Parabolic Torus Recurrent Core

1. Integration point: replace the transition block of a small state-space model or GRU used for long-horizon forecasting. Split its hidden state into phase coordinates \(\theta\in\mathbb{R}^{d}\) and transverse coordinates \(z\in\mathbb{R}^{r}\), with \(\theta\) represented by sine and cosine features. Use a fixed or learned frequency vector \(\omega\), a zero or nilpotent upper-triangular parabolic matrix \(A_0\), and inject the input through the decaying perturbation. 2. Pseudocode: set \(t=1\); at each step compute \(g=t^{-\alpha}\), then \(f_\theta,f_z=\mathrm{MLP}([\sin\theta,\cos\theta,z,u_t])\). Update \(\theta\leftarrow\theta+\Delta t[\omega+\varepsilon g f_\theta]\) and \(z\leftarrow z+\Delta t[A_0z+\varepsilon g f_z]\), and predict from \([\sin\theta,\cos\theta,z]\). Train the output head and perturbation networks by teacher forcing, using spectral normalization or clipping to bound their Jacobians. 3. Computed from the mechanism: the factor \(t^{-\alpha}\), the phase transport term, and the cumulative bound \(E(T)\). Estimated empirically: the residual-to-torus error \(r(T)\), effective frequency, and perturbation norm. 4. First cheap experiment: train this model and a matched GRU and linear SSM on sums of two incommensurate sinusoids with transient inputs, then evaluate 1000-step free rollouts. Sweep \(\alpha\in\{0.5,0.75,1,1.25,1.5\}\). The prediction is residual drift proportional to \(T^{1-\alpha}\) for \(\alpha<1\), proportional to \(\log T\) at \(\alpha=1\), and saturation for \(\alpha>1\). Fitted log-log slopes should be approximately \(1-\alpha\) below the boundary, while the constant-coupling control should show linear drift.