CFL-Optimized Palindromic Residual Block

Integrate this at the state-transition operation of a Hamiltonian RNN, linear or nonlinear SSM, or antisymmetric residual network. Represent the hidden state as two equal-width tensors h and e, and parameterize the coupling as W=R-R^T for a dense layer, or as a convolution paired with its adjoint. Do not materialize matrix exponentials: each exponential is a shear and requires one application of W. Set a=0.25 and execute the six assignments in the final formula. For nonlinear models, initially evaluate W at the current stage state only after validating the exactly linear version, so the stability mechanism is isolated. Estimate the operator norm every K optimizer steps with 3 to 5 power iterations: initialize a unit vector v, repeatedly compute v <- W^T W v / ||W^T W v||, and set L_hat=||Wv||. Use delta_t=min(delta_t_max, eta*12/(7*sqrt(d)*L_hat+epsilon)) with eta between 0.5 and 0.9. This is a conservative neural transfer of the paper's CFL bound, not a theorem for arbitrary W. First test a two-dimensional linear oscillatory sequence task with W=R-R^T, comparing Euler, standard leapfrog, and the palindromic block at equal W-application counts. Then test sequential MNIST or copy-memory with a small SSM. Record maximum stable step size, hidden-state norm drift over 10000 unforced steps, gradient norms, training loss versus operator calls, and long-horizon validation error. Success means a larger stable step and lower long-horizon error at equal compute; failure is no gain after accounting for the five W applications.