Periodic Perfect-Matching Router

Integrate this at the token-mixing operation of a small ViT, grid transformer, or 2D SSM. Let the input be X with shape [batch, height, width, d]. Choose a period r and a matching table containing one destination offset for each source residue class modulo r; enforce the matching constraint by sampling a permutation or by using Sinkhorn during initialization and then projecting to the nearest permutation. For every token (h,w), compute its routed partner using the periodic rule W(i,j)=W(i+r,j-r), gather that token, apply a shared linear map U for its orbit, and add the result to a local residual projection Bx. Pseudocode: res = B(X); for each residue edge e=(i,j): src = periodic_index(h,w,e); res += alpha[e] * U[e](gather(X,src)); Y = phi(res). Use one or a few matching layers before a dense MLP; optionally stack K independently shifted matchings so the receptive field grows while each layer remains O(HW d) rather than O((HW)^2 d). Compute the mathematical route exactly from the stored permutation; estimate no spectral quantities. First experiment: CIFAR-10 or ImageNet-100 with a 4-layer ViT at equal parameter count, comparing dense attention, local-window attention, and one to four matching layers. Measure validation accuracy, wall-clock throughput, and memory. Success is at least 2x lower attention-memory cost with no more than 1 percentage-point accuracy loss, or better accuracy than local windows at equal FLOPs.