Mutation-invariant interaction mixer

Integrate this into a small graph transformer or MoE router whose tokens or experts are vertices. Construct a signed Euler-like matrix \(E\in\mathbb{R}^{n\times n}\) from the current directed interaction graph: initialize \(E_{ii}=1-\rho\), set \(E_{ij}=\rho a_{ij}\) for a forward edge, and \(E_{ij}=-a_{ji}\) for a reverse edge, with \(a_{ij}\) a bounded sigmoid edge weight and \(\rho\in[0.2,0.8]\). Use \(E\) to mix node states, for example \(H_{\mathrm{mix}}=\operatorname{LayerNorm}(H+\alpha EH)\), or add \(\alpha E\) to attention logits before row normalization. When a router rewires edges, compute an invertible change-of-basis matrix \(M\); for an MVP use products of elementary matrices \(M=I+uv^{\top}\) with small norm and explicitly reject proposals with determinant magnitude below \(0.5\). Apply the corresponding hidden-state transform \(H'=M^{-1}H\), then train the new edge weights with the penalty \(\mathcal{L}_{\mathrm{cong}}=\|E'-M^{\top}EM\|_F^2\). The paper supplies the congruence law; the implementation estimates \(E\), \(M\), and edge weights from the current graph. Pseudocode: build \(E\); compute \(Z=H+\alpha EH\); periodically propose a graph mutation; construct \(M\); replace \((E,H)\) by \((E',M^{-1}H)\); optimize task loss plus \(\lambda\mathcal{L}_{\mathrm{cong}}\). First test on Cora or a synthetic long-range graph with a 2-layer GAT baseline and random edge rewiring every 100 steps. Measure validation accuracy after rewiring, loss spikes, and recovery time. Success means at least 20% smaller loss spikes and no accuracy loss relative to an equally sized non-invariant dynamic graph model.