Spectrally identifiable phaseless recurrent layer

1. Integration point: replace the hidden-state transition in a small complex-valued RNN or graph neural network with a fixed graph Schrödinger encoder. Let $h\in\mathbb{C}^n$ be the incoming hidden state, choose a connected graph Laplacian $\Delta_G$, and compute $K$ intensity observations $z_r=|U_rh|$ with $U_r=\exp(-i\tau_rH_Q)$ and $\tau_r=r\Delta t$. Concatenate $z_0,\ldots,z_{K-1}$ and feed them to a real MLP or classifier. 2. Pseudocode: initialize $H=\Delta_G+\operatorname{diag}(q)$; precompute $U=\operatorname{matrix\_exponential}(-i\Delta tH)$; set $x=h$; for $r=0,\ldots,K-1$, append $|x|$ to a list and set $x=Ux$; concatenate the list and apply the readout. Use $K=4$--$16$ and normalize each magnitude vector by its RMS. 3. During initialization, eigendecompose $H=\Phi\Lambda\Phi^T$, calculate $S=\Phi\odot\Phi$, reject potentials with small $\sigma_{\min}(S)$, reject near-collisions among all pair sums $\lambda_j+\lambda_k$, and reject eigenvector pairs with no overlapping support. In floating point, use thresholds such as $\sigma_{\min}(S)>10^{-2}$ and minimum pair-sum separation above $10^{-3}$ after spectral normalization. The uniqueness theorem is imported from the paper; the time grid and conditioning thresholds are empirical. 4. First experiment: compare a 2-layer complex graph-RNN using this encoder against a same-parameter real GRU and a generic magnitude-only encoder on synthetic graph-sequence classification. Generate random complex initial states and labels invariant to global phase, train at equal parameter count and FLOPs, and test robustness to random global phase and additive sensor noise. Success is higher accuracy and lower variance across global phases, ideally matching the full complex-state baseline while exposing only magnitude observations.