Symplectic Polar Transition Normalization

Use this as a recurrent or state-space transition normalization. Parameterize a base transition M_theta as a symplectic matrix, preferably with a product of differentiable symplectic shears or M_theta = exp(JH_theta), where H_theta is symmetric. At every optimizer step, or every K forward steps for speed, split M_theta into blocks a,b,c,d and compute D = a^T a + c^T c. Add epsilon I before the eigendecomposition, compute D^{1/2} and D^{-1/2}, and form G = (a+i c)D^{-1/2}. Construct the compact real embedding K(G). Reconstruct the remaining symplectic factor from the displayed positive block and symmetric shear, but replace D^{1/2} and D^{-1/2} by D^{beta/2} and D^{-beta/2}. Apply the resulting M_beta to the hidden state. Begin with beta=0.25 and linearly anneal to 1 over 20 percent of training, or learn beta through sigmoid(beta_raw) constrained to [0,1]. The paper determines the factorization and guarantees the shear is symmetric; the implementation estimates matrix functions numerically and should verify ||M_beta^T J M_beta-J||_F after reconstruction. Test on sequential MNIST and a length-512 synthetic harmonic-memory task against an unnormalized symplectic RNN and an orthogonal RNN. Measure exploding and vanishing hidden norms, maximum gradient norm, wall-clock cost, and validation accuracy. The desired signal is fewer divergence runs and better long-context accuracy at equal parameter count; reject the method if matrix-function overhead exceeds 20 percent or if short-context accuracy decreases by more than 1 percent.