Order-Derivative Fractional Regularizer

Apply the regularizer to an intermediate CNN feature map $U\in\mathbb{R}^{B\times C\times H\times W}$, preferably at 1/4 or 1/8 image resolution to limit overhead. Choose a finite offset set $\Delta=\{\delta:0<\|\delta\|_2\le R\}$, use reflect padding, and measure each offset distance in pixels as $r_\delta=\|\delta\|_2$. For each offset compute $D_\delta=U-\mathrm{shift}(U,\delta)$ and $a_\delta=(r_\delta+\epsilon)^{-d-2s}$ with $d=2$. Accumulate $E=\frac12\sum_{\delta\in\Delta}a_\delta\operatorname{mean}(D_\delta^2)$ and $G=\frac12\sum_{\delta\in\Delta}a_\delta(B-2\log(r_\delta+\epsilon))\operatorname{mean}(D_\delta^2)$, then add $\lambda G^2/(E+\epsilon)$ to the task loss. Pseudocode is: initialize $E=G=0$; for each offset, shift $U$, compute $D$, update $E\mathrel{+}=a_\delta\operatorname{mean}(D^2)/2$ and $G\mathrel{+}=a_\delta(B-2\log(r_\delta+\epsilon))\operatorname{mean}(D^2)/2$; return $\mathcal{L}_{task}+\lambda G^2/(E+\epsilon)$. The paper supplies the logarithmic derivative factor; the finite offset quadrature and cutoff radius $R$ are approximations. Initially set $B=0$ to isolate the logarithmic effect; optionally estimate $B$ by finite differences of the chosen discrete normalization, $B\approx[\log C_{d,s+h}-\log C_{d,s-h}]/(2h)$. Test on CIFAR-10 with ResNet-18 against no regularizer and ordinary fractional energy $E_s$, using $s\in\{0.25,0.5,0.75\}$ and $\lambda\in\{10^{-5},10^{-4},10^{-3}\}$. Log clean accuracy, CIFAR-10-C accuracy, feature energy $E$, the signed derivative $G$, gradient norms, and training-loss variance. Success means better corruption robustness or lower feature-scale drift without more than a 1% clean-accuracy loss or materially worse optimization stability.