The occurrence of high, persistent ice supersaturation inside and outside cold cirrus in the tropical tropopause layer (TTL) remains an enigma that is intensely debated as the "ice supersaturation puzzle". However, it was recently confirmed that observed supersaturations are consistent with very low ice crystal concentrations, which is incompatible with the idea that homogeneous freezing is the major method of ice formation in the TTL. Thus, the tropical tropopause "ice supersaturation puzzle" has become an "ice nucleation puzzle". To explain the low ice crystal concentrations, a number of mainly heterogeneous freezing methods have been proposed. Here, we reproduce in situ measurements of frequencies of occurrence of ice crystal concentrations by extensive model simulations, driven by the special dynamic conditions in the TTL, namely the superposition of slow large-scale updraughts with high-frequency short waves. From the simulations, it follows that the full range of observed ice crystal concentrations can be explained when the model results are composed from scenarios with consecutive heterogeneous and homogeneous ice formation and scenarios with pure homogeneous ice formation occurring in very slow (< 1 cm s⁻¹) and faster (> 1 cm s⁻¹) large-scale updraughts, respectively. This statistical analysis shows that about 80% of TTL cirrus can be explained by "classical" homogeneous ice nucleation, while the remaining 20% stem from heterogeneous and homogeneous freezing occurring within the same environment. The mechanism limiting ice crystal production via homogeneous freezing in an environment full of gravity waves is the shortness of the gravity waves, which stalls freezing events before a higher ice crystal concentration can be formed.