Log-Fourier Normal-Form Recurrent Cell

1. Integration point: implement a second-order continuous-time RNN or neural-ODE cell. For hidden vector h, use oscillator states x and v with x_dot = v and v_dot = -x^(2n+1) - sum over j<n of b_j(theta)x^(2j+1) - [sum over j<=m of a_j(theta)x^(2j+1)]v + W u. Let theta(t) = omega t + theta_0, and parameterize each a_j and b_j by learned Fourier coefficients. Start with scalar or two-dimensional oscillator blocks before scaling to a full hidden state. 2. Pseudocode: compute A = 1 + norm(h), set K = ceil(c_* log(A)), zero Fourier coefficients whose integer frequency satisfies norm(k)_1 > K, estimate denominators k dot omega, and apply q_hat_k <- q_hat_k - r_hat_k divided by (i k dot omega) only when |k dot omega| >= gamma norm(k)^(-tau). Integrate the corrected cell with RK4 or a split integrator. 3. Computed from the paper: the degree condition, logarithmic cutoff, Diophantine test, and analytic-tail prediction. Estimated empirically: the analyticity constant, correction size, hidden-state growth rate, and practical resonance threshold. 4. First cheap experiment: compare a vanilla tanh GRU, an unfiltered oscillator RNN, and this cell on the adding task and Copy-Memory, followed by autonomous 1,000- to 10,000-step rollouts. Sweep c_*, m, and n, using n = 2(m+1) as the predicted boundary. Measure maximum hidden norm, Jacobian-product norm, and rollout MSE. The quantitative signature is a sharp reduction in state-growth probability for n >= 2(m+1). The discarded-mode residual should scale approximately as A^(-rho c_*); fitting log residual against log A should give slope approximately -rho c_*. Choosing K below c_* log A should increase resonance spikes and long-horizon divergence.