Normal Damped-Rotation Recurrent Layer

Integrate this at the recurrent state transition of a tanh RNN or GRU-like minimal baseline. Choose hidden width d = 2K. Store K unconstrained frequency parameters omega_j and positive decay parameters gamma_j = softplus(rho_j) + gamma_min. Construct each 2-by-2 block B_j = exp(-gamma_j) [[cos(omega_j), -sin(omega_j)], [sin(omega_j), cos(omega_j)]]. Either use Q = I for the cheapest version or generate one fixed random Gaussian matrix and orthogonalize it once with QR; set W = Q blockdiag(B_j) Q^T. At every timestep compute preactivation a_t = W h_t + W_in x_t + b and h_{t+1} = tanh(a_t). For a trainable recurrent layer, optimize W_in, b, and optionally Q through a Cayley or QR parameterization, and optimize rho_j and omega_j; for a reservoir, freeze W, W_in, and b and train only a linear readout on [h_t, x_t, 1]. Pseudocode: initialize gamma_j and omega_j; build B_j; build W; for each batch and timestep, h = tanh(W @ h + W_in @ x + b); accumulate readout loss. The formula ||W||_2 = exp(-min gamma_j) is computed analytically and should be logged as the predicted linear stability margin; estimate the actual nonlinear Jacobian norm with occasional power iteration. First test on sequential MNIST and NARMA-10 using width 128, comparing vanilla tanh RNN, orthogonal RNN, and this layer at equal parameters and FLOPs. Success means fewer exploding or saturated trajectories, a lower validation loss at equal training steps, and stable performance when sequence length increases from 100 to 1000.