Mixed-norm Brascamp–Lieb product block

(1) Integrate the block at the fusion point of a small MLP or vision latent module. Let each minibatch provide two normalized latent coordinates u=(x,y), either on a fixed two-dimensional grid or as sampled points. Define four nonnegative branch networks fi(t)=softplus(hi(t)), with inputs t1=x+y, t2=-x+y, t3=y, and t4=x+3y. Compute g=f1(t1)^(2/7) f2(t2)^(2/7) f3(t3)^(2/7) f4(t4)^(1/7), then output the normalized feature g-tilde=g divided by M1^(2/7)M2^(2/7)M3^(2/7)M4^(1/7). (2) Estimate Mi by batch quadrature or a sample mean over scalar branch inputs, using epsilon=1e-6; use a stop-gradient version of Mi initially to prevent the branches from reducing their own normalization denominator. Concatenate g-tilde with the original latent vector and apply a learned linear projection. (3) Compute the empirical mixed norm by taking a mean over y raised to the 7/3 power, then a mean over x raised to the 7/4 power. The affine maps, fractional exponents, and target norm exponents come from the paper; quadrature and epsilon are implementation choices. (4) First test on a CIFAR-10 two-layer MLP with a two-dimensional bottleneck, comparing this block against ordinary multiplicative fusion plus LayerNorm and against additive fusion at matched parameter count. Measure activation mixed norms, gradient norms, training loss at equal FLOPs, validation accuracy, and robustness under a 10-fold learning-rate increase. Success means fewer activation or gradient spikes and equal or better accuracy without LayerNorm, or faster loss descent at matched compute.