Near universality of nonlinear transverse and radial velocity responses in spherical collapse with arbitrary radial profiles

The nonlinear relation between density and expansion is usually formulated for a homogeneous spherical top hat, which has a single local Hubble rate. A smooth spherical profile instead expands differently along ($H_{\parallel }$) and across ($H_{\perp }$) the radial direction-a direct signature of radial inhomogeneity. We show that, for growing-mode pressureless matter with a cosmological constant, the complete shellwise response is nevertheless fixed by one top-hat function. Given the local density contrast $\delta (t,r)$ and the enclosed contrast $\mathrm{\Delta }(t,r)$, a transverse response and its derivative determine $H_{\perp }$ and $H_{\parallel }$. The linear and second-order limits are algebraic maps whose only dynamical input is the usual growth rate $f$. An exact equal-age construction supplies the nonlinear response without integrating an evolution equation. We check that it reconstructs full $\mathrm{\Lambda }$LTB profiles to numerical precision. We also provide a derivative-aware, cosmology-independent three-term symbolic fit that requires only $f$, $\delta$, and $\mathrm{\Delta }$. Across a representative set of matter-curvature-redshift combinations and for a shell located in the compensated transition, the maximum relative errors are $0.3\mathrm{\%}$ and $0.7\mathrm{\%}$ in the transverse and radial responses, respectively. This compact formulation separates the production of a density profile from its expansion response and makes the effect of radial gradients explicit.