Charged Diffusive-Precession State Space

Integrate this as a replaceable mixer inside a small SSM, RNN, or spatiotemporal Transformer block. For an image, video, or grid sequence, FFT the input along spatial axes; for a plain 1D sequence, use a bank of learned frequencies \(\mu_{3,j}\) and omit \(\lVert\vec{k}\rVert^2\), or assign each bank element a fixed scale \(q_j\). For each channel group and frequency, maintain complex states \(h^{(1)},h^{(2)}\). Initialize \(\mu_{3,j}=\operatorname{softplus}(\theta_{\mu,j})\), \(D_j=\operatorname{softplus}(\theta_{D,j})\), compute \(z_1=\exp(-i\mu_{3,j}\Delta t)\) and \(z_2=\exp((-i\mu_{3,j}-D_jq^2)\Delta t)\), then execute the two recurrences in the displayed formula. Apply a learned real projection after inverse FFT and add a gated residual from the original block. Use both signs of \(\mu_3\) by pairing conjugate channels, guaranteeing real outputs. The paper supplies the pole locations and the stability condition \(D\ge0\); the network learns \(\Sigma\)-like gains, input/output matrices, and the mixture with the residual branch. Estimate \(q^2\) exactly from FFT frequencies, or use fixed logarithmically spaced scales for 1D data. First test on Copy, Adding, and Long Range Arena tasks using a 100k-token context, comparing equal-width S4/Mamba-style diagonal SSMs and a GRU. Track validation loss versus context length, gradient norm, and recurrent rollout stability. A successful signal is lower long-context loss or fewer divergence events at equal parameters and FLOPs; on video prediction, measure multistep error and whether the learned \(D\) orders low spatial frequencies as slowly decaying modes.