Quantization of Big Bang in crypto-Hermitian Heisenberg picture

A background-independent quantization of the Universe near its Big Bang singularity is considered using a drastically simplified toy model. Several conceptual issues are addressed. (1) The observable spatial-geometry characteristics of our empty-space expanding Universe is sampled by the time-dependent operator $Q=Q(t)$ of the distance between two space-attached observers (``Alice and Bob''). (2) For any pre-selected guess of the simple, non-covariant time-dependent observable $Q(t)$ one of the Kato's exceptional points (viz., $t=\tau_{(EP)}$) is postulated {\em real-valued}. This enables us to treat it as the time of Big Bang. (3) During our ``Eon'' (i.e., at all $t>\tau_{(EP)}$) the observability status of operator $Q(t)$ is mathematically guaranteed by its self-adjoint nature with respect to an {\em ad hoc} Hilbert-space metric $\Theta(t) \neq I$. (4) In adiabatic approximation (i.e., in Heisenberg picture) the passage of the Universe through its $t=\tau_{(EP)}$ singularity is interpreted as a quantum phase transition between the preceding and the present Eon. It is worth adding that in our model the widely accepted ``Big Bounce'' regularization of the classical Big Bang singularity after quantization gets replaced by the full-fledged quantum degeneracy. At the quantum-phase-transition singularity, operator $Q(\tau_{(EP)})$ becomes unobservable and acquires a non-diagonalizable Jordan-block structure.