Schur-Agler contractive gated layer

Integrate this at the transition map of a small GRU-like sequence model or at the residual branch of a Transformer. First normalize each input feature group x^(j) to [-1,1], compute three bounded scalar gates alpha_j(x)=tanh(w_j^T x+b_j), and form the block-diagonal X(x)=diag(alpha_1 I, alpha_2 I, alpha_3 I). Parameterize the colligation U=[[A,B],[C,D]] as a spectral-normalized matrix with ||U||_2<=1; enforce this by dividing U by max(1,sigma_max(U)) after every optimizer step, or by using two power iterations to estimate sigma_max. Evaluate y=Aq+Bp, where p=X(x)h and q=(I-DX(x))^{-1}Ch. Compute the inverse with 3-8 fixed-point iterations q_{k+1}=Ch+DX(x)q_k, which is stable when ||DX(x)||<1; alternatively solve directly for small hidden dimension. Use the layer as h_{t+1}=h_t+gamma y with gamma in [0,1], or replace the usual recurrent transition entirely. The paper's mathematics supplies the block contraction and Gram-kernel identity; the feature maps, dimensions, and colligation are learned. During training, sample pairs x_i,x_j and optionally add L_kernel=|< [1;X_i v_i],[1;X_j v_j]>-f_i\overline{f_j}-<v_i,v_j>|^2, with v_i learned intermediate features, but begin without this auxiliary loss. The first experiment should compare a 2-layer 128-unit vanilla RNN, spectrally normalized RNN, and this realization layer on sequential MNIST or copy-memory at equal parameter count. Measure exploding-gradient frequency, maximum hidden-state norm, validation accuracy, and loss under long sequences and distribution-shifted input scales. Success is lower gradient variance and no hidden-state blow-up at comparable accuracy, or longer stable sequence lengths at equal FLOPs.