Braid-word reversible mixer

Integrate the mixer at the token-mixing or channel-mixing line of a small Transformer or MLP, immediately before the feed-forward block or in place of a dense linear projection. Let the input be X in R^(B x n x d). Choose a braid word w = ((i_1,s_1),...,(i_k,s_k)), where i_t is an adjacent pair in {1,...,n-1} and s_t in {+1,-1} selects the generator or inverse. For each batch and feature dimension, apply this pseudocode: Y = X; for (i,s) in word: if s == +1: (Y[...,i],Y[...,i+1]) = (Y[...,i+1], -Y[...,i] + 2*Y[...,i+1]); else: (Y[...,i],Y[...,i+1]) = (2*Y[...,i] - Y[...,i+1], Y[...,i]); Y = Y / sqrt(2). Use the same scale convention in every branch and optionally learn one scalar gate g_t per crossing, initialized at zero, with Y <- Y + g_t*(cross(Y)-Y). For exact reversibility, omit gates and use the explicit inverse word in reverse order. The paper supplies the local update, inverse, braid identities, and invariant constant direction; empirical choices are word length, index pattern, sign pattern, normalization, and gating. Log the ratio ||Y||_2/||X||_2 and compute singular values of the composed operator on a probe matrix. First run on CIFAR-10 with a 6-layer, width-256 MLP-Mixer or a 4-layer, 8-head Transformer. Compare against a fixed permutation, depthwise convolution, butterfly mixing, and a dense learned n-by-n mixer at matched FLOPs. Measure accuracy, loss at equal optimizer steps, wall-clock throughput, activation-norm drift, and parameter count. Success means comparable or better accuracy with at least 2x lower mixer FLOPs and bounded activation norms; failure means norm explosion, severe rank deficiency, or slower loss descent than the permutation baseline.