Eigenvalue-Sensitivity Stability Margin

1. Exact integration point: use a recurrent cell, diagonal-plus-low-rank state-space model, or neural state-space predictor. During training, select representative hidden states from each minibatch and form the local Jacobian A = partial f_theta / partial h using automatic differentiation, Jacobian-vector products, or an explicit matrix for small hidden dimensions. Treat p as a low-dimensional perturbation such as input mean, input variance, recurrent gain, normalization scale, or a connectivity-mask amplitude. 2. Pseudocode: compute A; obtain the k eigenvalues closest to the unit circle and their left and right eigenvectors; estimate partial A / partial p using finite differences A(p plus epsilon) and A(p minus epsilon), or second-order autodiff; calculate each d lambda_i / d p with the eigenvector formula; compute m and delta_hat_c; add a penalty for m below a target m_0 and optionally a squared sensitivity penalty; backpropagate the task loss plus this penalty. 3. Computed versus estimated: eigenvalues and eigenvectors are computed locally, while derivatives of A may initially be estimated with two additional Jacobian evaluations. The crossing radius is a first-order prediction and must be checked by explicitly recomputing eigenvalues after perturbation. 4. First cheap experiment: compare vanilla tanh RNNs and sensitivity-regularized RNNs on Copy Memory and adding problems with identical hidden size and optimizer. Sweep recurrent-gain and input-statistics perturbations after training. The falsifiable signature is that the measured dominant-eigenvalue crossing occurs within 20 percent of delta_hat_c for sufficiently small perturbations, while the regularized model has a larger median predicted crossing radius and delayed long-horizon error explosion. Also plot the dominant eigenvalue modulus against perturbation size; the initial slope should match the sensitivity estimate.