Quasiperiodic Additive State Module

Insert the module before a lightweight output head in a sequence predictor, or replace part of a diagonal state-space model's recurrent state. For each sequence, initialize a scalar phase x_0 either to a learned value or deterministically from the sequence index, and initialize y_0=0. At each step compute x=(x+alpha) modulo 1, evaluate h=sum_j cos(2*pi*q_j*x)/q_j, and update y=y+h. Feed the token embedding u_t together with x, y, and optional sine and cosine phase channels into a two-layer gated head. A learned gate g can scale the injected state, for example z_t=MLP([u_t,g*y,x,phi_J(x)]), with g initialized near zero. Compute q_j once from the continued-fraction recurrence and cache them. The fixed recurrence is the paper's mathematics; gate strength, truncation J, phase initialization, and feature normalization are empirical. Keep alpha fixed for the first experiment. Test on delayed-copy, adding, and periodic-plus-noisy sequence tasks, followed by sequential MNIST or a small character-level language model. Compare against a GRU, a diagonal SSM, and sinusoidal positional encoding with matched state size. Measure validation loss versus sequence length, gradient norm variance, parameter count, and recurrent memory. Success is competitive short-context accuracy with slower degradation at long contexts and fewer learned recurrent parameters.