Four-Root Ternary Mixer

(1) Integration point: use this module as a replacement for a three-input fusion MLP. In a graph neural network, let x, y, and z in R^d be a center-node state and two sampled neighbor states. In a transformer, they can be three routed token features or three expert outputs. Apply LayerNorm and a shared projection P in R^(d by m), obtaining u=P norm(x), v=P norm(y), and w=P norm(z). Compute the four signed roots elementwise: t1=u+v-w, t2=u-v+w, t3=-u+v+w, and t4=-u-v-w. (2) Pseudocode: u,v,w = P(norm(x)), P(norm(y)), P(norm(z)); T = [u+v-w, u-v+w, -u+v+w, -u-v-w]; E[j] = sum(r=0..K, T[j]^r / factorial(r)); q = sum(i<j, E[i] * E[j]); h = W_out GELU(q) + W_skip x. The tensor q is the adapted truncated version of the paper's h(F_2). Start with K=2 or K=3. For numerical stability, use E_K(t_j divided by tau), where tau is a learned positive channelwise temperature initialized to 2. (3) The signed-root pattern and symmetric pair sum are taken from the paper; P, W_out, W_skip, and tau are learned. Estimate computational cost analytically from four evaluations of a degree-K polynomial and six elementwise products, and compare it with the baseline MLP at matched FLOPs. Test permutation invariance by randomly permuting the three inputs during training and evaluating all six permutations at validation time. (4) First cheap experiment: train a small graph network on ZINC, or a three-stream MLP on the synthetic target f(x,y,z)=sum_k x_k y_k z_k with added pairwise distractors. Compare K=2, 3, and 4 against a parameter-matched concatenation-plus-two-layer-MLP baseline. Success means lower validation error at equal parameters and FLOPs, or equal accuracy with at least 20 percent fewer parameters. Also record gradient norms and activation maxima; the truncated construction should avoid the instability of an unconstrained learned trilinear tensor.