Rotational-Twist Recurrent Layer

1. Integration point: implement this as a continuous-time recurrent or state-space block between embedding/input projections and the task readout. Store $N$ complex channels, represented in code by $2N$ real channels. Choose a directed circulant kernel with first row $a_0,\ldots,a_{N-1}$; compute its eigenvalues once with an FFT. Select a nonzero Fourier mode $q$ and initialize $\alpha$, $\beta$, and $\kappa$ so that $\mu_q$ is slightly positive while competing modes have negative linear growth. Use RK4 or a semimplicit solver over each token interval. 2. Pseudocode: compute $z\leftarrow z+\Delta t[(\alpha+i\omega)z-\beta|z|^2z+\kappa(Az-dz)+P x_t]$; optionally apply a mode penalty $L_{mode}=|\widehat z_q|^2/(\sum_p|\widehat z_p|^2+\epsilon)$ and a radius penalty $(|z|^2-R_q^2)^2$. The readout receives $[\operatorname{Re}z,\operatorname{Im}z]$. 3. Compute $\lambda_q$, $d$, $\mu_q$, $R_q$, and $\Omega_q$ analytically from the circulant kernel; estimate the actual transverse exponent $\nu_p$ empirically by propagating Jacobian-vector products or by fitting perturbation decay. 4. First cheap experiment: train this layer and an equal-parameter vanilla GRU on Copy Memory and sequential MNIST. Sweep $\kappa$ across the predicted threshold $\kappa_c$. With zero input after initialization, measure Fourier-mode amplitude, phase velocity, and perturbation decay. The predicted signature is no oscillation below $\mu_q=0$, $R_q$ growing with slope $1/2$ on a log-log plot above threshold, frequency $\Omega_q$ matching the eigenvalue formula within 10%, and exponential perturbation decay with rate approximately $\gamma$ when the selected attractor is stable. Also test that replacing the directed kernel by a symmetric one removes the graph-induced frequency shift because its eigenvalues are real.