Annealed-resonance recurrent dynamics

1. Integration point: use a stochastic RNN or state-space model with $M$ transition branches, $h_{t+1}=T_{\xi_t,\theta}(h_t)$, where $\xi_t$ is sampled independently at each sequence step or per training example. Start with affine branches $T_m(h)=\tanh(A_mh+B_mx+b_m)$, and first perform the diagnostic with linear recurrent branches. 2. Pseudocode: roll out independent branch samples, collect feature vectors $z_t=\phi(h_t)$, and fit an empirical annealed Koopman matrix by ridge regression, $\widehat K=\arg\min_K\sum_t\|z_{t+1}-Kz_t\|^2+\alpha\|K\|_F^2$. Include a constant feature, estimate its invariant mode, then project that mode out. Compute eigenvalues $\widehat\lambda_j$ and add $L_{\mathrm{res}}=\gamma[\max(0,\max_j|\widehat\lambda_j|-\rho_*)]^2$. For oscillatory memory, add a penalty matching selected poles $\rho_*e^{\pm i\theta_*}$. 3. Computed from the paper: the averaged-operator construction, iid composition rule $K^n$, and resonance-to-correlation correspondence. Estimated empirically: the finite-feature Koopman matrix, invariant distribution, eigenvalues, and correlation coefficients $c_j$. 4. First cheap experiment: train a 32-unit stochastic RNN on adding and delayed-copy tasks, comparing a shared deterministic RNN, unconstrained random branches, and resonance-constrained branches. Measure hidden-state autocorrelations and test loss versus lag. The falsifiable prediction is that, after removing the constant mode, correlation envelopes follow $\rho^n$ and have half-life $n_{1/2}=\log(1/2)/\log\rho$ within about 20%. Instability or growing correlations should appear when the estimated spectral radius crosses $1$, while targeted complex poles should produce a spectral peak near frequency $\theta_*/(2\pi)$.