Quantum Big Bang without fine-tuning in a toy-model

The question of possible physics before Big Bang (or after Big Crunch) is addressed via a schematic non-covariant simulation of the loss of observability of the Universe. Our model is drastically simplified by the reduction of its degrees of freedom to the mere finite number. The Hilbert space of states is then allowed time-dependent and singular at the critical time $t=t_c$. This option circumvents several traditional theoretical difficulties in a way illustrated via solvable examples. In particular, the unitary evolution of our toy-model quantum Universe is shown interruptible, without any fine-tuning, at the instant of its bang or collapse $t= t_c$.