Periodic CMV Unitary Recurrent Layer

(1) Integration point: use this transition as the recurrent matrix in an Elman RNN or as the state transition in a linear/nonlinear SSM. For a real implementation, either represent each complex coordinate with two real channels or restrict the cores to real 2D rotations. Store one parameter pair for each core in a period instead of storing an n-by-n matrix. For a real rotation, use C_k=[[cos(theta_k),-sin(theta_k)],[sin(theta_k),cos(theta_k)]]. For the complex parameterization, maintain unconstrained c_k and s_k and normalize them by r_k=sqrt(|c_k|^2+|s_k|^2+epsilon), setting c_k<-c_k/r_k and s_k<-s_k/r_k. (2) Pseudocode: input h_t, x_t, and period-core parameters; set v=h_t; for each odd-position core in M, apply its 2x2 unitary to adjacent coordinates; for each even-position core in L, apply its 2x2 unitary to the staggered adjacent coordinates; for the periodic version, apply the last core to the wrapped pair (v_n,v_1); compute h_{t+1}=phi(Uv+B x_t), or use h_{t+1}=Uv+phi(Bx_t) when strict linear norm preservation is preferred. If a cyclic core must be moved through an overlapping sequence, form the affected 3x3 product and refactor it as the turnover equation specifies using a small unitary QR or SVD, replacing the three old local cores by the three new cores. (3) Compute directly from the mathematics: the alternating support pattern, periodic parameter sharing, unitary local updates, and norm-preserving transition. Estimate empirically: hidden-state norm distributions, gradient norms, Jacobian singular values, and the best nonlinear placement or optional scalar gain. (4) First cheap experiment: train a width-256 sequential-MNIST or copy-memory model with a dense tanh RNN, a standard orthogonal RNN, and this CMV layer using periods 2, 8, 32, and 256. Record validation accuracy, loss versus sequence length, gradient norm variance, parameter count, peak memory, and wall-clock throughput. Success means stable gradients over longer sequences than the dense baseline, comparable or better accuracy than the orthogonal baseline, and approximately linear parameter and memory scaling. A stronger result would be a period-8 or period-32 model matching the full-width unitary baseline with substantially fewer trainable parameters.