KAM-Stabilized Quasiperiodic Recurrent Memory

Integrate the mechanism as a recurrent state with phase theta_t in [0,1) and scalar or low-dimensional transverse state z_t. Use theta_(t+1) = frac(theta_t + omega + lambda z_t + epsilon phi_psi(theta_t)) and z_(t+1) = lambda z_t + epsilon phi_psi(theta_t) + W_r r_t, where r_t is the input embedding and phi_psi is a small Fourier network. Maintain a learnable graph h_eta(theta), represented by K sine and cosine modes. For each rollout, compute the prediction loss, the graph residual L_graph = mean_t ||z_(t+1) - h_eta(theta_(t+1))||^2, and the invariance residual L_inv = mean_t ||h_eta(theta_(t+1)) - lambda h_eta(theta_t) - epsilon phi_psi(theta_t)||^2. Estimate a KAM correction from residual samples on a uniform phase grid: take an FFT, divide each nonzero Fourier coefficient by exp(2 pi i k omega) - 1, truncate at frequency K, and add the correction to h_eta. The paper supplies the twist-map equation, invariant-graph condition, Brjuno criterion, and small-divisor correction; the graph, effective perturbation norm, residual spectrum, and Lyapunov exponents are estimated during training. First test on delayed-sine and Mackey-Glass prediction with equal-size tanh-RNN, fixed-frequency phase RNN, and corrected phase RNN baselines. Sweep epsilon and compare golden-ratio omega = (sqrt(5)-1)/2 against near-rational omega = 13/21. Measure graph residual, prediction error at horizons 100 to 10000, and the largest transverse Lyapunov exponent. The predicted signature is a sharp loss of graph persistence when A_K(omega) = max_(1<=|k|<=K) |exp(2 pi i k omega) - 1|^(-1) becomes large; the critical epsilon should decrease approximately as 1/A_K(omega), while the golden-ratio module should remain bounded at larger epsilon than the near-rational module.