Erdelyi-Kober Log-Scale Mixer

Integrate the mixer at the cross-scale feature tensor, rather than inside the channel MLP. For a CNN, construct a feature pyramid with levels indexed by uniformly spaced log-scales \(s_i=s_0+i\Delta s\), project each level to the same channel width, and apply the operation independently per channel or after a learned \(1\times1\) projection. For a transformer, group tokens by resolution or patch scale and apply the same operation before the next attention block. Use this procedure: (1) choose a kernel length \(L\), spacing \(\Delta s\), and initial parameters \(\alpha=1\), \(\beta=1\), \(\gamma=0\); (2) compute \(u_j=j\Delta s\) and the unnormalized kernel weights \(q_j=\frac{\beta}{\Gamma(\alpha)}(1-e^{-\beta u_j})^{\alpha-1}e^{-\beta(\gamma+1)u_j}\); (3) avoid the singular endpoint when \(\alpha<1\) by using midpoint quadrature, replacing \(u_0\) with \(\Delta s/2\), or by analytically integrating the first bin; (4) normalize \(w_j=q_j/\sum_{\ell=0}^Lq_\ell\); (5) compute the causal grouped convolution \(Y_i=\sum_{j=0}^{\min(i,L)}w_jF_{i-j}\); and (6) return \(Z_i=F_i+\sigma(g)Y_i\), where \(g\) is a learned scalar or per-channel gate. Parameterize \(\alpha=\operatorname{softplus}(a)+\epsilon\), \(\beta=\operatorname{softplus}(b)+\epsilon\), and \(\gamma=-1+\operatorname{softplus}(c)\), ensuring integrability. The exact mathematical quantities are the analytic kernel and its mass; empirical choices are \(L\), \(\Delta s\), channel projections, and learned parameters. First test on CIFAR-100 with a small ResNet-FPN or ConvNeXt-Tiny using three resolution levels. Compare against nearest-neighbor plus \(1\times1\) fusion and a learned depthwise cross-scale convolution at matched FLOPs. Measure top-1 accuracy, robustness under input resizing, gradient variance, and wall-clock cost. Success is higher accuracy or resize robustness at equal FLOPs, with no loss spikes or unstable gradients.