Isochronous Emden recurrent cell

Integrate this as a drop-in recurrent or SSM layer. (1) Exact integration point: maintain two hidden tensors \(x_t,v_t\in\mathbb{R}^{B\times d}\) alongside the sequence state; use them to produce \(h_t=\operatorname{LayerNorm}(W_xx_t+W_vv_t+b)\), then combine \(h_t\) with the token embedding through a gated residual update. Initially choose fixed odd \(q=1\) or \(q=3\), rather than learning q. Parameterize \(\omega_i=\operatorname{softplus}(a_i)+\omega_{\min}\) and \(k_i=k_{\max}\tanh(c_i)\). (2) Pseudocode: x,v=x0,v0; for each token compute e=Embedding(token), u=W_u e, a=-(q+2)*k*x**q*v-k**2*x**(2*q+1)-omega**2*x; update (x1,v1) using one RK4 step for dx/dt=v, dv/dt=a; compute g=sigmoid(W_g[e,h(x1,v1)]); set x=(1-g)*x+g*x1 and v=(1-g)*v+g*v1; return h=LayerNorm(W_x*x+W_v*v+b). Apply the forcing term through the gate or readout first; only later test direct a += u. (3) Compute from the mathematics the acceleration and predicted period \(2\pi/\omega_i\). Estimate empirical phase using zero crossings of \(x_i\), and amplitude using \(\sqrt{x_i^2+(v_i/\omega_i)^2}\). Learn \(k_i,\omega_i\) through backpropagation through RK4, while penalizing excursions with \(\lambda\max(0,|x_i|-x_{\max})^2\). (4) First experiment: train a 64- or 128-dimensional model on copy-memory, sequential MNIST, and a small WikiText-2 character prediction task. Compare against a tanh RNN, a linear diagonal SSM, and the same architecture with \(k=0\). Measure validation loss versus sequence length, gradient norms, hidden-state norms, equal-FLOP throughput, and phase drift \(|\widehat{T}_i-2\pi/\omega_i|\). Success requires lower long-sequence loss and bounded states than the tanh RNN, with measured periods remaining nearly constant across a 10-fold amplitude range. Run an even-q=2 ablation; it should show decay and no sustained periodic memory, testing whether the odd-q structure is causally useful.