Correlation-controlled residual roughness

Integrate this as a depth-wise parameter generator for residual blocks, with candidate H selected before training. Begin with q=1 because it is easiest to implement and corresponds to fractional Brownian motion. For each H in {0.5, 0.6, 0.7, 0.8}, choose alpha=2(1-H), generate a stationary Gaussian vector g[0:L] with correlation approximately c*k^{-alpha}, standardize it, and assign W_l=sigma_W*g_l to scalar gates or channel-wise low-rank gates. To test Hermite rank separately, use w_l=He_q(g_l)/sqrt(q!) for q=1 or q=2, standardize empirically, and choose alpha=2(1-H)/q so the same H is obtained. Keep the marginal variance of W_l fixed across settings and use lambda=L^{-H}. Pseudocode: for each H,q, set alpha=2*(1-H)/q; g=sample_stationary_gaussian(alpha,L); z=hermite_q(g); z=(z-mean(z))/std(z); gate[l]=sigma_gate*z[l]; lambda=L**(-H); output[l]=input[l]+lambda*gate[l]*F_l(input[l]). Estimate achieved H with the displayed block-sum log-log estimator and independently estimate covariance decay; these are diagnostics, not learned quantities. First experiment: train 20-layer and 100-layer residual MLPs on MNIST or CIFAR-10 using iid, H=0.5, H=0.7, and H=0.8 gates with three seeds and matched FLOPs. Measure early loss slope, gradient-noise variance, activation drift, final accuracy, and learning-rate sensitivity. The hypothesis is that an intermediate H improves deep optimization or accuracy while keeping activations bounded; it is falsified if H has no reproducible effect after variance matching or if larger H consistently worsens trainability.