannot uniformly replace this coefficient precision.