Twisted-Shift Feature Mixer

Integrate the module before the token-mixing or channel-mixing projection in a small Transformer, SSM, or one-dimensional signal model. Let the graded feature dimension be n+1 and represent each complex feature with two real channels, giving width 2(n+1). Choose q=exp(i theta), with theta fixed from a known period or learned as a scalar parameter. For each grade j, implement multiplication by q^j using a 2 by 2 real rotation with entries cos(j theta), -sin(j theta), sin(j theta), and cos(j theta). Implement S as a sparse lower shift that copies grade j into grade j+1 and zeros the final grade. Parameterize the mixer as A=sum_r a_r S^r, where a_r can be scalar coefficients or learned 2 by 2 real blocks. Compute it without a dense matrix: set u=z and y=0; for r from 0 to R, add a_r times u to y, then replace u by S u. Optionally add phase-modulated taps b_r T u. Return LayerNorm(z+y) through a residual connection. The paper supplies the exact algebra; estimate theta, tap count R, coefficient initialization, and normalization empirically. Test first on Sequential MNIST, sinusoid forecasting, and a synthetic sequence task with phase transformation z_j mapped to q^j z_j. Compare against a dense mixer and standard SSM at matched width, parameters, and FLOPs. Success is lower phase-shifted test error, better extrapolation to unseen phases, or equal accuracy with fewer parameters. Monitor gradient norms and the commutation residual, which should remain near machine precision.