Eigenmode-Targeted Hidden-State Actuator Selection

Integrate this into a GRU, vanilla RNN, S4-like state-space layer, or recurrent transformer memory, treating hidden coordinates or groups of recurrent parameters as intervention channels. After training warmup, collect representative hidden trajectories and estimate the local Jacobian operator $A$ at several checkpoints. For small hidden states, form $A$ explicitly with automatic differentiation; for large states, use Arnoldi iteration and Jacobian-vector products to estimate the eigenvalue of largest modulus and its right eigenvector $v$. Define each intervention direction $b_i$ as a unit basis vector for a hidden coordinate, channel, head, or parameter block. Compute the score $s_i=\|Cv\|_2^2/(\varepsilon+\|b_i v_i\|_2^2)$ and select the top $k$ coordinates, optionally followed by one-swap local search over the binary subset. For a red-team test, inject $\Delta x_t=B_Sa_t$ with $a_t$ aligned to the left-eigenvector projection and measure target-output growth. For a defense, add a mode-gain penalty or apply coordinate damping $x_{t+1}\leftarrow x_{t+1}-\gamma D_Sx_t$, where $D_S$ is diagonal on selected coordinates. Compare with random selection, gradient saliency, magnitude pruning, and uniform spectral normalization under equal compute and parameter budgets. The paper-derived quantities are eigenmode extraction and subset ranking; Jacobian estimates, nonlinear rollout gain, and the target readout $C$ are measured empirically. Pre-register that amplification scales approximately as $(1-|\lambda|)^{-1}$ and that the selected top-$k$ set yields at least 2x the target amplification of a random set at identical $k$. Reject the transfer if the fitted log-amplification slope versus $-\log(1-|\lambda|)$ is below 0.5, or if eigenmode selection fails to beat random selection by 20% across three seeds.