Maslov Phase Budget for Symplectic Recurrence

Integrate this at the recurrent transition, not at the input projection. Build a symplectic hidden-state matrix M_t for each sequence step, for example from a Hamiltonian exponential M_t = exp(JH_t) with H_t symmetric, or from a learned symplectic matrix shared across time. Partition M_t into n-by-n blocks a_t,b_t,c_t,d_t. Compute D_t = a_t^T a_t + c_t^T c_t, obtain D_t^{-1/2} by an eigendecomposition for the first MVP or by Newton-Schulz iterations for larger models, and form the complex matrix G_t = (a_t+i c_t)D_t^{-1/2}. Convert G_t to its real symplectic embedding U_t = K(G_t). For t greater than zero, calculate q_t = Trace(J (U_t-U_{t-1}) U_{t-1}^{-1})/(2 Delta t); use a linear solve instead of explicitly forming U_{t-1}^{-1}. Add lambda_phase times q_t squared to the task loss and optionally lambda_tv times (q_t-q_{t-1}) squared. Do not detach U_t so gradients reach the recurrent parameters. The paper supplies the compact factor and trace identity; the engineer estimates q_t by finite differences and computes matrix functions numerically. Start with a 64-dimensional symplectic RNN on sequential MNIST and a copy-memory task of lengths 100, 500, and 1000. Compare vanilla tanh, orthogonal RNN, unregularized symplectic RNN, and the proposed loss at equal parameter count. Track gradient norm, hidden-state norm, q_t variance, and accuracy versus sequence length. Success means lower long-horizon gradient variance and at least 5 percentage points higher copy accuracy at length 1000 without more than 10 percent training slowdown.