Projective Spectral Anti-Flattening

1. Integration point: apply this to a GRU, linear state-space model, equilibrium layer, or repeatedly iterated residual MLP. Every $m$ training steps, estimate the local transition matrix around representative minibatch states. For a recurrent block, collect perturbation pairs $(\delta x_t,\delta x_{t+1})$ using small random probes and fit $A$ by least squares; for a low-dimensional SSM, use its explicit state matrix. Center and RMS-normalize the hidden state before the next iteration, while leaving the task output path unchanged. 2. Pseudocode: execute the block; collect probe pairs; compute $A\leftarrow\arg\min_A\sum_t\|\delta x_{t+1}-A\delta x_t\|_2^2$; estimate the two largest eigenvalue moduli $a_1\ge a_2$; add $\mathcal{L}_{\mathrm{spec}}$ to the task loss; optionally rescale the dominant Schur block whenever $a_1/a_2>\kappa_{\max}$. 3. The paper supplies the dominant-eigenvalue dichotomy and predicted alignment rate; the implementation estimates $A$, eigenvalues, and $q_{\min}$ empirically. 4. First cheap experiment: compare a vanilla GRU, GRU with RMS normalization, and GRU with the spectral penalty on sequential MNIST and the Copying Memory task. Sweep $a_1/a_2$. The quantitative prediction is that hidden-state participation ratio and cosine alignment will collapse with slope approximately $\log(a_2/a_1)$ once $a_1/a_2$ exceeds the selected threshold, while the controller keeps the ratio within 20% of $[1,\kappa_{\max}]$ and delays or prevents the collapse.