Jordan-Isometric Matrix Layer

Integrate this construction into a matrix-valued MLP or state-space layer. Represent each sample's hidden state as X with shape [batch,n,n] and output Y with shape [batch,k,k]. Start with alpha=1 and beta=1, so J(X) is the block diagonal matrix X direct-sum X^t of size 2n by 2n. For the cheapest exact-isometry baseline, set m=2n, T_1=T_2=I_{2n}, f=0, and draw V_1,V_2 as fixed unitary matrices from QR decompositions of Gaussian matrices. Compute J = block_diag(X, transpose(X)), then Z = conjugate_transpose(T1) @ J @ T2, and finally Y = V1 @ Z @ conjugate_transpose(V2). To train T_1,T_2, parameterize them by SVD and clamp singular values to [0,1] after every optimizer step; initially use identity matrices so the defect ranks are zero. If width expansion is desired, append a branch f(X) implemented by a linear matrix map and enforce ||f||<=1 using spectral normalization or power-iteration rescaling. The exact theorem supplies the isometry condition; the implementation should empirically estimate ||Phi|| by batched power iteration and verify ||Y||_op/||X||_op near one. The first experiment should train a 12-layer matrix MLP on CIFAR-10 after reshaping channel groups into n-by-n matrices. Compare this layer with equal-width unconstrained linear layers, spectral normalization, and orthogonal initialization. Record loss at equal FLOPs, final accuracy, activation-norm drift across depth, and gradient-norm variance. A successful result is bounded activation norms and lower gradient variance with equal or better accuracy; a second experiment should test long-horizon stability in a recurrent state update.