Low-Rank Radiation Resonant State Space

(1) Integration point: implement this as a recurrent or state-space block between the input projection and output projection. Map each input token \(x_t\in\mathbb{R}^{d_{in}}\) to resonant coordinates with \(Ux_t\). Maintain displacement \(h_t\in\mathbb{R}^m\) and velocity \(v_t\in\mathbb{R}^m\). Parameterize \(K=LL^{\mathsf T}+\kappa I_m\), with \(\kappa>0\), and parameterize damping as \(BB^{\mathsf T}\), where \(B\in\mathbb{R}^{m\times r}\) and \(r\ll m\). (2) Pseudocode: compute \(a_t=Ux_t\); compute \(p_t=B^{\mathsf T}v_t\); compute the regularized dark projector \(P_{dark}=I-B(B^{\mathsf T}B+\epsilon I)^{-1}B^{\mathsf T}\); update \(v_{t+1/2}=v_t+\Delta t(a_t-\delta Kh_t-\delta Bp_t-\eta P_{dark}v_t)\); update \(h_{t+1}=h_t+\Delta t v_{t+1/2}\); then use a linear or gated readout of \(h_{t+1}\), optionally with a residual path from \(x_t\). Use semi-implicit integration if explicit Euler becomes unstable. (3) The mathematical structure fixes the low-rank damping and bright/dark decomposition. Estimate empirically the rank \(r\), stiffness spectrum, \(\delta\), \(\eta\), and step size. Monitor the eigenvalues of the discrete transition matrix and the condition number of \(\delta K-\omega^2I_m-\mathrm{i}\delta\omega BB^{\mathsf T}\) over frequencies found by an FFT of training sequences. (4) First experiment: use \(m=64\), \(r=4\), and two layers on sequential MNIST, the Copying task, and a synthetic long-memory task requiring a delayed weighted sum. Compare against a same-width GRU and a diagonal SSM with matched parameter count and optimizer. Measure validation loss at equal FLOPs, gradient norm, hidden-state retention versus sequence length, and the largest discrete-transition eigenvalue. The idea succeeds if it improves long-context loss or memory retention without exploding gradients, and if low-rank damping outperforms both full-rank damping and no damping at equal parameter count.