Positive Mellin Mixture Gate

Use this as a drop-in replacement for a scalar gate that currently depends on an ordered control variable. Examples include timestep-dependent residual scaling in diffusion U-Nets, temperature-dependent attention output scaling, layer-index-dependent SSM gains, or curriculum-dependent loss weights. Given a batch feature vector $z$, compute a nonnegative base amplitude $b(z)=\operatorname{softplus}(\mathrm{MLP}(z))$, then compute $h(a)$ from $L=4$ to $16$ learned atoms. Parameterize $w_\ell=\operatorname{softplus}(\tilde w_\ell)$ and $s_\ell=\operatorname{softplus}(\tilde s_\ell)$; return gate $q(a,z)=\operatorname{sigmoid}(c+b(z)h(a))$ or residual multiplier $1+\rho b(z)h(a)$. Pseudocode: inputs $(z,a)$; $b=\operatorname{softplus}(MLP(z))$; for each atom calculate $r_\ell=-a\log1p(s_\ell)$; $h=\sum_\ell\exp(\log w_\ell+r_\ell)$ using log-sum-exp; apply the gate to the selected residual or attention branch. Initialize rates log-uniformly over a range corresponding to the observed $a$ interval and initialize weights so the mean gate is one. The mathematical guarantee is exact for the parameterized gate; rates and weights are learned empirically. Validate first on CIFAR-10 diffusion with a small U-Net, comparing a free two-layer timestep MLP against the Mellin gate at equal parameter count. Measure FID, denoising loss, gradient variance across timesteps, and extrapolation to unseen timesteps. Success is smoother timestep response, lower gradient variance, and equal or better FID without increasing sampling steps.