Phase-retrieval observability regularizer

1. Integration point: apply this to a learnable graph-dynamics layer, not to ordinary transformer weights. Parameterize $H_Q=\Delta_G+\operatorname{diag}(q)$ with trainable $q$, or use a low-dimensional learned diagonal potential added to a fixed graph Laplacian. The forward pass computes $U_r=\exp(-i\tau_rH_Q)$ and magnitude observations $|U_rh|$; add the observability penalty to the task loss. 2. Pseudocode: compute the symmetric eigendecomposition $(\lambda,\Phi)=\operatorname{eigh}(H_Q)$; set $S=\Phi\odot\Phi$; compute $L_c=-\log\det(S^TS+\varepsilon I)$; form every unordered pair sum $s_{ab}=\lambda_a+\lambda_b$ and accumulate $L_s=\operatorname{mean}_{p\ne q}\operatorname{softplus}((\delta-|s_p-s_q|)/\tau)^2$; compute $o_{k\ell}=\sum_jS_{jk}S_{j\ell}$ and $L_o=\sum_{k<\ell}\operatorname{softplus}(\rho-o_{k\ell})$; optimize $L=L_{task}+\alpha L_c+\beta L_s+\gamma L_o$. If autodiff through eigendecomposition is unstable, use a stabilized symmetric eigensolver or apply the penalty less frequently. 3. The paper supplies the structural targets $\det S\neq0$, pair-sum distinctness, and support overlap; $\delta$, $\rho$, $\alpha$, $\beta$, and $\gamma$ are engineering hyperparameters. Estimate conditioning during training using $\sigma_{\min}(S)$ and the minimum pair-sum gap, and monitor them beside task loss. 4. First cheap experiment: use a 20-node random connected graph and a 1-layer complex graph-RNN on synthetic phase-invariant sequence classification. Compare no regularizer, weight decay, and the proposed penalty at equal training budgets. Evaluate held-out accuracy, sensitivity to global phase, additive Gaussian noise on magnitudes, and failures caused by nearly repeated spectra. Success is a larger minimum pair-sum gap and $\sigma_{\min}(S)$, lower validation-loss variance across seeds, and improved accuracy at noisy or short-time observations without a significant increase in forward FLOPs.