Contractive Slow-State Cross-Coupled Reservoir

First, integrate the mechanism as a recurrent backbone for sequence prediction, a reservoir with a trained readout, or a latent transition model in a world model. Use two or more fast-slow cells, with cross-coupling initially entering only through K h_cross in the slow update. Parameterize A_s so its spectral radius is controlled: for a diagonal version use A_s = diag(μ tanh(r_j)); for a full version use A_s = μ Q diag(tanh(r_j)) Q⁻¹, with Q orthogonal. At every time step, compute the fast states, form a permutation or average of other-cell fast states, update the slow states, and then compute the readout from the concatenated fast and slow states. Log the estimated spectral radius of A_s, the empirical forcing magnitude C_t, the predicted radius C/(1−μ), actual state norms, and finite-time Lyapunov exponents. The paper supplies the contraction inequality and absorbing-radius prediction; the forcing bound, fixed points, Lyapunov exponents, and snap-back candidates must be estimated numerically. For a first cheap experiment, compare this model with a tanh RNN and GRU on Mackey-Glass prediction and permuted-MNIST, using matched parameter counts. Sweep μ in {0.8, 0.9, 0.95, 0.99, 1.01} and several values of ||K||. The quantitative prediction is that for μ < 1, state norms eventually remain near or below C/(1−μ), with radius scaling approximately as 1/(1−μ); for μ > 1, this uniform bound disappears. Increasing ||K|| should raise the largest Lyapunov exponent, with a qualitative transition near its measured zero crossing. If a snap-back candidate is found, its computed determinant magnitude should exceed one and coincide with positive long-run Lyapunov exponent.