Normal Spectral Linear Layer

Integrate this at the weight matrix of a linear transition in a state-space model, recurrent network, or distillation head. Use complex tensors for the first MVP; represent the trainable weight as a unitary matrix $U$ and complex eigenvalue vector $\lambda$, and do not materialize $A$ unless required. For an input state $h$, compute $z=U^*h$, multiply elementwise by $\lambda$, and return $Uh$ after replacing the latter with $U(\lambda\odot z)$. Parameterize each eigenvalue as $\lambda_i=\rho\tanh(r_i)e^{\mathrm{i}\theta_i}$, with $\rho\leq 1$ for a contractive transition. Initialize $U$ as a random unitary matrix from QR and initialize eigenvalues with small magnitudes. For Procrustes initialization, collect minibatch matrices $X=[h_1,\ldots,h_n]$ and target matrices $Y=[y_1,\ldots,y_n]$, then optimize $f=\|U\operatorname{diag}(\lambda)U^*X-Y\|_F^2$ for 100--500 steps. In each manifold step, obtain the autodiff Euclidean gradient $G_U$, form $\Omega=(U^*G_U-G_U^*U)/2$, and update $U\leftarrow U\exp(-\eta_U\Omega)$; this preserves $U^*U=I$ up to floating-point error. Update $\lambda$ and all other network parameters with AdamW. Re-orthogonalize $U$ with QR every 50 steps if numerical drift appears. The first experiment should compare this layer with an unconstrained dense transition and an orthogonal-only transition on a small linear-system-identification task and a copy-task RNN, matching parameter count and FLOPs. Track loss at equal optimization steps, hidden-state norm over long horizons, gradient-explosion frequency, and validation error. Success means lower initialization error, bounded long-horizon dynamics, or better validation loss at comparable throughput.