Scalene anticommuting three-token mixer

1. Integration point: insert the module between transformer blocks as a local three-token mixer. Partition a sequence into overlapping triples (x_i,x_j,x_k), project each token embedding to a channel width divisible by 2, and interpret each adjacent pair of channels as a two-dimensional algebra register. Apply the fixed matrices U=X and V=Z independently to the registers; their Kronecker products produce U_1U_2, V_1V_2, U_1U_3, V_1V_3, U_2U_3, and V_2V_3. The six scalar coefficients a,b,c,d,e,f are learned per layer, per head, or generated by a small coefficient MLP from the mean of the three token embeddings. 2. Pseudocode: for each triple h1,h2,h3, reshape each h_r to registers; compute A=a*kron(U1,U2)+b*kron(V1,V2), B=c*kron(U1,U3)+d*kron(V1,V3), and C=e*kron(U2,U3)+f*kron(V2,V3); form y_left=A@(B@(C@h)) and y_right=C@(B@(A@h)); use y=(y_left+y_right)/2, followed by a learned output projection, residual connection, and normalization. Since the algebra is exact, the path discrepancy d_cons=||y_left-y_right||^2 should be numerical roundoff only; optionally use it as a diagnostic rather than a loss. 3. The matrices X and Z and the commutation law come directly from the paper's anticommuting construction. The coefficients are learned empirically, and overlapping-triple aggregation can use mean pooling, learned edge weights, or a depthwise convolution over triple positions. To control cost, start with 2-channel registers and apply the mixer only to every fourth token triple, comparing against an equally sized local self-attention block. 4. First experiment: train a 6-layer, 128-dimensional causal language model on TinyStories or a synthetic parity/compositionality dataset. Compare a standard 3-token local attention block, an unconstrained six-parameter pair mixer, and this scalene mixer at equal parameter count and FLOPs. Measure validation perplexity, gradient norm variance, training loss at fixed updates, and sensitivity to learning rate. Success is faster loss descent or lower perplexity at equal compute, together with a path discrepancy below 1e-6 and reduced variance of gradients across random seeds. A second ablation should replace anticommuting X,Z by commuting generators; degradation would test whether the algebraic structure, rather than merely low rank, is responsible.