Frieze-consistent multiplicative feature block

(1) Integration point: insert the block after a Transformer or MLP hidden projection, or between two 1D convolution layers. Reshape a hidden vector h in R^d into U in R^{I x K}, with d=I K; k is a circular feature axis and i indexes internal interaction layers. Compute positive factors X=softplus(U)+epsilon, with epsilon=1e-4. Produce two feature grids y and x using y[i,k]=X[i,k]*X[i,(k+1) mod K]-1 and x[i,k]=X[i,k]*X[i-1,k], using a learned or fixed positive boundary X[-1,k]. Flatten concat(x,y), project back to the original hidden dimension, and add a residual connection to h. (2) Pseudocode: U=reshape(linear(h)); X=softplus(U)+eps; y=X*roll(X,-1,axis=k)-1; x=X*shift(X,+1,axis=i); residual=(x[1:]+1)*(x[:-1]+1)-y[:-1]*roll(y[:-1],+1,axis=k); output=h+linear(flatten(concat(x,y))); loss=task_loss+lambda_f*mean(residual**2). A stricter variant computes x[i+1,k]=y[i,k]*y[i,k-1]/(x[i,k]+1)-1 and uses the resulting x as the next layer state, with clipping if numerical instability occurs. (3) The exact mathematical quantities are the adjacent products and Y-diamond residual; no theorem constants are estimated. Empirically sweep lambda_f, K, and the number of layers, and monitor the fraction of negative y/x values, residual magnitude, activation variance, and gradient norm. (4) First experiment: train a 4-layer MLP and a small CIFAR-10 ConvNet with identical parameter counts, comparing a standard gated multiplicative block against this block at K=16 or 32. Use AdamW, identical initialization and learning-rate schedule, and three seeds. Success means lower gradient-norm variance, fewer NaNs or activation explosions, and either better validation accuracy at equal parameters or equal accuracy with at least 20% fewer hidden parameters.