Log-Fractional Scale-Derivative Layer

Integrate the layer after a convolutional feature map $X\in\mathbb{R}^{B\times C\times H\times W}$ and before the next normalization or residual block. Start with the linear $p=2$ version because it avoids the numerical difficulty of a principal-value nonlinear integral. Construct 2-D FFT frequency grids $\xi_x,\xi_y$, let $r_\xi=\sqrt{\xi_x^2+\xi_y^2}$, and define $m_s(\xi)=2(r_\xi+\epsilon)^{2s}\log(r_\xi+\epsilon)$ with the DC multiplier explicitly set to zero. Compute $Z=\mathrm{IFFT2}(m_s\odot\mathrm{FFT2}(X))$ independently for every batch item and channel, then return $X+\alpha Z$ or concatenate $[X,Z]$ followed by a learned $1\times1$ convolution. Use fixed $s\in\{0.25,0.5,0.75\}$ initially; an optional learned parameter is $s=0.1+0.8\,\sigma(\rho)$ per channel. The exact spectral multiplier comes from the paper's order derivative; FFT discretization, boundary padding, $\epsilon$, and the residual gate $\alpha$ are engineering choices. As a spatial cross-check, use offsets with weights $w_\delta=C_{d,s}(B_{d,s}-2\log\|\delta\|)\|\delta\|^{-d-2s}$ and apply $\sum_\delta w_\delta(X-\mathrm{shift}_\delta X)$. First test on CIFAR-10 with a small ResNet-18, comparing the baseline, a fractional layer with multiplier $|\xi|^{2s}$, and the proposed log-fractional layer at matched channels and training FLOPs. Measure clean accuracy, CIFAR-10-C accuracy, loss descent, gradient norms, and wall-clock cost. A useful success signal is at least 0.5 percentage points better corruption accuracy or faster convergence at equal final clean accuracy with less than 15% extra training time.