The fluctuations of proton multiplicity in heavy-ion collisions are the key observables in the search for the QCD critical point. In this work we present an approach to constraining the cumulants of proton number based on the analytical properties of the QCD equation of state in the vicinity of the critical point. We show that, under the assumption of local equilibrium, the features of the collision energy dependence, such as the peaks and the dips of the cumulants, are significantly constrained by the data on the Lee-Yang singularity structure available via Padé resummation of the lattice QCD data. Furthermore, we identify four topologically distinct scenarios, all within the uncertainty range of the Padé estimates for the non-universal mapping parameters, classified based on the location of the critical point and the slope of the chiral crossover curve with respect to the freeze-out curve. These different scenarios result in qualitatively different critical signatures, especially for the third factorial cumulant and thus could be potentially discriminated using the experimental data.