Single-Node Anti-Oscillation Anchor

Add the anchor after the normal recurrent or message-passing update, rather than modifying every learned weight. Select node or coordinate $a$ by estimating the dangerous oscillatory mode from two late iterates: $q=(h_K-h_{K-1})/\|h_K-h_{K-1}\|$. Choose the coordinate with largest $|q_i|$, or test a small candidate set. Set $h^{\mathrm{ref}}$ to an exponential moving average of recent states, a detached converged state, or a learned constant vector; detach it initially so the controller does not chase the oscillation. Apply $h\leftarrow F_\theta(h,x)-\kappa E_a(h-h^{\mathrm{ref}})$ at every inference iteration. For stronger control, estimate left and right eigenvectors with Arnoldi and select the node maximizing $|u_iv_i|$. The paper contributes the single-anchor stabilization mechanism; $J_c$, eigenvector sensitivity, and the required gain are measured or estimated in the neural system. First experiment: use a 16-node directed bipartite graph and recurrent GNN while sweeping message gain through the period-doubling transition. Compare no anchor, random anchor, and mode-selected anchor. Measure $A_K=\|h_K-h_{K-2}\|$, task error, and the eigenvalue near -1. The prediction is that $A_K$ collapses once $\rho(J_c)<1$, while the flip multiplier shifts linearly with gain with slope $-u_av_a$; agreement within 30% validates the approximation.