Why Geometric Continuity Emerges in Deep Neural Networks: Residual Connections and Rotational Symmetry Breaking

Weight matrices in deep networks exhibit geometric continuity---principal singular vectors of adjacent layers point in similar directions. While this property has been widely observed, its origin remains unexplained. Through experiments on toy MLPs and small transformers, we identify two mechanisms: residual connections create cross-layer gradient coherence that aligns weight updates across layers, and symmetry-breaking nonlinearities constrain all layers to a shared coordinate frame, preventing the rotation drift that would otherwise destabilize weight structure. Crucially, a nonlinear but rotation-preserving activation fails to retain continuity, isolating symmetry breaking---not nonlinearity itself---as the active ingredient. Activation and normalization play distinct roles: activation concentrates continuity in the leading singular direction, while normalization distributes it across multiple directions. In transformers, continuity is \emph{projection-specific}: Q, K, Gate, and Up (which read from the residual stream) develop input-space ($\mathbf{v}_1$) continuity; O and Down (which write to it) develop output-space ($\mathbf{u}_1$) continuity; V alone, lacking an adjacent nonlinearity, develops only low continuity.