Distributional cosmological quantities solve the paradox of soft singularity crossing

A flat Friedmann universe filled with a mixture of anti-Chaplygin gas and dust-like matter evolves into a future soft singularity, where despite infinite tidal forces the geodesics can be continued. In the singularity the pressure of the anti-Chaplygin gas diverges, while its energy density is zero. The dust energy density however does not vanish, neither does the Hubble parameter, which implies further expansion, if its evolution is to be continuous. If so, the energy density and the pressure of the anti-Chaplygin gas would become ill-defined, hence only a contraction would be allowed. Paradoxically, the universe in this cosmological model would have to expand and contract simultaneously. The paradox can be avoided by redefining the anti-Chaplygin gas in a distributional sense. Then the Hubble parameter could be mirrored to have a jump at the singularity, allowing for a subsequent contraction. With this modification the set of Friedmann, Raychaudhuri and continuity equations are all obeyed both at the singularity and in its vicinity.