arXiv:2610.23163v1 Announce Type: new Abstract: We develop stability and error estimates for multistep convolution quadrature (CQ) under general radial bounds on the Laplace symbol. Instead of the classical single-power condition $|F(z)|\leq M|z|^{-\mu}$, we assume $|F(z)|\leq C\Phi(|z|)$ and consider two classes of radial functions: monotone bounds satisfying a scaling condition and regularly varying bounds with locally bounded above and away from zero. For strongly $A(\alpha)$-stable multistep methods, we derive approximation and stability estimates for the CQ weights, as well as error bounds for power-type inputs $g_\beta(t)=t^{\beta-1}/\Gamma(\beta)$ with $\beta>0$. These estimates explicitly retain the dependence on the memory law through the factor $\Phi(t_n^{-1})$. The framework accommodates symbols whose behavior cannot be described by a global single-power. It also preserves slowly varying and multiscale factors that would be lost under a coarser power majorant. The theory is applied to fractional viscoelastic models, distributed-order operators, and logarithmic-power memory operators. Numerical experiments with backward Euler and second-order backward differentiation CQ confirm the expected temporal convergence orders for direct responses. Complementary evolution examples demonstrate the corresponding temporal behavior for smooth and nonsmooth initial data.