Calogero Spectral Barrier for Recurrent Dynamics

Integrate the method into a small recurrent network, linear state-space model, or neural ODE discretization. At every K training steps, collect representative hidden states from a minibatch and compute the transition Jacobian J with automatic differentiation; for a linear state-space model this is simply the learned transition matrix A. Compute its eigenvalues z_i exactly when the hidden dimension is below roughly 256. For larger systems, estimate leading eigenvalues with Arnoldi iteration and use a random subset of eigenvalue pairs. Add $\lambda_C R_C+\lambda_\rho R_\rho$ to the task loss, detach the spectrum between updates if eigenvalue gradients are unstable, and clip the total gradient. Pseudocode is: sample states; form J; eigvals = eig(J); collision = sum over i<j of g/(abs(eigvals[i]-eigvals[j])^2+eps^2); radius = max(abs(eigvals)); loss = task_loss + lambda_C*collision + lambda_rho*relu(radius-(1-delta))^2; backpropagate. The Calogero operator and inverse-square mechanism come from the paper; the Jacobian penalty, softening, and contraction term are engineering adaptations. First test on copy/addition and sequential MNIST with equal parameter, wall-clock, and optimizer budgets against the same model with only spectral-radius control, plus an ablation removing $R_C$. Pre-register that the measured divergence boundary occurs at spectral radius 1 within 20%, hidden perturbation norms decay with slope at least $\log(1-\delta)$ over long rollouts, and the number of eigenvalue pairs with spacing below a fixed threshold $s$ is reduced by at least 30% at fixed radius. Reject the transfer if the Calogero penalty does not reduce near-collision frequency, or if it improves neither rollout decay nor validation loss while costing more than 10% training time.