Hill-Floquet Regularization for Periodic RNNs

Use this after each optimizer step, or every 100 steps, for an RNN or SSM trained on a periodically repeated sequence with period T. First obtain a periodic hidden trajectory x_*(t_m) by unrolling one period and requiring x_*(T) to approximately equal x_*(0), or by applying a short shooting correction. If the architecture has constrained states, express it as A dot(x) = f(t,x) with a singular diagonal mask A; for an unconstrained model use A = I. Sample M at least 4N+1 time points, compute J_m = partial f(t_m,x_*(t_m))/partial x with autodiff or Jacobian-vector products, and FFT every Jacobian entry to estimate J_k. Assemble H_N(mu), solve the truncated generalized eigenvalue problem, and obtain the dominant Floquet exponents. For singular A, discard eigenvectors with large algebraic-constraint residuals and use the Drazin-inverse or consistency projector when constructing tangent evolution. Add beta times L_Hill to the task loss; if eigenvalues are poorly separated, stop-gradient through the monitor and use it to reduce the learning rate. The paper supplies the Hill construction and stability criterion; the orbit, Fourier coefficients, Jacobians, and truncation error are estimated numerically. First experiment: compare a 32-unit tanh GRU and diagonal SSM on periodically forced sine prediction, with T = 2 pi, M = 128, N = 8, Adam baseline, and Adam plus the penalty. Independently propagate tangent vectors for one period to estimate the monodromy multiplier. The prediction is decay for rho_F < 1 and divergence for rho_F > 1, with the measured boundary within 20% of the Hill prediction and less than 10% change in the dominant exponent when N is increased from 4 to 8.