Curvature Prevents Lossless Layer Fusion in Fréchet-Mean Networks

Layer fusion is exact for networks of Euclidean weighted averages: a hierarchy of averages reduces to one average with path-product weights. We ask whether the same compression is valid for intrinsic networks that aggregate manifold-valued representations with weighted Fr\'echet means. For a canonical depth-two, three-input block, we show that the Euclidean path-product weights are the only possible fixed fused weights and derive the leading fusion error. For inputs of radius $\text{eps}$ around $p$, the leading possible error is cubic in $\text{eps}$ and linear in the Riemann curvature tensor; it is nonzero on suitable probes whenever $R_p\ne 0$. Consequently, this block admits an exact input-independent one-layer replacement on a neighborhood if and only if that neighborhood is flat. We extend the calculation to arbitrary finite Fr\'echet-mean trees via a bottom-up cubic recursion, yielding an architecture-level certificate for when depth is locally observable. Controlled experiments on constant-curvature spaces validate the predicted scaling and coefficient over $366,996$ source-block evaluations and $1,792$ random trees. A 90-configuration teacher–student study further separates three notions often conflated in compression: forced path-product fusion, a globally learned fixed surrogate, and input-conditioned fusion. Learned fixed weights reduce the reported average error but remain inexact on the curved runs; a per-input oracle attains near-zero error on the tested space-form inputs, without implying pointwise representability on general manifolds. On BCI Competition IV-2a, three frozen replacements retain the nested SPD teacher's subject-mean cross-session balanced accuracy to within $0.18$ percentage points despite nonzero representation gaps; the task-specific FBCSP baseline remains stronger. These results identify curvature as a concrete barrier to lossless static compression while separating geometric fidelity from downstream accuracy.