Cosmic large-scale structure is described by evolving density fields, yet critical-point topology is usually analyzed through snapshots or threshold filtrations. We formulate the physical-time evolution of critical structures for a smooth, positive, coarse-grained density field using $f_{t,R}(𝐱)=ln[{\rho }_R(𝐱,t)/{\rho }_{\star }(t)]$. This monotone transformation preserves critical points, Morse indices, generic birth–death locations, and excursion-set topology. The resulting Morse–Cerf framework tracks critical-point worldlines, distinguishes Eulerian birth–death events, fixed-threshold events, and continuous level-set volume fluxes, and yields index-resolved balance laws and Kac–Rice intensities. As an analytically tractable proxy model, we use the Zel’dovich approximation while separating singularities of the Lagrangian map from Eulerian scalar-field bifurcations. The conditions $D(t){\lambda }_k(𝐪)=1$ define continuous collapse-crossing hypersurfaces, whereas local minima of $D_{c,k}=1/{\lambda }_k$ define discrete nucleation proxies. For Gaussian initial conditions, we derive one-scale collapse fractions and crossing fluxes from the Doroshkevich distribution and characterize the proxy catalogs numerically at the adopted grid resolution. Their finite-time populations show temporally concentrated activity and ordered median growth factors, ${\tilde{D}}_{1,R}^{nuc,fin}<{\tilde{D}}_{2,R}^{nuc,fin}<{\tilde{D}}_{3,R}^{nuc,fin}$, over the reported smoothing scales, while their abundances and widths vary with smoothing. These discrete-catalog results are a fixed-resolution proof of concept, with no continuum-resolution claim. Continuous crossing fluxes and discrete proxy rates are complementary observables and provide a controlled basis for measuring exact Eulerian event processes in nonlinear simulations.