Beyond Pairwise: Diagnosing Higher-Order Merge Failures via Hodge Decomposition

Model merging implicitly assumes pairwise mergeability extends to higher-order combinations. On 24 weight-matched PlainCNNs (CIFAR-10), this fails: over 90\% of pairwise-mergeable triples are not jointly mergeable ($\beta_1$ peaks at 122). Hodge decomposition of the merge-barrier signal separates incompatibility into gradient, curl, and harmonic components. Star-topology alignment leaves 25.5\% of energy in the harmonic component; adjusting merge weights within the fixed alignment cannot reduce it. Re-aligning along a minimum spanning tree weighted by harmonic energy drives this to 1.8\% and collapses $\beta_1$ from 75 to 5, at a cost of 15\% higher total barrier. The composition moves out of the harmonic component into gradient and curl, which existing merging strategies already address. Whether these patterns hold at larger scales remains open.