After performing extensive simulations with the code CoRSiKa, we have obtained an analytical description fitting with surprising accuracy the numerical densities up to distances larger than 5 km from the shower axis. This was achieved by using the hypergeometric formalism in place of the traditional NKG approach. The difficulty of cascade theory (validity limited to 3.5 Moliere radii), underlined with reason by the particle data group, is solved here, after overcoming the constraints of approximation B, to show that the distribution of lateral profiles at large distances is also correlated with the age parameter. This is an important step for a coherent interpretation of hybrid events recorded with both surface array and fluorescence telescopes, even with other information coming from Cerenkov or radio emission. A set of hypergeometric Gaussian functions, with a consistent relation between age parameter and total size, is proposed in the ultra-high-energy range (above 1 EeV) for electrons, muons and vertical equivalent muons.