Power-Preserving Formation GNN

Integrate this as a replacement for one or more residual message-passing blocks in a graph neural network. Let each node carry hidden state $h_i\in\mathbb{R}^d$ and let each graph edge $(i,j)$ carry an edge state or force $f_{ij}$. Build a learned formation operator $F_\theta(h)$ from relative node features, with one row per edge; its action can approximate edge differences, for example $F_\theta h=\{W_e(h_i-h_j)\}_e$. Construct the node-edge interconnection explicitly as $J_F=\begin{bmatrix}0&-F_\theta^T\\F_\theta&0\end{bmatrix}$, which is skew-symmetric by construction. Use energy $H_\theta(h)=\frac12h^TM_\theta^{-1}h+\sum_e\psi_{e,\theta}(f_e)$ with diagonal positive masses parameterized by softplus, and parameterize damping as $R_\theta=L_\theta L_\theta^T$ rather than learning an unrestricted matrix. The update is: compute $F$ from edges; compute $g=\nabla H$; compute $v=(J_F-R)g$; update $h\leftarrow h+\Delta t v$. Estimate $\lambda_{\max}$ by 5-10 power iterations and clip or adapt $\Delta t$ below $2/\lambda_{\max}(M^{-1/2}RM^{-1/2})$. The paper supplies the power-preserving interconnection and energy argument; the neural energy, edge feature map, and damping factors are learned. First test on Cora or a synthetic spring-graph node-classification task with 20-50 repeated layers, comparing an unconstrained GNN, an antisymmetric GNN, and this layer. Measure hidden energy $H_k$, gradient norms, accuracy, and failure rate while sweeping $\Delta t$. The prediction is that the unforced layer has non-increasing energy up to numerical error, and instability should appear near the predicted explicit-Euler boundary $\Delta t_c\approx2/\lambda_{\max}(M^{-1/2}RM^{-1/2})$; measured divergence should occur within 20% of this value, while the skew-only variant should show bounded but non-decaying energy.