Ramanujan Signed Ring Mixer

Integrate this as a fixed sequence mixer or graph-convolution layer for tensors H with shape [batch, n, d], where n is even and d is the feature width. Construct A_sigma H without storing a dense matrix: for each position i, compute M_i=H_{i-1}+H_{i+1}+(-1)^i(H_{i-2}+H_{i+2}), using circular indexing. In tensor code, create a parity vector p_i=(-1)^i and calculate M=roll(H,+1)+roll(H,-1)+p*(roll(H,+2)+roll(H,-2)); then calculate Y=activation(c*M@W+b). At every training step, estimate the spectral norm of W with one to three power-iteration steps, or use a cached estimate updated once per optimizer step. Set c=gamma/(2√2*||W||_2), initially gamma=0.9, so the fixed graph factor is the exact paper value rho(A_sigma)=2√2 and only the learned channel norm is estimated. Compare against the unsigned operator M_plus=roll(H,+1)+roll(H,-1)+roll(H,+2)+roll(H,-2), normalized with c=gamma/(4*||W||_2), and against a standard residual 1D convolution with the same parameter count. First test 12- and 24-layer models on ring-node label propagation, cyclic denoising, and a small sequence-copy task. Record loss versus optimizer steps and FLOPs, gradient norms by depth, activation variance, and validation accuracy. The hypothesis is that the signed mixer permits larger unnormalized mixing or deeper stable stacks; success is faster loss descent at equal FLOPs, fewer exploding or vanishing gradients, or higher accuracy at equal depth.