Asynchronous p-Mean Token Mixer

1. Integration point: insert the module between Transformer blocks as a token-mixing residual layer. Construct a fixed sparse graph on the T token positions, such as a 2-D grid for image patches, a ring plus local windows for text, or a k-nearest-neighbor graph in positional space. Apply the same operation independently to every hidden channel, optionally after a learned scalar gate and normalization. 2. Pseudocode: input X[T,d]; set Y=X; for k=1,...,K choose vertex v using a random, cyclic, or learned schedule; for every channel c solve g(a)=sum_{u in N(v)} |a-Y[u,c]|^(p-2)(a-Y[u,c])=0 by 8-12 bisection iterations on the interval [min_u Y[u,c], max_u Y[u,c]]; set Y[v,c]=a and leave all other rows unchanged; return X+alpha*(Y-X). Bisection is valid because g is monotone for p>1 and the minimizer lies in the neighbor-value interval. 3. Compute the p-mean numerically, while treating the paper's consensus-time scaling as a qualitative graph-design prediction rather than an exact neural-network guarantee. For differentiable implementation, backpropagate through the fixed bisection iterations, or use the implicit derivative da/dy_j=|a-y_j|^(p-2)/sum_l |a-y_l|^(p-2) away from coincident values. Track the token oscillation statistic osc(Y) and the norm of the residual update. 4. First cheap experiment: compare a 6-layer small Transformer using this mixer against local-window attention and full attention on CIFAR-10 ViT-Tiny or WikiText-2 with equal hidden size and matched training FLOPs. Sweep p in {1.25,1.5,2,3,4}, K, and alpha. Success means lower validation loss or higher accuracy than local averaging at equal FLOPs, with token-mixing cost scaling as O(KTd) rather than O(T^2d), and no instability in the oscillation statistic.